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433cd492b67f90dc1b3390eba38b08417fd9fc1597963748f190e2d254560b2a | from sympy import symbols
from sympy.tensor.array.expressions.array_expressions import ZeroArray
from sympy.testing.pytest import raises
def test_zero_array():
assert ZeroArray() == 0
assert ZeroArray().is_Integer
za = ZeroArray(3, 2, 4)
assert za.shape == (3, 2, 4)
za_e = za.as_explicit()
assert za_e.shape == (3, 2, 4)
m, n, k = symbols("m n k")
za = ZeroArray(m, n, k, 2)
assert za.shape == (m, n, k, 2)
raises(ValueError, lambda: za.as_explicit())
|
85fb8873fc929e82cd0b75370787530e159706510ac396d8bd197eb3daec127c | from sympy import MatrixSymbol, symbols, Identity
from sympy.codegen.array_utils import CodegenArrayTensorProduct, CodegenArrayPermuteDims
from sympy.tensor.array.expressions.arrayexpr_derivatives import array_derive
k = symbols("k")
I = Identity(k)
X = MatrixSymbol("X", k, k)
x = MatrixSymbol("x", k, 1)
A = MatrixSymbol("A", k, k)
B = MatrixSymbol("B", k, k)
C = MatrixSymbol("C", k, k)
D = MatrixSymbol("D", k, k)
def test_arrayexpr_derivatives1():
res = array_derive(X, X)
assert res == CodegenArrayPermuteDims(CodegenArrayTensorProduct(I, I), [0, 2, 1, 3])
cg = CodegenArrayTensorProduct(A, X, B)
res = array_derive(cg, X)
assert res == CodegenArrayPermuteDims(
CodegenArrayTensorProduct(I, A, I, B),
[0, 4, 2, 3, 1, 5, 6, 7])
|
7915a5f02a3f695312ef18942b68b0abcb2d7643016da30975b2fc29b149301f | from sympy.assumptions.ask import Q
from sympy.assumptions.refine import refine
from sympy.core.numbers import oo
from sympy.core.relational import Equality, Eq, Ne
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, symbols)
from sympy.functions import Piecewise
from sympy.functions.elementary.miscellaneous import Max, Min
from sympy.functions.elementary.trigonometric import sin
from sympy.sets.sets import (EmptySet, Interval, Union)
from sympy.simplify.simplify import simplify
from sympy.logic.boolalg import (
And, Boolean, Equivalent, ITE, Implies, Nand, Nor, Not, Or,
POSform, SOPform, Xor, Xnor, conjuncts, disjuncts,
distribute_or_over_and, distribute_and_over_or,
eliminate_implications, is_nnf, is_cnf, is_dnf, simplify_logic,
to_nnf, to_cnf, to_dnf, to_int_repr, bool_map, true, false,
BooleanAtom, is_literal, term_to_integer, integer_to_term,
truth_table, as_Boolean, to_anf, is_anf, distribute_xor_over_and,
anf_coeffs, ANFform, bool_minterm, bool_maxterm, bool_monomial,
_check_pair, _convert_to_varsSOP, _convert_to_varsPOS)
from sympy.assumptions.cnf import CNF
from sympy.testing.pytest import raises, XFAIL, slow
from sympy.utilities.iterables import cartes
from itertools import combinations, permutations
A, B, C, D = symbols('A:D')
a, b, c, d, e, w, x, y, z = symbols('a:e w:z')
def test_overloading():
"""Test that |, & are overloaded as expected"""
assert A & B == And(A, B)
assert A | B == Or(A, B)
assert (A & B) | C == Or(And(A, B), C)
assert A >> B == Implies(A, B)
assert A << B == Implies(B, A)
assert ~A == Not(A)
assert A ^ B == Xor(A, B)
def test_And():
assert And() is true
assert And(A) == A
assert And(True) is true
assert And(False) is false
assert And(True, True) is true
assert And(True, False) is false
assert And(False, False) is false
assert And(True, A) == A
assert And(False, A) is false
assert And(True, True, True) is true
assert And(True, True, A) == A
assert And(True, False, A) is false
assert And(1, A) == A
raises(TypeError, lambda: And(2, A))
raises(TypeError, lambda: And(A < 2, A))
assert And(A < 1, A >= 1) is false
e = A > 1
assert And(e, e.canonical) == e.canonical
g, l, ge, le = A > B, B < A, A >= B, B <= A
assert And(g, l, ge, le) == And(ge, g)
assert {And(*i) for i in permutations((l,g,le,ge))} == {And(ge, g)}
assert And(And(Eq(a, 0), Eq(b, 0)), And(Ne(a, 0), Eq(c, 0))) is false
def test_Or():
assert Or() is false
assert Or(A) == A
assert Or(True) is true
assert Or(False) is false
assert Or(True, True) is true
assert Or(True, False) is true
assert Or(False, False) is false
assert Or(True, A) is true
assert Or(False, A) == A
assert Or(True, False, False) is true
assert Or(True, False, A) is true
assert Or(False, False, A) == A
assert Or(1, A) is true
raises(TypeError, lambda: Or(2, A))
raises(TypeError, lambda: Or(A < 2, A))
assert Or(A < 1, A >= 1) is true
e = A > 1
assert Or(e, e.canonical) == e
g, l, ge, le = A > B, B < A, A >= B, B <= A
assert Or(g, l, ge, le) == Or(g, ge)
def test_Xor():
assert Xor() is false
assert Xor(A) == A
assert Xor(A, A) is false
assert Xor(True, A, A) is true
assert Xor(A, A, A, A, A) == A
assert Xor(True, False, False, A, B) == ~Xor(A, B)
assert Xor(True) is true
assert Xor(False) is false
assert Xor(True, True) is false
assert Xor(True, False) is true
assert Xor(False, False) is false
assert Xor(True, A) == ~A
assert Xor(False, A) == A
assert Xor(True, False, False) is true
assert Xor(True, False, A) == ~A
assert Xor(False, False, A) == A
assert isinstance(Xor(A, B), Xor)
assert Xor(A, B, Xor(C, D)) == Xor(A, B, C, D)
assert Xor(A, B, Xor(B, C)) == Xor(A, C)
assert Xor(A < 1, A >= 1, B) == Xor(0, 1, B) == Xor(1, 0, B)
e = A > 1
assert Xor(e, e.canonical) == Xor(0, 0) == Xor(1, 1)
def test_rewrite_as_And():
expr = x ^ y
assert expr.rewrite(And) == (x | y) & (~x | ~y)
def test_rewrite_as_Or():
expr = x ^ y
assert expr.rewrite(Or) == (x & ~y) | (y & ~x)
def test_rewrite_as_Nand():
expr = (y & z) | (z & ~w)
assert expr.rewrite(Nand) == ~(~(y & z) & ~(z & ~w))
def test_rewrite_as_Nor():
expr = z & (y | ~w)
assert expr.rewrite(Nor) == ~(~z | ~(y | ~w))
def test_Not():
raises(TypeError, lambda: Not(True, False))
assert Not(True) is false
assert Not(False) is true
assert Not(0) is true
assert Not(1) is false
assert Not(2) is false
def test_Nand():
assert Nand() is false
assert Nand(A) == ~A
assert Nand(True) is false
assert Nand(False) is true
assert Nand(True, True) is false
assert Nand(True, False) is true
assert Nand(False, False) is true
assert Nand(True, A) == ~A
assert Nand(False, A) is true
assert Nand(True, True, True) is false
assert Nand(True, True, A) == ~A
assert Nand(True, False, A) is true
def test_Nor():
assert Nor() is true
assert Nor(A) == ~A
assert Nor(True) is false
assert Nor(False) is true
assert Nor(True, True) is false
assert Nor(True, False) is false
assert Nor(False, False) is true
assert Nor(True, A) is false
assert Nor(False, A) == ~A
assert Nor(True, True, True) is false
assert Nor(True, True, A) is false
assert Nor(True, False, A) is false
def test_Xnor():
assert Xnor() is true
assert Xnor(A) == ~A
assert Xnor(A, A) is true
assert Xnor(True, A, A) is false
assert Xnor(A, A, A, A, A) == ~A
assert Xnor(True) is false
assert Xnor(False) is true
assert Xnor(True, True) is true
assert Xnor(True, False) is false
assert Xnor(False, False) is true
assert Xnor(True, A) == A
assert Xnor(False, A) == ~A
assert Xnor(True, False, False) is false
assert Xnor(True, False, A) == A
assert Xnor(False, False, A) == ~A
def test_Implies():
raises(ValueError, lambda: Implies(A, B, C))
assert Implies(True, True) is true
assert Implies(True, False) is false
assert Implies(False, True) is true
assert Implies(False, False) is true
assert Implies(0, A) is true
assert Implies(1, 1) is true
assert Implies(1, 0) is false
assert A >> B == B << A
assert (A < 1) >> (A >= 1) == (A >= 1)
assert (A < 1) >> (S.One > A) is true
assert A >> A is true
def test_Equivalent():
assert Equivalent(A, B) == Equivalent(B, A) == Equivalent(A, B, A)
assert Equivalent() is true
assert Equivalent(A, A) == Equivalent(A) is true
assert Equivalent(True, True) == Equivalent(False, False) is true
assert Equivalent(True, False) == Equivalent(False, True) is false
assert Equivalent(A, True) == A
assert Equivalent(A, False) == Not(A)
assert Equivalent(A, B, True) == A & B
assert Equivalent(A, B, False) == ~A & ~B
assert Equivalent(1, A) == A
assert Equivalent(0, A) == Not(A)
assert Equivalent(A, Equivalent(B, C)) != Equivalent(Equivalent(A, B), C)
assert Equivalent(A < 1, A >= 1) is false
assert Equivalent(A < 1, A >= 1, 0) is false
assert Equivalent(A < 1, A >= 1, 1) is false
assert Equivalent(A < 1, S.One > A) == Equivalent(1, 1) == Equivalent(0, 0)
assert Equivalent(Equality(A, B), Equality(B, A)) is true
def test_equals():
assert Not(Or(A, B)).equals(And(Not(A), Not(B))) is True
assert Equivalent(A, B).equals((A >> B) & (B >> A)) is True
assert ((A | ~B) & (~A | B)).equals((~A & ~B) | (A & B)) is True
assert (A >> B).equals(~A >> ~B) is False
assert (A >> (B >> A)).equals(A >> (C >> A)) is False
raises(NotImplementedError, lambda: (A & B).equals(A > B))
def test_simplification():
"""
Test working of simplification methods.
"""
set1 = [[0, 0, 1], [0, 1, 1], [1, 0, 0], [1, 1, 0]]
set2 = [[0, 0, 0], [0, 1, 0], [1, 0, 1], [1, 1, 1]]
assert SOPform([x, y, z], set1) == Or(And(Not(x), z), And(Not(z), x))
assert Not(SOPform([x, y, z], set2)) == \
Not(Or(And(Not(x), Not(z)), And(x, z)))
assert POSform([x, y, z], set1 + set2) is true
assert SOPform([x, y, z], set1 + set2) is true
assert SOPform([Dummy(), Dummy(), Dummy()], set1 + set2) is true
minterms = [[0, 0, 0, 1], [0, 0, 1, 1], [0, 1, 1, 1], [1, 0, 1, 1],
[1, 1, 1, 1]]
dontcares = [[0, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 1]]
assert (
SOPform([w, x, y, z], minterms, dontcares) ==
Or(And(y, z), And(Not(w), Not(x))))
assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)
minterms = [1, 3, 7, 11, 15]
dontcares = [0, 2, 5]
assert (
SOPform([w, x, y, z], minterms, dontcares) ==
Or(And(y, z), And(Not(w), Not(x))))
assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)
minterms = [1, [0, 0, 1, 1], 7, [1, 0, 1, 1],
[1, 1, 1, 1]]
dontcares = [0, [0, 0, 1, 0], 5]
assert (
SOPform([w, x, y, z], minterms, dontcares) ==
Or(And(y, z), And(Not(w), Not(x))))
assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)
minterms = [1, {y: 1, z: 1}]
dontcares = [0, [0, 0, 1, 0], 5]
assert (
SOPform([w, x, y, z], minterms, dontcares) ==
Or(And(y, z), And(Not(w), Not(x))))
assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)
minterms = [{y: 1, z: 1}, 1]
dontcares = [[0, 0, 0, 0]]
minterms = [[0, 0, 0]]
raises(ValueError, lambda: SOPform([w, x, y, z], minterms))
raises(ValueError, lambda: POSform([w, x, y, z], minterms))
raises(TypeError, lambda: POSform([w, x, y, z], ["abcdefg"]))
# test simplification
ans = And(A, Or(B, C))
assert simplify_logic(A & (B | C)) == ans
assert simplify_logic((A & B) | (A & C)) == ans
assert simplify_logic(Implies(A, B)) == Or(Not(A), B)
assert simplify_logic(Equivalent(A, B)) == \
Or(And(A, B), And(Not(A), Not(B)))
assert simplify_logic(And(Equality(A, 2), C)) == And(Equality(A, 2), C)
assert simplify_logic(And(Equality(A, 2), A)) is S.false
assert simplify_logic(And(Equality(A, 2), A)) == And(Equality(A, 2), A)
assert simplify_logic(And(Equality(A, B), C)) == And(Equality(A, B), C)
assert simplify_logic(Or(And(Equality(A, 3), B), And(Equality(A, 3), C))) \
== And(Equality(A, 3), Or(B, C))
b = (~x & ~y & ~z) | (~x & ~y & z)
e = And(A, b)
assert simplify_logic(e) == A & ~x & ~y
raises(ValueError, lambda: simplify_logic(A & (B | C), form='blabla'))
# Check that expressions with nine variables or more are not simplified
# (without the force-flag)
a, b, c, d, e, f, g, h, j = symbols('a b c d e f g h j')
expr = a & b & c & d & e & f & g & h & j | \
a & b & c & d & e & f & g & h & ~j
# This expression can be simplified to get rid of the j variables
assert simplify_logic(expr) == expr
# check input
ans = SOPform([x, y], [[1, 0]])
assert SOPform([x, y], [[1, 0]]) == ans
assert POSform([x, y], [[1, 0]]) == ans
raises(ValueError, lambda: SOPform([x], [[1]], [[1]]))
assert SOPform([x], [[1]], [[0]]) is true
assert SOPform([x], [[0]], [[1]]) is true
assert SOPform([x], [], []) is false
raises(ValueError, lambda: POSform([x], [[1]], [[1]]))
assert POSform([x], [[1]], [[0]]) is true
assert POSform([x], [[0]], [[1]]) is true
assert POSform([x], [], []) is false
# check working of simplify
assert simplify((A & B) | (A & C)) == And(A, Or(B, C))
assert simplify(And(x, Not(x))) == False
assert simplify(Or(x, Not(x))) == True
assert simplify(And(Eq(x, 0), Eq(x, y))) == And(Eq(x, 0), Eq(y, 0))
assert And(Eq(x - 1, 0), Eq(x, y)).simplify() == And(Eq(x, 1), Eq(y, 1))
assert And(Ne(x - 1, 0), Ne(x, y)).simplify() == And(Ne(x, 1), Ne(x, y))
assert And(Eq(x - 1, 0), Ne(x, y)).simplify() == And(Eq(x, 1), Ne(y, 1))
assert And(Eq(x - 1, 0), Eq(x, z + y), Eq(y + x, 0)).simplify(
) == And(Eq(x, 1), Eq(y, -1), Eq(z, 2))
assert And(Eq(x - 1, 0), Eq(x + 2, 3)).simplify() == Eq(x, 1)
assert And(Ne(x - 1, 0), Ne(x + 2, 3)).simplify() == Ne(x, 1)
assert And(Eq(x - 1, 0), Eq(x + 2, 2)).simplify() == False
assert And(Ne(x - 1, 0), Ne(x + 2, 2)).simplify(
) == And(Ne(x, 1), Ne(x, 0))
def test_bool_map():
"""
Test working of bool_map function.
"""
minterms = [[0, 0, 0, 1], [0, 0, 1, 1], [0, 1, 1, 1], [1, 0, 1, 1],
[1, 1, 1, 1]]
assert bool_map(Not(Not(a)), a) == (a, {a: a})
assert bool_map(SOPform([w, x, y, z], minterms),
POSform([w, x, y, z], minterms)) == \
(And(Or(Not(w), y), Or(Not(x), y), z), {x: x, w: w, z: z, y: y})
assert bool_map(SOPform([x, z, y], [[1, 0, 1]]),
SOPform([a, b, c], [[1, 0, 1]])) != False
function1 = SOPform([x, z, y], [[1, 0, 1], [0, 0, 1]])
function2 = SOPform([a, b, c], [[1, 0, 1], [1, 0, 0]])
assert bool_map(function1, function2) == \
(function1, {y: a, z: b})
assert bool_map(Xor(x, y), ~Xor(x, y)) == False
assert bool_map(And(x, y), Or(x, y)) is None
assert bool_map(And(x, y), And(x, y, z)) is None
# issue 16179
assert bool_map(Xor(x, y, z), ~Xor(x, y, z)) == False
assert bool_map(Xor(a, x, y, z), ~Xor(a, x, y, z)) == False
def test_bool_symbol():
"""Test that mixing symbols with boolean values
works as expected"""
assert And(A, True) == A
assert And(A, True, True) == A
assert And(A, False) is false
assert And(A, True, False) is false
assert Or(A, True) is true
assert Or(A, False) == A
def test_is_boolean():
assert isinstance(True, Boolean) is False
assert isinstance(true, Boolean) is True
assert 1 == True
assert 1 != true
assert (1 == true) is False
assert 0 == False
assert 0 != false
assert (0 == false) is False
assert true.is_Boolean is True
assert (A & B).is_Boolean
assert (A | B).is_Boolean
assert (~A).is_Boolean
assert (A ^ B).is_Boolean
assert A.is_Boolean != isinstance(A, Boolean)
assert isinstance(A, Boolean)
def test_subs():
assert (A & B).subs(A, True) == B
assert (A & B).subs(A, False) is false
assert (A & B).subs(B, True) == A
assert (A & B).subs(B, False) is false
assert (A & B).subs({A: True, B: True}) is true
assert (A | B).subs(A, True) is true
assert (A | B).subs(A, False) == B
assert (A | B).subs(B, True) is true
assert (A | B).subs(B, False) == A
assert (A | B).subs({A: True, B: True}) is true
"""
we test for axioms of boolean algebra
see https://en.wikipedia.org/wiki/Boolean_algebra_(structure)
"""
def test_commutative():
"""Test for commutativity of And and Or"""
A, B = map(Boolean, symbols('A,B'))
assert A & B == B & A
assert A | B == B | A
def test_and_associativity():
"""Test for associativity of And"""
assert (A & B) & C == A & (B & C)
def test_or_assicativity():
assert ((A | B) | C) == (A | (B | C))
def test_double_negation():
a = Boolean()
assert ~(~a) == a
# test methods
def test_eliminate_implications():
assert eliminate_implications(Implies(A, B, evaluate=False)) == (~A) | B
assert eliminate_implications(
A >> (C >> Not(B))) == Or(Or(Not(B), Not(C)), Not(A))
assert eliminate_implications(Equivalent(A, B, C, D)) == \
(~A | B) & (~B | C) & (~C | D) & (~D | A)
def test_conjuncts():
assert conjuncts(A & B & C) == {A, B, C}
assert conjuncts((A | B) & C) == {A | B, C}
assert conjuncts(A) == {A}
assert conjuncts(True) == {True}
assert conjuncts(False) == {False}
def test_disjuncts():
assert disjuncts(A | B | C) == {A, B, C}
assert disjuncts((A | B) & C) == {(A | B) & C}
assert disjuncts(A) == {A}
assert disjuncts(True) == {True}
assert disjuncts(False) == {False}
def test_distribute():
assert distribute_and_over_or(Or(And(A, B), C)) == And(Or(A, C), Or(B, C))
assert distribute_or_over_and(And(A, Or(B, C))) == Or(And(A, B), And(A, C))
assert distribute_xor_over_and(And(A, Xor(B, C))) == Xor(And(A, B), And(A, C))
def test_to_anf():
x, y, z = symbols('x,y,z')
assert to_anf(And(x, y)) == And(x, y)
assert to_anf(Or(x, y)) == Xor(x, y, And(x, y))
assert to_anf(Or(Implies(x, y), And(x, y), y)) == \
Xor(x, True, x & y, remove_true=False)
assert to_anf(Or(Nand(x, y), Nor(x, y), Xnor(x, y), Implies(x, y))) == True
assert to_anf(Or(x, Not(y), Nor(x,z), And(x, y), Nand(y, z))) == \
Xor(True, And(y, z), And(x, y, z), remove_true=False)
assert to_anf(Xor(x, y)) == Xor(x, y)
assert to_anf(Not(x)) == Xor(x, True, remove_true=False)
assert to_anf(Nand(x, y)) == Xor(True, And(x, y), remove_true=False)
assert to_anf(Nor(x, y)) == Xor(x, y, True, And(x, y), remove_true=False)
assert to_anf(Implies(x, y)) == Xor(x, True, And(x, y), remove_true=False)
assert to_anf(Equivalent(x, y)) == Xor(x, y, True, remove_true=False)
assert to_anf(Nand(x | y, x >> y), deep=False) == \
Xor(True, And(Or(x, y), Implies(x, y)), remove_true=False)
assert to_anf(Nor(x ^ y, x & y), deep=False) == \
Xor(True, Or(Xor(x, y), And(x, y)), remove_true=False)
def test_to_nnf():
assert to_nnf(true) is true
assert to_nnf(false) is false
assert to_nnf(A) == A
assert to_nnf(A | ~A | B) is true
assert to_nnf(A & ~A & B) is false
assert to_nnf(A >> B) == ~A | B
assert to_nnf(Equivalent(A, B, C)) == (~A | B) & (~B | C) & (~C | A)
assert to_nnf(A ^ B ^ C) == \
(A | B | C) & (~A | ~B | C) & (A | ~B | ~C) & (~A | B | ~C)
assert to_nnf(ITE(A, B, C)) == (~A | B) & (A | C)
assert to_nnf(Not(A | B | C)) == ~A & ~B & ~C
assert to_nnf(Not(A & B & C)) == ~A | ~B | ~C
assert to_nnf(Not(A >> B)) == A & ~B
assert to_nnf(Not(Equivalent(A, B, C))) == And(Or(A, B, C), Or(~A, ~B, ~C))
assert to_nnf(Not(A ^ B ^ C)) == \
(~A | B | C) & (A | ~B | C) & (A | B | ~C) & (~A | ~B | ~C)
assert to_nnf(Not(ITE(A, B, C))) == (~A | ~B) & (A | ~C)
assert to_nnf((A >> B) ^ (B >> A)) == (A & ~B) | (~A & B)
assert to_nnf((A >> B) ^ (B >> A), False) == \
(~A | ~B | A | B) & ((A & ~B) | (~A & B))
assert ITE(A, 1, 0).to_nnf() == A
assert ITE(A, 0, 1).to_nnf() == ~A
# although ITE can hold non-Boolean, it will complain if
# an attempt is made to convert the ITE to Boolean nnf
raises(TypeError, lambda: ITE(A < 1, [1], B).to_nnf())
def test_to_cnf():
assert to_cnf(~(B | C)) == And(Not(B), Not(C))
assert to_cnf((A & B) | C) == And(Or(A, C), Or(B, C))
assert to_cnf(A >> B) == (~A) | B
assert to_cnf(A >> (B & C)) == (~A | B) & (~A | C)
assert to_cnf(A & (B | C) | ~A & (B | C), True) == B | C
assert to_cnf(A & B) == And(A, B)
assert to_cnf(Equivalent(A, B)) == And(Or(A, Not(B)), Or(B, Not(A)))
assert to_cnf(Equivalent(A, B & C)) == \
(~A | B) & (~A | C) & (~B | ~C | A)
assert to_cnf(Equivalent(A, B | C), True) == \
And(Or(Not(B), A), Or(Not(C), A), Or(B, C, Not(A)))
assert to_cnf(A + 1) == A + 1
def test_issue_18904():
x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15 = symbols('x1:16')
eq = (( x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 ) |
( x1 & x2 & x3 & x4 & x5 & x6 & x7 & x10 & x9 ) |
( x1 & x11 & x3 & x12 & x5 & x13 & x14 & x15 & x9 ))
assert is_cnf(to_cnf(eq))
raises(ValueError, lambda: to_cnf(eq, simplify=True))
for f, t in zip((And, Or), (to_cnf, to_dnf)):
eq = f(x1, x2, x3, x4, x5, x6, x7, x8, x9)
raises(ValueError, lambda: to_cnf(eq, simplify=True))
assert t(eq, simplify=True, force=True) == eq
def test_issue_9949():
assert is_cnf(to_cnf((b > -5) | (a > 2) & (a < 4)))
def test_to_CNF():
assert CNF.CNF_to_cnf(CNF.to_CNF(~(B | C))) == to_cnf(~(B | C))
assert CNF.CNF_to_cnf(CNF.to_CNF((A & B) | C)) == to_cnf((A & B) | C)
assert CNF.CNF_to_cnf(CNF.to_CNF(A >> B)) == to_cnf(A >> B)
assert CNF.CNF_to_cnf(CNF.to_CNF(A >> (B & C))) == to_cnf(A >> (B & C))
assert CNF.CNF_to_cnf(CNF.to_CNF(A & (B | C) | ~A & (B | C))) == to_cnf(A & (B | C) | ~A & (B | C))
assert CNF.CNF_to_cnf(CNF.to_CNF(A & B)) == to_cnf(A & B)
def test_to_dnf():
assert to_dnf(~(B | C)) == And(Not(B), Not(C))
assert to_dnf(A & (B | C)) == Or(And(A, B), And(A, C))
assert to_dnf(A >> B) == (~A) | B
assert to_dnf(A >> (B & C)) == (~A) | (B & C)
assert to_dnf(A | B) == A | B
assert to_dnf(Equivalent(A, B), True) == \
Or(And(A, B), And(Not(A), Not(B)))
assert to_dnf(Equivalent(A, B & C), True) == \
Or(And(A, B, C), And(Not(A), Not(B)), And(Not(A), Not(C)))
assert to_dnf(A + 1) == A + 1
def test_to_int_repr():
x, y, z = map(Boolean, symbols('x,y,z'))
def sorted_recursive(arg):
try:
return sorted(sorted_recursive(x) for x in arg)
except TypeError: # arg is not a sequence
return arg
assert sorted_recursive(to_int_repr([x | y, z | x], [x, y, z])) == \
sorted_recursive([[1, 2], [1, 3]])
assert sorted_recursive(to_int_repr([x | y, z | ~x], [x, y, z])) == \
sorted_recursive([[1, 2], [3, -1]])
def test_is_anf():
x, y = symbols('x,y')
assert is_anf(true) is True
assert is_anf(false) is True
assert is_anf(x) is True
assert is_anf(And(x, y)) is True
assert is_anf(Xor(x, y, And(x, y))) is True
assert is_anf(Xor(x, y, Or(x, y))) is False
assert is_anf(Xor(Not(x), y)) is False
def test_is_nnf():
assert is_nnf(true) is True
assert is_nnf(A) is True
assert is_nnf(~A) is True
assert is_nnf(A & B) is True
assert is_nnf((A & B) | (~A & A) | (~B & B) | (~A & ~B), False) is True
assert is_nnf((A | B) & (~A | ~B)) is True
assert is_nnf(Not(Or(A, B))) is False
assert is_nnf(A ^ B) is False
assert is_nnf((A & B) | (~A & A) | (~B & B) | (~A & ~B), True) is False
def test_is_cnf():
assert is_cnf(x) is True
assert is_cnf(x | y | z) is True
assert is_cnf(x & y & z) is True
assert is_cnf((x | y) & z) is True
assert is_cnf((x & y) | z) is False
assert is_cnf(~(x & y) | z) is False
def test_is_dnf():
assert is_dnf(x) is True
assert is_dnf(x | y | z) is True
assert is_dnf(x & y & z) is True
assert is_dnf((x & y) | z) is True
assert is_dnf((x | y) & z) is False
assert is_dnf(~(x | y) & z) is False
def test_ITE():
A, B, C = symbols('A:C')
assert ITE(True, False, True) is false
assert ITE(True, True, False) is true
assert ITE(False, True, False) is false
assert ITE(False, False, True) is true
assert isinstance(ITE(A, B, C), ITE)
A = True
assert ITE(A, B, C) == B
A = False
assert ITE(A, B, C) == C
B = True
assert ITE(And(A, B), B, C) == C
assert ITE(Or(A, False), And(B, True), False) is false
assert ITE(x, A, B) == Not(x)
assert ITE(x, B, A) == x
assert ITE(1, x, y) == x
assert ITE(0, x, y) == y
raises(TypeError, lambda: ITE(2, x, y))
raises(TypeError, lambda: ITE(1, [], y))
raises(TypeError, lambda: ITE(1, (), y))
raises(TypeError, lambda: ITE(1, y, []))
assert ITE(1, 1, 1) is S.true
assert isinstance(ITE(1, 1, 1, evaluate=False), ITE)
raises(TypeError, lambda: ITE(x > 1, y, x))
assert ITE(Eq(x, True), y, x) == ITE(x, y, x)
assert ITE(Eq(x, False), y, x) == ITE(~x, y, x)
assert ITE(Ne(x, True), y, x) == ITE(~x, y, x)
assert ITE(Ne(x, False), y, x) == ITE(x, y, x)
assert ITE(Eq(S. true, x), y, x) == ITE(x, y, x)
assert ITE(Eq(S.false, x), y, x) == ITE(~x, y, x)
assert ITE(Ne(S.true, x), y, x) == ITE(~x, y, x)
assert ITE(Ne(S.false, x), y, x) == ITE(x, y, x)
# 0 and 1 in the context are not treated as True/False
# so the equality must always be False since dissimilar
# objects cannot be equal
assert ITE(Eq(x, 0), y, x) == x
assert ITE(Eq(x, 1), y, x) == x
assert ITE(Ne(x, 0), y, x) == y
assert ITE(Ne(x, 1), y, x) == y
assert ITE(Eq(x, 0), y, z).subs(x, 0) == y
assert ITE(Eq(x, 0), y, z).subs(x, 1) == z
raises(ValueError, lambda: ITE(x > 1, y, x, z))
def test_is_literal():
assert is_literal(True) is True
assert is_literal(False) is True
assert is_literal(A) is True
assert is_literal(~A) is True
assert is_literal(Or(A, B)) is False
assert is_literal(Q.zero(A)) is True
assert is_literal(Not(Q.zero(A))) is True
assert is_literal(Or(A, B)) is False
assert is_literal(And(Q.zero(A), Q.zero(B))) is False
assert is_literal(x < 3)
assert not is_literal(x + y < 3)
def test_operators():
# Mostly test __and__, __rand__, and so on
assert True & A == A & True == A
assert False & A == A & False == False
assert A & B == And(A, B)
assert True | A == A | True == True
assert False | A == A | False == A
assert A | B == Or(A, B)
assert ~A == Not(A)
assert True >> A == A << True == A
assert False >> A == A << False == True
assert A >> True == True << A == True
assert A >> False == False << A == ~A
assert A >> B == B << A == Implies(A, B)
assert True ^ A == A ^ True == ~A
assert False ^ A == A ^ False == A
assert A ^ B == Xor(A, B)
def test_true_false():
assert true is S.true
assert false is S.false
assert true is not True
assert false is not False
assert true
assert not false
assert true == True
assert false == False
assert not (true == False)
assert not (false == True)
assert not (true == false)
assert hash(true) == hash(True)
assert hash(false) == hash(False)
assert len({true, True}) == len({false, False}) == 1
assert isinstance(true, BooleanAtom)
assert isinstance(false, BooleanAtom)
# We don't want to subclass from bool, because bool subclasses from
# int. But operators like &, |, ^, <<, >>, and ~ act differently on 0 and
# 1 then we want them to on true and false. See the docstrings of the
# various And, Or, etc. functions for examples.
assert not isinstance(true, bool)
assert not isinstance(false, bool)
# Note: using 'is' comparison is important here. We want these to return
# true and false, not True and False
assert Not(true) is false
assert Not(True) is false
assert Not(false) is true
assert Not(False) is true
assert ~true is false
assert ~false is true
for T, F in cartes([True, true], [False, false]):
assert And(T, F) is false
assert And(F, T) is false
assert And(F, F) is false
assert And(T, T) is true
assert And(T, x) == x
assert And(F, x) is false
if not (T is True and F is False):
assert T & F is false
assert F & T is false
if F is not False:
assert F & F is false
if T is not True:
assert T & T is true
assert Or(T, F) is true
assert Or(F, T) is true
assert Or(F, F) is false
assert Or(T, T) is true
assert Or(T, x) is true
assert Or(F, x) == x
if not (T is True and F is False):
assert T | F is true
assert F | T is true
if F is not False:
assert F | F is false
if T is not True:
assert T | T is true
assert Xor(T, F) is true
assert Xor(F, T) is true
assert Xor(F, F) is false
assert Xor(T, T) is false
assert Xor(T, x) == ~x
assert Xor(F, x) == x
if not (T is True and F is False):
assert T ^ F is true
assert F ^ T is true
if F is not False:
assert F ^ F is false
if T is not True:
assert T ^ T is false
assert Nand(T, F) is true
assert Nand(F, T) is true
assert Nand(F, F) is true
assert Nand(T, T) is false
assert Nand(T, x) == ~x
assert Nand(F, x) is true
assert Nor(T, F) is false
assert Nor(F, T) is false
assert Nor(F, F) is true
assert Nor(T, T) is false
assert Nor(T, x) is false
assert Nor(F, x) == ~x
assert Implies(T, F) is false
assert Implies(F, T) is true
assert Implies(F, F) is true
assert Implies(T, T) is true
assert Implies(T, x) == x
assert Implies(F, x) is true
assert Implies(x, T) is true
assert Implies(x, F) == ~x
if not (T is True and F is False):
assert T >> F is false
assert F << T is false
assert F >> T is true
assert T << F is true
if F is not False:
assert F >> F is true
assert F << F is true
if T is not True:
assert T >> T is true
assert T << T is true
assert Equivalent(T, F) is false
assert Equivalent(F, T) is false
assert Equivalent(F, F) is true
assert Equivalent(T, T) is true
assert Equivalent(T, x) == x
assert Equivalent(F, x) == ~x
assert Equivalent(x, T) == x
assert Equivalent(x, F) == ~x
assert ITE(T, T, T) is true
assert ITE(T, T, F) is true
assert ITE(T, F, T) is false
assert ITE(T, F, F) is false
assert ITE(F, T, T) is true
assert ITE(F, T, F) is false
assert ITE(F, F, T) is true
assert ITE(F, F, F) is false
assert all(i.simplify(1, 2) is i for i in (S.true, S.false))
def test_bool_as_set():
assert ITE(y <= 0, False, y >= 1).as_set() == Interval(1, oo)
assert And(x <= 2, x >= -2).as_set() == Interval(-2, 2)
assert Or(x >= 2, x <= -2).as_set() == Interval(-oo, -2) + Interval(2, oo)
assert Not(x > 2).as_set() == Interval(-oo, 2)
# issue 10240
assert Not(And(x > 2, x < 3)).as_set() == \
Union(Interval(-oo, 2), Interval(3, oo))
assert true.as_set() == S.UniversalSet
assert false.as_set() == EmptySet()
assert x.as_set() == S.UniversalSet
assert And(Or(x < 1, x > 3), x < 2).as_set() == Interval.open(-oo, 1)
assert And(x < 1, sin(x) < 3).as_set() == (x < 1).as_set()
raises(NotImplementedError, lambda: (sin(x) < 1).as_set())
@XFAIL
def test_multivariate_bool_as_set():
x, y = symbols('x,y')
assert And(x >= 0, y >= 0).as_set() == Interval(0, oo)*Interval(0, oo)
assert Or(x >= 0, y >= 0).as_set() == S.Reals*S.Reals - \
Interval(-oo, 0, True, True)*Interval(-oo, 0, True, True)
def test_all_or_nothing():
x = symbols('x', extended_real=True)
args = x >= -oo, x <= oo
v = And(*args)
if v.func is And:
assert len(v.args) == len(args) - args.count(S.true)
else:
assert v == True
v = Or(*args)
if v.func is Or:
assert len(v.args) == 2
else:
assert v == True
def test_canonical_atoms():
assert true.canonical == true
assert false.canonical == false
def test_negated_atoms():
assert true.negated == false
assert false.negated == true
def test_issue_8777():
assert And(x > 2, x < oo).as_set() == Interval(2, oo, left_open=True)
assert And(x >= 1, x < oo).as_set() == Interval(1, oo)
assert (x < oo).as_set() == Interval(-oo, oo)
assert (x > -oo).as_set() == Interval(-oo, oo)
def test_issue_8975():
assert Or(And(-oo < x, x <= -2), And(2 <= x, x < oo)).as_set() == \
Interval(-oo, -2) + Interval(2, oo)
def test_term_to_integer():
assert term_to_integer([1, 0, 1, 0, 0, 1, 0]) == 82
assert term_to_integer('0010101000111001') == 10809
def test_integer_to_term():
assert integer_to_term(777) == [1, 1, 0, 0, 0, 0, 1, 0, 0, 1]
assert integer_to_term(123, 3) == [1, 1, 1, 1, 0, 1, 1]
assert integer_to_term(456, 16) == [0, 0, 0, 0, 0, 0, 0, 1,
1, 1, 0, 0, 1, 0, 0, 0]
def test_truth_table():
assert list(truth_table(And(x, y), [x, y], input=False)) == \
[False, False, False, True]
assert list(truth_table(x | y, [x, y], input=False)) == \
[False, True, True, True]
assert list(truth_table(x >> y, [x, y], input=False)) == \
[True, True, False, True]
assert list(truth_table(And(x, y), [x, y])) == \
[([0, 0], False), ([0, 1], False), ([1, 0], False), ([1, 1], True)]
def test_issue_8571():
for t in (S.true, S.false):
raises(TypeError, lambda: +t)
raises(TypeError, lambda: -t)
raises(TypeError, lambda: abs(t))
# use int(bool(t)) to get 0 or 1
raises(TypeError, lambda: int(t))
for o in [S.Zero, S.One, x]:
for _ in range(2):
raises(TypeError, lambda: o + t)
raises(TypeError, lambda: o - t)
raises(TypeError, lambda: o % t)
raises(TypeError, lambda: o*t)
raises(TypeError, lambda: o/t)
raises(TypeError, lambda: o**t)
o, t = t, o # do again in reversed order
def test_expand_relational():
n = symbols('n', negative=True)
p, q = symbols('p q', positive=True)
r = ((n + q*(-n/q + 1))/(q*(-n/q + 1)) < 0)
assert r is not S.false
assert r.expand() is S.false
assert (q > 0).expand() is S.true
def test_issue_12717():
assert S.true.is_Atom == True
assert S.false.is_Atom == True
def test_as_Boolean():
nz = symbols('nz', nonzero=True)
assert all(as_Boolean(i) is S.true for i in (True, S.true, 1, nz))
z = symbols('z', zero=True)
assert all(as_Boolean(i) is S.false for i in (False, S.false, 0, z))
assert all(as_Boolean(i) == i for i in (x, x < 0))
for i in (2, S(2), x + 1, []):
raises(TypeError, lambda: as_Boolean(i))
def test_binary_symbols():
assert ITE(x < 1, y, z).binary_symbols == {y, z}
for f in (Eq, Ne):
assert f(x, 1).binary_symbols == set()
assert f(x, True).binary_symbols == {x}
assert f(x, False).binary_symbols == {x}
assert S.true.binary_symbols == set()
assert S.false.binary_symbols == set()
assert x.binary_symbols == {x}
assert And(x, Eq(y, False), Eq(z, 1)).binary_symbols == {x, y}
assert Q.prime(x).binary_symbols == set()
assert Q.is_true(x < 1).binary_symbols == set()
assert Q.is_true(x).binary_symbols == {x}
assert Q.is_true(Eq(x, True)).binary_symbols == {x}
assert Q.prime(x).binary_symbols == set()
def test_BooleanFunction_diff():
assert And(x, y).diff(x) == Piecewise((0, Eq(y, False)), (1, True))
def test_issue_14700():
A, B, C, D, E, F, G, H = symbols('A B C D E F G H')
q = ((B & D & H & ~F) | (B & H & ~C & ~D) | (B & H & ~C & ~F) |
(B & H & ~D & ~G) | (B & H & ~F & ~G) | (C & G & ~B & ~D) |
(C & G & ~D & ~H) | (C & G & ~F & ~H) | (D & F & H & ~B) |
(D & F & ~G & ~H) | (B & D & F & ~C & ~H) | (D & E & F & ~B & ~C) |
(D & F & ~A & ~B & ~C) | (D & F & ~A & ~C & ~H) |
(A & B & D & F & ~E & ~H))
soldnf = ((B & D & H & ~F) | (D & F & H & ~B) | (B & H & ~C & ~D) |
(B & H & ~D & ~G) | (C & G & ~B & ~D) | (C & G & ~D & ~H) |
(C & G & ~F & ~H) | (D & F & ~G & ~H) | (D & E & F & ~C & ~H) |
(D & F & ~A & ~C & ~H) | (A & B & D & F & ~E & ~H))
solcnf = ((B | C | D) & (B | D | G) & (C | D | H) & (C | F | H) &
(D | G | H) & (F | G | H) & (B | F | ~D | ~H) &
(~B | ~D | ~F | ~H) & (D | ~B | ~C | ~G | ~H) &
(A | H | ~C | ~D | ~F | ~G) & (H | ~C | ~D | ~E | ~F | ~G) &
(B | E | H | ~A | ~D | ~F | ~G))
assert simplify_logic(q, "dnf") == soldnf
assert simplify_logic(q, "cnf") == solcnf
minterms = [[0, 1, 0, 0], [0, 1, 0, 1], [0, 1, 1, 0], [0, 1, 1, 1],
[0, 0, 1, 1], [1, 0, 1, 1]]
dontcares = [[1, 0, 0, 0], [1, 0, 0, 1], [1, 1, 0, 0], [1, 1, 0, 1]]
assert SOPform([w, x, y, z], minterms) == (x & ~w) | (y & z & ~x)
# Should not be more complicated with don't cares
assert SOPform([w, x, y, z], minterms, dontcares) == \
(x & ~w) | (y & z & ~x)
def test_relational_simplification():
w, x, y, z = symbols('w x y z', real=True)
d, e = symbols('d e', real=False)
# Test all combinations or sign and order
assert Or(x >= y, x < y).simplify() == S.true
assert Or(x >= y, y > x).simplify() == S.true
assert Or(x >= y, -x > -y).simplify() == S.true
assert Or(x >= y, -y < -x).simplify() == S.true
assert Or(-x <= -y, x < y).simplify() == S.true
assert Or(-x <= -y, -x > -y).simplify() == S.true
assert Or(-x <= -y, y > x).simplify() == S.true
assert Or(-x <= -y, -y < -x).simplify() == S.true
assert Or(y <= x, x < y).simplify() == S.true
assert Or(y <= x, y > x).simplify() == S.true
assert Or(y <= x, -x > -y).simplify() == S.true
assert Or(y <= x, -y < -x).simplify() == S.true
assert Or(-y >= -x, x < y).simplify() == S.true
assert Or(-y >= -x, y > x).simplify() == S.true
assert Or(-y >= -x, -x > -y).simplify() == S.true
assert Or(-y >= -x, -y < -x).simplify() == S.true
assert Or(x < y, x >= y).simplify() == S.true
assert Or(y > x, x >= y).simplify() == S.true
assert Or(-x > -y, x >= y).simplify() == S.true
assert Or(-y < -x, x >= y).simplify() == S.true
assert Or(x < y, -x <= -y).simplify() == S.true
assert Or(-x > -y, -x <= -y).simplify() == S.true
assert Or(y > x, -x <= -y).simplify() == S.true
assert Or(-y < -x, -x <= -y).simplify() == S.true
assert Or(x < y, y <= x).simplify() == S.true
assert Or(y > x, y <= x).simplify() == S.true
assert Or(-x > -y, y <= x).simplify() == S.true
assert Or(-y < -x, y <= x).simplify() == S.true
assert Or(x < y, -y >= -x).simplify() == S.true
assert Or(y > x, -y >= -x).simplify() == S.true
assert Or(-x > -y, -y >= -x).simplify() == S.true
assert Or(-y < -x, -y >= -x).simplify() == S.true
# Some other tests
assert Or(x >= y, w < z, x <= y).simplify() == S.true
assert And(x >= y, x < y).simplify() == S.false
assert Or(x >= y, Eq(y, x)).simplify() == (x >= y)
assert And(x >= y, Eq(y, x)).simplify() == Eq(x, y)
assert Or(Eq(x, y), x >= y, w < y, z < y).simplify() == \
Or(x >= y, y > Min(w, z))
assert And(Eq(x, y), x >= y, w < y, y >= z, z < y).simplify() == \
And(Eq(x, y), y > Max(w, z))
assert Or(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y).simplify() == \
(Eq(x, y) | (x >= 1) | (y > Min(2, z)))
assert And(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y).simplify() == \
(Eq(x, y) & (x >= 1) & (y >= 5) & (y > z))
assert (Eq(x, y) & Eq(d, e) & (x >= y) & (d >= e)).simplify() == \
(Eq(x, y) & Eq(d, e) & (d >= e))
assert And(Eq(x, y), Eq(x, -y)).simplify() == And(Eq(x, 0), Eq(y, 0))
assert Xor(x >= y, x <= y).simplify() == Ne(x, y)
@slow
def test_relational_simplification_numerically():
def test_simplification_numerically_function(original, simplified):
symb = original.free_symbols
n = len(symb)
valuelist = list(set(list(combinations(list(range(-(n-1), n))*n, n))))
for values in valuelist:
sublist = dict(zip(symb, values))
originalvalue = original.subs(sublist)
simplifiedvalue = simplified.subs(sublist)
assert originalvalue == simplifiedvalue, "Original: {}\nand"\
" simplified: {}\ndo not evaluate to the same value for {}"\
"".format(original, simplified, sublist)
w, x, y, z = symbols('w x y z', real=True)
d, e = symbols('d e', real=False)
expressions = (And(Eq(x, y), x >= y, w < y, y >= z, z < y),
And(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y),
Or(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y),
And(x >= y, Eq(y, x)),
Or(And(Eq(x, y), x >= y, w < y, Or(y >= z, z < y)),
And(Eq(x, y), x >= 1, 2 < y, y >= -1, z < y)),
(Eq(x, y) & Eq(d, e) & (x >= y) & (d >= e)),
)
for expression in expressions:
test_simplification_numerically_function(expression,
expression.simplify())
def test_relational_simplification_patterns_numerically():
from sympy.core import Wild
from sympy.logic.boolalg import simplify_patterns_and, \
simplify_patterns_or, simplify_patterns_xor
a = Wild('a')
b = Wild('b')
c = Wild('c')
symb = [a, b, c]
patternlists = [simplify_patterns_and(), simplify_patterns_or(),
simplify_patterns_xor()]
for patternlist in patternlists:
for pattern in patternlist:
original = pattern[0]
simplified = pattern[1]
valuelist = list(set(list(combinations(list(range(-2, 2))*3, 3))))
for values in valuelist:
sublist = dict(zip(symb, values))
originalvalue = original.subs(sublist)
simplifiedvalue = simplified.subs(sublist)
assert originalvalue == simplifiedvalue, "Original: {}\nand"\
" simplified: {}\ndo not evaluate to the same value for"\
"{}".format(original, simplified, sublist)
def test_issue_16803():
n = symbols('n')
# No simplification done, but should not raise an exception
assert ((n > 3) | (n < 0) | ((n > 0) & (n < 3))).simplify() == \
((n > 3) | (n < 0) | ((n > 0) & (n < 3)))
def test_issue_17530():
r = {x: oo, y: oo}
assert Or(x + y > 0, x - y < 0).subs(r)
assert not And(x + y < 0, x - y < 0).subs(r)
raises(TypeError, lambda: Or(x + y < 0, x - y < 0).subs(r))
raises(TypeError, lambda: And(x + y > 0, x - y < 0).subs(r))
raises(TypeError, lambda: And(x + y > 0, x - y < 0).subs(r))
def test_anf_coeffs():
assert anf_coeffs([1, 0]) == [1, 1]
assert anf_coeffs([0, 0, 0, 1]) == [0, 0, 0, 1]
assert anf_coeffs([0, 1, 1, 1]) == [0, 1, 1, 1]
assert anf_coeffs([1, 1, 1, 0]) == [1, 0, 0, 1]
assert anf_coeffs([1, 0, 0, 0]) == [1, 1, 1, 1]
assert anf_coeffs([1, 0, 0, 1]) == [1, 1, 1, 0]
assert anf_coeffs([1, 1, 0, 1]) == [1, 0, 1, 1]
def test_ANFform():
x, y = symbols('x,y')
assert ANFform([x], [1, 1]) == True
assert ANFform([x], [0, 0]) == False
assert ANFform([x], [1, 0]) == Xor(x, True, remove_true=False)
assert ANFform([x, y], [1, 1, 1, 0]) == \
Xor(True, And(x, y), remove_true=False)
def test_bool_minterm():
x, y = symbols('x,y')
assert bool_minterm(3, [x, y]) == And(x, y)
assert bool_minterm([1, 0], [x, y]) == And(Not(y), x)
def test_bool_maxterm():
x, y = symbols('x,y')
assert bool_maxterm(2, [x, y]) == Or(Not(x), y)
assert bool_maxterm([0, 1], [x, y]) == Or(Not(y), x)
def test_bool_monomial():
x, y = symbols('x,y')
assert bool_monomial(1, [x, y]) == y
assert bool_monomial([1, 1], [x, y]) == And(x, y)
def test_check_pair():
assert _check_pair([0, 1, 0], [0, 1, 1]) == 2
assert _check_pair([0, 1, 0], [1, 1, 1]) == -1
def test_convert_to_varsSOP():
assert _convert_to_varsSOP([0, 1, 0], [x, y, z]) == And(Not(x), y, Not(z))
assert _convert_to_varsSOP([3, 1, 0], [x, y, z]) == And(y, Not(z))
def test_convert_to_varsPOS():
assert _convert_to_varsPOS([0, 1, 0], [x, y, z]) == Or(x, Not(y), z)
assert _convert_to_varsPOS([3, 1, 0], [x, y, z]) == Or(Not(y), z)
def test_refine():
# relational
assert not refine(x < 0, ~Q.is_true(x < 0))
assert refine(x < 0, Q.is_true(x < 0))
assert refine(x < 0, Q.is_true(0 > x)) == True
assert refine(x < 0, Q.is_true(y < 0)) == (x < 0)
assert not refine(x <= 0, ~Q.is_true(x <= 0))
assert refine(x <= 0, Q.is_true(x <= 0))
assert refine(x <= 0, Q.is_true(0 >= x)) == True
assert refine(x <= 0, Q.is_true(y <= 0)) == (x <= 0)
assert not refine(x > 0, ~Q.is_true(x > 0))
assert refine(x > 0, Q.is_true(x > 0))
assert refine(x > 0, Q.is_true(0 < x)) == True
assert refine(x > 0, Q.is_true(y > 0)) == (x > 0)
assert not refine(x >= 0, ~Q.is_true(x >= 0))
assert refine(x >= 0, Q.is_true(x >= 0))
assert refine(x >= 0, Q.is_true(0 <= x)) == True
assert refine(x >= 0, Q.is_true(y >= 0)) == (x >= 0)
assert not refine(Eq(x, 0), ~Q.is_true(Eq(x, 0)))
assert refine(Eq(x, 0), Q.is_true(Eq(x, 0)))
assert refine(Eq(x, 0), Q.is_true(Eq(0, x))) == True
assert refine(Eq(x, 0), Q.is_true(Eq(y, 0))) == Eq(x, 0)
assert not refine(Ne(x, 0), ~Q.is_true(Ne(x, 0)))
assert refine(Ne(x, 0), Q.is_true(Ne(0, x))) == True
assert refine(Ne(x, 0), Q.is_true(Ne(x, 0)))
assert refine(Ne(x, 0), Q.is_true(Ne(y, 0))) == (Ne(x, 0))
# boolean functions
assert refine(And(x > 0, y > 0), Q.is_true(x > 0)) == (y > 0)
assert refine(And(x > 0, y > 0), Q.is_true(x > 0) & Q.is_true(y > 0)) == True
# predicates
assert refine(Q.positive(x), Q.positive(x)) == True
assert refine(Q.positive(x), Q.negative(x)) == False
assert refine(Q.positive(x), Q.real(x)) == Q.positive(x)
|
71594457c701a1d03c31d0d2f8c96118440d4f7dfacf361fbbbdfb3b5a1e3459 | from sympy import Rational, I, expand_mul, S, simplify, sqrt
from sympy.matrices.matrices import NonSquareMatrixError
from sympy.matrices import Matrix, zeros, eye, SparseMatrix
from sympy.abc import x, y, z
from sympy.testing.pytest import raises
from sympy.testing.matrices import allclose
def test_LUdecomp():
testmat = Matrix([[0, 2, 5, 3],
[3, 3, 7, 4],
[8, 4, 0, 2],
[-2, 6, 3, 4]])
L, U, p = testmat.LUdecomposition()
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - testmat == zeros(4)
testmat = Matrix([[6, -2, 7, 4],
[0, 3, 6, 7],
[1, -2, 7, 4],
[-9, 2, 6, 3]])
L, U, p = testmat.LUdecomposition()
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - testmat == zeros(4)
# non-square
testmat = Matrix([[1, 2, 3],
[4, 5, 6],
[7, 8, 9],
[10, 11, 12]])
L, U, p = testmat.LUdecomposition(rankcheck=False)
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - testmat == zeros(4, 3)
# square and singular
testmat = Matrix([[1, 2, 3],
[2, 4, 6],
[4, 5, 6]])
L, U, p = testmat.LUdecomposition(rankcheck=False)
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - testmat == zeros(3)
M = Matrix(((1, x, 1), (2, y, 0), (y, 0, z)))
L, U, p = M.LUdecomposition()
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - M == zeros(3)
mL = Matrix((
(1, 0, 0),
(2, 3, 0),
))
assert mL.is_lower is True
assert mL.is_upper is False
mU = Matrix((
(1, 2, 3),
(0, 4, 5),
))
assert mU.is_lower is False
assert mU.is_upper is True
# test FF LUdecomp
M = Matrix([[1, 3, 3],
[3, 2, 6],
[3, 2, 2]])
P, L, Dee, U = M.LUdecompositionFF()
assert P*M == L*Dee.inv()*U
M = Matrix([[1, 2, 3, 4],
[3, -1, 2, 3],
[3, 1, 3, -2],
[6, -1, 0, 2]])
P, L, Dee, U = M.LUdecompositionFF()
assert P*M == L*Dee.inv()*U
M = Matrix([[0, 0, 1],
[2, 3, 0],
[3, 1, 4]])
P, L, Dee, U = M.LUdecompositionFF()
assert P*M == L*Dee.inv()*U
# issue 15794
M = Matrix(
[[1, 2, 3],
[4, 5, 6],
[7, 8, 9]]
)
raises(ValueError, lambda : M.LUdecomposition_Simple(rankcheck=True))
def test_singular_value_decompositionD():
A = Matrix([[1, 2], [2, 1]])
U, S, V = A.singular_value_decomposition()
assert U * S * V.T == A
assert U.T * U == eye(U.cols)
assert V.T * V == eye(V.cols)
B = Matrix([[1, 2]])
U, S, V = B.singular_value_decomposition()
assert U * S * V.T == B
assert U.T * U == eye(U.cols)
assert V.T * V == eye(V.cols)
C = Matrix([
[1, 0, 0, 0, 2],
[0, 0, 3, 0, 0],
[0, 0, 0, 0, 0],
[0, 2, 0, 0, 0],
])
U, S, V = C.singular_value_decomposition()
assert U * S * V.T == C
assert U.T * U == eye(U.cols)
assert V.T * V == eye(V.cols)
D = Matrix([[Rational(1, 3), sqrt(2)], [0, Rational(1, 4)]])
U, S, V = D.singular_value_decomposition()
assert simplify(U.T * U) == eye(U.cols)
assert simplify(V.T * V) == eye(V.cols)
assert simplify(U * S * V.T) == D
def test_QR():
A = Matrix([[1, 2], [2, 3]])
Q, S = A.QRdecomposition()
R = Rational
assert Q == Matrix([
[ 5**R(-1, 2), (R(2)/5)*(R(1)/5)**R(-1, 2)],
[2*5**R(-1, 2), (-R(1)/5)*(R(1)/5)**R(-1, 2)]])
assert S == Matrix([[5**R(1, 2), 8*5**R(-1, 2)], [0, (R(1)/5)**R(1, 2)]])
assert Q*S == A
assert Q.T * Q == eye(2)
A = Matrix([[1, 1, 1], [1, 1, 3], [2, 3, 4]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[12, 0, -51], [6, 0, 167], [-4, 0, 24]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
def test_QR_non_square():
# Narrow (cols < rows) matrices
A = Matrix([[9, 0, 26], [12, 0, -7], [0, 4, 4], [0, -3, -3]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[1, -1, 4], [1, 4, -2], [1, 4, 2], [1, -1, 0]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix(2, 1, [1, 2])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
# Wide (cols > rows) matrices
A = Matrix([[1, 2, 3], [4, 5, 6]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[1, 2, 3, 4], [1, 4, 9, 16], [1, 8, 27, 64]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix(1, 2, [1, 2])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
def test_QR_trivial():
# Rank deficient matrices
A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[1, 1, 1], [2, 2, 2], [3, 3, 3], [4, 4, 4]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[1, 1, 1], [2, 2, 2], [3, 3, 3], [4, 4, 4]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
# Zero rank matrices
A = Matrix([[0, 0, 0]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0], [0, 0, 0]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0], [0, 0, 0]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
# Rank deficient matrices with zero norm from beginning columns
A = Matrix([[0, 0, 0], [1, 2, 3]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0, 0], [1, 2, 3, 4], [0, 0, 0, 0]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0, 0], [1, 2, 3, 4], [0, 0, 0, 0], [2, 4, 6, 8]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0], [0, 0, 0], [0, 0, 0], [1, 2, 3]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
def test_QR_float():
A = Matrix([[1, 1], [1, 1.01]])
Q, R = A.QRdecomposition()
assert allclose(Q * R, A)
assert allclose(Q * Q.T, Matrix.eye(2))
assert allclose(Q.T * Q, Matrix.eye(2))
A = Matrix([[1, 1], [1, 1.001]])
Q, R = A.QRdecomposition()
assert allclose(Q * R, A)
assert allclose(Q * Q.T, Matrix.eye(2))
assert allclose(Q.T * Q, Matrix.eye(2))
def test_LUdecomposition_Simple_iszerofunc():
# Test if callable passed to matrices.LUdecomposition_Simple() as iszerofunc keyword argument is used inside
# matrices.LUdecomposition_Simple()
magic_string = "I got passed in!"
def goofyiszero(value):
raise ValueError(magic_string)
try:
lu, p = Matrix([[1, 0], [0, 1]]).LUdecomposition_Simple(iszerofunc=goofyiszero)
except ValueError as err:
assert magic_string == err.args[0]
return
assert False
def test_LUdecomposition_iszerofunc():
# Test if callable passed to matrices.LUdecomposition() as iszerofunc keyword argument is used inside
# matrices.LUdecomposition_Simple()
magic_string = "I got passed in!"
def goofyiszero(value):
raise ValueError(magic_string)
try:
l, u, p = Matrix([[1, 0], [0, 1]]).LUdecomposition(iszerofunc=goofyiszero)
except ValueError as err:
assert magic_string == err.args[0]
return
assert False
def test_LDLdecomposition():
raises(NonSquareMatrixError, lambda: Matrix((1, 2)).LDLdecomposition())
raises(ValueError, lambda: Matrix(((1, 2), (3, 4))).LDLdecomposition())
raises(ValueError, lambda: Matrix(((5 + I, 0), (0, 1))).LDLdecomposition())
raises(ValueError, lambda: Matrix(((1, 5), (5, 1))).LDLdecomposition())
raises(ValueError, lambda: Matrix(((1, 2), (3, 4))).LDLdecomposition(hermitian=False))
A = Matrix(((1, 5), (5, 1)))
L, D = A.LDLdecomposition(hermitian=False)
assert L * D * L.T == A
A = Matrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
L, D = A.LDLdecomposition()
assert L * D * L.T == A
assert L.is_lower
assert L == Matrix([[1, 0, 0], [ Rational(3, 5), 1, 0], [Rational(-1, 5), Rational(1, 3), 1]])
assert D.is_diagonal()
assert D == Matrix([[25, 0, 0], [0, 9, 0], [0, 0, 9]])
A = Matrix(((4, -2*I, 2 + 2*I), (2*I, 2, -1 + I), (2 - 2*I, -1 - I, 11)))
L, D = A.LDLdecomposition()
assert expand_mul(L * D * L.H) == A
assert L.expand() == Matrix([[1, 0, 0], [I/2, 1, 0], [S.Half - I/2, 0, 1]])
assert D.expand() == Matrix(((4, 0, 0), (0, 1, 0), (0, 0, 9)))
raises(NonSquareMatrixError, lambda: SparseMatrix((1, 2)).LDLdecomposition())
raises(ValueError, lambda: SparseMatrix(((1, 2), (3, 4))).LDLdecomposition())
raises(ValueError, lambda: SparseMatrix(((5 + I, 0), (0, 1))).LDLdecomposition())
raises(ValueError, lambda: SparseMatrix(((1, 5), (5, 1))).LDLdecomposition())
raises(ValueError, lambda: SparseMatrix(((1, 2), (3, 4))).LDLdecomposition(hermitian=False))
A = SparseMatrix(((1, 5), (5, 1)))
L, D = A.LDLdecomposition(hermitian=False)
assert L * D * L.T == A
A = SparseMatrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
L, D = A.LDLdecomposition()
assert L * D * L.T == A
assert L.is_lower
assert L == Matrix([[1, 0, 0], [ Rational(3, 5), 1, 0], [Rational(-1, 5), Rational(1, 3), 1]])
assert D.is_diagonal()
assert D == Matrix([[25, 0, 0], [0, 9, 0], [0, 0, 9]])
A = SparseMatrix(((4, -2*I, 2 + 2*I), (2*I, 2, -1 + I), (2 - 2*I, -1 - I, 11)))
L, D = A.LDLdecomposition()
assert expand_mul(L * D * L.H) == A
assert L == Matrix(((1, 0, 0), (I/2, 1, 0), (S.Half - I/2, 0, 1)))
assert D == Matrix(((4, 0, 0), (0, 1, 0), (0, 0, 9)))
def test_pinv_succeeds_with_rank_decomposition_method():
# Test rank decomposition method of pseudoinverse succeeding
As = [Matrix([
[61, 89, 55, 20, 71, 0],
[62, 96, 85, 85, 16, 0],
[69, 56, 17, 4, 54, 0],
[10, 54, 91, 41, 71, 0],
[ 7, 30, 10, 48, 90, 0],
[0,0,0,0,0,0]])]
for A in As:
A_pinv = A.pinv(method="RD")
AAp = A * A_pinv
ApA = A_pinv * A
assert simplify(AAp * A) == A
assert simplify(ApA * A_pinv) == A_pinv
assert AAp.H == AAp
assert ApA.H == ApA
def test_rank_decomposition():
a = Matrix(0, 0, [])
c, f = a.rank_decomposition()
assert f.is_echelon
assert c.cols == f.rows == a.rank()
assert c * f == a
a = Matrix(1, 1, [5])
c, f = a.rank_decomposition()
assert f.is_echelon
assert c.cols == f.rows == a.rank()
assert c * f == a
a = Matrix(3, 3, [1, 2, 3, 1, 2, 3, 1, 2, 3])
c, f = a.rank_decomposition()
assert f.is_echelon
assert c.cols == f.rows == a.rank()
assert c * f == a
a = Matrix([
[0, 0, 1, 2, 2, -5, 3],
[-1, 5, 2, 2, 1, -7, 5],
[0, 0, -2, -3, -3, 8, -5],
[-1, 5, 0, -1, -2, 1, 0]])
c, f = a.rank_decomposition()
assert f.is_echelon
assert c.cols == f.rows == a.rank()
assert c * f == a
def test_upper_hessenberg_decomposition():
A = Matrix([
[1, 0, sqrt(3)],
[sqrt(2), Rational(1, 2), 2],
[1, Rational(1, 4), 3],
])
H, P = A.upper_hessenberg_decomposition()
assert simplify(P * P.H) == eye(P.cols)
assert simplify(P.H * P) == eye(P.cols)
assert H.is_upper_hessenberg
assert (simplify(P * H * P.H)) == A
B = Matrix([
[1, 2, 10],
[8, 2, 5],
[3, 12, 34],
])
H, P = B.upper_hessenberg_decomposition()
assert simplify(P * P.H) == eye(P.cols)
assert simplify(P.H * P) == eye(P.cols)
assert H.is_upper_hessenberg
assert simplify(P * H * P.H) == B
C = Matrix([
[1, sqrt(2), 2, 3],
[0, 5, 3, 4],
[1, 1, 4, sqrt(5)],
[0, 2, 2, 3]
])
H, P = C.upper_hessenberg_decomposition()
assert simplify(P * P.H) == eye(P.cols)
assert simplify(P.H * P) == eye(P.cols)
assert H.is_upper_hessenberg
assert simplify(P * H * P.H) == C
D = Matrix([
[1, 2, 3],
[-3, 5, 6],
[4, -8, 9],
])
H, P = D.upper_hessenberg_decomposition()
assert simplify(P * P.H) == eye(P.cols)
assert simplify(P.H * P) == eye(P.cols)
assert H.is_upper_hessenberg
assert simplify(P * H * P.H) == D
E = Matrix([
[1, 0, 0, 0],
[0, 1, 0, 0],
[1, 1, 0, 1],
[1, 1, 1, 0]
])
H, P = E.upper_hessenberg_decomposition()
assert simplify(P * P.H) == eye(P.cols)
assert simplify(P.H * P) == eye(P.cols)
assert H.is_upper_hessenberg
assert simplify(P * H * P.H) == E
|
52a9e85a295bf0f233f11323f9b526cc4eea0e280031c5f2119903585393805b | from sympy.matrices.expressions import MatrixExpr
from sympy import Q
class Factorization(MatrixExpr):
arg = property(lambda self: self.args[0])
shape = property(lambda self: self.arg.shape) # type: ignore
class LofLU(Factorization):
@property
def predicates(self):
return (Q.lower_triangular,)
class UofLU(Factorization):
@property
def predicates(self):
return (Q.upper_triangular,)
class LofCholesky(LofLU): pass
class UofCholesky(UofLU): pass
class QofQR(Factorization):
@property
def predicates(self):
return (Q.orthogonal,)
class RofQR(Factorization):
@property
def predicates(self):
return (Q.upper_triangular,)
class EigenVectors(Factorization):
@property
def predicates(self):
return (Q.orthogonal,)
class EigenValues(Factorization):
@property
def predicates(self):
return (Q.diagonal,)
class UofSVD(Factorization):
@property
def predicates(self):
return (Q.orthogonal,)
class SofSVD(Factorization):
@property
def predicates(self):
return (Q.diagonal,)
class VofSVD(Factorization):
@property
def predicates(self):
return (Q.orthogonal,)
def lu(expr):
return LofLU(expr), UofLU(expr)
def qr(expr):
return QofQR(expr), RofQR(expr)
def eig(expr):
return EigenValues(expr), EigenVectors(expr)
def svd(expr):
return UofSVD(expr), SofSVD(expr), VofSVD(expr)
|
6f0c02cf987cc7bd9ff674776d78875d6b862e0eb574427fe2fdd239067eb87f | from sympy.core.expr import unchanged
from sympy.sets import (ConditionSet, Intersection, FiniteSet,
EmptySet, Union, Contains, ImageSet)
from sympy import (Symbol, Eq, Ne, S, Abs, sin, asin, pi, Interval,
And, Mod, oo, Function, Lambda, symbols, Matrix, MatrixSymbol)
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))
|
2d971c13c98a8e09cf93064c7b0b28234dd349e81f3041ac13cd252812d157cb | #!/usr/bin/env python
"""Distutils based setup script for SymPy.
This uses Distutils (https://python.org/sigs/distutils-sig/) the standard
python mechanism for installing packages. Optionally, you can use
Setuptools (https://setuptools.readthedocs.io/en/latest/)
to automatically handle dependencies. For the easiest installation
just type the command (you'll probably need root privileges for that):
python setup.py install
This will install the library in the default location. For instructions on
how to customize the install procedure read the output of:
python setup.py --help install
In addition, there are some other commands:
python setup.py clean -> will clean all trash (*.pyc and stuff)
python setup.py test -> will run the complete test suite
python setup.py bench -> will run the complete benchmark suite
python setup.py audit -> will run pyflakes checker on source code
To get a full list of available commands, read the output of:
python setup.py --help-commands
Or, if all else fails, feel free to write to the sympy list at
[email protected] and ask for help.
"""
import sys
import os
import shutil
import glob
import subprocess
from distutils.command.sdist import sdist
min_mpmath_version = '0.19'
# This directory
dir_setup = os.path.dirname(os.path.realpath(__file__))
extra_kwargs = {}
try:
from setuptools import setup, Command
extra_kwargs['zip_safe'] = False
extra_kwargs['entry_points'] = {
'console_scripts': [
'isympy = isympy:main',
]
}
except ImportError:
from distutils.core import setup, Command
extra_kwargs['scripts'] = ['bin/isympy']
# handle mpmath deps in the hard way:
from distutils.version import LooseVersion
try:
import mpmath
if mpmath.__version__ < LooseVersion(min_mpmath_version):
raise ImportError
except ImportError:
print("Please install the mpmath package with a version >= %s"
% min_mpmath_version)
sys.exit(-1)
if sys.version_info < (3, 6):
print("SymPy requires Python 3.6 or newer. Python %d.%d detected"
% sys.version_info[:2])
sys.exit(-1)
# Check that this list is uptodate against the result of the command:
# python bin/generate_module_list.py
modules = [
'sympy.algebras',
'sympy.assumptions',
'sympy.assumptions.handlers',
'sympy.assumptions.predicates',
'sympy.assumptions.relation',
'sympy.benchmarks',
'sympy.calculus',
'sympy.categories',
'sympy.codegen',
'sympy.combinatorics',
'sympy.concrete',
'sympy.core',
'sympy.core.benchmarks',
'sympy.crypto',
'sympy.diffgeom',
'sympy.discrete',
'sympy.external',
'sympy.functions',
'sympy.functions.combinatorial',
'sympy.functions.elementary',
'sympy.functions.elementary.benchmarks',
'sympy.functions.special',
'sympy.functions.special.benchmarks',
'sympy.geometry',
'sympy.holonomic',
'sympy.integrals',
'sympy.integrals.benchmarks',
'sympy.integrals.rubi',
'sympy.integrals.rubi.parsetools',
'sympy.integrals.rubi.rubi_tests',
'sympy.integrals.rubi.rules',
'sympy.interactive',
'sympy.liealgebras',
'sympy.logic',
'sympy.logic.algorithms',
'sympy.logic.utilities',
'sympy.matrices',
'sympy.matrices.benchmarks',
'sympy.matrices.expressions',
'sympy.multipledispatch',
'sympy.ntheory',
'sympy.parsing',
'sympy.parsing.autolev',
'sympy.parsing.autolev._antlr',
'sympy.parsing.c',
'sympy.parsing.fortran',
'sympy.parsing.latex',
'sympy.parsing.latex._antlr',
'sympy.physics',
'sympy.physics.continuum_mechanics',
'sympy.physics.control',
'sympy.physics.hep',
'sympy.physics.mechanics',
'sympy.physics.optics',
'sympy.physics.quantum',
'sympy.physics.units',
'sympy.physics.units.definitions',
'sympy.physics.units.systems',
'sympy.physics.vector',
'sympy.plotting',
'sympy.plotting.intervalmath',
'sympy.plotting.pygletplot',
'sympy.polys',
'sympy.polys.agca',
'sympy.polys.benchmarks',
'sympy.polys.domains',
'sympy.polys.matrices',
'sympy.printing',
'sympy.printing.pretty',
'sympy.sandbox',
'sympy.series',
'sympy.series.benchmarks',
'sympy.sets',
'sympy.sets.handlers',
'sympy.simplify',
'sympy.solvers',
'sympy.solvers.benchmarks',
'sympy.solvers.diophantine',
'sympy.solvers.ode',
'sympy.stats',
'sympy.stats.sampling',
'sympy.strategies',
'sympy.strategies.branch',
'sympy.tensor',
'sympy.tensor.array',
'sympy.tensor.array.expressions',
'sympy.testing',
'sympy.unify',
'sympy.utilities',
'sympy.utilities._compilation',
'sympy.utilities.mathml',
'sympy.vector',
]
class audit(Command):
"""Audits SymPy's source code for following issues:
- Names which are used but not defined or used before they are defined.
- Names which are redefined without having been used.
"""
description = "Audit SymPy source with PyFlakes"
user_options = []
def initialize_options(self):
self.all = None
def finalize_options(self):
pass
def run(self):
import os
try:
import pyflakes.scripts.pyflakes as flakes
except ImportError:
print("In order to run the audit, you need to have PyFlakes installed.")
sys.exit(-1)
dirs = (os.path.join(*d) for d in (m.split('.') for m in modules))
warns = 0
for dir in dirs:
for filename in os.listdir(dir):
if filename.endswith('.py') and filename != '__init__.py':
warns += flakes.checkPath(os.path.join(dir, filename))
if warns > 0:
print("Audit finished with total %d warnings" % warns)
class clean(Command):
"""Cleans *.pyc and debian trashs, so you should get the same copy as
is in the VCS.
"""
description = "remove build files"
user_options = [("all", "a", "the same")]
def initialize_options(self):
self.all = None
def finalize_options(self):
pass
def run(self):
curr_dir = os.getcwd()
for root, dirs, files in os.walk(dir_setup):
for file in files:
if file.endswith('.pyc') and os.path.isfile:
os.remove(os.path.join(root, file))
os.chdir(dir_setup)
names = ["python-build-stamp-2.4", "MANIFEST", "build",
"dist", "doc/_build", "sample.tex"]
for f in names:
if os.path.isfile(f):
os.remove(f)
elif os.path.isdir(f):
shutil.rmtree(f)
for name in glob.glob(os.path.join(dir_setup, "doc", "src", "modules",
"physics", "vector", "*.pdf")):
if os.path.isfile(name):
os.remove(name)
os.chdir(curr_dir)
class test_sympy(Command):
"""Runs all tests under the sympy/ folder
"""
description = "run all tests and doctests; also see bin/test and bin/doctest"
user_options = [] # distutils complains if this is not here.
def __init__(self, *args):
self.args = args[0] # so we can pass it to other classes
Command.__init__(self, *args)
def initialize_options(self): # distutils wants this
pass
def finalize_options(self): # this too
pass
def run(self):
from sympy.utilities import runtests
runtests.run_all_tests()
class run_benchmarks(Command):
"""Runs all SymPy benchmarks"""
description = "run all benchmarks"
user_options = [] # distutils complains if this is not here.
def __init__(self, *args):
self.args = args[0] # so we can pass it to other classes
Command.__init__(self, *args)
def initialize_options(self): # distutils wants this
pass
def finalize_options(self): # this too
pass
# we use py.test like architecture:
#
# o collector -- collects benchmarks
# o runner -- executes benchmarks
# o presenter -- displays benchmarks results
#
# this is done in sympy.utilities.benchmarking on top of py.test
def run(self):
from sympy.utilities import benchmarking
benchmarking.main(['sympy'])
class antlr(Command):
"""Generate code with antlr4"""
description = "generate parser code from antlr grammars"
user_options = [] # distutils complains if this is not here.
def __init__(self, *args):
self.args = args[0] # so we can pass it to other classes
Command.__init__(self, *args)
def initialize_options(self): # distutils wants this
pass
def finalize_options(self): # this too
pass
def run(self):
from sympy.parsing.latex._build_latex_antlr import build_parser
if not build_parser():
sys.exit(-1)
class sdist_sympy(sdist):
def run(self):
# Fetch git commit hash and write down to commit_hash.txt before
# shipped in tarball.
commit_hash = None
commit_hash_filepath = 'doc/commit_hash.txt'
try:
commit_hash = \
subprocess.check_output(['git', 'rev-parse', 'HEAD'])
commit_hash = commit_hash.decode('ascii')
commit_hash = commit_hash.rstrip()
print('Commit hash found : {}.'.format(commit_hash))
print('Writing it to {}.'.format(commit_hash_filepath))
except:
pass
if commit_hash:
with open(commit_hash_filepath, 'w') as f:
f.write(commit_hash)
super(sdist_sympy, self).run()
try:
os.remove(commit_hash_filepath)
print(
'Successfully removed temporary file {}.'
.format(commit_hash_filepath))
except OSError as e:
print("Error deleting %s - %s." % (e.filename, e.strerror))
# Check that this list is uptodate against the result of the command:
# python bin/generate_test_list.py
tests = [
'sympy.algebras.tests',
'sympy.assumptions.tests',
'sympy.calculus.tests',
'sympy.categories.tests',
'sympy.codegen.tests',
'sympy.combinatorics.tests',
'sympy.concrete.tests',
'sympy.core.tests',
'sympy.crypto.tests',
'sympy.diffgeom.tests',
'sympy.discrete.tests',
'sympy.external.tests',
'sympy.functions.combinatorial.tests',
'sympy.functions.elementary.tests',
'sympy.functions.special.tests',
'sympy.geometry.tests',
'sympy.holonomic.tests',
'sympy.integrals.rubi.parsetools.tests',
'sympy.integrals.rubi.rubi_tests.tests',
'sympy.integrals.rubi.tests',
'sympy.integrals.tests',
'sympy.interactive.tests',
'sympy.liealgebras.tests',
'sympy.logic.tests',
'sympy.matrices.expressions.tests',
'sympy.matrices.tests',
'sympy.multipledispatch.tests',
'sympy.ntheory.tests',
'sympy.parsing.tests',
'sympy.physics.continuum_mechanics.tests',
'sympy.physics.control.tests',
'sympy.physics.hep.tests',
'sympy.physics.mechanics.tests',
'sympy.physics.optics.tests',
'sympy.physics.quantum.tests',
'sympy.physics.tests',
'sympy.physics.units.tests',
'sympy.physics.vector.tests',
'sympy.plotting.intervalmath.tests',
'sympy.plotting.pygletplot.tests',
'sympy.plotting.tests',
'sympy.polys.agca.tests',
'sympy.polys.domains.tests',
'sympy.polys.matrices.tests',
'sympy.polys.tests',
'sympy.printing.pretty.tests',
'sympy.printing.tests',
'sympy.sandbox.tests',
'sympy.series.tests',
'sympy.sets.tests',
'sympy.simplify.tests',
'sympy.solvers.diophantine.tests',
'sympy.solvers.ode.tests',
'sympy.solvers.tests',
'sympy.stats.sampling.tests',
'sympy.stats.tests',
'sympy.strategies.branch.tests',
'sympy.strategies.tests',
'sympy.tensor.array.expressions.tests',
'sympy.tensor.array.tests',
'sympy.tensor.tests',
'sympy.testing.tests',
'sympy.unify.tests',
'sympy.utilities._compilation.tests',
'sympy.utilities.tests',
'sympy.vector.tests',
]
with open(os.path.join(dir_setup, 'sympy', 'release.py')) as f:
# Defines __version__
exec(f.read())
if __name__ == '__main__':
setup(name='sympy',
version=__version__,
description='Computer algebra system (CAS) in Python',
author='SymPy development team',
author_email='[email protected]',
license='BSD',
keywords="Math CAS",
url='https://sympy.org',
py_modules=['isympy'],
packages=['sympy'] + modules + tests,
ext_modules=[],
package_data={
'sympy.utilities.mathml': ['data/*.xsl'],
'sympy.logic.benchmarks': ['input/*.cnf'],
'sympy.parsing.autolev': [
'*.g4', 'test-examples/*.al', 'test-examples/*.py',
'test-examples/pydy-example-repo/*.al',
'test-examples/pydy-example-repo/*.py',
'test-examples/README.txt',
],
'sympy.parsing.latex': ['*.txt', '*.g4'],
'sympy.integrals.rubi.parsetools': ['header.py.txt'],
'sympy.plotting.tests': ['test_region_*.png'],
},
data_files=[('share/man/man1', ['doc/man/isympy.1'])],
cmdclass={'test': test_sympy,
'bench': run_benchmarks,
'clean': clean,
'audit': audit,
'antlr': antlr,
'sdist': sdist_sympy,
},
python_requires='>=3.6',
classifiers=[
'License :: OSI Approved :: BSD License',
'Operating System :: OS Independent',
'Programming Language :: Python',
'Topic :: Scientific/Engineering',
'Topic :: Scientific/Engineering :: Mathematics',
'Topic :: Scientific/Engineering :: Physics',
'Programming Language :: Python :: 3',
'Programming Language :: Python :: 3.6',
'Programming Language :: Python :: 3.7',
'Programming Language :: Python :: 3.8',
'Programming Language :: Python :: 3 :: Only',
'Programming Language :: Python :: Implementation :: CPython',
'Programming Language :: Python :: Implementation :: PyPy',
],
install_requires=[
'mpmath>=%s' % min_mpmath_version,
],
**extra_kwargs
)
|
8a1170ef671d03018f13b5c73e9c7cb4561fe3255f9773c16e1f5069f155389b | #!/usr/bin/env python
"""
Run tests for specific packages that use optional dependencies.
The optional dependencies need to be installed before running this.
"""
# Add the local sympy to sys.path (needed for CI)
from get_sympy import path_hack
path_hack()
class TestsFailedError(Exception):
pass
print('Testing optional dependencies')
import sympy
test_list = [
# numpy
'*numpy*',
'sympy/core/',
'sympy/matrices/',
'sympy/physics/quantum/',
'sympy/utilities/tests/test_lambdify.py',
# scipy
'*scipy*',
# llvmlite
'*llvm*',
# aesara
'*aesara*',
# gmpy
'polys',
# autowrap
'*autowrap*',
# ipython
'*ipython*',
# antlr, lfortran, clang
'sympy/parsing/',
# matchpy
'*rubi*',
# codegen
'sympy/codegen/',
'sympy/utilities/tests/test_codegen',
'sympy/utilities/_compilation/tests/test_compilation',
# cloudpickle
'pickling',
# pycosat
'sympy/logic',
'sympy/assumptions',
#stats
'sympy/stats',
]
blacklist = [
'sympy/physics/quantum/tests/test_circuitplot.py',
]
doctest_list = [
# numpy
'sympy/matrices/',
'sympy/utilities/lambdify.py',
# scipy
'*scipy*',
# llvmlite
'*llvm*',
# aesara
'*aesara*',
# gmpy
'polys',
# autowrap
'*autowrap*',
# ipython
'*ipython*',
# antlr, lfortran, clang
'sympy/parsing/',
# matchpy
'*rubi*',
# codegen
'sympy/codegen/',
# pycosat
'sympy/logic',
'sympy/assumptions',
#stats
'sympy/stats',
]
if not (sympy.test(*test_list, blacklist=blacklist) and sympy.doctest(*doctest_list)):
raise TestsFailedError('Tests failed')
print('Testing MATPLOTLIB')
# Set matplotlib so that it works correctly in headless Travis. We have to do
# this here because it doesn't work after the sympy plotting module is
# imported.
import matplotlib
matplotlib.use("Agg")
import sympy
# Unfortunately, we have to use subprocess=False so that the above will be
# applied, so no hash randomization here.
if not (sympy.test('sympy/plotting', 'sympy/physics/quantum/tests/test_circuitplot.py',
subprocess=False) and sympy.doctest('sympy/plotting', subprocess=False)):
raise TestsFailedError('Tests failed')
print('Testing SYMENGINE')
import sympy
if not sympy.test('sympy/physics/mechanics'):
raise TestsFailedError('Tests failed')
if not sympy.test('sympy/liealgebras'):
raise TestsFailedError('Tests failed')
|
23abeff7d8e66e9abb6331eae5090933db11acd9d9abd078f8447ca1286ba3f1 | #!/usr/bin/env python
""" Update the ask_generated.py file
This must be run each time known_facts is changed
Should be run from sympy root directory
$ python bin/ask_update.py
"""
# hook in-tree SymPy into Python path, if possible
import os
import sys
isympy_path = os.path.abspath(__file__)
isympy_dir = os.path.dirname(isympy_path)
sympy_top = os.path.split(isympy_dir)[0]
sympy_dir = os.path.join(sympy_top, 'sympy')
if os.path.isdir(sympy_dir):
sys.path.insert(0, sympy_top)
from sympy.assumptions.facts import (compute_known_facts,
get_known_facts, get_known_facts_keys)
with open('sympy/assumptions/ask_generated.py', 'w') as f:
code = compute_known_facts(get_known_facts(), get_known_facts_keys())
f.write(code)
|
f107212b25a5acfc6fa8f261332cc877327938dfe87c5f8a22ee00d7e7dbf112 | """
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, 6):
raise ImportError("Python version 3.6 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)
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, 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, 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,
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, bottom_up,
nsimplify, FU, fu, sqrtdenest, cse, use, 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, 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,
Complexes)
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, postorder_traversal,
interactive_traversal, prefixes, postfixes, sift, topological_sort,
unflatten, has_dups, has_variety, reshape, default_sort_key, ordered,
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
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',
# 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', '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', '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',
'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', 'bottom_up',
'nsimplify', 'FU', 'fu', 'sqrtdenest', 'cse', 'use', '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', 'Reals', 'Naturals',
'Naturals0', 'UniversalSet', 'Integers', 'Rationals',
# 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', 'Complexes',
# 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', 'postorder_traversal',
'interactive_traversal', 'prefixes', 'postfixes', 'sift',
'topological_sort', 'unflatten', 'has_dups', 'has_variety', 'reshape',
'default_sort_key', 'ordered', 'rotations', 'filldedent', 'lambdify',
'source', 'threaded', 'xthreaded', 'public', 'memoize_property', 'test',
'doctest', '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',
# 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',
])
|
fa19c91743fcff8fdfef0d3b66a769e9a67387844bf540833828fd95edd03037 | import sys
sys._running_pytest = True # type: ignore
from distutils.version import LooseVersion as V
import pytest
from sympy.core.cache import clear_cache
import re
sp = re.compile(r'([0-9]+)/([1-9][0-9]*)')
def process_split(config, items):
split = config.getoption("--split")
if not split:
return
m = sp.match(split)
if not m:
raise ValueError("split must be a string of the form a/b "
"where a and b are ints.")
i, t = map(int, m.groups())
start, end = (i-1)*len(items)//t, i*len(items)//t
if i < t:
# remove elements from end of list first
del items[end:]
del items[:start]
def pytest_report_header(config):
from sympy.utilities.misc import ARCH
s = "architecture: %s\n" % ARCH
from sympy.core.cache import USE_CACHE
s += "cache: %s\n" % USE_CACHE
from sympy.external.gmpy import GROUND_TYPES, HAS_GMPY
version = ''
if GROUND_TYPES =='gmpy':
if HAS_GMPY == 1:
import gmpy
elif HAS_GMPY == 2:
import gmpy2 as gmpy
version = gmpy.version()
s += "ground types: %s %s\n" % (GROUND_TYPES, version)
return s
def pytest_terminal_summary(terminalreporter):
if (terminalreporter.stats.get('error', None) or
terminalreporter.stats.get('failed', None)):
terminalreporter.write_sep(
' ', 'DO *NOT* COMMIT!', red=True, bold=True)
def pytest_addoption(parser):
parser.addoption("--split", action="store", default="",
help="split tests")
def pytest_collection_modifyitems(config, items):
""" pytest hook. """
# handle splits
process_split(config, items)
@pytest.fixture(autouse=True, scope='module')
def file_clear_cache():
clear_cache()
@pytest.fixture(autouse=True, scope='module')
def check_disabled(request):
if getattr(request.module, 'disabled', False):
pytest.skip("test requirements not met.")
elif getattr(request.module, 'ipython', False):
# need to check version and options for ipython tests
if (V(pytest.__version__) < '2.6.3' and
pytest.config.getvalue('-s') != 'no'):
pytest.skip("run py.test with -s or upgrade to newer version.")
|
57f3ee2c395cc2c27f788a39ff27e8171d116349c44daa4c13477ae46f459cc8 | __version__ = "1.9.dev"
|
149291da1b99e9ff4a7f95dc84b8ff5988098475c3a0ef9c7c28f0865c0d648d | """
This module exports all latin and greek letters as Symbols, so you can
conveniently do
>>> from sympy.abc import x, y
instead of the slightly more clunky-looking
>>> from sympy import symbols
>>> x, y = symbols('x y')
Caveats
=======
1. As of the time of writing this, the names ``O``, ``S``, ``I``, ``N``,
``E``, and ``Q`` are colliding with names defined in SymPy. If you import them
from both ``sympy.abc`` and ``sympy``, the second import will "win".
This is an issue only for * imports, which should only be used for short-lived
code such as interactive sessions and throwaway scripts that do not survive
until the next SymPy upgrade, where ``sympy`` may contain a different set of
names.
2. This module does not define symbol names on demand, i.e.
``from sympy.abc import foo`` will be reported as an error because
``sympy.abc`` does not contain the name ``foo``. To get a symbol named ``foo``,
you still need to use ``Symbol('foo')`` or ``symbols('foo')``.
You can freely mix usage of ``sympy.abc`` and ``Symbol``/``symbols``, though
sticking with one and only one way to get the symbols does tend to make the code
more readable.
The module also defines some special names to help detect which names clash
with the default SymPy namespace.
``_clash1`` defines all the single letter variables that clash with
SymPy objects; ``_clash2`` defines the multi-letter clashing symbols;
and ``_clash`` is the union of both. These can be passed for ``locals``
during sympification if one desires Symbols rather than the non-Symbol
objects for those names.
Examples
========
>>> from sympy import S
>>> from sympy.abc import _clash1, _clash2, _clash
>>> S("Q & C", locals=_clash1)
C & Q
>>> S('pi(x)', locals=_clash2)
pi(x)
>>> S('pi(C, Q)', locals=_clash)
pi(C, Q)
"""
from typing import Any, Dict
import string
from .core import Symbol, symbols
from .core.alphabets import greeks
##### Symbol definitions #####
# Implementation note: The easiest way to avoid typos in the symbols()
# parameter is to copy it from the left-hand side of the assignment.
a, b, c, d, e, f, g, h, i, j = symbols('a, b, c, d, e, f, g, h, i, j')
k, l, m, n, o, p, q, r, s, t = symbols('k, l, m, n, o, p, q, r, s, t')
u, v, w, x, y, z = symbols('u, v, w, x, y, z')
A, B, C, D, E, F, G, H, I, J = symbols('A, B, C, D, E, F, G, H, I, J')
K, L, M, N, O, P, Q, R, S, T = symbols('K, L, M, N, O, P, Q, R, S, T')
U, V, W, X, Y, Z = symbols('U, V, W, X, Y, Z')
alpha, beta, gamma, delta = symbols('alpha, beta, gamma, delta')
epsilon, zeta, eta, theta = symbols('epsilon, zeta, eta, theta')
iota, kappa, lamda, mu = symbols('iota, kappa, lamda, mu')
nu, xi, omicron, pi = symbols('nu, xi, omicron, pi')
rho, sigma, tau, upsilon = symbols('rho, sigma, tau, upsilon')
phi, chi, psi, omega = symbols('phi, chi, psi, omega')
##### Clashing-symbols diagnostics #####
# We want to know which names in SymPy collide with those in here.
# This is mostly for diagnosing SymPy's namespace during SymPy development.
_latin = list(string.ascii_letters)
# OSINEQ should not be imported as they clash; gamma, pi and zeta clash, too
_greek = list(greeks) # make a copy, so we can mutate it
# Note: We import lamda since lambda is a reserved keyword in Python
_greek.remove("lambda")
_greek.append("lamda")
ns = {} # type: Dict[str, Any]
exec('from sympy import *', ns)
_clash1 = {}
_clash2 = {}
while ns:
_k, _ = ns.popitem()
if _k in _greek:
_clash2[_k] = Symbol(_k)
_greek.remove(_k)
elif _k in _latin:
_clash1[_k] = Symbol(_k)
_latin.remove(_k)
_clash = {}
_clash.update(_clash1)
_clash.update(_clash2)
del _latin, _greek, Symbol, _k
|
cc066e437b4fb8e1c534cf1fe53d6438830bba9bcfdb3ede844ddf9775711128 | #
# 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
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.ext.graphviz', 'matplotlib.sphinxext.plot_directive']
# 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
# MathJax file, which is free to use. See https://www.mathjax.org/#gettingstarted
# As explained in the link using latest.js will get the latest version even
# though it says 2.7.5.
mathjax_path = 'https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/latest.js?config=TeX-AMS_HTML-full'
# See https://www.sympy.org/sphinx-math-dollar/
mathjax_config = {
'tex2jax': {
'inlineMath': [ ["\\(","\\)"] ],
'displayMath': [["\\[","\\]"] ],
},
}
# 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'
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
|
4f4f78f6c59ad78a2d6e90a6fc57b280c6b7f2f9167f9b75db3c7b98b1dbee2f | """
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 import beta as beta_fn
from sympy import cos, sin, tan, atan, exp, besseli, besselj, besselk
from sympy import (log, sqrt, pi, S, Dummy, Interval, sympify, gamma, sign,
Piecewise, And, Eq, binomial, factorial, Sum, floor, Abs,
Lambda, Basic, lowergamma, erf, erfc, erfi, erfinv, I, asin,
hyper, uppergamma, sinh, Ne, expint, Rational, integrate)
from sympy.matrices import MatrixBase, MatrixExpr
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*(l*z)**(-k/2)*exp(-l**2/2 - z**2/2)*besseli(k/2 - 1, l*z)
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)
2**(-k/2)*z**(k/2 - 1)*exp(-z/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(-((-m + z)/s)**(-a))/s
>>> cdf(X)(z)
Piecewise((exp(-((-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", real=True, positive=True)
>>> beta = Symbol("beta", real=True, 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 + 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)
/1 y\ /2*y z\ / z\ / y 2*z \
|- - -|*|--- - -| + |1 - -|*|- - + --- - 1|
___ \2 2/ \ 3 3/ \ 2/ \ 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, MatrixBase, MatrixExpr)) and\
isinstance(std, (list, MatrixBase, MatrixExpr)):
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 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,))
|
8e2703415495ed0dedd3c9f935e88fe6fc1d067ec3a715d6d16cd5cfccdffbc0 | 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
from sympy import (Matrix, MatrixSymbol, S, Indexed, Basic, Tuple, Range,
Set, And, Eq, FiniteSet, ImmutableMatrix, Integer, igcd,
Lambda, Mul, Dummy, IndexedBase, Add, Interval, oo,
linsolve, eye, Or, Not, Intersection, factorial, Contains,
Union, Expr, Function, exp, cacheit, sqrt, pi, gamma,
Ge, Piecewise, Symbol, NonSquareMatrixError, EmptySet,
ceiling, MatrixBase, ConditionSet, ones, zeros, Identity,
Rational, Lt, Gt, Le, Ne, BlockMatrix, Sum)
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
__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.
"""
if isinstance(itr, (Tuple, set, FiniteSet)):
itr = 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 = 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 = Tuple(*(sympify(i) if isinstance(i, str) else i for i in itr))
except ValueError:
pass
else:
raise TypeError("%s is not an instance of list/tuple/set/Range/Tuple/FiniteSet." % (itr))
return itr
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)):
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])
@property
def _state_index(self) -> Range:
"""
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 isinstance(self.number_of_states, (int, 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 == None or \
given_condition == 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) or isinstance(condition, 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) or isinstance(condition, Lt):
upper = 1
if isinstance(condition, Gt) or isinstance(condition, Ge):
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 any((k not 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 isinstance(state, (int, 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) or isinstance(condition, 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) or isinstance(condition, Lt):
upper = 0
if isinstance(condition, Gt) or isinstance(condition, Ge):
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
FiniteSet(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):
# type: (Basic, tUnion[str, Symbol], tSequence, tUnion[MatrixBase, MatrixSymbol]) -> DiscreteMarkovChain
sym = _symbol_converter(sym)
state_space, trans_probs = MarkovProcess._sanity_checks(state_space, trans_probs)
obj = Basic.__new__(cls, sym, state_space, trans_probs)
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) -> tUnion[MatrixBase, MatrixSymbol]:
"""
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 = 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 = [[self._state_index[i] for i in class_] for class_ in classes]
return sympify(list(zip(classes, recurrence, 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]:
"""
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, ..., 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)
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]
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
FiniteSet(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 == 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
FiniteSet(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)
|
eec6d5bc8d123390661baffce0e7bcd657cc8815baafe3e60324e09f177c6765 | from sympy import S, Basic, exp, multigamma, pi
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)
}
dist_list = scipy_rv_map.keys()
if dist.__class__.__name__ not in dist_list:
return None
samples = []
if seed is None or isinstance(seed, int):
rand_state = numpy.random.default_rng(seed=seed)
else:
rand_state = seed
for _ in range(size[0]):
samp = scipy_rv_map[dist.__class__.__name__](dist, size[1] if len(size) > 1 else 1, rand_state)
samples.append(samp)
return samples
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 = {
}
dist_list = numpy_rv_map.keys()
if dist.__class__.__name__ not in dist_list:
return None
samples = []
import numpy
if seed is None or isinstance(seed, int):
rand_state = numpy.random.default_rng(seed=seed)
else:
rand_state = seed
for _ in range(size[0]):
samp = numpy_rv_map[dist.__class__.__name__](dist, size[1], rand_state)
samples.append(samp)
return samples
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))
}
dist_list = pymc3_rv_map.keys()
if dist.__class__.__name__ not in dist_list:
return None
with pymc3.Model():
pymc3_rv_map[dist.__class__.__name__](dist)
return pymc3.sample(size, chains=1, progressbar=False)[:]['X']
_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 = list(map(sympify, 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()
3**(-a)*b**(-2*a)*exp(Trace(Matrix([
[-2/3, 1/3],
[ 1/3, -2/3]])*X)/b)*Determinant(X)**(a - 3/2)/(sqrt(pi)*gamma(a)*gamma(a - 1/2))
>>> density(M)([[1, 0], [0, 1]]).doit()
3**(-a)*b**(-2*a)*exp(-4/(3*b))/(sqrt(pi)*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()
2**(-n)*3**(-n/2)*exp(Trace(Matrix([
[-1/3, 1/6],
[ 1/6, -1/3]])*X))*Determinant(X)**(n/2 - 3/2)/(sqrt(pi)*gamma(n/2)*gamma(n/2 - 1/2))
>>> density(W)([[1, 0], [0, 1]]).doit()
2**(-n)*3**(-n/2)*exp(-2/3)/(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 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)
pi**(-1.0)*gamma(v/2 + 1)*Determinant((Matrix([[-1, -2]]) + X)*(Matrix([
[-1],
[-2]]) + X.T) + Matrix([[1]]))**(-v/2 - 1)*Determinant(Matrix([[1]]))**(-1.0)*Determinant(Matrix([
[1, 0],
[0, 1]]))**(-0.5)/gamma(v/2)
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)
|
5a3b44adf2ff6b6587eb70e7e44dfc8e1692983adcf2e1eaf94640f70b6fa851 | import itertools
from sympy import (MatrixExpr, Expr, ShapeError, ZeroMatrix,
Add, Mul, MatMul, S, expand as _expand)
from sympy.stats.rv import RandomSymbol, is_random
from sympy.core.sympify import _sympify
from sympy.stats.symbolic_probability import Variance, Covariance, Expectation
class ExpectationMatrix(Expectation, MatrixExpr):
"""
Expectation of a random matrix expression.
Examples
========
>>> from sympy.stats import ExpectationMatrix, Normal
>>> from sympy.stats.rv import RandomMatrixSymbol
>>> from sympy import symbols, MatrixSymbol, Matrix
>>> k = symbols("k")
>>> A, B = MatrixSymbol("A", k, k), MatrixSymbol("B", k, k)
>>> X, Y = RandomMatrixSymbol("X", k, 1), RandomMatrixSymbol("Y", k, 1)
>>> ExpectationMatrix(X)
ExpectationMatrix(X)
>>> ExpectationMatrix(A*X).shape
(k, 1)
To expand the expectation in its expression, use ``expand()``:
>>> ExpectationMatrix(A*X + B*Y).expand()
A*ExpectationMatrix(X) + B*ExpectationMatrix(Y)
>>> ExpectationMatrix((X + Y)*(X - Y).T).expand()
ExpectationMatrix(X*X.T) - ExpectationMatrix(X*Y.T) + ExpectationMatrix(Y*X.T) - ExpectationMatrix(Y*Y.T)
To evaluate the ``ExpectationMatrix``, use ``doit()``:
>>> N11, N12 = Normal('N11', 11, 1), Normal('N12', 12, 1)
>>> N21, N22 = Normal('N21', 21, 1), Normal('N22', 22, 1)
>>> M11, M12 = Normal('M11', 1, 1), Normal('M12', 2, 1)
>>> M21, M22 = Normal('M21', 3, 1), Normal('M22', 4, 1)
>>> x1 = Matrix([[N11, N12], [N21, N22]])
>>> x2 = Matrix([[M11, M12], [M21, M22]])
>>> ExpectationMatrix(x1 + x2).doit()
Matrix([
[12, 14],
[24, 26]])
"""
def __new__(cls, expr, condition=None):
expr = _sympify(expr)
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._shape = expr.shape
obj._condition = condition
return obj
@property
def shape(self):
return self._shape
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, MatMul)):
rv = []
nonrv = []
postnon = []
for a in expr.args:
if is_random(a):
if rv:
rv.extend(postnon)
else:
nonrv.extend(postnon)
postnon = []
rv.append(a)
elif a.is_Matrix:
postnon.append(a)
else:
nonrv.append(a)
# In order to avoid infinite-looping (MatMul may call .doit() again),
# do not rebuild
if len(nonrv) == 0:
return self
return Mul.fromiter(nonrv)*Expectation(Mul.fromiter(rv),
condition=condition)*Mul.fromiter(postnon)
return self
class VarianceMatrix(Variance, MatrixExpr):
"""
Variance of a random matrix probability expression. Also known as
Covariance matrix, auto-covariance matrix, dispersion matrix,
or variance-covariance matrix.
Examples
========
>>> from sympy.stats import VarianceMatrix
>>> from sympy.stats.rv import RandomMatrixSymbol
>>> from sympy import symbols, MatrixSymbol
>>> k = symbols("k")
>>> A, B = MatrixSymbol("A", k, k), MatrixSymbol("B", k, k)
>>> X, Y = RandomMatrixSymbol("X", k, 1), RandomMatrixSymbol("Y", k, 1)
>>> VarianceMatrix(X)
VarianceMatrix(X)
>>> VarianceMatrix(X).shape
(k, k)
To expand the variance in its expression, use ``expand()``:
>>> VarianceMatrix(A*X).expand()
A*VarianceMatrix(X)*A.T
>>> VarianceMatrix(A*X + B*Y).expand()
2*A*CrossCovarianceMatrix(X, Y)*B.T + A*VarianceMatrix(X)*A.T + B*VarianceMatrix(Y)*B.T
"""
def __new__(cls, arg, condition=None):
arg = _sympify(arg)
if 1 not in arg.shape:
raise ShapeError("Expression is not a vector")
shape = (arg.shape[0], arg.shape[0]) if arg.shape[1] == 1 else (arg.shape[1], arg.shape[1])
if condition:
obj = Expr.__new__(cls, arg, condition)
else:
obj = Expr.__new__(cls, arg)
obj._shape = shape
obj._condition = condition
return obj
@property
def shape(self):
return self._shape
def expand(self, **hints):
arg = self.args[0]
condition = self._condition
if not is_random(arg):
return ZeroMatrix(*self.shape)
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, MatMul)):
nonrv = []
rv = []
for a in arg.args:
if is_random(a):
rv.append(a)
else:
nonrv.append(a)
if len(rv) == 0:
return ZeroMatrix(*self.shape)
# Avoid possible infinite loops with MatMul:
if len(nonrv) == 0:
return self
# Variance of many multiple matrix products is not implemented:
if len(rv) > 1:
return self
return Mul.fromiter(nonrv)*Variance(Mul.fromiter(rv),
condition)*(Mul.fromiter(nonrv)).transpose()
# this expression contains a RandomSymbol somehow:
return self
class CrossCovarianceMatrix(Covariance, MatrixExpr):
"""
Covariance of a random matrix probability expression.
Examples
========
>>> from sympy.stats import CrossCovarianceMatrix
>>> from sympy.stats.rv import RandomMatrixSymbol
>>> from sympy import symbols, MatrixSymbol
>>> k = symbols("k")
>>> A, B = MatrixSymbol("A", k, k), MatrixSymbol("B", k, k)
>>> C, D = MatrixSymbol("C", k, k), MatrixSymbol("D", k, k)
>>> X, Y = RandomMatrixSymbol("X", k, 1), RandomMatrixSymbol("Y", k, 1)
>>> Z, W = RandomMatrixSymbol("Z", k, 1), RandomMatrixSymbol("W", k, 1)
>>> CrossCovarianceMatrix(X, Y)
CrossCovarianceMatrix(X, Y)
>>> CrossCovarianceMatrix(X, Y).shape
(k, k)
To expand the covariance in its expression, use ``expand()``:
>>> CrossCovarianceMatrix(X + Y, Z).expand()
CrossCovarianceMatrix(X, Z) + CrossCovarianceMatrix(Y, Z)
>>> CrossCovarianceMatrix(A*X , Y).expand()
A*CrossCovarianceMatrix(X, Y)
>>> CrossCovarianceMatrix(A*X, B.T*Y).expand()
A*CrossCovarianceMatrix(X, Y)*B
>>> CrossCovarianceMatrix(A*X + B*Y, C.T*Z + D.T*W).expand()
A*CrossCovarianceMatrix(X, W)*D + A*CrossCovarianceMatrix(X, Z)*C + B*CrossCovarianceMatrix(Y, W)*D + B*CrossCovarianceMatrix(Y, Z)*C
"""
def __new__(cls, arg1, arg2, condition=None):
arg1 = _sympify(arg1)
arg2 = _sympify(arg2)
if (1 not in arg1.shape) or (1 not in arg2.shape) or (arg1.shape[1] != arg2.shape[1]):
raise ShapeError("Expression is not a vector")
shape = (arg1.shape[0], arg2.shape[0]) if arg1.shape[1] == 1 and arg2.shape[1] == 1 \
else (1, 1)
if condition:
obj = Expr.__new__(cls, arg1, arg2, condition)
else:
obj = Expr.__new__(cls, arg1, arg2)
obj._shape = shape
obj._condition = condition
return obj
@property
def shape(self):
return self._shape
def expand(self, **hints):
arg1 = self.args[0]
arg2 = self.args[1]
condition = self._condition
if arg1 == arg2:
return VarianceMatrix(arg1, condition).expand()
if not is_random(arg1) or not is_random(arg2):
return ZeroMatrix(*self.shape)
if isinstance(arg1, RandomSymbol) and isinstance(arg2, RandomSymbol):
return CrossCovarianceMatrix(arg1, arg2, condition)
coeff_rv_list1 = self._expand_single_argument(arg1.expand())
coeff_rv_list2 = self._expand_single_argument(arg2.expand())
addends = [a*CrossCovarianceMatrix(r1, r2, condition=condition)*b.transpose()
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, MatMul)):
outval.append(cls._get_mul_nonrv_rv_tuple(a))
elif is_random(a):
outval.append((S.One, a))
return outval
elif isinstance(expr, (Mul, MatMul)):
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))
|
0ce0452f6f466d8a39e201a081a652aa329e97135f5a8d5f4f990c3c59782721 | """
Contains
========
FlorySchulz
Geometric
Hermite
Logarithmic
NegativeBinomial
Poisson
Skellam
YuleSimon
Zeta
"""
from sympy import (Basic, factorial, exp, S, sympify, I, zeta, polylog, log, beta,
hyper, binomial, Piecewise, floor, besseli, sqrt, Sum, Dummy,
Lambda, Eq)
from sympy.stats.drv import SingleDiscreteDistribution, SingleDiscretePSpace
from sympy.stats.rv import _value_check, is_random
__all__ = ['FlorySchulz',
'Geometric',
'Hermite',
'Logarithmic',
'NegativeBinomial',
'Poisson',
'Skellam',
'YuleSimon',
'Zeta'
]
def rv(symbol, cls, *args, **kwargs):
args = list(map(sympify, args))
dist = cls(*args)
if kwargs.pop('check', True):
dist.check(*args)
pspace = SingleDiscretePSpace(symbol, dist)
if any(is_random(arg) for arg in args):
from sympy.stats.compound_rv import CompoundPSpace, CompoundDistribution
pspace = CompoundPSpace(symbol, CompoundDistribution(dist))
return pspace.value
class DiscreteDistributionHandmade(SingleDiscreteDistribution):
_argnames = ('pdf',)
def __new__(cls, pdf, set=S.Integers):
return Basic.__new__(cls, pdf, set)
@property
def set(self):
return self.args[1]
@staticmethod
def check(pdf, set):
x = Dummy('x')
val = Sum(pdf(x), (x, set._inf, set._sup)).doit()
_value_check(Eq(val, 1) != S.false, "The pdf is incorrect on the given set.")
def DiscreteRV(symbol, density, set=S.Integers, **kwargs):
"""
Create a Discrete Random Variable given the following:
Parameters
==========
symbol : Symbol
Represents name of the random variable.
density : Expression containing symbol
Represents probability density function.
set : set
Represents the region where the pdf is valid, by default is real line.
check : bool
If True, it will check whether the given density
integrates to 1 over the given set. If False, it
will not perform this check. Default is False.
Examples
========
>>> from sympy.stats import DiscreteRV, P, E
>>> from sympy import Rational, Symbol
>>> x = Symbol('x')
>>> n = 10
>>> density = Rational(1, 10)
>>> X = DiscreteRV(x, density, set=set(range(n)))
>>> E(X)
9/2
>>> P(X>3)
3/5
Returns
=======
RandomSymbol
"""
set = sympify(set)
pdf = Piecewise((density, set.as_relational(symbol)), (0, True))
pdf = Lambda(symbol, pdf)
# have a default of False while `rv` should have a default of True
kwargs['check'] = kwargs.pop('check', False)
return rv(symbol.name, DiscreteDistributionHandmade, pdf, set, **kwargs)
#-------------------------------------------------------------------------------
# Flory-Schulz distribution ------------------------------------------------------------
class FlorySchulzDistribution(SingleDiscreteDistribution):
_argnames = ('a',)
set = S.Naturals
@staticmethod
def check(a):
_value_check((0 < a, a < 1), "a must be between 0 and 1")
def pdf(self, k):
a = self.a
return (a**2 * k * (1 - a)**(k - 1))
def _characteristic_function(self, t):
a = self.a
return a**2*exp(I*t)/((1 + (a - 1)*exp(I*t))**2)
def _moment_generating_function(self, t):
a = self.a
return a**2*exp(t)/((1 + (a - 1)*exp(t))**2)
def FlorySchulz(name, a):
r"""
Create a discrete random variable with a FlorySchulz distribution.
The density of the FlorySchulz distribution is given by
.. math::
f(k) := (a^2) k (1 - a)^{k-1}
Parameters
==========
a
A real number between 0 and 1
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import density, E, variance, FlorySchulz
>>> from sympy import Symbol, S
>>> a = S.One / 5
>>> z = Symbol("z")
>>> X = FlorySchulz("x", a)
>>> density(X)(z)
(4/5)**(z - 1)*z/25
>>> E(X)
9
>>> variance(X)
40
References
==========
https://en.wikipedia.org/wiki/Flory%E2%80%93Schulz_distribution
"""
return rv(name, FlorySchulzDistribution, a)
#-------------------------------------------------------------------------------
# Geometric distribution ------------------------------------------------------------
class GeometricDistribution(SingleDiscreteDistribution):
_argnames = ('p',)
set = S.Naturals
@staticmethod
def check(p):
_value_check((0 < p, p <= 1), "p must be between 0 and 1")
def pdf(self, k):
return (1 - self.p)**(k - 1) * self.p
def _characteristic_function(self, t):
p = self.p
return p * exp(I*t) / (1 - (1 - p)*exp(I*t))
def _moment_generating_function(self, t):
p = self.p
return p * exp(t) / (1 - (1 - p) * exp(t))
def Geometric(name, p):
r"""
Create a discrete random variable with a Geometric distribution.
Explanation
===========
The density of the Geometric distribution is given by
.. math::
f(k) := p (1 - p)^{k - 1}
Parameters
==========
p: A probability between 0 and 1
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Geometric, density, E, variance
>>> from sympy import Symbol, S
>>> p = S.One / 5
>>> z = Symbol("z")
>>> X = Geometric("x", p)
>>> density(X)(z)
(4/5)**(z - 1)/5
>>> E(X)
5
>>> variance(X)
20
References
==========
.. [1] https://en.wikipedia.org/wiki/Geometric_distribution
.. [2] http://mathworld.wolfram.com/GeometricDistribution.html
"""
return rv(name, GeometricDistribution, p)
#-------------------------------------------------------------------------------
# Hermite distribution ---------------------------------------------------------
class HermiteDistribution(SingleDiscreteDistribution):
_argnames = ('a1', 'a2')
set = S.Naturals0
@staticmethod
def check(a1, a2):
_value_check(a1.is_nonnegative, 'Parameter a1 must be >= 0.')
_value_check(a2.is_nonnegative, 'Parameter a2 must be >= 0.')
def pdf(self, k):
a1, a2 = self.a1, self.a2
term1 = exp(-(a1 + a2))
j = Dummy("j", integer=True)
num = a1**(k - 2*j) * a2**j
den = factorial(k - 2*j) * factorial(j)
return term1 * Sum(num/den, (j, 0, k//2)).doit()
def _moment_generating_function(self, t):
a1, a2 = self.a1, self.a2
term1 = a1 * (exp(t) - 1)
term2 = a2 * (exp(2*t) - 1)
return exp(term1 + term2)
def _characteristic_function(self, t):
a1, a2 = self.a1, self.a2
term1 = a1 * (exp(I*t) - 1)
term2 = a2 * (exp(2*I*t) - 1)
return exp(term1 + term2)
def Hermite(name, a1, a2):
r"""
Create a discrete random variable with a Hermite distribution.
Explanation
===========
The density of the Hermite distribution is given by
.. math::
f(x):= e^{-a_1 -a_2}\sum_{j=0}^{\left \lfloor x/2 \right \rfloor}
\frac{a_{1}^{x-2j}a_{2}^{j}}{(x-2j)!j!}
Parameters
==========
a1: A Positive number greater than equal to 0.
a2: A Positive number greater than equal to 0.
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Hermite, density, E, variance
>>> from sympy import Symbol
>>> a1 = Symbol("a1", positive=True)
>>> a2 = Symbol("a2", positive=True)
>>> x = Symbol("x")
>>> H = Hermite("H", a1=5, a2=4)
>>> density(H)(2)
33*exp(-9)/2
>>> E(H)
13
>>> variance(H)
21
References
==========
.. [1] https://en.wikipedia.org/wiki/Hermite_distribution
"""
return rv(name, HermiteDistribution, a1, a2)
#-------------------------------------------------------------------------------
# Logarithmic distribution ------------------------------------------------------------
class LogarithmicDistribution(SingleDiscreteDistribution):
_argnames = ('p',)
set = S.Naturals
@staticmethod
def check(p):
_value_check((p > 0, p < 1), "p should be between 0 and 1")
def pdf(self, k):
p = self.p
return (-1) * p**k / (k * log(1 - p))
def _characteristic_function(self, t):
p = self.p
return log(1 - p * exp(I*t)) / log(1 - p)
def _moment_generating_function(self, t):
p = self.p
return log(1 - p * exp(t)) / log(1 - p)
def Logarithmic(name, p):
r"""
Create a discrete random variable with a Logarithmic distribution.
Explanation
===========
The density of the Logarithmic distribution is given by
.. math::
f(k) := \frac{-p^k}{k \ln{(1 - p)}}
Parameters
==========
p: A value between 0 and 1
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Logarithmic, density, E, variance
>>> from sympy import Symbol, S
>>> p = S.One / 5
>>> z = Symbol("z")
>>> X = Logarithmic("x", p)
>>> density(X)(z)
-5**(-z)/(z*log(4/5))
>>> E(X)
-1/(-4*log(5) + 8*log(2))
>>> variance(X)
-1/((-4*log(5) + 8*log(2))*(-2*log(5) + 4*log(2))) + 1/(-64*log(2)*log(5) + 64*log(2)**2 + 16*log(5)**2) - 10/(-32*log(5) + 64*log(2))
References
==========
.. [1] https://en.wikipedia.org/wiki/Logarithmic_distribution
.. [2] http://mathworld.wolfram.com/LogarithmicDistribution.html
"""
return rv(name, LogarithmicDistribution, p)
#-------------------------------------------------------------------------------
# Negative binomial distribution ------------------------------------------------------------
class NegativeBinomialDistribution(SingleDiscreteDistribution):
_argnames = ('r', 'p')
set = S.Naturals0
@staticmethod
def check(r, p):
_value_check(r > 0, 'r should be positive')
_value_check((p > 0, p < 1), 'p should be between 0 and 1')
def pdf(self, k):
r = self.r
p = self.p
return binomial(k + r - 1, k) * (1 - p)**r * p**k
def _characteristic_function(self, t):
r = self.r
p = self.p
return ((1 - p) / (1 - p * exp(I*t)))**r
def _moment_generating_function(self, t):
r = self.r
p = self.p
return ((1 - p) / (1 - p * exp(t)))**r
def NegativeBinomial(name, r, p):
r"""
Create a discrete random variable with a Negative Binomial distribution.
Explanation
===========
The density of the Negative Binomial distribution is given by
.. math::
f(k) := \binom{k + r - 1}{k} (1 - p)^r p^k
Parameters
==========
r: A positive value
p: A value between 0 and 1
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import NegativeBinomial, density, E, variance
>>> from sympy import Symbol, S
>>> r = 5
>>> p = S.One / 5
>>> z = Symbol("z")
>>> X = NegativeBinomial("x", r, p)
>>> density(X)(z)
1024*5**(-z)*binomial(z + 4, z)/3125
>>> E(X)
5/4
>>> variance(X)
25/16
References
==========
.. [1] https://en.wikipedia.org/wiki/Negative_binomial_distribution
.. [2] http://mathworld.wolfram.com/NegativeBinomialDistribution.html
"""
return rv(name, NegativeBinomialDistribution, r, p)
#-------------------------------------------------------------------------------
# Poisson distribution ------------------------------------------------------------
class PoissonDistribution(SingleDiscreteDistribution):
_argnames = ('lamda',)
set = S.Naturals0
@staticmethod
def check(lamda):
_value_check(lamda > 0, "Lambda must be positive")
def pdf(self, k):
return self.lamda**k / factorial(k) * exp(-self.lamda)
def _characteristic_function(self, t):
return exp(self.lamda * (exp(I*t) - 1))
def _moment_generating_function(self, t):
return exp(self.lamda * (exp(t) - 1))
def Poisson(name, lamda):
r"""
Create a discrete random variable with a Poisson distribution.
Explanation
===========
The density of the Poisson distribution is given by
.. math::
f(k) := \frac{\lambda^{k} e^{- \lambda}}{k!}
Parameters
==========
lamda: Positive number, a rate
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Poisson, density, E, variance
>>> from sympy import Symbol, simplify
>>> rate = Symbol("lambda", positive=True)
>>> z = Symbol("z")
>>> X = Poisson("x", rate)
>>> density(X)(z)
lambda**z*exp(-lambda)/factorial(z)
>>> E(X)
lambda
>>> simplify(variance(X))
lambda
References
==========
.. [1] https://en.wikipedia.org/wiki/Poisson_distribution
.. [2] http://mathworld.wolfram.com/PoissonDistribution.html
"""
return rv(name, PoissonDistribution, lamda)
# -----------------------------------------------------------------------------
# Skellam distribution --------------------------------------------------------
class SkellamDistribution(SingleDiscreteDistribution):
_argnames = ('mu1', 'mu2')
set = S.Integers
@staticmethod
def check(mu1, mu2):
_value_check(mu1 >= 0, 'Parameter mu1 must be >= 0')
_value_check(mu2 >= 0, 'Parameter mu2 must be >= 0')
def pdf(self, k):
(mu1, mu2) = (self.mu1, self.mu2)
term1 = exp(-(mu1 + mu2)) * (mu1 / mu2) ** (k / 2)
term2 = besseli(k, 2 * sqrt(mu1 * mu2))
return term1 * term2
def _cdf(self, x):
raise NotImplementedError(
"Skellam doesn't have closed form for the CDF.")
def _characteristic_function(self, t):
(mu1, mu2) = (self.mu1, self.mu2)
return exp(-(mu1 + mu2) + mu1 * exp(I * t) + mu2 * exp(-I * t))
def _moment_generating_function(self, t):
(mu1, mu2) = (self.mu1, self.mu2)
return exp(-(mu1 + mu2) + mu1 * exp(t) + mu2 * exp(-t))
def Skellam(name, mu1, mu2):
r"""
Create a discrete random variable with a Skellam distribution.
Explanation
===========
The Skellam is the distribution of the difference N1 - N2
of two statistically independent random variables N1 and N2
each Poisson-distributed with respective expected values mu1 and mu2.
The density of the Skellam distribution is given by
.. math::
f(k) := e^{-(\mu_1+\mu_2)}(\frac{\mu_1}{\mu_2})^{k/2}I_k(2\sqrt{\mu_1\mu_2})
Parameters
==========
mu1: A non-negative value
mu2: A non-negative value
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Skellam, density, E, variance
>>> from sympy import Symbol, pprint
>>> z = Symbol("z", integer=True)
>>> mu1 = Symbol("mu1", positive=True)
>>> mu2 = Symbol("mu2", positive=True)
>>> X = Skellam("x", mu1, mu2)
>>> pprint(density(X)(z), use_unicode=False)
z
-
2
/mu1\ -mu1 - mu2 / _____ _____\
|---| *e *besseli\z, 2*\/ mu1 *\/ mu2 /
\mu2/
>>> E(X)
mu1 - mu2
>>> variance(X).expand()
mu1 + mu2
References
==========
.. [1] https://en.wikipedia.org/wiki/Skellam_distribution
"""
return rv(name, SkellamDistribution, mu1, mu2)
#-------------------------------------------------------------------------------
# Yule-Simon distribution ------------------------------------------------------------
class YuleSimonDistribution(SingleDiscreteDistribution):
_argnames = ('rho',)
set = S.Naturals
@staticmethod
def check(rho):
_value_check(rho > 0, 'rho should be positive')
def pdf(self, k):
rho = self.rho
return rho * beta(k, rho + 1)
def _cdf(self, x):
return Piecewise((1 - floor(x) * beta(floor(x), self.rho + 1), x >= 1), (0, True))
def _characteristic_function(self, t):
rho = self.rho
return rho * hyper((1, 1), (rho + 2,), exp(I*t)) * exp(I*t) / (rho + 1)
def _moment_generating_function(self, t):
rho = self.rho
return rho * hyper((1, 1), (rho + 2,), exp(t)) * exp(t) / (rho + 1)
def YuleSimon(name, rho):
r"""
Create a discrete random variable with a Yule-Simon distribution.
Explanation
===========
The density of the Yule-Simon distribution is given by
.. math::
f(k) := \rho B(k, \rho + 1)
Parameters
==========
rho: A positive value
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import YuleSimon, density, E, variance
>>> from sympy import Symbol, simplify
>>> p = 5
>>> z = Symbol("z")
>>> X = YuleSimon("x", p)
>>> density(X)(z)
5*beta(z, 6)
>>> simplify(E(X))
5/4
>>> simplify(variance(X))
25/48
References
==========
.. [1] https://en.wikipedia.org/wiki/Yule%E2%80%93Simon_distribution
"""
return rv(name, YuleSimonDistribution, rho)
#-------------------------------------------------------------------------------
# Zeta distribution ------------------------------------------------------------
class ZetaDistribution(SingleDiscreteDistribution):
_argnames = ('s',)
set = S.Naturals
@staticmethod
def check(s):
_value_check(s > 1, 's should be greater than 1')
def pdf(self, k):
s = self.s
return 1 / (k**s * zeta(s))
def _characteristic_function(self, t):
return polylog(self.s, exp(I*t)) / zeta(self.s)
def _moment_generating_function(self, t):
return polylog(self.s, exp(t)) / zeta(self.s)
def Zeta(name, s):
r"""
Create a discrete random variable with a Zeta distribution.
Explanation
===========
The density of the Zeta distribution is given by
.. math::
f(k) := \frac{1}{k^s \zeta{(s)}}
Parameters
==========
s: A value greater than 1
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Zeta, density, E, variance
>>> from sympy import Symbol
>>> s = 5
>>> z = Symbol("z")
>>> X = Zeta("x", s)
>>> density(X)(z)
1/(z**5*zeta(5))
>>> E(X)
pi**4/(90*zeta(5))
>>> variance(X)
-pi**8/(8100*zeta(5)**2) + zeta(3)/zeta(5)
References
==========
.. [1] https://en.wikipedia.org/wiki/Zeta_distribution
"""
return rv(name, ZetaDistribution, s)
|
0c8026ebbcb743c84fe8053423c7881ce6955a5a5395cda02cedf0629f519dab | from sympy.sets import FiniteSet
from sympy import (sqrt, log, exp, FallingFactorial, Rational, Eq, Dummy,
piecewise_fold, solveset, Integral)
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, finite=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, finite=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, finite=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, finite=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, finite=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)
FiniteSet(3)
>>> D = Die('D')
>>> median(D)
FiniteSet(3, 4)
References
==========
.. [1] https://en.wikipedia.org/wiki/Median#Probability_distributions
"""
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) or isinstance(pspace(X), 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
|
34fe072665e7a5569da38da9998d4e002baa45e9c5bcf61ee992e1046c40314e | """
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 import (Basic, S, Expr, Symbol, Tuple, And, Add, Eq, lambdify, Or,
Equality, Lambda, sympify, Dummy, Ne, KroneckerDelta,
DiracDelta, Mul, Indexed, MatrixSymbol, Function)
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
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):
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})
if any([pspace(rv).is_Continuous for rv in rvs]):
from sympy.stats.crv import (ConditionalContinuousDomain,
ContinuousPSpace)
space = ContinuousPSpace
domain = ConditionalContinuousDomain(self.domain, condition)
elif any([pspace(rv).is_Discrete for rv in rvs]):
from sympy.stats.drv import (ConditionalDiscreteDomain,
DiscretePSpace)
space = DiscretePSpace
domain = ConditionalDiscreteDomain(self.domain, condition)
elif all([pspace(rv).is_Finite for rv in rvs]):
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, Equality) 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])
@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)
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``
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 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) # doctest: +SKIP
>>> next(die_roll) # doctest: +SKIP
6
>>> N = Normal('N', 3, 4) # Continuous Random Variable
>>> samp = next(sample(N)) # doctest: +SKIP
>>> samp in N.pspace.domain.set # doctest: +SKIP
True
>>> samp = next(sample(N, N>0)) # doctest: +SKIP
>>> samp > 0 # doctest: +SKIP
True
>>> samp_list = next(sample(N, size=4)) # doctest: +SKIP
>>> [sam in N.pspace.domain.set for sam in samp_list] # doctest: +SKIP
[True, True, True, True]
>>> G = Geometric('G', 0.5) # Discrete Random Variable
>>> samp_list = next(sample(G, size=3)) # doctest: +SKIP
>>> samp_list # doctest: +SKIP
array([10, 4, 1])
>>> [sam in G.pspace.domain.set for sam in samp_list] # doctest: +SKIP
[True, True, True]
>>> MN = Normal("MN", [3, 4], [[2, 1], [1, 2]]) # Joint Random Variable
>>> samp_list = next(sample(MN, size=4)) # doctest: +SKIP
>>> samp_list # doctest: +SKIP
array([[4.22564264, 3.23364418],
[3.41002011, 4.60090908],
[3.76151866, 4.77617143],
[4.71440865, 2.65714157]])
>>> [tuple(sam) in MN.pspace.domain.set for sam in samp_list] # doctest: +SKIP
[True, True, True, True]
Returns
=======
sample: iterator object
iterator object containing the sample/samples of given expr
"""
### TODO: Remove the user warnings in the future releases
message = ("The return type of sample has been changed to return an "
"iterator object since version 1.7. For more information see "
"https://github.com/sympy/sympy/issues/19061")
warnings.warn(filldedent(message))
return sample_iter(expr, condition, size=size, library=library,
numsamples=numsamples, seed=seed)
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
np = import_module('numpy')
if np:
rand_state = np.random.default_rng(seed=seed)
else:
rand_state = None
while count < numsamples:
d = ps.sample(size=size, library=library, seed=rand_state) # a dictionary that maps RVs to values
args = [d[rv] 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
samps = do_sample_scipy(self, size, seed)
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
samps = do_sample_numpy(self, size, rand_state)
elif library == 'pymc3':
from sympy.stats.sampling.sample_pymc3 import do_sample_pymc3
import pymc3
with pymc3.Model():
if do_sample_pymc3(self):
samps = pymc3.sample(size, chains=1, progressbar=False, random_seed=seed)[:]['X']
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
"""
from sympy.core.compatibility import iterable
from sympy.core.logic import fuzzy_and
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()
|
b1a66941e5b49bdca9252b93e1e6b87c617a1fd5739fd2bc1372be4dba551121 | from sympy import (Basic, sympify, symbols, Dummy, Lambda, summation,
Piecewise, S, cacheit, Sum, exp, I, Ne, Eq, poly,
series, factorial, And, floor)
from sympy.polys.polyerrors import PolynomialError
from sympy.stats.crv import reduce_rational_inequalities_wrap
from sympy.stats.rv import (NamedArgsMixin, SinglePSpace, SingleDomain,
random_symbols, PSpace, ConditionalDomain, RandomDomain,
ProductDomain, Distribution)
from sympy.stats.symbolic_probability import Probability
from sympy.sets.fancysets import Range, FiniteSet
from sympy.sets.sets import Union
from sympy.sets.contains import Contains
from sympy.utilities import filldedent
from sympy.core.sympify import _sympify
class DiscreteDistribution(Distribution):
def __call__(self, *args):
return self.pdf(*args)
class SingleDiscreteDistribution(DiscreteDistribution, NamedArgsMixin):
""" Discrete distribution of a single variable.
Serves as superclass for PoissonDistribution etc....
Provides methods for pdf, cdf, and sampling
See Also:
sympy.stats.crv_types.*
"""
set = S.Integers
def __new__(cls, *args):
args = list(map(sympify, args))
return Basic.__new__(cls, *args)
@staticmethod
def check(*args):
pass
@cacheit
def compute_cdf(self, **kwargs):
""" Compute the CDF from the PDF.
Returns a Lambda.
"""
x = symbols('x', integer=True, cls=Dummy)
z = symbols('z', real=True, cls=Dummy)
left_bound = self.set.inf
# CDF is integral of PDF from left bound to z
pdf = self.pdf(x)
cdf = summation(pdf, (x, left_bound, floor(z)), **kwargs)
# CDF Ensure that CDF left of left_bound is zero
cdf = Piecewise((cdf, z >= left_bound), (0, True))
return Lambda(z, cdf)
def _cdf(self, x):
return None
def cdf(self, x, **kwargs):
""" Cumulative density function """
if not kwargs:
cdf = self._cdf(x)
if cdf is not None:
return cdf
return self.compute_cdf(**kwargs)(x)
@cacheit
def compute_characteristic_function(self, **kwargs):
""" Compute the characteristic function from the PDF.
Returns a Lambda.
"""
x, t = symbols('x, t', real=True, cls=Dummy)
pdf = self.pdf(x)
cf = summation(exp(I*t*x)*pdf, (x, self.set.inf, self.set.sup))
return Lambda(t, cf)
def _characteristic_function(self, t):
return None
def characteristic_function(self, t, **kwargs):
""" Characteristic function """
if not kwargs:
cf = self._characteristic_function(t)
if cf is not None:
return cf
return self.compute_characteristic_function(**kwargs)(t)
@cacheit
def compute_moment_generating_function(self, **kwargs):
t = Dummy('t', real=True)
x = Dummy('x', integer=True)
pdf = self.pdf(x)
mgf = summation(exp(t*x)*pdf, (x, self.set.inf, self.set.sup))
return Lambda(t, mgf)
def _moment_generating_function(self, t):
return None
def moment_generating_function(self, t, **kwargs):
if not kwargs:
mgf = self._moment_generating_function(t)
if mgf is not None:
return mgf
return self.compute_moment_generating_function(**kwargs)(t)
@cacheit
def compute_quantile(self, **kwargs):
""" Compute the Quantile from the PDF.
Returns a Lambda.
"""
x = Dummy('x', integer=True)
p = Dummy('p', real=True)
left_bound = self.set.inf
pdf = self.pdf(x)
cdf = summation(pdf, (x, left_bound, x), **kwargs)
set = ((x, p <= cdf), )
return Lambda(p, Piecewise(*set))
def _quantile(self, x):
return None
def quantile(self, x, **kwargs):
""" Cumulative density function """
if not kwargs:
quantile = self._quantile(x)
if quantile is not None:
return quantile
return self.compute_quantile(**kwargs)(x)
def expectation(self, expr, var, evaluate=True, **kwargs):
""" Expectation of expression over distribution """
# TODO: support discrete sets with non integer stepsizes
if evaluate:
try:
p = poly(expr, var)
t = Dummy('t', real=True)
mgf = self.moment_generating_function(t)
deg = p.degree()
taylor = poly(series(mgf, t, 0, deg + 1).removeO(), t)
result = 0
for k in range(deg+1):
result += p.coeff_monomial(var ** k) * taylor.coeff_monomial(t ** k) * factorial(k)
return result
except PolynomialError:
return summation(expr * self.pdf(var),
(var, self.set.inf, self.set.sup), **kwargs)
else:
return Sum(expr * self.pdf(var),
(var, self.set.inf, self.set.sup), **kwargs)
def __call__(self, *args):
return self.pdf(*args)
class DiscreteDomain(RandomDomain):
"""
A domain with discrete support with step size one.
Represented using symbols and Range.
"""
is_Discrete = True
class SingleDiscreteDomain(DiscreteDomain, SingleDomain):
def as_boolean(self):
return Contains(self.symbol, self.set)
class ConditionalDiscreteDomain(DiscreteDomain, ConditionalDomain):
"""
Domain with discrete support of step size one, that is restricted by
some condition.
"""
@property
def set(self):
rv = self.symbols
if len(self.symbols) > 1:
raise NotImplementedError(filldedent('''
Multivariate conditional domains are not yet implemented.'''))
rv = list(rv)[0]
return reduce_rational_inequalities_wrap(self.condition,
rv).intersect(self.fulldomain.set)
class DiscretePSpace(PSpace):
is_real = True
is_Discrete = True
@property
def pdf(self):
return self.density(*self.symbols)
def where(self, condition):
rvs = random_symbols(condition)
assert all(r.symbol in self.symbols for r in rvs)
if len(rvs) > 1:
raise NotImplementedError(filldedent('''Multivariate discrete
random variables are not yet supported.'''))
conditional_domain = reduce_rational_inequalities_wrap(condition,
rvs[0])
conditional_domain = conditional_domain.intersect(self.domain.set)
return SingleDiscreteDomain(rvs[0].symbol, conditional_domain)
def probability(self, condition):
complement = isinstance(condition, Ne)
if complement:
condition = Eq(condition.args[0], condition.args[1])
try:
_domain = self.where(condition).set
if condition == False or _domain is S.EmptySet:
return S.Zero
if condition == True or _domain == self.domain.set:
return S.One
prob = self.eval_prob(_domain)
except NotImplementedError:
from sympy.stats.rv import density
expr = condition.lhs - condition.rhs
dens = density(expr)
if not isinstance(dens, DiscreteDistribution):
from sympy.stats.drv_types import DiscreteDistributionHandmade
dens = DiscreteDistributionHandmade(dens)
z = Dummy('z', real=True)
space = SingleDiscretePSpace(z, dens)
prob = space.probability(condition.__class__(space.value, 0))
if prob is None:
prob = Probability(condition)
return prob if not complement else S.One - prob
def eval_prob(self, _domain):
sym = list(self.symbols)[0]
if isinstance(_domain, Range):
n = symbols('n', integer=True)
inf, sup, step = (r for r in _domain.args)
summand = ((self.pdf).replace(
sym, n*step))
rv = summation(summand,
(n, inf/step, (sup)/step - 1)).doit()
return rv
elif isinstance(_domain, FiniteSet):
pdf = Lambda(sym, self.pdf)
rv = sum(pdf(x) for x in _domain)
return rv
elif isinstance(_domain, Union):
rv = sum(self.eval_prob(x) for x in _domain.args)
return rv
def conditional_space(self, condition):
# XXX: Converting from set to tuple. The order matters to Lambda
# though so we should be starting with a set...
density = Lambda(tuple(self.symbols), self.pdf/self.probability(condition))
condition = condition.xreplace({rv: rv.symbol for rv in self.values})
domain = ConditionalDiscreteDomain(self.domain, condition)
return DiscretePSpace(domain, density)
class ProductDiscreteDomain(ProductDomain, DiscreteDomain):
def as_boolean(self):
return And(*[domain.as_boolean for domain in self.domains])
class SingleDiscretePSpace(DiscretePSpace, SinglePSpace):
""" Discrete probability space over a single univariate variable """
is_real = True
@property
def set(self):
return self.distribution.set
@property
def domain(self):
return SingleDiscreteDomain(self.symbol, self.set)
def sample(self, size=(), library='scipy', seed=None):
"""
Internal sample method.
Returns dictionary mapping RandomSymbol to realization value.
"""
return {self.value: self.distribution.sample(size, library=library, seed=seed)}
def compute_expectation(self, expr, rvs=None, evaluate=True, **kwargs):
rvs = rvs or (self.value,)
if self.value not in rvs:
return expr
expr = _sympify(expr)
expr = expr.xreplace({rv: rv.symbol for rv in rvs})
x = self.value.symbol
try:
return self.distribution.expectation(expr, x, evaluate=evaluate,
**kwargs)
except NotImplementedError:
return Sum(expr * self.pdf, (x, self.set.inf, self.set.sup),
**kwargs)
def compute_cdf(self, expr, **kwargs):
if expr == self.value:
x = Dummy("x", real=True)
return Lambda(x, self.distribution.cdf(x, **kwargs))
else:
raise NotImplementedError()
def compute_density(self, expr, **kwargs):
if expr == self.value:
return self.distribution
raise NotImplementedError()
def compute_characteristic_function(self, expr, **kwargs):
if expr == self.value:
t = Dummy("t", real=True)
return Lambda(t, self.distribution.characteristic_function(t, **kwargs))
else:
raise NotImplementedError()
def compute_moment_generating_function(self, expr, **kwargs):
if expr == self.value:
t = Dummy("t", real=True)
return Lambda(t, self.distribution.moment_generating_function(t, **kwargs))
else:
raise NotImplementedError()
def compute_quantile(self, expr, **kwargs):
if expr == self.value:
p = Dummy("p", real=True)
return Lambda(p, self.distribution.quantile(p, **kwargs))
else:
raise NotImplementedError()
|
51ab2f096371e763ae338ee2a5ea243a72495f92bc94151df26632f14a81d736 | """
Continuous Random Variables Module
See Also
========
sympy.stats.crv_types
sympy.stats.rv
sympy.stats.frv
"""
from sympy import (Interval, Intersection, symbols, sympify, Dummy, nan,
Integral, And, Or, Piecewise, cacheit, integrate, oo, Lambda,
Basic, S, exp, I, FiniteSet, Ne, Eq, Union, poly, series, factorial)
from sympy.core.function import PoleError
from sympy.functions.special.delta_functions import DiracDelta
from sympy.polys.polyerrors import PolynomialError
from sympy.solvers.solveset import solveset
from sympy.solvers.inequalities import reduce_rational_inequalities
from sympy.core.sympify import _sympify
from sympy.stats.rv import (RandomDomain, SingleDomain, ConditionalDomain, is_random,
ProductDomain, PSpace, SinglePSpace, random_symbols, NamedArgsMixin, Distribution)
class ContinuousDomain(RandomDomain):
"""
A domain with continuous support
Represented using symbols and Intervals.
"""
is_Continuous = True
def as_boolean(self):
raise NotImplementedError("Not Implemented for generic Domains")
class SingleContinuousDomain(ContinuousDomain, SingleDomain):
"""
A univariate domain with continuous support
Represented using a single symbol and interval.
"""
def compute_expectation(self, expr, variables=None, **kwargs):
if variables is None:
variables = self.symbols
if not variables:
return expr
if frozenset(variables) != frozenset(self.symbols):
raise ValueError("Values should be equal")
# assumes only intervals
return Integral(expr, (self.symbol, self.set), **kwargs)
def as_boolean(self):
return self.set.as_relational(self.symbol)
class ProductContinuousDomain(ProductDomain, ContinuousDomain):
"""
A collection of independent domains with continuous support
"""
def compute_expectation(self, expr, variables=None, **kwargs):
if variables is None:
variables = self.symbols
for domain in self.domains:
domain_vars = frozenset(variables) & frozenset(domain.symbols)
if domain_vars:
expr = domain.compute_expectation(expr, domain_vars, **kwargs)
return expr
def as_boolean(self):
return And(*[domain.as_boolean() for domain in self.domains])
class ConditionalContinuousDomain(ContinuousDomain, ConditionalDomain):
"""
A domain with continuous support that has been further restricted by a
condition such as $x > 3$.
"""
def compute_expectation(self, expr, variables=None, **kwargs):
if variables is None:
variables = self.symbols
if not variables:
return expr
# Extract the full integral
fullintgrl = self.fulldomain.compute_expectation(expr, variables)
# separate into integrand and limits
integrand, limits = fullintgrl.function, list(fullintgrl.limits)
conditions = [self.condition]
while conditions:
cond = conditions.pop()
if cond.is_Boolean:
if isinstance(cond, And):
conditions.extend(cond.args)
elif isinstance(cond, Or):
raise NotImplementedError("Or not implemented here")
elif cond.is_Relational:
if cond.is_Equality:
# Add the appropriate Delta to the integrand
integrand *= DiracDelta(cond.lhs - cond.rhs)
else:
symbols = cond.free_symbols & set(self.symbols)
if len(symbols) != 1: # Can't handle x > y
raise NotImplementedError(
"Multivariate Inequalities not yet implemented")
# Can handle x > 0
symbol = symbols.pop()
# Find the limit with x, such as (x, -oo, oo)
for i, limit in enumerate(limits):
if limit[0] == symbol:
# Make condition into an Interval like [0, oo]
cintvl = reduce_rational_inequalities_wrap(
cond, symbol)
# Make limit into an Interval like [-oo, oo]
lintvl = Interval(limit[1], limit[2])
# Intersect them to get [0, oo]
intvl = cintvl.intersect(lintvl)
# Put back into limits list
limits[i] = (symbol, intvl.left, intvl.right)
else:
raise TypeError(
"Condition %s is not a relational or Boolean" % cond)
return Integral(integrand, *limits, **kwargs)
def as_boolean(self):
return And(self.fulldomain.as_boolean(), self.condition)
@property
def set(self):
if len(self.symbols) == 1:
return (self.fulldomain.set & reduce_rational_inequalities_wrap(
self.condition, tuple(self.symbols)[0]))
else:
raise NotImplementedError(
"Set of Conditional Domain not Implemented")
class ContinuousDistribution(Distribution):
def __call__(self, *args):
return self.pdf(*args)
class SingleContinuousDistribution(ContinuousDistribution, NamedArgsMixin):
""" Continuous distribution of a single variable.
Explanation
===========
Serves as superclass for Normal/Exponential/UniformDistribution etc....
Represented by parameters for each of the specific classes. E.g
NormalDistribution is represented by a mean and standard deviation.
Provides methods for pdf, cdf, and sampling.
See Also
========
sympy.stats.crv_types.*
"""
set = Interval(-oo, oo)
def __new__(cls, *args):
args = list(map(sympify, args))
return Basic.__new__(cls, *args)
@staticmethod
def check(*args):
pass
@cacheit
def compute_cdf(self, **kwargs):
""" Compute the CDF from the PDF.
Returns a Lambda.
"""
x, z = symbols('x, z', real=True, cls=Dummy)
left_bound = self.set.start
# CDF is integral of PDF from left bound to z
pdf = self.pdf(x)
cdf = integrate(pdf.doit(), (x, left_bound, z), **kwargs)
# CDF Ensure that CDF left of left_bound is zero
cdf = Piecewise((cdf, z >= left_bound), (0, True))
return Lambda(z, cdf)
def _cdf(self, x):
return None
def cdf(self, x, **kwargs):
""" Cumulative density function """
if len(kwargs) == 0:
cdf = self._cdf(x)
if cdf is not None:
return cdf
return self.compute_cdf(**kwargs)(x)
@cacheit
def compute_characteristic_function(self, **kwargs):
""" Compute the characteristic function from the PDF.
Returns a Lambda.
"""
x, t = symbols('x, t', real=True, cls=Dummy)
pdf = self.pdf(x)
cf = integrate(exp(I*t*x)*pdf, (x, self.set))
return Lambda(t, cf)
def _characteristic_function(self, t):
return None
def characteristic_function(self, t, **kwargs):
""" Characteristic function """
if len(kwargs) == 0:
cf = self._characteristic_function(t)
if cf is not None:
return cf
return self.compute_characteristic_function(**kwargs)(t)
@cacheit
def compute_moment_generating_function(self, **kwargs):
""" Compute the moment generating function from the PDF.
Returns a Lambda.
"""
x, t = symbols('x, t', real=True, cls=Dummy)
pdf = self.pdf(x)
mgf = integrate(exp(t * x) * pdf, (x, self.set))
return Lambda(t, mgf)
def _moment_generating_function(self, t):
return None
def moment_generating_function(self, t, **kwargs):
""" Moment generating function """
if not kwargs:
mgf = self._moment_generating_function(t)
if mgf is not None:
return mgf
return self.compute_moment_generating_function(**kwargs)(t)
def expectation(self, expr, var, evaluate=True, **kwargs):
""" Expectation of expression over distribution """
if evaluate:
try:
p = poly(expr, var)
if p.is_zero:
return S.Zero
t = Dummy('t', real=True)
mgf = self._moment_generating_function(t)
if mgf is None:
return integrate(expr * self.pdf(var), (var, self.set), **kwargs)
deg = p.degree()
taylor = poly(series(mgf, t, 0, deg + 1).removeO(), t)
result = 0
for k in range(deg+1):
result += p.coeff_monomial(var ** k) * taylor.coeff_monomial(t ** k) * factorial(k)
return result
except PolynomialError:
return integrate(expr * self.pdf(var), (var, self.set), **kwargs)
else:
return Integral(expr * self.pdf(var), (var, self.set), **kwargs)
@cacheit
def compute_quantile(self, **kwargs):
""" Compute the Quantile from the PDF.
Returns a Lambda.
"""
x, p = symbols('x, p', real=True, cls=Dummy)
left_bound = self.set.start
pdf = self.pdf(x)
cdf = integrate(pdf, (x, left_bound, x), **kwargs)
quantile = solveset(cdf - p, x, self.set)
return Lambda(p, Piecewise((quantile, (p >= 0) & (p <= 1) ), (nan, True)))
def _quantile(self, x):
return None
def quantile(self, x, **kwargs):
""" Cumulative density function """
if len(kwargs) == 0:
quantile = self._quantile(x)
if quantile is not None:
return quantile
return self.compute_quantile(**kwargs)(x)
class ContinuousPSpace(PSpace):
""" Continuous Probability Space
Represents the likelihood of an event space defined over a continuum.
Represented with a ContinuousDomain and a PDF (Lambda-Like)
"""
is_Continuous = True
is_real = True
@property
def pdf(self):
return self.density(*self.domain.symbols)
def compute_expectation(self, expr, rvs=None, evaluate=False, **kwargs):
if rvs is None:
rvs = self.values
else:
rvs = frozenset(rvs)
expr = expr.xreplace({rv: rv.symbol for rv in rvs})
domain_symbols = frozenset(rv.symbol for rv in rvs)
return self.domain.compute_expectation(self.pdf * expr,
domain_symbols, **kwargs)
def compute_density(self, expr, **kwargs):
# Common case Density(X) where X in self.values
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.compute_expectation(self.pdf, symbols, **kwargs)
return Lambda(expr.symbol, pdf)
z = Dummy('z', real=True)
return Lambda(z, self.compute_expectation(DiracDelta(expr - z), **kwargs))
@cacheit
def compute_cdf(self, expr, **kwargs):
if not self.domain.set.is_Interval:
raise ValueError(
"CDF not well defined on multivariate expressions")
d = self.compute_density(expr, **kwargs)
x, z = symbols('x, z', real=True, cls=Dummy)
left_bound = self.domain.set.start
# CDF is integral of PDF from left bound to z
cdf = integrate(d(x), (x, left_bound, z), **kwargs)
# CDF Ensure that CDF left of left_bound is zero
cdf = Piecewise((cdf, z >= left_bound), (0, True))
return Lambda(z, cdf)
@cacheit
def compute_characteristic_function(self, expr, **kwargs):
if not self.domain.set.is_Interval:
raise NotImplementedError("Characteristic function of multivariate expressions not implemented")
d = self.compute_density(expr, **kwargs)
x, t = symbols('x, t', real=True, cls=Dummy)
cf = integrate(exp(I*t*x)*d(x), (x, -oo, oo), **kwargs)
return Lambda(t, cf)
@cacheit
def compute_moment_generating_function(self, expr, **kwargs):
if not self.domain.set.is_Interval:
raise NotImplementedError("Moment generating function of multivariate expressions not implemented")
d = self.compute_density(expr, **kwargs)
x, t = symbols('x, t', real=True, cls=Dummy)
mgf = integrate(exp(t * x) * d(x), (x, -oo, oo), **kwargs)
return Lambda(t, mgf)
@cacheit
def compute_quantile(self, expr, **kwargs):
if not self.domain.set.is_Interval:
raise ValueError(
"Quantile not well defined on multivariate expressions")
d = self.compute_cdf(expr, **kwargs)
x = Dummy('x', real=True)
p = Dummy('p', positive=True)
quantile = solveset(d(x) - p, x, self.set)
return Lambda(p, quantile)
def probability(self, condition, **kwargs):
z = Dummy('z', real=True)
cond_inv = False
if isinstance(condition, Ne):
condition = Eq(condition.args[0], condition.args[1])
cond_inv = True
# Univariate case can be handled by where
try:
domain = self.where(condition)
rv = [rv for rv in self.values if rv.symbol == domain.symbol][0]
# Integrate out all other random variables
pdf = self.compute_density(rv, **kwargs)
# return S.Zero if `domain` is empty set
if domain.set is S.EmptySet or isinstance(domain.set, FiniteSet):
return S.Zero if not cond_inv else S.One
if isinstance(domain.set, Union):
return sum(
Integral(pdf(z), (z, subset), **kwargs) for subset in
domain.set.args if isinstance(subset, Interval))
# Integrate out the last variable over the special domain
return Integral(pdf(z), (z, domain.set), **kwargs)
# Other cases can be turned into univariate case
# by computing a density handled by density computation
except NotImplementedError:
from sympy.stats.rv import density
expr = condition.lhs - condition.rhs
if not is_random(expr):
dens = self.density
comp = condition.rhs
else:
dens = density(expr, **kwargs)
comp = 0
if not isinstance(dens, ContinuousDistribution):
from sympy.stats.crv_types import ContinuousDistributionHandmade
dens = ContinuousDistributionHandmade(dens, set=self.domain.set)
# Turn problem into univariate case
space = SingleContinuousPSpace(z, dens)
result = space.probability(condition.__class__(space.value, comp))
return result if not cond_inv else S.One - result
def where(self, condition):
rvs = frozenset(random_symbols(condition))
if not (len(rvs) == 1 and rvs.issubset(self.values)):
raise NotImplementedError(
"Multiple continuous random variables not supported")
rv = tuple(rvs)[0]
interval = reduce_rational_inequalities_wrap(condition, rv)
interval = interval.intersect(self.domain.set)
return SingleContinuousDomain(rv.symbol, interval)
def conditional_space(self, condition, normalize=True, **kwargs):
condition = condition.xreplace({rv: rv.symbol for rv in self.values})
domain = ConditionalContinuousDomain(self.domain, condition)
if normalize:
# create a clone of the variable to
# make sure that variables in nested integrals are different
# from the variables outside the integral
# this makes sure that they are evaluated separately
# and in the correct order
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 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 ContinuousPSpace(domain, density)
class SingleContinuousPSpace(ContinuousPSpace, SinglePSpace):
"""
A continuous probability space over a single univariate variable.
These consist of a Symbol and a SingleContinuousDistribution
This class is normally accessed through the various random variable
functions, Normal, Exponential, Uniform, etc....
"""
@property
def set(self):
return self.distribution.set
@property
def domain(self):
return SingleContinuousDomain(sympify(self.symbol), self.set)
def sample(self, size=(), library='scipy', seed=None):
"""
Internal sample method.
Returns dictionary mapping RandomSymbol to realization value.
"""
return {self.value: self.distribution.sample(size, library=library, seed=seed)}
def compute_expectation(self, expr, rvs=None, evaluate=False, **kwargs):
rvs = rvs or (self.value,)
if self.value not in rvs:
return expr
expr = _sympify(expr)
expr = expr.xreplace({rv: rv.symbol for rv in rvs})
x = self.value.symbol
try:
return self.distribution.expectation(expr, x, evaluate=evaluate, **kwargs)
except PoleError:
return Integral(expr * self.pdf, (x, self.set), **kwargs)
def compute_cdf(self, expr, **kwargs):
if expr == self.value:
z = Dummy("z", real=True)
return Lambda(z, self.distribution.cdf(z, **kwargs))
else:
return ContinuousPSpace.compute_cdf(self, expr, **kwargs)
def compute_characteristic_function(self, expr, **kwargs):
if expr == self.value:
t = Dummy("t", real=True)
return Lambda(t, self.distribution.characteristic_function(t, **kwargs))
else:
return ContinuousPSpace.compute_characteristic_function(self, expr, **kwargs)
def compute_moment_generating_function(self, expr, **kwargs):
if expr == self.value:
t = Dummy("t", real=True)
return Lambda(t, self.distribution.moment_generating_function(t, **kwargs))
else:
return ContinuousPSpace.compute_moment_generating_function(self, expr, **kwargs)
def compute_density(self, expr, **kwargs):
# https://en.wikipedia.org/wiki/Random_variable#Functions_of_random_variables
if expr == self.value:
return self.density
y = Dummy('y', real=True)
gs = solveset(expr - y, self.value, S.Reals)
if isinstance(gs, Intersection) and S.Reals in gs.args:
gs = list(gs.args[1])
if not gs:
raise ValueError("Can not solve %s for %s"%(expr, self.value))
fx = self.compute_density(self.value)
fy = sum(fx(g) * abs(g.diff(y)) for g in gs)
return Lambda(y, fy)
def compute_quantile(self, expr, **kwargs):
if expr == self.value:
p = Dummy("p", real=True)
return Lambda(p, self.distribution.quantile(p, **kwargs))
else:
return ContinuousPSpace.compute_quantile(self, expr, **kwargs)
def _reduce_inequalities(conditions, var, **kwargs):
try:
return reduce_rational_inequalities(conditions, var, **kwargs)
except PolynomialError:
raise ValueError("Reduction of condition failed %s\n" % conditions[0])
def reduce_rational_inequalities_wrap(condition, var):
if condition.is_Relational:
return _reduce_inequalities([[condition]], var, relational=False)
if isinstance(condition, Or):
return Union(*[_reduce_inequalities([[arg]], var, relational=False)
for arg in condition.args])
if isinstance(condition, And):
intervals = [_reduce_inequalities([[arg]], var, relational=False)
for arg in condition.args]
I = intervals[0]
for i in intervals:
I = I.intersect(i)
return I
|
19d062d085a7d3a14680898fd7d5a21e8c052bf9372e7dfa01af1bbec59ca5cf | from sympy import (Basic, exp, pi, Lambda, Trace, S, MatrixSymbol, Integral,
gamma, Product, Dummy, Sum, Abs, IndexedBase, I)
from sympy.core.sympify import _sympify
from sympy.stats.rv import _symbol_converter, Density, RandomMatrixSymbol, is_random
from sympy.stats.joint_rv_types import JointDistributionHandmade
from sympy.stats.random_matrix import RandomMatrixPSpace
from sympy.tensor.array import ArrayComprehension
__all__ = [
'CircularEnsemble',
'CircularUnitaryEnsemble',
'CircularOrthogonalEnsemble',
'CircularSymplecticEnsemble',
'GaussianEnsemble',
'GaussianUnitaryEnsemble',
'GaussianOrthogonalEnsemble',
'GaussianSymplecticEnsemble',
'joint_eigen_distribution',
'JointEigenDistribution',
'level_spacing_distribution'
]
@is_random.register(RandomMatrixSymbol)
def _(x):
return True
class RandomMatrixEnsembleModel(Basic):
"""
Base class for random matrix ensembles.
It acts as an umbrella and contains
the methods common to all the ensembles
defined in sympy.stats.random_matrix_models.
"""
def __new__(cls, sym, dim=None):
sym, dim = _symbol_converter(sym), _sympify(dim)
if dim.is_integer == False:
raise ValueError("Dimension of the random matrices must be "
"integers, received %s instead."%(dim))
return Basic.__new__(cls, sym, dim)
symbol = property(lambda self: self.args[0])
dimension = property(lambda self: self.args[1])
def density(self, expr):
return Density(expr)
def __call__(self, expr):
return self.density(expr)
class GaussianEnsembleModel(RandomMatrixEnsembleModel):
"""
Abstract class for Gaussian ensembles.
Contains the properties common to all the
gaussian ensembles.
References
==========
.. [1] https://en.wikipedia.org/wiki/Random_matrix#Gaussian_ensembles
.. [2] https://arxiv.org/pdf/1712.07903.pdf
"""
def _compute_normalization_constant(self, beta, n):
"""
Helper function for computing normalization
constant for joint probability density of eigen
values of Gaussian ensembles.
References
==========
.. [1] https://en.wikipedia.org/wiki/Selberg_integral#Mehta's_integral
"""
n = S(n)
prod_term = lambda j: gamma(1 + beta*S(j)/2)/gamma(S.One + beta/S(2))
j = Dummy('j', integer=True, positive=True)
term1 = Product(prod_term(j), (j, 1, n)).doit()
term2 = (2/(beta*n))**(beta*n*(n - 1)/4 + n/2)
term3 = (2*pi)**(n/2)
return term1 * term2 * term3
def _compute_joint_eigen_distribution(self, beta):
"""
Helper function for computing the joint
probability distribution of eigen values
of the random matrix.
"""
n = self.dimension
Zbn = self._compute_normalization_constant(beta, n)
l = IndexedBase('l')
i = Dummy('i', integer=True, positive=True)
j = Dummy('j', integer=True, positive=True)
k = Dummy('k', integer=True, positive=True)
term1 = exp((-S(n)/2) * Sum(l[k]**2, (k, 1, n)).doit())
sub_term = Lambda(i, Product(Abs(l[j] - l[i])**beta, (j, i + 1, n)))
term2 = Product(sub_term(i).doit(), (i, 1, n - 1)).doit()
syms = ArrayComprehension(l[k], (k, 1, n)).doit()
return Lambda(tuple(syms), (term1 * term2)/Zbn)
class GaussianUnitaryEnsembleModel(GaussianEnsembleModel):
@property
def normalization_constant(self):
n = self.dimension
return 2**(S(n)/2) * pi**(S(n**2)/2)
def density(self, expr):
n, ZGUE = self.dimension, self.normalization_constant
h_pspace = RandomMatrixPSpace('P', model=self)
H = RandomMatrixSymbol('H', n, n, pspace=h_pspace)
return Lambda(H, exp(-S(n)/2 * Trace(H**2))/ZGUE)(expr)
def joint_eigen_distribution(self):
return self._compute_joint_eigen_distribution(S(2))
def level_spacing_distribution(self):
s = Dummy('s')
f = (32/pi**2)*(s**2)*exp((-4/pi)*s**2)
return Lambda(s, f)
class GaussianOrthogonalEnsembleModel(GaussianEnsembleModel):
@property
def normalization_constant(self):
n = self.dimension
_H = MatrixSymbol('_H', n, n)
return Integral(exp(-S(n)/4 * Trace(_H**2)))
def density(self, expr):
n, ZGOE = self.dimension, self.normalization_constant
h_pspace = RandomMatrixPSpace('P', model=self)
H = RandomMatrixSymbol('H', n, n, pspace=h_pspace)
return Lambda(H, exp(-S(n)/4 * Trace(H**2))/ZGOE)(expr)
def joint_eigen_distribution(self):
return self._compute_joint_eigen_distribution(S.One)
def level_spacing_distribution(self):
s = Dummy('s')
f = (pi/2)*s*exp((-pi/4)*s**2)
return Lambda(s, f)
class GaussianSymplecticEnsembleModel(GaussianEnsembleModel):
@property
def normalization_constant(self):
n = self.dimension
_H = MatrixSymbol('_H', n, n)
return Integral(exp(-S(n) * Trace(_H**2)))
def density(self, expr):
n, ZGSE = self.dimension, self.normalization_constant
h_pspace = RandomMatrixPSpace('P', model=self)
H = RandomMatrixSymbol('H', n, n, pspace=h_pspace)
return Lambda(H, exp(-S(n) * Trace(H**2))/ZGSE)(expr)
def joint_eigen_distribution(self):
return self._compute_joint_eigen_distribution(S(4))
def level_spacing_distribution(self):
s = Dummy('s')
f = ((S(2)**18)/((S(3)**6)*(pi**3)))*(s**4)*exp((-64/(9*pi))*s**2)
return Lambda(s, f)
def GaussianEnsemble(sym, dim):
sym, dim = _symbol_converter(sym), _sympify(dim)
model = GaussianEnsembleModel(sym, dim)
rmp = RandomMatrixPSpace(sym, model=model)
return RandomMatrixSymbol(sym, dim, dim, pspace=rmp)
def GaussianUnitaryEnsemble(sym, dim):
"""
Represents Gaussian Unitary Ensembles.
Examples
========
>>> from sympy.stats import GaussianUnitaryEnsemble as GUE, density
>>> from sympy import MatrixSymbol
>>> G = GUE('U', 2)
>>> X = MatrixSymbol('X', 2, 2)
>>> density(G)(X)
exp(-Trace(X**2))/(2*pi**2)
"""
sym, dim = _symbol_converter(sym), _sympify(dim)
model = GaussianUnitaryEnsembleModel(sym, dim)
rmp = RandomMatrixPSpace(sym, model=model)
return RandomMatrixSymbol(sym, dim, dim, pspace=rmp)
def GaussianOrthogonalEnsemble(sym, dim):
"""
Represents Gaussian Orthogonal Ensembles.
Examples
========
>>> from sympy.stats import GaussianOrthogonalEnsemble as GOE, density
>>> from sympy import MatrixSymbol
>>> G = GOE('U', 2)
>>> X = MatrixSymbol('X', 2, 2)
>>> density(G)(X)
exp(-Trace(X**2)/2)/Integral(exp(-Trace(_H**2)/2), _H)
"""
sym, dim = _symbol_converter(sym), _sympify(dim)
model = GaussianOrthogonalEnsembleModel(sym, dim)
rmp = RandomMatrixPSpace(sym, model=model)
return RandomMatrixSymbol(sym, dim, dim, pspace=rmp)
def GaussianSymplecticEnsemble(sym, dim):
"""
Represents Gaussian Symplectic Ensembles.
Examples
========
>>> from sympy.stats import GaussianSymplecticEnsemble as GSE, density
>>> from sympy import MatrixSymbol
>>> G = GSE('U', 2)
>>> X = MatrixSymbol('X', 2, 2)
>>> density(G)(X)
exp(-2*Trace(X**2))/Integral(exp(-2*Trace(_H**2)), _H)
"""
sym, dim = _symbol_converter(sym), _sympify(dim)
model = GaussianSymplecticEnsembleModel(sym, dim)
rmp = RandomMatrixPSpace(sym, model=model)
return RandomMatrixSymbol(sym, dim, dim, pspace=rmp)
class CircularEnsembleModel(RandomMatrixEnsembleModel):
"""
Abstract class for Circular ensembles.
Contains the properties and methods
common to all the circular ensembles.
References
==========
.. [1] https://en.wikipedia.org/wiki/Circular_ensemble
"""
def density(self, expr):
# TODO : Add support for Lie groups(as extensions of sympy.diffgeom)
# and define measures on them
raise NotImplementedError("Support for Haar measure hasn't been "
"implemented yet, therefore the density of "
"%s cannot be computed."%(self))
def _compute_joint_eigen_distribution(self, beta):
"""
Helper function to compute the joint distribution of phases
of the complex eigen values of matrices belonging to any
circular ensembles.
"""
n = self.dimension
Zbn = ((2*pi)**n)*(gamma(beta*n/2 + 1)/S(gamma(beta/2 + 1))**n)
t = IndexedBase('t')
i, j, k = (Dummy('i', integer=True), Dummy('j', integer=True),
Dummy('k', integer=True))
syms = ArrayComprehension(t[i], (i, 1, n)).doit()
f = Product(Product(Abs(exp(I*t[k]) - exp(I*t[j]))**beta, (j, k + 1, n)).doit(),
(k, 1, n - 1)).doit()
return Lambda(tuple(syms), f/Zbn)
class CircularUnitaryEnsembleModel(CircularEnsembleModel):
def joint_eigen_distribution(self):
return self._compute_joint_eigen_distribution(S(2))
class CircularOrthogonalEnsembleModel(CircularEnsembleModel):
def joint_eigen_distribution(self):
return self._compute_joint_eigen_distribution(S.One)
class CircularSymplecticEnsembleModel(CircularEnsembleModel):
def joint_eigen_distribution(self):
return self._compute_joint_eigen_distribution(S(4))
def CircularEnsemble(sym, dim):
sym, dim = _symbol_converter(sym), _sympify(dim)
model = CircularEnsembleModel(sym, dim)
rmp = RandomMatrixPSpace(sym, model=model)
return RandomMatrixSymbol(sym, dim, dim, pspace=rmp)
def CircularUnitaryEnsemble(sym, dim):
"""
Represents Cicular Unitary Ensembles.
Examples
========
>>> from sympy.stats import CircularUnitaryEnsemble as CUE
>>> from sympy.stats import joint_eigen_distribution
>>> C = CUE('U', 1)
>>> joint_eigen_distribution(C)
Lambda(t[1], Product(Abs(exp(I*t[_j]) - exp(I*t[_k]))**2, (_j, _k + 1, 1), (_k, 1, 0))/(2*pi))
Note
====
As can be seen above in the example, density of CiruclarUnitaryEnsemble
is not evaluated becuase the exact definition is based on haar measure of
unitary group which is not unique.
"""
sym, dim = _symbol_converter(sym), _sympify(dim)
model = CircularUnitaryEnsembleModel(sym, dim)
rmp = RandomMatrixPSpace(sym, model=model)
return RandomMatrixSymbol(sym, dim, dim, pspace=rmp)
def CircularOrthogonalEnsemble(sym, dim):
"""
Represents Cicular Orthogonal Ensembles.
Examples
========
>>> from sympy.stats import CircularOrthogonalEnsemble as COE
>>> from sympy.stats import joint_eigen_distribution
>>> C = COE('O', 1)
>>> joint_eigen_distribution(C)
Lambda(t[1], Product(Abs(exp(I*t[_j]) - exp(I*t[_k])), (_j, _k + 1, 1), (_k, 1, 0))/(2*pi))
Note
====
As can be seen above in the example, density of CiruclarOrthogonalEnsemble
is not evaluated becuase the exact definition is based on haar measure of
unitary group which is not unique.
"""
sym, dim = _symbol_converter(sym), _sympify(dim)
model = CircularOrthogonalEnsembleModel(sym, dim)
rmp = RandomMatrixPSpace(sym, model=model)
return RandomMatrixSymbol(sym, dim, dim, pspace=rmp)
def CircularSymplecticEnsemble(sym, dim):
"""
Represents Cicular Symplectic Ensembles.
Examples
========
>>> from sympy.stats import CircularSymplecticEnsemble as CSE
>>> from sympy.stats import joint_eigen_distribution
>>> C = CSE('S', 1)
>>> joint_eigen_distribution(C)
Lambda(t[1], Product(Abs(exp(I*t[_j]) - exp(I*t[_k]))**4, (_j, _k + 1, 1), (_k, 1, 0))/(2*pi))
Note
====
As can be seen above in the example, density of CiruclarSymplecticEnsemble
is not evaluated becuase the exact definition is based on haar measure of
unitary group which is not unique.
"""
sym, dim = _symbol_converter(sym), _sympify(dim)
model = CircularSymplecticEnsembleModel(sym, dim)
rmp = RandomMatrixPSpace(sym, model=model)
return RandomMatrixSymbol(sym, dim, dim, pspace=rmp)
def joint_eigen_distribution(mat):
"""
For obtaining joint probability distribution
of eigen values of random matrix.
Parameters
==========
mat: RandomMatrixSymbol
The matrix symbol whose eigen values are to be considered.
Returns
=======
Lambda
Examples
========
>>> from sympy.stats import GaussianUnitaryEnsemble as GUE
>>> from sympy.stats import joint_eigen_distribution
>>> U = GUE('U', 2)
>>> joint_eigen_distribution(U)
Lambda((l[1], l[2]), exp(-l[1]**2 - l[2]**2)*Product(Abs(l[_i] - l[_j])**2, (_j, _i + 1, 2), (_i, 1, 1))/pi)
"""
if not isinstance(mat, RandomMatrixSymbol):
raise ValueError("%s is not of type, RandomMatrixSymbol."%(mat))
return mat.pspace.model.joint_eigen_distribution()
def JointEigenDistribution(mat):
"""
Creates joint distribution of eigen values of matrices with random
expressions.
Parameters
==========
mat: Matrix
The matrix under consideration.
Returns
=======
JointDistributionHandmade
Examples
========
>>> from sympy.stats import Normal, JointEigenDistribution
>>> from sympy import Matrix
>>> A = [[Normal('A00', 0, 1), Normal('A01', 0, 1)],
... [Normal('A10', 0, 1), Normal('A11', 0, 1)]]
>>> JointEigenDistribution(Matrix(A))
JointDistributionHandmade(-sqrt(A00**2 - 2*A00*A11 + 4*A01*A10 + A11**2)/2
+ A00/2 + A11/2, sqrt(A00**2 - 2*A00*A11 + 4*A01*A10 + A11**2)/2 + A00/2 + A11/2)
"""
eigenvals = mat.eigenvals(multiple=True)
if any(not is_random(eigenval) for eigenval in set(eigenvals)):
raise ValueError("Eigen values don't have any random expression, "
"joint distribution cannot be generated.")
return JointDistributionHandmade(*eigenvals)
def level_spacing_distribution(mat):
"""
For obtaining distribution of level spacings.
Parameters
==========
mat: RandomMatrixSymbol
The random matrix symbol whose eigen values are
to be considered for finding the level spacings.
Returns
=======
Lambda
Examples
========
>>> from sympy.stats import GaussianUnitaryEnsemble as GUE
>>> from sympy.stats import level_spacing_distribution
>>> U = GUE('U', 2)
>>> level_spacing_distribution(U)
Lambda(_s, 32*_s**2*exp(-4*_s**2/pi)/pi**2)
References
==========
.. [1] https://en.wikipedia.org/wiki/Random_matrix#Distribution_of_level_spacings
"""
return mat.pspace.model.level_spacing_distribution()
|
bf6b7b8bf185e168c5c7a3bf91cf64c440fcbcc64fa3cecafc2619319017ac89 | from sympy import Basic
from sympy.stats.joint_rv import ProductPSpace
from sympy.stats.rv import ProductDomain, _symbol_converter, Distribution
class StochasticPSpace(ProductPSpace):
"""
Represents probability space of stochastic processes
and their random variables. Contains mechanics to do
computations for queries of stochastic processes.
Explanation
===========
Initialized by symbol, the specific process and
distribution(optional) if the random indexed symbols
of the process follows any specific distribution, like,
in Bernoulli Process, each random indexed symbol follows
Bernoulli distribution. For processes with memory, this
parameter should not be passed.
"""
def __new__(cls, sym, process, distribution=None):
sym = _symbol_converter(sym)
from sympy.stats.stochastic_process_types import StochasticProcess
if not isinstance(process, StochasticProcess):
raise TypeError("`process` must be an instance of StochasticProcess.")
if distribution is None:
distribution = Distribution()
return Basic.__new__(cls, sym, process, distribution)
@property
def process(self):
"""
The associated stochastic process.
"""
return self.args[1]
@property
def domain(self):
return ProductDomain(self.process.index_set,
self.process.state_space)
@property
def symbol(self):
return self.args[0]
@property
def distribution(self):
return self.args[2]
def probability(self, condition, given_condition=None, evaluate=True, **kwargs):
"""
Transfers the task of handling queries to the specific stochastic
process because every process has their own logic of handling such
queries.
"""
return self.process.probability(condition, given_condition, evaluate, **kwargs)
def compute_expectation(self, expr, condition=None, evaluate=True, **kwargs):
"""
Transfers the task of handling queries to the specific stochastic
process because every process has their own logic of handling such
queries.
"""
return self.process.expectation(expr, condition, evaluate, **kwargs)
|
f0d7a31e6240f521921e1641f47591bac55435b5dd9ef49b05a53aa31bfbf8a1 | """
Generating and counting primes.
"""
import random
from bisect import bisect
from itertools import count
# Using arrays for sieving instead of lists greatly reduces
# memory consumption
from array import array as _array
from sympy import Function, S
from sympy.core.compatibility import as_int
from .primetest import isprime
def _azeros(n):
return _array('l', [0]*n)
def _aset(*v):
return _array('l', v)
def _arange(a, b):
return _array('l', range(a, b))
class Sieve:
"""An infinite list of prime numbers, implemented as a dynamically
growing sieve of Eratosthenes. When a lookup is requested involving
an odd number that has not been sieved, the sieve is automatically
extended up to that number.
Examples
========
>>> from sympy import sieve
>>> sieve._reset() # this line for doctest only
>>> 25 in sieve
False
>>> sieve._list
array('l', [2, 3, 5, 7, 11, 13, 17, 19, 23])
"""
# data shared (and updated) by all Sieve instances
def __init__(self):
self._n = 6
self._list = _aset(2, 3, 5, 7, 11, 13) # primes
self._tlist = _aset(0, 1, 1, 2, 2, 4) # totient
self._mlist = _aset(0, 1, -1, -1, 0, -1) # mobius
assert all(len(i) == self._n for i in (self._list, self._tlist, self._mlist))
def __repr__(self):
return ("<%s sieve (%i): %i, %i, %i, ... %i, %i\n"
"%s sieve (%i): %i, %i, %i, ... %i, %i\n"
"%s sieve (%i): %i, %i, %i, ... %i, %i>") % (
'prime', len(self._list),
self._list[0], self._list[1], self._list[2],
self._list[-2], self._list[-1],
'totient', len(self._tlist),
self._tlist[0], self._tlist[1],
self._tlist[2], self._tlist[-2], self._tlist[-1],
'mobius', len(self._mlist),
self._mlist[0], self._mlist[1],
self._mlist[2], self._mlist[-2], self._mlist[-1])
def _reset(self, prime=None, totient=None, mobius=None):
"""Reset all caches (default). To reset one or more set the
desired keyword to True."""
if all(i is None for i in (prime, totient, mobius)):
prime = totient = mobius = True
if prime:
self._list = self._list[:self._n]
if totient:
self._tlist = self._tlist[:self._n]
if mobius:
self._mlist = self._mlist[:self._n]
def extend(self, n):
"""Grow the sieve to cover all primes <= n (a real number).
Examples
========
>>> from sympy import sieve
>>> sieve._reset() # this line for doctest only
>>> sieve.extend(30)
>>> sieve[10] == 29
True
"""
n = int(n)
if n <= self._list[-1]:
return
# We need to sieve against all bases up to sqrt(n).
# This is a recursive call that will do nothing if there are enough
# known bases already.
maxbase = int(n**0.5) + 1
self.extend(maxbase)
# Create a new sieve starting from sqrt(n)
begin = self._list[-1] + 1
newsieve = _arange(begin, n + 1)
# Now eliminate all multiples of primes in [2, sqrt(n)]
for p in self.primerange(maxbase):
# Start counting at a multiple of p, offsetting
# the index to account for the new sieve's base index
startindex = (-begin) % p
for i in range(startindex, len(newsieve), p):
newsieve[i] = 0
# Merge the sieves
self._list += _array('l', [x for x in newsieve if x])
def extend_to_no(self, i):
"""Extend to include the ith prime number.
Parameters
==========
i : integer
Examples
========
>>> from sympy import sieve
>>> sieve._reset() # this line for doctest only
>>> sieve.extend_to_no(9)
>>> sieve._list
array('l', [2, 3, 5, 7, 11, 13, 17, 19, 23])
Notes
=====
The list is extended by 50% if it is too short, so it is
likely that it will be longer than requested.
"""
i = as_int(i)
while len(self._list) < i:
self.extend(int(self._list[-1] * 1.5))
def primerange(self, a, b=None):
"""Generate all prime numbers in the range [2, a) or [a, b).
Examples
========
>>> from sympy import sieve, prime
All primes less than 19:
>>> print([i for i in sieve.primerange(19)])
[2, 3, 5, 7, 11, 13, 17]
All primes greater than or equal to 7 and less than 19:
>>> print([i for i in sieve.primerange(7, 19)])
[7, 11, 13, 17]
All primes through the 10th prime
>>> list(sieve.primerange(prime(10) + 1))
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
"""
from sympy.functions.elementary.integers import ceiling
# wrapping ceiling in as_int will raise an error if there was a problem
# determining whether the expression was exactly an integer or not
if b is None:
b = as_int(ceiling(a))
a = 2
else:
a = max(2, as_int(ceiling(a)))
b = as_int(ceiling(b))
if a >= b:
return
self.extend(b)
i = self.search(a)[1]
maxi = len(self._list) + 1
while i < maxi:
p = self._list[i - 1]
if p < b:
yield p
i += 1
else:
return
def totientrange(self, a, b):
"""Generate all totient numbers for the range [a, b).
Examples
========
>>> from sympy import sieve
>>> print([i for i in sieve.totientrange(7, 18)])
[6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16]
"""
from sympy.functions.elementary.integers import ceiling
# wrapping ceiling in as_int will raise an error if there was a problem
# determining whether the expression was exactly an integer or not
a = max(1, as_int(ceiling(a)))
b = as_int(ceiling(b))
n = len(self._tlist)
if a >= b:
return
elif b <= n:
for i in range(a, b):
yield self._tlist[i]
else:
self._tlist += _arange(n, b)
for i in range(1, n):
ti = self._tlist[i]
startindex = (n + i - 1) // i * i
for j in range(startindex, b, i):
self._tlist[j] -= ti
if i >= a:
yield ti
for i in range(n, b):
ti = self._tlist[i]
for j in range(2 * i, b, i):
self._tlist[j] -= ti
if i >= a:
yield ti
def mobiusrange(self, a, b):
"""Generate all mobius numbers for the range [a, b).
Parameters
==========
a : integer
First number in range
b : integer
First number outside of range
Examples
========
>>> from sympy import sieve
>>> print([i for i in sieve.mobiusrange(7, 18)])
[-1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1]
"""
from sympy.functions.elementary.integers import ceiling
# wrapping ceiling in as_int will raise an error if there was a problem
# determining whether the expression was exactly an integer or not
a = max(1, as_int(ceiling(a)))
b = as_int(ceiling(b))
n = len(self._mlist)
if a >= b:
return
elif b <= n:
for i in range(a, b):
yield self._mlist[i]
else:
self._mlist += _azeros(b - n)
for i in range(1, n):
mi = self._mlist[i]
startindex = (n + i - 1) // i * i
for j in range(startindex, b, i):
self._mlist[j] -= mi
if i >= a:
yield mi
for i in range(n, b):
mi = self._mlist[i]
for j in range(2 * i, b, i):
self._mlist[j] -= mi
if i >= a:
yield mi
def search(self, n):
"""Return the indices i, j of the primes that bound n.
If n is prime then i == j.
Although n can be an expression, if ceiling cannot convert
it to an integer then an n error will be raised.
Examples
========
>>> from sympy import sieve
>>> sieve.search(25)
(9, 10)
>>> sieve.search(23)
(9, 9)
"""
from sympy.functions.elementary.integers import ceiling
# wrapping ceiling in as_int will raise an error if there was a problem
# determining whether the expression was exactly an integer or not
test = as_int(ceiling(n))
n = as_int(n)
if n < 2:
raise ValueError("n should be >= 2 but got: %s" % n)
if n > self._list[-1]:
self.extend(n)
b = bisect(self._list, n)
if self._list[b - 1] == test:
return b, b
else:
return b, b + 1
def __contains__(self, n):
try:
n = as_int(n)
assert n >= 2
except (ValueError, AssertionError):
return False
if n % 2 == 0:
return n == 2
a, b = self.search(n)
return a == b
def __iter__(self):
for n in count(1):
yield self[n]
def __getitem__(self, n):
"""Return the nth prime number"""
if isinstance(n, slice):
self.extend_to_no(n.stop)
# Python 2.7 slices have 0 instead of None for start, so
# we can't default to 1.
start = n.start if n.start is not None else 0
if start < 1:
# sieve[:5] would be empty (starting at -1), let's
# just be explicit and raise.
raise IndexError("Sieve indices start at 1.")
return self._list[start - 1:n.stop - 1:n.step]
else:
if n < 1:
# offset is one, so forbid explicit access to sieve[0]
# (would surprisingly return the last one).
raise IndexError("Sieve indices start at 1.")
n = as_int(n)
self.extend_to_no(n)
return self._list[n - 1]
# Generate a global object for repeated use in trial division etc
sieve = Sieve()
def prime(nth):
""" Return the nth prime, with the primes indexed as prime(1) = 2,
prime(2) = 3, etc.... The nth prime is approximately n*log(n).
Logarithmic integral of x is a pretty nice approximation for number of
primes <= x, i.e.
li(x) ~ pi(x)
In fact, for the numbers we are concerned about( x<1e11 ),
li(x) - pi(x) < 50000
Also,
li(x) > pi(x) can be safely assumed for the numbers which
can be evaluated by this function.
Here, we find the least integer m such that li(m) > n using binary search.
Now pi(m-1) < li(m-1) <= n,
We find pi(m - 1) using primepi function.
Starting from m, we have to find n - pi(m-1) more primes.
For the inputs this implementation can handle, we will have to test
primality for at max about 10**5 numbers, to get our answer.
Examples
========
>>> from sympy import prime
>>> prime(10)
29
>>> prime(1)
2
>>> prime(100000)
1299709
See Also
========
sympy.ntheory.primetest.isprime : Test if n is prime
primerange : Generate all primes in a given range
primepi : Return the number of primes less than or equal to n
References
==========
.. [1] https://en.wikipedia.org/wiki/Prime_number_theorem#Table_of_.CF.80.28x.29.2C_x_.2F_log_x.2C_and_li.28x.29
.. [2] https://en.wikipedia.org/wiki/Prime_number_theorem#Approximations_for_the_nth_prime_number
.. [3] https://en.wikipedia.org/wiki/Skewes%27_number
"""
n = as_int(nth)
if n < 1:
raise ValueError("nth must be a positive integer; prime(1) == 2")
if n <= len(sieve._list):
return sieve[n]
from sympy.functions.special.error_functions import li
from sympy.functions.elementary.exponential import log
a = 2 # Lower bound for binary search
b = int(n*(log(n) + log(log(n)))) # Upper bound for the search.
while a < b:
mid = (a + b) >> 1
if li(mid) > n:
b = mid
else:
a = mid + 1
n_primes = primepi(a - 1)
while n_primes < n:
if isprime(a):
n_primes += 1
a += 1
return a - 1
class primepi(Function):
""" Represents the prime counting function pi(n) = the number
of prime numbers less than or equal to n.
Algorithm Description:
In sieve method, we remove all multiples of prime p
except p itself.
Let phi(i,j) be the number of integers 2 <= k <= i
which remain after sieving from primes less than
or equal to j.
Clearly, pi(n) = phi(n, sqrt(n))
If j is not a prime,
phi(i,j) = phi(i, j - 1)
if j is a prime,
We remove all numbers(except j) whose
smallest prime factor is j.
Let x= j*a be such a number, where 2 <= a<= i / j
Now, after sieving from primes <= j - 1,
a must remain
(because x, and hence a has no prime factor <= j - 1)
Clearly, there are phi(i / j, j - 1) such a
which remain on sieving from primes <= j - 1
Now, if a is a prime less than equal to j - 1,
x= j*a has smallest prime factor = a, and
has already been removed(by sieving from a).
So, we don't need to remove it again.
(Note: there will be pi(j - 1) such x)
Thus, number of x, that will be removed are:
phi(i / j, j - 1) - phi(j - 1, j - 1)
(Note that pi(j - 1) = phi(j - 1, j - 1))
=> phi(i,j) = phi(i, j - 1) - phi(i / j, j - 1) + phi(j - 1, j - 1)
So,following recursion is used and implemented as dp:
phi(a, b) = phi(a, b - 1), if b is not a prime
phi(a, b) = phi(a, b-1)-phi(a / b, b-1) + phi(b-1, b-1), if b is prime
Clearly a is always of the form floor(n / k),
which can take at most 2*sqrt(n) values.
Two arrays arr1,arr2 are maintained
arr1[i] = phi(i, j),
arr2[i] = phi(n // i, j)
Finally the answer is arr2[1]
Examples
========
>>> from sympy import primepi, prime, prevprime, isprime
>>> primepi(25)
9
So there are 9 primes less than or equal to 25. Is 25 prime?
>>> isprime(25)
False
It isn't. So the first prime less than 25 must be the
9th prime:
>>> prevprime(25) == prime(9)
True
See Also
========
sympy.ntheory.primetest.isprime : Test if n is prime
primerange : Generate all primes in a given range
prime : Return the nth prime
"""
@classmethod
def eval(cls, n):
if n is S.Infinity:
return S.Infinity
if n is S.NegativeInfinity:
return S.Zero
try:
n = int(n)
except TypeError:
if n.is_real == False or n is S.NaN:
raise ValueError("n must be real")
return
if n < 2:
return S.Zero
if n <= sieve._list[-1]:
return S(sieve.search(n)[0])
lim = int(n ** 0.5)
lim -= 1
lim = max(lim, 0)
while lim * lim <= n:
lim += 1
lim -= 1
arr1 = [0] * (lim + 1)
arr2 = [0] * (lim + 1)
for i in range(1, lim + 1):
arr1[i] = i - 1
arr2[i] = n // i - 1
for i in range(2, lim + 1):
# Presently, arr1[k]=phi(k,i - 1),
# arr2[k] = phi(n // k,i - 1)
if arr1[i] == arr1[i - 1]:
continue
p = arr1[i - 1]
for j in range(1, min(n // (i * i), lim) + 1):
st = i * j
if st <= lim:
arr2[j] -= arr2[st] - p
else:
arr2[j] -= arr1[n // st] - p
lim2 = min(lim, i * i - 1)
for j in range(lim, lim2, -1):
arr1[j] -= arr1[j // i] - p
return S(arr2[1])
def nextprime(n, ith=1):
""" Return the ith prime greater than n.
i must be an integer.
Notes
=====
Potential primes are located at 6*j +/- 1. This
property is used during searching.
>>> from sympy import nextprime
>>> [(i, nextprime(i)) for i in range(10, 15)]
[(10, 11), (11, 13), (12, 13), (13, 17), (14, 17)]
>>> nextprime(2, ith=2) # the 2nd prime after 2
5
See Also
========
prevprime : Return the largest prime smaller than n
primerange : Generate all primes in a given range
"""
n = int(n)
i = as_int(ith)
if i > 1:
pr = n
j = 1
while 1:
pr = nextprime(pr)
j += 1
if j > i:
break
return pr
if n < 2:
return 2
if n < 7:
return {2: 3, 3: 5, 4: 5, 5: 7, 6: 7}[n]
if n <= sieve._list[-2]:
l, u = sieve.search(n)
if l == u:
return sieve[u + 1]
else:
return sieve[u]
nn = 6*(n//6)
if nn == n:
n += 1
if isprime(n):
return n
n += 4
elif n - nn == 5:
n += 2
if isprime(n):
return n
n += 4
else:
n = nn + 5
while 1:
if isprime(n):
return n
n += 2
if isprime(n):
return n
n += 4
def prevprime(n):
""" Return the largest prime smaller than n.
Notes
=====
Potential primes are located at 6*j +/- 1. This
property is used during searching.
>>> from sympy import prevprime
>>> [(i, prevprime(i)) for i in range(10, 15)]
[(10, 7), (11, 7), (12, 11), (13, 11), (14, 13)]
See Also
========
nextprime : Return the ith prime greater than n
primerange : Generates all primes in a given range
"""
from sympy.functions.elementary.integers import ceiling
# wrapping ceiling in as_int will raise an error if there was a problem
# determining whether the expression was exactly an integer or not
n = as_int(ceiling(n))
if n < 3:
raise ValueError("no preceding primes")
if n < 8:
return {3: 2, 4: 3, 5: 3, 6: 5, 7: 5}[n]
if n <= sieve._list[-1]:
l, u = sieve.search(n)
if l == u:
return sieve[l-1]
else:
return sieve[l]
nn = 6*(n//6)
if n - nn <= 1:
n = nn - 1
if isprime(n):
return n
n -= 4
else:
n = nn + 1
while 1:
if isprime(n):
return n
n -= 2
if isprime(n):
return n
n -= 4
def primerange(a, b=None):
""" Generate a list of all prime numbers in the range [2, a),
or [a, b).
If the range exists in the default sieve, the values will
be returned from there; otherwise values will be returned
but will not modify the sieve.
Examples
========
>>> from sympy import primerange, prime
All primes less than 19:
>>> list(primerange(19))
[2, 3, 5, 7, 11, 13, 17]
All primes greater than or equal to 7 and less than 19:
>>> list(primerange(7, 19))
[7, 11, 13, 17]
All primes through the 10th prime
>>> list(primerange(prime(10) + 1))
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
The Sieve method, primerange, is generally faster but it will
occupy more memory as the sieve stores values. The default
instance of Sieve, named sieve, can be used:
>>> from sympy import sieve
>>> list(sieve.primerange(1, 30))
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
Notes
=====
Some famous conjectures about the occurrence of primes in a given
range are [1]:
- Twin primes: though often not, the following will give 2 primes
an infinite number of times:
primerange(6*n - 1, 6*n + 2)
- Legendre's: the following always yields at least one prime
primerange(n**2, (n+1)**2+1)
- Bertrand's (proven): there is always a prime in the range
primerange(n, 2*n)
- Brocard's: there are at least four primes in the range
primerange(prime(n)**2, prime(n+1)**2)
The average gap between primes is log(n) [2]; the gap between
primes can be arbitrarily large since sequences of composite
numbers are arbitrarily large, e.g. the numbers in the sequence
n! + 2, n! + 3 ... n! + n are all composite.
See Also
========
prime : Return the nth prime
nextprime : Return the ith prime greater than n
prevprime : Return the largest prime smaller than n
randprime : Returns a random prime in a given range
primorial : Returns the product of primes based on condition
Sieve.primerange : return range from already computed primes
or extend the sieve to contain the requested
range.
References
==========
.. [1] https://en.wikipedia.org/wiki/Prime_number
.. [2] http://primes.utm.edu/notes/gaps.html
"""
from sympy.functions.elementary.integers import ceiling
if b is None:
a, b = 2, a
if a >= b:
return
# if we already have the range, return it
if b <= sieve._list[-1]:
yield from sieve.primerange(a, b)
return
# otherwise compute, without storing, the desired range.
# wrapping ceiling in as_int will raise an error if there was a problem
# determining whether the expression was exactly an integer or not
a = as_int(ceiling(a)) - 1
b = as_int(ceiling(b))
while 1:
a = nextprime(a)
if a < b:
yield a
else:
return
def randprime(a, b):
""" Return a random prime number in the range [a, b).
Bertrand's postulate assures that
randprime(a, 2*a) will always succeed for a > 1.
Examples
========
>>> from sympy import randprime, isprime
>>> randprime(1, 30) #doctest: +SKIP
13
>>> isprime(randprime(1, 30))
True
See Also
========
primerange : Generate all primes in a given range
References
==========
.. [1] https://en.wikipedia.org/wiki/Bertrand's_postulate
"""
if a >= b:
return
a, b = map(int, (a, b))
n = random.randint(a - 1, b)
p = nextprime(n)
if p >= b:
p = prevprime(b)
if p < a:
raise ValueError("no primes exist in the specified range")
return p
def primorial(n, nth=True):
"""
Returns the product of the first n primes (default) or
the primes less than or equal to n (when ``nth=False``).
Examples
========
>>> from sympy.ntheory.generate import primorial, primerange
>>> from sympy import factorint, Mul, primefactors, sqrt
>>> primorial(4) # the first 4 primes are 2, 3, 5, 7
210
>>> primorial(4, nth=False) # primes <= 4 are 2 and 3
6
>>> primorial(1)
2
>>> primorial(1, nth=False)
1
>>> primorial(sqrt(101), nth=False)
210
One can argue that the primes are infinite since if you take
a set of primes and multiply them together (e.g. the primorial) and
then add or subtract 1, the result cannot be divided by any of the
original factors, hence either 1 or more new primes must divide this
product of primes.
In this case, the number itself is a new prime:
>>> factorint(primorial(4) + 1)
{211: 1}
In this case two new primes are the factors:
>>> factorint(primorial(4) - 1)
{11: 1, 19: 1}
Here, some primes smaller and larger than the primes multiplied together
are obtained:
>>> p = list(primerange(10, 20))
>>> sorted(set(primefactors(Mul(*p) + 1)).difference(set(p)))
[2, 5, 31, 149]
See Also
========
primerange : Generate all primes in a given range
"""
if nth:
n = as_int(n)
else:
n = int(n)
if n < 1:
raise ValueError("primorial argument must be >= 1")
p = 1
if nth:
for i in range(1, n + 1):
p *= prime(i)
else:
for i in primerange(2, n + 1):
p *= i
return p
def cycle_length(f, x0, nmax=None, values=False):
"""For a given iterated sequence, return a generator that gives
the length of the iterated cycle (lambda) and the length of terms
before the cycle begins (mu); if ``values`` is True then the
terms of the sequence will be returned instead. The sequence is
started with value ``x0``.
Note: more than the first lambda + mu terms may be returned and this
is the cost of cycle detection with Brent's method; there are, however,
generally less terms calculated than would have been calculated if the
proper ending point were determined, e.g. by using Floyd's method.
>>> from sympy.ntheory.generate import cycle_length
This will yield successive values of i <-- func(i):
>>> def iter(func, i):
... while 1:
... ii = func(i)
... yield ii
... i = ii
...
A function is defined:
>>> func = lambda i: (i**2 + 1) % 51
and given a seed of 4 and the mu and lambda terms calculated:
>>> next(cycle_length(func, 4))
(6, 2)
We can see what is meant by looking at the output:
>>> n = cycle_length(func, 4, values=True)
>>> list(ni for ni in n)
[17, 35, 2, 5, 26, 14, 44, 50, 2, 5, 26, 14]
There are 6 repeating values after the first 2.
If a sequence is suspected of being longer than you might wish, ``nmax``
can be used to exit early (and mu will be returned as None):
>>> next(cycle_length(func, 4, nmax = 4))
(4, None)
>>> [ni for ni in cycle_length(func, 4, nmax = 4, values=True)]
[17, 35, 2, 5]
Code modified from:
https://en.wikipedia.org/wiki/Cycle_detection.
"""
nmax = int(nmax or 0)
# main phase: search successive powers of two
power = lam = 1
tortoise, hare = x0, f(x0) # f(x0) is the element/node next to x0.
i = 0
while tortoise != hare and (not nmax or i < nmax):
i += 1
if power == lam: # time to start a new power of two?
tortoise = hare
power *= 2
lam = 0
if values:
yield hare
hare = f(hare)
lam += 1
if nmax and i == nmax:
if values:
return
else:
yield nmax, None
return
if not values:
# Find the position of the first repetition of length lambda
mu = 0
tortoise = hare = x0
for i in range(lam):
hare = f(hare)
while tortoise != hare:
tortoise = f(tortoise)
hare = f(hare)
mu += 1
if mu:
mu -= 1
yield lam, mu
def composite(nth):
""" Return the nth composite number, with the composite numbers indexed as
composite(1) = 4, composite(2) = 6, etc....
Examples
========
>>> from sympy import composite
>>> composite(36)
52
>>> composite(1)
4
>>> composite(17737)
20000
See Also
========
sympy.ntheory.primetest.isprime : Test if n is prime
primerange : Generate all primes in a given range
primepi : Return the number of primes less than or equal to n
prime : Return the nth prime
compositepi : Return the number of positive composite numbers less than or equal to n
"""
n = as_int(nth)
if n < 1:
raise ValueError("nth must be a positive integer; composite(1) == 4")
composite_arr = [4, 6, 8, 9, 10, 12, 14, 15, 16, 18]
if n <= 10:
return composite_arr[n - 1]
a, b = 4, sieve._list[-1]
if n <= b - primepi(b) - 1:
while a < b - 1:
mid = (a + b) >> 1
if mid - primepi(mid) - 1 > n:
b = mid
else:
a = mid
if isprime(a):
a -= 1
return a
from sympy.functions.special.error_functions import li
from sympy.functions.elementary.exponential import log
a = 4 # Lower bound for binary search
b = int(n*(log(n) + log(log(n)))) # Upper bound for the search.
while a < b:
mid = (a + b) >> 1
if mid - li(mid) - 1 > n:
b = mid
else:
a = mid + 1
n_composites = a - primepi(a) - 1
while n_composites > n:
if not isprime(a):
n_composites -= 1
a -= 1
if isprime(a):
a -= 1
return a
def compositepi(n):
""" Return the number of positive composite numbers less than or equal to n.
The first positive composite is 4, i.e. compositepi(4) = 1.
Examples
========
>>> from sympy import compositepi
>>> compositepi(25)
15
>>> compositepi(1000)
831
See Also
========
sympy.ntheory.primetest.isprime : Test if n is prime
primerange : Generate all primes in a given range
prime : Return the nth prime
primepi : Return the number of primes less than or equal to n
composite : Return the nth composite number
"""
n = int(n)
if n < 4:
return 0
return n - primepi(n) - 1
|
c4b80764de5e5da194c37d1310e20c0a07d21d27bf6ea13107276f3b3147b871 | from sympy.core.mul import Mul
from sympy.core.singleton import S
from sympy.concrete.expr_with_intlimits import ExprWithIntLimits
from sympy.core.exprtools import factor_terms
from sympy.functions.elementary.exponential import exp, log
from sympy.polys import quo, roots
from sympy.simplify import powsimp
from sympy.core.function import Derivative
from sympy.core.symbol import Dummy, Symbol
class Product(ExprWithIntLimits):
r"""
Represents unevaluated products.
Explanation
===========
``Product`` represents a finite or infinite product, 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 product.
Finite products
===============
For finite products (and products with symbolic limits assumed to be finite)
we follow the analogue of the summation convention described by Karr [1],
especially definition 3 of section 1.4. The product:
.. math::
\prod_{m \leq i < n} f(i)
has *the obvious meaning* for `m < n`, namely:
.. math::
\prod_{m \leq i < n} f(i) = f(m) f(m+1) \cdot \ldots \cdot f(n-2) f(n-1)
with the upper limit value `f(n)` excluded. The product over an empty set is
one if and only if `m = n`:
.. math::
\prod_{m \leq i < n} f(i) = 1 \quad \mathrm{for} \quad m = n
Finally, for all other products over empty sets we assume the following
definition:
.. math::
\prod_{m \leq i < n} f(i) = \frac{1}{\prod_{n \leq i < m} f(i)} \quad \mathrm{for} \quad m > n
It is important to note that above we define all products with the upper
limit being exclusive. This is in contrast to the usual mathematical notation,
but does not affect the product convention. Indeed we have:
.. math::
\prod_{m \leq i < n} f(i) = \prod_{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 a, b, i, k, m, n, x
>>> from sympy import Product, oo
>>> Product(k, (k, 1, m))
Product(k, (k, 1, m))
>>> Product(k, (k, 1, m)).doit()
factorial(m)
>>> Product(k**2,(k, 1, m))
Product(k**2, (k, 1, m))
>>> Product(k**2,(k, 1, m)).doit()
factorial(m)**2
Wallis' product for pi:
>>> W = Product(2*i/(2*i-1) * 2*i/(2*i+1), (i, 1, oo))
>>> W
Product(4*i**2/((2*i - 1)*(2*i + 1)), (i, 1, oo))
Direct computation currently fails:
>>> W.doit()
Product(4*i**2/((2*i - 1)*(2*i + 1)), (i, 1, oo))
But we can approach the infinite product by a limit of finite products:
>>> from sympy import limit
>>> W2 = Product(2*i/(2*i-1)*2*i/(2*i+1), (i, 1, n))
>>> W2
Product(4*i**2/((2*i - 1)*(2*i + 1)), (i, 1, n))
>>> W2e = W2.doit()
>>> W2e
2**(-2*n)*4**n*factorial(n)**2/(RisingFactorial(1/2, n)*RisingFactorial(3/2, n))
>>> limit(W2e, n, oo)
pi/2
By the same formula we can compute sin(pi/2):
>>> from sympy import combsimp, pi, gamma, simplify
>>> P = pi * x * Product(1 - x**2/k**2, (k, 1, n))
>>> P = P.subs(x, pi/2)
>>> P
pi**2*Product(1 - pi**2/(4*k**2), (k, 1, n))/2
>>> Pe = P.doit()
>>> Pe
pi**2*RisingFactorial(1 - pi/2, n)*RisingFactorial(1 + pi/2, n)/(2*factorial(n)**2)
>>> limit(Pe, n, oo).gammasimp()
sin(pi**2/2)
>>> Pe.rewrite(gamma)
(-1)**n*pi**2*gamma(pi/2)*gamma(n + 1 + pi/2)/(2*gamma(1 + pi/2)*gamma(-n + pi/2)*gamma(n + 1)**2)
Products with the lower limit being larger than the upper one:
>>> Product(1/i, (i, 6, 1)).doit()
120
>>> Product(i, (i, 2, 5)).doit()
120
The empty product:
>>> Product(i, (i, n, n-1)).doit()
1
An example showing that the symbolic result of a product is still
valid for seemingly nonsensical values of the limits. Then the Karr
convention allows us to give a perfectly valid interpretation to
those products by interchanging the limits according to the above rules:
>>> P = Product(2, (i, 10, n)).doit()
>>> P
2**(n - 9)
>>> P.subs(n, 5)
1/16
>>> Product(2, (i, 10, 5)).doit()
1/16
>>> 1/Product(2, (i, 6, 9)).doit()
1/16
An explicit example of the Karr summation convention applied to products:
>>> P1 = Product(x, (i, a, b)).doit()
>>> P1
x**(-a + b + 1)
>>> P2 = Product(x, (i, b+1, a-1)).doit()
>>> P2
x**(a - b - 1)
>>> simplify(P1 * P2)
1
And another one:
>>> P1 = Product(i, (i, b, a)).doit()
>>> P1
RisingFactorial(b, a - b + 1)
>>> P2 = Product(i, (i, a+1, b-1)).doit()
>>> P2
RisingFactorial(a + 1, -a + b - 1)
>>> P1 * P2
RisingFactorial(b, a - b + 1)*RisingFactorial(a + 1, -a + b - 1)
>>> combsimp(P1 * P2)
1
See Also
========
Sum, summation
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/Multiplication#Capital_Pi_notation
.. [3] https://en.wikipedia.org/wiki/Empty_product
"""
__slots__ = ('is_commutative',)
def __new__(cls, function, *symbols, **assumptions):
obj = ExprWithIntLimits.__new__(cls, function, *symbols, **assumptions)
return obj
def _eval_rewrite_as_Sum(self, *args, **kwargs):
from sympy.concrete.summations import Sum
return exp(Sum(log(self.function), *self.limits))
@property
def term(self):
return self._args[0]
function = term
def _eval_is_zero(self):
if self.has_empty_sequence:
return False
z = self.term.is_zero
if z is True:
return True
if self.has_finite_limits:
# A Product is zero only if its term is zero assuming finite limits.
return z
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_empty_sequence:
return True
if self.function.is_positive and self.has_finite_limits:
return True
def _eval_is_nonnegative(self):
if self.has_empty_sequence:
return True
if self.function.is_nonnegative and self.has_finite_limits:
return True
def _eval_is_extended_nonnegative(self):
if self.has_empty_sequence:
return True
if self.function.is_extended_nonnegative:
return True
def _eval_is_extended_nonpositive(self):
if self.has_empty_sequence:
return True
def _eval_is_finite(self):
if self.has_finite_limits and self.function.is_finite:
return True
def doit(self, **hints):
# first make sure any definite limits have product
# variables with matching assumptions
reps = {}
for xab in self.limits:
# Must be imported here to avoid circular imports
from .summations import _dummy_with_inherited_properties_concrete
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 type(did) is tuple: # when separate=True
did = tuple([i.xreplace(undo) for i in did])
else:
did = did.xreplace(undo)
return did
f = self.function
for index, limit in enumerate(self.limits):
i, a, b = limit
dif = b - a
if dif.is_integer and dif.is_negative:
a, b = b + 1, a - 1
f = 1 / f
g = self._eval_product(f, (i, a, b))
if g in (None, S.NaN):
return self.func(powsimp(f), *self.limits[index:])
else:
f = g
if hints.get('deep', True):
return f.doit(**hints)
else:
return powsimp(f)
def _eval_adjoint(self):
if self.is_commutative:
return self.func(self.function.adjoint(), *self.limits)
return None
def _eval_conjugate(self):
return self.func(self.function.conjugate(), *self.limits)
def _eval_product(self, term, limits):
from sympy.concrete.delta import deltaproduct, _has_simple_delta
from sympy.concrete.summations import summation
from sympy.functions import KroneckerDelta, RisingFactorial
(k, a, n) = limits
if k not in term.free_symbols:
if (term - 1).is_zero:
return S.One
return term**(n - a + 1)
if a == n:
return term.subs(k, a)
if term.has(KroneckerDelta) and _has_simple_delta(term, limits[0]):
return deltaproduct(term, limits)
dif = n - a
definite = dif.is_Integer
if definite and (dif < 100):
return self._eval_product_direct(term, limits)
elif term.is_polynomial(k):
poly = term.as_poly(k)
A = B = Q = S.One
all_roots = roots(poly)
M = 0
for r, m in all_roots.items():
M += m
A *= RisingFactorial(a - r, n - a + 1)**m
Q *= (n - r)**m
if M < poly.degree():
arg = quo(poly, Q.as_poly(k))
B = self.func(arg, (k, a, n)).doit()
return poly.LC()**(n - a + 1) * A * B
elif term.is_Add:
factored = factor_terms(term, fraction=True)
if factored.is_Mul:
return self._eval_product(factored, (k, a, n))
elif term.is_Mul:
# Factor in part without the summation variable and part with
without_k, with_k = term.as_coeff_mul(k)
if len(with_k) >= 2:
# More than one term including k, so still a multiplication
exclude, include = [], []
for t in with_k:
p = self._eval_product(t, (k, a, n))
if p is not None:
exclude.append(p)
else:
include.append(t)
if not exclude:
return None
else:
arg = term._new_rawargs(*include)
A = Mul(*exclude)
B = self.func(arg, (k, a, n)).doit()
return without_k**(n - a + 1)*A * B
else:
# Just a single term
p = self._eval_product(with_k[0], (k, a, n))
if p is None:
p = self.func(with_k[0], (k, a, n)).doit()
return without_k**(n - a + 1)*p
elif term.is_Pow:
if not term.base.has(k):
s = summation(term.exp, (k, a, n))
return term.base**s
elif not term.exp.has(k):
p = self._eval_product(term.base, (k, a, n))
if p is not None:
return p**term.exp
elif isinstance(term, Product):
evaluated = term.doit()
f = self._eval_product(evaluated, limits)
if f is None:
return self.func(evaluated, limits)
else:
return f
if definite:
return self._eval_product_direct(term, limits)
def _eval_simplify(self, **kwargs):
from sympy.simplify.simplify import product_simplify
rv = product_simplify(self)
return rv.doit() if kwargs['doit'] else rv
def _eval_transpose(self):
if self.is_commutative:
return self.func(self.function.transpose(), *self.limits)
return None
def _eval_product_direct(self, term, limits):
(k, a, n) = limits
return Mul(*[term.subs(k, a + i) for i in range(n - a + 1)])
def _eval_derivative(self, x):
from sympy.concrete.summations import Sum
if isinstance(x, Symbol) and x not in self.free_symbols:
return S.Zero
f, limits = self.function, list(self.limits)
limit = limits.pop(-1)
if limits:
f = self.func(f, *limits)
i, a, b = limit
if x in a.free_symbols or x in b.free_symbols:
return None
h = Dummy()
rv = Sum( Product(f, (i, a, h - 1)) * Product(f, (i, h + 1, b)) * Derivative(f, x, evaluate=True).subs(i, h), (h, a, b))
return rv
def is_convergent(self):
r"""
See docs of :obj:`.Sum.is_convergent()` for explanation of convergence
in SymPy.
Explanation
===========
The infinite product:
.. math::
\prod_{1 \leq i < \infty} f(i)
is defined by the sequence of partial products:
.. math::
\prod_{i=1}^{n} f(i) = f(1) f(2) \cdots f(n)
as n increases without bound. The product converges to a non-zero
value if and only if the sum:
.. math::
\sum_{1 \leq i < \infty} \log{f(n)}
converges.
Examples
========
>>> from sympy import Product, Symbol, cos, pi, exp, oo
>>> n = Symbol('n', integer=True)
>>> Product(n/(n + 1), (n, 1, oo)).is_convergent()
False
>>> Product(1/n**2, (n, 1, oo)).is_convergent()
False
>>> Product(cos(pi/n), (n, 1, oo)).is_convergent()
True
>>> Product(exp(-n**2), (n, 1, oo)).is_convergent()
False
References
==========
.. [1] https://en.wikipedia.org/wiki/Infinite_product
"""
from sympy.concrete.summations import Sum
sequence_term = self.function
log_sum = log(sequence_term)
lim = self.limits
try:
is_conv = Sum(log_sum, *lim).is_convergent()
except NotImplementedError:
if Sum(sequence_term - 1, *lim).is_absolutely_convergent() is S.true:
return S.true
raise NotImplementedError("The algorithm to find the product convergence of %s "
"is not yet implemented" % (sequence_term))
return is_conv
def reverse_order(expr, *indices):
"""
Reverse the order of a limit in a Product.
Explanation
===========
``reverse_order(expr, *indices)`` reverses some limits in the expression
``expr`` 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 gamma, Product, simplify, Sum
>>> from sympy.abc import x, y, a, b, c, d
>>> P = Product(x, (x, a, b))
>>> Pr = P.reverse_order(x)
>>> Pr
Product(1/x, (x, b + 1, a - 1))
>>> Pr = Pr.doit()
>>> Pr
1/RisingFactorial(b + 1, a - b - 1)
>>> simplify(Pr.rewrite(gamma))
Piecewise((gamma(b + 1)/gamma(a), b > -1), ((-1)**(-a + b + 1)*gamma(1 - a)/gamma(-b), True))
>>> P = P.doit()
>>> P
RisingFactorial(a, -a + b + 1)
>>> simplify(P.rewrite(gamma))
Piecewise((gamma(b + 1)/gamma(a), a > 0), ((-1)**(-a + b + 1)*gamma(1 - a)/gamma(-b), True))
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*y, (x, a, b), (y, c, d))
>>> S
Sum(x*y, (x, a, b), (y, c, d))
>>> S0 = S.reverse_order(0)
>>> S0
Sum(-x*y, (x, b + 1, a - 1), (y, c, d))
>>> S1 = S0.reverse_order(1)
>>> S1
Sum(x*y, (x, b + 1, a - 1), (y, 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] = expr.index(indx)
e = 1
limits = []
for i, limit in enumerate(expr.limits):
l = limit
if i in l_indices:
e = -e
l = (limit[0], limit[2] + 1, limit[1] - 1)
limits.append(l)
return Product(expr.function ** e, *limits)
def product(*args, **kwargs):
r"""
Compute the product.
Explanation
===========
The notation for symbols is similar to the notation used in Sum or
Integral. product(f, (i, a, b)) computes the product of f with
respect to i from a to b, i.e.,
::
b
_____
product(f(n), (i, a, b)) = | | f(n)
| |
i = a
If it cannot compute the product, it returns an unevaluated Product object.
Repeated products can be computed by introducing additional symbols tuples::
Examples
========
>>> from sympy import product, symbols
>>> i, n, m, k = symbols('i n m k', integer=True)
>>> product(i, (i, 1, k))
factorial(k)
>>> product(m, (i, 1, k))
m**k
>>> product(i, (i, 1, k), (k, 1, n))
Product(factorial(k), (k, 1, n))
"""
prod = Product(*args, **kwargs)
if isinstance(prod, Product):
return prod.doit(deep=False)
else:
return prod
|
0a0b2808d6cee6671199b3cbc488e7901159a088c48d60985476e3d985d0e39b | from sympy.calculus.singularities import is_decreasing
from sympy.calculus.util import AccumulationBounds
from sympy.concrete.expr_with_limits import AddWithLimits
from sympy.concrete.expr_with_intlimits import ExprWithIntLimits
from sympy.concrete.gosper import gosper_sum
from sympy.core.add import Add
from sympy.core.function import Derivative
from sympy.core.mul import Mul
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import Dummy, Wild, Symbol
from sympy.functions.special.zeta_functions import zeta
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import cot, csc
from sympy.logic.boolalg import And
from sympy.polys import apart, together
from sympy.polys.polyerrors import PolynomialError, PolificationFailed
from sympy.polys.polytools import parallel_poly_from_expr
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
from sympy.simplify import denom
from sympy.simplify.combsimp import combsimp
from sympy.simplify.powsimp import powsimp
from sympy.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',)
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 type(did) is 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):
from sympy.simplify.simplify import factor_sum, sum_combine
from sympy.core.function import expand
from sympy.core.mul import Mul
# 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 can not 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()
"""
from sympy import Interval, Integral, log, symbols, simplify
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((-1)**(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.
"""
from sympy.functions import bernoulli, factorial
from sympy.integrals import Integral
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 (eps and term and abs(term.evalf(3)) < eps) or (k > n):
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):
from sympy.concrete.delta import deltasummation, _has_simple_delta
from sympy.functions import KroneckerDelta
(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):
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.
"""
from sympy.core import Add
(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):
from sympy.functions import harmonic, bernoulli
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 not a is S.NegativeInfinity) or \
(a is S.NegativeInfinity and not b is 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)):
from sympy import ordered, Tuple
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 not r 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. """
from sympy.functions import hyper
from sympy.simplify import hyperexpand, hypersimp, fraction, simplify
from sympy.polys.polytools import Poly, factor
from sympy.core.numbers import Float
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):
from sympy.simplify.simplify import nsimplify
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):
from sympy.logic.boolalg import And
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 $\deg(g(n)) - \deg(f(n)) >= 2$,
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 / (-1)**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 = (-1)**i * ((-1)**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)
|
a8a622283cf0ecd0e32467b96d4a667aa9bc3d18a16de2ae1de96b41a69b82e7 | """
PythonMPQ: Rational number type based on Python integers.
This class is intended as a pure Python fallback for when gmpy2 is not
installed. If gmpy2 is installed then its mpq type will be used instead. The
mpq type is around 20x faster. We could just use the stdlib Fraction class
here but that is slower:
from fractions import Fraction
from sympy.external.pythonmpq import PythonMPQ
nums = range(1000)
dens = range(5, 1005)
rats = [Fraction(n, d) for n, d in zip(nums, dens)]
sum(rats) # <--- 24 milliseconds
rats = [PythonMPQ(n, d) for n, d in zip(nums, dens)]
sum(rats) # <--- 7 milliseconds
Both mpq and Fraction have some awkward features like the behaviour of
division with // and %:
>>> from fractions import Fraction
>>> Fraction(2, 3) % Fraction(1, 4)
1/6
For the QQ domain we don't want this behaviour because there should be no
remainder when dividing rational numbers. SymPy doesn't make use of this
aspect of mpq when gmpy2 is installed. Since this class is a fallback for that
case we don't bother implementing e.g. __mod__ so that we can be sure we
aren't using it when gmpy2 is installed either.
"""
import operator
from math import gcd
from decimal import Decimal
from fractions import Fraction
import sys
# Used for __hash__
_PyHASH_MODULUS = sys.hash_info.modulus
_PyHASH_INF = sys.hash_info.inf
class PythonMPQ:
"""Rational number implementation that is intended to be compatible with
gmpy2's mpq.
Also slightly faster than fractions.Fraction.
PythonMPQ should be treated as immutable although no effort is made to
prevent mutation (since that might slow down calculations).
"""
__slots__ = ('numerator', 'denominator')
def __new__(cls, numerator, denominator=None):
"""Construct PythonMPQ with gcd computation and checks"""
if denominator is not None:
#
# PythonMPQ(n, d): require n and d to be int and d != 0
#
if isinstance(numerator, int) and isinstance(denominator, int):
# This is the slow part:
divisor = gcd(numerator, denominator)
numerator //= divisor
denominator //= divisor
return cls._new_check(numerator, denominator)
else:
#
# PythonMPQ(q)
#
# Here q can be PythonMPQ, int, Decimal, float, Fraction or str
#
if isinstance(numerator, int):
return cls._new(numerator, 1)
elif isinstance(numerator, PythonMPQ):
return cls._new(numerator.numerator, numerator.denominator)
# Let Fraction handle Decimal/float conversion and str parsing
if isinstance(numerator, (Decimal, float, str)):
numerator = Fraction(numerator)
if isinstance(numerator, Fraction):
return cls._new(numerator.numerator, numerator.denominator)
#
# Reject everything else. This is more strict than mpq which allows
# things like mpq(Fraction, Fraction) or mpq(Decimal, any). The mpq
# behaviour is somewhat inconsistent so we choose to accept only a
# more strict subset of what mpq allows.
#
raise TypeError("PythonMPQ() requires numeric or string argument")
@classmethod
def _new_check(cls, numerator, denominator):
"""Construct PythonMPQ, check divide by zero and canonicalize signs"""
if not denominator:
raise ZeroDivisionError(f'Zero divisor {numerator}/{denominator}')
elif denominator < 0:
numerator = -numerator
denominator = -denominator
return cls._new(numerator, denominator)
@classmethod
def _new(cls, numerator, denominator):
"""Construct PythonMPQ efficiently (no checks)"""
obj = super().__new__(cls)
obj.numerator = numerator
obj.denominator = denominator
return obj
def __int__(self):
"""Convert to int (truncates towards zero)"""
p, q = self.numerator, self.denominator
if p < 0:
return -(-p//q)
return p//q
def __float__(self):
"""Convert to float (approximately)"""
return self.numerator / self.denominator
def __bool__(self):
"""True/False if nonzero/zero"""
return bool(self.numerator)
def __eq__(self, other):
"""Compare equal with PythonMPQ, int, float, Decimal or Fraction"""
if isinstance(other, PythonMPQ):
return (self.numerator == other.numerator
and self.denominator == other.denominator)
elif isinstance(other, self._compatible_types):
return self.__eq__(PythonMPQ(other))
else:
return NotImplemented
# The hashing algorithm for Fraction changed in Python 3.8
if sys.version_info >= (3, 8):
#
# Hash for Python 3.8 onwards
#
def __hash__(self):
"""hash - same as mpq/Fraction"""
try:
dinv = pow(self.denominator, -1, _PyHASH_MODULUS)
except ValueError:
hash_ = _PyHASH_INF
else:
hash_ = hash(hash(abs(self.numerator)) * dinv)
result = hash_ if self.numerator >= 0 else -hash_
return -2 if result == -1 else result
else:
#
# Hash for Python < 3.7
#
def __hash__(self):
"""hash - same as mpq/Fraction"""
# This is from fractions.py in the stdlib.
dinv = pow(self.denominator, _PyHASH_MODULUS - 2, _PyHASH_MODULUS)
if not dinv:
hash_ = _PyHASH_INF
else:
hash_ = abs(self.numerator) * dinv % _PyHASH_MODULUS
result = hash_ if self >= 0 else -hash_
return -2 if result == -1 else result
def __reduce__(self):
"""Deconstruct for pickling"""
return type(self), (self.numerator, self.denominator)
def __str__(self):
"""Convert to string"""
if self.denominator != 1:
return f"{self.numerator}/{self.denominator}"
else:
return f"{self.numerator}"
def __repr__(self):
"""Convert to string"""
return f"MPQ({self.numerator},{self.denominator})"
def _cmp(self, other, op):
"""Helper for lt/le/gt/ge"""
if not isinstance(other, self._compatible_types):
return NotImplemented
lhs = self.numerator * other.denominator
rhs = other.numerator * self.denominator
return op(lhs, rhs)
def __lt__(self, other):
"""self < other"""
return self._cmp(other, operator.lt)
def __le__(self, other):
"""self <= other"""
return self._cmp(other, operator.le)
def __gt__(self, other):
"""self > other"""
return self._cmp(other, operator.gt)
def __ge__(self, other):
"""self >= other"""
return self._cmp(other, operator.ge)
def __abs__(self):
"""abs(q)"""
return self._new(abs(self.numerator), self.denominator)
def __pos__(self):
"""+q"""
return self
def __neg__(self):
"""-q"""
return self._new(-self.numerator, self.denominator)
def __add__(self, other):
"""q1 + q2"""
if isinstance(other, PythonMPQ):
#
# This is much faster than the naive method used in the stdlib
# fractions module. Not sure where this method comes from
# though...
#
# Compare timings for something like:
# nums = range(1000)
# rats = [PythonMPQ(n, d) for n, d in zip(nums[:-5], nums[5:])]
# sum(rats) # <-- time this
#
ap, aq = self.numerator, self.denominator
bp, bq = other.numerator, other.denominator
g = gcd(aq, bq)
if g == 1:
p = ap*bq + aq*bp
q = bq*aq
else:
q1, q2 = aq//g, bq//g
p, q = ap*q2 + bp*q1, q1*q2
g2 = gcd(p, g)
p, q = (p // g2), q * (g // g2)
elif isinstance(other, int):
p = self.numerator + self.denominator * other
q = self.denominator
else:
return NotImplemented
return self._new(p, q)
def __radd__(self, other):
"""z1 + q2"""
if isinstance(other, int):
p = self.numerator + self.denominator * other
q = self.denominator
return self._new(p, q)
else:
return NotImplemented
def __sub__(self ,other):
"""q1 - q2"""
if isinstance(other, PythonMPQ):
ap, aq = self.numerator, self.denominator
bp, bq = other.numerator, other.denominator
g = gcd(aq, bq)
if g == 1:
p = ap*bq - aq*bp
q = bq*aq
else:
q1, q2 = aq//g, bq//g
p, q = ap*q2 - bp*q1, q1*q2
g2 = gcd(p, g)
p, q = (p // g2), q * (g // g2)
elif isinstance(other, int):
p = self.numerator - self.denominator*other
q = self.denominator
else:
return NotImplemented
return self._new(p, q)
def __rsub__(self, other):
"""z1 - q2"""
if isinstance(other, int):
p = self.denominator * other - self.numerator
q = self.denominator
return self._new(p, q)
else:
return NotImplemented
def __mul__(self, other):
"""q1 * q2"""
if isinstance(other, PythonMPQ):
ap, aq = self.numerator, self.denominator
bp, bq = other.numerator, other.denominator
x1 = gcd(ap, bq)
x2 = gcd(bp, aq)
p, q = ((ap//x1)*(bp//x2), (aq//x2)*(bq//x1))
elif isinstance(other, int):
x = gcd(other, self.denominator)
p = self.numerator*(other//x)
q = self.denominator//x
else:
return NotImplemented
return self._new(p, q)
def __rmul__(self, other):
"""z1 * q2"""
if isinstance(other, int):
x = gcd(self.denominator, other)
p = self.numerator*(other//x)
q = self.denominator//x
return self._new(p, q)
else:
return NotImplemented
def __pow__(self, exp):
"""q ** z"""
p, q = self.numerator, self.denominator
if exp < 0:
p, q, exp = q, p, -exp
return self._new_check(p**exp, q**exp)
def __truediv__(self, other):
"""q1 / q2"""
if isinstance(other, PythonMPQ):
ap, aq = self.numerator, self.denominator
bp, bq = other.numerator, other.denominator
x1 = gcd(ap, bp)
x2 = gcd(bq, aq)
p, q = ((ap//x1)*(bq//x2), (aq//x2)*(bp//x1))
elif isinstance(other, int):
x = gcd(other, self.numerator)
p = self.numerator//x
q = self.denominator*(other//x)
else:
return NotImplemented
return self._new_check(p, q)
def __rtruediv__(self, other):
"""z / q"""
if isinstance(other, int):
x = gcd(self.numerator, other)
p = self.denominator*(other//x)
q = self.numerator//x
return self._new_check(p, q)
else:
return NotImplemented
#
# These are the types that PythonMPQ will interoperate with for operations
# and comparisons such as ==, + etc. We define this down here so that we can
# include PythonMPQ in the list as well.
#
PythonMPQ._compatible_types = (PythonMPQ, int, Decimal, Fraction)
|
3be848077e87a1a9ef12d3b2b006d9a1e80424776fb107d0618cac295d4a8e3f | import os
import mpmath.libmp as mlib
from sympy.external import import_module
__all__ = [
# GROUND_TYPES is either 'gmpy' or 'python' depending on which is used. If
# gmpy is installed then it will be used unless the environment variable
# SYMPY_GROUND_TYPES is set to something other than 'auto', 'gmpy', or
# 'gmpy2'.
'GROUND_TYPES',
# If HAS_GMPY is 0, no supported version of gmpy is available. Otherwise,
# HAS_GMPY will be 2 for gmpy2 if GROUND_TYPES is 'gmpy'. It used to be
# possible for HAS_GMPY to be 1 for gmpy but gmpy is no longer supported.
'HAS_GMPY',
# SYMPY_INTS is a tuple containing the base types for valid integer types.
# This is either (int,) or (int, type(mpz(0))) depending on GROUND_TYPES.
'SYMPY_INTS',
# MPQ is either gmpy.mpq or the Python equivalent from
# sympy.external.pythonmpq
'MPQ',
# MPZ is either gmpy.mpz or int.
'MPZ',
# Either the gmpy or the mpmath function
'factorial',
# isqrt from gmpy or mpmath
'sqrt',
]
#
# SYMPY_GROUND_TYPES can be gmpy, gmpy2, python or auto
#
GROUND_TYPES = os.environ.get('SYMPY_GROUND_TYPES', 'auto').lower()
#
# Try to import gmpy2 by default. If gmpy or gmpy2 is specified in
# SYMPY_GROUND_TYPES then warn if gmpy2 is not found. In all cases there is a
# fallback based on pure Python int and PythonMPQ that should still work fine.
#
if GROUND_TYPES in ('auto', 'gmpy', 'gmpy2'):
# Actually import gmpy2
gmpy = import_module('gmpy2', min_module_version='2.0.0',
module_version_attr='version', module_version_attr_call_args=())
# Warn if user explicitly asked for gmpy but it isn't available.
if gmpy is None and GROUND_TYPES in ('gmpy', 'gmpy2'):
from warnings import warn
warn("gmpy library is not installed, switching to 'python' ground types")
elif GROUND_TYPES == 'python':
# The user asked for python so ignore gmpy2 module.
gmpy = None
else:
# Invalid value for SYMPY_GROUND_TYPES. Ignore the gmpy2 module.
from warnings import warn
warn("SYMPY_GROUND_TYPES environment variable unrecognised. "
"Should be 'python', 'auto', 'gmpy', or 'gmpy2'")
gmpy = None
#
# At this point gmpy will be None if gmpy2 was not successfully imported or if
# the environment variable SYMPY_GROUND_TYPES was set to 'python' (or some
# unrecognised value). The two blocks below define the values exported by this
# module in each case.
#
if gmpy is not None:
HAS_GMPY = 2
GROUND_TYPES = 'gmpy'
SYMPY_INTS = (int, type(gmpy.mpz(0)))
MPZ = gmpy.mpz
MPQ = gmpy.mpq
factorial = gmpy.fac
sqrt = gmpy.isqrt
else:
from .pythonmpq import PythonMPQ
HAS_GMPY = 0
GROUND_TYPES = 'python'
SYMPY_INTS = (int,)
MPZ = int
MPQ = PythonMPQ
factorial = mlib.ifac
sqrt = mlib.isqrt
|
86872904f741e9add7f3686115a3857fabaa06849c524fb7fe481c98a7fcb91a | """
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
don't 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 import cacheit
from sympy.core import Basic, S, oo, I, Dummy, Wild, Mul
from sympy.functions import log, exp
from sympy.series.order import Order
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 not key 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.")
return mrv(e.args[0], x)
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.]
"""
from sympy import sign as _sign
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):
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.
"""
from sympy.polys import cancel
for t in e.lseries(x, logx=logx):
t = cancel(t)
if t.has(exp) and t.has(log):
t = powdenest(t)
if t.simplify():
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)
e_up = moveup([e], x)[0]
exps_up = moveup([exps], x)[0]
# NOTE: there is no need to move this down!
e = e_up
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", real=True, positive=True)
f, logw = rewrite(exps, Omega, x, w)
series = calculate_series(f, w, logx=logw)
return series.leadterm(w)
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 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.before = []
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.
"""
from sympy import ilcm
if not isinstance(Omega, SubsSet):
raise TypeError("Omega should be an instance of SubsSet")
if len(Omega) == 0:
raise ValueError("Length can not 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)
|
897a231bb6d3ebd948183a99b0fb9b965922adf7ac65af89db3dfb2b65228ef1 | 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.compatibility import iterable, ordered, as_int
from sympy.core.parameters import global_parameters
from sympy.core.function import (expand_log, count_ops, _mexpand, _coeff_isneg,
nfloat, expand_mul)
from sympy.core.numbers import Float, I, pi, Rational, Integer
from sympy.core.relational import Relational
from sympy.core.rules import Transform
from sympy.core.sympify import _sympify
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
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 besselj, besseli, besselk, jn, bessely
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.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.iterables import has_variety, sift
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
e = sub_post(sub_pre(expr))
if not isinstance(e, (Expr, Relational)) or e.is_Atom:
return e
if e.is_Add:
return e.func(*[signsimp(a, evaluate) for a in e.args])
if evaluate:
e = e.xreplace({m: -(-m) for m in e.atoms(Mul) if -(-m) != m})
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 don't 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 can't 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 don't
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
_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)
from sympy.simplify.hyperexpand import hyperexpand
from sympy.functions.special.bessel import BesselBase
from sympy import Sum, Product, Integral
from sympy.functions.elementary.complexes import sign
# 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)
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
from sympy.physics.units.util import quantity_simplify
if expr.has(Quantity):
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
from sympy.core.function import expand
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) or isinstance(temp, 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
from sympy import Mul
#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 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 not k 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 walk(e, *target):
"""Iterate through the args that are the given types (target) and
return a list of the args that were traversed; arguments
that are not of the specified types are not traversed.
Examples
========
>>> from sympy.simplify.simplify import walk
>>> from sympy import Min, Max
>>> from sympy.abc import x, y, z
>>> list(walk(Min(x, Max(y, Min(1, z))), Min))
[Min(x, Max(y, Min(1, z)))]
>>> list(walk(Min(x, Max(y, Min(1, z))), Min, Max))
[Min(x, Max(y, Min(1, z))), Max(y, Min(1, z)), Min(1, z)]
See Also
========
bottom_up
"""
if isinstance(e, target):
yield e
for i in e.args:
yield from walk(i, *target)
def bottom_up(rv, F, atoms=False, nonbasic=False):
"""Apply ``F`` to all expressions in an expression tree from the
bottom up. If ``atoms`` is True, apply ``F`` even if there are no args;
if ``nonbasic`` is True, try to apply ``F`` to non-Basic objects.
"""
args = getattr(rv, 'args', None)
if args is not None:
if args:
args = tuple([bottom_up(a, F, atoms, nonbasic) for a in args])
if args != rv.args:
rv = rv.func(*args)
rv = F(rv)
elif atoms:
rv = F(rv)
else:
if nonbasic:
try:
rv = F(rv)
except TypeError:
pass
return rv
def 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):
from sympy.utilities.iterables import subsets
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 0*m
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 not xv 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 == inf or fl == -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 = Integer(0)
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 _coeff_isneg(expr):
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 don't 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)
from sympy.core.exprtools import factor_nc
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 _coeff_isneg(a):
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 _coeff_isneg(ai):
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
|
a89cf952500226bbe7280af9bd9118a386cf5bd1c955d78e5605dee475a985d6 | from collections import defaultdict
from sympy import SYMPY_DEBUG
from sympy.core import expand_power_base, sympify, Add, S, Mul, Derivative, Pow, symbols, expand_mul
from sympy.core.add import _unevaluated_Add
from sympy.core.compatibility import iterable, ordered, default_sort_key
from sympy.core.parameters import global_parameters
from sympy.core.exprtools import Factors, gcd_terms
from sympy.core.function import _mexpand
from sympy.core.mul import _keep_coeff, _unevaluated_Mul
from sympy.core.numbers import Rational
from sympy.functions import exp, sqrt, log
from sympy.functions.elementary.complexes import Abs
from sympy.polys import gcd
from sympy.simplify.sqrtdenest import sqrtdenest
def collect(expr, syms, func=None, evaluate=None, exact=False, distribute_order_term=True):
"""
Collect additive terms of an expression.
Explanation
===========
This function collects additive terms of an expression with respect
to a list of expression up to powers with rational exponents. By the
term symbol here are meant arbitrary expressions, which can contain
powers, products, sums etc. In other words symbol is a pattern which
will be searched for in the expression's terms.
The input expression is not expanded by :func:`collect`, so user is
expected to provide an expression is an appropriate form. This makes
:func:`collect` more predictable as there is no magic happening behind the
scenes. However, it is important to note, that powers of products are
converted to products of powers using the :func:`~.expand_power_base`
function.
There are two possible types of output. First, if ``evaluate`` flag is
set, this function will return an expression with collected terms or
else it will return a dictionary with expressions up to rational powers
as keys and collected coefficients as values.
Examples
========
>>> from sympy import S, collect, expand, factor, Wild
>>> from sympy.abc import a, b, c, x, y
This function can collect symbolic coefficients in polynomials or
rational expressions. It will manage to find all integer or rational
powers of collection variable::
>>> collect(a*x**2 + b*x**2 + a*x - b*x + c, x)
c + x**2*(a + b) + x*(a - b)
The same result can be achieved in dictionary form::
>>> d = collect(a*x**2 + b*x**2 + a*x - b*x + c, x, evaluate=False)
>>> d[x**2]
a + b
>>> d[x]
a - b
>>> d[S.One]
c
You can also work with multivariate polynomials. However, remember that
this function is greedy so it will care only about a single symbol at time,
in specification order::
>>> collect(x**2 + y*x**2 + x*y + y + a*y, [x, y])
x**2*(y + 1) + x*y + y*(a + 1)
Also more complicated expressions can be used as patterns::
>>> from sympy import sin, log
>>> collect(a*sin(2*x) + b*sin(2*x), sin(2*x))
(a + b)*sin(2*x)
>>> collect(a*x*log(x) + b*(x*log(x)), x*log(x))
x*(a + b)*log(x)
You can use wildcards in the pattern::
>>> w = Wild('w1')
>>> collect(a*x**y - b*x**y, w**y)
x**y*(a - b)
It is also possible to work with symbolic powers, although it has more
complicated behavior, because in this case power's base and symbolic part
of the exponent are treated as a single symbol::
>>> collect(a*x**c + b*x**c, x)
a*x**c + b*x**c
>>> collect(a*x**c + b*x**c, x**c)
x**c*(a + b)
However if you incorporate rationals to the exponents, then you will get
well known behavior::
>>> collect(a*x**(2*c) + b*x**(2*c), x**c)
x**(2*c)*(a + b)
Note also that all previously stated facts about :func:`collect` function
apply to the exponential function, so you can get::
>>> from sympy import exp
>>> collect(a*exp(2*x) + b*exp(2*x), exp(x))
(a + b)*exp(2*x)
If you are interested only in collecting specific powers of some symbols
then set ``exact`` flag in arguments::
>>> collect(a*x**7 + b*x**7, x, exact=True)
a*x**7 + b*x**7
>>> collect(a*x**7 + b*x**7, x**7, exact=True)
x**7*(a + b)
You can also apply this function to differential equations, where
derivatives of arbitrary order can be collected. Note that if you
collect with respect to a function or a derivative of a function, all
derivatives of that function will also be collected. Use
``exact=True`` to prevent this from happening::
>>> from sympy import Derivative as D, collect, Function
>>> f = Function('f') (x)
>>> collect(a*D(f,x) + b*D(f,x), D(f,x))
(a + b)*Derivative(f(x), x)
>>> collect(a*D(D(f,x),x) + b*D(D(f,x),x), f)
(a + b)*Derivative(f(x), (x, 2))
>>> collect(a*D(D(f,x),x) + b*D(D(f,x),x), D(f,x), exact=True)
a*Derivative(f(x), (x, 2)) + b*Derivative(f(x), (x, 2))
>>> collect(a*D(f,x) + b*D(f,x) + a*f + b*f, f)
(a + b)*f(x) + (a + b)*Derivative(f(x), x)
Or you can even match both derivative order and exponent at the same time::
>>> collect(a*D(D(f,x),x)**2 + b*D(D(f,x),x)**2, D(f,x))
(a + b)*Derivative(f(x), (x, 2))**2
Finally, you can apply a function to each of the collected coefficients.
For example you can factorize symbolic coefficients of polynomial::
>>> f = expand((x + a + 1)**3)
>>> collect(f, x, factor)
x**3 + 3*x**2*(a + 1) + 3*x*(a + 1)**2 + (a + 1)**3
.. note:: Arguments are expected to be in expanded form, so you might have
to call :func:`~.expand` prior to calling this function.
See Also
========
collect_const, collect_sqrt, rcollect
"""
from sympy.core.assumptions import assumptions
from sympy.utilities.iterables import sift
from sympy.core.symbol import Dummy, Wild
expr = sympify(expr)
syms = [sympify(i) for i in (syms if iterable(syms) else [syms])]
# replace syms[i] if it is not x, -x or has Wild symbols
cond = lambda x: x.is_Symbol or (-x).is_Symbol or bool(
x.atoms(Wild))
_, nonsyms = sift(syms, cond, binary=True)
if nonsyms:
reps = dict(zip(nonsyms, [Dummy(**assumptions(i)) for i in nonsyms]))
syms = [reps.get(s, s) for s in syms]
rv = collect(expr.subs(reps), syms,
func=func, evaluate=evaluate, exact=exact,
distribute_order_term=distribute_order_term)
urep = {v: k for k, v in reps.items()}
if not isinstance(rv, dict):
return rv.xreplace(urep)
else:
return {urep.get(k, k).xreplace(urep): v.xreplace(urep)
for k, v in rv.items()}
if evaluate is None:
evaluate = global_parameters.evaluate
def make_expression(terms):
product = []
for term, rat, sym, deriv in terms:
if deriv is not None:
var, order = deriv
while order > 0:
term, order = Derivative(term, var), order - 1
if sym is None:
if rat is S.One:
product.append(term)
else:
product.append(Pow(term, rat))
else:
product.append(Pow(term, rat*sym))
return Mul(*product)
def parse_derivative(deriv):
# scan derivatives tower in the input expression and return
# underlying function and maximal differentiation order
expr, sym, order = deriv.expr, deriv.variables[0], 1
for s in deriv.variables[1:]:
if s == sym:
order += 1
else:
raise NotImplementedError(
'Improve MV Derivative support in collect')
while isinstance(expr, Derivative):
s0 = expr.variables[0]
for s in expr.variables:
if s != s0:
raise NotImplementedError(
'Improve MV Derivative support in collect')
if s0 == sym:
expr, order = expr.expr, order + len(expr.variables)
else:
break
return expr, (sym, Rational(order))
def parse_term(expr):
"""Parses expression expr and outputs tuple (sexpr, rat_expo,
sym_expo, deriv)
where:
- sexpr is the base expression
- rat_expo is the rational exponent that sexpr is raised to
- sym_expo is the symbolic exponent that sexpr is raised to
- deriv contains the derivatives the the expression
For example, the output of x would be (x, 1, None, None)
the output of 2**x would be (2, 1, x, None).
"""
rat_expo, sym_expo = S.One, None
sexpr, deriv = expr, None
if expr.is_Pow:
if isinstance(expr.base, Derivative):
sexpr, deriv = parse_derivative(expr.base)
else:
sexpr = expr.base
if expr.base == S.Exp1:
arg = expr.exp
if arg.is_Rational:
sexpr, rat_expo = S.Exp1, arg
elif arg.is_Mul:
coeff, tail = arg.as_coeff_Mul(rational=True)
sexpr, rat_expo = exp(tail), coeff
elif expr.exp.is_Number:
rat_expo = expr.exp
else:
coeff, tail = expr.exp.as_coeff_Mul()
if coeff.is_Number:
rat_expo, sym_expo = coeff, tail
else:
sym_expo = expr.exp
elif isinstance(expr, exp):
arg = expr.exp
if arg.is_Rational:
sexpr, rat_expo = S.Exp1, arg
elif arg.is_Mul:
coeff, tail = arg.as_coeff_Mul(rational=True)
sexpr, rat_expo = exp(tail), coeff
elif isinstance(expr, Derivative):
sexpr, deriv = parse_derivative(expr)
return sexpr, rat_expo, sym_expo, deriv
def parse_expression(terms, pattern):
"""Parse terms searching for a pattern.
Terms is a list of tuples as returned by parse_terms;
Pattern is an expression treated as a product of factors.
"""
pattern = Mul.make_args(pattern)
if len(terms) < len(pattern):
# pattern is longer than matched product
# so no chance for positive parsing result
return None
else:
pattern = [parse_term(elem) for elem in pattern]
terms = terms[:] # need a copy
elems, common_expo, has_deriv = [], None, False
for elem, e_rat, e_sym, e_ord in pattern:
if elem.is_Number and e_rat == 1 and e_sym is None:
# a constant is a match for everything
continue
for j in range(len(terms)):
if terms[j] is None:
continue
term, t_rat, t_sym, t_ord = terms[j]
# keeping track of whether one of the terms had
# a derivative or not as this will require rebuilding
# the expression later
if t_ord is not None:
has_deriv = True
if (term.match(elem) is not None and
(t_sym == e_sym or t_sym is not None and
e_sym is not None and
t_sym.match(e_sym) is not None)):
if exact is False:
# we don't have to be exact so find common exponent
# for both expression's term and pattern's element
expo = t_rat / e_rat
if common_expo is None:
# first time
common_expo = expo
else:
# common exponent was negotiated before so
# there is no chance for a pattern match unless
# common and current exponents are equal
if common_expo != expo:
common_expo = 1
else:
# we ought to be exact so all fields of
# interest must match in every details
if e_rat != t_rat or e_ord != t_ord:
continue
# found common term so remove it from the expression
# and try to match next element in the pattern
elems.append(terms[j])
terms[j] = None
break
else:
# pattern element not found
return None
return [_f for _f in terms if _f], elems, common_expo, has_deriv
if evaluate:
if expr.is_Add:
o = expr.getO() or 0
expr = expr.func(*[
collect(a, syms, func, True, exact, distribute_order_term)
for a in expr.args if a != o]) + o
elif expr.is_Mul:
return expr.func(*[
collect(term, syms, func, True, exact, distribute_order_term)
for term in expr.args])
elif expr.is_Pow:
b = collect(
expr.base, syms, func, True, exact, distribute_order_term)
return Pow(b, expr.exp)
syms = [expand_power_base(i, deep=False) for i in syms]
order_term = None
if distribute_order_term:
order_term = expr.getO()
if order_term is not None:
if order_term.has(*syms):
order_term = None
else:
expr = expr.removeO()
summa = [expand_power_base(i, deep=False) for i in Add.make_args(expr)]
collected, disliked = defaultdict(list), S.Zero
for product in summa:
c, nc = product.args_cnc(split_1=False)
args = list(ordered(c)) + nc
terms = [parse_term(i) for i in args]
small_first = True
for symbol in syms:
if SYMPY_DEBUG:
print("DEBUG: parsing of expression %s with symbol %s " % (
str(terms), str(symbol))
)
if isinstance(symbol, Derivative) and small_first:
terms = list(reversed(terms))
small_first = not small_first
result = parse_expression(terms, symbol)
if SYMPY_DEBUG:
print("DEBUG: returned %s" % str(result))
if result is not None:
if not symbol.is_commutative:
raise AttributeError("Can not collect noncommutative symbol")
terms, elems, common_expo, has_deriv = result
# when there was derivative in current pattern we
# will need to rebuild its expression from scratch
if not has_deriv:
margs = []
for elem in elems:
if elem[2] is None:
e = elem[1]
else:
e = elem[1]*elem[2]
margs.append(Pow(elem[0], e))
index = Mul(*margs)
else:
index = make_expression(elems)
terms = expand_power_base(make_expression(terms), deep=False)
index = expand_power_base(index, deep=False)
collected[index].append(terms)
break
else:
# none of the patterns matched
disliked += product
# add terms now for each key
collected = {k: Add(*v) for k, v in collected.items()}
if disliked is not S.Zero:
collected[S.One] = disliked
if order_term is not None:
for key, val in collected.items():
collected[key] = val + order_term
if func is not None:
collected = {
key: func(val) for key, val in collected.items()}
if evaluate:
return Add(*[key*val for key, val in collected.items()])
else:
return collected
def rcollect(expr, *vars):
"""
Recursively collect sums in an expression.
Examples
========
>>> from sympy.simplify import rcollect
>>> from sympy.abc import x, y
>>> expr = (x**2*y + x*y + x + y)/(x + y)
>>> rcollect(expr, y)
(x + y*(x**2 + x + 1))/(x + y)
See Also
========
collect, collect_const, collect_sqrt
"""
if expr.is_Atom or not expr.has(*vars):
return expr
else:
expr = expr.__class__(*[rcollect(arg, *vars) for arg in expr.args])
if expr.is_Add:
return collect(expr, vars)
else:
return expr
def collect_sqrt(expr, evaluate=None):
"""Return expr with terms having common square roots collected together.
If ``evaluate`` is False a count indicating the number of sqrt-containing
terms will be returned and, if non-zero, the terms of the Add will be
returned, else the expression itself will be returned as a single term.
If ``evaluate`` is True, the expression with any collected terms will be
returned.
Note: since I = sqrt(-1), it is collected, too.
Examples
========
>>> from sympy import sqrt
>>> from sympy.simplify.radsimp import collect_sqrt
>>> from sympy.abc import a, b
>>> r2, r3, r5 = [sqrt(i) for i in [2, 3, 5]]
>>> collect_sqrt(a*r2 + b*r2)
sqrt(2)*(a + b)
>>> collect_sqrt(a*r2 + b*r2 + a*r3 + b*r3)
sqrt(2)*(a + b) + sqrt(3)*(a + b)
>>> collect_sqrt(a*r2 + b*r2 + a*r3 + b*r5)
sqrt(3)*a + sqrt(5)*b + sqrt(2)*(a + b)
If evaluate is False then the arguments will be sorted and
returned as a list and a count of the number of sqrt-containing
terms will be returned:
>>> collect_sqrt(a*r2 + b*r2 + a*r3 + b*r5, evaluate=False)
((sqrt(3)*a, sqrt(5)*b, sqrt(2)*(a + b)), 3)
>>> collect_sqrt(a*sqrt(2) + b, evaluate=False)
((b, sqrt(2)*a), 1)
>>> collect_sqrt(a + b, evaluate=False)
((a + b,), 0)
See Also
========
collect, collect_const, rcollect
"""
if evaluate is None:
evaluate = global_parameters.evaluate
# this step will help to standardize any complex arguments
# of sqrts
coeff, expr = expr.as_content_primitive()
vars = set()
for a in Add.make_args(expr):
for m in a.args_cnc()[0]:
if m.is_number and (
m.is_Pow and m.exp.is_Rational and m.exp.q == 2 or
m is S.ImaginaryUnit):
vars.add(m)
# we only want radicals, so exclude Number handling; in this case
# d will be evaluated
d = collect_const(expr, *vars, Numbers=False)
hit = expr != d
if not evaluate:
nrad = 0
# make the evaluated args canonical
args = list(ordered(Add.make_args(d)))
for i, m in enumerate(args):
c, nc = m.args_cnc()
for ci in c:
# XXX should this be restricted to ci.is_number as above?
if ci.is_Pow and ci.exp.is_Rational and ci.exp.q == 2 or \
ci is S.ImaginaryUnit:
nrad += 1
break
args[i] *= coeff
if not (hit or nrad):
args = [Add(*args)]
return tuple(args), nrad
return coeff*d
def collect_abs(expr):
"""Return ``expr`` with arguments of multiple Abs in a term collected
under a single instance.
Examples
========
>>> from sympy.simplify.radsimp import collect_abs
>>> from sympy.abc import x
>>> collect_abs(abs(x + 1)/abs(x**2 - 1))
Abs((x + 1)/(x**2 - 1))
>>> collect_abs(abs(1/x))
Abs(1/x)
"""
def _abs(mul):
from sympy.core.mul import _mulsort
c, nc = mul.args_cnc()
a = []
o = []
for i in c:
if isinstance(i, Abs):
a.append(i.args[0])
elif isinstance(i, Pow) and isinstance(i.base, Abs) and i.exp.is_real:
a.append(i.base.args[0]**i.exp)
else:
o.append(i)
if len(a) < 2 and not any(i.exp.is_negative for i in a if isinstance(i, Pow)):
return mul
absarg = Mul(*a)
A = Abs(absarg)
args = [A]
args.extend(o)
if not A.has(Abs):
args.extend(nc)
return Mul(*args)
if not isinstance(A, Abs):
# reevaluate and make it unevaluated
A = Abs(absarg, evaluate=False)
args[0] = A
_mulsort(args)
args.extend(nc) # nc always go last
return Mul._from_args(args, is_commutative=not nc)
return expr.replace(
lambda x: isinstance(x, Mul),
lambda x: _abs(x)).replace(
lambda x: isinstance(x, Pow),
lambda x: _abs(x))
def collect_const(expr, *vars, Numbers=True):
"""A non-greedy collection of terms with similar number coefficients in
an Add expr. If ``vars`` is given then only those constants will be
targeted. Although any Number can also be targeted, if this is not
desired set ``Numbers=False`` and no Float or Rational will be collected.
Parameters
==========
expr : sympy expression
This parameter defines the expression the expression from which
terms with similar coefficients are to be collected. A non-Add
expression is returned as it is.
vars : variable length collection of Numbers, optional
Specifies the constants to target for collection. Can be multiple in
number.
Numbers : bool
Specifies to target all instance of
:class:`sympy.core.numbers.Number` class. If ``Numbers=False``, then
no Float or Rational will be collected.
Returns
=======
expr : Expr
Returns an expression with similar coefficient terms collected.
Examples
========
>>> from sympy import sqrt
>>> from sympy.abc import s, x, y, z
>>> from sympy.simplify.radsimp import collect_const
>>> collect_const(sqrt(3) + sqrt(3)*(1 + sqrt(2)))
sqrt(3)*(sqrt(2) + 2)
>>> collect_const(sqrt(3)*s + sqrt(7)*s + sqrt(3) + sqrt(7))
(sqrt(3) + sqrt(7))*(s + 1)
>>> s = sqrt(2) + 2
>>> collect_const(sqrt(3)*s + sqrt(3) + sqrt(7)*s + sqrt(7))
(sqrt(2) + 3)*(sqrt(3) + sqrt(7))
>>> collect_const(sqrt(3)*s + sqrt(3) + sqrt(7)*s + sqrt(7), sqrt(3))
sqrt(7) + sqrt(3)*(sqrt(2) + 3) + sqrt(7)*(sqrt(2) + 2)
The collection is sign-sensitive, giving higher precedence to the
unsigned values:
>>> collect_const(x - y - z)
x - (y + z)
>>> collect_const(-y - z)
-(y + z)
>>> collect_const(2*x - 2*y - 2*z, 2)
2*(x - y - z)
>>> collect_const(2*x - 2*y - 2*z, -2)
2*x - 2*(y + z)
See Also
========
collect, collect_sqrt, rcollect
"""
if not expr.is_Add:
return expr
recurse = False
if not vars:
recurse = True
vars = set()
for a in expr.args:
for m in Mul.make_args(a):
if m.is_number:
vars.add(m)
else:
vars = sympify(vars)
if not Numbers:
vars = [v for v in vars if not v.is_Number]
vars = list(ordered(vars))
for v in vars:
terms = defaultdict(list)
Fv = Factors(v)
for m in Add.make_args(expr):
f = Factors(m)
q, r = f.div(Fv)
if r.is_one:
# only accept this as a true factor if
# it didn't change an exponent from an Integer
# to a non-Integer, e.g. 2/sqrt(2) -> sqrt(2)
# -- we aren't looking for this sort of change
fwas = f.factors.copy()
fnow = q.factors
if not any(k in fwas and fwas[k].is_Integer and not
fnow[k].is_Integer for k in fnow):
terms[v].append(q.as_expr())
continue
terms[S.One].append(m)
args = []
hit = False
uneval = False
for k in ordered(terms):
v = terms[k]
if k is S.One:
args.extend(v)
continue
if len(v) > 1:
v = Add(*v)
hit = True
if recurse and v != expr:
vars.append(v)
else:
v = v[0]
# be careful not to let uneval become True unless
# it must be because it's going to be more expensive
# to rebuild the expression as an unevaluated one
if Numbers and k.is_Number and v.is_Add:
args.append(_keep_coeff(k, v, sign=True))
uneval = True
else:
args.append(k*v)
if hit:
if uneval:
expr = _unevaluated_Add(*args)
else:
expr = Add(*args)
if not expr.is_Add:
break
return expr
def radsimp(expr, symbolic=True, max_terms=4):
r"""
Rationalize the denominator by removing square roots.
Explanation
===========
The expression returned from radsimp must be used with caution
since if the denominator contains symbols, it will be possible to make
substitutions that violate the assumptions of the simplification process:
that for a denominator matching a + b*sqrt(c), a != +/-b*sqrt(c). (If
there are no symbols, this assumptions is made valid by collecting terms
of sqrt(c) so the match variable ``a`` does not contain ``sqrt(c)``.) If
you do not want the simplification to occur for symbolic denominators, set
``symbolic`` to False.
If there are more than ``max_terms`` radical terms then the expression is
returned unchanged.
Examples
========
>>> from sympy import radsimp, sqrt, Symbol, pprint
>>> from sympy import factor_terms, fraction, signsimp
>>> from sympy.simplify.radsimp import collect_sqrt
>>> from sympy.abc import a, b, c
>>> radsimp(1/(2 + sqrt(2)))
(2 - sqrt(2))/2
>>> x,y = map(Symbol, 'xy')
>>> e = ((2 + 2*sqrt(2))*x + (2 + sqrt(8))*y)/(2 + sqrt(2))
>>> radsimp(e)
sqrt(2)*(x + y)
No simplification beyond removal of the gcd is done. One might
want to polish the result a little, however, by collecting
square root terms:
>>> r2 = sqrt(2)
>>> r5 = sqrt(5)
>>> ans = radsimp(1/(y*r2 + x*r2 + a*r5 + b*r5)); pprint(ans)
___ ___ ___ ___
\/ 5 *a + \/ 5 *b - \/ 2 *x - \/ 2 *y
------------------------------------------
2 2 2 2
5*a + 10*a*b + 5*b - 2*x - 4*x*y - 2*y
>>> n, d = fraction(ans)
>>> pprint(factor_terms(signsimp(collect_sqrt(n))/d, radical=True))
___ ___
\/ 5 *(a + b) - \/ 2 *(x + y)
------------------------------------------
2 2 2 2
5*a + 10*a*b + 5*b - 2*x - 4*x*y - 2*y
If radicals in the denominator cannot be removed or there is no denominator,
the original expression will be returned.
>>> radsimp(sqrt(2)*x + sqrt(2))
sqrt(2)*x + sqrt(2)
Results with symbols will not always be valid for all substitutions:
>>> eq = 1/(a + b*sqrt(c))
>>> eq.subs(a, b*sqrt(c))
1/(2*b*sqrt(c))
>>> radsimp(eq).subs(a, b*sqrt(c))
nan
If ``symbolic=False``, symbolic denominators will not be transformed (but
numeric denominators will still be processed):
>>> radsimp(eq, symbolic=False)
1/(a + b*sqrt(c))
"""
from sympy.simplify.simplify import signsimp
syms = symbols("a:d A:D")
def _num(rterms):
# return the multiplier that will simplify the expression described
# by rterms [(sqrt arg, coeff), ... ]
a, b, c, d, A, B, C, D = syms
if len(rterms) == 2:
reps = dict(list(zip([A, a, B, b], [j for i in rterms for j in i])))
return (
sqrt(A)*a - sqrt(B)*b).xreplace(reps)
if len(rterms) == 3:
reps = dict(list(zip([A, a, B, b, C, c], [j for i in rterms for j in i])))
return (
(sqrt(A)*a + sqrt(B)*b - sqrt(C)*c)*(2*sqrt(A)*sqrt(B)*a*b - A*a**2 -
B*b**2 + C*c**2)).xreplace(reps)
elif len(rterms) == 4:
reps = dict(list(zip([A, a, B, b, C, c, D, d], [j for i in rterms for j in i])))
return ((sqrt(A)*a + sqrt(B)*b - sqrt(C)*c - sqrt(D)*d)*(2*sqrt(A)*sqrt(B)*a*b
- A*a**2 - B*b**2 - 2*sqrt(C)*sqrt(D)*c*d + C*c**2 +
D*d**2)*(-8*sqrt(A)*sqrt(B)*sqrt(C)*sqrt(D)*a*b*c*d + A**2*a**4 -
2*A*B*a**2*b**2 - 2*A*C*a**2*c**2 - 2*A*D*a**2*d**2 + B**2*b**4 -
2*B*C*b**2*c**2 - 2*B*D*b**2*d**2 + C**2*c**4 - 2*C*D*c**2*d**2 +
D**2*d**4)).xreplace(reps)
elif len(rterms) == 1:
return sqrt(rterms[0][0])
else:
raise NotImplementedError
def ispow2(d, log2=False):
if not d.is_Pow:
return False
e = d.exp
if e.is_Rational and e.q == 2 or symbolic and denom(e) == 2:
return True
if log2:
q = 1
if e.is_Rational:
q = e.q
elif symbolic:
d = denom(e)
if d.is_Integer:
q = d
if q != 1 and log(q, 2).is_Integer:
return True
return False
def handle(expr):
# Handle first reduces to the case
# expr = 1/d, where d is an add, or d is base**p/2.
# We do this by recursively calling handle on each piece.
from sympy.simplify.simplify import nsimplify
n, d = fraction(expr)
if expr.is_Atom or (d.is_Atom and n.is_Atom):
return expr
elif not n.is_Atom:
n = n.func(*[handle(a) for a in n.args])
return _unevaluated_Mul(n, handle(1/d))
elif n is not S.One:
return _unevaluated_Mul(n, handle(1/d))
elif d.is_Mul:
return _unevaluated_Mul(*[handle(1/d) for d in d.args])
# By this step, expr is 1/d, and d is not a mul.
if not symbolic and d.free_symbols:
return expr
if ispow2(d):
d2 = sqrtdenest(sqrt(d.base))**numer(d.exp)
if d2 != d:
return handle(1/d2)
elif d.is_Pow and (d.exp.is_integer or d.base.is_positive):
# (1/d**i) = (1/d)**i
return handle(1/d.base)**d.exp
if not (d.is_Add or ispow2(d)):
return 1/d.func(*[handle(a) for a in d.args])
# handle 1/d treating d as an Add (though it may not be)
keep = True # keep changes that are made
# flatten it and collect radicals after checking for special
# conditions
d = _mexpand(d)
# did it change?
if d.is_Atom:
return 1/d
# is it a number that might be handled easily?
if d.is_number:
_d = nsimplify(d)
if _d.is_Number and _d.equals(d):
return 1/_d
while True:
# collect similar terms
collected = defaultdict(list)
for m in Add.make_args(d): # d might have become non-Add
p2 = []
other = []
for i in Mul.make_args(m):
if ispow2(i, log2=True):
p2.append(i.base if i.exp is S.Half else i.base**(2*i.exp))
elif i is S.ImaginaryUnit:
p2.append(S.NegativeOne)
else:
other.append(i)
collected[tuple(ordered(p2))].append(Mul(*other))
rterms = list(ordered(list(collected.items())))
rterms = [(Mul(*i), Add(*j)) for i, j in rterms]
nrad = len(rterms) - (1 if rterms[0][0] is S.One else 0)
if nrad < 1:
break
elif nrad > max_terms:
# there may have been invalid operations leading to this point
# so don't keep changes, e.g. this expression is troublesome
# in collecting terms so as not to raise the issue of 2834:
# r = sqrt(sqrt(5) + 5)
# eq = 1/(sqrt(5)*r + 2*sqrt(5)*sqrt(-sqrt(5) + 5) + 5*r)
keep = False
break
if len(rterms) > 4:
# in general, only 4 terms can be removed with repeated squaring
# but other considerations can guide selection of radical terms
# so that radicals are removed
if all([x.is_Integer and (y**2).is_Rational for x, y in rterms]):
nd, d = rad_rationalize(S.One, Add._from_args(
[sqrt(x)*y for x, y in rterms]))
n *= nd
else:
# is there anything else that might be attempted?
keep = False
break
from sympy.simplify.powsimp import powsimp, powdenest
num = powsimp(_num(rterms))
n *= num
d *= num
d = powdenest(_mexpand(d), force=symbolic)
if d.is_Atom:
break
if not keep:
return expr
return _unevaluated_Mul(n, 1/d)
coeff, expr = expr.as_coeff_Add()
expr = expr.normal()
old = fraction(expr)
n, d = fraction(handle(expr))
if old != (n, d):
if not d.is_Atom:
was = (n, d)
n = signsimp(n, evaluate=False)
d = signsimp(d, evaluate=False)
u = Factors(_unevaluated_Mul(n, 1/d))
u = _unevaluated_Mul(*[k**v for k, v in u.factors.items()])
n, d = fraction(u)
if old == (n, d):
n, d = was
n = expand_mul(n)
if d.is_Number or d.is_Add:
n2, d2 = fraction(gcd_terms(_unevaluated_Mul(n, 1/d)))
if d2.is_Number or (d2.count_ops() <= d.count_ops()):
n, d = [signsimp(i) for i in (n2, d2)]
if n.is_Mul and n.args[0].is_Number:
n = n.func(*n.args)
return coeff + _unevaluated_Mul(n, 1/d)
def rad_rationalize(num, den):
"""
Rationalize ``num/den`` by removing square roots in the denominator;
num and den are sum of terms whose squares are positive rationals.
Examples
========
>>> from sympy import sqrt
>>> from sympy.simplify.radsimp import rad_rationalize
>>> rad_rationalize(sqrt(3), 1 + sqrt(2)/3)
(-sqrt(3) + sqrt(6)/3, -7/9)
"""
if not den.is_Add:
return num, den
g, a, b = split_surds(den)
a = a*sqrt(g)
num = _mexpand((a - b)*num)
den = _mexpand(a**2 - b**2)
return rad_rationalize(num, den)
def fraction(expr, exact=False):
"""Returns a pair with expression's numerator and denominator.
If the given expression is not a fraction then this function
will return the tuple (expr, 1).
This function will not make any attempt to simplify nested
fractions or to do any term rewriting at all.
If only one of the numerator/denominator pair is needed then
use numer(expr) or denom(expr) functions respectively.
>>> from sympy import fraction, Rational, Symbol
>>> from sympy.abc import x, y
>>> fraction(x/y)
(x, y)
>>> fraction(x)
(x, 1)
>>> fraction(1/y**2)
(1, y**2)
>>> fraction(x*y/2)
(x*y, 2)
>>> fraction(Rational(1, 2))
(1, 2)
This function will also work fine with assumptions:
>>> k = Symbol('k', negative=True)
>>> fraction(x * y**k)
(x, y**(-k))
If we know nothing about sign of some exponent and ``exact``
flag is unset, then structure this exponent's structure will
be analyzed and pretty fraction will be returned:
>>> from sympy import exp, Mul
>>> fraction(2*x**(-y))
(2, x**y)
>>> fraction(exp(-x))
(1, exp(x))
>>> fraction(exp(-x), exact=True)
(exp(-x), 1)
The ``exact`` flag will also keep any unevaluated Muls from
being evaluated:
>>> u = Mul(2, x + 1, evaluate=False)
>>> fraction(u)
(2*x + 2, 1)
>>> fraction(u, exact=True)
(2*(x + 1), 1)
"""
expr = sympify(expr)
numer, denom = [], []
for term in Mul.make_args(expr):
if term.is_commutative and (term.is_Pow or isinstance(term, exp)):
b, ex = term.as_base_exp()
if ex.is_negative:
if ex is S.NegativeOne:
denom.append(b)
elif exact:
if ex.is_constant():
denom.append(Pow(b, -ex))
else:
numer.append(term)
else:
denom.append(Pow(b, -ex))
elif ex.is_positive:
numer.append(term)
elif not exact and ex.is_Mul:
n, d = term.as_numer_denom()
if n != 1:
numer.append(n)
denom.append(d)
else:
numer.append(term)
elif term.is_Rational and not term.is_Integer:
if term.p != 1:
numer.append(term.p)
denom.append(term.q)
else:
numer.append(term)
return Mul(*numer, evaluate=not exact), Mul(*denom, evaluate=not exact)
def numer(expr):
return fraction(expr)[0]
def denom(expr):
return fraction(expr)[1]
def fraction_expand(expr, **hints):
return expr.expand(frac=True, **hints)
def numer_expand(expr, **hints):
a, b = fraction(expr)
return a.expand(numer=True, **hints) / b
def denom_expand(expr, **hints):
a, b = fraction(expr)
return a / b.expand(denom=True, **hints)
expand_numer = numer_expand
expand_denom = denom_expand
expand_fraction = fraction_expand
def split_surds(expr):
"""
Split an expression with terms whose squares are positive rationals
into a sum of terms whose surds squared have gcd equal to g
and a sum of terms with surds squared prime with g.
Examples
========
>>> from sympy import sqrt
>>> from sympy.simplify.radsimp import split_surds
>>> split_surds(3*sqrt(3) + sqrt(5)/7 + sqrt(6) + sqrt(10) + sqrt(15))
(3, sqrt(2) + sqrt(5) + 3, sqrt(5)/7 + sqrt(10))
"""
args = sorted(expr.args, key=default_sort_key)
coeff_muls = [x.as_coeff_Mul() for x in args]
surds = [x[1]**2 for x in coeff_muls if x[1].is_Pow]
surds.sort(key=default_sort_key)
g, b1, b2 = _split_gcd(*surds)
g2 = g
if not b2 and len(b1) >= 2:
b1n = [x/g for x in b1]
b1n = [x for x in b1n if x != 1]
# only a common factor has been factored; split again
g1, b1n, b2 = _split_gcd(*b1n)
g2 = g*g1
a1v, a2v = [], []
for c, s in coeff_muls:
if s.is_Pow and s.exp == S.Half:
s1 = s.base
if s1 in b1:
a1v.append(c*sqrt(s1/g2))
else:
a2v.append(c*s)
else:
a2v.append(c*s)
a = Add(*a1v)
b = Add(*a2v)
return g2, a, b
def _split_gcd(*a):
"""
Split the list of integers ``a`` into a list of integers, ``a1`` having
``g = gcd(a1)``, and a list ``a2`` whose elements are not divisible by
``g``. Returns ``g, a1, a2``.
Examples
========
>>> from sympy.simplify.radsimp import _split_gcd
>>> _split_gcd(55, 35, 22, 14, 77, 10)
(5, [55, 35, 10], [22, 14, 77])
"""
g = a[0]
b1 = [g]
b2 = []
for x in a[1:]:
g1 = gcd(g, x)
if g1 == 1:
b2.append(x)
else:
g = g1
b1.append(x)
return g, b1, b2
|
cab5982a15b90e4752cba6bbddc6f8b8d30ceee3e5793fab2d0f9c8832c1bee6 | from collections import defaultdict
from functools import reduce
from sympy.core import (sympify, Basic, S, Expr, expand_mul, factor_terms,
Mul, Dummy, igcd, FunctionClass, Add, symbols, Wild, expand)
from sympy.core.cache import cacheit
from sympy.core.compatibility import iterable, SYMPY_INTS
from sympy.core.function import count_ops, _mexpand
from sympy.core.numbers import I, Integer
from sympy.functions import sin, cos, exp, cosh, tanh, sinh, tan, cot, coth
from sympy.functions.elementary.hyperbolic import HyperbolicFunction
from sympy.functions.elementary.trigonometric import TrigonometricFunction
from sympy.polys import Poly, factor, cancel, parallel_poly_from_expr
from sympy.polys.domains import ZZ
from sympy.polys.polyerrors import PolificationFailed
from sympy.polys.polytools import groebner
from sympy.simplify.cse_main import cse
from sympy.strategies.core import identity
from sympy.strategies.tree import greedy
from sympy.utilities.misc import debug
def trigsimp_groebner(expr, hints=[], quick=False, order="grlex",
polynomial=False):
"""
Simplify trigonometric expressions using a groebner basis algorithm.
Explanation
===========
This routine takes a fraction involving trigonometric or hyperbolic
expressions, and tries to simplify it. The primary metric is the
total degree. Some attempts are made to choose the simplest possible
expression of the minimal degree, but this is non-rigorous, and also
very slow (see the ``quick=True`` option).
If ``polynomial`` is set to True, instead of simplifying numerator and
denominator together, this function just brings numerator and denominator
into a canonical form. This is much faster, but has potentially worse
results. However, if the input is a polynomial, then the result is
guaranteed to be an equivalent polynomial of minimal degree.
The most important option is hints. Its entries can be any of the
following:
- a natural number
- a function
- an iterable of the form (func, var1, var2, ...)
- anything else, interpreted as a generator
A number is used to indicate that the search space should be increased.
A function is used to indicate that said function is likely to occur in a
simplified expression.
An iterable is used indicate that func(var1 + var2 + ...) is likely to
occur in a simplified .
An additional generator also indicates that it is likely to occur.
(See examples below).
This routine carries out various computationally intensive algorithms.
The option ``quick=True`` can be used to suppress one particularly slow
step (at the expense of potentially more complicated results, but never at
the expense of increased total degree).
Examples
========
>>> from sympy.abc import x, y
>>> from sympy import sin, tan, cos, sinh, cosh, tanh
>>> from sympy.simplify.trigsimp import trigsimp_groebner
Suppose you want to simplify ``sin(x)*cos(x)``. Naively, nothing happens:
>>> ex = sin(x)*cos(x)
>>> trigsimp_groebner(ex)
sin(x)*cos(x)
This is because ``trigsimp_groebner`` only looks for a simplification
involving just ``sin(x)`` and ``cos(x)``. You can tell it to also try
``2*x`` by passing ``hints=[2]``:
>>> trigsimp_groebner(ex, hints=[2])
sin(2*x)/2
>>> trigsimp_groebner(sin(x)**2 - cos(x)**2, hints=[2])
-cos(2*x)
Increasing the search space this way can quickly become expensive. A much
faster way is to give a specific expression that is likely to occur:
>>> trigsimp_groebner(ex, hints=[sin(2*x)])
sin(2*x)/2
Hyperbolic expressions are similarly supported:
>>> trigsimp_groebner(sinh(2*x)/sinh(x))
2*cosh(x)
Note how no hints had to be passed, since the expression already involved
``2*x``.
The tangent function is also supported. You can either pass ``tan`` in the
hints, to indicate that tan should be tried whenever cosine or sine are,
or you can pass a specific generator:
>>> trigsimp_groebner(sin(x)/cos(x), hints=[tan])
tan(x)
>>> trigsimp_groebner(sinh(x)/cosh(x), hints=[tanh(x)])
tanh(x)
Finally, you can use the iterable form to suggest that angle sum formulae
should be tried:
>>> ex = (tan(x) + tan(y))/(1 - tan(x)*tan(y))
>>> trigsimp_groebner(ex, hints=[(tan, x, y)])
tan(x + y)
"""
# TODO
# - preprocess by replacing everything by funcs we can handle
# - optionally use cot instead of tan
# - more intelligent hinting.
# For example, if the ideal is small, and we have sin(x), sin(y),
# add sin(x + y) automatically... ?
# - algebraic numbers ...
# - expressions of lowest degree are not distinguished properly
# e.g. 1 - sin(x)**2
# - we could try to order the generators intelligently, so as to influence
# which monomials appear in the quotient basis
# THEORY
# ------
# Ratsimpmodprime above can be used to "simplify" a rational function
# modulo a prime ideal. "Simplify" mainly means finding an equivalent
# expression of lower total degree.
#
# We intend to use this to simplify trigonometric functions. To do that,
# we need to decide (a) which ring to use, and (b) modulo which ideal to
# simplify. In practice, (a) means settling on a list of "generators"
# a, b, c, ..., such that the fraction we want to simplify is a rational
# function in a, b, c, ..., with coefficients in ZZ (integers).
# (2) means that we have to decide what relations to impose on the
# generators. There are two practical problems:
# (1) The ideal has to be *prime* (a technical term).
# (2) The relations have to be polynomials in the generators.
#
# We typically have two kinds of generators:
# - trigonometric expressions, like sin(x), cos(5*x), etc
# - "everything else", like gamma(x), pi, etc.
#
# Since this function is trigsimp, we will concentrate on what to do with
# trigonometric expressions. We can also simplify hyperbolic expressions,
# but the extensions should be clear.
#
# One crucial point is that all *other* generators really should behave
# like indeterminates. In particular if (say) "I" is one of them, then
# in fact I**2 + 1 = 0 and we may and will compute non-sensical
# expressions. However, we can work with a dummy and add the relation
# I**2 + 1 = 0 to our ideal, then substitute back in the end.
#
# Now regarding trigonometric generators. We split them into groups,
# according to the argument of the trigonometric functions. We want to
# organise this in such a way that most trigonometric identities apply in
# the same group. For example, given sin(x), cos(2*x) and cos(y), we would
# group as [sin(x), cos(2*x)] and [cos(y)].
#
# Our prime ideal will be built in three steps:
# (1) For each group, compute a "geometrically prime" ideal of relations.
# Geometrically prime means that it generates a prime ideal in
# CC[gens], not just ZZ[gens].
# (2) Take the union of all the generators of the ideals for all groups.
# By the geometric primality condition, this is still prime.
# (3) Add further inter-group relations which preserve primality.
#
# Step (1) works as follows. We will isolate common factors in the
# argument, so that all our generators are of the form sin(n*x), cos(n*x)
# or tan(n*x), with n an integer. Suppose first there are no tan terms.
# The ideal [sin(x)**2 + cos(x)**2 - 1] is geometrically prime, since
# X**2 + Y**2 - 1 is irreducible over CC.
# Now, if we have a generator sin(n*x), than we can, using trig identities,
# express sin(n*x) as a polynomial in sin(x) and cos(x). We can add this
# relation to the ideal, preserving geometric primality, since the quotient
# ring is unchanged.
# Thus we have treated all sin and cos terms.
# For tan(n*x), we add a relation tan(n*x)*cos(n*x) - sin(n*x) = 0.
# (This requires of course that we already have relations for cos(n*x) and
# sin(n*x).) It is not obvious, but it seems that this preserves geometric
# primality.
# XXX A real proof would be nice. HELP!
# Sketch that <S**2 + C**2 - 1, C*T - S> is a prime ideal of
# CC[S, C, T]:
# - it suffices to show that the projective closure in CP**3 is
# irreducible
# - using the half-angle substitutions, we can express sin(x), tan(x),
# cos(x) as rational functions in tan(x/2)
# - from this, we get a rational map from CP**1 to our curve
# - this is a morphism, hence the curve is prime
#
# Step (2) is trivial.
#
# Step (3) works by adding selected relations of the form
# sin(x + y) - sin(x)*cos(y) - sin(y)*cos(x), etc. Geometric primality is
# preserved by the same argument as before.
def parse_hints(hints):
"""Split hints into (n, funcs, iterables, gens)."""
n = 1
funcs, iterables, gens = [], [], []
for e in hints:
if isinstance(e, (SYMPY_INTS, Integer)):
n = e
elif isinstance(e, FunctionClass):
funcs.append(e)
elif iterable(e):
iterables.append((e[0], e[1:]))
# XXX sin(x+2y)?
# Note: we go through polys so e.g.
# sin(-x) -> -sin(x) -> sin(x)
gens.extend(parallel_poly_from_expr(
[e[0](x) for x in e[1:]] + [e[0](Add(*e[1:]))])[1].gens)
else:
gens.append(e)
return n, funcs, iterables, gens
def build_ideal(x, terms):
"""
Build generators for our ideal. ``Terms`` is an iterable with elements of
the form (fn, coeff), indicating that we have a generator fn(coeff*x).
If any of the terms is trigonometric, sin(x) and cos(x) are guaranteed
to appear in terms. Similarly for hyperbolic functions. For tan(n*x),
sin(n*x) and cos(n*x) are guaranteed.
"""
I = []
y = Dummy('y')
for fn, coeff in terms:
for c, s, t, rel in (
[cos, sin, tan, cos(x)**2 + sin(x)**2 - 1],
[cosh, sinh, tanh, cosh(x)**2 - sinh(x)**2 - 1]):
if coeff == 1 and fn in [c, s]:
I.append(rel)
elif fn == t:
I.append(t(coeff*x)*c(coeff*x) - s(coeff*x))
elif fn in [c, s]:
cn = fn(coeff*y).expand(trig=True).subs(y, x)
I.append(fn(coeff*x) - cn)
return list(set(I))
def analyse_gens(gens, hints):
"""
Analyse the generators ``gens``, using the hints ``hints``.
The meaning of ``hints`` is described in the main docstring.
Return a new list of generators, and also the ideal we should
work with.
"""
# First parse the hints
n, funcs, iterables, extragens = parse_hints(hints)
debug('n=%s' % n, 'funcs:', funcs, 'iterables:',
iterables, 'extragens:', extragens)
# We just add the extragens to gens and analyse them as before
gens = list(gens)
gens.extend(extragens)
# remove duplicates
funcs = list(set(funcs))
iterables = list(set(iterables))
gens = list(set(gens))
# all the functions we can do anything with
allfuncs = {sin, cos, tan, sinh, cosh, tanh}
# sin(3*x) -> ((3, x), sin)
trigterms = [(g.args[0].as_coeff_mul(), g.func) for g in gens
if g.func in allfuncs]
# Our list of new generators - start with anything that we cannot
# work with (i.e. is not a trigonometric term)
freegens = [g for g in gens if g.func not in allfuncs]
newgens = []
trigdict = {}
for (coeff, var), fn in trigterms:
trigdict.setdefault(var, []).append((coeff, fn))
res = [] # the ideal
for key, val in trigdict.items():
# We have now assembeled a dictionary. Its keys are common
# arguments in trigonometric expressions, and values are lists of
# pairs (fn, coeff). x0, (fn, coeff) in trigdict means that we
# need to deal with fn(coeff*x0). We take the rational gcd of the
# coeffs, call it ``gcd``. We then use x = x0/gcd as "base symbol",
# all other arguments are integral multiples thereof.
# We will build an ideal which works with sin(x), cos(x).
# If hint tan is provided, also work with tan(x). Moreover, if
# n > 1, also work with sin(k*x) for k <= n, and similarly for cos
# (and tan if the hint is provided). Finally, any generators which
# the ideal does not work with but we need to accommodate (either
# because it was in expr or because it was provided as a hint)
# we also build into the ideal.
# This selection process is expressed in the list ``terms``.
# build_ideal then generates the actual relations in our ideal,
# from this list.
fns = [x[1] for x in val]
val = [x[0] for x in val]
gcd = reduce(igcd, val)
terms = [(fn, v/gcd) for (fn, v) in zip(fns, val)]
fs = set(funcs + fns)
for c, s, t in ([cos, sin, tan], [cosh, sinh, tanh]):
if any(x in fs for x in (c, s, t)):
fs.add(c)
fs.add(s)
for fn in fs:
for k in range(1, n + 1):
terms.append((fn, k))
extra = []
for fn, v in terms:
if fn == tan:
extra.append((sin, v))
extra.append((cos, v))
if fn in [sin, cos] and tan in fs:
extra.append((tan, v))
if fn == tanh:
extra.append((sinh, v))
extra.append((cosh, v))
if fn in [sinh, cosh] and tanh in fs:
extra.append((tanh, v))
terms.extend(extra)
x = gcd*Mul(*key)
r = build_ideal(x, terms)
res.extend(r)
newgens.extend({fn(v*x) for fn, v in terms})
# Add generators for compound expressions from iterables
for fn, args in iterables:
if fn == tan:
# Tan expressions are recovered from sin and cos.
iterables.extend([(sin, args), (cos, args)])
elif fn == tanh:
# Tanh expressions are recovered from sihn and cosh.
iterables.extend([(sinh, args), (cosh, args)])
else:
dummys = symbols('d:%i' % len(args), cls=Dummy)
expr = fn( Add(*dummys)).expand(trig=True).subs(list(zip(dummys, args)))
res.append(fn(Add(*args)) - expr)
if myI in gens:
res.append(myI**2 + 1)
freegens.remove(myI)
newgens.append(myI)
return res, freegens, newgens
myI = Dummy('I')
expr = expr.subs(S.ImaginaryUnit, myI)
subs = [(myI, S.ImaginaryUnit)]
num, denom = cancel(expr).as_numer_denom()
try:
(pnum, pdenom), opt = parallel_poly_from_expr([num, denom])
except PolificationFailed:
return expr
debug('initial gens:', opt.gens)
ideal, freegens, gens = analyse_gens(opt.gens, hints)
debug('ideal:', ideal)
debug('new gens:', gens, " -- len", len(gens))
debug('free gens:', freegens, " -- len", len(gens))
# NOTE we force the domain to be ZZ to stop polys from injecting generators
# (which is usually a sign of a bug in the way we build the ideal)
if not gens:
return expr
G = groebner(ideal, order=order, gens=gens, domain=ZZ)
debug('groebner basis:', list(G), " -- len", len(G))
# If our fraction is a polynomial in the free generators, simplify all
# coefficients separately:
from sympy.simplify.ratsimp import ratsimpmodprime
if freegens and pdenom.has_only_gens(*set(gens).intersection(pdenom.gens)):
num = Poly(num, gens=gens+freegens).eject(*gens)
res = []
for monom, coeff in num.terms():
ourgens = set(parallel_poly_from_expr([coeff, denom])[1].gens)
# We compute the transitive closure of all generators that can
# be reached from our generators through relations in the ideal.
changed = True
while changed:
changed = False
for p in ideal:
p = Poly(p)
if not ourgens.issuperset(p.gens) and \
not p.has_only_gens(*set(p.gens).difference(ourgens)):
changed = True
ourgens.update(p.exclude().gens)
# NOTE preserve order!
realgens = [x for x in gens if x in ourgens]
# The generators of the ideal have now been (implicitly) split
# into two groups: those involving ourgens and those that don't.
# Since we took the transitive closure above, these two groups
# live in subgrings generated by a *disjoint* set of variables.
# Any sensible groebner basis algorithm will preserve this disjoint
# structure (i.e. the elements of the groebner basis can be split
# similarly), and and the two subsets of the groebner basis then
# form groebner bases by themselves. (For the smaller generating
# sets, of course.)
ourG = [g.as_expr() for g in G.polys if
g.has_only_gens(*ourgens.intersection(g.gens))]
res.append(Mul(*[a**b for a, b in zip(freegens, monom)]) * \
ratsimpmodprime(coeff/denom, ourG, order=order,
gens=realgens, quick=quick, domain=ZZ,
polynomial=polynomial).subs(subs))
return Add(*res)
# NOTE The following is simpler and has less assumptions on the
# groebner basis algorithm. If the above turns out to be broken,
# use this.
return Add(*[Mul(*[a**b for a, b in zip(freegens, monom)]) * \
ratsimpmodprime(coeff/denom, list(G), order=order,
gens=gens, quick=quick, domain=ZZ)
for monom, coeff in num.terms()])
else:
return ratsimpmodprime(
expr, list(G), order=order, gens=freegens+gens,
quick=quick, domain=ZZ, polynomial=polynomial).subs(subs)
_trigs = (TrigonometricFunction, HyperbolicFunction)
def trigsimp(expr, **opts):
"""
reduces expression by using known trig identities
Explanation
===========
method:
- Determine the method to use. Valid choices are 'matching' (default),
'groebner', 'combined', and 'fu'. If 'matching', simplify the
expression recursively by targeting common patterns. If 'groebner', apply
an experimental groebner basis algorithm. In this case further options
are forwarded to ``trigsimp_groebner``, please refer to its docstring.
If 'combined', first run the groebner basis algorithm with small
default parameters, then run the 'matching' algorithm. 'fu' runs the
collection of trigonometric transformations described by Fu, et al.
(see the `fu` docstring).
Examples
========
>>> from sympy import trigsimp, sin, cos, log
>>> from sympy.abc import x
>>> e = 2*sin(x)**2 + 2*cos(x)**2
>>> trigsimp(e)
2
Simplification occurs wherever trigonometric functions are located.
>>> trigsimp(log(e))
log(2)
Using `method="groebner"` (or `"combined"`) might lead to greater
simplification.
The old trigsimp routine can be accessed as with method 'old'.
>>> from sympy import coth, tanh
>>> t = 3*tanh(x)**7 - 2/coth(x)**7
>>> trigsimp(t, method='old') == t
True
>>> trigsimp(t)
tanh(x)**7
"""
from sympy.simplify.fu import fu
expr = sympify(expr)
_eval_trigsimp = getattr(expr, '_eval_trigsimp', None)
if _eval_trigsimp is not None:
return _eval_trigsimp(**opts)
old = opts.pop('old', False)
if not old:
opts.pop('deep', None)
opts.pop('recursive', None)
method = opts.pop('method', 'matching')
else:
method = 'old'
def groebnersimp(ex, **opts):
def traverse(e):
if e.is_Atom:
return e
args = [traverse(x) for x in e.args]
if e.is_Function or e.is_Pow:
args = [trigsimp_groebner(x, **opts) for x in args]
return e.func(*args)
new = traverse(ex)
if not isinstance(new, Expr):
return new
return trigsimp_groebner(new, **opts)
trigsimpfunc = {
'fu': (lambda x: fu(x, **opts)),
'matching': (lambda x: futrig(x)),
'groebner': (lambda x: groebnersimp(x, **opts)),
'combined': (lambda x: futrig(groebnersimp(x,
polynomial=True, hints=[2, tan]))),
'old': lambda x: trigsimp_old(x, **opts),
}[method]
return trigsimpfunc(expr)
def exptrigsimp(expr):
"""
Simplifies exponential / trigonometric / hyperbolic functions.
Examples
========
>>> from sympy import exptrigsimp, exp, cosh, sinh
>>> from sympy.abc import z
>>> exptrigsimp(exp(z) + exp(-z))
2*cosh(z)
>>> exptrigsimp(cosh(z) - sinh(z))
exp(-z)
"""
from sympy.simplify.fu import hyper_as_trig, TR2i
from sympy.simplify.simplify import bottom_up
def exp_trig(e):
# select the better of e, and e rewritten in terms of exp or trig
# functions
choices = [e]
if e.has(*_trigs):
choices.append(e.rewrite(exp))
choices.append(e.rewrite(cos))
return min(*choices, key=count_ops)
newexpr = bottom_up(expr, exp_trig)
def f(rv):
if not rv.is_Mul:
return rv
commutative_part, noncommutative_part = rv.args_cnc()
# Since as_powers_dict loses order information,
# if there is more than one noncommutative factor,
# it should only be used to simplify the commutative part.
if (len(noncommutative_part) > 1):
return f(Mul(*commutative_part))*Mul(*noncommutative_part)
rvd = rv.as_powers_dict()
newd = rvd.copy()
def signlog(expr, sign=1):
if expr is S.Exp1:
return sign, 1
elif isinstance(expr, exp) or (expr.is_Pow and expr.base == S.Exp1):
return sign, expr.exp
elif sign == 1:
return signlog(-expr, sign=-1)
else:
return None, None
ee = rvd[S.Exp1]
for k in rvd:
if k.is_Add and len(k.args) == 2:
# k == c*(1 + sign*E**x)
c = k.args[0]
sign, x = signlog(k.args[1]/c)
if not x:
continue
m = rvd[k]
newd[k] -= m
if ee == -x*m/2:
# sinh and cosh
newd[S.Exp1] -= ee
ee = 0
if sign == 1:
newd[2*c*cosh(x/2)] += m
else:
newd[-2*c*sinh(x/2)] += m
elif newd[1 - sign*S.Exp1**x] == -m:
# tanh
del newd[1 - sign*S.Exp1**x]
if sign == 1:
newd[-c/tanh(x/2)] += m
else:
newd[-c*tanh(x/2)] += m
else:
newd[1 + sign*S.Exp1**x] += m
newd[c] += m
return Mul(*[k**newd[k] for k in newd])
newexpr = bottom_up(newexpr, f)
# sin/cos and sinh/cosh ratios to tan and tanh, respectively
if newexpr.has(HyperbolicFunction):
e, f = hyper_as_trig(newexpr)
newexpr = f(TR2i(e))
if newexpr.has(TrigonometricFunction):
newexpr = TR2i(newexpr)
# can we ever generate an I where there was none previously?
if not (newexpr.has(I) and not expr.has(I)):
expr = newexpr
return expr
#-------------------- the old trigsimp routines ---------------------
def trigsimp_old(expr, *, first=True, **opts):
"""
Reduces expression by using known trig identities.
Notes
=====
deep:
- Apply trigsimp inside all objects with arguments
recursive:
- Use common subexpression elimination (cse()) and apply
trigsimp recursively (this is quite expensive if the
expression is large)
method:
- Determine the method to use. Valid choices are 'matching' (default),
'groebner', 'combined', 'fu' and 'futrig'. If 'matching', simplify the
expression recursively by pattern matching. If 'groebner', apply an
experimental groebner basis algorithm. In this case further options
are forwarded to ``trigsimp_groebner``, please refer to its docstring.
If 'combined', first run the groebner basis algorithm with small
default parameters, then run the 'matching' algorithm. 'fu' runs the
collection of trigonometric transformations described by Fu, et al.
(see the `fu` docstring) while `futrig` runs a subset of Fu-transforms
that mimic the behavior of `trigsimp`.
compare:
- show input and output from `trigsimp` and `futrig` when different,
but returns the `trigsimp` value.
Examples
========
>>> from sympy import trigsimp, sin, cos, log, cot
>>> from sympy.abc import x
>>> e = 2*sin(x)**2 + 2*cos(x)**2
>>> trigsimp(e, old=True)
2
>>> trigsimp(log(e), old=True)
log(2*sin(x)**2 + 2*cos(x)**2)
>>> trigsimp(log(e), deep=True, old=True)
log(2)
Using `method="groebner"` (or `"combined"`) can sometimes lead to a lot
more simplification:
>>> e = (-sin(x) + 1)/cos(x) + cos(x)/(-sin(x) + 1)
>>> trigsimp(e, old=True)
(1 - sin(x))/cos(x) + cos(x)/(1 - sin(x))
>>> trigsimp(e, method="groebner", old=True)
2/cos(x)
>>> trigsimp(1/cot(x)**2, compare=True, old=True)
futrig: tan(x)**2
cot(x)**(-2)
"""
old = expr
if first:
if not expr.has(*_trigs):
return expr
trigsyms = set().union(*[t.free_symbols for t in expr.atoms(*_trigs)])
if len(trigsyms) > 1:
from sympy.simplify.simplify import separatevars
d = separatevars(expr)
if d.is_Mul:
d = separatevars(d, dict=True) or d
if isinstance(d, dict):
expr = 1
for k, v in d.items():
# remove hollow factoring
was = v
v = expand_mul(v)
opts['first'] = False
vnew = trigsimp(v, **opts)
if vnew == v:
vnew = was
expr *= vnew
old = expr
else:
if d.is_Add:
for s in trigsyms:
r, e = expr.as_independent(s)
if r:
opts['first'] = False
expr = r + trigsimp(e, **opts)
if not expr.is_Add:
break
old = expr
recursive = opts.pop('recursive', False)
deep = opts.pop('deep', False)
method = opts.pop('method', 'matching')
def groebnersimp(ex, deep, **opts):
def traverse(e):
if e.is_Atom:
return e
args = [traverse(x) for x in e.args]
if e.is_Function or e.is_Pow:
args = [trigsimp_groebner(x, **opts) for x in args]
return e.func(*args)
if deep:
ex = traverse(ex)
return trigsimp_groebner(ex, **opts)
trigsimpfunc = {
'matching': (lambda x, d: _trigsimp(x, d)),
'groebner': (lambda x, d: groebnersimp(x, d, **opts)),
'combined': (lambda x, d: _trigsimp(groebnersimp(x,
d, polynomial=True, hints=[2, tan]),
d))
}[method]
if recursive:
w, g = cse(expr)
g = trigsimpfunc(g[0], deep)
for sub in reversed(w):
g = g.subs(sub[0], sub[1])
g = trigsimpfunc(g, deep)
result = g
else:
result = trigsimpfunc(expr, deep)
if opts.get('compare', False):
f = futrig(old)
if f != result:
print('\tfutrig:', f)
return result
def _dotrig(a, b):
"""Helper to tell whether ``a`` and ``b`` have the same sorts
of symbols in them -- no need to test hyperbolic patterns against
expressions that have no hyperbolics in them."""
return a.func == b.func and (
a.has(TrigonometricFunction) and b.has(TrigonometricFunction) or
a.has(HyperbolicFunction) and b.has(HyperbolicFunction))
_trigpat = None
def _trigpats():
global _trigpat
a, b, c = symbols('a b c', cls=Wild)
d = Wild('d', commutative=False)
# for the simplifications like sinh/cosh -> tanh:
# DO NOT REORDER THE FIRST 14 since these are assumed to be in this
# order in _match_div_rewrite.
matchers_division = (
(a*sin(b)**c/cos(b)**c, a*tan(b)**c, sin(b), cos(b)),
(a*tan(b)**c*cos(b)**c, a*sin(b)**c, sin(b), cos(b)),
(a*cot(b)**c*sin(b)**c, a*cos(b)**c, sin(b), cos(b)),
(a*tan(b)**c/sin(b)**c, a/cos(b)**c, sin(b), cos(b)),
(a*cot(b)**c/cos(b)**c, a/sin(b)**c, sin(b), cos(b)),
(a*cot(b)**c*tan(b)**c, a, sin(b), cos(b)),
(a*(cos(b) + 1)**c*(cos(b) - 1)**c,
a*(-sin(b)**2)**c, cos(b) + 1, cos(b) - 1),
(a*(sin(b) + 1)**c*(sin(b) - 1)**c,
a*(-cos(b)**2)**c, sin(b) + 1, sin(b) - 1),
(a*sinh(b)**c/cosh(b)**c, a*tanh(b)**c, S.One, S.One),
(a*tanh(b)**c*cosh(b)**c, a*sinh(b)**c, S.One, S.One),
(a*coth(b)**c*sinh(b)**c, a*cosh(b)**c, S.One, S.One),
(a*tanh(b)**c/sinh(b)**c, a/cosh(b)**c, S.One, S.One),
(a*coth(b)**c/cosh(b)**c, a/sinh(b)**c, S.One, S.One),
(a*coth(b)**c*tanh(b)**c, a, S.One, S.One),
(c*(tanh(a) + tanh(b))/(1 + tanh(a)*tanh(b)),
tanh(a + b)*c, S.One, S.One),
)
matchers_add = (
(c*sin(a)*cos(b) + c*cos(a)*sin(b) + d, sin(a + b)*c + d),
(c*cos(a)*cos(b) - c*sin(a)*sin(b) + d, cos(a + b)*c + d),
(c*sin(a)*cos(b) - c*cos(a)*sin(b) + d, sin(a - b)*c + d),
(c*cos(a)*cos(b) + c*sin(a)*sin(b) + d, cos(a - b)*c + d),
(c*sinh(a)*cosh(b) + c*sinh(b)*cosh(a) + d, sinh(a + b)*c + d),
(c*cosh(a)*cosh(b) + c*sinh(a)*sinh(b) + d, cosh(a + b)*c + d),
)
# for cos(x)**2 + sin(x)**2 -> 1
matchers_identity = (
(a*sin(b)**2, a - a*cos(b)**2),
(a*tan(b)**2, a*(1/cos(b))**2 - a),
(a*cot(b)**2, a*(1/sin(b))**2 - a),
(a*sin(b + c), a*(sin(b)*cos(c) + sin(c)*cos(b))),
(a*cos(b + c), a*(cos(b)*cos(c) - sin(b)*sin(c))),
(a*tan(b + c), a*((tan(b) + tan(c))/(1 - tan(b)*tan(c)))),
(a*sinh(b)**2, a*cosh(b)**2 - a),
(a*tanh(b)**2, a - a*(1/cosh(b))**2),
(a*coth(b)**2, a + a*(1/sinh(b))**2),
(a*sinh(b + c), a*(sinh(b)*cosh(c) + sinh(c)*cosh(b))),
(a*cosh(b + c), a*(cosh(b)*cosh(c) + sinh(b)*sinh(c))),
(a*tanh(b + c), a*((tanh(b) + tanh(c))/(1 + tanh(b)*tanh(c)))),
)
# Reduce any lingering artifacts, such as sin(x)**2 changing
# to 1-cos(x)**2 when sin(x)**2 was "simpler"
artifacts = (
(a - a*cos(b)**2 + c, a*sin(b)**2 + c, cos),
(a - a*(1/cos(b))**2 + c, -a*tan(b)**2 + c, cos),
(a - a*(1/sin(b))**2 + c, -a*cot(b)**2 + c, sin),
(a - a*cosh(b)**2 + c, -a*sinh(b)**2 + c, cosh),
(a - a*(1/cosh(b))**2 + c, a*tanh(b)**2 + c, cosh),
(a + a*(1/sinh(b))**2 + c, a*coth(b)**2 + c, sinh),
# same as above but with noncommutative prefactor
(a*d - a*d*cos(b)**2 + c, a*d*sin(b)**2 + c, cos),
(a*d - a*d*(1/cos(b))**2 + c, -a*d*tan(b)**2 + c, cos),
(a*d - a*d*(1/sin(b))**2 + c, -a*d*cot(b)**2 + c, sin),
(a*d - a*d*cosh(b)**2 + c, -a*d*sinh(b)**2 + c, cosh),
(a*d - a*d*(1/cosh(b))**2 + c, a*d*tanh(b)**2 + c, cosh),
(a*d + a*d*(1/sinh(b))**2 + c, a*d*coth(b)**2 + c, sinh),
)
_trigpat = (a, b, c, d, matchers_division, matchers_add,
matchers_identity, artifacts)
return _trigpat
def _replace_mul_fpowxgpow(expr, f, g, rexp, h, rexph):
"""Helper for _match_div_rewrite.
Replace f(b_)**c_*g(b_)**(rexp(c_)) with h(b)**rexph(c) if f(b_)
and g(b_) are both positive or if c_ is an integer.
"""
# assert expr.is_Mul and expr.is_commutative and f != g
fargs = defaultdict(int)
gargs = defaultdict(int)
args = []
for x in expr.args:
if x.is_Pow or x.func in (f, g):
b, e = x.as_base_exp()
if b.is_positive or e.is_integer:
if b.func == f:
fargs[b.args[0]] += e
continue
elif b.func == g:
gargs[b.args[0]] += e
continue
args.append(x)
common = set(fargs) & set(gargs)
hit = False
while common:
key = common.pop()
fe = fargs.pop(key)
ge = gargs.pop(key)
if fe == rexp(ge):
args.append(h(key)**rexph(fe))
hit = True
else:
fargs[key] = fe
gargs[key] = ge
if not hit:
return expr
while fargs:
key, e = fargs.popitem()
args.append(f(key)**e)
while gargs:
key, e = gargs.popitem()
args.append(g(key)**e)
return Mul(*args)
_idn = lambda x: x
_midn = lambda x: -x
_one = lambda x: S.One
def _match_div_rewrite(expr, i):
"""helper for __trigsimp"""
if i == 0:
expr = _replace_mul_fpowxgpow(expr, sin, cos,
_midn, tan, _idn)
elif i == 1:
expr = _replace_mul_fpowxgpow(expr, tan, cos,
_idn, sin, _idn)
elif i == 2:
expr = _replace_mul_fpowxgpow(expr, cot, sin,
_idn, cos, _idn)
elif i == 3:
expr = _replace_mul_fpowxgpow(expr, tan, sin,
_midn, cos, _midn)
elif i == 4:
expr = _replace_mul_fpowxgpow(expr, cot, cos,
_midn, sin, _midn)
elif i == 5:
expr = _replace_mul_fpowxgpow(expr, cot, tan,
_idn, _one, _idn)
# i in (6, 7) is skipped
elif i == 8:
expr = _replace_mul_fpowxgpow(expr, sinh, cosh,
_midn, tanh, _idn)
elif i == 9:
expr = _replace_mul_fpowxgpow(expr, tanh, cosh,
_idn, sinh, _idn)
elif i == 10:
expr = _replace_mul_fpowxgpow(expr, coth, sinh,
_idn, cosh, _idn)
elif i == 11:
expr = _replace_mul_fpowxgpow(expr, tanh, sinh,
_midn, cosh, _midn)
elif i == 12:
expr = _replace_mul_fpowxgpow(expr, coth, cosh,
_midn, sinh, _midn)
elif i == 13:
expr = _replace_mul_fpowxgpow(expr, coth, tanh,
_idn, _one, _idn)
else:
return None
return expr
def _trigsimp(expr, deep=False):
# protect the cache from non-trig patterns; we only allow
# trig patterns to enter the cache
if expr.has(*_trigs):
return __trigsimp(expr, deep)
return expr
@cacheit
def __trigsimp(expr, deep=False):
"""recursive helper for trigsimp"""
from sympy.simplify.fu import TR10i
if _trigpat is None:
_trigpats()
a, b, c, d, matchers_division, matchers_add, \
matchers_identity, artifacts = _trigpat
if expr.is_Mul:
# do some simplifications like sin/cos -> tan:
if not expr.is_commutative:
com, nc = expr.args_cnc()
expr = _trigsimp(Mul._from_args(com), deep)*Mul._from_args(nc)
else:
for i, (pattern, simp, ok1, ok2) in enumerate(matchers_division):
if not _dotrig(expr, pattern):
continue
newexpr = _match_div_rewrite(expr, i)
if newexpr is not None:
if newexpr != expr:
expr = newexpr
break
else:
continue
# use SymPy matching instead
res = expr.match(pattern)
if res and res.get(c, 0):
if not res[c].is_integer:
ok = ok1.subs(res)
if not ok.is_positive:
continue
ok = ok2.subs(res)
if not ok.is_positive:
continue
# if "a" contains any of trig or hyperbolic funcs with
# argument "b" then skip the simplification
if any(w.args[0] == res[b] for w in res[a].atoms(
TrigonometricFunction, HyperbolicFunction)):
continue
# simplify and finish:
expr = simp.subs(res)
break # process below
if expr.is_Add:
args = []
for term in expr.args:
if not term.is_commutative:
com, nc = term.args_cnc()
nc = Mul._from_args(nc)
term = Mul._from_args(com)
else:
nc = S.One
term = _trigsimp(term, deep)
for pattern, result in matchers_identity:
res = term.match(pattern)
if res is not None:
term = result.subs(res)
break
args.append(term*nc)
if args != expr.args:
expr = Add(*args)
expr = min(expr, expand(expr), key=count_ops)
if expr.is_Add:
for pattern, result in matchers_add:
if not _dotrig(expr, pattern):
continue
expr = TR10i(expr)
if expr.has(HyperbolicFunction):
res = expr.match(pattern)
# if "d" contains any trig or hyperbolic funcs with
# argument "a" or "b" then skip the simplification;
# this isn't perfect -- see tests
if res is None or not (a in res and b in res) or any(
w.args[0] in (res[a], res[b]) for w in res[d].atoms(
TrigonometricFunction, HyperbolicFunction)):
continue
expr = result.subs(res)
break
# Reduce any lingering artifacts, such as sin(x)**2 changing
# to 1 - cos(x)**2 when sin(x)**2 was "simpler"
for pattern, result, ex in artifacts:
if not _dotrig(expr, pattern):
continue
# Substitute a new wild that excludes some function(s)
# to help influence a better match. This is because
# sometimes, for example, 'a' would match sec(x)**2
a_t = Wild('a', exclude=[ex])
pattern = pattern.subs(a, a_t)
result = result.subs(a, a_t)
m = expr.match(pattern)
was = None
while m and was != expr:
was = expr
if m[a_t] == 0 or \
-m[a_t] in m[c].args or m[a_t] + m[c] == 0:
break
if d in m and m[a_t]*m[d] + m[c] == 0:
break
expr = result.subs(m)
m = expr.match(pattern)
m.setdefault(c, S.Zero)
elif expr.is_Mul or expr.is_Pow or deep and expr.args:
expr = expr.func(*[_trigsimp(a, deep) for a in expr.args])
try:
if not expr.has(*_trigs):
raise TypeError
e = expr.atoms(exp)
new = expr.rewrite(exp, deep=deep)
if new == e:
raise TypeError
fnew = factor(new)
if fnew != new:
new = sorted([new, factor(new)], key=count_ops)[0]
# if all exp that were introduced disappeared then accept it
if not (new.atoms(exp) - e):
expr = new
except TypeError:
pass
return expr
#------------------- end of old trigsimp routines --------------------
def futrig(e, *, hyper=True, **kwargs):
"""Return simplified ``e`` using Fu-like transformations.
This is not the "Fu" algorithm. This is called by default
from ``trigsimp``. By default, hyperbolics subexpressions
will be simplified, but this can be disabled by setting
``hyper=False``.
Examples
========
>>> from sympy import trigsimp, tan, sinh, tanh
>>> from sympy.simplify.trigsimp import futrig
>>> from sympy.abc import x
>>> trigsimp(1/tan(x)**2)
tan(x)**(-2)
>>> futrig(sinh(x)/tanh(x))
cosh(x)
"""
from sympy.simplify.fu import hyper_as_trig
from sympy.simplify.simplify import bottom_up
e = sympify(e)
if not isinstance(e, Basic):
return e
if not e.args:
return e
old = e
e = bottom_up(e, _futrig)
if hyper and e.has(HyperbolicFunction):
e, f = hyper_as_trig(e)
e = f(bottom_up(e, _futrig))
if e != old and e.is_Mul and e.args[0].is_Rational:
# redistribute leading coeff on 2-arg Add
e = Mul(*e.as_coeff_Mul())
return e
def _futrig(e):
"""Helper for futrig."""
from sympy.simplify.fu import (
TR1, TR2, TR3, TR2i, TR10, L, TR10i,
TR8, TR6, TR15, TR16, TR111, TR5, TRmorrie, TR11, _TR11, TR14, TR22,
TR12)
from sympy.core.compatibility import _nodes
if not e.has(TrigonometricFunction):
return e
if e.is_Mul:
coeff, e = e.as_independent(TrigonometricFunction)
else:
coeff = None
Lops = lambda x: (L(x), x.count_ops(), _nodes(x), len(x.args), x.is_Add)
trigs = lambda x: x.has(TrigonometricFunction)
tree = [identity,
(
TR3, # canonical angles
TR1, # sec-csc -> cos-sin
TR12, # expand tan of sum
lambda x: _eapply(factor, x, trigs),
TR2, # tan-cot -> sin-cos
[identity, lambda x: _eapply(_mexpand, x, trigs)],
TR2i, # sin-cos ratio -> tan
lambda x: _eapply(lambda i: factor(i.normal()), x, trigs),
TR14, # factored identities
TR5, # sin-pow -> cos_pow
TR10, # sin-cos of sums -> sin-cos prod
TR11, _TR11, TR6, # reduce double angles and rewrite cos pows
lambda x: _eapply(factor, x, trigs),
TR14, # factored powers of identities
[identity, lambda x: _eapply(_mexpand, x, trigs)],
TR10i, # sin-cos products > sin-cos of sums
TRmorrie,
[identity, TR8], # sin-cos products -> sin-cos of sums
[identity, lambda x: TR2i(TR2(x))], # tan -> sin-cos -> tan
[
lambda x: _eapply(expand_mul, TR5(x), trigs),
lambda x: _eapply(
expand_mul, TR15(x), trigs)], # pos/neg powers of sin
[
lambda x: _eapply(expand_mul, TR6(x), trigs),
lambda x: _eapply(
expand_mul, TR16(x), trigs)], # pos/neg powers of cos
TR111, # tan, sin, cos to neg power -> cot, csc, sec
[identity, TR2i], # sin-cos ratio to tan
[identity, lambda x: _eapply(
expand_mul, TR22(x), trigs)], # tan-cot to sec-csc
TR1, TR2, TR2i,
[identity, lambda x: _eapply(
factor_terms, TR12(x), trigs)], # expand tan of sum
)]
e = greedy(tree, objective=Lops)(e)
if coeff is not None:
e = coeff * e
return e
def _is_Expr(e):
"""_eapply helper to tell whether ``e`` and all its args
are Exprs."""
from sympy import Derivative
if isinstance(e, Derivative):
return _is_Expr(e.expr)
if not isinstance(e, Expr):
return False
return all(_is_Expr(i) for i in e.args)
def _eapply(func, e, cond=None):
"""Apply ``func`` to ``e`` if all args are Exprs else only
apply it to those args that *are* Exprs."""
if not isinstance(e, Expr):
return e
if _is_Expr(e) or not e.args:
return func(e)
return e.func(*[
_eapply(func, ei) if (cond is None or cond(ei)) else ei
for ei in e.args])
|
cb24db7b8dccf3d4ec445abfe79aa810d0af406ddaa39d3d771d15d84f42de44 | from collections import defaultdict
from sympy.core.add import Add
from sympy.core.basic import S
from sympy.core.compatibility import ordered
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.symbol import Dummy
from sympy.core.sympify import sympify
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.simplify.simplify import bottom_up
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)
(cos(x) + 1)**(-a)*sin(x)**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[rv.func](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
----------------------------------------------------
cos(a) 0 1/2 sqrt(2)/2 sqrt(3)/2 1
sin(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[a.func].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[a.base.func].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 not rv.func 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 not rv.func 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 not t.func 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[b.func].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))
"""
from sympy import factor
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[a.func].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)*(-1)**((n-1)/2)*Add(*[binomial(n, k)*
(-1)**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)*(-1)**(n/2)*Add(*[binomial(n, k)*
(-1)**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)))
|
0f9c5796687500bc63f52c8275d51d8266a59e2bca0742b20900b3619c9e1e1d | from sympy.core import S, Basic, Dict, Symbol, Tuple, sympify
from sympy.core.compatibility import iterable
from sympy.core.symbol import Str
from sympy.sets import Set, FiniteSet, EmptySet
class Class(Set):
r"""
The base class for any kind of class in the set-theoretic sense.
Explanation
===========
In axiomatic set theories, everything is a class. A class which
can be a member of another class is a set. A class which is not a
member of another class is a proper class. The class `\{1, 2\}`
is a set; the class of all sets is a proper class.
This class is essentially a synonym for :class:`sympy.core.Set`.
The goal of this class is to assure easier migration to the
eventual proper implementation of set theory.
"""
is_proper = False
class Object(Symbol):
"""
The base class for any kind of object in an abstract category.
Explanation
===========
While technically any instance of :class:`~.Basic` will do, this
class is the recommended way to create abstract objects in
abstract categories.
"""
class Morphism(Basic):
"""
The base class for any morphism in an abstract category.
Explanation
===========
In abstract categories, a morphism is an arrow between two
category objects. The object where the arrow starts is called the
domain, while the object where the arrow ends is called the
codomain.
Two morphisms between the same pair of objects are considered to
be the same morphisms. To distinguish between morphisms between
the same objects use :class:`NamedMorphism`.
It is prohibited to instantiate this class. Use one of the
derived classes instead.
See Also
========
IdentityMorphism, NamedMorphism, CompositeMorphism
"""
def __new__(cls, domain, codomain):
raise(NotImplementedError(
"Cannot instantiate Morphism. Use derived classes instead."))
@property
def domain(self):
"""
Returns the domain of the morphism.
Examples
========
>>> from sympy.categories import Object, NamedMorphism
>>> A = Object("A")
>>> B = Object("B")
>>> f = NamedMorphism(A, B, "f")
>>> f.domain
Object("A")
"""
return self.args[0]
@property
def codomain(self):
"""
Returns the codomain of the morphism.
Examples
========
>>> from sympy.categories import Object, NamedMorphism
>>> A = Object("A")
>>> B = Object("B")
>>> f = NamedMorphism(A, B, "f")
>>> f.codomain
Object("B")
"""
return self.args[1]
def compose(self, other):
r"""
Composes self with the supplied morphism.
The order of elements in the composition is the usual order,
i.e., to construct `g\circ f` use ``g.compose(f)``.
Examples
========
>>> from sympy.categories import Object, NamedMorphism
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = NamedMorphism(A, B, "f")
>>> g = NamedMorphism(B, C, "g")
>>> g * f
CompositeMorphism((NamedMorphism(Object("A"), Object("B"), "f"),
NamedMorphism(Object("B"), Object("C"), "g")))
>>> (g * f).domain
Object("A")
>>> (g * f).codomain
Object("C")
"""
return CompositeMorphism(other, self)
def __mul__(self, other):
r"""
Composes self with the supplied morphism.
The semantics of this operation is given by the following
equation: ``g * f == g.compose(f)`` for composable morphisms
``g`` and ``f``.
See Also
========
compose
"""
return self.compose(other)
class IdentityMorphism(Morphism):
"""
Represents an identity morphism.
Explanation
===========
An identity morphism is a morphism with equal domain and codomain,
which acts as an identity with respect to composition.
Examples
========
>>> from sympy.categories import Object, NamedMorphism, IdentityMorphism
>>> A = Object("A")
>>> B = Object("B")
>>> f = NamedMorphism(A, B, "f")
>>> id_A = IdentityMorphism(A)
>>> id_B = IdentityMorphism(B)
>>> f * id_A == f
True
>>> id_B * f == f
True
See Also
========
Morphism
"""
def __new__(cls, domain):
return Basic.__new__(cls, domain)
@property
def codomain(self):
return self.domain
class NamedMorphism(Morphism):
"""
Represents a morphism which has a name.
Explanation
===========
Names are used to distinguish between morphisms which have the
same domain and codomain: two named morphisms are equal if they
have the same domains, codomains, and names.
Examples
========
>>> from sympy.categories import Object, NamedMorphism
>>> A = Object("A")
>>> B = Object("B")
>>> f = NamedMorphism(A, B, "f")
>>> f
NamedMorphism(Object("A"), Object("B"), "f")
>>> f.name
'f'
See Also
========
Morphism
"""
def __new__(cls, domain, codomain, name):
if not name:
raise ValueError("Empty morphism names not allowed.")
if not isinstance(name, Str):
name = Str(name)
return Basic.__new__(cls, domain, codomain, name)
@property
def name(self):
"""
Returns the name of the morphism.
Examples
========
>>> from sympy.categories import Object, NamedMorphism
>>> A = Object("A")
>>> B = Object("B")
>>> f = NamedMorphism(A, B, "f")
>>> f.name
'f'
"""
return self.args[2].name
class CompositeMorphism(Morphism):
r"""
Represents a morphism which is a composition of other morphisms.
Explanation
===========
Two composite morphisms are equal if the morphisms they were
obtained from (components) are the same and were listed in the
same order.
The arguments to the constructor for this class should be listed
in diagram order: to obtain the composition `g\circ f` from the
instances of :class:`Morphism` ``g`` and ``f`` use
``CompositeMorphism(f, g)``.
Examples
========
>>> from sympy.categories import Object, NamedMorphism, CompositeMorphism
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = NamedMorphism(A, B, "f")
>>> g = NamedMorphism(B, C, "g")
>>> g * f
CompositeMorphism((NamedMorphism(Object("A"), Object("B"), "f"),
NamedMorphism(Object("B"), Object("C"), "g")))
>>> CompositeMorphism(f, g) == g * f
True
"""
@staticmethod
def _add_morphism(t, morphism):
"""
Intelligently adds ``morphism`` to tuple ``t``.
Explanation
===========
If ``morphism`` is a composite morphism, its components are
added to the tuple. If ``morphism`` is an identity, nothing
is added to the tuple.
No composability checks are performed.
"""
if isinstance(morphism, CompositeMorphism):
# ``morphism`` is a composite morphism; we have to
# denest its components.
return t + morphism.components
elif isinstance(morphism, IdentityMorphism):
# ``morphism`` is an identity. Nothing happens.
return t
else:
return t + Tuple(morphism)
def __new__(cls, *components):
if components and not isinstance(components[0], Morphism):
# Maybe the user has explicitly supplied a list of
# morphisms.
return CompositeMorphism.__new__(cls, *components[0])
normalised_components = Tuple()
for current, following in zip(components, components[1:]):
if not isinstance(current, Morphism) or \
not isinstance(following, Morphism):
raise TypeError("All components must be morphisms.")
if current.codomain != following.domain:
raise ValueError("Uncomposable morphisms.")
normalised_components = CompositeMorphism._add_morphism(
normalised_components, current)
# We haven't added the last morphism to the list of normalised
# components. Add it now.
normalised_components = CompositeMorphism._add_morphism(
normalised_components, components[-1])
if not normalised_components:
# If ``normalised_components`` is empty, only identities
# were supplied. Since they all were composable, they are
# all the same identities.
return components[0]
elif len(normalised_components) == 1:
# No sense to construct a whole CompositeMorphism.
return normalised_components[0]
return Basic.__new__(cls, normalised_components)
@property
def components(self):
"""
Returns the components of this composite morphism.
Examples
========
>>> from sympy.categories import Object, NamedMorphism
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = NamedMorphism(A, B, "f")
>>> g = NamedMorphism(B, C, "g")
>>> (g * f).components
(NamedMorphism(Object("A"), Object("B"), "f"),
NamedMorphism(Object("B"), Object("C"), "g"))
"""
return self.args[0]
@property
def domain(self):
"""
Returns the domain of this composite morphism.
The domain of the composite morphism is the domain of its
first component.
Examples
========
>>> from sympy.categories import Object, NamedMorphism
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = NamedMorphism(A, B, "f")
>>> g = NamedMorphism(B, C, "g")
>>> (g * f).domain
Object("A")
"""
return self.components[0].domain
@property
def codomain(self):
"""
Returns the codomain of this composite morphism.
The codomain of the composite morphism is the codomain of its
last component.
Examples
========
>>> from sympy.categories import Object, NamedMorphism
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = NamedMorphism(A, B, "f")
>>> g = NamedMorphism(B, C, "g")
>>> (g * f).codomain
Object("C")
"""
return self.components[-1].codomain
def flatten(self, new_name):
"""
Forgets the composite structure of this morphism.
Explanation
===========
If ``new_name`` is not empty, returns a :class:`NamedMorphism`
with the supplied name, otherwise returns a :class:`Morphism`.
In both cases the domain of the new morphism is the domain of
this composite morphism and the codomain of the new morphism
is the codomain of this composite morphism.
Examples
========
>>> from sympy.categories import Object, NamedMorphism
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = NamedMorphism(A, B, "f")
>>> g = NamedMorphism(B, C, "g")
>>> (g * f).flatten("h")
NamedMorphism(Object("A"), Object("C"), "h")
"""
return NamedMorphism(self.domain, self.codomain, new_name)
class Category(Basic):
r"""
An (abstract) category.
Explanation
===========
A category [JoyOfCats] is a quadruple `\mbox{K} = (O, \hom, id,
\circ)` consisting of
* a (set-theoretical) class `O`, whose members are called
`K`-objects,
* for each pair `(A, B)` of `K`-objects, a set `\hom(A, B)` whose
members are called `K`-morphisms from `A` to `B`,
* for a each `K`-object `A`, a morphism `id:A\rightarrow A`,
called the `K`-identity of `A`,
* a composition law `\circ` associating with every `K`-morphisms
`f:A\rightarrow B` and `g:B\rightarrow C` a `K`-morphism `g\circ
f:A\rightarrow C`, called the composite of `f` and `g`.
Composition is associative, `K`-identities are identities with
respect to composition, and the sets `\hom(A, B)` are pairwise
disjoint.
This class knows nothing about its objects and morphisms.
Concrete cases of (abstract) categories should be implemented as
classes derived from this one.
Certain instances of :class:`Diagram` can be asserted to be
commutative in a :class:`Category` by supplying the argument
``commutative_diagrams`` in the constructor.
Examples
========
>>> from sympy.categories import Object, NamedMorphism, Diagram, Category
>>> from sympy import FiniteSet
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = NamedMorphism(A, B, "f")
>>> g = NamedMorphism(B, C, "g")
>>> d = Diagram([f, g])
>>> K = Category("K", commutative_diagrams=[d])
>>> K.commutative_diagrams == FiniteSet(d)
True
See Also
========
Diagram
"""
def __new__(cls, name, objects=EmptySet, commutative_diagrams=EmptySet):
if not name:
raise ValueError("A Category cannot have an empty name.")
if not isinstance(name, Str):
name = Str(name)
if not isinstance(objects, Class):
objects = Class(objects)
new_category = Basic.__new__(cls, name, objects,
FiniteSet(*commutative_diagrams))
return new_category
@property
def name(self):
"""
Returns the name of this category.
Examples
========
>>> from sympy.categories import Category
>>> K = Category("K")
>>> K.name
'K'
"""
return self.args[0].name
@property
def objects(self):
"""
Returns the class of objects of this category.
Examples
========
>>> from sympy.categories import Object, Category
>>> from sympy import FiniteSet
>>> A = Object("A")
>>> B = Object("B")
>>> K = Category("K", FiniteSet(A, B))
>>> K.objects
Class(FiniteSet(Object("A"), Object("B")))
"""
return self.args[1]
@property
def commutative_diagrams(self):
"""
Returns the :class:`~.FiniteSet` of diagrams which are known to
be commutative in this category.
Examples
========
>>> from sympy.categories import Object, NamedMorphism, Diagram, Category
>>> from sympy import FiniteSet
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = NamedMorphism(A, B, "f")
>>> g = NamedMorphism(B, C, "g")
>>> d = Diagram([f, g])
>>> K = Category("K", commutative_diagrams=[d])
>>> K.commutative_diagrams == FiniteSet(d)
True
"""
return self.args[2]
def hom(self, A, B):
raise NotImplementedError(
"hom-sets are not implemented in Category.")
def all_morphisms(self):
raise NotImplementedError(
"Obtaining the class of morphisms is not implemented in Category.")
class Diagram(Basic):
r"""
Represents a diagram in a certain category.
Explanation
===========
Informally, a diagram is a collection of objects of a category and
certain morphisms between them. A diagram is still a monoid with
respect to morphism composition; i.e., identity morphisms, as well
as all composites of morphisms included in the diagram belong to
the diagram. For a more formal approach to this notion see
[Pare1970].
The components of composite morphisms are also added to the
diagram. No properties are assigned to such morphisms by default.
A commutative diagram is often accompanied by a statement of the
following kind: "if such morphisms with such properties exist,
then such morphisms which such properties exist and the diagram is
commutative". To represent this, an instance of :class:`Diagram`
includes a collection of morphisms which are the premises and
another collection of conclusions. ``premises`` and
``conclusions`` associate morphisms belonging to the corresponding
categories with the :class:`~.FiniteSet`'s of their properties.
The set of properties of a composite morphism is the intersection
of the sets of properties of its components. The domain and
codomain of a conclusion morphism should be among the domains and
codomains of the morphisms listed as the premises of a diagram.
No checks are carried out of whether the supplied object and
morphisms do belong to one and the same category.
Examples
========
>>> from sympy.categories import Object, NamedMorphism, Diagram
>>> from sympy import pprint, default_sort_key
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = NamedMorphism(A, B, "f")
>>> g = NamedMorphism(B, C, "g")
>>> d = Diagram([f, g])
>>> premises_keys = sorted(d.premises.keys(), key=default_sort_key)
>>> pprint(premises_keys, use_unicode=False)
[g*f:A-->C, id:A-->A, id:B-->B, id:C-->C, f:A-->B, g:B-->C]
>>> pprint(d.premises, use_unicode=False)
{g*f:A-->C: EmptySet, id:A-->A: EmptySet, id:B-->B: EmptySet, id:C-->C: EmptyS
et, f:A-->B: EmptySet, g:B-->C: EmptySet}
>>> d = Diagram([f, g], {g * f: "unique"})
>>> pprint(d.conclusions,use_unicode=False)
{g*f:A-->C: {unique}}
References
==========
[Pare1970] B. Pareigis: Categories and functors. Academic Press,
1970.
"""
@staticmethod
def _set_dict_union(dictionary, key, value):
"""
If ``key`` is in ``dictionary``, set the new value of ``key``
to be the union between the old value and ``value``.
Otherwise, set the value of ``key`` to ``value.
Returns ``True`` if the key already was in the dictionary and
``False`` otherwise.
"""
if key in dictionary:
dictionary[key] = dictionary[key] | value
return True
else:
dictionary[key] = value
return False
@staticmethod
def _add_morphism_closure(morphisms, morphism, props, add_identities=True,
recurse_composites=True):
"""
Adds a morphism and its attributes to the supplied dictionary
``morphisms``. If ``add_identities`` is True, also adds the
identity morphisms for the domain and the codomain of
``morphism``.
"""
if not Diagram._set_dict_union(morphisms, morphism, props):
# We have just added a new morphism.
if isinstance(morphism, IdentityMorphism):
if props:
# Properties for identity morphisms don't really
# make sense, because very much is known about
# identity morphisms already, so much that they
# are trivial. Having properties for identity
# morphisms would only be confusing.
raise ValueError(
"Instances of IdentityMorphism cannot have properties.")
return
if add_identities:
empty = EmptySet
id_dom = IdentityMorphism(morphism.domain)
id_cod = IdentityMorphism(morphism.codomain)
Diagram._set_dict_union(morphisms, id_dom, empty)
Diagram._set_dict_union(morphisms, id_cod, empty)
for existing_morphism, existing_props in list(morphisms.items()):
new_props = existing_props & props
if morphism.domain == existing_morphism.codomain:
left = morphism * existing_morphism
Diagram._set_dict_union(morphisms, left, new_props)
if morphism.codomain == existing_morphism.domain:
right = existing_morphism * morphism
Diagram._set_dict_union(morphisms, right, new_props)
if isinstance(morphism, CompositeMorphism) and recurse_composites:
# This is a composite morphism, add its components as
# well.
empty = EmptySet
for component in morphism.components:
Diagram._add_morphism_closure(morphisms, component, empty,
add_identities)
def __new__(cls, *args):
"""
Construct a new instance of Diagram.
Explanation
===========
If no arguments are supplied, an empty diagram is created.
If at least an argument is supplied, ``args[0]`` is
interpreted as the premises of the diagram. If ``args[0]`` is
a list, it is interpreted as a list of :class:`Morphism`'s, in
which each :class:`Morphism` has an empty set of properties.
If ``args[0]`` is a Python dictionary or a :class:`Dict`, it
is interpreted as a dictionary associating to some
:class:`Morphism`'s some properties.
If at least two arguments are supplied ``args[1]`` is
interpreted as the conclusions of the diagram. The type of
``args[1]`` is interpreted in exactly the same way as the type
of ``args[0]``. If only one argument is supplied, the diagram
has no conclusions.
Examples
========
>>> from sympy.categories import Object, NamedMorphism
>>> from sympy.categories import IdentityMorphism, Diagram
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = NamedMorphism(A, B, "f")
>>> g = NamedMorphism(B, C, "g")
>>> d = Diagram([f, g])
>>> IdentityMorphism(A) in d.premises.keys()
True
>>> g * f in d.premises.keys()
True
>>> d = Diagram([f, g], {g * f: "unique"})
>>> d.conclusions[g * f]
FiniteSet(unique)
"""
premises = {}
conclusions = {}
# Here we will keep track of the objects which appear in the
# premises.
objects = EmptySet
if len(args) >= 1:
# We've got some premises in the arguments.
premises_arg = args[0]
if isinstance(premises_arg, list):
# The user has supplied a list of morphisms, none of
# which have any attributes.
empty = EmptySet
for morphism in premises_arg:
objects |= FiniteSet(morphism.domain, morphism.codomain)
Diagram._add_morphism_closure(premises, morphism, empty)
elif isinstance(premises_arg, dict) or isinstance(premises_arg, Dict):
# The user has supplied a dictionary of morphisms and
# their properties.
for morphism, props in premises_arg.items():
objects |= FiniteSet(morphism.domain, morphism.codomain)
Diagram._add_morphism_closure(
premises, morphism, FiniteSet(*props) if iterable(props) else FiniteSet(props))
if len(args) >= 2:
# We also have some conclusions.
conclusions_arg = args[1]
if isinstance(conclusions_arg, list):
# The user has supplied a list of morphisms, none of
# which have any attributes.
empty = EmptySet
for morphism in conclusions_arg:
# Check that no new objects appear in conclusions.
if ((sympify(objects.contains(morphism.domain)) is S.true) and
(sympify(objects.contains(morphism.codomain)) is S.true)):
# No need to add identities and recurse
# composites this time.
Diagram._add_morphism_closure(
conclusions, morphism, empty, add_identities=False,
recurse_composites=False)
elif isinstance(conclusions_arg, dict) or \
isinstance(conclusions_arg, Dict):
# The user has supplied a dictionary of morphisms and
# their properties.
for morphism, props in conclusions_arg.items():
# Check that no new objects appear in conclusions.
if (morphism.domain in objects) and \
(morphism.codomain in objects):
# No need to add identities and recurse
# composites this time.
Diagram._add_morphism_closure(
conclusions, morphism, FiniteSet(*props) if iterable(props) else FiniteSet(props),
add_identities=False, recurse_composites=False)
return Basic.__new__(cls, Dict(premises), Dict(conclusions), objects)
@property
def premises(self):
"""
Returns the premises of this diagram.
Examples
========
>>> from sympy.categories import Object, NamedMorphism
>>> from sympy.categories import IdentityMorphism, Diagram
>>> from sympy import pretty
>>> A = Object("A")
>>> B = Object("B")
>>> f = NamedMorphism(A, B, "f")
>>> id_A = IdentityMorphism(A)
>>> id_B = IdentityMorphism(B)
>>> d = Diagram([f])
>>> print(pretty(d.premises, use_unicode=False))
{id:A-->A: EmptySet, id:B-->B: EmptySet, f:A-->B: EmptySet}
"""
return self.args[0]
@property
def conclusions(self):
"""
Returns the conclusions of this diagram.
Examples
========
>>> from sympy.categories import Object, NamedMorphism
>>> from sympy.categories import IdentityMorphism, Diagram
>>> from sympy import FiniteSet
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = NamedMorphism(A, B, "f")
>>> g = NamedMorphism(B, C, "g")
>>> d = Diagram([f, g])
>>> IdentityMorphism(A) in d.premises.keys()
True
>>> g * f in d.premises.keys()
True
>>> d = Diagram([f, g], {g * f: "unique"})
>>> d.conclusions[g * f] == FiniteSet("unique")
True
"""
return self.args[1]
@property
def objects(self):
"""
Returns the :class:`~.FiniteSet` of objects that appear in this
diagram.
Examples
========
>>> from sympy.categories import Object, NamedMorphism, Diagram
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = NamedMorphism(A, B, "f")
>>> g = NamedMorphism(B, C, "g")
>>> d = Diagram([f, g])
>>> d.objects
FiniteSet(Object("A"), Object("B"), Object("C"))
"""
return self.args[2]
def hom(self, A, B):
"""
Returns a 2-tuple of sets of morphisms between objects ``A`` and
``B``: one set of morphisms listed as premises, and the other set
of morphisms listed as conclusions.
Examples
========
>>> from sympy.categories import Object, NamedMorphism, Diagram
>>> from sympy import pretty
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = NamedMorphism(A, B, "f")
>>> g = NamedMorphism(B, C, "g")
>>> d = Diagram([f, g], {g * f: "unique"})
>>> print(pretty(d.hom(A, C), use_unicode=False))
({g*f:A-->C}, {g*f:A-->C})
See Also
========
Object, Morphism
"""
premises = EmptySet
conclusions = EmptySet
for morphism in self.premises.keys():
if (morphism.domain == A) and (morphism.codomain == B):
premises |= FiniteSet(morphism)
for morphism in self.conclusions.keys():
if (morphism.domain == A) and (morphism.codomain == B):
conclusions |= FiniteSet(morphism)
return (premises, conclusions)
def is_subdiagram(self, diagram):
"""
Checks whether ``diagram`` is a subdiagram of ``self``.
Diagram `D'` is a subdiagram of `D` if all premises
(conclusions) of `D'` are contained in the premises
(conclusions) of `D`. The morphisms contained
both in `D'` and `D` should have the same properties for `D'`
to be a subdiagram of `D`.
Examples
========
>>> from sympy.categories import Object, NamedMorphism, Diagram
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = NamedMorphism(A, B, "f")
>>> g = NamedMorphism(B, C, "g")
>>> d = Diagram([f, g], {g * f: "unique"})
>>> d1 = Diagram([f])
>>> d.is_subdiagram(d1)
True
>>> d1.is_subdiagram(d)
False
"""
premises = all([(m in self.premises) and
(diagram.premises[m] == self.premises[m])
for m in diagram.premises])
if not premises:
return False
conclusions = all([(m in self.conclusions) and
(diagram.conclusions[m] == self.conclusions[m])
for m in diagram.conclusions])
# Premises is surely ``True`` here.
return conclusions
def subdiagram_from_objects(self, objects):
"""
If ``objects`` is a subset of the objects of ``self``, returns
a diagram which has as premises all those premises of ``self``
which have a domains and codomains in ``objects``, likewise
for conclusions. Properties are preserved.
Examples
========
>>> from sympy.categories import Object, NamedMorphism, Diagram
>>> from sympy import FiniteSet
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = NamedMorphism(A, B, "f")
>>> g = NamedMorphism(B, C, "g")
>>> d = Diagram([f, g], {f: "unique", g*f: "veryunique"})
>>> d1 = d.subdiagram_from_objects(FiniteSet(A, B))
>>> d1 == Diagram([f], {f: "unique"})
True
"""
if not objects.is_subset(self.objects):
raise ValueError(
"Supplied objects should all belong to the diagram.")
new_premises = {}
for morphism, props in self.premises.items():
if ((sympify(objects.contains(morphism.domain)) is S.true) and
(sympify(objects.contains(morphism.codomain)) is S.true)):
new_premises[morphism] = props
new_conclusions = {}
for morphism, props in self.conclusions.items():
if ((sympify(objects.contains(morphism.domain)) is S.true) and
(sympify(objects.contains(morphism.codomain)) is S.true)):
new_conclusions[morphism] = props
return Diagram(new_premises, new_conclusions)
|
b415c17ae57f52c47cb1c3b3fb89a09db3fceaa52fad5b5625d3cd2148f07ba3 | from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct, ArrayContraction, ArrayAdd, \
ArrayDiagonal
from sympy.tensor.array.expressions.conv_array_to_matrix import convert_array_to_matrix
from sympy.tensor.array.expressions.conv_indexed_to_array import convert_indexed_to_array
from sympy.tensor.array.expressions.conv_matrix_to_array import convert_matrix_to_array
from sympy.utilities.exceptions import SymPyDeprecationWarning
SymPyDeprecationWarning(
feature="Array expressions inside the codegen module",
useinstead="Experimental module in sympy.tensor.array.expressions",
issue=20996,
deprecated_since_version="1.8"
).warn()
CodegenArrayTensorProduct = ArrayTensorProduct
CodegenArrayContraction = ArrayContraction
CodegenArrayElementwiseAdd = ArrayAdd
CodegenArrayDiagonal = ArrayDiagonal
recognize_matrix_expression = convert_array_to_matrix
parse_matrix_expression = convert_matrix_to_array
parse_indexed_expression = convert_indexed_to_array
|
4111cc678c42b60b1b41562099e6577e0b500b27f4ae21471f92c434475837ce | """
Additional AST nodes for operations on matrices. The nodes in this module
are meant to represent optimization of matrix expressions within codegen's
target languages that cannot be represented by SymPy expressions.
As an example, we can use :meth:`sympy.codegen.rewriting.optimize` and the
``matin_opt`` optimization provided in :mod:`sympy.codegen.rewriting` to
transform matrix multiplication under certain assumptions:
>>> from sympy import symbols, MatrixSymbol
>>> n = symbols('n', integer=True)
>>> A = MatrixSymbol('A', n, n)
>>> x = MatrixSymbol('x', n, 1)
>>> expr = A**(-1) * x
>>> from sympy.assumptions import assuming, Q
>>> from sympy.codegen.rewriting import matinv_opt, optimize
>>> with assuming(Q.fullrank(A)):
... optimize(expr, [matinv_opt])
MatrixSolve(A, vector=x)
"""
from .ast import Token
from sympy.matrices import MatrixExpr
from sympy.core.sympify import sympify
class MatrixSolve(Token, MatrixExpr):
"""Represents an operation to solve a linear matrix equation.
Parameters
==========
matrix : MatrixSymbol
Matrix representing the coefficients of variables in the linear
equation. This matrix must be square and full-rank (i.e. all columns must
be linearly independent) for the solving operation to be valid.
vector : MatrixSymbol
One-column matrix representing the solutions to the equations
represented in ``matrix``.
Examples
========
>>> from sympy import symbols, MatrixSymbol
>>> from sympy.codegen.matrix_nodes import MatrixSolve
>>> n = symbols('n', integer=True)
>>> A = MatrixSymbol('A', n, n)
>>> x = MatrixSymbol('x', n, 1)
>>> from sympy.printing.numpy import NumPyPrinter
>>> NumPyPrinter().doprint(MatrixSolve(A, x))
'numpy.linalg.solve(A, x)'
>>> from sympy.printing import octave_code
>>> octave_code(MatrixSolve(A, x))
'A \\\\ x'
"""
__slots__ = ('matrix', 'vector')
_construct_matrix = staticmethod(sympify)
@property
def shape(self):
return self.vector.shape
|
1706fbd1638f6a2058f036645cd6f6955dab7d4ed3f9703bba92aa9d4824aa33 | """
The contents of this file are the return value of
``sympy.assumptions.ask.compute_known_facts``.
Do NOT manually edit this file.
Instead, run ./bin/ask_update.py.
"""
from sympy.core.cache import cacheit
from sympy.assumptions.cnf import Literal
from sympy.assumptions.ask import Q
@cacheit
def get_all_known_facts():
"""
Known facts as CNF clauses. Used by satask.
"""
return {
frozenset((Literal(Q.algebraic, False), Literal(Q.imaginary, True), Literal(Q.transcendental, False))),
frozenset((Literal(Q.algebraic, False), Literal(Q.negative, True), Literal(Q.transcendental, False))),
frozenset((Literal(Q.algebraic, False), Literal(Q.positive, True), Literal(Q.transcendental, False))),
frozenset((Literal(Q.algebraic, False), Literal(Q.rational, True))),
frozenset((Literal(Q.algebraic, False), Literal(Q.transcendental, False), Literal(Q.zero, True))),
frozenset((Literal(Q.algebraic, True), Literal(Q.finite, False))),
frozenset((Literal(Q.algebraic, True), Literal(Q.transcendental, True))),
frozenset((Literal(Q.antihermitian, False), Literal(Q.hermitian, False), Literal(Q.zero, True))),
frozenset((Literal(Q.antihermitian, False), Literal(Q.imaginary, True))),
frozenset((Literal(Q.commutative, False), Literal(Q.finite, True))),
frozenset((Literal(Q.commutative, False), Literal(Q.infinite, True))),
frozenset((Literal(Q.complex_elements, False), Literal(Q.real_elements, True))),
frozenset((Literal(Q.composite, False), Literal(Q.even, True), Literal(Q.positive, True), Literal(Q.prime, False))),
frozenset((Literal(Q.composite, True), Literal(Q.even, False), Literal(Q.odd, False))),
frozenset((Literal(Q.composite, True), Literal(Q.positive, False))),
frozenset((Literal(Q.composite, True), Literal(Q.prime, True))),
frozenset((Literal(Q.diagonal, False), Literal(Q.lower_triangular, True), Literal(Q.upper_triangular, True))),
frozenset((Literal(Q.diagonal, True), Literal(Q.lower_triangular, False))),
frozenset((Literal(Q.diagonal, True), Literal(Q.normal, False))),
frozenset((Literal(Q.diagonal, True), Literal(Q.symmetric, False))),
frozenset((Literal(Q.diagonal, True), Literal(Q.upper_triangular, False))),
frozenset((Literal(Q.even, False), Literal(Q.odd, False), Literal(Q.prime, True))),
frozenset((Literal(Q.even, False), Literal(Q.zero, True))),
frozenset((Literal(Q.even, True), Literal(Q.odd, True))),
frozenset((Literal(Q.even, True), Literal(Q.rational, False))),
frozenset((Literal(Q.finite, False), Literal(Q.transcendental, True))),
frozenset((Literal(Q.finite, True), Literal(Q.infinite, True))),
frozenset((Literal(Q.fullrank, False), Literal(Q.invertible, True))),
frozenset((Literal(Q.fullrank, True), Literal(Q.invertible, False), Literal(Q.square, True))),
frozenset((Literal(Q.hermitian, False), Literal(Q.negative, True))),
frozenset((Literal(Q.hermitian, False), Literal(Q.positive, True))),
frozenset((Literal(Q.hermitian, False), Literal(Q.zero, True))),
frozenset((Literal(Q.imaginary, True), Literal(Q.negative, True))),
frozenset((Literal(Q.imaginary, True), Literal(Q.positive, True))),
frozenset((Literal(Q.imaginary, True), Literal(Q.zero, True))),
frozenset((Literal(Q.infinite, False), Literal(Q.negative_infinite, True))),
frozenset((Literal(Q.infinite, False), Literal(Q.positive_infinite, True))),
frozenset((Literal(Q.integer_elements, True), Literal(Q.real_elements, False))),
frozenset((Literal(Q.invertible, False), Literal(Q.positive_definite, True))),
frozenset((Literal(Q.invertible, False), Literal(Q.singular, False))),
frozenset((Literal(Q.invertible, False), Literal(Q.unitary, True))),
frozenset((Literal(Q.invertible, True), Literal(Q.singular, True))),
frozenset((Literal(Q.invertible, True), Literal(Q.square, False))),
frozenset((Literal(Q.irrational, False), Literal(Q.negative, True), Literal(Q.rational, False))),
frozenset((Literal(Q.irrational, False), Literal(Q.positive, True), Literal(Q.rational, False))),
frozenset((Literal(Q.irrational, False), Literal(Q.rational, False), Literal(Q.zero, True))),
frozenset((Literal(Q.irrational, True), Literal(Q.negative, False), Literal(Q.positive, False), Literal(Q.zero, False))),
frozenset((Literal(Q.irrational, True), Literal(Q.rational, True))),
frozenset((Literal(Q.lower_triangular, False), Literal(Q.triangular, True), Literal(Q.upper_triangular, False))),
frozenset((Literal(Q.lower_triangular, True), Literal(Q.triangular, False))),
frozenset((Literal(Q.negative, False), Literal(Q.positive, False), Literal(Q.rational, True), Literal(Q.zero, False))),
frozenset((Literal(Q.negative, True), Literal(Q.positive, True))),
frozenset((Literal(Q.negative, True), Literal(Q.zero, True))),
frozenset((Literal(Q.negative_infinite, True), Literal(Q.positive_infinite, True))),
frozenset((Literal(Q.normal, False), Literal(Q.unitary, True))),
frozenset((Literal(Q.normal, True), Literal(Q.square, False))),
frozenset((Literal(Q.odd, True), Literal(Q.rational, False))),
frozenset((Literal(Q.orthogonal, False), Literal(Q.real_elements, True), Literal(Q.unitary, True))),
frozenset((Literal(Q.orthogonal, True), Literal(Q.positive_definite, False))),
frozenset((Literal(Q.orthogonal, True), Literal(Q.unitary, False))),
frozenset((Literal(Q.positive, False), Literal(Q.prime, True))),
frozenset((Literal(Q.positive, True), Literal(Q.zero, True))),
frozenset((Literal(Q.square, False), Literal(Q.symmetric, True))),
frozenset((Literal(Q.triangular, False), Literal(Q.unit_triangular, True))),
frozenset((Literal(Q.triangular, False), Literal(Q.upper_triangular, True)))
}
# -{ Known facts in compressed sets }-
@cacheit
def get_known_facts_dict():
"""
Logical implication as dictionary. Key implies every item in its value.
Used for quick lookup of single facts.
"""
return {
Q.algebraic: set([Q.algebraic, Q.commutative, Q.finite, ~Q.infinite,
~Q.negative_infinite, ~Q.positive_infinite,
~Q.transcendental]),
Q.antihermitian: set([Q.antihermitian]),
Q.commutative: set([Q.commutative]),
Q.complex_elements: set([Q.complex_elements]),
Q.composite: set([Q.algebraic, Q.commutative, Q.composite, Q.finite,
Q.hermitian, Q.positive, Q.rational, ~Q.imaginary,
~Q.infinite, ~Q.irrational, ~Q.negative, ~Q.negative_infinite,
~Q.positive_infinite, ~Q.prime, ~Q.transcendental, ~Q.zero]),
Q.diagonal: set([Q.diagonal, Q.lower_triangular, Q.normal, Q.square,
Q.symmetric, Q.triangular, Q.upper_triangular]),
Q.even: set([Q.algebraic, Q.commutative, Q.even, Q.finite,
Q.hermitian, Q.rational, ~Q.imaginary, ~Q.infinite,
~Q.irrational, ~Q.negative_infinite, ~Q.odd,
~Q.positive_infinite, ~Q.transcendental]),
Q.finite: set([Q.commutative, Q.finite, ~Q.infinite,
~Q.negative_infinite, ~Q.positive_infinite]),
Q.fullrank: set([Q.fullrank]),
Q.hermitian: set([Q.hermitian]),
Q.imaginary: set([Q.antihermitian, Q.commutative, Q.finite,
Q.imaginary, ~Q.composite, ~Q.even, ~Q.infinite,
~Q.irrational, ~Q.negative, ~Q.negative_infinite, ~Q.odd,
~Q.positive, ~Q.positive_infinite, ~Q.prime, ~Q.rational,
~Q.zero]),
Q.infinite: set([Q.commutative, Q.infinite, ~Q.algebraic,
~Q.composite, ~Q.even, ~Q.finite, ~Q.imaginary, ~Q.irrational,
~Q.negative, ~Q.odd, ~Q.positive, ~Q.prime, ~Q.rational,
~Q.transcendental, ~Q.zero]),
Q.integer_elements: set([Q.complex_elements, Q.integer_elements,
Q.real_elements]),
Q.invertible: set([Q.fullrank, Q.invertible, Q.square, ~Q.singular]),
Q.irrational: set([Q.commutative, Q.finite, Q.hermitian, Q.irrational,
~Q.composite, ~Q.even, ~Q.imaginary, ~Q.infinite,
~Q.negative_infinite, ~Q.odd, ~Q.positive_infinite, ~Q.prime,
~Q.rational, ~Q.zero]),
Q.is_true: set([Q.is_true]),
Q.lower_triangular: set([Q.lower_triangular, Q.triangular]),
Q.negative: set([Q.commutative, Q.finite, Q.hermitian, Q.negative,
~Q.composite, ~Q.imaginary, ~Q.infinite, ~Q.negative_infinite,
~Q.positive, ~Q.positive_infinite, ~Q.prime, ~Q.zero]),
Q.negative_infinite: set([Q.commutative, Q.infinite,
Q.negative_infinite, ~Q.algebraic, ~Q.composite, ~Q.even,
~Q.finite, ~Q.imaginary, ~Q.irrational, ~Q.negative, ~Q.odd,
~Q.positive, ~Q.positive_infinite, ~Q.prime, ~Q.rational,
~Q.transcendental, ~Q.zero]),
Q.normal: set([Q.normal, Q.square]),
Q.odd: set([Q.algebraic, Q.commutative, Q.finite, Q.hermitian, Q.odd,
Q.rational, ~Q.even, ~Q.imaginary, ~Q.infinite, ~Q.irrational,
~Q.negative_infinite, ~Q.positive_infinite, ~Q.transcendental,
~Q.zero]),
Q.orthogonal: set([Q.fullrank, Q.invertible, Q.normal, Q.orthogonal,
Q.positive_definite, Q.square, Q.unitary, ~Q.singular]),
Q.positive: set([Q.commutative, Q.finite, Q.hermitian, Q.positive,
~Q.imaginary, ~Q.infinite, ~Q.negative, ~Q.negative_infinite,
~Q.positive_infinite, ~Q.zero]),
Q.positive_definite: set([Q.fullrank, Q.invertible,
Q.positive_definite, Q.square, ~Q.singular]),
Q.positive_infinite: set([Q.commutative, Q.infinite,
Q.positive_infinite, ~Q.algebraic, ~Q.composite, ~Q.even,
~Q.finite, ~Q.imaginary, ~Q.irrational, ~Q.negative,
~Q.negative_infinite, ~Q.odd, ~Q.positive, ~Q.prime,
~Q.rational, ~Q.transcendental, ~Q.zero]),
Q.prime: set([Q.algebraic, Q.commutative, Q.finite, Q.hermitian,
Q.positive, Q.prime, Q.rational, ~Q.composite, ~Q.imaginary,
~Q.infinite, ~Q.irrational, ~Q.negative, ~Q.negative_infinite,
~Q.positive_infinite, ~Q.transcendental, ~Q.zero]),
Q.rational: set([Q.algebraic, Q.commutative, Q.finite, Q.hermitian,
Q.rational, ~Q.imaginary, ~Q.infinite, ~Q.irrational,
~Q.negative_infinite, ~Q.positive_infinite,
~Q.transcendental]),
Q.real_elements: set([Q.complex_elements, Q.real_elements]),
Q.singular: set([Q.singular, ~Q.invertible, ~Q.orthogonal,
~Q.positive_definite, ~Q.unitary]),
Q.square: set([Q.square]),
Q.symmetric: set([Q.square, Q.symmetric]),
Q.transcendental: set([Q.commutative, Q.finite, Q.transcendental,
~Q.algebraic, ~Q.composite, ~Q.even, ~Q.infinite,
~Q.negative_infinite, ~Q.odd, ~Q.positive_infinite, ~Q.prime,
~Q.rational, ~Q.zero]),
Q.triangular: set([Q.triangular]),
Q.unit_triangular: set([Q.triangular, Q.unit_triangular]),
Q.unitary: set([Q.fullrank, Q.invertible, Q.normal, Q.square,
Q.unitary, ~Q.singular]),
Q.upper_triangular: set([Q.triangular, Q.upper_triangular]),
Q.zero: set([Q.algebraic, Q.commutative, Q.even, Q.finite,
Q.hermitian, Q.rational, Q.zero, ~Q.composite, ~Q.imaginary,
~Q.infinite, ~Q.irrational, ~Q.negative, ~Q.negative_infinite,
~Q.odd, ~Q.positive, ~Q.positive_infinite, ~Q.prime,
~Q.transcendental]),
}
|
888f2b07056321efda00586380721f0980645940ab98ea86e31e7c9b58578627 | from typing import Dict, Callable
from sympy.core import S, Add, Expr, Basic, Mul
from sympy.logic.boolalg import Boolean
from sympy.assumptions import ask, Q # type: ignore
def refine(expr, assumptions=True):
"""
Simplify an expression using assumptions.
Explanation
===========
Unlike :func:`~.simplify()` which performs structural simplification
without any assumption, this function transforms the expression into
the form which is only valid under certain assumptions. Note that
``simplify()`` is generally not done in refining process.
Refining boolean expression involves reducing it to ``S.true`` or
``S.false``. Unlike :func:`~.ask()`, the expression will not be reduced
if the truth value cannot be determined.
Examples
========
>>> from sympy import refine, sqrt, Q
>>> from sympy.abc import x
>>> refine(sqrt(x**2), Q.real(x))
Abs(x)
>>> refine(sqrt(x**2), Q.positive(x))
x
>>> refine(Q.real(x), Q.positive(x))
True
>>> refine(Q.positive(x), Q.real(x))
Q.positive(x)
See Also
========
sympy.simplify.simplify.simplify : Structural simplification without assumptions.
sympy.assumptions.ask.ask : Query for boolean expressions using assumptions.
"""
if not isinstance(expr, Basic):
return expr
if not expr.is_Atom:
args = [refine(arg, assumptions) for arg in expr.args]
# TODO: this will probably not work with Integral or Polynomial
expr = expr.func(*args)
if hasattr(expr, '_eval_refine'):
ref_expr = expr._eval_refine(assumptions)
if ref_expr is not None:
return ref_expr
name = expr.__class__.__name__
handler = handlers_dict.get(name, None)
if handler is None:
return expr
new_expr = handler(expr, assumptions)
if (new_expr is None) or (expr == new_expr):
return expr
if not isinstance(new_expr, Expr):
return new_expr
return refine(new_expr, assumptions)
def refine_abs(expr, assumptions):
"""
Handler for the absolute value.
Examples
========
>>> from sympy import Q, Abs
>>> from sympy.assumptions.refine import refine_abs
>>> from sympy.abc import x
>>> refine_abs(Abs(x), Q.real(x))
>>> refine_abs(Abs(x), Q.positive(x))
x
>>> refine_abs(Abs(x), Q.negative(x))
-x
"""
from sympy.core.logic import fuzzy_not
from sympy import Abs
arg = expr.args[0]
if ask(Q.real(arg), assumptions) and \
fuzzy_not(ask(Q.negative(arg), assumptions)):
# if it's nonnegative
return arg
if ask(Q.negative(arg), assumptions):
return -arg
# arg is Mul
if isinstance(arg, Mul):
r = [refine(abs(a), assumptions) for a in arg.args]
non_abs = []
in_abs = []
for i in r:
if isinstance(i, Abs):
in_abs.append(i.args[0])
else:
non_abs.append(i)
return Mul(*non_abs) * Abs(Mul(*in_abs))
def refine_Pow(expr, assumptions):
"""
Handler for instances of Pow.
Examples
========
>>> from sympy import Q
>>> from sympy.assumptions.refine import refine_Pow
>>> from sympy.abc import x,y,z
>>> refine_Pow((-1)**x, Q.real(x))
>>> refine_Pow((-1)**x, Q.even(x))
1
>>> refine_Pow((-1)**x, Q.odd(x))
-1
For powers of -1, even parts of the exponent can be simplified:
>>> refine_Pow((-1)**(x+y), Q.even(x))
(-1)**y
>>> refine_Pow((-1)**(x+y+z), Q.odd(x) & Q.odd(z))
(-1)**y
>>> refine_Pow((-1)**(x+y+2), Q.odd(x))
(-1)**(y + 1)
>>> refine_Pow((-1)**(x+3), True)
(-1)**(x + 1)
"""
from sympy.core import Pow, Rational
from sympy.functions.elementary.complexes import Abs
from sympy.functions import sign
if isinstance(expr.base, Abs):
if ask(Q.real(expr.base.args[0]), assumptions) and \
ask(Q.even(expr.exp), assumptions):
return expr.base.args[0] ** expr.exp
if ask(Q.real(expr.base), assumptions):
if expr.base.is_number:
if ask(Q.even(expr.exp), assumptions):
return abs(expr.base) ** expr.exp
if ask(Q.odd(expr.exp), assumptions):
return sign(expr.base) * abs(expr.base) ** expr.exp
if isinstance(expr.exp, Rational):
if type(expr.base) is Pow:
return abs(expr.base.base) ** (expr.base.exp * expr.exp)
if expr.base is S.NegativeOne:
if expr.exp.is_Add:
old = expr
# For powers of (-1) we can remove
# - even terms
# - pairs of odd terms
# - a single odd term + 1
# - A numerical constant N can be replaced with mod(N,2)
coeff, terms = expr.exp.as_coeff_add()
terms = set(terms)
even_terms = set()
odd_terms = set()
initial_number_of_terms = len(terms)
for t in terms:
if ask(Q.even(t), assumptions):
even_terms.add(t)
elif ask(Q.odd(t), assumptions):
odd_terms.add(t)
terms -= even_terms
if len(odd_terms) % 2:
terms -= odd_terms
new_coeff = (coeff + S.One) % 2
else:
terms -= odd_terms
new_coeff = coeff % 2
if new_coeff != coeff or len(terms) < initial_number_of_terms:
terms.add(new_coeff)
expr = expr.base**(Add(*terms))
# Handle (-1)**((-1)**n/2 + m/2)
e2 = 2*expr.exp
if ask(Q.even(e2), assumptions):
if e2.could_extract_minus_sign():
e2 *= expr.base
if e2.is_Add:
i, p = e2.as_two_terms()
if p.is_Pow and p.base is S.NegativeOne:
if ask(Q.integer(p.exp), assumptions):
i = (i + 1)/2
if ask(Q.even(i), assumptions):
return expr.base**p.exp
elif ask(Q.odd(i), assumptions):
return expr.base**(p.exp + 1)
else:
return expr.base**(p.exp + i)
if old != expr:
return expr
def refine_atan2(expr, assumptions):
"""
Handler for the atan2 function.
Examples
========
>>> from sympy import Q, atan2
>>> from sympy.assumptions.refine import refine_atan2
>>> from sympy.abc import x, y
>>> refine_atan2(atan2(y,x), Q.real(y) & Q.positive(x))
atan(y/x)
>>> refine_atan2(atan2(y,x), Q.negative(y) & Q.negative(x))
atan(y/x) - pi
>>> refine_atan2(atan2(y,x), Q.positive(y) & Q.negative(x))
atan(y/x) + pi
>>> refine_atan2(atan2(y,x), Q.zero(y) & Q.negative(x))
pi
>>> refine_atan2(atan2(y,x), Q.positive(y) & Q.zero(x))
pi/2
>>> refine_atan2(atan2(y,x), Q.negative(y) & Q.zero(x))
-pi/2
>>> refine_atan2(atan2(y,x), Q.zero(y) & Q.zero(x))
nan
"""
from sympy.functions.elementary.trigonometric import atan
from sympy.core import S
y, x = expr.args
if ask(Q.real(y) & Q.positive(x), assumptions):
return atan(y / x)
elif ask(Q.negative(y) & Q.negative(x), assumptions):
return atan(y / x) - S.Pi
elif ask(Q.positive(y) & Q.negative(x), assumptions):
return atan(y / x) + S.Pi
elif ask(Q.zero(y) & Q.negative(x), assumptions):
return S.Pi
elif ask(Q.positive(y) & Q.zero(x), assumptions):
return S.Pi/2
elif ask(Q.negative(y) & Q.zero(x), assumptions):
return -S.Pi/2
elif ask(Q.zero(y) & Q.zero(x), assumptions):
return S.NaN
else:
return expr
def refine_re(expr, assumptions):
"""
Handler for real part.
Examples
========
>>> from sympy.assumptions.refine import refine_re
>>> from sympy import Q, re
>>> from sympy.abc import x
>>> refine_re(re(x), Q.real(x))
x
>>> refine_re(re(x), Q.imaginary(x))
0
"""
arg = expr.args[0]
if ask(Q.real(arg), assumptions):
return arg
if ask(Q.imaginary(arg), assumptions):
return S.Zero
return _refine_reim(expr, assumptions)
def refine_im(expr, assumptions):
"""
Handler for imaginary part.
Explanation
===========
>>> from sympy.assumptions.refine import refine_im
>>> from sympy import Q, im
>>> from sympy.abc import x
>>> refine_im(im(x), Q.real(x))
0
>>> refine_im(im(x), Q.imaginary(x))
-I*x
"""
arg = expr.args[0]
if ask(Q.real(arg), assumptions):
return S.Zero
if ask(Q.imaginary(arg), assumptions):
return - S.ImaginaryUnit * arg
return _refine_reim(expr, assumptions)
def refine_arg(expr, assumptions):
"""
Handler for complex argument
Explanation
===========
>>> from sympy.assumptions.refine import refine_arg
>>> from sympy import Q, arg
>>> from sympy.abc import x
>>> refine_arg(arg(x), Q.positive(x))
0
>>> refine_arg(arg(x), Q.negative(x))
pi
"""
rg = expr.args[0]
if ask(Q.positive(rg), assumptions):
return S.Zero
if ask(Q.negative(rg), assumptions):
return S.Pi
return None
def _refine_reim(expr, assumptions):
# Helper function for refine_re & refine_im
expanded = expr.expand(complex = True)
if expanded != expr:
refined = refine(expanded, assumptions)
if refined != expanded:
return refined
# Best to leave the expression as is
return None
def refine_sign(expr, assumptions):
"""
Handler for sign.
Examples
========
>>> from sympy.assumptions.refine import refine_sign
>>> from sympy import Symbol, Q, sign, im
>>> x = Symbol('x', real = True)
>>> expr = sign(x)
>>> refine_sign(expr, Q.positive(x) & Q.nonzero(x))
1
>>> refine_sign(expr, Q.negative(x) & Q.nonzero(x))
-1
>>> refine_sign(expr, Q.zero(x))
0
>>> y = Symbol('y', imaginary = True)
>>> expr = sign(y)
>>> refine_sign(expr, Q.positive(im(y)))
I
>>> refine_sign(expr, Q.negative(im(y)))
-I
"""
arg = expr.args[0]
if ask(Q.zero(arg), assumptions):
return S.Zero
if ask(Q.real(arg)):
if ask(Q.positive(arg), assumptions):
return S.One
if ask(Q.negative(arg), assumptions):
return S.NegativeOne
if ask(Q.imaginary(arg)):
arg_re, arg_im = arg.as_real_imag()
if ask(Q.positive(arg_im), assumptions):
return S.ImaginaryUnit
if ask(Q.negative(arg_im), assumptions):
return -S.ImaginaryUnit
return expr
def refine_matrixelement(expr, assumptions):
"""
Handler for symmetric part.
Examples
========
>>> from sympy.assumptions.refine import refine_matrixelement
>>> from sympy import Q
>>> from sympy.matrices.expressions.matexpr import MatrixSymbol
>>> X = MatrixSymbol('X', 3, 3)
>>> refine_matrixelement(X[0, 1], Q.symmetric(X))
X[0, 1]
>>> refine_matrixelement(X[1, 0], Q.symmetric(X))
X[0, 1]
"""
from sympy.matrices.expressions.matexpr import MatrixElement
matrix, i, j = expr.args
if ask(Q.symmetric(matrix), assumptions):
if (i - j).could_extract_minus_sign():
return expr
return MatrixElement(matrix, j, i)
handlers_dict = {
'Abs': refine_abs,
'Pow': refine_Pow,
'atan2': refine_atan2,
're': refine_re,
'im': refine_im,
'arg': refine_arg,
'sign': refine_sign,
'MatrixElement': refine_matrixelement
} # type: Dict[str, Callable[[Expr, Boolean], Expr]]
|
3c54bc7634ea88d6c654c8e0ecf68f8c47f09a3c9955f8a85e2455ceaa80dc4d | """
A module to implement logical predicates and assumption system.
"""
from .assume import (
AppliedPredicate, Predicate, AssumptionsContext, assuming,
global_assumptions
)
from .ask import Q, ask, register_handler, remove_handler
from .refine import refine
from .relation import BinaryRelation, AppliedBinaryRelation
__all__ = [
'AppliedPredicate', 'Predicate', 'AssumptionsContext', 'assuming',
'global_assumptions', 'Q', 'ask', 'register_handler', 'remove_handler',
'refine',
'BinaryRelation', 'AppliedBinaryRelation'
]
|
0f210aa2e4011c7cdd883d2cb648a2fc90ec5bf57cba461047d69dcd0ac62952 | """
Functions and wrapper object to call assumption property and predicate
query with same syntax.
In SymPy, there are two assumption systems. Old assumption system is
defined in sympy/core/assumptions, and it can be accessed by attribute
such as ``x.is_even``. New assumption system is definded in
sympy/assumptions, and it can be accessed by predicates such as
``Q.even(x)``.
Old assumption is fast, while new assumptions can freely take local facts.
In general, old assumption is used in evaluation method and new assumption
is used in refinement method.
In most cases, both evaluation and refinement follow the same process, and
the only difference is which assumption system is used. This module provides
``is_[...]()`` functions and ``AssumptionsWrapper()`` class which allows
using two systems with same syntax so that parallel code implementation can be
avoided.
Examples
========
For multiple use, use ``AssumptionsWrapper()``.
>>> from sympy import Q, Symbol
>>> from sympy.assumptions.wrapper import AssumptionsWrapper
>>> x = Symbol('x')
>>> _x = AssumptionsWrapper(x, Q.even(x))
>>> _x.is_integer
True
>>> _x.is_odd
False
For single use, use ``is_[...]()`` functions.
>>> from sympy.assumptions.wrapper import is_infinite
>>> a = Symbol('a')
>>> print(is_infinite(a))
None
>>> is_infinite(a, Q.finite(a))
False
"""
from sympy.assumptions import ask, Q
from sympy.core.assumptions import (_assume_defined, as_property,
ManagedProperties)
from sympy.core.basic import Basic
class AssumptionsWrapperMeta(ManagedProperties):
"""
Metaclass to give _eval_is_[...] attributes to AssumptionsWrapper
"""
def __init__(cls, *args, **kws):
for fact in _assume_defined:
pname = "_eval_%s" % as_property(fact)
setattr(cls, pname, make_eval_method(fact))
super().__init__(cls, *args, **kws)
def make_eval_method(fact):
def getit(self):
try:
pred = getattr(Q, fact)
ret = ask(pred(self.expr), self.assumptions)
return ret
except AttributeError:
return None
return getit
# we subclass Basic to use the fact deduction and caching
class AssumptionsWrapper(Basic, metaclass=AssumptionsWrapperMeta):
"""
Wrapper over ``Basic`` instances to call predicate query by
``.is_[...]`` property
Parameters
==========
expr : Basic
assumptions : Boolean, optional
Examples
========
>>> from sympy import Q, Symbol
>>> from sympy.assumptions.wrapper import AssumptionsWrapper
>>> x = Symbol('x', even=True)
>>> AssumptionsWrapper(x).is_integer
True
>>> y = Symbol('y')
>>> AssumptionsWrapper(y, Q.even(y)).is_integer
True
With ``AssumptionsWrapper``, both evaluation and refinement can be supported
by single implementation.
>>> from sympy import Function
>>> class MyAbs(Function):
... @classmethod
... def eval(cls, x, assumptions=True):
... _x = AssumptionsWrapper(x, assumptions)
... if _x.is_nonnegative:
... return x
... if _x.is_negative:
... return -x
... def _eval_refine(self, assumptions):
... return MyAbs.eval(self.args[0], assumptions)
>>> MyAbs(x)
MyAbs(x)
>>> MyAbs(x).refine(Q.positive(x))
x
>>> MyAbs(Symbol('y', negative=True))
-y
"""
def __new__(cls, expr, assumptions=None):
if assumptions is None:
return expr
obj = super().__new__(cls, expr, assumptions)
obj.expr = expr
obj.assumptions = assumptions
return obj
# one shot functions which are faster than AssumptionsWrapper
def is_infinite(obj, assumptions=None):
if assumptions is None:
return obj.is_infinite
return ask(Q.infinite(obj), assumptions)
def is_extended_real(obj, assumptions=None):
if assumptions is None:
return obj.is_extended_real
return ask(Q.extended_real(obj), assumptions)
def is_extended_nonnegative(obj, assumptions=None):
if assumptions is None:
return obj.is_extended_nonnegative
return ask(Q.extended_nonnegative(obj), assumptions)
|
f318642be315ff0f0352c1ef85ae62a82b28cc49de5b5fe4725b89fd0deb4ba7 | """Module for querying SymPy objects about assumptions."""
from sympy.assumptions.assume import (global_assumptions, Predicate,
AppliedPredicate)
from sympy.assumptions.cnf import CNF, EncodedCNF, Literal
from sympy.core import sympify
from sympy.core.kind import BooleanKind
from sympy.core.relational import Eq, Ne, Gt, Lt, Ge, Le
from sympy.logic.boolalg import Not
from sympy.logic.inference import satisfiable
from sympy.utilities.decorator import memoize_property
from sympy.utilities.exceptions import SymPyDeprecationWarning
# Memoization is necessary for the properties of AssumptionKeys to
# ensure that only one object of Predicate objects are created.
# This is because assumption handlers are registered on those objects.
class AssumptionKeys:
"""
This class contains all the supported keys by ``ask``.
It should be accessed via the instance ``sympy.Q``.
"""
# DO NOT add methods or properties other than predicate keys.
# SAT solver checks the properties of Q and use them to compute the
# fact system. Non-predicate attributes will break this.
@memoize_property
def hermitian(self):
from .handlers.sets import HermitianPredicate
return HermitianPredicate()
@memoize_property
def antihermitian(self):
from .handlers.sets import AntihermitianPredicate
return AntihermitianPredicate()
@memoize_property
def real(self):
from .handlers.sets import RealPredicate
return RealPredicate()
@memoize_property
def extended_real(self):
from .handlers.sets import ExtendedRealPredicate
return ExtendedRealPredicate()
@memoize_property
def imaginary(self):
from .handlers.sets import ImaginaryPredicate
return ImaginaryPredicate()
@memoize_property
def complex(self):
from .handlers.sets import ComplexPredicate
return ComplexPredicate()
@memoize_property
def algebraic(self):
from .handlers.sets import AlgebraicPredicate
return AlgebraicPredicate()
@memoize_property
def transcendental(self):
from .predicates.sets import TranscendentalPredicate
return TranscendentalPredicate()
@memoize_property
def integer(self):
from .handlers.sets import IntegerPredicate
return IntegerPredicate()
@memoize_property
def rational(self):
from .handlers.sets import RationalPredicate
return RationalPredicate()
@memoize_property
def irrational(self):
from .handlers.sets import IrrationalPredicate
return IrrationalPredicate()
@memoize_property
def finite(self):
from .handlers.calculus import FinitePredicate
return FinitePredicate()
@memoize_property
def infinite(self):
from .handlers.calculus import InfinitePredicate
return InfinitePredicate()
@memoize_property
def positive_infinite(self):
from .handlers.calculus import PositiveInfinitePredicate
return PositiveInfinitePredicate()
@memoize_property
def negative_infinite(self):
from .handlers.calculus import NegativeInfinitePredicate
return NegativeInfinitePredicate()
@memoize_property
def positive(self):
from .handlers.order import PositivePredicate
return PositivePredicate()
@memoize_property
def negative(self):
from .handlers.order import NegativePredicate
return NegativePredicate()
@memoize_property
def zero(self):
from .handlers.order import ZeroPredicate
return ZeroPredicate()
@memoize_property
def extended_positive(self):
from .handlers.order import ExtendedPositivePredicate
return ExtendedPositivePredicate()
@memoize_property
def extended_negative(self):
from .handlers.order import ExtendedNegativePredicate
return ExtendedNegativePredicate()
@memoize_property
def nonzero(self):
from .handlers.order import NonZeroPredicate
return NonZeroPredicate()
@memoize_property
def nonpositive(self):
from .handlers.order import NonPositivePredicate
return NonPositivePredicate()
@memoize_property
def nonnegative(self):
from .handlers.order import NonNegativePredicate
return NonNegativePredicate()
@memoize_property
def extended_nonzero(self):
from .handlers.order import ExtendedNonZeroPredicate
return ExtendedNonZeroPredicate()
@memoize_property
def extended_nonpositive(self):
from .handlers.order import ExtendedNonPositivePredicate
return ExtendedNonPositivePredicate()
@memoize_property
def extended_nonnegative(self):
from .handlers.order import ExtendedNonNegativePredicate
return ExtendedNonNegativePredicate()
@memoize_property
def even(self):
from .handlers.ntheory import EvenPredicate
return EvenPredicate()
@memoize_property
def odd(self):
from .handlers.ntheory import OddPredicate
return OddPredicate()
@memoize_property
def prime(self):
from .handlers.ntheory import PrimePredicate
return PrimePredicate()
@memoize_property
def composite(self):
from .handlers.ntheory import CompositePredicate
return CompositePredicate()
@memoize_property
def commutative(self):
from .handlers.common import CommutativePredicate
return CommutativePredicate()
@memoize_property
def is_true(self):
from .handlers.common import IsTruePredicate
return IsTruePredicate()
@memoize_property
def symmetric(self):
from .handlers.matrices import SymmetricPredicate
return SymmetricPredicate()
@memoize_property
def invertible(self):
from .handlers.matrices import InvertiblePredicate
return InvertiblePredicate()
@memoize_property
def orthogonal(self):
from .handlers.matrices import OrthogonalPredicate
return OrthogonalPredicate()
@memoize_property
def unitary(self):
from .handlers.matrices import UnitaryPredicate
return UnitaryPredicate()
@memoize_property
def positive_definite(self):
from .handlers.matrices import PositiveDefinitePredicate
return PositiveDefinitePredicate()
@memoize_property
def upper_triangular(self):
from .handlers.matrices import UpperTriangularPredicate
return UpperTriangularPredicate()
@memoize_property
def lower_triangular(self):
from .handlers.matrices import LowerTriangularPredicate
return LowerTriangularPredicate()
@memoize_property
def diagonal(self):
from .handlers.matrices import DiagonalPredicate
return DiagonalPredicate()
@memoize_property
def fullrank(self):
from .handlers.matrices import FullRankPredicate
return FullRankPredicate()
@memoize_property
def square(self):
from .handlers.matrices import SquarePredicate
return SquarePredicate()
@memoize_property
def integer_elements(self):
from .handlers.matrices import IntegerElementsPredicate
return IntegerElementsPredicate()
@memoize_property
def real_elements(self):
from .handlers.matrices import RealElementsPredicate
return RealElementsPredicate()
@memoize_property
def complex_elements(self):
from .handlers.matrices import ComplexElementsPredicate
return ComplexElementsPredicate()
@memoize_property
def singular(self):
from .predicates.matrices import SingularPredicate
return SingularPredicate()
@memoize_property
def normal(self):
from .predicates.matrices import NormalPredicate
return NormalPredicate()
@memoize_property
def triangular(self):
from .predicates.matrices import TriangularPredicate
return TriangularPredicate()
@memoize_property
def unit_triangular(self):
from .predicates.matrices import UnitTriangularPredicate
return UnitTriangularPredicate()
@memoize_property
def eq(self):
from .relation.equality import EqualityPredicate
return EqualityPredicate()
@memoize_property
def ne(self):
from .relation.equality import UnequalityPredicate
return UnequalityPredicate()
@memoize_property
def gt(self):
from .relation.equality import StrictGreaterThanPredicate
return StrictGreaterThanPredicate()
@memoize_property
def ge(self):
from .relation.equality import GreaterThanPredicate
return GreaterThanPredicate()
@memoize_property
def lt(self):
from .relation.equality import StrictLessThanPredicate
return StrictLessThanPredicate()
@memoize_property
def le(self):
from .relation.equality import LessThanPredicate
return LessThanPredicate()
Q = AssumptionKeys()
def _extract_all_facts(assump, exprs):
"""
Extract all relevant assumptions from *assump* with respect to given *exprs*.
Parameters
==========
assump : sympy.assumptions.cnf.CNF
exprs : tuple of expressions
Returns
=======
sympy.assumptions.cnf.CNF
Examples
========
>>> from sympy import Q
>>> from sympy.assumptions.cnf import CNF
>>> from sympy.assumptions.ask import _extract_all_facts
>>> from sympy.abc import x, y
>>> assump = CNF.from_prop(Q.positive(x) & Q.integer(y))
>>> exprs = (x,)
>>> cnf = _extract_all_facts(assump, exprs)
>>> cnf.clauses
{frozenset({Literal(Q.positive, False)})}
"""
facts = set()
for clause in assump.clauses:
args = []
for literal in clause:
if isinstance(literal.lit, AppliedPredicate) and len(literal.lit.arguments) == 1:
if literal.lit.arg in exprs:
# Add literal if it has matching in it
args.append(Literal(literal.lit.function, literal.is_Not))
else:
# If any of the literals doesn't have matching expr don't add the whole clause.
break
else:
if args:
facts.add(frozenset(args))
return CNF(facts)
def ask(proposition, assumptions=True, context=global_assumptions):
"""
Function to evaluate the proposition with assumptions.
Explanation
===========
This function evaluates the proposition to ``True`` or ``False`` if
the truth value can be determined. If not, it returns ``None``.
It should be discerned from :func:`~.refine()` which, when applied to a
proposition, simplifies the argument to symbolic ``Boolean`` instead of
Python built-in ``True``, ``False`` or ``None``.
**Syntax**
* ask(proposition)
Evaluate the *proposition* in global assumption context.
* ask(proposition, assumptions)
Evaluate the *proposition* with respect to *assumptions* in
global assumption context.
Parameters
==========
proposition : Any boolean expression.
Proposition which will be evaluated to boolean value. If this is
not ``AppliedPredicate``, it will be wrapped by ``Q.is_true``.
assumptions : Any boolean expression, optional.
Local assumptions to evaluate the *proposition*.
context : AssumptionsContext, optional.
Default assumptions to evaluate the *proposition*. By default,
this is ``sympy.assumptions.global_assumptions`` variable.
Returns
=======
``True``, ``False``, or ``None``
Raises
======
TypeError : *proposition* or *assumptions* is not valid logical expression.
ValueError : assumptions are inconsistent.
Examples
========
>>> from sympy import ask, Q, pi
>>> from sympy.abc import x, y
>>> ask(Q.rational(pi))
False
>>> ask(Q.even(x*y), Q.even(x) & Q.integer(y))
True
>>> ask(Q.prime(4*x), Q.integer(x))
False
If the truth value cannot be determined, ``None`` will be returned.
>>> print(ask(Q.odd(3*x))) # cannot determine unless we know x
None
``ValueError`` is raised if assumptions are inconsistent.
>>> ask(Q.integer(x), Q.even(x) & Q.odd(x))
Traceback (most recent call last):
...
ValueError: inconsistent assumptions Q.even(x) & Q.odd(x)
Notes
=====
Relations in assumptions are not implemented (yet), so the following
will not give a meaningful result.
>>> ask(Q.positive(x), x > 0)
It is however a work in progress.
See Also
========
sympy.assumptions.refine.refine : Simplification using assumptions.
Proposition is not reduced to ``None`` if the truth value cannot
be determined.
"""
from sympy.assumptions.satask import satask
proposition = sympify(proposition)
assumptions = sympify(assumptions)
if isinstance(proposition, Predicate) or proposition.kind is not BooleanKind:
raise TypeError("proposition must be a valid logical expression")
if isinstance(assumptions, Predicate) or assumptions.kind is not BooleanKind:
raise TypeError("assumptions must be a valid logical expression")
binrelpreds = {Eq: Q.eq, Ne: Q.ne, Gt: Q.gt, Lt: Q.lt, Ge: Q.ge, Le: Q.le}
if isinstance(proposition, AppliedPredicate):
key, args = proposition.function, proposition.arguments
elif proposition.func in binrelpreds:
key, args = binrelpreds[proposition.func], proposition.args
else:
key, args = Q.is_true, (proposition,)
# convert local and global assumptions to CNF
assump_cnf = CNF.from_prop(assumptions)
assump_cnf.extend(context)
# extract the relevant facts from assumptions with respect to args
local_facts = _extract_all_facts(assump_cnf, args)
# convert default facts and assumed facts to encoded CNF
known_facts_cnf = get_all_known_facts()
enc_cnf = EncodedCNF()
enc_cnf.from_cnf(CNF(known_facts_cnf))
enc_cnf.add_from_cnf(local_facts)
# check the satisfiability of given assumptions
if local_facts.clauses and satisfiable(enc_cnf) is False:
raise ValueError("inconsistent assumptions %s" % assumptions)
# quick computation for single fact
res = _ask_single_fact(key, local_facts)
if res is not None:
return res
# direct resolution method, no logic
res = key(*args)._eval_ask(assumptions)
if res is not None:
return bool(res)
# using satask (still costly)
res = satask(proposition, assumptions=assumptions, context=context)
return res
def _ask_single_fact(key, local_facts):
"""
Compute the truth value of single predicate using assumptions.
Parameters
==========
key : sympy.assumptions.assume.Predicate
Proposition predicate.
local_facts : sympy.assumptions.cnf.CNF
Local assumption in CNF form.
Returns
=======
``True``, ``False`` or ``None``
Examples
========
>>> from sympy import Q
>>> from sympy.assumptions.cnf import CNF
>>> from sympy.assumptions.ask import _ask_single_fact
If prerequisite of proposition is rejected by the assumption,
return ``False``.
>>> key, assump = Q.zero, ~Q.zero
>>> local_facts = CNF.from_prop(assump)
>>> _ask_single_fact(key, local_facts)
False
>>> key, assump = Q.zero, ~Q.even
>>> local_facts = CNF.from_prop(assump)
>>> _ask_single_fact(key, local_facts)
False
If assumption implies the proposition, return ``True``.
>>> key, assump = Q.even, Q.zero
>>> local_facts = CNF.from_prop(assump)
>>> _ask_single_fact(key, local_facts)
True
If proposition rejects the assumption, return ``False``.
>>> key, assump = Q.even, Q.odd
>>> local_facts = CNF.from_prop(assump)
>>> _ask_single_fact(key, local_facts)
False
"""
if local_facts.clauses:
known_facts_dict = get_known_facts_dict()
if len(local_facts.clauses) == 1:
cl, = local_facts.clauses
if len(cl) == 1:
f, = cl
if f.is_Not and f.arg in known_facts_dict.get(key, []):
# the prerequisite of proposition is rejected
return False
for clause in local_facts.clauses:
if len(clause) == 1:
f, = clause
fdict = known_facts_dict.get(f.arg, None) if not f.is_Not else None
if fdict is None:
pass
elif key in fdict:
# assumption implies the proposition
return True
elif Not(key) in fdict:
# proposition rejects the assumption
return False
return None
def register_handler(key, handler):
"""
Register a handler in the ask system. key must be a string and handler a
class inheriting from AskHandler.
.. deprecated:: 1.8.
Use multipledispatch handler instead. See :obj:`~.Predicate`.
"""
SymPyDeprecationWarning(
feature="register_handler() function",
useinstead="multipledispatch handler of Predicate",
issue=20873,
deprecated_since_version="1.8"
).warn()
if isinstance(key, Predicate):
key = key.name.name
Qkey = getattr(Q, key, None)
if Qkey is not None:
Qkey.add_handler(handler)
else:
setattr(Q, key, Predicate(key, handlers=[handler]))
def remove_handler(key, handler):
"""
Removes a handler from the ask system. Same syntax as register_handler
.. deprecated:: 1.8.
Use multipledispatch handler instead. See :obj:`~.Predicate`.
"""
SymPyDeprecationWarning(
feature="remove_handler() function",
useinstead="multipledispatch handler of Predicate",
issue=20873,
deprecated_since_version="1.8"
).warn()
if isinstance(key, Predicate):
key = key.name.name
getattr(Q, key).remove_handler(handler)
from sympy.assumptions.ask_generated import (
get_known_facts_dict, get_all_known_facts)
|
74d4ccb74b7df07b701a7737b4a3747c47a0a88b35f53c97eb990e51be70d1ec | """
Module to evaluate the proposition with assumptions using SAT algorithm.
"""
from sympy import Symbol, S
from sympy.assumptions.ask_generated import get_all_known_facts
from sympy.assumptions.assume import global_assumptions, AppliedPredicate
from sympy.assumptions.sathandlers import class_fact_registry
from sympy.core import oo
from sympy.logic.inference import satisfiable
from sympy.assumptions.cnf import CNF, EncodedCNF
def satask(proposition, assumptions=True, context=global_assumptions,
use_known_facts=True, iterations=oo):
"""
Function to evaluate the proposition with assumptions using SAT algorithm.
This function extracts every fact relevant to the expressions composing
proposition and assumptions. For example, if a predicate containing
``Abs(x)`` is proposed, then ``Q.zero(Abs(x)) | Q.positive(Abs(x))``
will be found and passed to SAT solver because ``Q.nonnegative`` is
registered as a fact for ``Abs``.
Proposition is evaluated to ``True`` or ``False`` if the truth value can be
determined. If not, ``None`` is returned.
Parameters
==========
proposition : Any boolean expression.
Proposition which will be evaluated to boolean value.
assumptions : Any boolean expression, optional.
Local assumptions to evaluate the *proposition*.
context : AssumptionsContext, optional.
Default assumptions to evaluate the *proposition*. By default,
this is ``sympy.assumptions.global_assumptions`` variable.
use_known_facts : bool, optional.
If ``True``, facts from ``sympy.assumptions.ask_generated`` module
are passed to SAT solver as well.
iterations : int, optional.
Number of times that relevant facts are recursively extracted.
Default is infinite times until no new fact is found.
Returns
=======
``True``, ``False``, or ``None``
Examples
========
>>> from sympy import Abs, Q
>>> from sympy.assumptions.satask import satask
>>> from sympy.abc import x
>>> satask(Q.zero(Abs(x)), Q.zero(x))
True
"""
props = CNF.from_prop(proposition)
_props = CNF.from_prop(~proposition)
assumptions = CNF.from_prop(assumptions)
context_cnf = CNF()
if context:
context_cnf = context_cnf.extend(context)
sat = get_all_relevant_facts(props, assumptions, context_cnf,
use_known_facts=use_known_facts, iterations=iterations)
sat.add_from_cnf(assumptions)
if context:
sat.add_from_cnf(context_cnf)
return check_satisfiability(props, _props, sat)
def check_satisfiability(prop, _prop, factbase):
sat_true = factbase.copy()
sat_false = factbase.copy()
sat_true.add_from_cnf(prop)
sat_false.add_from_cnf(_prop)
can_be_true = satisfiable(sat_true)
can_be_false = satisfiable(sat_false)
if can_be_true and can_be_false:
return None
if can_be_true and not can_be_false:
return True
if not can_be_true and can_be_false:
return False
if not can_be_true and not can_be_false:
# TODO: Run additional checks to see which combination of the
# assumptions, global_assumptions, and relevant_facts are
# inconsistent.
raise ValueError("Inconsistent assumptions")
def extract_predargs(proposition, assumptions=None, context=None):
"""
Extract every expression in the argument of predicates from *proposition*,
*assumptions* and *context*.
Parameters
==========
proposition : sympy.assumptions.cnf.CNF
assumptions : sympy.assumptions.cnf.CNF, optional.
context : sympy.assumptions.cnf.CNF, optional.
CNF generated from assumptions context.
Examples
========
>>> from sympy import Q, Abs
>>> from sympy.assumptions.cnf import CNF
>>> from sympy.assumptions.satask import extract_predargs
>>> from sympy.abc import x, y
>>> props = CNF.from_prop(Q.zero(Abs(x*y)))
>>> assump = CNF.from_prop(Q.zero(x) & Q.zero(y))
>>> extract_predargs(props, assump)
{x, y, Abs(x*y)}
"""
req_keys = find_symbols(proposition)
keys = proposition.all_predicates()
# XXX: We need this since True/False are not Basic
lkeys = set()
if assumptions:
lkeys |= assumptions.all_predicates()
if context:
lkeys |= context.all_predicates()
lkeys = lkeys - {S.true, S.false}
tmp_keys = None
while tmp_keys != set():
tmp = set()
for l in lkeys:
syms = find_symbols(l)
if (syms & req_keys) != set():
tmp |= syms
tmp_keys = tmp - req_keys
req_keys |= tmp_keys
keys |= {l for l in lkeys if find_symbols(l) & req_keys != set()}
exprs = set()
for key in keys:
if isinstance(key, AppliedPredicate):
exprs |= set(key.arguments)
else:
exprs.add(key)
return exprs
def find_symbols(pred):
"""
Find every :obj:`~.Symbol` in *pred*.
Parameters
==========
pred : sympy.assumptions.cnf.CNF, or any Expr.
"""
if isinstance(pred, CNF):
symbols = set()
for a in pred.all_predicates():
symbols |= find_symbols(a)
return symbols
return pred.atoms(Symbol)
def get_relevant_clsfacts(exprs, relevant_facts=None):
"""
Extract relevant facts from the items in *exprs*. Facts are defined in
``assumptions.sathandlers`` module.
This function is recursively called by ``get_all_relevant_facts()``.
Parameters
==========
exprs : set
Expressions whose relevant facts are searched.
relevant_facts : sympy.assumptions.cnf.CNF, optional.
Pre-discovered relevant facts.
Returns
=======
exprs : set
Candidates for next relevant fact searching.
relevant_facts : sympy.assumptions.cnf.CNF
Updated relevant facts.
Examples
========
Here, we will see how facts relevant to ``Abs(x*y)`` are recursively
extracted. On the first run, set containing the expression is passed
without pre-discovered relevant facts. The result is a set containig
candidates for next run, and ``CNF()`` instance containing facts
which are relevant to ``Abs`` and its argument.
>>> from sympy import Abs
>>> from sympy.assumptions.satask import get_relevant_clsfacts
>>> from sympy.abc import x, y
>>> exprs = {Abs(x*y)}
>>> exprs, facts = get_relevant_clsfacts(exprs)
>>> exprs
{x*y}
>>> facts.clauses #doctest: +SKIP
{frozenset({Literal(Q.odd(Abs(x*y)), False), Literal(Q.odd(x*y), True)}),
frozenset({Literal(Q.zero(Abs(x*y)), False), Literal(Q.zero(x*y), True)}),
frozenset({Literal(Q.even(Abs(x*y)), False), Literal(Q.even(x*y), True)}),
frozenset({Literal(Q.zero(Abs(x*y)), True), Literal(Q.zero(x*y), False)}),
frozenset({Literal(Q.even(Abs(x*y)), False),
Literal(Q.odd(Abs(x*y)), False),
Literal(Q.odd(x*y), True)}),
frozenset({Literal(Q.even(Abs(x*y)), False),
Literal(Q.even(x*y), True),
Literal(Q.odd(Abs(x*y)), False)}),
frozenset({Literal(Q.positive(Abs(x*y)), False),
Literal(Q.zero(Abs(x*y)), False)})}
We pass the first run's results to the second run, and get the expressions
for next run and updated facts.
>>> exprs, facts = get_relevant_clsfacts(exprs, relevant_facts=facts)
>>> exprs
{x, y}
On final run, no more candidate is returned thus we know that all
relevant facts are successfully retrieved.
>>> exprs, facts = get_relevant_clsfacts(exprs, relevant_facts=facts)
>>> exprs
set()
"""
if not relevant_facts:
relevant_facts = CNF()
newexprs = set()
for expr in exprs:
for fact in class_fact_registry(expr):
newfact = CNF.to_CNF(fact)
relevant_facts = relevant_facts._and(newfact)
for key in newfact.all_predicates():
if isinstance(key, AppliedPredicate):
newexprs |= set(key.arguments)
return newexprs - exprs, relevant_facts
def get_all_relevant_facts(proposition, assumptions, context,
use_known_facts=True, iterations=oo):
"""
Extract all relevant facts from *proposition* and *assumptions*.
This function extracts the facts by recursively calling
``get_relevant_clsfacts()``. Extracted facts are converted to
``EncodedCNF`` and returned.
Parameters
==========
proposition : sympy.assumptions.cnf.CNF
CNF generated from proposition expression.
assumptions : sympy.assumptions.cnf.CNF
CNF generated from assumption expression.
context : sympy.assumptions.cnf.CNF
CNF generated from assumptions context.
use_known_facts : bool, optional.
If ``True``, facts from ``sympy.assumptions.ask_generated`` module
are encoded as well.
iterations : int, optional.
Number of times that relevant facts are recursively extracted.
Default is infinite times until no new fact is found.
Returns
=======
sympy.assumptions.cnf.EncodedCNF
Examples
========
>>> from sympy import Q
>>> from sympy.assumptions.cnf import CNF
>>> from sympy.assumptions.satask import get_all_relevant_facts
>>> from sympy.abc import x, y
>>> props = CNF.from_prop(Q.nonzero(x*y))
>>> assump = CNF.from_prop(Q.nonzero(x))
>>> context = CNF.from_prop(Q.nonzero(y))
>>> get_all_relevant_facts(props, assump, context) #doctest: +SKIP
<sympy.assumptions.cnf.EncodedCNF at 0x7f09faa6ccd0>
"""
# The relevant facts might introduce new keys, e.g., Q.zero(x*y) will
# introduce the keys Q.zero(x) and Q.zero(y), so we need to run it until
# we stop getting new things. Hopefully this strategy won't lead to an
# infinite loop in the future.
i = 0
relevant_facts = CNF()
all_exprs = set()
while True:
if i == 0:
exprs = extract_predargs(proposition, assumptions, context)
all_exprs |= exprs
exprs, relevant_facts = get_relevant_clsfacts(exprs, relevant_facts)
i += 1
if i >= iterations:
break
if not exprs:
break
if use_known_facts:
known_facts_CNF = CNF()
known_facts_CNF.add_clauses(get_all_known_facts())
kf_encoded = EncodedCNF()
kf_encoded.from_cnf(known_facts_CNF)
def translate_literal(lit, delta):
if lit > 0:
return lit + delta
else:
return lit - delta
def translate_data(data, delta):
return [{translate_literal(i, delta) for i in clause} for clause in data]
data = []
symbols = []
n_lit = len(kf_encoded.symbols)
for i, expr in enumerate(all_exprs):
symbols += [pred(expr) for pred in kf_encoded.symbols]
data += translate_data(kf_encoded.data, i * n_lit)
encoding = dict(list(zip(symbols, range(1, len(symbols)+1))))
ctx = EncodedCNF(data, encoding)
else:
ctx = EncodedCNF()
ctx.add_from_cnf(relevant_facts)
return ctx
|
7b754cd7fadd68235c6047aad54e81327052c73cf9dd6217c987023f836e1912 | """
Known facts in assumptions module.
This module defines the facts in ``get_known_facts()``, and supports functions
to generate the contents in ``sympy.assumptions.ask_generated`` file.
"""
from sympy.core.cache import cacheit
from sympy.assumptions import Q
from sympy.assumptions.cnf import CNF
from sympy.logic.boolalg import (to_cnf, And, Not, Implies, Equivalent)
from sympy.logic.inference import satisfiable
@cacheit
def get_composite_predicates():
# To reduce the complexity of sat solver, these predicates never goes into facts
# but are transformed into the combination of primitive predicates.
return {
Q.real : Q.negative | Q.zero | Q.positive,
Q.integer : Q.even | Q.odd,
Q.nonpositive : Q.negative | Q.zero,
Q.nonzero : Q.negative | Q.positive,
Q.nonnegative : Q.zero | Q.positive,
Q.extended_real : Q.negative_infinite | Q.negative | Q.zero | Q.positive | Q.positive_infinite,
Q.extended_positive: Q.positive | Q.positive_infinite,
Q.extended_negative: Q.negative | Q.negative_infinite,
Q.extended_nonzero: Q.negative_infinite | Q.negative | Q.positive | Q.positive_infinite,
Q.extended_nonpositive: Q.negative_infinite | Q.negative | Q.zero,
Q.extended_nonnegative: Q.zero | Q.positive | Q.positive_infinite,
Q.complex : Q.algebraic | Q.transcendental
}
@cacheit
def get_known_facts():
# We build the facts starting with primitive predicates.
# DO NOT include the predicates in get_composite_predicates()'s keys here!
return And(
# primitive predicates exclude each other
Implies(Q.negative_infinite, ~Q.positive_infinite),
Implies(Q.negative, ~Q.zero & ~Q.positive),
Implies(Q.positive, ~Q.zero),
# build real line and complex plane
Implies(Q.negative | Q.zero | Q.positive, ~Q.imaginary),
Implies(Q.negative | Q.zero | Q.positive | Q.imaginary, Q.algebraic | Q.transcendental),
# other subsets of complex
Implies(Q.transcendental, ~Q.algebraic),
Implies(Q.irrational, ~Q.rational),
Equivalent(Q.rational | Q.irrational, Q.negative | Q.zero | Q.positive),
Implies(Q.rational, Q.algebraic),
# integers
Implies(Q.even, ~Q.odd),
Implies(Q.even | Q.odd, Q.rational),
Implies(Q.zero, Q.even),
Implies(Q.composite, ~Q.prime),
Implies(Q.composite | Q.prime, (Q.even | Q.odd) & Q.positive),
Implies(Q.even & Q.positive & ~Q.prime, Q.composite),
# hermitian and antihermitian
Implies(Q.negative | Q.zero | Q.positive, Q.hermitian),
Implies(Q.imaginary, Q.antihermitian),
Implies(Q.zero, Q.hermitian | Q.antihermitian),
# define finity and infinity, and build extended real line
Implies(Q.infinite, ~Q.finite),
Implies(Q.algebraic | Q.transcendental, Q.finite),
Implies(Q.negative_infinite | Q.positive_infinite, Q.infinite),
# commutativity
Implies(Q.finite | Q.infinite, Q.commutative),
# matrices
Implies(Q.orthogonal, Q.positive_definite),
Implies(Q.orthogonal, Q.unitary),
Implies(Q.unitary & Q.real_elements, Q.orthogonal),
Implies(Q.unitary, Q.normal),
Implies(Q.unitary, Q.invertible),
Implies(Q.normal, Q.square),
Implies(Q.diagonal, Q.normal),
Implies(Q.positive_definite, Q.invertible),
Implies(Q.diagonal, Q.upper_triangular),
Implies(Q.diagonal, Q.lower_triangular),
Implies(Q.lower_triangular, Q.triangular),
Implies(Q.upper_triangular, Q.triangular),
Implies(Q.triangular, Q.upper_triangular | Q.lower_triangular),
Implies(Q.upper_triangular & Q.lower_triangular, Q.diagonal),
Implies(Q.diagonal, Q.symmetric),
Implies(Q.unit_triangular, Q.triangular),
Implies(Q.invertible, Q.fullrank),
Implies(Q.invertible, Q.square),
Implies(Q.symmetric, Q.square),
Implies(Q.fullrank & Q.square, Q.invertible),
Equivalent(Q.invertible, ~Q.singular),
Implies(Q.integer_elements, Q.real_elements),
Implies(Q.real_elements, Q.complex_elements),
)
@cacheit
def get_known_facts_keys():
exclude = set()
for pred in get_composite_predicates():
exclude.add(pred)
for pred in [Q.eq, Q.ne, Q.gt, Q.lt, Q.ge, Q.le]:
# sat does not support polyadic predicates yet
exclude.add(pred)
result = []
for attr in Q.__class__.__dict__:
if attr.startswith('__'):
continue
pred = getattr(Q, attr)
if pred in exclude:
continue
result.append(pred)
return result
def compute_known_facts(known_facts, known_facts_keys):
"""Compute the various forms of knowledge compilation used by the
assumptions system.
Explanation
===========
This function is typically applied to the results of the ``get_known_facts``
and ``get_known_facts_keys`` functions defined at the bottom of
this file.
"""
from textwrap import dedent, wrap
fact_string = dedent('''\
"""
The contents of this file are the return value of
``sympy.assumptions.ask.compute_known_facts``.
Do NOT manually edit this file.
Instead, run ./bin/ask_update.py.
"""
from sympy.core.cache import cacheit
from sympy.assumptions.cnf import Literal
from sympy.assumptions.ask import Q
@cacheit
def get_all_known_facts():
"""
Known facts as CNF clauses. Used by satask.
"""
return {
%s
}
# -{ Known facts in compressed sets }-
@cacheit
def get_known_facts_dict():
"""
Logical implication as dictionary. Key implies every item in its value.
Used for quick lookup of single facts.
"""
return {
%s
}
''')
# Compute the known facts in CNF form for logical inference
LINE = ",\n "
HANG = ' '*8
cnf = to_cnf(known_facts)
cnf_ = CNF.to_CNF(known_facts)
p = LINE.join(sorted(['frozenset((' + ', '.join(str(lit) for lit in sorted(clause, key=str)) +'))' for clause in cnf_.clauses]))
mapping = single_fact_lookup(known_facts_keys, cnf)
items = sorted(mapping.items(), key=str)
keys = [str(i[0]) for i in items]
values = ['set(%s)' % sorted(i[1], key=str) for i in items]
m = LINE.join(['\n'.join(
wrap("{}: {}".format(k, v),
subsequent_indent=HANG,
break_long_words=False))
for k, v in zip(keys, values)]) + ','
return fact_string % (p, m)
def single_fact_lookup(known_facts_keys, known_facts_cnf):
# Return the dictionary for quick lookup of single fact
mapping = {}
for key in known_facts_keys:
mapping[key] = {key}
for other_key in known_facts_keys:
if other_key != key:
if ask_full_inference(other_key, key, known_facts_cnf):
mapping[key].add(other_key)
if ask_full_inference(~other_key, key, known_facts_cnf):
mapping[key].add(~other_key)
return mapping
def ask_full_inference(proposition, assumptions, known_facts_cnf):
"""
Method for inferring properties about objects.
"""
if not satisfiable(And(known_facts_cnf, assumptions, proposition)):
return False
if not satisfiable(And(known_facts_cnf, assumptions, Not(proposition))):
return True
return None
|
437df5797f39236f05ae3e2020d71183502b55b47b26a4dd59943ca42c39fb8f | """A module which implements predicates and assumption context."""
from contextlib import contextmanager
import inspect
from sympy.core.assumptions import ManagedProperties
from sympy.core.symbol import Str
from sympy.core.sympify import _sympify
from sympy.logic.boolalg import Boolean, false, true
from sympy.multipledispatch.dispatcher import Dispatcher, str_signature
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.utilities.iterables import is_sequence
from sympy.utilities.source import get_class
class AssumptionsContext(set):
"""
Set containing default assumptions which are applied to the ``ask()``
function.
Explanation
===========
This is used to represent global assumptions, but you can also use this
class to create your own local assumptions contexts. It is basically a thin
wrapper to Python's set, so see its documentation for advanced usage.
Examples
========
The default assumption context is ``global_assumptions``, which is initially empty:
>>> from sympy import ask, Q
>>> from sympy.assumptions import global_assumptions
>>> global_assumptions
AssumptionsContext()
You can add default assumptions:
>>> from sympy.abc import x
>>> global_assumptions.add(Q.real(x))
>>> global_assumptions
AssumptionsContext({Q.real(x)})
>>> ask(Q.real(x))
True
And remove them:
>>> global_assumptions.remove(Q.real(x))
>>> print(ask(Q.real(x)))
None
The ``clear()`` method removes every assumption:
>>> global_assumptions.add(Q.positive(x))
>>> global_assumptions
AssumptionsContext({Q.positive(x)})
>>> global_assumptions.clear()
>>> global_assumptions
AssumptionsContext()
See Also
========
assuming
"""
def add(self, *assumptions):
"""Add assumptions."""
for a in assumptions:
super().add(a)
def _sympystr(self, printer):
if not self:
return "%s()" % self.__class__.__name__
return "{}({})".format(self.__class__.__name__, printer._print_set(self))
global_assumptions = AssumptionsContext()
class AppliedPredicate(Boolean):
"""
The class of expressions resulting from applying ``Predicate`` to
the arguments. ``AppliedPredicate`` merely wraps its argument and
remain unevaluated. To evaluate it, use the ``ask()`` function.
Examples
========
>>> from sympy import Q, ask
>>> Q.integer(1)
Q.integer(1)
The ``function`` attribute returns the predicate, and the ``arguments``
attribute returns the tuple of arguments.
>>> type(Q.integer(1))
<class 'sympy.assumptions.assume.AppliedPredicate'>
>>> Q.integer(1).function
Q.integer
>>> Q.integer(1).arguments
(1,)
Applied predicates can be evaluated to a boolean value with ``ask``:
>>> ask(Q.integer(1))
True
"""
__slots__ = ()
def __new__(cls, predicate, *args):
if not isinstance(predicate, Predicate):
raise TypeError("%s is not a Predicate." % predicate)
args = map(_sympify, args)
return super().__new__(cls, predicate, *args)
@property
def arg(self):
"""
Return the expression used by this assumption.
Examples
========
>>> from sympy import Q, Symbol
>>> x = Symbol('x')
>>> a = Q.integer(x + 1)
>>> a.arg
x + 1
"""
# Will be deprecated
args = self._args
if len(args) == 2:
# backwards compatibility
return args[1]
raise TypeError("'arg' property is allowed only for unary predicates.")
@property
def function(self):
"""
Return the predicate.
"""
# Will be changed to self.args[0] after args overridding is removed
return self._args[0]
@property
def arguments(self):
"""
Return the arguments which are applied to the predicate.
"""
# Will be changed to self.args[1:] after args overridding is removed
return self._args[1:]
def _eval_ask(self, assumptions):
return self.function.eval(self.arguments, assumptions)
@property
def binary_symbols(self):
from .ask import Q
if self.function == Q.is_true:
i = self.arguments[0]
if i.is_Boolean or i.is_Symbol:
return i.binary_symbols
if self.function in (Q.eq, Q.ne):
if true in self.arguments or false in self.arguments:
if self.arguments[0].is_Symbol:
return {self.arguments[0]}
elif self.arguments[1].is_Symbol:
return {self.arguments[1]}
return set()
class PredicateMeta(ManagedProperties):
def __new__(cls, clsname, bases, dct):
# If handler is not defined, assign empty dispatcher.
if "handler" not in dct:
name = f"Ask{clsname.capitalize()}Handler"
handler = Dispatcher(name, doc="Handler for key %s" % name)
dct["handler"] = handler
dct["_orig_doc"] = dct.get("__doc__", "")
return super().__new__(cls, clsname, bases, dct)
@property
def __doc__(cls):
handler = cls.handler
doc = cls._orig_doc
if cls is not Predicate and handler is not None:
doc += "Handler\n"
doc += " =======\n\n"
# Append the handler's doc without breaking sphinx documentation.
docs = [" Multiply dispatched method: %s" % handler.name]
if handler.doc:
for line in handler.doc.splitlines():
if not line:
continue
docs.append(" %s" % line)
other = []
for sig in handler.ordering[::-1]:
func = handler.funcs[sig]
if func.__doc__:
s = ' Inputs: <%s>' % str_signature(sig)
lines = []
for line in func.__doc__.splitlines():
lines.append(" %s" % line)
s += "\n".join(lines)
docs.append(s)
else:
other.append(str_signature(sig))
if other:
othersig = " Other signatures:"
for line in other:
othersig += "\n * %s" % line
docs.append(othersig)
doc += '\n\n'.join(docs)
return doc
class Predicate(Boolean, metaclass=PredicateMeta):
"""
Base class for mathematical predicates. It also serves as a
constructor for undefined predicate objects.
Explanation
===========
Predicate is a function that returns a boolean value [1].
Predicate function is object, and it is instance of predicate class.
When a predicate is applied to arguments, ``AppliedPredicate``
instance is returned. This merely wraps the argument and remain
unevaluated. To obtain the truth value of applied predicate, use the
function ``ask``.
Evaluation of predicate is done by multiple dispatching. You can
register new handler to the predicate to support new types.
Every predicate in SymPy can be accessed via the property of ``Q``.
For example, ``Q.even`` returns the predicate which checks if the
argument is even number.
To define a predicate which can be evaluated, you must subclass this
class, make an instance of it, and register it to ``Q``. After then,
dispatch the handler by argument types.
If you directly construct predicate using this class, you will get
``UndefinedPredicate`` which cannot be dispatched. This is useful
when you are building boolean expressions which do not need to be
evaluated.
Examples
========
Applying and evaluating to boolean value:
>>> from sympy import Q, ask
>>> ask(Q.prime(7))
True
You can define a new predicate by subclassing and dispatching. Here,
we define a predicate for sexy primes [2] as an example.
>>> from sympy import Predicate, Integer
>>> class SexyPrimePredicate(Predicate):
... name = "sexyprime"
>>> Q.sexyprime = SexyPrimePredicate()
>>> @Q.sexyprime.register(Integer, Integer)
... def _(int1, int2, assumptions):
... args = sorted([int1, int2])
... if not all(ask(Q.prime(a), assumptions) for a in args):
... return False
... return args[1] - args[0] == 6
>>> ask(Q.sexyprime(5, 11))
True
Direct constructing returns ``UndefinedPredicate``, which can be
applied but cannot be dispatched.
>>> from sympy import Predicate, Integer
>>> Q.P = Predicate("P")
>>> type(Q.P)
<class 'sympy.assumptions.assume.UndefinedPredicate'>
>>> Q.P(1)
Q.P(1)
>>> Q.P.register(Integer)(lambda expr, assump: True)
Traceback (most recent call last):
...
TypeError: <class 'sympy.assumptions.assume.UndefinedPredicate'> cannot be dispatched.
References
==========
.. [1] https://en.wikipedia.org/wiki/Predicate_(mathematical_logic)
.. [2] https://en.wikipedia.org/wiki/Sexy_prime
"""
is_Atom = True
def __new__(cls, *args, **kwargs):
if cls is Predicate:
return UndefinedPredicate(*args, **kwargs)
obj = super().__new__(cls, *args)
return obj
@property
def name(self):
# May be overridden
return type(self).__name__
@classmethod
def register(cls, *types, **kwargs):
"""
Register the signature to the handler.
"""
if cls.handler is None:
raise TypeError("%s cannot be dispatched." % type(cls))
return cls.handler.register(*types, **kwargs)
@classmethod
def register_many(cls, *types, **kwargs):
"""
Register multiple signatures to same handler.
"""
def _(func):
for t in types:
if not is_sequence(t):
t = (t,) # for convenience, allow passing `type` to mean `(type,)`
cls.register(*t, **kwargs)(func)
return _
def __call__(self, *args):
return AppliedPredicate(self, *args)
def eval(self, args, assumptions=True):
"""
Evaluate ``self(*args)`` under the given assumptions.
This uses only direct resolution methods, not logical inference.
"""
result = None
try:
result = self.handler(*args, assumptions=assumptions)
except NotImplementedError:
pass
return result
def _eval_refine(self, assumptions):
# When Predicate is no longer Boolean, delete this method
return self
class UndefinedPredicate(Predicate):
"""
Predicate without handler.
Explanation
===========
This predicate is generated by using ``Predicate`` directly for
construction. It does not have a handler, and evaluating this with
arguments is done by SAT solver.
Examples
========
>>> from sympy import Predicate, Q
>>> Q.P = Predicate('P')
>>> Q.P.func
<class 'sympy.assumptions.assume.UndefinedPredicate'>
>>> Q.P.name
Str('P')
"""
handler = None
def __new__(cls, name, handlers=None):
# "handlers" parameter supports old design
if not isinstance(name, Str):
name = Str(name)
obj = super(Boolean, cls).__new__(cls, name)
obj.handlers = handlers or []
return obj
@property
def name(self):
return self.args[0]
def _hashable_content(self):
return (self.name,)
def __getnewargs__(self):
return (self.name,)
def __call__(self, expr):
return AppliedPredicate(self, expr)
def add_handler(self, handler):
SymPyDeprecationWarning(
feature="Predicate.add_handler() method",
useinstead="multipledispatch handler of Predicate",
issue=20873,
deprecated_since_version="1.8"
).warn()
self.handlers.append(handler)
def remove_handler(self, handler):
SymPyDeprecationWarning(
feature="Predicate.remove_handler() method",
useinstead="multipledispatch handler of Predicate",
issue=20873,
deprecated_since_version="1.8"
).warn()
self.handlers.remove(handler)
def eval(self, args, assumptions=True):
# Support for deprecated design
# When old design is removed, this will always return None
SymPyDeprecationWarning(
feature="Evaluating UndefinedPredicate",
useinstead="multipledispatch handler of Predicate",
issue=20873,
deprecated_since_version="1.8"
).warn()
expr, = args
res, _res = None, None
mro = inspect.getmro(type(expr))
for handler in self.handlers:
cls = get_class(handler)
for subclass in mro:
eval_ = getattr(cls, subclass.__name__, None)
if eval_ is None:
continue
res = eval_(expr, assumptions)
# Do not stop if value returned is None
# Try to check for higher classes
if res is None:
continue
if _res is None:
_res = res
elif res is None:
# since first resolutor was conclusive, we keep that value
res = _res
else:
# only check consistency if both resolutors have concluded
if _res != res:
raise ValueError('incompatible resolutors')
break
return res
@contextmanager
def assuming(*assumptions):
"""
Context manager for assumptions.
Examples
========
>>> from sympy.assumptions import assuming, Q, ask
>>> from sympy.abc import x, y
>>> print(ask(Q.integer(x + y)))
None
>>> with assuming(Q.integer(x), Q.integer(y)):
... print(ask(Q.integer(x + y)))
True
"""
old_global_assumptions = global_assumptions.copy()
global_assumptions.update(assumptions)
try:
yield
finally:
global_assumptions.clear()
global_assumptions.update(old_global_assumptions)
|
69a3a6c8227e4a7f37b779a3c225177eaa8e649780ce131a2faf2f66fca0daed | """
The classes used here are for the internal use of assumptions system
only and should not be used anywhere else as these don't possess the
signatures common to SymPy objects. For general use of logic constructs
please refer to sympy.logic classes And, Or, Not, etc.
"""
from itertools import combinations, product
from sympy import S, Nor, Nand, Xor, Implies, Equivalent, ITE
from sympy.core.relational import Eq, Ne, Gt, Lt, Ge, Le
from sympy.logic.boolalg import Or, And, Not, Xnor
from itertools import zip_longest
class Literal:
"""
The smallest element of a CNF object.
Parameters
==========
lit : Boolean expression
is_Not : bool
Examples
========
>>> from sympy import Q
>>> from sympy.assumptions.cnf import Literal
>>> from sympy.abc import x
>>> Literal(Q.even(x))
Literal(Q.even(x), False)
>>> Literal(~Q.even(x))
Literal(Q.even(x), True)
"""
def __new__(cls, lit, is_Not=False):
if isinstance(lit, Not):
lit = lit.args[0]
is_Not = True
elif isinstance(lit, (AND, OR, Literal)):
return ~lit if is_Not else lit
obj = super().__new__(cls)
obj.lit = lit
obj.is_Not = is_Not
return obj
@property
def arg(self):
return self.lit
def rcall(self, expr):
if callable(self.lit):
lit = self.lit(expr)
else:
try:
lit = self.lit.apply(expr)
except AttributeError:
lit = self.lit.rcall(expr)
return type(self)(lit, self.is_Not)
def __invert__(self):
is_Not = not self.is_Not
return Literal(self.lit, is_Not)
def __str__(self):
return '{}({}, {})'.format(type(self).__name__, self.lit, self.is_Not)
__repr__ = __str__
def __eq__(self, other):
return self.arg == other.arg and self.is_Not == other.is_Not
def __hash__(self):
h = hash((type(self).__name__, self.arg, self.is_Not))
return h
class OR:
"""
A low-level implementation for Or
"""
def __init__(self, *args):
self._args = args
@property
def args(self):
return sorted(self._args, key=str)
def rcall(self, expr):
return type(self)(*[arg.rcall(expr)
for arg in self._args
])
def __invert__(self):
return AND(*[~arg for arg in self._args])
def __hash__(self):
return hash((type(self).__name__,) + tuple(self.args))
def __eq__(self, other):
return self.args == other.args
def __str__(self):
s = '(' + ' | '.join([str(arg) for arg in self.args]) + ')'
return s
__repr__ = __str__
class AND:
"""
A low-level implementation for And
"""
def __init__(self, *args):
self._args = args
def __invert__(self):
return OR(*[~arg for arg in self._args])
@property
def args(self):
return sorted(self._args, key=str)
def rcall(self, expr):
return type(self)(*[arg.rcall(expr)
for arg in self._args
])
def __hash__(self):
return hash((type(self).__name__,) + tuple(self.args))
def __eq__(self, other):
return self.args == other.args
def __str__(self):
s = '('+' & '.join([str(arg) for arg in self.args])+')'
return s
__repr__ = __str__
def to_NNF(expr, composite_map={}):
"""
Generates the Negation Normal Form of any boolean expression in terms
of AND, OR, and Literal objects.
Examples
========
>>> from sympy import Q, Eq
>>> from sympy.assumptions.cnf import to_NNF
>>> from sympy.abc import x, y
>>> expr = Q.even(x) & ~Q.positive(x)
>>> to_NNF(expr)
(Literal(Q.even(x), False) & Literal(Q.positive(x), True))
Supported boolean objects are converted to corresponding predicates.
>>> to_NNF(Eq(x, y))
Literal(Q.eq(x, y), False)
If ``composite_map`` argument is given, ``to_NNF`` decomposes the
specified predicate into a combination of primitive predicates.
>>> cmap = {Q.nonpositive: Q.negative | Q.zero}
>>> to_NNF(Q.nonpositive, cmap)
(Literal(Q.negative, False) | Literal(Q.zero, False))
>>> to_NNF(Q.nonpositive(x), cmap)
(Literal(Q.negative(x), False) | Literal(Q.zero(x), False))
"""
from sympy.assumptions.ask import Q
from sympy.assumptions.assume import AppliedPredicate, Predicate
binrelpreds = {Eq: Q.eq, Ne: Q.ne, Gt: Q.gt, Lt: Q.lt, Ge: Q.ge, Le: Q.le}
if type(expr) in binrelpreds:
pred = binrelpreds[type(expr)]
expr = pred(*expr.args)
if isinstance(expr, Not):
arg = expr.args[0]
tmp = to_NNF(arg, composite_map) # Strategy: negate the NNF of expr
return ~tmp
if isinstance(expr, Or):
return OR(*[to_NNF(x, composite_map) for x in Or.make_args(expr)])
if isinstance(expr, And):
return AND(*[to_NNF(x, composite_map) for x in And.make_args(expr)])
if isinstance(expr, Nand):
tmp = AND(*[to_NNF(x, composite_map) for x in expr.args])
return ~tmp
if isinstance(expr, Nor):
tmp = OR(*[to_NNF(x, composite_map) for x in expr.args])
return ~tmp
if isinstance(expr, Xor):
cnfs = []
for i in range(0, len(expr.args) + 1, 2):
for neg in combinations(expr.args, i):
clause = [~to_NNF(s, composite_map) if s in neg else to_NNF(s, composite_map)
for s in expr.args]
cnfs.append(OR(*clause))
return AND(*cnfs)
if isinstance(expr, Xnor):
cnfs = []
for i in range(0, len(expr.args) + 1, 2):
for neg in combinations(expr.args, i):
clause = [~to_NNF(s, composite_map) if s in neg else to_NNF(s, composite_map)
for s in expr.args]
cnfs.append(OR(*clause))
return ~AND(*cnfs)
if isinstance(expr, Implies):
L, R = to_NNF(expr.args[0], composite_map), to_NNF(expr.args[1], composite_map)
return OR(~L, R)
if isinstance(expr, Equivalent):
cnfs = []
for a, b in zip_longest(expr.args, expr.args[1:], fillvalue=expr.args[0]):
a = to_NNF(a, composite_map)
b = to_NNF(b, composite_map)
cnfs.append(OR(~a, b))
return AND(*cnfs)
if isinstance(expr, ITE):
L = to_NNF(expr.args[0], composite_map)
M = to_NNF(expr.args[1], composite_map)
R = to_NNF(expr.args[2], composite_map)
return AND(OR(~L, M), OR(L, R))
if isinstance(expr, AppliedPredicate):
pred, args = expr.function, expr.arguments
newpred = composite_map.get(pred, None)
if newpred is not None:
return to_NNF(newpred.rcall(*args), composite_map)
if isinstance(expr, Predicate):
newpred = composite_map.get(expr, None)
if newpred is not None:
return to_NNF(newpred, composite_map)
return Literal(expr)
def distribute_AND_over_OR(expr):
"""
Distributes AND over OR in the NNF expression.
Returns the result( Conjunctive Normal Form of expression)
as a CNF object.
"""
if not isinstance(expr, (AND, OR)):
tmp = set()
tmp.add(frozenset((expr,)))
return CNF(tmp)
if isinstance(expr, OR):
return CNF.all_or(*[distribute_AND_over_OR(arg)
for arg in expr._args])
if isinstance(expr, AND):
return CNF.all_and(*[distribute_AND_over_OR(arg)
for arg in expr._args])
class CNF:
"""
Class to represent CNF of a Boolean expression.
Consists of set of clauses, which themselves are stored as
frozenset of Literal objects.
Examples
========
>>> from sympy import Q
>>> from sympy.assumptions.cnf import CNF
>>> from sympy.abc import x
>>> cnf = CNF.from_prop(Q.real(x) & ~Q.zero(x))
>>> cnf.clauses
{frozenset({Literal(Q.zero(x), True)}),
frozenset({Literal(Q.negative(x), False),
Literal(Q.positive(x), False), Literal(Q.zero(x), False)})}
"""
def __init__(self, clauses=None):
if not clauses:
clauses = set()
self.clauses = clauses
def add(self, prop):
clauses = CNF.to_CNF(prop).clauses
self.add_clauses(clauses)
def __str__(self):
s = ' & '.join(
['(' + ' | '.join([str(lit) for lit in clause]) +')'
for clause in self.clauses]
)
return s
def extend(self, props):
for p in props:
self.add(p)
return self
def copy(self):
return CNF(set(self.clauses))
def add_clauses(self, clauses):
self.clauses |= clauses
@classmethod
def from_prop(cls, prop):
res = cls()
res.add(prop)
return res
def __iand__(self, other):
self.add_clauses(other.clauses)
return self
def all_predicates(self):
predicates = set()
for c in self.clauses:
predicates |= {arg.lit for arg in c}
return predicates
def _or(self, cnf):
clauses = set()
for a, b in product(self.clauses, cnf.clauses):
tmp = set(a)
for t in b:
tmp.add(t)
clauses.add(frozenset(tmp))
return CNF(clauses)
def _and(self, cnf):
clauses = self.clauses.union(cnf.clauses)
return CNF(clauses)
def _not(self):
clss = list(self.clauses)
ll = set()
for x in clss[-1]:
ll.add(frozenset((~x,)))
ll = CNF(ll)
for rest in clss[:-1]:
p = set()
for x in rest:
p.add(frozenset((~x,)))
ll = ll._or(CNF(p))
return ll
def rcall(self, expr):
clause_list = list()
for clause in self.clauses:
lits = [arg.rcall(expr) for arg in clause]
clause_list.append(OR(*lits))
expr = AND(*clause_list)
return distribute_AND_over_OR(expr)
@classmethod
def all_or(cls, *cnfs):
b = cnfs[0].copy()
for rest in cnfs[1:]:
b = b._or(rest)
return b
@classmethod
def all_and(cls, *cnfs):
b = cnfs[0].copy()
for rest in cnfs[1:]:
b = b._and(rest)
return b
@classmethod
def to_CNF(cls, expr):
from sympy.assumptions.facts import get_composite_predicates
expr = to_NNF(expr, get_composite_predicates())
expr = distribute_AND_over_OR(expr)
return expr
@classmethod
def CNF_to_cnf(cls, cnf):
"""
Converts CNF object to SymPy's boolean expression
retaining the form of expression.
"""
def remove_literal(arg):
return Not(arg.lit) if arg.is_Not else arg.lit
return And(*(Or(*(remove_literal(arg) for arg in clause)) for clause in cnf.clauses))
class EncodedCNF:
"""
Class for encoding the CNF expression.
"""
def __init__(self, data=None, encoding=None):
if not data and not encoding:
data = list()
encoding = dict()
self.data = data
self.encoding = encoding
self._symbols = list(encoding.keys())
def from_cnf(self, cnf):
self._symbols = list(cnf.all_predicates())
n = len(self._symbols)
self.encoding = dict(list(zip(self._symbols, list(range(1, n + 1)))))
self.data = [self.encode(clause) for clause in cnf.clauses]
@property
def symbols(self):
return self._symbols
@property
def variables(self):
return range(1, len(self._symbols) + 1)
def copy(self):
new_data = [set(clause) for clause in self.data]
return EncodedCNF(new_data, dict(self.encoding))
def add_prop(self, prop):
cnf = CNF.from_prop(prop)
self.add_from_cnf(cnf)
def add_from_cnf(self, cnf):
clauses = [self.encode(clause) for clause in cnf.clauses]
self.data += clauses
def encode_arg(self, arg):
literal = arg.lit
value = self.encoding.get(literal, None)
if value is None:
n = len(self._symbols)
self._symbols.append(literal)
value = self.encoding[literal] = n + 1
if arg.is_Not:
return -value
else:
return value
def encode(self, clause):
return {self.encode_arg(arg) if not arg.lit == S.false else 0 for arg in clause}
|
7db11e8cecfec6f48a61aec1e648d87b23f0db4da248e29e2967b25728f00e6c | from collections import defaultdict
from sympy.assumptions.ask import Q
from sympy.core import (Add, Mul, Pow, Number, NumberSymbol, Symbol)
from sympy.core.numbers import ImaginaryUnit
from sympy.functions.elementary.complexes import Abs
from sympy.logic.boolalg import (Equivalent, And, Or, Implies)
from sympy.matrices.expressions import MatMul
# APIs here may be subject to change
### Helper functions ###
def allarg(symbol, fact, expr):
"""
Apply all argument of the expression to the fact structure.
Parameters
==========
symbol : Symbol
A placeholder symbol.
fact : Boolean
Resulting ``Boolean`` expression.
expr : Expr
Examples
========
>>> from sympy import Q
>>> from sympy.assumptions.sathandlers import allarg
>>> from sympy.abc import x, y
>>> allarg(x, Q.negative(x) | Q.positive(x), x*y)
(Q.negative(x) | Q.positive(x)) & (Q.negative(y) | Q.positive(y))
"""
return And(*[fact.subs(symbol, arg) for arg in expr.args])
def anyarg(symbol, fact, expr):
"""
Apply any argument of the expression to the fact structure.
Parameters
==========
symbol : Symbol
A placeholder symbol.
fact : Boolean
Resulting ``Boolean`` expression.
expr : Expr
Examples
========
>>> from sympy import Q
>>> from sympy.assumptions.sathandlers import anyarg
>>> from sympy.abc import x, y
>>> anyarg(x, Q.negative(x) & Q.positive(x), x*y)
(Q.negative(x) & Q.positive(x)) | (Q.negative(y) & Q.positive(y))
"""
return Or(*[fact.subs(symbol, arg) for arg in expr.args])
def exactlyonearg(symbol, fact, expr):
"""
Apply exactly one argument of the expression to the fact structure.
Parameters
==========
symbol : Symbol
A placeholder symbol.
fact : Boolean
Resulting ``Boolean`` expression.
expr : Expr
Examples
========
>>> from sympy import Q
>>> from sympy.assumptions.sathandlers import exactlyonearg
>>> from sympy.abc import x, y
>>> exactlyonearg(x, Q.positive(x), x*y)
(Q.positive(x) & ~Q.positive(y)) | (Q.positive(y) & ~Q.positive(x))
"""
pred_args = [fact.subs(symbol, arg) for arg in expr.args]
res = Or(*[And(pred_args[i], *[~lit for lit in pred_args[:i] +
pred_args[i+1:]]) for i in range(len(pred_args))])
return res
### Fact registry ###
class ClassFactRegistry:
"""
Register handlers against classes.
Explanation
===========
``register`` method registers the handler function for a class. Here,
handler function should return a single fact. ``multiregister`` method
registers the handler function for multiple classes. Here, handler function
should return a container of multiple facts.
``registry(expr)`` returns a set of facts for *expr*.
Examples
========
Here, we register the facts for ``Abs``.
>>> from sympy import Abs, Q
>>> from sympy.logic.boolalg import Equivalent
>>> from sympy.assumptions.sathandlers import ClassFactRegistry
>>> reg = ClassFactRegistry()
>>> @reg.register(Abs)
... def f1(expr):
... return Q.nonnegative(expr)
>>> @reg.register(Abs)
... def f2(expr):
... arg = expr.args[0]
... return Equivalent(~Q.zero(arg), ~Q.zero(expr))
Calling the registry with expression returns the defined facts for the
expression.
>>> from sympy.abc import x
>>> reg(Abs(x))
{Q.nonnegative(Abs(x)), Equivalent(~Q.zero(x), ~Q.zero(Abs(x)))}
Multiple facts can be registered at once by ``multiregister`` method.
>>> reg2 = ClassFactRegistry()
>>> @reg2.multiregister(Abs)
... def _(expr):
... arg = expr.args[0]
... return [Q.even(arg) >> Q.even(expr), Q.odd(arg) >> Q.odd(expr)]
>>> reg2(Abs(x))
{Implies(Q.even(x), Q.even(Abs(x))), Implies(Q.odd(x), Q.odd(Abs(x)))}
"""
def __init__(self):
self.singlefacts = defaultdict(frozenset)
self.multifacts = defaultdict(frozenset)
def register(self, cls):
def _(func):
self.singlefacts[cls] |= {func}
return func
return _
def multiregister(self, *classes):
def _(func):
for cls in classes:
self.multifacts[cls] |= {func}
return func
return _
def __getitem__(self, key):
ret1 = self.singlefacts[key]
for k in self.singlefacts:
if issubclass(key, k):
ret1 |= self.singlefacts[k]
ret2 = self.multifacts[key]
for k in self.multifacts:
if issubclass(key, k):
ret2 |= self.multifacts[k]
return ret1, ret2
def __call__(self, expr):
ret = set()
handlers1, handlers2 = self[expr.func]
for h in handlers1:
ret.add(h(expr))
for h in handlers2:
ret.update(h(expr))
return ret
class_fact_registry = ClassFactRegistry()
### Class fact registration ###
x = Symbol('x')
## Abs ##
@class_fact_registry.multiregister(Abs)
def _(expr):
arg = expr.args[0]
return [Q.nonnegative(expr),
Equivalent(~Q.zero(arg), ~Q.zero(expr)),
Q.even(arg) >> Q.even(expr),
Q.odd(arg) >> Q.odd(expr),
Q.integer(arg) >> Q.integer(expr),
]
### Add ##
@class_fact_registry.multiregister(Add)
def _(expr):
return [allarg(x, Q.positive(x), expr) >> Q.positive(expr),
allarg(x, Q.negative(x), expr) >> Q.negative(expr),
allarg(x, Q.real(x), expr) >> Q.real(expr),
allarg(x, Q.rational(x), expr) >> Q.rational(expr),
allarg(x, Q.integer(x), expr) >> Q.integer(expr),
exactlyonearg(x, ~Q.integer(x), expr) >> ~Q.integer(expr),
]
@class_fact_registry.register(Add)
def _(expr):
allarg_real = allarg(x, Q.real(x), expr)
onearg_irrational = exactlyonearg(x, Q.irrational(x), expr)
return Implies(allarg_real, Implies(onearg_irrational, Q.irrational(expr)))
### Mul ###
@class_fact_registry.multiregister(Mul)
def _(expr):
return [Equivalent(Q.zero(expr), anyarg(x, Q.zero(x), expr)),
allarg(x, Q.positive(x), expr) >> Q.positive(expr),
allarg(x, Q.real(x), expr) >> Q.real(expr),
allarg(x, Q.rational(x), expr) >> Q.rational(expr),
allarg(x, Q.integer(x), expr) >> Q.integer(expr),
exactlyonearg(x, ~Q.rational(x), expr) >> ~Q.integer(expr),
allarg(x, Q.commutative(x), expr) >> Q.commutative(expr),
]
@class_fact_registry.register(Mul)
def _(expr):
# Implicitly assumes Mul has more than one arg
# Would be allarg(x, Q.prime(x) | Q.composite(x)) except 1 is composite
# More advanced prime assumptions will require inequalities, as 1 provides
# a corner case.
allarg_prime = allarg(x, Q.prime(x), expr)
return Implies(allarg_prime, ~Q.prime(expr))
@class_fact_registry.register(Mul)
def _(expr):
# General Case: Odd number of imaginary args implies mul is imaginary(To be implemented)
allarg_imag_or_real = allarg(x, Q.imaginary(x) | Q.real(x), expr)
onearg_imaginary = exactlyonearg(x, Q.imaginary(x), expr)
return Implies(allarg_imag_or_real, Implies(onearg_imaginary, Q.imaginary(expr)))
@class_fact_registry.register(Mul)
def _(expr):
allarg_real = allarg(x, Q.real(x), expr)
onearg_irrational = exactlyonearg(x, Q.irrational(x), expr)
return Implies(allarg_real, Implies(onearg_irrational, Q.irrational(expr)))
@class_fact_registry.register(Mul)
def _(expr):
# Including the integer qualification means we don't need to add any facts
# for odd, since the assumptions already know that every integer is
# exactly one of even or odd.
allarg_integer = allarg(x, Q.integer(x), expr)
anyarg_even = anyarg(x, Q.even(x), expr)
return Implies(allarg_integer, Equivalent(anyarg_even, Q.even(expr)))
### MatMul ###
@class_fact_registry.register(MatMul)
def _(expr):
allarg_square = allarg(x, Q.square(x), expr)
allarg_invertible = allarg(x, Q.invertible(x), expr)
return Implies(allarg_square, Equivalent(Q.invertible(expr), allarg_invertible))
### Pow ###
@class_fact_registry.multiregister(Pow)
def _(expr):
base, exp = expr.base, expr.exp
return [
(Q.real(base) & Q.even(exp) & Q.nonnegative(exp)) >> Q.nonnegative(expr),
(Q.nonnegative(base) & Q.odd(exp) & Q.nonnegative(exp)) >> Q.nonnegative(expr),
(Q.nonpositive(base) & Q.odd(exp) & Q.nonnegative(exp)) >> Q.nonpositive(expr),
Equivalent(Q.zero(expr), Q.zero(base) & Q.positive(exp))
]
### Numbers ###
_old_assump_getters = {
Q.positive: lambda o: o.is_positive,
Q.zero: lambda o: o.is_zero,
Q.negative: lambda o: o.is_negative,
Q.rational: lambda o: o.is_rational,
Q.irrational: lambda o: o.is_irrational,
Q.even: lambda o: o.is_even,
Q.odd: lambda o: o.is_odd,
Q.imaginary: lambda o: o.is_imaginary,
Q.prime: lambda o: o.is_prime,
Q.composite: lambda o: o.is_composite,
}
@class_fact_registry.multiregister(Number, NumberSymbol, ImaginaryUnit)
def _(expr):
ret = []
for p, getter in _old_assump_getters.items():
pred = p(expr)
prop = getter(expr)
if prop is not None:
ret.append(Equivalent(pred, prop))
return ret
|
99e717a76001dcda41893e8b6f020030f0c5236d8d0abeba91461398f13984ee | """A functions module, includes all the standard functions.
Combinatorial - factorial, fibonacci, harmonic, bernoulli...
Elementary - hyperbolic, trigonometric, exponential, floor and ceiling, sqrt...
Special - gamma, zeta,spherical harmonics...
"""
from sympy.functions.combinatorial.factorials import (factorial, factorial2,
rf, ff, binomial, RisingFactorial, FallingFactorial, subfactorial)
from sympy.functions.combinatorial.numbers import (carmichael, fibonacci, lucas, tribonacci,
harmonic, bernoulli, bell, euler, catalan, genocchi, partition, motzkin)
from sympy.functions.elementary.miscellaneous import (sqrt, root, Min, Max,
Id, real_root, cbrt)
from sympy.functions.elementary.complexes import (re, im, sign, Abs,
conjugate, arg, polar_lift, periodic_argument, unbranched_argument,
principal_branch, transpose, adjoint, polarify, unpolarify)
from sympy.functions.elementary.trigonometric import (sin, cos, tan,
sec, csc, cot, sinc, asin, acos, atan, asec, acsc, acot, atan2)
from sympy.functions.elementary.exponential import (exp_polar, exp, log,
LambertW)
from sympy.functions.elementary.hyperbolic import (sinh, cosh, tanh, coth,
sech, csch, asinh, acosh, atanh, acoth, asech, acsch)
from sympy.functions.elementary.integers import floor, ceiling, frac
from sympy.functions.elementary.piecewise import Piecewise, piecewise_fold
from sympy.functions.special.error_functions import (erf, erfc, erfi, erf2,
erfinv, erfcinv, erf2inv, Ei, expint, E1, li, Li, Si, Ci, Shi, Chi,
fresnels, fresnelc)
from sympy.functions.special.gamma_functions import (gamma, lowergamma,
uppergamma, polygamma, loggamma, digamma, trigamma, multigamma)
from sympy.functions.special.zeta_functions import (dirichlet_eta, zeta,
lerchphi, polylog, stieltjes, riemann_xi)
from sympy.functions.special.tensor_functions import (Eijk, LeviCivita,
KroneckerDelta)
from sympy.functions.special.singularity_functions import SingularityFunction
from sympy.functions.special.delta_functions import DiracDelta, Heaviside
from sympy.functions.special.bsplines import bspline_basis, bspline_basis_set, interpolating_spline
from sympy.functions.special.bessel import (besselj, bessely, besseli, besselk,
hankel1, hankel2, jn, yn, jn_zeros, hn1, hn2, airyai, airybi, airyaiprime, airybiprime, marcumq)
from sympy.functions.special.hyper import hyper, meijerg, appellf1
from sympy.functions.special.polynomials import (legendre, assoc_legendre,
hermite, chebyshevt, chebyshevu, chebyshevu_root, chebyshevt_root,
laguerre, assoc_laguerre, gegenbauer, jacobi, jacobi_normalized)
from sympy.functions.special.spherical_harmonics import Ynm, Ynm_c, Znm
from sympy.functions.special.elliptic_integrals import (elliptic_k,
elliptic_f, elliptic_e, elliptic_pi)
from sympy.functions.special.beta_functions import beta, betainc, betainc_regularized
from sympy.functions.special.mathieu_functions import (mathieus, mathieuc,
mathieusprime, mathieucprime)
ln = log
__all__ = [
'factorial', 'factorial2', 'rf', 'ff', 'binomial', 'RisingFactorial',
'FallingFactorial', 'subfactorial',
'carmichael', 'fibonacci', 'lucas', 'motzkin', 'tribonacci', 'harmonic',
'bernoulli', 'bell', 'euler', 'catalan', 'genocchi', 'partition',
'sqrt', 'root', 'Min', 'Max', 'Id', 'real_root', 'cbrt',
'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', 'riemann_xi',
'Eijk', 'LeviCivita', 'KroneckerDelta',
'SingularityFunction',
'DiracDelta', 'Heaviside',
'bspline_basis', 'bspline_basis_set', 'interpolating_spline',
'besselj', 'bessely', 'besseli', 'besselk', 'hankel1', 'hankel2', 'jn',
'yn', 'jn_zeros', 'hn1', 'hn2', 'airyai', 'airybi', 'airyaiprime',
'airybiprime', 'marcumq',
'hyper', 'meijerg', 'appellf1',
'legendre', 'assoc_legendre', 'hermite', 'chebyshevt', 'chebyshevu',
'chebyshevu_root', 'chebyshevt_root', 'laguerre', 'assoc_laguerre',
'gegenbauer', 'jacobi', 'jacobi_normalized',
'Ynm', 'Ynm_c', 'Znm',
'elliptic_k', 'elliptic_f', 'elliptic_e', 'elliptic_pi',
'beta', 'betainc', 'betainc_regularized',
'mathieus', 'mathieuc', 'mathieusprime', 'mathieucprime',
]
|
f472476c85b313d0d17047e25b797c4ac639f283676ff9cff3ec2b94abbc45d9 | from sympy.core.add import Add
from sympy.core.compatibility import ordered
from sympy.core.function import expand_log
from sympy.core.power import Pow
from sympy.core.singleton import S
from sympy.core.symbol import Dummy
from sympy.functions.elementary.exponential import (LambertW, exp, log)
from sympy.functions.elementary.miscellaneous import root
from sympy.polys.polyroots import roots
from sympy.polys.polytools import Poly, factor
from sympy.core.function import _mexpand
from sympy.simplify.simplify import separatevars
from sympy.simplify.radsimp import collect
from sympy.simplify.simplify import powsimp
from sympy.solvers.solvers import solve, _invert
from sympy.utilities.iterables import uniq
def _filtered_gens(poly, symbol):
"""process the generators of ``poly``, returning the set of generators that
have ``symbol``. If there are two generators that are inverses of each other,
prefer the one that has no denominator.
Examples
========
>>> from sympy.solvers.bivariate import _filtered_gens
>>> from sympy import Poly, exp
>>> from sympy.abc import x
>>> _filtered_gens(Poly(x + 1/x + exp(x)), x)
{x, exp(x)}
"""
gens = {g for g in poly.gens if symbol in g.free_symbols}
for g in list(gens):
ag = 1/g
if g in gens and ag in gens:
if ag.as_numer_denom()[1] is not S.One:
g = ag
gens.remove(g)
return gens
def _mostfunc(lhs, func, X=None):
"""Returns the term in lhs which contains the most of the
func-type things e.g. log(log(x)) wins over log(x) if both terms appear.
``func`` can be a function (exp, log, etc...) or any other SymPy object,
like Pow.
If ``X`` is not ``None``, then the function returns the term composed with the
most ``func`` having the specified variable.
Examples
========
>>> from sympy.solvers.bivariate import _mostfunc
>>> from sympy.functions.elementary.exponential import exp
>>> from sympy.abc import x, y
>>> _mostfunc(exp(x) + exp(exp(x) + 2), exp)
exp(exp(x) + 2)
>>> _mostfunc(exp(x) + exp(exp(y) + 2), exp)
exp(exp(y) + 2)
>>> _mostfunc(exp(x) + exp(exp(y) + 2), exp, x)
exp(x)
>>> _mostfunc(x, exp, x) is None
True
>>> _mostfunc(exp(x) + exp(x*y), exp, x)
exp(x)
"""
fterms = [tmp for tmp in lhs.atoms(func) if (not X or
X.is_Symbol and X in tmp.free_symbols or
not X.is_Symbol and tmp.has(X))]
if len(fterms) == 1:
return fterms[0]
elif fterms:
return max(list(ordered(fterms)), key=lambda x: x.count(func))
return None
def _linab(arg, symbol):
"""Return ``a, b, X`` assuming ``arg`` can be written as ``a*X + b``
where ``X`` is a symbol-dependent factor and ``a`` and ``b`` are
independent of ``symbol``.
Examples
========
>>> from sympy.functions.elementary.exponential import exp
>>> from sympy.solvers.bivariate import _linab
>>> from sympy.abc import x, y
>>> from sympy import S
>>> _linab(S(2), x)
(2, 0, 1)
>>> _linab(2*x, x)
(2, 0, x)
>>> _linab(y + y*x + 2*x, x)
(y + 2, y, x)
>>> _linab(3 + 2*exp(x), x)
(2, 3, exp(x))
"""
from sympy.core.exprtools import factor_terms
arg = factor_terms(arg.expand())
ind, dep = arg.as_independent(symbol)
if arg.is_Mul and dep.is_Add:
a, b, x = _linab(dep, symbol)
return ind*a, ind*b, x
if not arg.is_Add:
b = 0
a, x = ind, dep
else:
b = ind
a, x = separatevars(dep).as_independent(symbol, as_Add=False)
if x.could_extract_minus_sign():
a = -a
x = -x
return a, b, x
def _lambert(eq, x):
"""
Given an expression assumed to be in the form
``F(X, a..f) = a*log(b*X + c) + d*X + f = 0``
where X = g(x) and x = g^-1(X), return the Lambert solution,
``x = g^-1(-c/b + (a/d)*W(d/(a*b)*exp(c*d/a/b)*exp(-f/a)))``.
"""
eq = _mexpand(expand_log(eq))
mainlog = _mostfunc(eq, log, x)
if not mainlog:
return [] # violated assumptions
other = eq.subs(mainlog, 0)
if isinstance(-other, log):
eq = (eq - other).subs(mainlog, mainlog.args[0])
mainlog = mainlog.args[0]
if not isinstance(mainlog, log):
return [] # violated assumptions
other = -(-other).args[0]
eq += other
if not x in other.free_symbols:
return [] # violated assumptions
d, f, X2 = _linab(other, x)
logterm = collect(eq - other, mainlog)
a = logterm.as_coefficient(mainlog)
if a is None or x in a.free_symbols:
return [] # violated assumptions
logarg = mainlog.args[0]
b, c, X1 = _linab(logarg, x)
if X1 != X2:
return [] # violated assumptions
# invert the generator X1 so we have x(u)
u = Dummy('rhs')
xusolns = solve(X1 - u, x)
# There are infinitely many branches for LambertW
# but only branches for k = -1 and 0 might be real. The k = 0
# branch is real and the k = -1 branch is real if the LambertW argumen
# in in range [-1/e, 0]. Since `solve` does not return infinite
# solutions we will only include the -1 branch if it tests as real.
# Otherwise, inclusion of any LambertW in the solution indicates to
# the user that there are imaginary solutions corresponding to
# different k values.
lambert_real_branches = [-1, 0]
sol = []
# solution of the given Lambert equation is like
# sol = -c/b + (a/d)*LambertW(arg, k),
# where arg = d/(a*b)*exp((c*d-b*f)/a/b) and k in lambert_real_branches.
# Instead of considering the single arg, `d/(a*b)*exp((c*d-b*f)/a/b)`,
# the individual `p` roots obtained when writing `exp((c*d-b*f)/a/b)`
# as `exp(A/p) = exp(A)**(1/p)`, where `p` is an Integer, are used.
# calculating args for LambertW
num, den = ((c*d-b*f)/a/b).as_numer_denom()
p, den = den.as_coeff_Mul()
e = exp(num/den)
t = Dummy('t')
args = [d/(a*b)*t for t in roots(t**p - e, t).keys()]
# calculating solutions from args
for arg in args:
for k in lambert_real_branches:
w = LambertW(arg, k)
if k and not w.is_real:
continue
rhs = -c/b + (a/d)*w
for xu in xusolns:
sol.append(xu.subs(u, rhs))
return sol
def _solve_lambert(f, symbol, gens):
"""Return solution to ``f`` if it is a Lambert-type expression
else raise NotImplementedError.
For ``f(X, a..f) = a*log(b*X + c) + d*X - f = 0`` the solution
for ``X`` is ``X = -c/b + (a/d)*W(d/(a*b)*exp(c*d/a/b)*exp(f/a))``.
There are a variety of forms for `f(X, a..f)` as enumerated below:
1a1)
if B**B = R for R not in [0, 1] (since those cases would already
be solved before getting here) then log of both sides gives
log(B) + log(log(B)) = log(log(R)) and
X = log(B), a = 1, b = 1, c = 0, d = 1, f = log(log(R))
1a2)
if B*(b*log(B) + c)**a = R then log of both sides gives
log(B) + a*log(b*log(B) + c) = log(R) and
X = log(B), d=1, f=log(R)
1b)
if a*log(b*B + c) + d*B = R and
X = B, f = R
2a)
if (b*B + c)*exp(d*B + g) = R then log of both sides gives
log(b*B + c) + d*B + g = log(R) and
X = B, a = 1, f = log(R) - g
2b)
if g*exp(d*B + h) - b*B = c then the log form is
log(g) + d*B + h - log(b*B + c) = 0 and
X = B, a = -1, f = -h - log(g)
3)
if d*p**(a*B + g) - b*B = c then the log form is
log(d) + (a*B + g)*log(p) - log(b*B + c) = 0 and
X = B, a = -1, d = a*log(p), f = -log(d) - g*log(p)
"""
def _solve_even_degree_expr(expr, t, symbol):
"""Return the unique solutions of equations derived from
``expr`` by replacing ``t`` with ``+/- symbol``.
Parameters
==========
expr : Expr
The expression which includes a dummy variable t to be
replaced with +symbol and -symbol.
symbol : Symbol
The symbol for which a solution is being sought.
Returns
=======
List of unique solution of the two equations generated by
replacing ``t`` with positive and negative ``symbol``.
Notes
=====
If ``expr = 2*log(t) + x/2` then solutions for
``2*log(x) + x/2 = 0`` and ``2*log(-x) + x/2 = 0`` are
returned by this function. Though this may seem
counter-intuitive, one must note that the ``expr`` being
solved here has been derived from a different expression. For
an expression like ``eq = x**2*g(x) = 1``, if we take the
log of both sides we obtain ``log(x**2) + log(g(x)) = 0``. If
x is positive then this simplifies to
``2*log(x) + log(g(x)) = 0``; the Lambert-solving routines will
return solutions for this, but we must also consider the
solutions for ``2*log(-x) + log(g(x))`` since those must also
be a solution of ``eq`` which has the same value when the ``x``
in ``x**2`` is negated. If `g(x)` does not have even powers of
symbol then we don't want to replace the ``x`` there with
``-x``. So the role of the ``t`` in the expression received by
this function is to mark where ``+/-x`` should be inserted
before obtaining the Lambert solutions.
"""
nlhs, plhs = [
expr.xreplace({t: sgn*symbol}) for sgn in (-1, 1)]
sols = _solve_lambert(nlhs, symbol, gens)
if plhs != nlhs:
sols.extend(_solve_lambert(plhs, symbol, gens))
# uniq is needed for a case like
# 2*log(t) - log(-z**2) + log(z + log(x) + log(z))
# where subtituting t with +/-x gives all the same solution;
# uniq, rather than list(set()), is used to maintain canonical
# order
return list(uniq(sols))
nrhs, lhs = f.as_independent(symbol, as_Add=True)
rhs = -nrhs
lamcheck = [tmp for tmp in gens
if (tmp.func in [exp, log] or
(tmp.is_Pow and symbol in tmp.exp.free_symbols))]
if not lamcheck:
raise NotImplementedError()
if lhs.is_Add or lhs.is_Mul:
# replacing all even_degrees of symbol with dummy variable t
# since these will need special handling; non-Add/Mul do not
# need this handling
t = Dummy('t', **symbol.assumptions0)
lhs = lhs.replace(
lambda i: # find symbol**even
i.is_Pow and i.base == symbol and i.exp.is_even,
lambda i: # replace t**even
t**i.exp)
if lhs.is_Add and lhs.has(t):
t_indep = lhs.subs(t, 0)
t_term = lhs - t_indep
_rhs = rhs - t_indep
if not t_term.is_Add and _rhs and not (
t_term.has(S.ComplexInfinity, S.NaN)):
eq = expand_log(log(t_term) - log(_rhs))
return _solve_even_degree_expr(eq, t, symbol)
elif lhs.is_Mul and rhs:
# this needs to happen whether t is present or not
lhs = expand_log(log(lhs), force=True)
rhs = log(rhs)
if lhs.has(t) and lhs.is_Add:
# it expanded from Mul to Add
eq = lhs - rhs
return _solve_even_degree_expr(eq, t, symbol)
# restore symbol in lhs
lhs = lhs.xreplace({t: symbol})
lhs = powsimp(factor(lhs, deep=True))
# make sure we have inverted as completely as possible
r = Dummy()
i, lhs = _invert(lhs - r, symbol)
rhs = i.xreplace({r: rhs})
# For the first forms:
#
# 1a1) B**B = R will arrive here as B*log(B) = log(R)
# lhs is Mul so take log of both sides:
# log(B) + log(log(B)) = log(log(R))
# 1a2) B*(b*log(B) + c)**a = R will arrive unchanged so
# lhs is Mul, so take log of both sides:
# log(B) + a*log(b*log(B) + c) = log(R)
# 1b) d*log(a*B + b) + c*B = R will arrive unchanged so
# lhs is Add, so isolate c*B and expand log of both sides:
# log(c) + log(B) = log(R - d*log(a*B + b))
soln = []
if not soln:
mainlog = _mostfunc(lhs, log, symbol)
if mainlog:
if lhs.is_Mul and rhs != 0:
soln = _lambert(log(lhs) - log(rhs), symbol)
elif lhs.is_Add:
other = lhs.subs(mainlog, 0)
if other and not other.is_Add and [
tmp for tmp in other.atoms(Pow)
if symbol in tmp.free_symbols]:
if not rhs:
diff = log(other) - log(other - lhs)
else:
diff = log(lhs - other) - log(rhs - other)
soln = _lambert(expand_log(diff), symbol)
else:
#it's ready to go
soln = _lambert(lhs - rhs, symbol)
# For the next forms,
#
# collect on main exp
# 2a) (b*B + c)*exp(d*B + g) = R
# lhs is mul, so take log of both sides:
# log(b*B + c) + d*B = log(R) - g
# 2b) g*exp(d*B + h) - b*B = R
# lhs is add, so add b*B to both sides,
# take the log of both sides and rearrange to give
# log(R + b*B) - d*B = log(g) + h
if not soln:
mainexp = _mostfunc(lhs, exp, symbol)
if mainexp:
lhs = collect(lhs, mainexp)
if lhs.is_Mul and rhs != 0:
soln = _lambert(expand_log(log(lhs) - log(rhs)), symbol)
elif lhs.is_Add:
# move all but mainexp-containing term to rhs
other = lhs.subs(mainexp, 0)
mainterm = lhs - other
rhs = rhs - other
if (mainterm.could_extract_minus_sign() and
rhs.could_extract_minus_sign()):
mainterm *= -1
rhs *= -1
diff = log(mainterm) - log(rhs)
soln = _lambert(expand_log(diff), symbol)
# For the last form:
#
# 3) d*p**(a*B + g) - b*B = c
# collect on main pow, add b*B to both sides,
# take log of both sides and rearrange to give
# a*B*log(p) - log(b*B + c) = -log(d) - g*log(p)
if not soln:
mainpow = _mostfunc(lhs, Pow, symbol)
if mainpow and symbol in mainpow.exp.free_symbols:
lhs = collect(lhs, mainpow)
if lhs.is_Mul and rhs != 0:
# b*B = 0
soln = _lambert(expand_log(log(lhs) - log(rhs)), symbol)
elif lhs.is_Add:
# move all but mainpow-containing term to rhs
other = lhs.subs(mainpow, 0)
mainterm = lhs - other
rhs = rhs - other
diff = log(mainterm) - log(rhs)
soln = _lambert(expand_log(diff), symbol)
if not soln:
raise NotImplementedError('%s does not appear to have a solution in '
'terms of LambertW' % f)
return list(ordered(soln))
def bivariate_type(f, x, y, *, first=True):
"""Given an expression, f, 3 tests will be done to see what type
of composite bivariate it might be, options for u(x, y) are::
x*y
x+y
x*y+x
x*y+y
If it matches one of these types, ``u(x, y)``, ``P(u)`` and dummy
variable ``u`` will be returned. Solving ``P(u)`` for ``u`` and
equating the solutions to ``u(x, y)`` and then solving for ``x`` or
``y`` is equivalent to solving the original expression for ``x`` or
``y``. If ``x`` and ``y`` represent two functions in the same
variable, e.g. ``x = g(t)`` and ``y = h(t)``, then if ``u(x, y) - p``
can be solved for ``t`` then these represent the solutions to
``P(u) = 0`` when ``p`` are the solutions of ``P(u) = 0``.
Only positive values of ``u`` are considered.
Examples
========
>>> from sympy.solvers.solvers import solve
>>> from sympy.solvers.bivariate import bivariate_type
>>> from sympy.abc import x, y
>>> eq = (x**2 - 3).subs(x, x + y)
>>> bivariate_type(eq, x, y)
(x + y, _u**2 - 3, _u)
>>> uxy, pu, u = _
>>> usol = solve(pu, u); usol
[sqrt(3)]
>>> [solve(uxy - s) for s in solve(pu, u)]
[[{x: -y + sqrt(3)}]]
>>> all(eq.subs(s).equals(0) for sol in _ for s in sol)
True
"""
u = Dummy('u', positive=True)
if first:
p = Poly(f, x, y)
f = p.as_expr()
_x = Dummy()
_y = Dummy()
rv = bivariate_type(Poly(f.subs({x: _x, y: _y}), _x, _y), _x, _y, first=False)
if rv:
reps = {_x: x, _y: y}
return rv[0].xreplace(reps), rv[1].xreplace(reps), rv[2]
return
p = f
f = p.as_expr()
# f(x*y)
args = Add.make_args(p.as_expr())
new = []
for a in args:
a = _mexpand(a.subs(x, u/y))
free = a.free_symbols
if x in free or y in free:
break
new.append(a)
else:
return x*y, Add(*new), u
def ok(f, v, c):
new = _mexpand(f.subs(v, c))
free = new.free_symbols
return None if (x in free or y in free) else new
# f(a*x + b*y)
new = []
d = p.degree(x)
if p.degree(y) == d:
a = root(p.coeff_monomial(x**d), d)
b = root(p.coeff_monomial(y**d), d)
new = ok(f, x, (u - b*y)/a)
if new is not None:
return a*x + b*y, new, u
# f(a*x*y + b*y)
new = []
d = p.degree(x)
if p.degree(y) == d:
for itry in range(2):
a = root(p.coeff_monomial(x**d*y**d), d)
b = root(p.coeff_monomial(y**d), d)
new = ok(f, x, (u - b*y)/a/y)
if new is not None:
return a*x*y + b*y, new, u
x, y = y, x
|
15a3238d6fd662a46b9da917d80969eaa818698261f6a34b88ce95c4b8f47648 | from sympy.core import (Function, Pow, sympify, Expr)
from sympy.core.relational import Relational
from sympy.polys import Poly, decompose
from sympy.utilities.misc import func_name
def decompogen(f, symbol):
"""
Computes General functional decomposition of ``f``.
Given an expression ``f``, returns a list ``[f_1, f_2, ..., f_n]``,
where::
f = f_1 o f_2 o ... f_n = f_1(f_2(... f_n))
Note: This is a General decomposition function. It also decomposes
Polynomials. For only Polynomial decomposition see ``decompose`` in polys.
Examples
========
>>> from sympy.solvers.decompogen import decompogen
>>> from sympy.abc import x
>>> from sympy import sqrt, sin, cos
>>> decompogen(sin(cos(x)), x)
[sin(x), cos(x)]
>>> decompogen(sin(x)**2 + sin(x) + 1, x)
[x**2 + x + 1, sin(x)]
>>> decompogen(sqrt(6*x**2 - 5), x)
[sqrt(x), 6*x**2 - 5]
>>> decompogen(sin(sqrt(cos(x**2 + 1))), x)
[sin(x), sqrt(x), cos(x), x**2 + 1]
>>> decompogen(x**4 + 2*x**3 - x - 1, x)
[x**2 - x - 1, x**2 + x]
"""
f = sympify(f)
if not isinstance(f, Expr) or isinstance(f, Relational):
raise TypeError('expecting Expr but got: `%s`' % func_name(f))
if symbol not in f.free_symbols:
return [f]
result = []
# ===== Simple Functions ===== #
if isinstance(f, (Function, Pow)):
from sympy import S
if f.is_Pow and f.base == S.Exp1:
arg = f.exp
else:
arg = f.args[0]
if arg == symbol:
return [f]
result += [f.subs(arg, symbol)] + decompogen(arg, symbol)
return result
# ===== Convert to Polynomial ===== #
fp = Poly(f)
gens = list(filter(lambda x: symbol in x.free_symbols , fp.gens))
if len(gens) == 1 and gens[0] != symbol:
f1 = f.subs(gens[0], symbol)
f2 = gens[0]
result += [f1] + decompogen(f2, symbol)
return result
# ===== Polynomial decompose() ====== #
try:
result += decompose(f)
return result
except ValueError:
return [f]
def compogen(g_s, symbol):
"""
Returns the composition of functions.
Given a list of functions ``g_s``, returns their composition ``f``,
where:
f = g_1 o g_2 o .. o g_n
Note: This is a General composition function. It also composes Polynomials.
For only Polynomial composition see ``compose`` in polys.
Examples
========
>>> from sympy.solvers.decompogen import compogen
>>> from sympy.abc import x
>>> from sympy import sqrt, sin, cos
>>> compogen([sin(x), cos(x)], x)
sin(cos(x))
>>> compogen([x**2 + x + 1, sin(x)], x)
sin(x)**2 + sin(x) + 1
>>> compogen([sqrt(x), 6*x**2 - 5], x)
sqrt(6*x**2 - 5)
>>> compogen([sin(x), sqrt(x), cos(x), x**2 + 1], x)
sin(sqrt(cos(x**2 + 1)))
>>> compogen([x**2 - x - 1, x**2 + x], x)
-x**2 - x + (x**2 + x)**2 - 1
"""
if len(g_s) == 1:
return g_s[0]
foo = g_s[0].subs(symbol, g_s[1])
if len(g_s) == 2:
return foo
return compogen([foo] + g_s[2:], symbol)
|
ed1c7df1b49fa905cc592e0af67c02eafaff908f954b0c81f86be749d9f817c6 | """
This module contains functions to:
- solve a single equation for a single variable, in any domain either real or complex.
- solve a single transcendental equation for a single variable in any domain either real or complex.
(currently supports solving in real domain only)
- solve a system of linear equations with N variables and M equations.
- solve a system of Non Linear Equations with N variables and M equations
"""
from sympy.core.sympify import sympify
from sympy.core import (S, Pow, Dummy, pi, Expr, Wild, Mul, Equality,
Add)
from sympy.core.containers import Tuple
from sympy.core.numbers import I, Number, Rational, oo
from sympy.core.function import (Lambda, expand_complex, AppliedUndef,
expand_log)
from sympy.core.mod import Mod
from sympy.core.numbers import igcd
from sympy.core.relational import Eq, Ne, Relational
from sympy.core.symbol import Symbol, _uniquely_named_symbol
from sympy.core.sympify import _sympify
from sympy.simplify.simplify import simplify, fraction, trigsimp
from sympy.simplify import powdenest, logcombine
from sympy.functions import (log, Abs, tan, cot, sin, cos, sec, csc, exp,
acos, asin, acsc, asec, arg,
piecewise_fold, Piecewise)
from sympy.functions.elementary.trigonometric import (TrigonometricFunction,
HyperbolicFunction)
from sympy.functions.elementary.miscellaneous import real_root
from sympy.logic.boolalg import And
from sympy.sets import (FiniteSet, EmptySet, imageset, Interval, Intersection,
Union, ConditionSet, ImageSet, Complement, Contains)
from sympy.sets.sets import Set, ProductSet
from sympy.matrices import Matrix, MatrixBase
from sympy.ntheory import totient
from sympy.ntheory.factor_ import divisors
from sympy.ntheory.residue_ntheory import discrete_log, nthroot_mod
from sympy.polys import (roots, Poly, degree, together, PolynomialError,
RootOf, factor, lcm, gcd)
from sympy.polys.polyerrors import CoercionFailed
from sympy.polys.polytools import invert
from sympy.polys.solvers import (sympy_eqs_to_ring, solve_lin_sys,
PolyNonlinearError)
from sympy.polys.matrices.linsolve import _linsolve
from sympy.solvers.solvers import (checksol, denoms, unrad,
_simple_dens, recast_to_symbols)
from sympy.solvers.polysys import solve_poly_system
from sympy.solvers.inequalities import solve_univariate_inequality
from sympy.utilities import filldedent
from sympy.utilities.iterables import numbered_symbols, has_dups
from sympy.calculus.util import periodicity, continuous_domain
from sympy.core.compatibility import ordered, default_sort_key, is_sequence
from types import GeneratorType
from collections import defaultdict
class NonlinearError(ValueError):
"""Raised when unexpectedly encountering nonlinear equations"""
pass
_rc = Dummy("R", real=True), Dummy("C", complex=True)
def _masked(f, *atoms):
"""Return ``f``, with all objects given by ``atoms`` replaced with
Dummy symbols, ``d``, and the list of replacements, ``(d, e)``,
where ``e`` is an object of type given by ``atoms`` in which
any other instances of atoms have been recursively replaced with
Dummy symbols, too. The tuples are ordered so that if they are
applied in sequence, the origin ``f`` will be restored.
Examples
========
>>> from sympy import cos
>>> from sympy.abc import x
>>> from sympy.solvers.solveset import _masked
>>> f = cos(cos(x) + 1)
>>> f, reps = _masked(cos(1 + cos(x)), cos)
>>> f
_a1
>>> reps
[(_a1, cos(_a0 + 1)), (_a0, cos(x))]
>>> for d, e in reps:
... f = f.xreplace({d: e})
>>> f
cos(cos(x) + 1)
"""
sym = numbered_symbols('a', cls=Dummy, real=True)
mask = []
for a in ordered(f.atoms(*atoms)):
for i in mask:
a = a.replace(*i)
mask.append((a, next(sym)))
for i, (o, n) in enumerate(mask):
f = f.replace(o, n)
mask[i] = (n, o)
mask = list(reversed(mask))
return f, mask
def _invert(f_x, y, x, domain=S.Complexes):
r"""
Reduce the complex valued equation ``f(x) = y`` to a set of equations
``{g(x) = h_1(y), g(x) = h_2(y), ..., g(x) = h_n(y) }`` where ``g(x)`` is
a simpler function than ``f(x)``. The return value is a tuple ``(g(x),
set_h)``, where ``g(x)`` is a function of ``x`` and ``set_h`` is
the set of function ``{h_1(y), h_2(y), ..., h_n(y)}``.
Here, ``y`` is not necessarily a symbol.
The ``set_h`` contains the functions, along with the information
about the domain in which they are valid, through set
operations. For instance, if ``y = Abs(x) - n`` is inverted
in the real domain, then ``set_h`` is not simply
`{-n, n}` as the nature of `n` is unknown; rather, it is:
`Intersection([0, oo) {n}) U Intersection((-oo, 0], {-n})`
By default, the complex domain is used which means that inverting even
seemingly simple functions like ``exp(x)`` will give very different
results from those obtained in the real domain.
(In the case of ``exp(x)``, the inversion via ``log`` is multi-valued
in the complex domain, having infinitely many branches.)
If you are working with real values only (or you are not sure which
function to use) you should probably set the domain to
``S.Reals`` (or use `invert\_real` which does that automatically).
Examples
========
>>> from sympy.solvers.solveset import invert_complex, invert_real
>>> from sympy.abc import x, y
>>> from sympy import exp
When does exp(x) == y?
>>> invert_complex(exp(x), y, x)
(x, ImageSet(Lambda(_n, I*(2*_n*pi + arg(y)) + log(Abs(y))), Integers))
>>> invert_real(exp(x), y, x)
(x, Intersection(FiniteSet(log(y)), Reals))
When does exp(x) == 1?
>>> invert_complex(exp(x), 1, x)
(x, ImageSet(Lambda(_n, 2*_n*I*pi), Integers))
>>> invert_real(exp(x), 1, x)
(x, FiniteSet(0))
See Also
========
invert_real, invert_complex
"""
x = sympify(x)
if not x.is_Symbol:
raise ValueError("x must be a symbol")
f_x = sympify(f_x)
if x not in f_x.free_symbols:
raise ValueError("Inverse of constant function doesn't exist")
y = sympify(y)
if x in y.free_symbols:
raise ValueError("y should be independent of x ")
if domain.is_subset(S.Reals):
x1, s = _invert_real(f_x, FiniteSet(y), x)
else:
x1, s = _invert_complex(f_x, FiniteSet(y), x)
if not isinstance(s, FiniteSet) or x1 != x:
return x1, s
# Avoid adding gratuitous intersections with S.Complexes. Actual
# conditions should be handled by the respective inverters.
if domain is S.Complexes:
return x1, s
else:
return x1, s.intersection(domain)
invert_complex = _invert
def invert_real(f_x, y, x, domain=S.Reals):
"""
Inverts a real-valued function. Same as _invert, but sets
the domain to ``S.Reals`` before inverting.
"""
return _invert(f_x, y, x, domain)
def _invert_real(f, g_ys, symbol):
"""Helper function for _invert."""
if f == symbol:
return (f, g_ys)
n = Dummy('n', real=True)
if isinstance(f, exp) or (f.is_Pow and f.base == S.Exp1):
return _invert_real(f.exp,
imageset(Lambda(n, log(n)), g_ys),
symbol)
if hasattr(f, 'inverse') and f.inverse() is not None and not isinstance(f, (
TrigonometricFunction,
HyperbolicFunction,
)):
if len(f.args) > 1:
raise ValueError("Only functions with one argument are supported.")
return _invert_real(f.args[0],
imageset(Lambda(n, f.inverse()(n)), g_ys),
symbol)
if isinstance(f, Abs):
return _invert_abs(f.args[0], g_ys, symbol)
if f.is_Add:
# f = g + h
g, h = f.as_independent(symbol)
if g is not S.Zero:
return _invert_real(h, imageset(Lambda(n, n - g), g_ys), symbol)
if f.is_Mul:
# f = g*h
g, h = f.as_independent(symbol)
if g is not S.One:
return _invert_real(h, imageset(Lambda(n, n/g), g_ys), symbol)
if f.is_Pow:
base, expo = f.args
base_has_sym = base.has(symbol)
expo_has_sym = expo.has(symbol)
if not expo_has_sym:
if expo.is_rational:
num, den = expo.as_numer_denom()
if den % 2 == 0 and num % 2 == 1 and den.is_zero is False:
root = Lambda(n, real_root(n, expo))
g_ys_pos = g_ys & Interval(0, oo)
res = imageset(root, g_ys_pos)
base_positive = solveset(base >= 0, symbol, S.Reals)
_inv, _set = _invert_real(base, res, symbol)
return (_inv, _set.intersect(base_positive))
if den % 2 == 1:
root = Lambda(n, real_root(n, expo))
res = imageset(root, g_ys)
if num % 2 == 0:
neg_res = imageset(Lambda(n, -n), res)
return _invert_real(base, res + neg_res, symbol)
if num % 2 == 1:
return _invert_real(base, res, symbol)
elif expo.is_irrational:
root = Lambda(n, real_root(n, expo))
g_ys_pos = g_ys & Interval(0, oo)
res = imageset(root, g_ys_pos)
return _invert_real(base, res, symbol)
else:
# indeterminate exponent, e.g. Float or parity of
# num, den of rational could not be determined
pass # use default return
if not base_has_sym:
rhs = g_ys.args[0]
if base.is_positive:
return _invert_real(expo,
imageset(Lambda(n, log(n, base, evaluate=False)), g_ys), symbol)
elif base.is_negative:
from sympy.core.power import integer_log
s, b = integer_log(rhs, base)
if b:
return _invert_real(expo, FiniteSet(s), symbol)
else:
return _invert_real(expo, S.EmptySet, symbol)
elif base.is_zero:
one = Eq(rhs, 1)
if one == S.true:
# special case: 0**x - 1
return _invert_real(expo, FiniteSet(0), symbol)
elif one == S.false:
return _invert_real(expo, S.EmptySet, symbol)
if isinstance(f, TrigonometricFunction):
if isinstance(g_ys, FiniteSet):
def inv(trig):
if isinstance(f, (sin, csc)):
F = asin if isinstance(f, sin) else acsc
return (lambda a: n*pi + (-1)**n*F(a),)
if isinstance(f, (cos, sec)):
F = acos if isinstance(f, cos) else asec
return (
lambda a: 2*n*pi + F(a),
lambda a: 2*n*pi - F(a),)
if isinstance(f, (tan, cot)):
return (lambda a: n*pi + f.inverse()(a),)
n = Dummy('n', integer=True)
invs = S.EmptySet
for L in inv(f):
invs += Union(*[imageset(Lambda(n, L(g)), S.Integers) for g in g_ys])
return _invert_real(f.args[0], invs, symbol)
return (f, g_ys)
def _invert_complex(f, g_ys, symbol):
"""Helper function for _invert."""
if f == symbol:
return (f, g_ys)
n = Dummy('n')
if f.is_Add:
# f = g + h
g, h = f.as_independent(symbol)
if g is not S.Zero:
return _invert_complex(h, imageset(Lambda(n, n - g), g_ys), symbol)
if f.is_Mul:
# f = g*h
g, h = f.as_independent(symbol)
if g is not S.One:
if g in {S.NegativeInfinity, S.ComplexInfinity, S.Infinity}:
return (h, S.EmptySet)
return _invert_complex(h, imageset(Lambda(n, n/g), g_ys), symbol)
if f.is_Pow:
base, expo = f.args
# special case: g**r = 0
# Could be improved like `_invert_real` to handle more general cases.
if expo.is_Rational and g_ys == FiniteSet(0):
if expo.is_positive:
return _invert_complex(base, g_ys, symbol)
if hasattr(f, 'inverse') and f.inverse() is not None and \
not isinstance(f, TrigonometricFunction) and \
not isinstance(f, HyperbolicFunction) and \
not isinstance(f, exp):
if len(f.args) > 1:
raise ValueError("Only functions with one argument are supported.")
return _invert_complex(f.args[0],
imageset(Lambda(n, f.inverse()(n)), g_ys), symbol)
if isinstance(f, exp) or (f.is_Pow and f.base == S.Exp1):
if isinstance(g_ys, ImageSet):
# can solve upto `(d*exp(exp(...(exp(a*x + b))...) + c)` format.
# Further can be improved to `(d*exp(exp(...(exp(a*x**n + b*x**(n-1) + ... + f))...) + c)`.
g_ys_expr = g_ys.lamda.expr
g_ys_vars = g_ys.lamda.variables
k = Dummy('k{}'.format(len(g_ys_vars)))
g_ys_vars_1 = (k,) + g_ys_vars
exp_invs = Union(*[imageset(Lambda((g_ys_vars_1,), (I*(2*k*pi + arg(g_ys_expr))
+ log(Abs(g_ys_expr)))), S.Integers**(len(g_ys_vars_1)))])
elif isinstance(g_ys, FiniteSet):
exp_invs = Union(*[imageset(Lambda(n, I*(2*n*pi + arg(g_y)) +
log(Abs(g_y))), S.Integers)
for g_y in g_ys if g_y != 0])
return _invert_complex(f.exp, exp_invs, symbol)
return (f, g_ys)
def _invert_abs(f, g_ys, symbol):
"""Helper function for inverting absolute value functions.
Returns the complete result of inverting an absolute value
function along with the conditions which must also be satisfied.
If it is certain that all these conditions are met, a `FiniteSet`
of all possible solutions is returned. If any condition cannot be
satisfied, an `EmptySet` is returned. Otherwise, a `ConditionSet`
of the solutions, with all the required conditions specified, is
returned.
"""
if not g_ys.is_FiniteSet:
# this could be used for FiniteSet, but the
# results are more compact if they aren't, e.g.
# ConditionSet(x, Contains(n, Interval(0, oo)), {-n, n}) vs
# Union(Intersection(Interval(0, oo), {n}), Intersection(Interval(-oo, 0), {-n}))
# for the solution of abs(x) - n
pos = Intersection(g_ys, Interval(0, S.Infinity))
parg = _invert_real(f, pos, symbol)
narg = _invert_real(-f, pos, symbol)
if parg[0] != narg[0]:
raise NotImplementedError
return parg[0], Union(narg[1], parg[1])
# check conditions: all these must be true. If any are unknown
# then return them as conditions which must be satisfied
unknown = []
for a in g_ys.args:
ok = a.is_nonnegative if a.is_Number else a.is_positive
if ok is None:
unknown.append(a)
elif not ok:
return symbol, S.EmptySet
if unknown:
conditions = And(*[Contains(i, Interval(0, oo))
for i in unknown])
else:
conditions = True
n = Dummy('n', real=True)
# this is slightly different than above: instead of solving
# +/-f on positive values, here we solve for f on +/- g_ys
g_x, values = _invert_real(f, Union(
imageset(Lambda(n, n), g_ys),
imageset(Lambda(n, -n), g_ys)), symbol)
return g_x, ConditionSet(g_x, conditions, values)
def domain_check(f, symbol, p):
"""Returns False if point p is infinite or any subexpression of f
is infinite or becomes so after replacing symbol with p. If none of
these conditions is met then True will be returned.
Examples
========
>>> from sympy import Mul, oo
>>> from sympy.abc import x
>>> from sympy.solvers.solveset import domain_check
>>> g = 1/(1 + (1/(x + 1))**2)
>>> domain_check(g, x, -1)
False
>>> domain_check(x**2, x, 0)
True
>>> domain_check(1/x, x, oo)
False
* The function relies on the assumption that the original form
of the equation has not been changed by automatic simplification.
>>> domain_check(x/x, x, 0) # x/x is automatically simplified to 1
True
* To deal with automatic evaluations use evaluate=False:
>>> domain_check(Mul(x, 1/x, evaluate=False), x, 0)
False
"""
f, p = sympify(f), sympify(p)
if p.is_infinite:
return False
return _domain_check(f, symbol, p)
def _domain_check(f, symbol, p):
# helper for domain check
if f.is_Atom and f.is_finite:
return True
elif f.subs(symbol, p).is_infinite:
return False
elif isinstance(f, Piecewise):
# Check the cases of the Piecewise in turn. There might be invalid
# expressions in later cases that don't apply e.g.
# solveset(Piecewise((0, Eq(x, 0)), (1/x, True)), x)
for expr, cond in f.args:
condsubs = cond.subs(symbol, p)
if condsubs is S.false:
continue
elif condsubs is S.true:
return _domain_check(expr, symbol, p)
else:
# We don't know which case of the Piecewise holds. On this
# basis we cannot decide whether any solution is in or out of
# the domain. Ideally this function would allow returning a
# symbolic condition for the validity of the solution that
# could be handled in the calling code. In the mean time we'll
# give this particular solution the benefit of the doubt and
# let it pass.
return True
else:
# TODO : We should not blindly recurse through all args of arbitrary expressions like this
return all([_domain_check(g, symbol, p)
for g in f.args])
def _is_finite_with_finite_vars(f, domain=S.Complexes):
"""
Return True if the given expression is finite. For symbols that
don't assign a value for `complex` and/or `real`, the domain will
be used to assign a value; symbols that don't assign a value
for `finite` will be made finite. All other assumptions are
left unmodified.
"""
def assumptions(s):
A = s.assumptions0
A.setdefault('finite', A.get('finite', True))
if domain.is_subset(S.Reals):
# if this gets set it will make complex=True, too
A.setdefault('real', True)
else:
# don't change 'real' because being complex implies
# nothing about being real
A.setdefault('complex', True)
return A
reps = {s: Dummy(**assumptions(s)) for s in f.free_symbols}
return f.xreplace(reps).is_finite
def _is_function_class_equation(func_class, f, symbol):
""" Tests whether the equation is an equation of the given function class.
The given equation belongs to the given function class if it is
comprised of functions of the function class which are multiplied by
or added to expressions independent of the symbol. In addition, the
arguments of all such functions must be linear in the symbol as well.
Examples
========
>>> from sympy.solvers.solveset import _is_function_class_equation
>>> from sympy import tan, sin, tanh, sinh, exp
>>> from sympy.abc import x
>>> from sympy.functions.elementary.trigonometric import (TrigonometricFunction,
... HyperbolicFunction)
>>> _is_function_class_equation(TrigonometricFunction, exp(x) + tan(x), x)
False
>>> _is_function_class_equation(TrigonometricFunction, tan(x) + sin(x), x)
True
>>> _is_function_class_equation(TrigonometricFunction, tan(x**2), x)
False
>>> _is_function_class_equation(TrigonometricFunction, tan(x + 2), x)
True
>>> _is_function_class_equation(HyperbolicFunction, tanh(x) + sinh(x), x)
True
"""
if f.is_Mul or f.is_Add:
return all(_is_function_class_equation(func_class, arg, symbol)
for arg in f.args)
if f.is_Pow:
if not f.exp.has(symbol):
return _is_function_class_equation(func_class, f.base, symbol)
else:
return False
if not f.has(symbol):
return True
if isinstance(f, func_class):
try:
g = Poly(f.args[0], symbol)
return g.degree() <= 1
except PolynomialError:
return False
else:
return False
def _solve_as_rational(f, symbol, domain):
""" solve rational functions"""
from sympy.core.function import _mexpand
f = together(_mexpand(f, recursive=True), deep=True)
g, h = fraction(f)
if not h.has(symbol):
try:
return _solve_as_poly(g, symbol, domain)
except NotImplementedError:
# The polynomial formed from g could end up having
# coefficients in a ring over which finding roots
# isn't implemented yet, e.g. ZZ[a] for some symbol a
return ConditionSet(symbol, Eq(f, 0), domain)
except CoercionFailed:
# contained oo, zoo or nan
return S.EmptySet
else:
valid_solns = _solveset(g, symbol, domain)
invalid_solns = _solveset(h, symbol, domain)
return valid_solns - invalid_solns
class _SolveTrig1Error(Exception):
"""Raised when _solve_trig1 heuristics do not apply"""
def _solve_trig(f, symbol, domain):
"""Function to call other helpers to solve trigonometric equations """
sol = None
try:
sol = _solve_trig1(f, symbol, domain)
except _SolveTrig1Error:
try:
sol = _solve_trig2(f, symbol, domain)
except ValueError:
raise NotImplementedError(filldedent('''
Solution to this kind of trigonometric equations
is yet to be implemented'''))
return sol
def _solve_trig1(f, symbol, domain):
"""Primary solver for trigonometric and hyperbolic equations
Returns either the solution set as a ConditionSet (auto-evaluated to a
union of ImageSets if no variables besides 'symbol' are involved) or
raises _SolveTrig1Error if f == 0 can't be solved.
Notes
=====
Algorithm:
1. Do a change of variable x -> mu*x in arguments to trigonometric and
hyperbolic functions, in order to reduce them to small integers. (This
step is crucial to keep the degrees of the polynomials of step 4 low.)
2. Rewrite trigonometric/hyperbolic functions as exponentials.
3. Proceed to a 2nd change of variable, replacing exp(I*x) or exp(x) by y.
4. Solve the resulting rational equation.
5. Use invert_complex or invert_real to return to the original variable.
6. If the coefficients of 'symbol' were symbolic in nature, add the
necessary consistency conditions in a ConditionSet.
"""
# Prepare change of variable
x = Dummy('x')
if _is_function_class_equation(HyperbolicFunction, f, symbol):
cov = exp(x)
inverter = invert_real if domain.is_subset(S.Reals) else invert_complex
else:
cov = exp(I*x)
inverter = invert_complex
f = trigsimp(f)
f_original = f
trig_functions = f.atoms(TrigonometricFunction, HyperbolicFunction)
trig_arguments = [e.args[0] for e in trig_functions]
# trigsimp may have reduced the equation to an expression
# that is independent of 'symbol' (e.g. cos**2+sin**2)
if not any(a.has(symbol) for a in trig_arguments):
return solveset(f_original, symbol, domain)
denominators = []
numerators = []
for ar in trig_arguments:
try:
poly_ar = Poly(ar, symbol)
except PolynomialError:
raise _SolveTrig1Error("trig argument is not a polynomial")
if poly_ar.degree() > 1: # degree >1 still bad
raise _SolveTrig1Error("degree of variable must not exceed one")
if poly_ar.degree() == 0: # degree 0, don't care
continue
c = poly_ar.all_coeffs()[0] # got the coefficient of 'symbol'
numerators.append(fraction(c)[0])
denominators.append(fraction(c)[1])
mu = lcm(denominators)/gcd(numerators)
f = f.subs(symbol, mu*x)
f = f.rewrite(exp)
f = together(f)
g, h = fraction(f)
y = Dummy('y')
g, h = g.expand(), h.expand()
g, h = g.subs(cov, y), h.subs(cov, y)
if g.has(x) or h.has(x):
raise _SolveTrig1Error("change of variable not possible")
solns = solveset_complex(g, y) - solveset_complex(h, y)
if isinstance(solns, ConditionSet):
raise _SolveTrig1Error("polynomial has ConditionSet solution")
if isinstance(solns, FiniteSet):
if any(isinstance(s, RootOf) for s in solns):
raise _SolveTrig1Error("polynomial results in RootOf object")
# revert the change of variable
cov = cov.subs(x, symbol/mu)
result = Union(*[inverter(cov, s, symbol)[1] for s in solns])
# In case of symbolic coefficients, the solution set is only valid
# if numerator and denominator of mu are non-zero.
if mu.has(Symbol):
syms = (mu).atoms(Symbol)
munum, muden = fraction(mu)
condnum = munum.as_independent(*syms, as_Add=False)[1]
condden = muden.as_independent(*syms, as_Add=False)[1]
cond = And(Ne(condnum, 0), Ne(condden, 0))
else:
cond = True
# Actual conditions are returned as part of the ConditionSet. Adding an
# intersection with C would only complicate some solution sets due to
# current limitations of intersection code. (e.g. #19154)
if domain is S.Complexes:
# This is a slight abuse of ConditionSet. Ideally this should
# be some kind of "PiecewiseSet". (See #19507 discussion)
return ConditionSet(symbol, cond, result)
else:
return ConditionSet(symbol, cond, Intersection(result, domain))
elif solns is S.EmptySet:
return S.EmptySet
else:
raise _SolveTrig1Error("polynomial solutions must form FiniteSet")
def _solve_trig2(f, symbol, domain):
"""Secondary helper to solve trigonometric equations,
called when first helper fails """
from sympy import ilcm, expand_trig, degree
f = trigsimp(f)
f_original = f
trig_functions = f.atoms(sin, cos, tan, sec, cot, csc)
trig_arguments = [e.args[0] for e in trig_functions]
denominators = []
numerators = []
# todo: This solver can be extended to hyperbolics if the
# analogous change of variable to tanh (instead of tan)
# is used.
if not trig_functions:
return ConditionSet(symbol, Eq(f_original, 0), domain)
# todo: The pre-processing below (extraction of numerators, denominators,
# gcd, lcm, mu, etc.) should be updated to the enhanced version in
# _solve_trig1. (See #19507)
for ar in trig_arguments:
try:
poly_ar = Poly(ar, symbol)
except PolynomialError:
raise ValueError("give up, we can't solve if this is not a polynomial in x")
if poly_ar.degree() > 1: # degree >1 still bad
raise ValueError("degree of variable inside polynomial should not exceed one")
if poly_ar.degree() == 0: # degree 0, don't care
continue
c = poly_ar.all_coeffs()[0] # got the coefficient of 'symbol'
try:
numerators.append(Rational(c).p)
denominators.append(Rational(c).q)
except TypeError:
return ConditionSet(symbol, Eq(f_original, 0), domain)
x = Dummy('x')
# ilcm() and igcd() require more than one argument
if len(numerators) > 1:
mu = Rational(2)*ilcm(*denominators)/igcd(*numerators)
else:
assert len(numerators) == 1
mu = Rational(2)*denominators[0]/numerators[0]
f = f.subs(symbol, mu*x)
f = f.rewrite(tan)
f = expand_trig(f)
f = together(f)
g, h = fraction(f)
y = Dummy('y')
g, h = g.expand(), h.expand()
g, h = g.subs(tan(x), y), h.subs(tan(x), y)
if g.has(x) or h.has(x):
return ConditionSet(symbol, Eq(f_original, 0), domain)
solns = solveset(g, y, S.Reals) - solveset(h, y, S.Reals)
if isinstance(solns, FiniteSet):
result = Union(*[invert_real(tan(symbol/mu), s, symbol)[1]
for s in solns])
dsol = invert_real(tan(symbol/mu), oo, symbol)[1]
if degree(h) > degree(g): # If degree(denom)>degree(num) then there
result = Union(result, dsol) # would be another sol at Lim(denom-->oo)
return Intersection(result, domain)
elif solns is S.EmptySet:
return S.EmptySet
else:
return ConditionSet(symbol, Eq(f_original, 0), S.Reals)
def _solve_as_poly(f, symbol, domain=S.Complexes):
"""
Solve the equation using polynomial techniques if it already is a
polynomial equation or, with a change of variables, can be made so.
"""
result = None
if f.is_polynomial(symbol):
solns = roots(f, symbol, cubics=True, quartics=True,
quintics=True, domain='EX')
num_roots = sum(solns.values())
if degree(f, symbol) <= num_roots:
result = FiniteSet(*solns.keys())
else:
poly = Poly(f, symbol)
solns = poly.all_roots()
if poly.degree() <= len(solns):
result = FiniteSet(*solns)
else:
result = ConditionSet(symbol, Eq(f, 0), domain)
else:
poly = Poly(f)
if poly is None:
result = ConditionSet(symbol, Eq(f, 0), domain)
gens = [g for g in poly.gens if g.has(symbol)]
if len(gens) == 1:
poly = Poly(poly, gens[0])
gen = poly.gen
deg = poly.degree()
poly = Poly(poly.as_expr(), poly.gen, composite=True)
poly_solns = FiniteSet(*roots(poly, cubics=True, quartics=True,
quintics=True).keys())
if len(poly_solns) < deg:
result = ConditionSet(symbol, Eq(f, 0), domain)
if gen != symbol:
y = Dummy('y')
inverter = invert_real if domain.is_subset(S.Reals) else invert_complex
lhs, rhs_s = inverter(gen, y, symbol)
if lhs == symbol:
result = Union(*[rhs_s.subs(y, s) for s in poly_solns])
else:
result = ConditionSet(symbol, Eq(f, 0), domain)
else:
result = ConditionSet(symbol, Eq(f, 0), domain)
if result is not None:
if isinstance(result, FiniteSet):
# this is to simplify solutions like -sqrt(-I) to sqrt(2)/2
# - sqrt(2)*I/2. We are not expanding for solution with symbols
# or undefined functions because that makes the solution more complicated.
# For example, expand_complex(a) returns re(a) + I*im(a)
if all([s.atoms(Symbol, AppliedUndef) == set() and not isinstance(s, RootOf)
for s in result]):
s = Dummy('s')
result = imageset(Lambda(s, expand_complex(s)), result)
if isinstance(result, FiniteSet) and domain != S.Complexes:
# Avoid adding gratuitous intersections with S.Complexes. Actual
# conditions should be handled elsewhere.
result = result.intersection(domain)
return result
else:
return ConditionSet(symbol, Eq(f, 0), domain)
def _solve_radical(f, unradf, symbol, solveset_solver):
""" Helper function to solve equations with radicals """
res = unradf
eq, cov = res if res else (f, [])
if not cov:
result = solveset_solver(eq, symbol) - \
Union(*[solveset_solver(g, symbol) for g in denoms(f, symbol)])
else:
y, yeq = cov
if not solveset_solver(y - I, y):
yreal = Dummy('yreal', real=True)
yeq = yeq.xreplace({y: yreal})
eq = eq.xreplace({y: yreal})
y = yreal
g_y_s = solveset_solver(yeq, symbol)
f_y_sols = solveset_solver(eq, y)
result = Union(*[imageset(Lambda(y, g_y), f_y_sols)
for g_y in g_y_s])
if isinstance(result, Complement) or isinstance(result,ConditionSet):
solution_set = result
else:
f_set = [] # solutions for FiniteSet
c_set = [] # solutions for ConditionSet
for s in result:
if checksol(f, symbol, s):
f_set.append(s)
else:
c_set.append(s)
solution_set = FiniteSet(*f_set) + ConditionSet(symbol, Eq(f, 0), FiniteSet(*c_set))
return solution_set
def _solve_abs(f, symbol, domain):
""" Helper function to solve equation involving absolute value function """
if not domain.is_subset(S.Reals):
raise ValueError(filldedent('''
Absolute values cannot be inverted in the
complex domain.'''))
p, q, r = Wild('p'), Wild('q'), Wild('r')
pattern_match = f.match(p*Abs(q) + r) or {}
f_p, f_q, f_r = [pattern_match.get(i, S.Zero) for i in (p, q, r)]
if not (f_p.is_zero or f_q.is_zero):
domain = continuous_domain(f_q, symbol, domain)
q_pos_cond = solve_univariate_inequality(f_q >= 0, symbol,
relational=False, domain=domain, continuous=True)
q_neg_cond = q_pos_cond.complement(domain)
sols_q_pos = solveset_real(f_p*f_q + f_r,
symbol).intersect(q_pos_cond)
sols_q_neg = solveset_real(f_p*(-f_q) + f_r,
symbol).intersect(q_neg_cond)
return Union(sols_q_pos, sols_q_neg)
else:
return ConditionSet(symbol, Eq(f, 0), domain)
def solve_decomposition(f, symbol, domain):
"""
Function to solve equations via the principle of "Decomposition
and Rewriting".
Examples
========
>>> from sympy import exp, sin, Symbol, pprint, S
>>> from sympy.solvers.solveset import solve_decomposition as sd
>>> x = Symbol('x')
>>> f1 = exp(2*x) - 3*exp(x) + 2
>>> sd(f1, x, S.Reals)
FiniteSet(0, log(2))
>>> f2 = sin(x)**2 + 2*sin(x) + 1
>>> pprint(sd(f2, x, S.Reals), use_unicode=False)
3*pi
{2*n*pi + ---- | n in Integers}
2
>>> f3 = sin(x + 2)
>>> pprint(sd(f3, x, S.Reals), use_unicode=False)
{2*n*pi - 2 | n in Integers} U {2*n*pi - 2 + pi | n in Integers}
"""
from sympy.solvers.decompogen import decompogen
from sympy.calculus.util import function_range
# decompose the given function
g_s = decompogen(f, symbol)
# `y_s` represents the set of values for which the function `g` is to be
# solved.
# `solutions` represent the solutions of the equations `g = y_s` or
# `g = 0` depending on the type of `y_s`.
# As we are interested in solving the equation: f = 0
y_s = FiniteSet(0)
for g in g_s:
frange = function_range(g, symbol, domain)
y_s = Intersection(frange, y_s)
result = S.EmptySet
if isinstance(y_s, FiniteSet):
for y in y_s:
solutions = solveset(Eq(g, y), symbol, domain)
if not isinstance(solutions, ConditionSet):
result += solutions
else:
if isinstance(y_s, ImageSet):
iter_iset = (y_s,)
elif isinstance(y_s, Union):
iter_iset = y_s.args
elif y_s is EmptySet:
# y_s is not in the range of g in g_s, so no solution exists
#in the given domain
return EmptySet
for iset in iter_iset:
new_solutions = solveset(Eq(iset.lamda.expr, g), symbol, domain)
dummy_var = tuple(iset.lamda.expr.free_symbols)[0]
(base_set,) = iset.base_sets
if isinstance(new_solutions, FiniteSet):
new_exprs = new_solutions
elif isinstance(new_solutions, Intersection):
if isinstance(new_solutions.args[1], FiniteSet):
new_exprs = new_solutions.args[1]
for new_expr in new_exprs:
result += ImageSet(Lambda(dummy_var, new_expr), base_set)
if result is S.EmptySet:
return ConditionSet(symbol, Eq(f, 0), domain)
y_s = result
return y_s
def _solveset(f, symbol, domain, _check=False):
"""Helper for solveset to return a result from an expression
that has already been sympify'ed and is known to contain the
given symbol."""
# _check controls whether the answer is checked or not
from sympy.simplify.simplify import signsimp
from sympy.logic.boolalg import BooleanTrue
if isinstance(f, BooleanTrue):
return domain
orig_f = f
if f.is_Mul:
coeff, f = f.as_independent(symbol, as_Add=False)
if coeff in {S.ComplexInfinity, S.NegativeInfinity, S.Infinity}:
f = together(orig_f)
elif f.is_Add:
a, h = f.as_independent(symbol)
m, h = h.as_independent(symbol, as_Add=False)
if m not in {S.ComplexInfinity, S.Zero, S.Infinity,
S.NegativeInfinity}:
f = a/m + h # XXX condition `m != 0` should be added to soln
# assign the solvers to use
solver = lambda f, x, domain=domain: _solveset(f, x, domain)
inverter = lambda f, rhs, symbol: _invert(f, rhs, symbol, domain)
result = EmptySet
if f.expand().is_zero:
return domain
elif not f.has(symbol):
return EmptySet
elif f.is_Mul and all(_is_finite_with_finite_vars(m, domain)
for m in f.args):
# if f(x) and g(x) are both finite we can say that the solution of
# f(x)*g(x) == 0 is same as Union(f(x) == 0, g(x) == 0) is not true in
# general. g(x) can grow to infinitely large for the values where
# f(x) == 0. To be sure that we are not silently allowing any
# wrong solutions we are using this technique only if both f and g are
# finite for a finite input.
result = Union(*[solver(m, symbol) for m in f.args])
elif _is_function_class_equation(TrigonometricFunction, f, symbol) or \
_is_function_class_equation(HyperbolicFunction, f, symbol):
result = _solve_trig(f, symbol, domain)
elif isinstance(f, arg):
a = f.args[0]
result = solveset_real(a > 0, symbol)
elif f.is_Piecewise:
expr_set_pairs = f.as_expr_set_pairs(domain)
for (expr, in_set) in expr_set_pairs:
if in_set.is_Relational:
in_set = in_set.as_set()
solns = solver(expr, symbol, in_set)
result += solns
elif isinstance(f, Eq):
result = solver(Add(f.lhs, - f.rhs, evaluate=False), symbol, domain)
elif f.is_Relational:
try:
result = solve_univariate_inequality(
f, symbol, domain=domain, relational=False)
except NotImplementedError:
result = ConditionSet(symbol, f, domain)
return result
elif _is_modular(f, symbol):
result = _solve_modular(f, symbol, domain)
else:
lhs, rhs_s = inverter(f, 0, symbol)
if lhs == symbol:
# do some very minimal simplification since
# repeated inversion may have left the result
# in a state that other solvers (e.g. poly)
# would have simplified; this is done here
# rather than in the inverter since here it
# is only done once whereas there it would
# be repeated for each step of the inversion
if isinstance(rhs_s, FiniteSet):
rhs_s = FiniteSet(*[Mul(*
signsimp(i).as_content_primitive())
for i in rhs_s])
result = rhs_s
elif isinstance(rhs_s, FiniteSet):
for equation in [lhs - rhs for rhs in rhs_s]:
if equation == f:
u = unrad(f)
if u:
result += _solve_radical(equation, u,
symbol,
solver)
elif equation.has(Abs):
result += _solve_abs(f, symbol, domain)
else:
result_rational = _solve_as_rational(equation, symbol, domain)
if not isinstance(result_rational, ConditionSet):
result += result_rational
else:
# may be a transcendental type equation
t_result = _transolve(equation, symbol, domain)
if isinstance(t_result, ConditionSet):
# might need factoring; this is expensive so we
# have delayed until now. To avoid recursion
# errors look for a non-trivial factoring into
# a product of symbol dependent terms; I think
# that something that factors as a Pow would
# have already been recognized by now.
factored = equation.factor()
if factored.is_Mul and equation != factored:
_, dep = factored.as_independent(symbol)
if not dep.is_Add:
# non-trivial factoring of equation
# but use form with constants
# in case they need special handling
t_result = solver(factored, symbol)
result += t_result
else:
result += solver(equation, symbol)
elif rhs_s is not S.EmptySet:
result = ConditionSet(symbol, Eq(f, 0), domain)
if isinstance(result, ConditionSet):
if isinstance(f, Expr):
num, den = f.as_numer_denom()
else:
num, den = f, S.One
if den.has(symbol):
_result = _solveset(num, symbol, domain)
if not isinstance(_result, ConditionSet):
singularities = _solveset(den, symbol, domain)
result = _result - singularities
if _check:
if isinstance(result, ConditionSet):
# it wasn't solved or has enumerated all conditions
# -- leave it alone
return result
# whittle away all but the symbol-containing core
# to use this for testing
if isinstance(orig_f, Expr):
fx = orig_f.as_independent(symbol, as_Add=True)[1]
fx = fx.as_independent(symbol, as_Add=False)[1]
else:
fx = orig_f
if isinstance(result, FiniteSet):
# check the result for invalid solutions
result = FiniteSet(*[s for s in result
if isinstance(s, RootOf)
or domain_check(fx, symbol, s)])
return result
def _is_modular(f, symbol):
"""
Helper function to check below mentioned types of modular equations.
``A - Mod(B, C) = 0``
A -> This can or cannot be a function of symbol.
B -> This is surely a function of symbol.
C -> It is an integer.
Parameters
==========
f : Expr
The equation to be checked.
symbol : Symbol
The concerned variable for which the equation is to be checked.
Examples
========
>>> from sympy import symbols, exp, Mod
>>> from sympy.solvers.solveset import _is_modular as check
>>> x, y = symbols('x y')
>>> check(Mod(x, 3) - 1, x)
True
>>> check(Mod(x, 3) - 1, y)
False
>>> check(Mod(x, 3)**2 - 5, x)
False
>>> check(Mod(x, 3)**2 - y, x)
False
>>> check(exp(Mod(x, 3)) - 1, x)
False
>>> check(Mod(3, y) - 1, y)
False
"""
if not f.has(Mod):
return False
# extract modterms from f.
modterms = list(f.atoms(Mod))
return (len(modterms) == 1 and # only one Mod should be present
modterms[0].args[0].has(symbol) and # B-> function of symbol
modterms[0].args[1].is_integer and # C-> to be an integer.
any(isinstance(term, Mod)
for term in list(_term_factors(f))) # free from other funcs
)
def _invert_modular(modterm, rhs, n, symbol):
"""
Helper function to invert modular equation.
``Mod(a, m) - rhs = 0``
Generally it is inverted as (a, ImageSet(Lambda(n, m*n + rhs), S.Integers)).
More simplified form will be returned if possible.
If it is not invertible then (modterm, rhs) is returned.
The following cases arise while inverting equation ``Mod(a, m) - rhs = 0``:
1. If a is symbol then m*n + rhs is the required solution.
2. If a is an instance of ``Add`` then we try to find two symbol independent
parts of a and the symbol independent part gets tranferred to the other
side and again the ``_invert_modular`` is called on the symbol
dependent part.
3. If a is an instance of ``Mul`` then same as we done in ``Add`` we separate
out the symbol dependent and symbol independent parts and transfer the
symbol independent part to the rhs with the help of invert and again the
``_invert_modular`` is called on the symbol dependent part.
4. If a is an instance of ``Pow`` then two cases arise as following:
- If a is of type (symbol_indep)**(symbol_dep) then the remainder is
evaluated with the help of discrete_log function and then the least
period is being found out with the help of totient function.
period*n + remainder is the required solution in this case.
For reference: (https://en.wikipedia.org/wiki/Euler's_theorem)
- If a is of type (symbol_dep)**(symbol_indep) then we try to find all
primitive solutions list with the help of nthroot_mod function.
m*n + rem is the general solution where rem belongs to solutions list
from nthroot_mod function.
Parameters
==========
modterm, rhs : Expr
The modular equation to be inverted, ``modterm - rhs = 0``
symbol : Symbol
The variable in the equation to be inverted.
n : Dummy
Dummy variable for output g_n.
Returns
=======
A tuple (f_x, g_n) is being returned where f_x is modular independent function
of symbol and g_n being set of values f_x can have.
Examples
========
>>> from sympy import symbols, exp, Mod, Dummy, S
>>> from sympy.solvers.solveset import _invert_modular as invert_modular
>>> x, y = symbols('x y')
>>> n = Dummy('n')
>>> invert_modular(Mod(exp(x), 7), S(5), n, x)
(Mod(exp(x), 7), 5)
>>> invert_modular(Mod(x, 7), S(5), n, x)
(x, ImageSet(Lambda(_n, 7*_n + 5), Integers))
>>> invert_modular(Mod(3*x + 8, 7), S(5), n, x)
(x, ImageSet(Lambda(_n, 7*_n + 6), Integers))
>>> invert_modular(Mod(x**4, 7), S(5), n, x)
(x, EmptySet)
>>> invert_modular(Mod(2**(x**2 + x + 1), 7), S(2), n, x)
(x**2 + x + 1, ImageSet(Lambda(_n, 3*_n + 1), Naturals0))
"""
a, m = modterm.args
if rhs.is_real is False or any(term.is_real is False
for term in list(_term_factors(a))):
# Check for complex arguments
return modterm, rhs
if abs(rhs) >= abs(m):
# if rhs has value greater than value of m.
return symbol, EmptySet
if a == symbol:
return symbol, ImageSet(Lambda(n, m*n + rhs), S.Integers)
if a.is_Add:
# g + h = a
g, h = a.as_independent(symbol)
if g is not S.Zero:
x_indep_term = rhs - Mod(g, m)
return _invert_modular(Mod(h, m), Mod(x_indep_term, m), n, symbol)
if a.is_Mul:
# g*h = a
g, h = a.as_independent(symbol)
if g is not S.One:
x_indep_term = rhs*invert(g, m)
return _invert_modular(Mod(h, m), Mod(x_indep_term, m), n, symbol)
if a.is_Pow:
# base**expo = a
base, expo = a.args
if expo.has(symbol) and not base.has(symbol):
# remainder -> solution independent of n of equation.
# m, rhs are made coprime by dividing igcd(m, rhs)
try:
remainder = discrete_log(m / igcd(m, rhs), rhs, a.base)
except ValueError: # log does not exist
return modterm, rhs
# period -> coefficient of n in the solution and also referred as
# the least period of expo in which it is repeats itself.
# (a**(totient(m)) - 1) divides m. Here is link of theorem:
# (https://en.wikipedia.org/wiki/Euler's_theorem)
period = totient(m)
for p in divisors(period):
# there might a lesser period exist than totient(m).
if pow(a.base, p, m / igcd(m, a.base)) == 1:
period = p
break
# recursion is not applied here since _invert_modular is currently
# not smart enough to handle infinite rhs as here expo has infinite
# rhs = ImageSet(Lambda(n, period*n + remainder), S.Naturals0).
return expo, ImageSet(Lambda(n, period*n + remainder), S.Naturals0)
elif base.has(symbol) and not expo.has(symbol):
try:
remainder_list = nthroot_mod(rhs, expo, m, all_roots=True)
if remainder_list == []:
return symbol, EmptySet
except (ValueError, NotImplementedError):
return modterm, rhs
g_n = EmptySet
for rem in remainder_list:
g_n += ImageSet(Lambda(n, m*n + rem), S.Integers)
return base, g_n
return modterm, rhs
def _solve_modular(f, symbol, domain):
r"""
Helper function for solving modular equations of type ``A - Mod(B, C) = 0``,
where A can or cannot be a function of symbol, B is surely a function of
symbol and C is an integer.
Currently ``_solve_modular`` is only able to solve cases
where A is not a function of symbol.
Parameters
==========
f : Expr
The modular equation to be solved, ``f = 0``
symbol : Symbol
The variable in the equation to be solved.
domain : Set
A set over which the equation is solved. It has to be a subset of
Integers.
Returns
=======
A set of integer solutions satisfying the given modular equation.
A ``ConditionSet`` if the equation is unsolvable.
Examples
========
>>> from sympy.solvers.solveset import _solve_modular as solve_modulo
>>> from sympy import S, Symbol, sin, Intersection, Interval
>>> from sympy.core.mod import Mod
>>> x = Symbol('x')
>>> solve_modulo(Mod(5*x - 8, 7) - 3, x, S.Integers)
ImageSet(Lambda(_n, 7*_n + 5), Integers)
>>> solve_modulo(Mod(5*x - 8, 7) - 3, x, S.Reals) # domain should be subset of integers.
ConditionSet(x, Eq(Mod(5*x + 6, 7) - 3, 0), Reals)
>>> solve_modulo(-7 + Mod(x, 5), x, S.Integers)
EmptySet
>>> solve_modulo(Mod(12**x, 21) - 18, x, S.Integers)
ImageSet(Lambda(_n, 6*_n + 2), Naturals0)
>>> solve_modulo(Mod(sin(x), 7) - 3, x, S.Integers) # not solvable
ConditionSet(x, Eq(Mod(sin(x), 7) - 3, 0), Integers)
>>> solve_modulo(3 - Mod(x, 5), x, Intersection(S.Integers, Interval(0, 100)))
Intersection(ImageSet(Lambda(_n, 5*_n + 3), Integers), Range(0, 101, 1))
"""
# extract modterm and g_y from f
unsolved_result = ConditionSet(symbol, Eq(f, 0), domain)
modterm = list(f.atoms(Mod))[0]
rhs = -S.One*(f.subs(modterm, S.Zero))
if f.as_coefficients_dict()[modterm].is_negative:
# checks if coefficient of modterm is negative in main equation.
rhs *= -S.One
if not domain.is_subset(S.Integers):
return unsolved_result
if rhs.has(symbol):
# TODO Case: A-> function of symbol, can be extended here
# in future.
return unsolved_result
n = Dummy('n', integer=True)
f_x, g_n = _invert_modular(modterm, rhs, n, symbol)
if f_x == modterm and g_n == rhs:
return unsolved_result
if f_x == symbol:
if domain is not S.Integers:
return domain.intersect(g_n)
return g_n
if isinstance(g_n, ImageSet):
lamda_expr = g_n.lamda.expr
lamda_vars = g_n.lamda.variables
base_sets = g_n.base_sets
sol_set = _solveset(f_x - lamda_expr, symbol, S.Integers)
if isinstance(sol_set, FiniteSet):
tmp_sol = EmptySet
for sol in sol_set:
tmp_sol += ImageSet(Lambda(lamda_vars, sol), *base_sets)
sol_set = tmp_sol
else:
sol_set = ImageSet(Lambda(lamda_vars, sol_set), *base_sets)
return domain.intersect(sol_set)
return unsolved_result
def _term_factors(f):
"""
Iterator to get the factors of all terms present
in the given equation.
Parameters
==========
f : Expr
Equation that needs to be addressed
Returns
=======
Factors of all terms present in the equation.
Examples
========
>>> from sympy import symbols
>>> from sympy.solvers.solveset import _term_factors
>>> x = symbols('x')
>>> list(_term_factors(-2 - x**2 + x*(x + 1)))
[-2, -1, x**2, x, x + 1]
"""
for add_arg in Add.make_args(f):
yield from Mul.make_args(add_arg)
def _solve_exponential(lhs, rhs, symbol, domain):
r"""
Helper function for solving (supported) exponential equations.
Exponential equations are the sum of (currently) at most
two terms with one or both of them having a power with a
symbol-dependent exponent.
For example
.. math:: 5^{2x + 3} - 5^{3x - 1}
.. math:: 4^{5 - 9x} - e^{2 - x}
Parameters
==========
lhs, rhs : Expr
The exponential equation to be solved, `lhs = rhs`
symbol : Symbol
The variable in which the equation is solved
domain : Set
A set over which the equation is solved.
Returns
=======
A set of solutions satisfying the given equation.
A ``ConditionSet`` if the equation is unsolvable or
if the assumptions are not properly defined, in that case
a different style of ``ConditionSet`` is returned having the
solution(s) of the equation with the desired assumptions.
Examples
========
>>> from sympy.solvers.solveset import _solve_exponential as solve_expo
>>> from sympy import symbols, S
>>> x = symbols('x', real=True)
>>> a, b = symbols('a b')
>>> solve_expo(2**x + 3**x - 5**x, 0, x, S.Reals) # not solvable
ConditionSet(x, Eq(2**x + 3**x - 5**x, 0), Reals)
>>> solve_expo(a**x - b**x, 0, x, S.Reals) # solvable but incorrect assumptions
ConditionSet(x, (a > 0) & (b > 0), FiniteSet(0))
>>> solve_expo(3**(2*x) - 2**(x + 3), 0, x, S.Reals)
FiniteSet(-3*log(2)/(-2*log(3) + log(2)))
>>> solve_expo(2**x - 4**x, 0, x, S.Reals)
FiniteSet(0)
* Proof of correctness of the method
The logarithm function is the inverse of the exponential function.
The defining relation between exponentiation and logarithm is:
.. math:: {\log_b x} = y \enspace if \enspace b^y = x
Therefore if we are given an equation with exponent terms, we can
convert every term to its corresponding logarithmic form. This is
achieved by taking logarithms and expanding the equation using
logarithmic identities so that it can easily be handled by ``solveset``.
For example:
.. math:: 3^{2x} = 2^{x + 3}
Taking log both sides will reduce the equation to
.. math:: (2x)\log(3) = (x + 3)\log(2)
This form can be easily handed by ``solveset``.
"""
unsolved_result = ConditionSet(symbol, Eq(lhs - rhs, 0), domain)
newlhs = powdenest(lhs)
if lhs != newlhs:
# it may also be advantageous to factor the new expr
neweq = factor(newlhs - rhs)
if neweq != (lhs - rhs):
return _solveset(neweq, symbol, domain) # try again with _solveset
if not (isinstance(lhs, Add) and len(lhs.args) == 2):
# solving for the sum of more than two powers is possible
# but not yet implemented
return unsolved_result
if rhs != 0:
return unsolved_result
a, b = list(ordered(lhs.args))
a_term = a.as_independent(symbol)[1]
b_term = b.as_independent(symbol)[1]
a_base, a_exp = a_term.as_base_exp()
b_base, b_exp = b_term.as_base_exp()
from sympy.functions.elementary.complexes import im
if domain.is_subset(S.Reals):
conditions = And(
a_base > 0,
b_base > 0,
Eq(im(a_exp), 0),
Eq(im(b_exp), 0))
else:
conditions = And(
Ne(a_base, 0),
Ne(b_base, 0))
L, R = map(lambda i: expand_log(log(i), force=True), (a, -b))
solutions = _solveset(L - R, symbol, domain)
return ConditionSet(symbol, conditions, solutions)
def _is_exponential(f, symbol):
r"""
Return ``True`` if one or more terms contain ``symbol`` only in
exponents, else ``False``.
Parameters
==========
f : Expr
The equation to be checked
symbol : Symbol
The variable in which the equation is checked
Examples
========
>>> from sympy import symbols, cos, exp
>>> from sympy.solvers.solveset import _is_exponential as check
>>> x, y = symbols('x y')
>>> check(y, y)
False
>>> check(x**y - 1, y)
True
>>> check(x**y*2**y - 1, y)
True
>>> check(exp(x + 3) + 3**x, x)
True
>>> check(cos(2**x), x)
False
* Philosophy behind the helper
The function extracts each term of the equation and checks if it is
of exponential form w.r.t ``symbol``.
"""
rv = False
for expr_arg in _term_factors(f):
if symbol not in expr_arg.free_symbols:
continue
if (isinstance(expr_arg, Pow) and
symbol not in expr_arg.base.free_symbols or
isinstance(expr_arg, exp)):
rv = True # symbol in exponent
else:
return False # dependent on symbol in non-exponential way
return rv
def _solve_logarithm(lhs, rhs, symbol, domain):
r"""
Helper to solve logarithmic equations which are reducible
to a single instance of `\log`.
Logarithmic equations are (currently) the equations that contains
`\log` terms which can be reduced to a single `\log` term or
a constant using various logarithmic identities.
For example:
.. math:: \log(x) + \log(x - 4)
can be reduced to:
.. math:: \log(x(x - 4))
Parameters
==========
lhs, rhs : Expr
The logarithmic equation to be solved, `lhs = rhs`
symbol : Symbol
The variable in which the equation is solved
domain : Set
A set over which the equation is solved.
Returns
=======
A set of solutions satisfying the given equation.
A ``ConditionSet`` if the equation is unsolvable.
Examples
========
>>> from sympy import symbols, log, S
>>> from sympy.solvers.solveset import _solve_logarithm as solve_log
>>> x = symbols('x')
>>> f = log(x - 3) + log(x + 3)
>>> solve_log(f, 0, x, S.Reals)
FiniteSet(sqrt(10), -sqrt(10))
* Proof of correctness
A logarithm is another way to write exponent and is defined by
.. math:: {\log_b x} = y \enspace if \enspace b^y = x
When one side of the equation contains a single logarithm, the
equation can be solved by rewriting the equation as an equivalent
exponential equation as defined above. But if one side contains
more than one logarithm, we need to use the properties of logarithm
to condense it into a single logarithm.
Take for example
.. math:: \log(2x) - 15 = 0
contains single logarithm, therefore we can directly rewrite it to
exponential form as
.. math:: x = \frac{e^{15}}{2}
But if the equation has more than one logarithm as
.. math:: \log(x - 3) + \log(x + 3) = 0
we use logarithmic identities to convert it into a reduced form
Using,
.. math:: \log(a) + \log(b) = \log(ab)
the equation becomes,
.. math:: \log((x - 3)(x + 3))
This equation contains one logarithm and can be solved by rewriting
to exponents.
"""
new_lhs = logcombine(lhs, force=True)
new_f = new_lhs - rhs
return _solveset(new_f, symbol, domain)
def _is_logarithmic(f, symbol):
r"""
Return ``True`` if the equation is in the form
`a\log(f(x)) + b\log(g(x)) + ... + c` else ``False``.
Parameters
==========
f : Expr
The equation to be checked
symbol : Symbol
The variable in which the equation is checked
Returns
=======
``True`` if the equation is logarithmic otherwise ``False``.
Examples
========
>>> from sympy import symbols, tan, log
>>> from sympy.solvers.solveset import _is_logarithmic as check
>>> x, y = symbols('x y')
>>> check(log(x + 2) - log(x + 3), x)
True
>>> check(tan(log(2*x)), x)
False
>>> check(x*log(x), x)
False
>>> check(x + log(x), x)
False
>>> check(y + log(x), x)
True
* Philosophy behind the helper
The function extracts each term and checks whether it is
logarithmic w.r.t ``symbol``.
"""
rv = False
for term in Add.make_args(f):
saw_log = False
for term_arg in Mul.make_args(term):
if symbol not in term_arg.free_symbols:
continue
if isinstance(term_arg, log):
if saw_log:
return False # more than one log in term
saw_log = True
else:
return False # dependent on symbol in non-log way
if saw_log:
rv = True
return rv
def _is_lambert(f, symbol):
r"""
If this returns ``False`` then the Lambert solver (``_solve_lambert``) will not be called.
Explanation
===========
Quick check for cases that the Lambert solver might be able to handle.
1. Equations containing more than two operands and `symbol`s involving any of
`Pow`, `exp`, `HyperbolicFunction`,`TrigonometricFunction`, `log` terms.
2. In `Pow`, `exp` the exponent should have `symbol` whereas for
`HyperbolicFunction`,`TrigonometricFunction`, `log` should contain `symbol`.
3. For `HyperbolicFunction`,`TrigonometricFunction` the number of trigonometric functions in
equation should be less than number of symbols. (since `A*cos(x) + B*sin(x) - c`
is not the Lambert type).
Some forms of lambert equations are:
1. X**X = C
2. X*(B*log(X) + D)**A = C
3. A*log(B*X + A) + d*X = C
4. (B*X + A)*exp(d*X + g) = C
5. g*exp(B*X + h) - B*X = C
6. A*D**(E*X + g) - B*X = C
7. A*cos(X) + B*sin(X) - D*X = C
8. A*cosh(X) + B*sinh(X) - D*X = C
Where X is any variable,
A, B, C, D, E are any constants,
g, h are linear functions or log terms.
Parameters
==========
f : Expr
The equation to be checked
symbol : Symbol
The variable in which the equation is checked
Returns
=======
If this returns ``False`` then the Lambert solver (``_solve_lambert``) will not be called.
Examples
========
>>> from sympy.solvers.solveset import _is_lambert
>>> from sympy import symbols, cosh, sinh, log
>>> x = symbols('x')
>>> _is_lambert(3*log(x) - x*log(3), x)
True
>>> _is_lambert(log(log(x - 3)) + log(x-3), x)
True
>>> _is_lambert(cosh(x) - sinh(x), x)
False
>>> _is_lambert((x**2 - 2*x + 1).subs(x, (log(x) + 3*x)**2 - 1), x)
True
See Also
========
_solve_lambert
"""
term_factors = list(_term_factors(f.expand()))
# total number of symbols in equation
no_of_symbols = len([arg for arg in term_factors if arg.has(symbol)])
# total number of trigonometric terms in equation
no_of_trig = len([arg for arg in term_factors \
if arg.has(HyperbolicFunction, TrigonometricFunction)])
if f.is_Add and no_of_symbols >= 2:
# `log`, `HyperbolicFunction`, `TrigonometricFunction` should have symbols
# and no_of_trig < no_of_symbols
lambert_funcs = (log, HyperbolicFunction, TrigonometricFunction)
if any(isinstance(arg, lambert_funcs)\
for arg in term_factors if arg.has(symbol)):
if no_of_trig < no_of_symbols:
return True
# here, `Pow`, `exp` exponent should have symbols
elif any(isinstance(arg, (Pow, exp)) \
for arg in term_factors if (arg.as_base_exp()[1]).has(symbol)):
return True
return False
def _transolve(f, symbol, domain):
r"""
Function to solve transcendental equations. It is a helper to
``solveset`` and should be used internally. ``_transolve``
currently supports the following class of equations:
- Exponential equations
- Logarithmic equations
Parameters
==========
f : Any transcendental equation that needs to be solved.
This needs to be an expression, which is assumed
to be equal to ``0``.
symbol : The variable for which the equation is solved.
This needs to be of class ``Symbol``.
domain : A set over which the equation is solved.
This needs to be of class ``Set``.
Returns
=======
Set
A set of values for ``symbol`` for which ``f`` is equal to
zero. An ``EmptySet`` is returned if ``f`` does not have solutions
in respective domain. A ``ConditionSet`` is returned as unsolved
object if algorithms to evaluate complete solution are not
yet implemented.
How to use ``_transolve``
=========================
``_transolve`` should not be used as an independent function, because
it assumes that the equation (``f``) and the ``symbol`` comes from
``solveset`` and might have undergone a few modification(s).
To use ``_transolve`` as an independent function the equation (``f``)
and the ``symbol`` should be passed as they would have been by
``solveset``.
Examples
========
>>> from sympy.solvers.solveset import _transolve as transolve
>>> from sympy.solvers.solvers import _tsolve as tsolve
>>> from sympy import symbols, S, pprint
>>> x = symbols('x', real=True) # assumption added
>>> transolve(5**(x - 3) - 3**(2*x + 1), x, S.Reals)
FiniteSet(-(log(3) + 3*log(5))/(-log(5) + 2*log(3)))
How ``_transolve`` works
========================
``_transolve`` uses two types of helper functions to solve equations
of a particular class:
Identifying helpers: To determine whether a given equation
belongs to a certain class of equation or not. Returns either
``True`` or ``False``.
Solving helpers: Once an equation is identified, a corresponding
helper either solves the equation or returns a form of the equation
that ``solveset`` might better be able to handle.
* Philosophy behind the module
The purpose of ``_transolve`` is to take equations which are not
already polynomial in their generator(s) and to either recast them
as such through a valid transformation or to solve them outright.
A pair of helper functions for each class of supported
transcendental functions are employed for this purpose. One
identifies the transcendental form of an equation and the other
either solves it or recasts it into a tractable form that can be
solved by ``solveset``.
For example, an equation in the form `ab^{f(x)} - cd^{g(x)} = 0`
can be transformed to
`\log(a) + f(x)\log(b) - \log(c) - g(x)\log(d) = 0`
(under certain assumptions) and this can be solved with ``solveset``
if `f(x)` and `g(x)` are in polynomial form.
How ``_transolve`` is better than ``_tsolve``
=============================================
1) Better output
``_transolve`` provides expressions in a more simplified form.
Consider a simple exponential equation
>>> f = 3**(2*x) - 2**(x + 3)
>>> pprint(transolve(f, x, S.Reals), use_unicode=False)
-3*log(2)
{------------------}
-2*log(3) + log(2)
>>> pprint(tsolve(f, x), use_unicode=False)
/ 3 \
| --------|
| log(2/9)|
[-log\2 /]
2) Extensible
The API of ``_transolve`` is designed such that it is easily
extensible, i.e. the code that solves a given class of
equations is encapsulated in a helper and not mixed in with
the code of ``_transolve`` itself.
3) Modular
``_transolve`` is designed to be modular i.e, for every class of
equation a separate helper for identification and solving is
implemented. This makes it easy to change or modify any of the
method implemented directly in the helpers without interfering
with the actual structure of the API.
4) Faster Computation
Solving equation via ``_transolve`` is much faster as compared to
``_tsolve``. In ``solve``, attempts are made computing every possibility
to get the solutions. This series of attempts makes solving a bit
slow. In ``_transolve``, computation begins only after a particular
type of equation is identified.
How to add new class of equations
=================================
Adding a new class of equation solver is a three-step procedure:
- Identify the type of the equations
Determine the type of the class of equations to which they belong:
it could be of ``Add``, ``Pow``, etc. types. Separate internal functions
are used for each type. Write identification and solving helpers
and use them from within the routine for the given type of equation
(after adding it, if necessary). Something like:
.. code-block:: python
def add_type(lhs, rhs, x):
....
if _is_exponential(lhs, x):
new_eq = _solve_exponential(lhs, rhs, x)
....
rhs, lhs = eq.as_independent(x)
if lhs.is_Add:
result = add_type(lhs, rhs, x)
- Define the identification helper.
- Define the solving helper.
Apart from this, a few other things needs to be taken care while
adding an equation solver:
- Naming conventions:
Name of the identification helper should be as
``_is_class`` where class will be the name or abbreviation
of the class of equation. The solving helper will be named as
``_solve_class``.
For example: for exponential equations it becomes
``_is_exponential`` and ``_solve_expo``.
- The identifying helpers should take two input parameters,
the equation to be checked and the variable for which a solution
is being sought, while solving helpers would require an additional
domain parameter.
- Be sure to consider corner cases.
- Add tests for each helper.
- Add a docstring to your helper that describes the method
implemented.
The documentation of the helpers should identify:
- the purpose of the helper,
- the method used to identify and solve the equation,
- a proof of correctness
- the return values of the helpers
"""
def add_type(lhs, rhs, symbol, domain):
"""
Helper for ``_transolve`` to handle equations of
``Add`` type, i.e. equations taking the form as
``a*f(x) + b*g(x) + .... = c``.
For example: 4**x + 8**x = 0
"""
result = ConditionSet(symbol, Eq(lhs - rhs, 0), domain)
# check if it is exponential type equation
if _is_exponential(lhs, symbol):
result = _solve_exponential(lhs, rhs, symbol, domain)
# check if it is logarithmic type equation
elif _is_logarithmic(lhs, symbol):
result = _solve_logarithm(lhs, rhs, symbol, domain)
return result
result = ConditionSet(symbol, Eq(f, 0), domain)
# invert_complex handles the call to the desired inverter based
# on the domain specified.
lhs, rhs_s = invert_complex(f, 0, symbol, domain)
if isinstance(rhs_s, FiniteSet):
assert (len(rhs_s.args)) == 1
rhs = rhs_s.args[0]
if lhs.is_Add:
result = add_type(lhs, rhs, symbol, domain)
else:
result = rhs_s
return result
def solveset(f, symbol=None, domain=S.Complexes):
r"""Solves a given inequality or equation with set as output
Parameters
==========
f : Expr or a relational.
The target equation or inequality
symbol : Symbol
The variable for which the equation is solved
domain : Set
The domain over which the equation is solved
Returns
=======
Set
A set of values for `symbol` for which `f` is True or is equal to
zero. An `EmptySet` is returned if `f` is False or nonzero.
A `ConditionSet` is returned as unsolved object if algorithms
to evaluate complete solution are not yet implemented.
`solveset` claims to be complete in the solution set that it returns.
Raises
======
NotImplementedError
The algorithms to solve inequalities in complex domain are
not yet implemented.
ValueError
The input is not valid.
RuntimeError
It is a bug, please report to the github issue tracker.
Notes
=====
Python interprets 0 and 1 as False and True, respectively, but
in this function they refer to solutions of an expression. So 0 and 1
return the Domain and EmptySet, respectively, while True and False
return the opposite (as they are assumed to be solutions of relational
expressions).
See Also
========
solveset_real: solver for real domain
solveset_complex: solver for complex domain
Examples
========
>>> from sympy import exp, sin, Symbol, pprint, S, Eq
>>> from sympy.solvers.solveset import solveset, solveset_real
* The default domain is complex. Not specifying a domain will lead
to the solving of the equation in the complex domain (and this
is not affected by the assumptions on the symbol):
>>> x = Symbol('x')
>>> pprint(solveset(exp(x) - 1, x), use_unicode=False)
{2*n*I*pi | n in Integers}
>>> x = Symbol('x', real=True)
>>> pprint(solveset(exp(x) - 1, x), use_unicode=False)
{2*n*I*pi | n in Integers}
* If you want to use `solveset` to solve the equation in the
real domain, provide a real domain. (Using ``solveset_real``
does this automatically.)
>>> R = S.Reals
>>> x = Symbol('x')
>>> solveset(exp(x) - 1, x, R)
FiniteSet(0)
>>> solveset_real(exp(x) - 1, x)
FiniteSet(0)
The solution is unaffected by assumptions on the symbol:
>>> p = Symbol('p', positive=True)
>>> pprint(solveset(p**2 - 4))
{-2, 2}
When a conditionSet is returned, symbols with assumptions that
would alter the set are replaced with more generic symbols:
>>> i = Symbol('i', imaginary=True)
>>> solveset(Eq(i**2 + i*sin(i), 1), i, domain=S.Reals)
ConditionSet(_R, Eq(_R**2 + _R*sin(_R) - 1, 0), Reals)
* Inequalities can be solved over the real domain only. Use of a complex
domain leads to a NotImplementedError.
>>> solveset(exp(x) > 1, x, R)
Interval.open(0, oo)
"""
f = sympify(f)
symbol = sympify(symbol)
if f is S.true:
return domain
if f is S.false:
return S.EmptySet
if not isinstance(f, (Expr, Relational, Number)):
raise ValueError("%s is not a valid SymPy expression" % f)
if not isinstance(symbol, (Expr, Relational)) and symbol is not None:
raise ValueError("%s is not a valid SymPy symbol" % (symbol,))
if not isinstance(domain, Set):
raise ValueError("%s is not a valid domain" %(domain))
free_symbols = f.free_symbols
if symbol is None and not free_symbols:
b = Eq(f, 0)
if b is S.true:
return domain
elif b is S.false:
return S.EmptySet
else:
raise NotImplementedError(filldedent('''
relationship between value and 0 is unknown: %s''' % b))
if symbol is None:
if len(free_symbols) == 1:
symbol = free_symbols.pop()
elif free_symbols:
raise ValueError(filldedent('''
The independent variable must be specified for a
multivariate equation.'''))
elif not isinstance(symbol, Symbol):
f, s, swap = recast_to_symbols([f], [symbol])
# the xreplace will be needed if a ConditionSet is returned
return solveset(f[0], s[0], domain).xreplace(swap)
# solveset should ignore assumptions on symbols
if symbol not in _rc:
x = _rc[0] if domain.is_subset(S.Reals) else _rc[1]
rv = solveset(f.xreplace({symbol: x}), x, domain)
# try to use the original symbol if possible
try:
_rv = rv.xreplace({x: symbol})
except TypeError:
_rv = rv
if rv.dummy_eq(_rv):
rv = _rv
return rv
# Abs has its own handling method which avoids the
# rewriting property that the first piece of abs(x)
# is for x >= 0 and the 2nd piece for x < 0 -- solutions
# can look better if the 2nd condition is x <= 0. Since
# the solution is a set, duplication of results is not
# an issue, e.g. {y, -y} when y is 0 will be {0}
f, mask = _masked(f, Abs)
f = f.rewrite(Piecewise) # everything that's not an Abs
for d, e in mask:
# everything *in* an Abs
e = e.func(e.args[0].rewrite(Piecewise))
f = f.xreplace({d: e})
f = piecewise_fold(f)
return _solveset(f, symbol, domain, _check=True)
def solveset_real(f, symbol):
return solveset(f, symbol, S.Reals)
def solveset_complex(f, symbol):
return solveset(f, symbol, S.Complexes)
def _solveset_multi(eqs, syms, domains):
'''Basic implementation of a multivariate solveset.
For internal use (not ready for public consumption)'''
rep = {}
for sym, dom in zip(syms, domains):
if dom is S.Reals:
rep[sym] = Symbol(sym.name, real=True)
eqs = [eq.subs(rep) for eq in eqs]
syms = [sym.subs(rep) for sym in syms]
syms = tuple(syms)
if len(eqs) == 0:
return ProductSet(*domains)
if len(syms) == 1:
sym = syms[0]
domain = domains[0]
solsets = [solveset(eq, sym, domain) for eq in eqs]
solset = Intersection(*solsets)
return ImageSet(Lambda((sym,), (sym,)), solset).doit()
eqs = sorted(eqs, key=lambda eq: len(eq.free_symbols & set(syms)))
for n in range(len(eqs)):
sols = []
all_handled = True
for sym in syms:
if sym not in eqs[n].free_symbols:
continue
sol = solveset(eqs[n], sym, domains[syms.index(sym)])
if isinstance(sol, FiniteSet):
i = syms.index(sym)
symsp = syms[:i] + syms[i+1:]
domainsp = domains[:i] + domains[i+1:]
eqsp = eqs[:n] + eqs[n+1:]
for s in sol:
eqsp_sub = [eq.subs(sym, s) for eq in eqsp]
sol_others = _solveset_multi(eqsp_sub, symsp, domainsp)
fun = Lambda((symsp,), symsp[:i] + (s,) + symsp[i:])
sols.append(ImageSet(fun, sol_others).doit())
else:
all_handled = False
if all_handled:
return Union(*sols)
def solvify(f, symbol, domain):
"""Solves an equation using solveset and returns the solution in accordance
with the `solve` output API.
Returns
=======
We classify the output based on the type of solution returned by `solveset`.
Solution | Output
----------------------------------------
FiniteSet | list
ImageSet, | list (if `f` is periodic)
Union |
Union | list (with FiniteSet)
EmptySet | empty list
Others | None
Raises
======
NotImplementedError
A ConditionSet is the input.
Examples
========
>>> from sympy.solvers.solveset import solvify
>>> from sympy.abc import x
>>> from sympy import S, tan, sin, exp
>>> solvify(x**2 - 9, x, S.Reals)
[-3, 3]
>>> solvify(sin(x) - 1, x, S.Reals)
[pi/2]
>>> solvify(tan(x), x, S.Reals)
[0]
>>> solvify(exp(x) - 1, x, S.Complexes)
>>> solvify(exp(x) - 1, x, S.Reals)
[0]
"""
solution_set = solveset(f, symbol, domain)
result = None
if solution_set is S.EmptySet:
result = []
elif isinstance(solution_set, ConditionSet):
raise NotImplementedError('solveset is unable to solve this equation.')
elif isinstance(solution_set, FiniteSet):
result = list(solution_set)
else:
period = periodicity(f, symbol)
if period is not None:
solutions = S.EmptySet
iter_solutions = ()
if isinstance(solution_set, ImageSet):
iter_solutions = (solution_set,)
elif isinstance(solution_set, Union):
if all(isinstance(i, ImageSet) for i in solution_set.args):
iter_solutions = solution_set.args
for solution in iter_solutions:
solutions += solution.intersect(Interval(0, period, False, True))
if isinstance(solutions, FiniteSet):
result = list(solutions)
else:
solution = solution_set.intersect(domain)
if isinstance(solution, Union):
# concerned about only FiniteSet with Union but not about ImageSet
# if required could be extend
if any(isinstance(i, FiniteSet) for i in solution.args):
result = [sol for soln in solution.args \
for sol in soln.args if isinstance(soln,FiniteSet)]
else:
return None
elif isinstance(solution, FiniteSet):
result += solution
return result
###############################################################################
################################ LINSOLVE #####################################
###############################################################################
def linear_coeffs(eq, *syms, **_kw):
"""Return a list whose elements are the coefficients of the
corresponding symbols in the sum of terms in ``eq``.
The additive constant is returned as the last element of the
list.
Raises
======
NonlinearError
The equation contains a nonlinear term
Examples
========
>>> from sympy.solvers.solveset import linear_coeffs
>>> from sympy.abc import x, y, z
>>> linear_coeffs(3*x + 2*y - 1, x, y)
[3, 2, -1]
It is not necessary to expand the expression:
>>> linear_coeffs(x + y*(z*(x*3 + 2) + 3), x)
[3*y*z + 1, y*(2*z + 3)]
But if there are nonlinear or cross terms -- even if they would
cancel after simplification -- an error is raised so the situation
does not pass silently past the caller's attention:
>>> eq = 1/x*(x - 1) + 1/x
>>> linear_coeffs(eq.expand(), x)
[0, 1]
>>> linear_coeffs(eq, x)
Traceback (most recent call last):
...
NonlinearError: nonlinear term encountered: 1/x
>>> linear_coeffs(x*(y + 1) - x*y, x, y)
Traceback (most recent call last):
...
NonlinearError: nonlinear term encountered: x*(y + 1)
"""
d = defaultdict(list)
eq = _sympify(eq)
symset = set(syms)
has = eq.free_symbols & symset
if not has:
return [S.Zero]*len(syms) + [eq]
c, terms = eq.as_coeff_add(*has)
d[0].extend(Add.make_args(c))
for t in terms:
m, f = t.as_coeff_mul(*has)
if len(f) != 1:
break
f = f[0]
if f in symset:
d[f].append(m)
elif f.is_Add:
d1 = linear_coeffs(f, *has, **{'dict': True})
d[0].append(m*d1.pop(0))
for xf, vf in d1.items():
d[xf].append(m*vf)
else:
break
else:
for k, v in d.items():
d[k] = Add(*v)
if not _kw:
return [d.get(s, S.Zero) for s in syms] + [d[0]]
return d # default is still list but this won't matter
raise NonlinearError('nonlinear term encountered: %s' % t)
def linear_eq_to_matrix(equations, *symbols):
r"""
Converts a given System of Equations into Matrix form.
Here `equations` must be a linear system of equations in
`symbols`. Element M[i, j] corresponds to the coefficient
of the jth symbol in the ith equation.
The Matrix form corresponds to the augmented matrix form.
For example:
.. math:: 4x + 2y + 3z = 1
.. math:: 3x + y + z = -6
.. math:: 2x + 4y + 9z = 2
This system would return `A` & `b` as given below:
::
[ 4 2 3 ] [ 1 ]
A = [ 3 1 1 ] b = [-6 ]
[ 2 4 9 ] [ 2 ]
The only simplification performed is to convert
`Eq(a, b) -> a - b`.
Raises
======
NonlinearError
The equations contain a nonlinear term.
ValueError
The symbols are not given or are not unique.
Examples
========
>>> from sympy import linear_eq_to_matrix, symbols
>>> c, x, y, z = symbols('c, x, y, z')
The coefficients (numerical or symbolic) of the symbols will
be returned as matrices:
>>> eqns = [c*x + z - 1 - c, y + z, x - y]
>>> A, b = linear_eq_to_matrix(eqns, [x, y, z])
>>> A
Matrix([
[c, 0, 1],
[0, 1, 1],
[1, -1, 0]])
>>> b
Matrix([
[c + 1],
[ 0],
[ 0]])
This routine does not simplify expressions and will raise an error
if nonlinearity is encountered:
>>> eqns = [
... (x**2 - 3*x)/(x - 3) - 3,
... y**2 - 3*y - y*(y - 4) + x - 4]
>>> linear_eq_to_matrix(eqns, [x, y])
Traceback (most recent call last):
...
NonlinearError:
The term (x**2 - 3*x)/(x - 3) is nonlinear in {x, y}
Simplifying these equations will discard the removable singularity
in the first, reveal the linear structure of the second:
>>> [e.simplify() for e in eqns]
[x - 3, x + y - 4]
Any such simplification needed to eliminate nonlinear terms must
be done before calling this routine.
"""
if not symbols:
raise ValueError(filldedent('''
Symbols must be given, for which coefficients
are to be found.
'''))
if hasattr(symbols[0], '__iter__'):
symbols = symbols[0]
for i in symbols:
if not isinstance(i, Symbol):
raise ValueError(filldedent('''
Expecting a Symbol but got %s
''' % i))
if has_dups(symbols):
raise ValueError('Symbols must be unique')
equations = sympify(equations)
if isinstance(equations, MatrixBase):
equations = list(equations)
elif isinstance(equations, (Expr, Eq)):
equations = [equations]
elif not is_sequence(equations):
raise ValueError(filldedent('''
Equation(s) must be given as a sequence, Expr,
Eq or Matrix.
'''))
A, b = [], []
for i, f in enumerate(equations):
if isinstance(f, Equality):
f = f.rewrite(Add, evaluate=False)
coeff_list = linear_coeffs(f, *symbols)
b.append(-coeff_list.pop())
A.append(coeff_list)
A, b = map(Matrix, (A, b))
return A, b
def linsolve(system, *symbols):
r"""
Solve system of N linear equations with M variables; both
underdetermined and overdetermined systems are supported.
The possible number of solutions is zero, one or infinite.
Zero solutions throws a ValueError, whereas infinite
solutions are represented parametrically in terms of the given
symbols. For unique solution a FiniteSet of ordered tuples
is returned.
All Standard input formats are supported:
For the given set of Equations, the respective input types
are given below:
.. math:: 3x + 2y - z = 1
.. math:: 2x - 2y + 4z = -2
.. math:: 2x - y + 2z = 0
* Augmented Matrix Form, `system` given below:
::
[3 2 -1 1]
system = [2 -2 4 -2]
[2 -1 2 0]
* List Of Equations Form
`system = [3x + 2y - z - 1, 2x - 2y + 4z + 2, 2x - y + 2z]`
* Input A & b Matrix Form (from Ax = b) are given as below:
::
[3 2 -1 ] [ 1 ]
A = [2 -2 4 ] b = [ -2 ]
[2 -1 2 ] [ 0 ]
`system = (A, b)`
Symbols can always be passed but are actually only needed
when 1) a system of equations is being passed and 2) the
system is passed as an underdetermined matrix and one wants
to control the name of the free variables in the result.
An error is raised if no symbols are used for case 1, but if
no symbols are provided for case 2, internally generated symbols
will be provided. When providing symbols for case 2, there should
be at least as many symbols are there are columns in matrix A.
The algorithm used here is Gauss-Jordan elimination, which
results, after elimination, in a row echelon form matrix.
Returns
=======
A FiniteSet containing an ordered tuple of values for the
unknowns for which the `system` has a solution. (Wrapping
the tuple in FiniteSet is used to maintain a consistent
output format throughout solveset.)
Returns EmptySet, if the linear system is inconsistent.
Raises
======
ValueError
The input is not valid.
The symbols are not given.
Examples
========
>>> from sympy import Matrix, linsolve, symbols
>>> x, y, z = symbols("x, y, z")
>>> A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 10]])
>>> b = Matrix([3, 6, 9])
>>> A
Matrix([
[1, 2, 3],
[4, 5, 6],
[7, 8, 10]])
>>> b
Matrix([
[3],
[6],
[9]])
>>> linsolve((A, b), [x, y, z])
FiniteSet((-1, 2, 0))
* Parametric Solution: In case the system is underdetermined, the
function will return a parametric solution in terms of the given
symbols. Those that are free will be returned unchanged. e.g. in
the system below, `z` is returned as the solution for variable z;
it can take on any value.
>>> A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
>>> b = Matrix([3, 6, 9])
>>> linsolve((A, b), x, y, z)
FiniteSet((z - 1, 2 - 2*z, z))
If no symbols are given, internally generated symbols will be used.
The `tau0` in the 3rd position indicates (as before) that the 3rd
variable -- whatever it's named -- can take on any value:
>>> linsolve((A, b))
FiniteSet((tau0 - 1, 2 - 2*tau0, tau0))
* List of Equations as input
>>> Eqns = [3*x + 2*y - z - 1, 2*x - 2*y + 4*z + 2, - x + y/2 - z]
>>> linsolve(Eqns, x, y, z)
FiniteSet((1, -2, -2))
* Augmented Matrix as input
>>> aug = Matrix([[2, 1, 3, 1], [2, 6, 8, 3], [6, 8, 18, 5]])
>>> aug
Matrix([
[2, 1, 3, 1],
[2, 6, 8, 3],
[6, 8, 18, 5]])
>>> linsolve(aug, x, y, z)
FiniteSet((3/10, 2/5, 0))
* Solve for symbolic coefficients
>>> a, b, c, d, e, f = symbols('a, b, c, d, e, f')
>>> eqns = [a*x + b*y - c, d*x + e*y - f]
>>> linsolve(eqns, x, y)
FiniteSet(((-b*f + c*e)/(a*e - b*d), (a*f - c*d)/(a*e - b*d)))
* A degenerate system returns solution as set of given
symbols.
>>> system = Matrix(([0, 0, 0], [0, 0, 0], [0, 0, 0]))
>>> linsolve(system, x, y)
FiniteSet((x, y))
* For an empty system linsolve returns empty set
>>> linsolve([], x)
EmptySet
* An error is raised if, after expansion, any nonlinearity
is detected:
>>> linsolve([x*(1/x - 1), (y - 1)**2 - y**2 + 1], x, y)
FiniteSet((1, 1))
>>> linsolve([x**2 - 1], x)
Traceback (most recent call last):
...
NonlinearError:
nonlinear term encountered: x**2
"""
if not system:
return S.EmptySet
# If second argument is an iterable
if symbols and hasattr(symbols[0], '__iter__'):
symbols = symbols[0]
sym_gen = isinstance(symbols, GeneratorType)
b = None # if we don't get b the input was bad
syms_needed_msg = None
# unpack system
if hasattr(system, '__iter__'):
# 1). (A, b)
if len(system) == 2 and isinstance(system[0], MatrixBase):
A, b = system
# 2). (eq1, eq2, ...)
if not isinstance(system[0], MatrixBase):
if sym_gen or not symbols:
raise ValueError(filldedent('''
When passing a system of equations, the explicit
symbols for which a solution is being sought must
be given as a sequence, too.
'''))
#
# Pass to the sparse solver implemented in polys. It is important
# that we do not attempt to convert the equations to a matrix
# because that would be very inefficient for large sparse systems
# of equations.
#
eqs = system
eqs = [sympify(eq) for eq in eqs]
try:
sol = _linsolve(eqs, symbols)
except PolyNonlinearError as exc:
# e.g. cos(x) contains an element of the set of generators
raise NonlinearError(str(exc))
if sol is None:
return S.EmptySet
sol = FiniteSet(Tuple(*(sol.get(sym, sym) for sym in symbols)))
return sol
elif isinstance(system, MatrixBase) and not (
symbols and not isinstance(symbols, GeneratorType) and
isinstance(symbols[0], MatrixBase)):
# 3). A augmented with b
A, b = system[:, :-1], system[:, -1:]
if b is None:
raise ValueError("Invalid arguments")
syms_needed_msg = syms_needed_msg or 'columns of A'
if sym_gen:
symbols = [next(symbols) for i in range(A.cols)]
if any(set(symbols) & (A.free_symbols | b.free_symbols)):
raise ValueError(filldedent('''
At least one of the symbols provided
already appears in the system to be solved.
One way to avoid this is to use Dummy symbols in
the generator, e.g. numbered_symbols('%s', cls=Dummy)
''' % symbols[0].name.rstrip('1234567890')))
if not symbols:
symbols = [Dummy() for _ in range(A.cols)]
name = _uniquely_named_symbol('tau', (A, b),
compare=lambda i: str(i).rstrip('1234567890')).name
gen = numbered_symbols(name)
else:
gen = None
# This is just a wrapper for solve_lin_sys
eqs = []
rows = A.tolist()
for rowi, bi in zip(rows, b):
terms = [elem * sym for elem, sym in zip(rowi, symbols) if elem]
terms.append(-bi)
eqs.append(Add(*terms))
eqs, ring = sympy_eqs_to_ring(eqs, symbols)
sol = solve_lin_sys(eqs, ring, _raw=False)
if sol is None:
return S.EmptySet
#sol = {sym:val for sym, val in sol.items() if sym != val}
sol = FiniteSet(Tuple(*(sol.get(sym, sym) for sym in symbols)))
if gen is not None:
solsym = sol.free_symbols
rep = {sym: next(gen) for sym in symbols if sym in solsym}
sol = sol.subs(rep)
return sol
##############################################################################
# ------------------------------nonlinsolve ---------------------------------#
##############################################################################
def _return_conditionset(eqs, symbols):
# return conditionset
eqs = (Eq(lhs, 0) for lhs in eqs)
condition_set = ConditionSet(
Tuple(*symbols), And(*eqs), S.Complexes**len(symbols))
return condition_set
def substitution(system, symbols, result=[{}], known_symbols=[],
exclude=[], all_symbols=None):
r"""
Solves the `system` using substitution method. It is used in
`nonlinsolve`. This will be called from `nonlinsolve` when any
equation(s) is non polynomial equation.
Parameters
==========
system : list of equations
The target system of equations
symbols : list of symbols to be solved.
The variable(s) for which the system is solved
known_symbols : list of solved symbols
Values are known for these variable(s)
result : An empty list or list of dict
If No symbol values is known then empty list otherwise
symbol as keys and corresponding value in dict.
exclude : Set of expression.
Mostly denominator expression(s) of the equations of the system.
Final solution should not satisfy these expressions.
all_symbols : known_symbols + symbols(unsolved).
Returns
=======
A FiniteSet of ordered tuple of values of `all_symbols` for which the
`system` has solution. Order of values in the tuple is same as symbols
present in the parameter `all_symbols`. If parameter `all_symbols` is None
then same as symbols present in the parameter `symbols`.
Please note that general FiniteSet is unordered, the solution returned
here is not simply a FiniteSet of solutions, rather it is a FiniteSet of
ordered tuple, i.e. the first & only argument to FiniteSet is a tuple of
solutions, which is ordered, & hence the returned solution is ordered.
Also note that solution could also have been returned as an ordered tuple,
FiniteSet is just a wrapper `{}` around the tuple. It has no other
significance except for the fact it is just used to maintain a consistent
output format throughout the solveset.
Raises
======
ValueError
The input is not valid.
The symbols are not given.
AttributeError
The input symbols are not `Symbol` type.
Examples
========
>>> from sympy.core.symbol import symbols
>>> x, y = symbols('x, y', real=True)
>>> from sympy.solvers.solveset import substitution
>>> substitution([x + y], [x], [{y: 1}], [y], set([]), [x, y])
FiniteSet((-1, 1))
* when you want soln should not satisfy eq `x + 1 = 0`
>>> substitution([x + y], [x], [{y: 1}], [y], set([x + 1]), [y, x])
EmptySet
>>> substitution([x + y], [x], [{y: 1}], [y], set([x - 1]), [y, x])
FiniteSet((1, -1))
>>> substitution([x + y - 1, y - x**2 + 5], [x, y])
FiniteSet((-3, 4), (2, -1))
* Returns both real and complex solution
>>> x, y, z = symbols('x, y, z')
>>> from sympy import exp, sin
>>> substitution([exp(x) - sin(y), y**2 - 4], [x, y])
FiniteSet((ImageSet(Lambda(_n, 2*_n*I*pi + log(sin(2))), Integers), 2),
(ImageSet(Lambda(_n, I*(2*_n*pi + pi) + log(sin(2))), Integers), -2))
>>> eqs = [z**2 + exp(2*x) - sin(y), -3 + exp(-y)]
>>> substitution(eqs, [y, z])
FiniteSet((-log(3), sqrt(-exp(2*x) - sin(log(3)))),
(-log(3), -sqrt(-exp(2*x) - sin(log(3)))),
(ImageSet(Lambda(_n, 2*_n*I*pi - log(3)), Integers),
ImageSet(Lambda(_n, sqrt(-exp(2*x) + sin(2*_n*I*pi - log(3)))), Integers)),
(ImageSet(Lambda(_n, 2*_n*I*pi - log(3)), Integers),
ImageSet(Lambda(_n, -sqrt(-exp(2*x) + sin(2*_n*I*pi - log(3)))), Integers)))
"""
from sympy import Complement
from sympy.core.compatibility import is_sequence
if not system:
return S.EmptySet
if not symbols:
msg = ('Symbols must be given, for which solution of the '
'system is to be found.')
raise ValueError(filldedent(msg))
if not is_sequence(symbols):
msg = ('symbols should be given as a sequence, e.g. a list.'
'Not type %s: %s')
raise TypeError(filldedent(msg % (type(symbols), symbols)))
if not getattr(symbols[0], 'is_Symbol', False):
msg = ('Iterable of symbols must be given as '
'second argument, not type %s: %s')
raise ValueError(filldedent(msg % (type(symbols[0]), symbols[0])))
# By default `all_symbols` will be same as `symbols`
if all_symbols is None:
all_symbols = symbols
old_result = result
# storing complements and intersection for particular symbol
complements = {}
intersections = {}
# when total_solveset_call equals total_conditionset
# it means that solveset failed to solve all eqs.
total_conditionset = -1
total_solveset_call = -1
def _unsolved_syms(eq, sort=False):
"""Returns the unsolved symbol present
in the equation `eq`.
"""
free = eq.free_symbols
unsolved = (free - set(known_symbols)) & set(all_symbols)
if sort:
unsolved = list(unsolved)
unsolved.sort(key=default_sort_key)
return unsolved
# end of _unsolved_syms()
# sort such that equation with the fewest potential symbols is first.
# means eq with less number of variable first in the list.
eqs_in_better_order = list(
ordered(system, lambda _: len(_unsolved_syms(_))))
def add_intersection_complement(result, intersection_dict, complement_dict):
# If solveset has returned some intersection/complement
# for any symbol, it will be added in the final solution.
final_result = []
for res in result:
res_copy = res
for key_res, value_res in res.items():
intersect_set, complement_set = None, None
for key_sym, value_sym in intersection_dict.items():
if key_sym == key_res:
intersect_set = value_sym
for key_sym, value_sym in complement_dict.items():
if key_sym == key_res:
complement_set = value_sym
if intersect_set or complement_set:
new_value = FiniteSet(value_res)
if intersect_set and intersect_set != S.Complexes:
new_value = Intersection(new_value, intersect_set)
if complement_set:
new_value = Complement(new_value, complement_set)
if new_value is S.EmptySet:
res_copy = None
break
elif new_value.is_FiniteSet and len(new_value) == 1:
res_copy[key_res] = set(new_value).pop()
else:
res_copy[key_res] = new_value
if res_copy is not None:
final_result.append(res_copy)
return final_result
# end of def add_intersection_complement()
def _extract_main_soln(sym, sol, soln_imageset):
"""Separate the Complements, Intersections, ImageSet lambda expr and
its base_set. This function returns the unmasks sol from different classes
of sets and also returns the appended ImageSet elements in a
soln_imageset (dict: where key as unmasked element and value as ImageSet).
"""
# if there is union, then need to check
# Complement, Intersection, Imageset.
# Order should not be changed.
if isinstance(sol, ConditionSet):
# extracts any solution in ConditionSet
sol = sol.base_set
if isinstance(sol, Complement):
# extract solution and complement
complements[sym] = sol.args[1]
sol = sol.args[0]
# complement will be added at the end
# using `add_intersection_complement` method
# if there is union of Imageset or other in soln.
# no testcase is written for this if block
if isinstance(sol, Union):
sol_args = sol.args
sol = S.EmptySet
# We need in sequence so append finteset elements
# and then imageset or other.
for sol_arg2 in sol_args:
if isinstance(sol_arg2, FiniteSet):
sol += sol_arg2
else:
# ImageSet, Intersection, complement then
# append them directly
sol += FiniteSet(sol_arg2)
if isinstance(sol, Intersection):
# Interval/Set will be at 0th index always
if sol.args[0] not in (S.Reals, S.Complexes):
# Sometimes solveset returns soln with intersection
# S.Reals or S.Complexes. We don't consider that
# intersection.
intersections[sym] = sol.args[0]
sol = sol.args[1]
# after intersection and complement Imageset should
# be checked.
if isinstance(sol, ImageSet):
soln_imagest = sol
expr2 = sol.lamda.expr
sol = FiniteSet(expr2)
soln_imageset[expr2] = soln_imagest
if not isinstance(sol, FiniteSet):
sol = FiniteSet(sol)
return sol, soln_imageset
# end of def _extract_main_soln()
# helper function for _append_new_soln
def _check_exclude(rnew, imgset_yes):
rnew_ = rnew
if imgset_yes:
# replace all dummy variables (Imageset lambda variables)
# with zero before `checksol`. Considering fundamental soln
# for `checksol`.
rnew_copy = rnew.copy()
dummy_n = imgset_yes[0]
for key_res, value_res in rnew_copy.items():
rnew_copy[key_res] = value_res.subs(dummy_n, 0)
rnew_ = rnew_copy
# satisfy_exclude == true if it satisfies the expr of `exclude` list.
try:
# something like : `Mod(-log(3), 2*I*pi)` can't be
# simplified right now, so `checksol` returns `TypeError`.
# when this issue is fixed this try block should be
# removed. Mod(-log(3), 2*I*pi) == -log(3)
satisfy_exclude = any(
checksol(d, rnew_) for d in exclude)
except TypeError:
satisfy_exclude = None
return satisfy_exclude
# end of def _check_exclude()
# helper function for _append_new_soln
def _restore_imgset(rnew, original_imageset, newresult):
restore_sym = set(rnew.keys()) & \
set(original_imageset.keys())
for key_sym in restore_sym:
img = original_imageset[key_sym]
rnew[key_sym] = img
if rnew not in newresult:
newresult.append(rnew)
# end of def _restore_imgset()
def _append_eq(eq, result, res, delete_soln, n=None):
u = Dummy('u')
if n:
eq = eq.subs(n, 0)
satisfy = checksol(u, u, eq, minimal=True)
if satisfy is False:
delete_soln = True
res = {}
else:
result.append(res)
return result, res, delete_soln
def _append_new_soln(rnew, sym, sol, imgset_yes, soln_imageset,
original_imageset, newresult, eq=None):
"""If `rnew` (A dict <symbol: soln>) contains valid soln
append it to `newresult` list.
`imgset_yes` is (base, dummy_var) if there was imageset in previously
calculated result(otherwise empty tuple). `original_imageset` is dict
of imageset expr and imageset from this result.
`soln_imageset` dict of imageset expr and imageset of new soln.
"""
satisfy_exclude = _check_exclude(rnew, imgset_yes)
delete_soln = False
# soln should not satisfy expr present in `exclude` list.
if not satisfy_exclude:
local_n = None
# if it is imageset
if imgset_yes:
local_n = imgset_yes[0]
base = imgset_yes[1]
if sym and sol:
# when `sym` and `sol` is `None` means no new
# soln. In that case we will append rnew directly after
# substituting original imagesets in rnew values if present
# (second last line of this function using _restore_imgset)
dummy_list = list(sol.atoms(Dummy))
# use one dummy `n` which is in
# previous imageset
local_n_list = [
local_n for i in range(
0, len(dummy_list))]
dummy_zip = zip(dummy_list, local_n_list)
lam = Lambda(local_n, sol.subs(dummy_zip))
rnew[sym] = ImageSet(lam, base)
if eq is not None:
newresult, rnew, delete_soln = _append_eq(
eq, newresult, rnew, delete_soln, local_n)
elif eq is not None:
newresult, rnew, delete_soln = _append_eq(
eq, newresult, rnew, delete_soln)
elif sol in soln_imageset.keys():
rnew[sym] = soln_imageset[sol]
# restore original imageset
_restore_imgset(rnew, original_imageset, newresult)
else:
newresult.append(rnew)
elif satisfy_exclude:
delete_soln = True
rnew = {}
_restore_imgset(rnew, original_imageset, newresult)
return newresult, delete_soln
# end of def _append_new_soln()
def _new_order_result(result, eq):
# separate first, second priority. `res` that makes `eq` value equals
# to zero, should be used first then other result(second priority).
# If it is not done then we may miss some soln.
first_priority = []
second_priority = []
for res in result:
if not any(isinstance(val, ImageSet) for val in res.values()):
if eq.subs(res) == 0:
first_priority.append(res)
else:
second_priority.append(res)
if first_priority or second_priority:
return first_priority + second_priority
return result
def _solve_using_known_values(result, solver):
"""Solves the system using already known solution
(result contains the dict <symbol: value>).
solver is `solveset_complex` or `solveset_real`.
"""
# stores imageset <expr: imageset(Lambda(n, expr), base)>.
soln_imageset = {}
total_solvest_call = 0
total_conditionst = 0
# sort such that equation with the fewest potential symbols is first.
# means eq with less variable first
for index, eq in enumerate(eqs_in_better_order):
newresult = []
original_imageset = {}
# if imageset expr is used to solve other symbol
imgset_yes = False
result = _new_order_result(result, eq)
for res in result:
got_symbol = set() # symbols solved in one iteration
# find the imageset and use its expr.
for key_res, value_res in res.items():
if isinstance(value_res, ImageSet):
res[key_res] = value_res.lamda.expr
original_imageset[key_res] = value_res
dummy_n = value_res.lamda.expr.atoms(Dummy).pop()
(base,) = value_res.base_sets
imgset_yes = (dummy_n, base)
# update eq with everything that is known so far
eq2 = eq.subs(res).expand()
unsolved_syms = _unsolved_syms(eq2, sort=True)
if not unsolved_syms:
if res:
newresult, delete_res = _append_new_soln(
res, None, None, imgset_yes, soln_imageset,
original_imageset, newresult, eq2)
if delete_res:
# `delete_res` is true, means substituting `res` in
# eq2 doesn't return `zero` or deleting the `res`
# (a soln) since it staisfies expr of `exclude`
# list.
result.remove(res)
continue # skip as it's independent of desired symbols
depen1, depen2 = (eq2.rewrite(Add)).as_independent(*unsolved_syms)
if (depen1.has(Abs) or depen2.has(Abs)) and solver == solveset_complex:
# Absolute values cannot be inverted in the
# complex domain
continue
soln_imageset = {}
for sym in unsolved_syms:
not_solvable = False
try:
soln = solver(eq2, sym)
total_solvest_call += 1
soln_new = S.EmptySet
if isinstance(soln, Complement):
# separate solution and complement
complements[sym] = soln.args[1]
soln = soln.args[0]
# complement will be added at the end
if isinstance(soln, Intersection):
# Interval will be at 0th index always
if soln.args[0] != Interval(-oo, oo):
# sometimes solveset returns soln
# with intersection S.Reals, to confirm that
# soln is in domain=S.Reals
intersections[sym] = soln.args[0]
soln_new += soln.args[1]
soln = soln_new if soln_new else soln
if index > 0 and solver == solveset_real:
# one symbol's real soln , another symbol may have
# corresponding complex soln.
if not isinstance(soln, (ImageSet, ConditionSet)):
soln += solveset_complex(eq2, sym)
except NotImplementedError:
# If sovleset is not able to solve equation `eq2`. Next
# time we may get soln using next equation `eq2`
continue
if isinstance(soln, ConditionSet):
if soln.base_set in (S.Reals, S.Complexes):
soln = S.EmptySet
# don't do `continue` we may get soln
# in terms of other symbol(s)
not_solvable = True
total_conditionst += 1
else:
soln = soln.base_set
if soln is not S.EmptySet:
soln, soln_imageset = _extract_main_soln(
sym, soln, soln_imageset)
for sol in soln:
# sol is not a `Union` since we checked it
# before this loop
sol, soln_imageset = _extract_main_soln(
sym, sol, soln_imageset)
sol = set(sol).pop()
free = sol.free_symbols
if got_symbol and any([
ss in free for ss in got_symbol
]):
# sol depends on previously solved symbols
# then continue
continue
rnew = res.copy()
# put each solution in res and append the new result
# in the new result list (solution for symbol `s`)
# along with old results.
for k, v in res.items():
if isinstance(v, Expr):
# if any unsolved symbol is present
# Then subs known value
rnew[k] = v.subs(sym, sol)
# and add this new solution
if sol in soln_imageset.keys():
# replace all lambda variables with 0.
imgst = soln_imageset[sol]
rnew[sym] = imgst.lamda(
*[0 for i in range(0, len(
imgst.lamda.variables))])
else:
rnew[sym] = sol
newresult, delete_res = _append_new_soln(
rnew, sym, sol, imgset_yes, soln_imageset,
original_imageset, newresult)
if delete_res:
# deleting the `res` (a soln) since it staisfies
# eq of `exclude` list
result.remove(res)
# solution got for sym
if not not_solvable:
got_symbol.add(sym)
# next time use this new soln
if newresult:
result = newresult
return result, total_solvest_call, total_conditionst
# end def _solve_using_know_values()
new_result_real, solve_call1, cnd_call1 = _solve_using_known_values(
old_result, solveset_real)
new_result_complex, solve_call2, cnd_call2 = _solve_using_known_values(
old_result, solveset_complex)
# If total_solveset_call is equal to total_conditionset
# then solveset failed to solve all of the equations.
# In this case we return a ConditionSet here.
total_conditionset += (cnd_call1 + cnd_call2)
total_solveset_call += (solve_call1 + solve_call2)
if total_conditionset == total_solveset_call and total_solveset_call != -1:
return _return_conditionset(eqs_in_better_order, all_symbols)
# don't keep duplicate solutions
filtered_complex = []
for i in list(new_result_complex):
for j in list(new_result_real):
if i.keys() != j.keys():
continue
if all(a.dummy_eq(b) for a, b in zip(i.values(), j.values()) \
if type(a) != int or type(b) != int):
break
else:
filtered_complex.append(i)
# overall result
result = new_result_real + filtered_complex
result_all_variables = []
result_infinite = []
for res in result:
if not res:
# means {None : None}
continue
# If length < len(all_symbols) means infinite soln.
# Some or all the soln is dependent on 1 symbol.
# eg. {x: y+2} then final soln {x: y+2, y: y}
if len(res) < len(all_symbols):
solved_symbols = res.keys()
unsolved = list(filter(
lambda x: x not in solved_symbols, all_symbols))
for unsolved_sym in unsolved:
res[unsolved_sym] = unsolved_sym
result_infinite.append(res)
if res not in result_all_variables:
result_all_variables.append(res)
if result_infinite:
# we have general soln
# eg : [{x: -1, y : 1}, {x : -y , y: y}] then
# return [{x : -y, y : y}]
result_all_variables = result_infinite
if intersections or complements:
result_all_variables = add_intersection_complement(
result_all_variables, intersections, complements)
# convert to ordered tuple
result = S.EmptySet
for r in result_all_variables:
temp = [r[symb] for symb in all_symbols]
result += FiniteSet(tuple(temp))
return result
# end of def substitution()
def _solveset_work(system, symbols):
soln = solveset(system[0], symbols[0])
if isinstance(soln, FiniteSet):
_soln = FiniteSet(*[tuple((s,)) for s in soln])
return _soln
else:
return FiniteSet(tuple(FiniteSet(soln)))
def _handle_positive_dimensional(polys, symbols, denominators):
from sympy.polys.polytools import groebner
# substitution method where new system is groebner basis of the system
_symbols = list(symbols)
_symbols.sort(key=default_sort_key)
basis = groebner(polys, _symbols, polys=True)
new_system = []
for poly_eq in basis:
new_system.append(poly_eq.as_expr())
result = [{}]
result = substitution(
new_system, symbols, result, [],
denominators)
return result
# end of def _handle_positive_dimensional()
def _handle_zero_dimensional(polys, symbols, system):
# solve 0 dimensional poly system using `solve_poly_system`
result = solve_poly_system(polys, *symbols)
# May be some extra soln is added because
# we used `unrad` in `_separate_poly_nonpoly`, so
# need to check and remove if it is not a soln.
result_update = S.EmptySet
for res in result:
dict_sym_value = dict(list(zip(symbols, res)))
if all(checksol(eq, dict_sym_value) for eq in system):
result_update += FiniteSet(res)
return result_update
# end of def _handle_zero_dimensional()
def _separate_poly_nonpoly(system, symbols):
polys = []
polys_expr = []
nonpolys = []
denominators = set()
poly = None
for eq in system:
# Store denom expression if it contains symbol
denominators.update(_simple_dens(eq, symbols))
# try to remove sqrt and rational power
without_radicals = unrad(simplify(eq))
if without_radicals:
eq_unrad, cov = without_radicals
if not cov:
eq = eq_unrad
if isinstance(eq, Expr):
eq = eq.as_numer_denom()[0]
poly = eq.as_poly(*symbols, extension=True)
elif simplify(eq).is_number:
continue
if poly is not None:
polys.append(poly)
polys_expr.append(poly.as_expr())
else:
nonpolys.append(eq)
return polys, polys_expr, nonpolys, denominators
# end of def _separate_poly_nonpoly()
def nonlinsolve(system, *symbols):
r"""
Solve system of N nonlinear equations with M variables, which means both
under and overdetermined systems are supported. Positive dimensional
system is also supported (A system with infinitely many solutions is said
to be positive-dimensional). In Positive dimensional system solution will
be dependent on at least one symbol. Returns both real solution
and complex solution(If system have). The possible number of solutions
is zero, one or infinite.
Parameters
==========
system : list of equations
The target system of equations
symbols : list of Symbols
symbols should be given as a sequence eg. list
Returns
=======
A FiniteSet of ordered tuple of values of `symbols` for which the `system`
has solution. Order of values in the tuple is same as symbols present in
the parameter `symbols`.
Please note that general FiniteSet is unordered, the solution returned
here is not simply a FiniteSet of solutions, rather it is a FiniteSet of
ordered tuple, i.e. the first & only argument to FiniteSet is a tuple of
solutions, which is ordered, & hence the returned solution is ordered.
Also note that solution could also have been returned as an ordered tuple,
FiniteSet is just a wrapper `{}` around the tuple. It has no other
significance except for the fact it is just used to maintain a consistent
output format throughout the solveset.
For the given set of Equations, the respective input types
are given below:
.. math:: x*y - 1 = 0
.. math:: 4*x**2 + y**2 - 5 = 0
`system = [x*y - 1, 4*x**2 + y**2 - 5]`
`symbols = [x, y]`
Raises
======
ValueError
The input is not valid.
The symbols are not given.
AttributeError
The input symbols are not `Symbol` type.
Examples
========
>>> from sympy.core.symbol import symbols
>>> from sympy.solvers.solveset import nonlinsolve
>>> x, y, z = symbols('x, y, z', real=True)
>>> nonlinsolve([x*y - 1, 4*x**2 + y**2 - 5], [x, y])
FiniteSet((-1, -1), (-1/2, -2), (1/2, 2), (1, 1))
1. Positive dimensional system and complements:
>>> from sympy import pprint
>>> from sympy.polys.polytools import is_zero_dimensional
>>> a, b, c, d = symbols('a, b, c, d', extended_real=True)
>>> eq1 = a + b + c + d
>>> eq2 = a*b + b*c + c*d + d*a
>>> eq3 = a*b*c + b*c*d + c*d*a + d*a*b
>>> eq4 = a*b*c*d - 1
>>> system = [eq1, eq2, eq3, eq4]
>>> is_zero_dimensional(system)
False
>>> pprint(nonlinsolve(system, [a, b, c, d]), use_unicode=False)
-1 1 1 -1
{(---, -d, -, {d} \ {0}), (-, -d, ---, {d} \ {0})}
d d d d
>>> nonlinsolve([(x+y)**2 - 4, x + y - 2], [x, y])
FiniteSet((2 - y, y))
2. If some of the equations are non-polynomial then `nonlinsolve`
will call the `substitution` function and return real and complex solutions,
if present.
>>> from sympy import exp, sin
>>> nonlinsolve([exp(x) - sin(y), y**2 - 4], [x, y])
FiniteSet((ImageSet(Lambda(_n, 2*_n*I*pi + log(sin(2))), Integers), 2),
(ImageSet(Lambda(_n, I*(2*_n*pi + pi) + log(sin(2))), Integers), -2))
3. If system is non-linear polynomial and zero-dimensional then it
returns both solution (real and complex solutions, if present) using
`solve_poly_system`:
>>> from sympy import sqrt
>>> nonlinsolve([x**2 - 2*y**2 -2, x*y - 2], [x, y])
FiniteSet((-2, -1), (2, 1), (-sqrt(2)*I, sqrt(2)*I), (sqrt(2)*I, -sqrt(2)*I))
4. `nonlinsolve` can solve some linear (zero or positive dimensional)
system (because it uses the `groebner` function to get the
groebner basis and then uses the `substitution` function basis as the
new `system`). But it is not recommended to solve linear system using
`nonlinsolve`, because `linsolve` is better for general linear systems.
>>> nonlinsolve([x + 2*y -z - 3, x - y - 4*z + 9 , y + z - 4], [x, y, z])
FiniteSet((3*z - 5, 4 - z, z))
5. System having polynomial equations and only real solution is
solved using `solve_poly_system`:
>>> e1 = sqrt(x**2 + y**2) - 10
>>> e2 = sqrt(y**2 + (-x + 10)**2) - 3
>>> nonlinsolve((e1, e2), (x, y))
FiniteSet((191/20, -3*sqrt(391)/20), (191/20, 3*sqrt(391)/20))
>>> nonlinsolve([x**2 + 2/y - 2, x + y - 3], [x, y])
FiniteSet((1, 2), (1 - sqrt(5), 2 + sqrt(5)), (1 + sqrt(5), 2 - sqrt(5)))
>>> nonlinsolve([x**2 + 2/y - 2, x + y - 3], [y, x])
FiniteSet((2, 1), (2 - sqrt(5), 1 + sqrt(5)), (2 + sqrt(5), 1 - sqrt(5)))
6. It is better to use symbols instead of Trigonometric Function or
Function (e.g. replace `sin(x)` with symbol, replace `f(x)` with symbol
and so on. Get soln from `nonlinsolve` and then using `solveset` get
the value of `x`)
How nonlinsolve is better than old solver `_solve_system` :
===========================================================
1. A positive dimensional system solver : nonlinsolve can return
solution for positive dimensional system. It finds the
Groebner Basis of the positive dimensional system(calling it as
basis) then we can start solving equation(having least number of
variable first in the basis) using solveset and substituting that
solved solutions into other equation(of basis) to get solution in
terms of minimum variables. Here the important thing is how we
are substituting the known values and in which equations.
2. Real and Complex both solutions : nonlinsolve returns both real
and complex solution. If all the equations in the system are polynomial
then using `solve_poly_system` both real and complex solution is returned.
If all the equations in the system are not polynomial equation then goes to
`substitution` method with this polynomial and non polynomial equation(s),
to solve for unsolved variables. Here to solve for particular variable
solveset_real and solveset_complex is used. For both real and complex
solution function `_solve_using_know_values` is used inside `substitution`
function.(`substitution` function will be called when there is any non
polynomial equation(s) is present). When solution is valid then add its
general solution in the final result.
3. Complement and Intersection will be added if any : nonlinsolve maintains
dict for complements and Intersections. If solveset find complements or/and
Intersection with any Interval or set during the execution of
`substitution` function ,then complement or/and Intersection for that
variable is added before returning final solution.
"""
from sympy.polys.polytools import is_zero_dimensional
if not system:
return S.EmptySet
if not symbols:
msg = ('Symbols must be given, for which solution of the '
'system is to be found.')
raise ValueError(filldedent(msg))
if hasattr(symbols[0], '__iter__'):
symbols = symbols[0]
if not is_sequence(symbols) or not symbols:
msg = ('Symbols must be given, for which solution of the '
'system is to be found.')
raise IndexError(filldedent(msg))
system, symbols, swap = recast_to_symbols(system, symbols)
if swap:
soln = nonlinsolve(system, symbols)
return FiniteSet(*[tuple(i.xreplace(swap) for i in s) for s in soln])
if len(system) == 1 and len(symbols) == 1:
return _solveset_work(system, symbols)
# main code of def nonlinsolve() starts from here
polys, polys_expr, nonpolys, denominators = _separate_poly_nonpoly(
system, symbols)
if len(symbols) == len(polys):
# If all the equations in the system are poly
if is_zero_dimensional(polys, symbols):
# finite number of soln (Zero dimensional system)
try:
return _handle_zero_dimensional(polys, symbols, system)
except NotImplementedError:
# Right now it doesn't fail for any polynomial system of
# equation. If `solve_poly_system` fails then `substitution`
# method will handle it.
result = substitution(
polys_expr, symbols, exclude=denominators)
return result
# positive dimensional system
res = _handle_positive_dimensional(polys, symbols, denominators)
if res is EmptySet and any(not p.domain.is_Exact for p in polys):
raise NotImplementedError("Equation not in exact domain. Try converting to rational")
else:
return res
else:
# If all the equations are not polynomial.
# Use `substitution` method for the system
result = substitution(
polys_expr + nonpolys, symbols, exclude=denominators)
return result
|
0c2e4c196ccbbefbb6731dbe1d93e08ee8a2816553440cb6c5fcec2f6d908b32 | """Tools for solving inequalities and systems of inequalities. """
from sympy.core import Symbol, Dummy, sympify
from sympy.core.compatibility import iterable
from sympy.core.exprtools import factor_terms
from sympy.core.relational import Relational, Eq, Ge, Lt
from sympy.sets import Interval
from sympy.sets.sets import FiniteSet, Union, EmptySet, Intersection
from sympy.core.singleton import S
from sympy.core.function import expand_mul
from sympy.functions import Abs
from sympy.logic import And
from sympy.polys import Poly, PolynomialError, parallel_poly_from_expr
from sympy.polys.polyutils import _nsort
from sympy.utilities.iterables import sift
from sympy.utilities.misc import filldedent
def solve_poly_inequality(poly, rel):
"""Solve a polynomial inequality with rational coefficients.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> from sympy.solvers.inequalities import solve_poly_inequality
>>> solve_poly_inequality(Poly(x, x, domain='ZZ'), '==')
[FiniteSet(0)]
>>> solve_poly_inequality(Poly(x**2 - 1, x, domain='ZZ'), '!=')
[Interval.open(-oo, -1), Interval.open(-1, 1), Interval.open(1, oo)]
>>> solve_poly_inequality(Poly(x**2 - 1, x, domain='ZZ'), '==')
[FiniteSet(-1), FiniteSet(1)]
See Also
========
solve_poly_inequalities
"""
if not isinstance(poly, Poly):
raise ValueError(
'For efficiency reasons, `poly` should be a Poly instance')
if poly.as_expr().is_number:
t = Relational(poly.as_expr(), 0, rel)
if t is S.true:
return [S.Reals]
elif t is S.false:
return [S.EmptySet]
else:
raise NotImplementedError(
"could not determine truth value of %s" % t)
reals, intervals = poly.real_roots(multiple=False), []
if rel == '==':
for root, _ in reals:
interval = Interval(root, root)
intervals.append(interval)
elif rel == '!=':
left = S.NegativeInfinity
for right, _ in reals + [(S.Infinity, 1)]:
interval = Interval(left, right, True, True)
intervals.append(interval)
left = right
else:
if poly.LC() > 0:
sign = +1
else:
sign = -1
eq_sign, equal = None, False
if rel == '>':
eq_sign = +1
elif rel == '<':
eq_sign = -1
elif rel == '>=':
eq_sign, equal = +1, True
elif rel == '<=':
eq_sign, equal = -1, True
else:
raise ValueError("'%s' is not a valid relation" % rel)
right, right_open = S.Infinity, True
for left, multiplicity in reversed(reals):
if multiplicity % 2:
if sign == eq_sign:
intervals.insert(
0, Interval(left, right, not equal, right_open))
sign, right, right_open = -sign, left, not equal
else:
if sign == eq_sign and not equal:
intervals.insert(
0, Interval(left, right, True, right_open))
right, right_open = left, True
elif sign != eq_sign and equal:
intervals.insert(0, Interval(left, left))
if sign == eq_sign:
intervals.insert(
0, Interval(S.NegativeInfinity, right, True, right_open))
return intervals
def solve_poly_inequalities(polys):
"""Solve polynomial inequalities with rational coefficients.
Examples
========
>>> from sympy.solvers.inequalities import solve_poly_inequalities
>>> from sympy.polys import Poly
>>> from sympy.abc import x
>>> solve_poly_inequalities(((
... Poly(x**2 - 3), ">"), (
... Poly(-x**2 + 1), ">")))
Union(Interval.open(-oo, -sqrt(3)), Interval.open(-1, 1), Interval.open(sqrt(3), oo))
"""
from sympy import Union
return Union(*[s for p in polys for s in solve_poly_inequality(*p)])
def solve_rational_inequalities(eqs):
"""Solve a system of rational inequalities with rational coefficients.
Examples
========
>>> from sympy.abc import x
>>> from sympy import Poly
>>> from sympy.solvers.inequalities import solve_rational_inequalities
>>> solve_rational_inequalities([[
... ((Poly(-x + 1), Poly(1, x)), '>='),
... ((Poly(-x + 1), Poly(1, x)), '<=')]])
FiniteSet(1)
>>> solve_rational_inequalities([[
... ((Poly(x), Poly(1, x)), '!='),
... ((Poly(-x + 1), Poly(1, x)), '>=')]])
Union(Interval.open(-oo, 0), Interval.Lopen(0, 1))
See Also
========
solve_poly_inequality
"""
result = S.EmptySet
for _eqs in eqs:
if not _eqs:
continue
global_intervals = [Interval(S.NegativeInfinity, S.Infinity)]
for (numer, denom), rel in _eqs:
numer_intervals = solve_poly_inequality(numer*denom, rel)
denom_intervals = solve_poly_inequality(denom, '==')
intervals = []
for numer_interval in numer_intervals:
for global_interval in global_intervals:
interval = numer_interval.intersect(global_interval)
if interval is not S.EmptySet:
intervals.append(interval)
global_intervals = intervals
intervals = []
for global_interval in global_intervals:
for denom_interval in denom_intervals:
global_interval -= denom_interval
if global_interval is not S.EmptySet:
intervals.append(global_interval)
global_intervals = intervals
if not global_intervals:
break
for interval in global_intervals:
result = result.union(interval)
return result
def reduce_rational_inequalities(exprs, gen, relational=True):
"""Reduce a system of rational inequalities with rational coefficients.
Examples
========
>>> from sympy import Symbol
>>> from sympy.solvers.inequalities import reduce_rational_inequalities
>>> x = Symbol('x', real=True)
>>> reduce_rational_inequalities([[x**2 <= 0]], x)
Eq(x, 0)
>>> reduce_rational_inequalities([[x + 2 > 0]], x)
-2 < x
>>> reduce_rational_inequalities([[(x + 2, ">")]], x)
-2 < x
>>> reduce_rational_inequalities([[x + 2]], x)
Eq(x, -2)
This function find the non-infinite solution set so if the unknown symbol
is declared as extended real rather than real then the result may include
finiteness conditions:
>>> y = Symbol('y', extended_real=True)
>>> reduce_rational_inequalities([[y + 2 > 0]], y)
(-2 < y) & (y < oo)
"""
exact = True
eqs = []
solution = S.Reals if exprs else S.EmptySet
for _exprs in exprs:
_eqs = []
for expr in _exprs:
if isinstance(expr, tuple):
expr, rel = expr
else:
if expr.is_Relational:
expr, rel = expr.lhs - expr.rhs, expr.rel_op
else:
expr, rel = expr, '=='
if expr is S.true:
numer, denom, rel = S.Zero, S.One, '=='
elif expr is S.false:
numer, denom, rel = S.One, S.One, '=='
else:
numer, denom = expr.together().as_numer_denom()
try:
(numer, denom), opt = parallel_poly_from_expr(
(numer, denom), gen)
except PolynomialError:
raise PolynomialError(filldedent('''
only polynomials and rational functions are
supported in this context.
'''))
if not opt.domain.is_Exact:
numer, denom, exact = numer.to_exact(), denom.to_exact(), False
domain = opt.domain.get_exact()
if not (domain.is_ZZ or domain.is_QQ):
expr = numer/denom
expr = Relational(expr, 0, rel)
solution &= solve_univariate_inequality(expr, gen, relational=False)
else:
_eqs.append(((numer, denom), rel))
if _eqs:
eqs.append(_eqs)
if eqs:
solution &= solve_rational_inequalities(eqs)
exclude = solve_rational_inequalities([[((d, d.one), '==')
for i in eqs for ((n, d), _) in i if d.has(gen)]])
solution -= exclude
if not exact and solution:
solution = solution.evalf()
if relational:
solution = solution.as_relational(gen)
return solution
def reduce_abs_inequality(expr, rel, gen):
"""Reduce an inequality with nested absolute values.
Examples
========
>>> from sympy import Abs, Symbol
>>> from sympy.solvers.inequalities import reduce_abs_inequality
>>> x = Symbol('x', real=True)
>>> reduce_abs_inequality(Abs(x - 5) - 3, '<', x)
(2 < x) & (x < 8)
>>> reduce_abs_inequality(Abs(x + 2)*3 - 13, '<', x)
(-19/3 < x) & (x < 7/3)
See Also
========
reduce_abs_inequalities
"""
if gen.is_extended_real is False:
raise TypeError(filldedent('''
can't solve inequalities with absolute values containing
non-real variables.
'''))
def _bottom_up_scan(expr):
exprs = []
if expr.is_Add or expr.is_Mul:
op = expr.func
for arg in expr.args:
_exprs = _bottom_up_scan(arg)
if not exprs:
exprs = _exprs
else:
args = []
for expr, conds in exprs:
for _expr, _conds in _exprs:
args.append((op(expr, _expr), conds + _conds))
exprs = args
elif expr.is_Pow:
n = expr.exp
if not n.is_Integer:
raise ValueError("Only Integer Powers are allowed on Abs.")
_exprs = _bottom_up_scan(expr.base)
for expr, conds in _exprs:
exprs.append((expr**n, conds))
elif isinstance(expr, Abs):
_exprs = _bottom_up_scan(expr.args[0])
for expr, conds in _exprs:
exprs.append(( expr, conds + [Ge(expr, 0)]))
exprs.append((-expr, conds + [Lt(expr, 0)]))
else:
exprs = [(expr, [])]
return exprs
exprs = _bottom_up_scan(expr)
mapping = {'<': '>', '<=': '>='}
inequalities = []
for expr, conds in exprs:
if rel not in mapping.keys():
expr = Relational( expr, 0, rel)
else:
expr = Relational(-expr, 0, mapping[rel])
inequalities.append([expr] + conds)
return reduce_rational_inequalities(inequalities, gen)
def reduce_abs_inequalities(exprs, gen):
"""Reduce a system of inequalities with nested absolute values.
Examples
========
>>> from sympy import Abs, Symbol
>>> from sympy.solvers.inequalities import reduce_abs_inequalities
>>> x = Symbol('x', extended_real=True)
>>> reduce_abs_inequalities([(Abs(3*x - 5) - 7, '<'),
... (Abs(x + 25) - 13, '>')], x)
(-2/3 < x) & (x < 4) & (((-oo < x) & (x < -38)) | ((-12 < x) & (x < oo)))
>>> reduce_abs_inequalities([(Abs(x - 4) + Abs(3*x - 5) - 7, '<')], x)
(1/2 < x) & (x < 4)
See Also
========
reduce_abs_inequality
"""
return And(*[ reduce_abs_inequality(expr, rel, gen)
for expr, rel in exprs ])
def solve_univariate_inequality(expr, gen, relational=True, domain=S.Reals, continuous=False):
"""Solves a real univariate inequality.
Parameters
==========
expr : Relational
The target inequality
gen : Symbol
The variable for which the inequality is solved
relational : bool
A Relational type output is expected or not
domain : Set
The domain over which the equation is solved
continuous: bool
True if expr is known to be continuous over the given domain
(and so continuous_domain() doesn't need to be called on it)
Raises
======
NotImplementedError
The solution of the inequality cannot be determined due to limitation
in :func:`sympy.solvers.solveset.solvify`.
Notes
=====
Currently, we cannot solve all the inequalities due to limitations in
:func:`sympy.solvers.solveset.solvify`. Also, the solution returned for trigonometric inequalities
are restricted in its periodic interval.
See Also
========
sympy.solvers.solveset.solvify: solver returning solveset solutions with solve's output API
Examples
========
>>> from sympy.solvers.inequalities import solve_univariate_inequality
>>> from sympy import Symbol, sin, Interval, S
>>> x = Symbol('x')
>>> solve_univariate_inequality(x**2 >= 4, x)
((2 <= x) & (x < oo)) | ((x <= -2) & (-oo < x))
>>> solve_univariate_inequality(x**2 >= 4, x, relational=False)
Union(Interval(-oo, -2), Interval(2, oo))
>>> domain = Interval(0, S.Infinity)
>>> solve_univariate_inequality(x**2 >= 4, x, False, domain)
Interval(2, oo)
>>> solve_univariate_inequality(sin(x) > 0, x, relational=False)
Interval.open(0, pi)
"""
from sympy import im
from sympy.calculus.util import (continuous_domain, periodicity,
function_range)
from sympy.solvers.solvers import denoms
from sympy.solvers.solveset import solvify, solveset
if domain.is_subset(S.Reals) is False:
raise NotImplementedError(filldedent('''
Inequalities in the complex domain are
not supported. Try the real domain by
setting domain=S.Reals'''))
elif domain is not S.Reals:
rv = solve_univariate_inequality(
expr, gen, relational=False, continuous=continuous).intersection(domain)
if relational:
rv = rv.as_relational(gen)
return rv
else:
pass # continue with attempt to solve in Real domain
# This keeps the function independent of the assumptions about `gen`.
# `solveset` makes sure this function is called only when the domain is
# real.
_gen = gen
_domain = domain
if gen.is_extended_real is False:
rv = S.EmptySet
return rv if not relational else rv.as_relational(_gen)
elif gen.is_extended_real is None:
gen = Dummy('gen', extended_real=True)
try:
expr = expr.xreplace({_gen: gen})
except TypeError:
raise TypeError(filldedent('''
When gen is real, the relational has a complex part
which leads to an invalid comparison like I < 0.
'''))
rv = None
if expr is S.true:
rv = domain
elif expr is S.false:
rv = S.EmptySet
else:
e = expr.lhs - expr.rhs
period = periodicity(e, gen)
if period == S.Zero:
e = expand_mul(e)
const = expr.func(e, 0)
if const is S.true:
rv = domain
elif const is S.false:
rv = S.EmptySet
elif period is not None:
frange = function_range(e, gen, domain)
rel = expr.rel_op
if rel == '<' or rel == '<=':
if expr.func(frange.sup, 0):
rv = domain
elif not expr.func(frange.inf, 0):
rv = S.EmptySet
elif rel == '>' or rel == '>=':
if expr.func(frange.inf, 0):
rv = domain
elif not expr.func(frange.sup, 0):
rv = S.EmptySet
inf, sup = domain.inf, domain.sup
if sup - inf is S.Infinity:
domain = Interval(0, period, False, True).intersect(_domain)
_domain = domain
if rv is None:
n, d = e.as_numer_denom()
try:
if gen not in n.free_symbols and len(e.free_symbols) > 1:
raise ValueError
# this might raise ValueError on its own
# or it might give None...
solns = solvify(e, gen, domain)
if solns is None:
# in which case we raise ValueError
raise ValueError
except (ValueError, NotImplementedError):
# replace gen with generic x since it's
# univariate anyway
raise NotImplementedError(filldedent('''
The inequality, %s, cannot be solved using
solve_univariate_inequality.
''' % expr.subs(gen, Symbol('x'))))
expanded_e = expand_mul(e)
def valid(x):
# this is used to see if gen=x satisfies the
# relational by substituting it into the
# expanded form and testing against 0, e.g.
# if expr = x*(x + 1) < 2 then e = x*(x + 1) - 2
# and expanded_e = x**2 + x - 2; the test is
# whether a given value of x satisfies
# x**2 + x - 2 < 0
#
# expanded_e, expr and gen used from enclosing scope
v = expanded_e.subs(gen, expand_mul(x))
try:
r = expr.func(v, 0)
except TypeError:
r = S.false
if r in (S.true, S.false):
return r
if v.is_extended_real is False:
return S.false
else:
v = v.n(2)
if v.is_comparable:
return expr.func(v, 0)
# not comparable or couldn't be evaluated
raise NotImplementedError(
'relationship did not evaluate: %s' % r)
singularities = []
for d in denoms(expr, gen):
singularities.extend(solvify(d, gen, domain))
if not continuous:
domain = continuous_domain(expanded_e, gen, domain)
include_x = '=' in expr.rel_op and expr.rel_op != '!='
try:
discontinuities = set(domain.boundary -
FiniteSet(domain.inf, domain.sup))
# remove points that are not between inf and sup of domain
critical_points = FiniteSet(*(solns + singularities + list(
discontinuities))).intersection(
Interval(domain.inf, domain.sup,
domain.inf not in domain, domain.sup not in domain))
if all(r.is_number for r in critical_points):
reals = _nsort(critical_points, separated=True)[0]
else:
sifted = sift(critical_points, lambda x: x.is_extended_real)
if sifted[None]:
# there were some roots that weren't known
# to be real
raise NotImplementedError
try:
reals = sifted[True]
if len(reals) > 1:
reals = list(sorted(reals))
except TypeError:
raise NotImplementedError
except NotImplementedError:
raise NotImplementedError('sorting of these roots is not supported')
# If expr contains imaginary coefficients, only take real
# values of x for which the imaginary part is 0
make_real = S.Reals
if im(expanded_e) != S.Zero:
check = True
im_sol = FiniteSet()
try:
a = solveset(im(expanded_e), gen, domain)
if not isinstance(a, Interval):
for z in a:
if z not in singularities and valid(z) and z.is_extended_real:
im_sol += FiniteSet(z)
else:
start, end = a.inf, a.sup
for z in _nsort(critical_points + FiniteSet(end)):
valid_start = valid(start)
if start != end:
valid_z = valid(z)
pt = _pt(start, z)
if pt not in singularities and pt.is_extended_real and valid(pt):
if valid_start and valid_z:
im_sol += Interval(start, z)
elif valid_start:
im_sol += Interval.Ropen(start, z)
elif valid_z:
im_sol += Interval.Lopen(start, z)
else:
im_sol += Interval.open(start, z)
start = z
for s in singularities:
im_sol -= FiniteSet(s)
except (TypeError):
im_sol = S.Reals
check = False
if isinstance(im_sol, EmptySet):
raise ValueError(filldedent('''
%s contains imaginary parts which cannot be
made 0 for any value of %s satisfying the
inequality, leading to relations like I < 0.
''' % (expr.subs(gen, _gen), _gen)))
make_real = make_real.intersect(im_sol)
sol_sets = [S.EmptySet]
start = domain.inf
if start in domain and valid(start) and start.is_finite:
sol_sets.append(FiniteSet(start))
for x in reals:
end = x
if valid(_pt(start, end)):
sol_sets.append(Interval(start, end, True, True))
if x in singularities:
singularities.remove(x)
else:
if x in discontinuities:
discontinuities.remove(x)
_valid = valid(x)
else: # it's a solution
_valid = include_x
if _valid:
sol_sets.append(FiniteSet(x))
start = end
end = domain.sup
if end in domain and valid(end) and end.is_finite:
sol_sets.append(FiniteSet(end))
if valid(_pt(start, end)):
sol_sets.append(Interval.open(start, end))
if im(expanded_e) != S.Zero and check:
rv = (make_real).intersect(_domain)
else:
rv = Intersection(
(Union(*sol_sets)), make_real, _domain).subs(gen, _gen)
return rv if not relational else rv.as_relational(_gen)
def _pt(start, end):
"""Return a point between start and end"""
if not start.is_infinite and not end.is_infinite:
pt = (start + end)/2
elif start.is_infinite and end.is_infinite:
pt = S.Zero
else:
if (start.is_infinite and start.is_extended_positive is None or
end.is_infinite and end.is_extended_positive is None):
raise ValueError('cannot proceed with unsigned infinite values')
if (end.is_infinite and end.is_extended_negative or
start.is_infinite and start.is_extended_positive):
start, end = end, start
# if possible, use a multiple of self which has
# better behavior when checking assumptions than
# an expression obtained by adding or subtracting 1
if end.is_infinite:
if start.is_extended_positive:
pt = start*2
elif start.is_extended_negative:
pt = start*S.Half
else:
pt = start + 1
elif start.is_infinite:
if end.is_extended_positive:
pt = end*S.Half
elif end.is_extended_negative:
pt = end*2
else:
pt = end - 1
return pt
def _solve_inequality(ie, s, linear=False):
"""Return the inequality with s isolated on the left, if possible.
If the relationship is non-linear, a solution involving And or Or
may be returned. False or True are returned if the relationship
is never True or always True, respectively.
If `linear` is True (default is False) an `s`-dependent expression
will be isolated on the left, if possible
but it will not be solved for `s` unless the expression is linear
in `s`. Furthermore, only "safe" operations which don't change the
sense of the relationship are applied: no division by an unsigned
value is attempted unless the relationship involves Eq or Ne and
no division by a value not known to be nonzero is ever attempted.
Examples
========
>>> from sympy import Eq, Symbol
>>> from sympy.solvers.inequalities import _solve_inequality as f
>>> from sympy.abc import x, y
For linear expressions, the symbol can be isolated:
>>> f(x - 2 < 0, x)
x < 2
>>> f(-x - 6 < x, x)
x > -3
Sometimes nonlinear relationships will be False
>>> f(x**2 + 4 < 0, x)
False
Or they may involve more than one region of values:
>>> f(x**2 - 4 < 0, x)
(-2 < x) & (x < 2)
To restrict the solution to a relational, set linear=True
and only the x-dependent portion will be isolated on the left:
>>> f(x**2 - 4 < 0, x, linear=True)
x**2 < 4
Division of only nonzero quantities is allowed, so x cannot
be isolated by dividing by y:
>>> y.is_nonzero is None # it is unknown whether it is 0 or not
True
>>> f(x*y < 1, x)
x*y < 1
And while an equality (or inequality) still holds after dividing by a
non-zero quantity
>>> nz = Symbol('nz', nonzero=True)
>>> f(Eq(x*nz, 1), x)
Eq(x, 1/nz)
the sign must be known for other inequalities involving > or <:
>>> f(x*nz <= 1, x)
nz*x <= 1
>>> p = Symbol('p', positive=True)
>>> f(x*p <= 1, x)
x <= 1/p
When there are denominators in the original expression that
are removed by expansion, conditions for them will be returned
as part of the result:
>>> f(x < x*(2/x - 1), x)
(x < 1) & Ne(x, 0)
"""
from sympy.solvers.solvers import denoms
if s not in ie.free_symbols:
return ie
if ie.rhs == s:
ie = ie.reversed
if ie.lhs == s and s not in ie.rhs.free_symbols:
return ie
def classify(ie, s, i):
# return True or False if ie evaluates when substituting s with
# i else None (if unevaluated) or NaN (when there is an error
# in evaluating)
try:
v = ie.subs(s, i)
if v is S.NaN:
return v
elif v not in (True, False):
return
return v
except TypeError:
return S.NaN
rv = None
oo = S.Infinity
expr = ie.lhs - ie.rhs
try:
p = Poly(expr, s)
if p.degree() == 0:
rv = ie.func(p.as_expr(), 0)
elif not linear and p.degree() > 1:
# handle in except clause
raise NotImplementedError
except (PolynomialError, NotImplementedError):
if not linear:
try:
rv = reduce_rational_inequalities([[ie]], s)
except PolynomialError:
rv = solve_univariate_inequality(ie, s)
# remove restrictions wrt +/-oo that may have been
# applied when using sets to simplify the relationship
okoo = classify(ie, s, oo)
if okoo is S.true and classify(rv, s, oo) is S.false:
rv = rv.subs(s < oo, True)
oknoo = classify(ie, s, -oo)
if (oknoo is S.true and
classify(rv, s, -oo) is S.false):
rv = rv.subs(-oo < s, True)
rv = rv.subs(s > -oo, True)
if rv is S.true:
rv = (s <= oo) if okoo is S.true else (s < oo)
if oknoo is not S.true:
rv = And(-oo < s, rv)
else:
p = Poly(expr)
conds = []
if rv is None:
e = p.as_expr() # this is in expanded form
# Do a safe inversion of e, moving non-s terms
# to the rhs and dividing by a nonzero factor if
# the relational is Eq/Ne; for other relationals
# the sign must also be positive or negative
rhs = 0
b, ax = e.as_independent(s, as_Add=True)
e -= b
rhs -= b
ef = factor_terms(e)
a, e = ef.as_independent(s, as_Add=False)
if (a.is_zero != False or # don't divide by potential 0
a.is_negative ==
a.is_positive is None and # if sign is not known then
ie.rel_op not in ('!=', '==')): # reject if not Eq/Ne
e = ef
a = S.One
rhs /= a
if a.is_positive:
rv = ie.func(e, rhs)
else:
rv = ie.reversed.func(e, rhs)
# return conditions under which the value is
# valid, too.
beginning_denoms = denoms(ie.lhs) | denoms(ie.rhs)
current_denoms = denoms(rv)
for d in beginning_denoms - current_denoms:
c = _solve_inequality(Eq(d, 0), s, linear=linear)
if isinstance(c, Eq) and c.lhs == s:
if classify(rv, s, c.rhs) is S.true:
# rv is permitting this value but it shouldn't
conds.append(~c)
for i in (-oo, oo):
if (classify(rv, s, i) is S.true and
classify(ie, s, i) is not S.true):
conds.append(s < i if i is oo else i < s)
conds.append(rv)
return And(*conds)
def _reduce_inequalities(inequalities, symbols):
# helper for reduce_inequalities
poly_part, abs_part = {}, {}
other = []
for inequality in inequalities:
expr, rel = inequality.lhs, inequality.rel_op # rhs is 0
# check for gens using atoms which is more strict than free_symbols to
# guard against EX domain which won't be handled by
# reduce_rational_inequalities
gens = expr.atoms(Symbol)
if len(gens) == 1:
gen = gens.pop()
else:
common = expr.free_symbols & symbols
if len(common) == 1:
gen = common.pop()
other.append(_solve_inequality(Relational(expr, 0, rel), gen))
continue
else:
raise NotImplementedError(filldedent('''
inequality has more than one symbol of interest.
'''))
if expr.is_polynomial(gen):
poly_part.setdefault(gen, []).append((expr, rel))
else:
components = expr.find(lambda u:
u.has(gen) and (
u.is_Function or u.is_Pow and not u.exp.is_Integer))
if components and all(isinstance(i, Abs) for i in components):
abs_part.setdefault(gen, []).append((expr, rel))
else:
other.append(_solve_inequality(Relational(expr, 0, rel), gen))
poly_reduced = []
abs_reduced = []
for gen, exprs in poly_part.items():
poly_reduced.append(reduce_rational_inequalities([exprs], gen))
for gen, exprs in abs_part.items():
abs_reduced.append(reduce_abs_inequalities(exprs, gen))
return And(*(poly_reduced + abs_reduced + other))
def reduce_inequalities(inequalities, symbols=[]):
"""Reduce a system of inequalities with rational coefficients.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.solvers.inequalities import reduce_inequalities
>>> reduce_inequalities(0 <= x + 3, [])
(-3 <= x) & (x < oo)
>>> reduce_inequalities(0 <= x + y*2 - 1, [x])
(x < oo) & (x >= 1 - 2*y)
"""
if not iterable(inequalities):
inequalities = [inequalities]
inequalities = [sympify(i) for i in inequalities]
gens = set().union(*[i.free_symbols for i in inequalities])
if not iterable(symbols):
symbols = [symbols]
symbols = (set(symbols) or gens) & gens
if any(i.is_extended_real is False for i in symbols):
raise TypeError(filldedent('''
inequalities cannot contain symbols that are not real.
'''))
# make vanilla symbol real
recast = {i: Dummy(i.name, extended_real=True)
for i in gens if i.is_extended_real is None}
inequalities = [i.xreplace(recast) for i in inequalities]
symbols = {i.xreplace(recast) for i in symbols}
# prefilter
keep = []
for i in inequalities:
if isinstance(i, Relational):
i = i.func(i.lhs.as_expr() - i.rhs.as_expr(), 0)
elif i not in (True, False):
i = Eq(i, 0)
if i == True:
continue
elif i == False:
return S.false
if i.lhs.is_number:
raise NotImplementedError(
"could not determine truth value of %s" % i)
keep.append(i)
inequalities = keep
del keep
# solve system
rv = _reduce_inequalities(inequalities, symbols)
# restore original symbols and return
return rv.xreplace({v: k for k, v in recast.items()})
|
93be5e3a84048fef238e29d39f2b033c4572b06a861f4d09dd3ef015a30ee9ba | """
This module contain solvers for all kinds of equations:
- algebraic or transcendental, use solve()
- recurrence, use rsolve()
- differential, use dsolve()
- nonlinear (numerically), use nsolve()
(you will need a good starting point)
"""
from sympy import divisors, binomial, expand_func
from sympy.core.assumptions import check_assumptions
from sympy.core.compatibility import (iterable, is_sequence, ordered,
default_sort_key)
from sympy.core.sympify import sympify
from sympy.core import (S, Add, Symbol, Equality, Dummy, Expr, Mul,
Pow, Unequality, Wild)
from sympy.core.exprtools import factor_terms
from sympy.core.function import (expand_mul, expand_log,
Derivative, AppliedUndef, UndefinedFunction, nfloat,
Function, expand_power_exp, _mexpand, expand)
from sympy.integrals.integrals import Integral
from sympy.core.numbers import ilcm, Float, Rational
from sympy.core.relational import Relational
from sympy.core.logic import fuzzy_not
from sympy.core.power import integer_log
from sympy.logic.boolalg import And, Or, BooleanAtom
from sympy.core.basic import preorder_traversal
from sympy.functions import (log, exp, LambertW, cos, sin, tan, acos, asin, atan,
Abs, re, im, arg, sqrt, atan2)
from sympy.functions.elementary.trigonometric import (TrigonometricFunction,
HyperbolicFunction)
from sympy.simplify import (simplify, collect, powsimp, posify, # type: ignore
powdenest, nsimplify, denom, logcombine, sqrtdenest, fraction,
separatevars)
from sympy.simplify.sqrtdenest import sqrt_depth
from sympy.simplify.fu import TR1, TR2i
from sympy.matrices.common import NonInvertibleMatrixError
from sympy.matrices import Matrix, zeros
from sympy.polys import roots, cancel, factor, Poly
from sympy.polys.polyerrors import GeneratorsNeeded, PolynomialError
from sympy.polys.solvers import sympy_eqs_to_ring, solve_lin_sys
from sympy.functions.elementary.piecewise import piecewise_fold, Piecewise
from sympy.utilities.lambdify import lambdify
from sympy.utilities.misc import filldedent
from sympy.utilities.iterables import (cartes, connected_components, flatten,
generate_bell, uniq, sift)
from sympy.utilities.decorator import conserve_mpmath_dps
from mpmath import findroot
from sympy.solvers.polysys import solve_poly_system
from sympy.solvers.inequalities import reduce_inequalities
from types import GeneratorType
from collections import defaultdict
import warnings
def recast_to_symbols(eqs, symbols):
"""
Return (e, s, d) where e and s are versions of *eqs* and
*symbols* in which any non-Symbol objects in *symbols* have
been replaced with generic Dummy symbols and d is a dictionary
that can be used to restore the original expressions.
Examples
========
>>> from sympy.solvers.solvers import recast_to_symbols
>>> from sympy import symbols, Function
>>> x, y = symbols('x y')
>>> fx = Function('f')(x)
>>> eqs, syms = [fx + 1, x, y], [fx, y]
>>> e, s, d = recast_to_symbols(eqs, syms); (e, s, d)
([_X0 + 1, x, y], [_X0, y], {_X0: f(x)})
The original equations and symbols can be restored using d:
>>> assert [i.xreplace(d) for i in eqs] == eqs
>>> assert [d.get(i, i) for i in s] == syms
"""
if not iterable(eqs) and iterable(symbols):
raise ValueError('Both eqs and symbols must be iterable')
new_symbols = list(symbols)
swap_sym = {}
for i, s in enumerate(symbols):
if not isinstance(s, Symbol) and s not in swap_sym:
swap_sym[s] = Dummy('X%d' % i)
new_symbols[i] = swap_sym[s]
new_f = []
for i in eqs:
isubs = getattr(i, 'subs', None)
if isubs is not None:
new_f.append(isubs(swap_sym))
else:
new_f.append(i)
swap_sym = {v: k for k, v in swap_sym.items()}
return new_f, new_symbols, swap_sym
def _ispow(e):
"""Return True if e is a Pow or is exp."""
return isinstance(e, Expr) and (e.is_Pow or isinstance(e, exp))
def _simple_dens(f, symbols):
# when checking if a denominator is zero, we can just check the
# base of powers with nonzero exponents since if the base is zero
# the power will be zero, too. To keep it simple and fast, we
# limit simplification to exponents that are Numbers
dens = set()
for d in denoms(f, symbols):
if d.is_Pow and d.exp.is_Number:
if d.exp.is_zero:
continue # foo**0 is never 0
d = d.base
dens.add(d)
return dens
def denoms(eq, *symbols):
"""
Return (recursively) set of all denominators that appear in *eq*
that contain any symbol in *symbols*; if *symbols* are not
provided then all denominators will be returned.
Examples
========
>>> from sympy.solvers.solvers import denoms
>>> from sympy.abc import x, y, z
>>> denoms(x/y)
{y}
>>> denoms(x/(y*z))
{y, z}
>>> denoms(3/x + y/z)
{x, z}
>>> denoms(x/2 + y/z)
{2, z}
If *symbols* are provided then only denominators containing
those symbols will be returned:
>>> denoms(1/x + 1/y + 1/z, y, z)
{y, z}
"""
pot = preorder_traversal(eq)
dens = set()
for p in pot:
# Here p might be Tuple or Relational
# Expr subtrees (e.g. lhs and rhs) will be traversed after by pot
if not isinstance(p, Expr):
continue
den = denom(p)
if den is S.One:
continue
for d in Mul.make_args(den):
dens.add(d)
if not symbols:
return dens
elif len(symbols) == 1:
if iterable(symbols[0]):
symbols = symbols[0]
rv = []
for d in dens:
free = d.free_symbols
if any(s in free for s in symbols):
rv.append(d)
return set(rv)
def checksol(f, symbol, sol=None, **flags):
"""
Checks whether sol is a solution of equation f == 0.
Explanation
===========
Input can be either a single symbol and corresponding value
or a dictionary of symbols and values. When given as a dictionary
and flag ``simplify=True``, the values in the dictionary will be
simplified. *f* can be a single equation or an iterable of equations.
A solution must satisfy all equations in *f* to be considered valid;
if a solution does not satisfy any equation, False is returned; if one or
more checks are inconclusive (and none are False) then None is returned.
Examples
========
>>> from sympy import symbols
>>> from sympy.solvers import checksol
>>> x, y = symbols('x,y')
>>> checksol(x**4 - 1, x, 1)
True
>>> checksol(x**4 - 1, x, 0)
False
>>> checksol(x**2 + y**2 - 5**2, {x: 3, y: 4})
True
To check if an expression is zero using ``checksol()``, pass it
as *f* and send an empty dictionary for *symbol*:
>>> checksol(x**2 + x - x*(x + 1), {})
True
None is returned if ``checksol()`` could not conclude.
flags:
'numerical=True (default)'
do a fast numerical check if ``f`` has only one symbol.
'minimal=True (default is False)'
a very fast, minimal testing.
'warn=True (default is False)'
show a warning if checksol() could not conclude.
'simplify=True (default)'
simplify solution before substituting into function and
simplify the function before trying specific simplifications
'force=True (default is False)'
make positive all symbols without assumptions regarding sign.
"""
from sympy.physics.units import Unit
minimal = flags.get('minimal', False)
if sol is not None:
sol = {symbol: sol}
elif isinstance(symbol, dict):
sol = symbol
else:
msg = 'Expecting (sym, val) or ({sym: val}, None) but got (%s, %s)'
raise ValueError(msg % (symbol, sol))
if iterable(f):
if not f:
raise ValueError('no functions to check')
rv = True
for fi in f:
check = checksol(fi, sol, **flags)
if check:
continue
if check is False:
return False
rv = None # don't return, wait to see if there's a False
return rv
if isinstance(f, Poly):
f = f.as_expr()
elif isinstance(f, (Equality, Unequality)):
if f.rhs in (S.true, S.false):
f = f.reversed
B, E = f.args
if isinstance(B, BooleanAtom):
f = f.subs(sol)
if not f.is_Boolean:
return
else:
f = f.rewrite(Add, evaluate=False)
if isinstance(f, BooleanAtom):
return bool(f)
elif not f.is_Relational and not f:
return True
if sol and not f.free_symbols & set(sol.keys()):
# if f(y) == 0, x=3 does not set f(y) to zero...nor does it not
return None
illegal = {S.NaN,
S.ComplexInfinity,
S.Infinity,
S.NegativeInfinity}
if any(sympify(v).atoms() & illegal for k, v in sol.items()):
return False
was = f
attempt = -1
numerical = flags.get('numerical', True)
while 1:
attempt += 1
if attempt == 0:
val = f.subs(sol)
if isinstance(val, Mul):
val = val.as_independent(Unit)[0]
if val.atoms() & illegal:
return False
elif attempt == 1:
if not val.is_number:
if not val.is_constant(*list(sol.keys()), simplify=not minimal):
return False
# there are free symbols -- simple expansion might work
_, val = val.as_content_primitive()
val = _mexpand(val.as_numer_denom()[0], recursive=True)
elif attempt == 2:
if minimal:
return
if flags.get('simplify', True):
for k in sol:
sol[k] = simplify(sol[k])
# start over without the failed expanded form, possibly
# with a simplified solution
val = simplify(f.subs(sol))
if flags.get('force', True):
val, reps = posify(val)
# expansion may work now, so try again and check
exval = _mexpand(val, recursive=True)
if exval.is_number:
# we can decide now
val = exval
else:
# if there are no radicals and no functions then this can't be
# zero anymore -- can it?
pot = preorder_traversal(expand_mul(val))
seen = set()
saw_pow_func = False
for p in pot:
if p in seen:
continue
seen.add(p)
if p.is_Pow and not p.exp.is_Integer:
saw_pow_func = True
elif p.is_Function:
saw_pow_func = True
elif isinstance(p, UndefinedFunction):
saw_pow_func = True
if saw_pow_func:
break
if saw_pow_func is False:
return False
if flags.get('force', True):
# don't do a zero check with the positive assumptions in place
val = val.subs(reps)
nz = fuzzy_not(val.is_zero)
if nz is not None:
# issue 5673: nz may be True even when False
# so these are just hacks to keep a false positive
# from being returned
# HACK 1: LambertW (issue 5673)
if val.is_number and val.has(LambertW):
# don't eval this to verify solution since if we got here,
# numerical must be False
return None
# add other HACKs here if necessary, otherwise we assume
# the nz value is correct
return not nz
break
if val == was:
continue
elif val.is_Rational:
return val == 0
if numerical and val.is_number:
return (abs(val.n(18).n(12, chop=True)) < 1e-9) is S.true
was = val
if flags.get('warn', False):
warnings.warn("\n\tWarning: could not verify solution %s." % sol)
# returns None if it can't conclude
# TODO: improve solution testing
def solve(f, *symbols, **flags):
r"""
Algebraically solves equations and systems of equations.
Explanation
===========
Currently supported:
- polynomial
- transcendental
- piecewise combinations of the above
- systems of linear and polynomial equations
- systems containing relational expressions
Examples
========
The output varies according to the input and can be seen by example:
>>> from sympy import solve, Poly, Eq, Function, exp
>>> from sympy.abc import x, y, z, a, b
>>> f = Function('f')
Boolean or univariate Relational:
>>> solve(x < 3)
(-oo < x) & (x < 3)
To always get a list of solution mappings, use flag dict=True:
>>> solve(x - 3, dict=True)
[{x: 3}]
>>> sol = solve([x - 3, y - 1], dict=True)
>>> sol
[{x: 3, y: 1}]
>>> sol[0][x]
3
>>> sol[0][y]
1
To get a list of *symbols* and set of solution(s) use flag set=True:
>>> solve([x**2 - 3, y - 1], set=True)
([x, y], {(-sqrt(3), 1), (sqrt(3), 1)})
Single expression and single symbol that is in the expression:
>>> solve(x - y, x)
[y]
>>> solve(x - 3, x)
[3]
>>> solve(Eq(x, 3), x)
[3]
>>> solve(Poly(x - 3), x)
[3]
>>> solve(x**2 - y**2, x, set=True)
([x], {(-y,), (y,)})
>>> solve(x**4 - 1, x, set=True)
([x], {(-1,), (1,), (-I,), (I,)})
Single expression with no symbol that is in the expression:
>>> solve(3, x)
[]
>>> solve(x - 3, y)
[]
Single expression with no symbol given. In this case, all free *symbols*
will be selected as potential *symbols* to solve for. If the equation is
univariate then a list of solutions is returned; otherwise - as is the case
when *symbols* are given as an iterable of length greater than 1 - a list of
mappings will be returned:
>>> solve(x - 3)
[3]
>>> solve(x**2 - y**2)
[{x: -y}, {x: y}]
>>> solve(z**2*x**2 - z**2*y**2)
[{x: -y}, {x: y}, {z: 0}]
>>> solve(z**2*x - z**2*y**2)
[{x: y**2}, {z: 0}]
When an object other than a Symbol is given as a symbol, it is
isolated algebraically and an implicit solution may be obtained.
This is mostly provided as a convenience to save you from replacing
the object with a Symbol and solving for that Symbol. It will only
work if the specified object can be replaced with a Symbol using the
subs method:
>>> solve(f(x) - x, f(x))
[x]
>>> solve(f(x).diff(x) - f(x) - x, f(x).diff(x))
[x + f(x)]
>>> solve(f(x).diff(x) - f(x) - x, f(x))
[-x + Derivative(f(x), x)]
>>> solve(x + exp(x)**2, exp(x), set=True)
([exp(x)], {(-sqrt(-x),), (sqrt(-x),)})
>>> from sympy import Indexed, IndexedBase, Tuple, sqrt
>>> A = IndexedBase('A')
>>> eqs = Tuple(A[1] + A[2] - 3, A[1] - A[2] + 1)
>>> solve(eqs, eqs.atoms(Indexed))
{A[1]: 1, A[2]: 2}
* To solve for a symbol implicitly, use implicit=True:
>>> solve(x + exp(x), x)
[-LambertW(1)]
>>> solve(x + exp(x), x, implicit=True)
[-exp(x)]
* It is possible to solve for anything that can be targeted with
subs:
>>> solve(x + 2 + sqrt(3), x + 2)
[-sqrt(3)]
>>> solve((x + 2 + sqrt(3), x + 4 + y), y, x + 2)
{y: -2 + sqrt(3), x + 2: -sqrt(3)}
* Nothing heroic is done in this implicit solving so you may end up
with a symbol still in the solution:
>>> eqs = (x*y + 3*y + sqrt(3), x + 4 + y)
>>> solve(eqs, y, x + 2)
{y: -sqrt(3)/(x + 3), x + 2: -2*x/(x + 3) - 6/(x + 3) + sqrt(3)/(x + 3)}
>>> solve(eqs, y*x, x)
{x: -y - 4, x*y: -3*y - sqrt(3)}
* If you attempt to solve for a number remember that the number
you have obtained does not necessarily mean that the value is
equivalent to the expression obtained:
>>> solve(sqrt(2) - 1, 1)
[sqrt(2)]
>>> solve(x - y + 1, 1) # /!\ -1 is targeted, too
[x/(y - 1)]
>>> [_.subs(z, -1) for _ in solve((x - y + 1).subs(-1, z), 1)]
[-x + y]
* To solve for a function within a derivative, use ``dsolve``.
Single expression and more than one symbol:
* When there is a linear solution:
>>> solve(x - y**2, x, y)
[(y**2, y)]
>>> solve(x**2 - y, x, y)
[(x, x**2)]
>>> solve(x**2 - y, x, y, dict=True)
[{y: x**2}]
* When undetermined coefficients are identified:
* That are linear:
>>> solve((a + b)*x - b + 2, a, b)
{a: -2, b: 2}
* That are nonlinear:
>>> solve((a + b)*x - b**2 + 2, a, b, set=True)
([a, b], {(-sqrt(2), sqrt(2)), (sqrt(2), -sqrt(2))})
* If there is no linear solution, then the first successful
attempt for a nonlinear solution will be returned:
>>> solve(x**2 - y**2, x, y, dict=True)
[{x: -y}, {x: y}]
>>> solve(x**2 - y**2/exp(x), x, y, dict=True)
[{x: 2*LambertW(-y/2)}, {x: 2*LambertW(y/2)}]
>>> solve(x**2 - y**2/exp(x), y, x)
[(-x*sqrt(exp(x)), x), (x*sqrt(exp(x)), x)]
Iterable of one or more of the above:
* Involving relationals or bools:
>>> solve([x < 3, x - 2])
Eq(x, 2)
>>> solve([x > 3, x - 2])
False
* When the system is linear:
* With a solution:
>>> solve([x - 3], x)
{x: 3}
>>> solve((x + 5*y - 2, -3*x + 6*y - 15), x, y)
{x: -3, y: 1}
>>> solve((x + 5*y - 2, -3*x + 6*y - 15), x, y, z)
{x: -3, y: 1}
>>> solve((x + 5*y - 2, -3*x + 6*y - z), z, x, y)
{x: 2 - 5*y, z: 21*y - 6}
* Without a solution:
>>> solve([x + 3, x - 3])
[]
* When the system is not linear:
>>> solve([x**2 + y -2, y**2 - 4], x, y, set=True)
([x, y], {(-2, -2), (0, 2), (2, -2)})
* If no *symbols* are given, all free *symbols* will be selected and a
list of mappings returned:
>>> solve([x - 2, x**2 + y])
[{x: 2, y: -4}]
>>> solve([x - 2, x**2 + f(x)], {f(x), x})
[{x: 2, f(x): -4}]
* If any equation does not depend on the symbol(s) given, it will be
eliminated from the equation set and an answer may be given
implicitly in terms of variables that were not of interest:
>>> solve([x - y, y - 3], x)
{x: y}
**Additional Examples**
``solve()`` with check=True (default) will run through the symbol tags to
elimate unwanted solutions. If no assumptions are included, all possible
solutions will be returned:
>>> from sympy import Symbol, solve
>>> x = Symbol("x")
>>> solve(x**2 - 1)
[-1, 1]
By using the positive tag, only one solution will be returned:
>>> pos = Symbol("pos", positive=True)
>>> solve(pos**2 - 1)
[1]
Assumptions are not checked when ``solve()`` input involves
relationals or bools.
When the solutions are checked, those that make any denominator zero
are automatically excluded. If you do not want to exclude such solutions,
then use the check=False option:
>>> from sympy import sin, limit
>>> solve(sin(x)/x) # 0 is excluded
[pi]
If check=False, then a solution to the numerator being zero is found: x = 0.
In this case, this is a spurious solution since $\sin(x)/x$ has the well
known limit (without dicontinuity) of 1 at x = 0:
>>> solve(sin(x)/x, check=False)
[0, pi]
In the following case, however, the limit exists and is equal to the
value of x = 0 that is excluded when check=True:
>>> eq = x**2*(1/x - z**2/x)
>>> solve(eq, x)
[]
>>> solve(eq, x, check=False)
[0]
>>> limit(eq, x, 0, '-')
0
>>> limit(eq, x, 0, '+')
0
**Disabling High-Order Explicit Solutions**
When solving polynomial expressions, you might not want explicit solutions
(which can be quite long). If the expression is univariate, ``CRootOf``
instances will be returned instead:
>>> solve(x**3 - x + 1)
[-1/((-1/2 - sqrt(3)*I/2)*(3*sqrt(69)/2 + 27/2)**(1/3)) - (-1/2 -
sqrt(3)*I/2)*(3*sqrt(69)/2 + 27/2)**(1/3)/3, -(-1/2 +
sqrt(3)*I/2)*(3*sqrt(69)/2 + 27/2)**(1/3)/3 - 1/((-1/2 +
sqrt(3)*I/2)*(3*sqrt(69)/2 + 27/2)**(1/3)), -(3*sqrt(69)/2 +
27/2)**(1/3)/3 - 1/(3*sqrt(69)/2 + 27/2)**(1/3)]
>>> solve(x**3 - x + 1, cubics=False)
[CRootOf(x**3 - x + 1, 0),
CRootOf(x**3 - x + 1, 1),
CRootOf(x**3 - x + 1, 2)]
If the expression is multivariate, no solution might be returned:
>>> solve(x**3 - x + a, x, cubics=False)
[]
Sometimes solutions will be obtained even when a flag is False because the
expression could be factored. In the following example, the equation can
be factored as the product of a linear and a quadratic factor so explicit
solutions (which did not require solving a cubic expression) are obtained:
>>> eq = x**3 + 3*x**2 + x - 1
>>> solve(eq, cubics=False)
[-1, -1 + sqrt(2), -sqrt(2) - 1]
**Solving Equations Involving Radicals**
Because of SymPy's use of the principle root, some solutions
to radical equations will be missed unless check=False:
>>> from sympy import root
>>> eq = root(x**3 - 3*x**2, 3) + 1 - x
>>> solve(eq)
[]
>>> solve(eq, check=False)
[1/3]
In the above example, there is only a single solution to the
equation. Other expressions will yield spurious roots which
must be checked manually; roots which give a negative argument
to odd-powered radicals will also need special checking:
>>> from sympy import real_root, S
>>> eq = root(x, 3) - root(x, 5) + S(1)/7
>>> solve(eq) # this gives 2 solutions but misses a 3rd
[CRootOf(7*x**5 - 7*x**3 + 1, 1)**15,
CRootOf(7*x**5 - 7*x**3 + 1, 2)**15]
>>> sol = solve(eq, check=False)
>>> [abs(eq.subs(x,i).n(2)) for i in sol]
[0.48, 0.e-110, 0.e-110, 0.052, 0.052]
The first solution is negative so ``real_root`` must be used to see that it
satisfies the expression:
>>> abs(real_root(eq.subs(x, sol[0])).n(2))
0.e-110
If the roots of the equation are not real then more care will be
necessary to find the roots, especially for higher order equations.
Consider the following expression:
>>> expr = root(x, 3) - root(x, 5)
We will construct a known value for this expression at x = 3 by selecting
the 1-th root for each radical:
>>> expr1 = root(x, 3, 1) - root(x, 5, 1)
>>> v = expr1.subs(x, -3)
The ``solve`` function is unable to find any exact roots to this equation:
>>> eq = Eq(expr, v); eq1 = Eq(expr1, v)
>>> solve(eq, check=False), solve(eq1, check=False)
([], [])
The function ``unrad``, however, can be used to get a form of the equation
for which numerical roots can be found:
>>> from sympy.solvers.solvers import unrad
>>> from sympy import nroots
>>> e, (p, cov) = unrad(eq)
>>> pvals = nroots(e)
>>> inversion = solve(cov, x)[0]
>>> xvals = [inversion.subs(p, i) for i in pvals]
Although ``eq`` or ``eq1`` could have been used to find ``xvals``, the
solution can only be verified with ``expr1``:
>>> z = expr - v
>>> [xi.n(chop=1e-9) for xi in xvals if abs(z.subs(x, xi).n()) < 1e-9]
[]
>>> z1 = expr1 - v
>>> [xi.n(chop=1e-9) for xi in xvals if abs(z1.subs(x, xi).n()) < 1e-9]
[-3.0]
Parameters
==========
f :
- a single Expr or Poly that must be zero
- an Equality
- a Relational expression
- a Boolean
- iterable of one or more of the above
symbols : (object(s) to solve for) specified as
- none given (other non-numeric objects will be used)
- single symbol
- denested list of symbols
(e.g., ``solve(f, x, y)``)
- ordered iterable of symbols
(e.g., ``solve(f, [x, y])``)
flags :
dict=True (default is False)
Return list (perhaps empty) of solution mappings.
set=True (default is False)
Return list of symbols and set of tuple(s) of solution(s).
exclude=[] (default)
Do not try to solve for any of the free symbols in exclude;
if expressions are given, the free symbols in them will
be extracted automatically.
check=True (default)
If False, do not do any testing of solutions. This can be
useful if you want to include solutions that make any
denominator zero.
numerical=True (default)
Do a fast numerical check if *f* has only one symbol.
minimal=True (default is False)
A very fast, minimal testing.
warn=True (default is False)
Show a warning if ``checksol()`` could not conclude.
simplify=True (default)
Simplify all but polynomials of order 3 or greater before
returning them and (if check is not False) use the
general simplify function on the solutions and the
expression obtained when they are substituted into the
function which should be zero.
force=True (default is False)
Make positive all symbols without assumptions regarding sign.
rational=True (default)
Recast Floats as Rational; if this option is not used, the
system containing Floats may fail to solve because of issues
with polys. If rational=None, Floats will be recast as
rationals but the answer will be recast as Floats. If the
flag is False then nothing will be done to the Floats.
manual=True (default is False)
Do not use the polys/matrix method to solve a system of
equations, solve them one at a time as you might "manually."
implicit=True (default is False)
Allows ``solve`` to return a solution for a pattern in terms of
other functions that contain that pattern; this is only
needed if the pattern is inside of some invertible function
like cos, exp, ect.
particular=True (default is False)
Instructs ``solve`` to try to find a particular solution to a linear
system with as many zeros as possible; this is very expensive.
quick=True (default is False)
When using particular=True, use a fast heuristic to find a
solution with many zeros (instead of using the very slow method
guaranteed to find the largest number of zeros possible).
cubics=True (default)
Return explicit solutions when cubic expressions are encountered.
quartics=True (default)
Return explicit solutions when quartic expressions are encountered.
quintics=True (default)
Return explicit solutions (if possible) when quintic expressions
are encountered.
See Also
========
rsolve: For solving recurrence relationships
dsolve: For solving differential equations
"""
# keeping track of how f was passed since if it is a list
# a dictionary of results will be returned.
###########################################################################
def _sympified_list(w):
return list(map(sympify, w if iterable(w) else [w]))
bare_f = not iterable(f)
ordered_symbols = (symbols and
symbols[0] and
(isinstance(symbols[0], Symbol) or
is_sequence(symbols[0],
include=GeneratorType)
)
)
f, symbols = (_sympified_list(w) for w in [f, symbols])
if isinstance(f, list):
f = [s for s in f if s is not S.true and s is not True]
implicit = flags.get('implicit', False)
# preprocess symbol(s)
###########################################################################
if not symbols:
# get symbols from equations
symbols = set().union(*[fi.free_symbols for fi in f])
if len(symbols) < len(f):
for fi in f:
pot = preorder_traversal(fi)
for p in pot:
if isinstance(p, AppliedUndef):
flags['dict'] = True # better show symbols
symbols.add(p)
pot.skip() # don't go any deeper
symbols = list(symbols)
ordered_symbols = False
elif len(symbols) == 1 and iterable(symbols[0]):
symbols = symbols[0]
# remove symbols the user is not interested in
exclude = flags.pop('exclude', set())
if exclude:
if isinstance(exclude, Expr):
exclude = [exclude]
exclude = set().union(*[e.free_symbols for e in sympify(exclude)])
symbols = [s for s in symbols if s not in exclude]
# preprocess equation(s)
###########################################################################
for i, fi in enumerate(f):
if isinstance(fi, (Equality, Unequality)):
if 'ImmutableDenseMatrix' in [type(a).__name__ for a in fi.args]:
fi = fi.lhs - fi.rhs
else:
L, R = fi.args
if isinstance(R, BooleanAtom):
L, R = R, L
if isinstance(L, BooleanAtom):
if isinstance(fi, Unequality):
L = ~L
if R.is_Relational:
fi = ~R if L is S.false else R
elif R.is_Symbol:
return L
elif R.is_Boolean and (~R).is_Symbol:
return ~L
else:
raise NotImplementedError(filldedent('''
Unanticipated argument of Eq when other arg
is True or False.
'''))
else:
fi = fi.rewrite(Add, evaluate=False)
f[i] = fi
if fi.is_Relational:
return reduce_inequalities(f, symbols=symbols)
if isinstance(fi, Poly):
f[i] = fi.as_expr()
# rewrite hyperbolics in terms of exp
f[i] = f[i].replace(lambda w: isinstance(w, HyperbolicFunction) and \
(len(w.free_symbols & set(symbols)) > 0), lambda w: w.rewrite(exp))
# if we have a Matrix, we need to iterate over its elements again
if f[i].is_Matrix:
bare_f = False
f.extend(list(f[i]))
f[i] = S.Zero
# if we can split it into real and imaginary parts then do so
freei = f[i].free_symbols
if freei and all(s.is_extended_real or s.is_imaginary for s in freei):
fr, fi = f[i].as_real_imag()
# accept as long as new re, im, arg or atan2 are not introduced
had = f[i].atoms(re, im, arg, atan2)
if fr and fi and fr != fi and not any(
i.atoms(re, im, arg, atan2) - had for i in (fr, fi)):
if bare_f:
bare_f = False
f[i: i + 1] = [fr, fi]
# real/imag handling -----------------------------
if any(isinstance(fi, (bool, BooleanAtom)) for fi in f):
if flags.get('set', False):
return [], set()
return []
for i, fi in enumerate(f):
# Abs
while True:
was = fi
fi = fi.replace(Abs, lambda arg:
separatevars(Abs(arg)).rewrite(Piecewise) if arg.has(*symbols)
else Abs(arg))
if was == fi:
break
for e in fi.find(Abs):
if e.has(*symbols):
raise NotImplementedError('solving %s when the argument '
'is not real or imaginary.' % e)
# arg
fi = fi.replace(arg, lambda a: arg(a).rewrite(atan2).rewrite(atan))
# save changes
f[i] = fi
# see if re(s) or im(s) appear
freim = [fi for fi in f if fi.has(re, im)]
if freim:
irf = []
for s in symbols:
if s.is_real or s.is_imaginary:
continue # neither re(x) nor im(x) will appear
# if re(s) or im(s) appear, the auxiliary equation must be present
if any(fi.has(re(s), im(s)) for fi in freim):
irf.append((s, re(s) + S.ImaginaryUnit*im(s)))
if irf:
for s, rhs in irf:
for i, fi in enumerate(f):
f[i] = fi.xreplace({s: rhs})
f.append(s - rhs)
symbols.extend([re(s), im(s)])
if bare_f:
bare_f = False
flags['dict'] = True
# end of real/imag handling -----------------------------
symbols = list(uniq(symbols))
if not ordered_symbols:
# we do this to make the results returned canonical in case f
# contains a system of nonlinear equations; all other cases should
# be unambiguous
symbols = sorted(symbols, key=default_sort_key)
# we can solve for non-symbol entities by replacing them with Dummy symbols
f, symbols, swap_sym = recast_to_symbols(f, symbols)
# this is needed in the next two events
symset = set(symbols)
# get rid of equations that have no symbols of interest; we don't
# try to solve them because the user didn't ask and they might be
# hard to solve; this means that solutions may be given in terms
# of the eliminated equations e.g. solve((x-y, y-3), x) -> {x: y}
newf = []
for fi in f:
# let the solver handle equations that..
# - have no symbols but are expressions
# - have symbols of interest
# - have no symbols of interest but are constant
# but when an expression is not constant and has no symbols of
# interest, it can't change what we obtain for a solution from
# the remaining equations so we don't include it; and if it's
# zero it can be removed and if it's not zero, there is no
# solution for the equation set as a whole
#
# The reason for doing this filtering is to allow an answer
# to be obtained to queries like solve((x - y, y), x); without
# this mod the return value is []
ok = False
if fi.free_symbols & symset:
ok = True
else:
if fi.is_number:
if fi.is_Number:
if fi.is_zero:
continue
return []
ok = True
else:
if fi.is_constant():
ok = True
if ok:
newf.append(fi)
if not newf:
return []
f = newf
del newf
# mask off any Object that we aren't going to invert: Derivative,
# Integral, etc... so that solving for anything that they contain will
# give an implicit solution
seen = set()
non_inverts = set()
for fi in f:
pot = preorder_traversal(fi)
for p in pot:
if not isinstance(p, Expr) or isinstance(p, Piecewise):
pass
elif (isinstance(p, bool) or
not p.args or
p in symset or
p.is_Add or p.is_Mul or
p.is_Pow and not implicit or
p.is_Function and not implicit) and p.func not in (re, im):
continue
elif not p in seen:
seen.add(p)
if p.free_symbols & symset:
non_inverts.add(p)
else:
continue
pot.skip()
del seen
non_inverts = dict(list(zip(non_inverts, [Dummy() for _ in non_inverts])))
f = [fi.subs(non_inverts) for fi in f]
# Both xreplace and subs are needed below: xreplace to force substitution
# inside Derivative, subs to handle non-straightforward substitutions
non_inverts = [(v, k.xreplace(swap_sym).subs(swap_sym)) for k, v in non_inverts.items()]
# rationalize Floats
floats = False
if flags.get('rational', True) is not False:
for i, fi in enumerate(f):
if fi.has(Float):
floats = True
f[i] = nsimplify(fi, rational=True)
# capture any denominators before rewriting since
# they may disappear after the rewrite, e.g. issue 14779
flags['_denominators'] = _simple_dens(f[0], symbols)
# Any embedded piecewise functions need to be brought out to the
# top level so that the appropriate strategy gets selected.
# However, this is necessary only if one of the piecewise
# functions depends on one of the symbols we are solving for.
def _has_piecewise(e):
if e.is_Piecewise:
return e.has(*symbols)
return any([_has_piecewise(a) for a in e.args])
for i, fi in enumerate(f):
if _has_piecewise(fi):
f[i] = piecewise_fold(fi)
#
# try to get a solution
###########################################################################
if bare_f:
solution = _solve(f[0], *symbols, **flags)
else:
solution = _solve_system(f, symbols, **flags)
#
# postprocessing
###########################################################################
# Restore masked-off objects
if non_inverts:
def _do_dict(solution):
return {k: v.subs(non_inverts) for k, v in
solution.items()}
for i in range(1):
if isinstance(solution, dict):
solution = _do_dict(solution)
break
elif solution and isinstance(solution, list):
if isinstance(solution[0], dict):
solution = [_do_dict(s) for s in solution]
break
elif isinstance(solution[0], tuple):
solution = [tuple([v.subs(non_inverts) for v in s]) for s
in solution]
break
else:
solution = [v.subs(non_inverts) for v in solution]
break
elif not solution:
break
else:
raise NotImplementedError(filldedent('''
no handling of %s was implemented''' % solution))
# Restore original "symbols" if a dictionary is returned.
# This is not necessary for
# - the single univariate equation case
# since the symbol will have been removed from the solution;
# - the nonlinear poly_system since that only supports zero-dimensional
# systems and those results come back as a list
#
# ** unless there were Derivatives with the symbols, but those were handled
# above.
if swap_sym:
symbols = [swap_sym.get(k, k) for k in symbols]
if isinstance(solution, dict):
solution = {swap_sym.get(k, k): v.subs(swap_sym)
for k, v in solution.items()}
elif solution and isinstance(solution, list) and isinstance(solution[0], dict):
for i, sol in enumerate(solution):
solution[i] = {swap_sym.get(k, k): v.subs(swap_sym)
for k, v in sol.items()}
# undo the dictionary solutions returned when the system was only partially
# solved with poly-system if all symbols are present
if (
not flags.get('dict', False) and
solution and
ordered_symbols and
not isinstance(solution, dict) and
all(isinstance(sol, dict) for sol in solution)
):
solution = [tuple([r.get(s, s) for s in symbols]) for r in solution]
# Get assumptions about symbols, to filter solutions.
# Note that if assumptions about a solution can't be verified, it is still
# returned.
check = flags.get('check', True)
# restore floats
if floats and solution and flags.get('rational', None) is None:
solution = nfloat(solution, exponent=False)
if check and solution: # assumption checking
warn = flags.get('warn', False)
got_None = [] # solutions for which one or more symbols gave None
no_False = [] # solutions for which no symbols gave False
if isinstance(solution, tuple):
# this has already been checked and is in as_set form
return solution
elif isinstance(solution, list):
if isinstance(solution[0], tuple):
for sol in solution:
for symb, val in zip(symbols, sol):
test = check_assumptions(val, **symb.assumptions0)
if test is False:
break
if test is None:
got_None.append(sol)
else:
no_False.append(sol)
elif isinstance(solution[0], dict):
for sol in solution:
a_None = False
for symb, val in sol.items():
test = check_assumptions(val, **symb.assumptions0)
if test:
continue
if test is False:
break
a_None = True
else:
no_False.append(sol)
if a_None:
got_None.append(sol)
else: # list of expressions
for sol in solution:
test = check_assumptions(sol, **symbols[0].assumptions0)
if test is False:
continue
no_False.append(sol)
if test is None:
got_None.append(sol)
elif isinstance(solution, dict):
a_None = False
for symb, val in solution.items():
test = check_assumptions(val, **symb.assumptions0)
if test:
continue
if test is False:
no_False = None
break
a_None = True
else:
no_False = solution
if a_None:
got_None.append(solution)
elif isinstance(solution, (Relational, And, Or)):
if len(symbols) != 1:
raise ValueError("Length should be 1")
if warn and symbols[0].assumptions0:
warnings.warn(filldedent("""
\tWarning: assumptions about variable '%s' are
not handled currently.""" % symbols[0]))
# TODO: check also variable assumptions for inequalities
else:
raise TypeError('Unrecognized solution') # improve the checker
solution = no_False
if warn and got_None:
warnings.warn(filldedent("""
\tWarning: assumptions concerning following solution(s)
can't be checked:""" + '\n\t' +
', '.join(str(s) for s in got_None)))
#
# done
###########################################################################
as_dict = flags.get('dict', False)
as_set = flags.get('set', False)
if not as_set and isinstance(solution, list):
# Make sure that a list of solutions is ordered in a canonical way.
solution.sort(key=default_sort_key)
if not as_dict and not as_set:
return solution or []
# return a list of mappings or []
if not solution:
solution = []
else:
if isinstance(solution, dict):
solution = [solution]
elif iterable(solution[0]):
solution = [dict(list(zip(symbols, s))) for s in solution]
elif isinstance(solution[0], dict):
pass
else:
if len(symbols) != 1:
raise ValueError("Length should be 1")
solution = [{symbols[0]: s} for s in solution]
if as_dict:
return solution
assert as_set
if not solution:
return [], set()
k = list(ordered(solution[0].keys()))
return k, {tuple([s[ki] for ki in k]) for s in solution}
def _solve(f, *symbols, **flags):
"""
Return a checked solution for *f* in terms of one or more of the
symbols. A list should be returned except for the case when a linear
undetermined-coefficients equation is encountered (in which case
a dictionary is returned).
If no method is implemented to solve the equation, a NotImplementedError
will be raised. In the case that conversion of an expression to a Poly
gives None a ValueError will be raised.
"""
not_impl_msg = "No algorithms are implemented to solve equation %s"
if len(symbols) != 1:
soln = None
free = f.free_symbols
ex = free - set(symbols)
if len(ex) != 1:
ind, dep = f.as_independent(*symbols)
ex = ind.free_symbols & dep.free_symbols
if len(ex) == 1:
ex = ex.pop()
try:
# soln may come back as dict, list of dicts or tuples, or
# tuple of symbol list and set of solution tuples
soln = solve_undetermined_coeffs(f, symbols, ex, **flags)
except NotImplementedError:
pass
if soln:
if flags.get('simplify', True):
if isinstance(soln, dict):
for k in soln:
soln[k] = simplify(soln[k])
elif isinstance(soln, list):
if isinstance(soln[0], dict):
for d in soln:
for k in d:
d[k] = simplify(d[k])
elif isinstance(soln[0], tuple):
soln = [tuple(simplify(i) for i in j) for j in soln]
else:
raise TypeError('unrecognized args in list')
elif isinstance(soln, tuple):
sym, sols = soln
soln = sym, {tuple(simplify(i) for i in j) for j in sols}
else:
raise TypeError('unrecognized solution type')
return soln
# find first successful solution
failed = []
got_s = set()
result = []
for s in symbols:
xi, v = solve_linear(f, symbols=[s])
if xi == s:
# no need to check but we should simplify if desired
if flags.get('simplify', True):
v = simplify(v)
vfree = v.free_symbols
if got_s and any([ss in vfree for ss in got_s]):
# sol depends on previously solved symbols: discard it
continue
got_s.add(xi)
result.append({xi: v})
elif xi: # there might be a non-linear solution if xi is not 0
failed.append(s)
if not failed:
return result
for s in failed:
try:
soln = _solve(f, s, **flags)
for sol in soln:
if got_s and any([ss in sol.free_symbols for ss in got_s]):
# sol depends on previously solved symbols: discard it
continue
got_s.add(s)
result.append({s: sol})
except NotImplementedError:
continue
if got_s:
return result
else:
raise NotImplementedError(not_impl_msg % f)
symbol = symbols[0]
#expand binomials only if it has the unknown symbol
f = f.replace(lambda e: isinstance(e, binomial) and e.has(symbol),
lambda e: expand_func(e))
# /!\ capture this flag then set it to False so that no checking in
# recursive calls will be done; only the final answer is checked
flags['check'] = checkdens = check = flags.pop('check', True)
# build up solutions if f is a Mul
if f.is_Mul:
result = set()
for m in f.args:
if m in {S.NegativeInfinity, S.ComplexInfinity, S.Infinity}:
result = set()
break
soln = _solve(m, symbol, **flags)
result.update(set(soln))
result = list(result)
if check:
# all solutions have been checked but now we must
# check that the solutions do not set denominators
# in any factor to zero
dens = flags.get('_denominators', _simple_dens(f, symbols))
result = [s for s in result if
all(not checksol(den, {symbol: s}, **flags) for den in
dens)]
# set flags for quick exit at end; solutions for each
# factor were already checked and simplified
check = False
flags['simplify'] = False
elif f.is_Piecewise:
result = set()
for i, (expr, cond) in enumerate(f.args):
if expr.is_zero:
raise NotImplementedError(
'solve cannot represent interval solutions')
candidates = _solve(expr, symbol, **flags)
# the explicit condition for this expr is the current cond
# and none of the previous conditions
args = [~c for _, c in f.args[:i]] + [cond]
cond = And(*args)
for candidate in candidates:
if candidate in result:
# an unconditional value was already there
continue
try:
v = cond.subs(symbol, candidate)
_eval_simplify = getattr(v, '_eval_simplify', None)
if _eval_simplify is not None:
# unconditionally take the simpification of v
v = _eval_simplify(ratio=2, measure=lambda x: 1)
except TypeError:
# incompatible type with condition(s)
continue
if v == False:
continue
if v == True:
result.add(candidate)
else:
result.add(Piecewise(
(candidate, v),
(S.NaN, True)))
# set flags for quick exit at end; solutions for each
# piece were already checked and simplified
check = False
flags['simplify'] = False
else:
# first see if it really depends on symbol and whether there
# is only a linear solution
f_num, sol = solve_linear(f, symbols=symbols)
if f_num.is_zero or sol is S.NaN:
return []
elif f_num.is_Symbol:
# no need to check but simplify if desired
if flags.get('simplify', True):
sol = simplify(sol)
return [sol]
poly = None
# check for a single non-symbol generator
dums = f_num.atoms(Dummy)
D = f_num.replace(
lambda i: isinstance(i, Add) and symbol in i.free_symbols,
lambda i: Dummy())
if not D.is_Dummy:
dgen = D.atoms(Dummy) - dums
if len(dgen) == 1:
d = dgen.pop()
w = Wild('g')
gen = f_num.match(D.xreplace({d: w}))[w]
spart = gen.as_independent(symbol)[1].as_base_exp()[0]
if spart == symbol:
try:
poly = Poly(f_num, spart)
except PolynomialError:
pass
result = False # no solution was obtained
msg = '' # there is no failure message
# Poly is generally robust enough to convert anything to
# a polynomial and tell us the different generators that it
# contains, so we will inspect the generators identified by
# polys to figure out what to do.
# try to identify a single generator that will allow us to solve this
# as a polynomial, followed (perhaps) by a change of variables if the
# generator is not a symbol
try:
if poly is None:
poly = Poly(f_num)
if poly is None:
raise ValueError('could not convert %s to Poly' % f_num)
except GeneratorsNeeded:
simplified_f = simplify(f_num)
if simplified_f != f_num:
return _solve(simplified_f, symbol, **flags)
raise ValueError('expression appears to be a constant')
gens = [g for g in poly.gens if g.has(symbol)]
def _as_base_q(x):
"""Return (b**e, q) for x = b**(p*e/q) where p/q is the leading
Rational of the exponent of x, e.g. exp(-2*x/3) -> (exp(x), 3)
"""
b, e = x.as_base_exp()
if e.is_Rational:
return b, e.q
if not e.is_Mul:
return x, 1
c, ee = e.as_coeff_Mul()
if c.is_Rational and c is not S.One: # c could be a Float
return b**ee, c.q
return x, 1
if len(gens) > 1:
# If there is more than one generator, it could be that the
# generators have the same base but different powers, e.g.
# >>> Poly(exp(x) + 1/exp(x))
# Poly(exp(-x) + exp(x), exp(-x), exp(x), domain='ZZ')
#
# If unrad was not disabled then there should be no rational
# exponents appearing as in
# >>> Poly(sqrt(x) + sqrt(sqrt(x)))
# Poly(sqrt(x) + x**(1/4), sqrt(x), x**(1/4), domain='ZZ')
bases, qs = list(zip(*[_as_base_q(g) for g in gens]))
bases = set(bases)
if len(bases) > 1 or not all(q == 1 for q in qs):
funcs = {b for b in bases if b.is_Function}
trig = {_ for _ in funcs if
isinstance(_, TrigonometricFunction)}
other = funcs - trig
if not other and len(funcs.intersection(trig)) > 1:
newf = None
if f_num.is_Add and len(f_num.args) == 2:
# check for sin(x)**p = cos(x)**p
_args = f_num.args
t = a, b = [i.atoms(Function).intersection(
trig) for i in _args]
if all(len(i) == 1 for i in t):
a, b = [i.pop() for i in t]
if isinstance(a, cos):
a, b = b, a
_args = _args[::-1]
if isinstance(a, sin) and isinstance(b, cos
) and a.args[0] == b.args[0]:
# sin(x) + cos(x) = 0 -> tan(x) + 1 = 0
newf, _d = (TR2i(_args[0]/_args[1]) + 1
).as_numer_denom()
if not _d.is_Number:
newf = None
if newf is None:
newf = TR1(f_num).rewrite(tan)
if newf != f_num:
# don't check the rewritten form --check
# solutions in the un-rewritten form below
flags['check'] = False
result = _solve(newf, symbol, **flags)
flags['check'] = check
# just a simple case - see if replacement of single function
# clears all symbol-dependent functions, e.g.
# log(x) - log(log(x) - 1) - 3 can be solved even though it has
# two generators.
if result is False and funcs:
funcs = list(ordered(funcs)) # put shallowest function first
f1 = funcs[0]
t = Dummy('t')
# perform the substitution
ftry = f_num.subs(f1, t)
# if no Functions left, we can proceed with usual solve
if not ftry.has(symbol):
cv_sols = _solve(ftry, t, **flags)
cv_inv = _solve(t - f1, symbol, **flags)[0]
sols = list()
for sol in cv_sols:
sols.append(cv_inv.subs(t, sol))
result = list(ordered(sols))
if result is False:
msg = 'multiple generators %s' % gens
else:
# e.g. case where gens are exp(x), exp(-x)
u = bases.pop()
t = Dummy('t')
inv = _solve(u - t, symbol, **flags)
if isinstance(u, (Pow, exp)):
# this will be resolved by factor in _tsolve but we might
# as well try a simple expansion here to get things in
# order so something like the following will work now without
# having to factor:
#
# >>> eq = (exp(I*(-x-2))+exp(I*(x+2)))
# >>> eq.subs(exp(x),y) # fails
# exp(I*(-x - 2)) + exp(I*(x + 2))
# >>> eq.expand().subs(exp(x),y) # works
# y**I*exp(2*I) + y**(-I)*exp(-2*I)
def _expand(p):
b, e = p.as_base_exp()
e = expand_mul(e)
return expand_power_exp(b**e)
ftry = f_num.replace(
lambda w: w.is_Pow or isinstance(w, exp),
_expand).subs(u, t)
if not ftry.has(symbol):
soln = _solve(ftry, t, **flags)
sols = list()
for sol in soln:
for i in inv:
sols.append(i.subs(t, sol))
result = list(ordered(sols))
elif len(gens) == 1:
# There is only one generator that we are interested in, but
# there may have been more than one generator identified by
# polys (e.g. for symbols other than the one we are interested
# in) so recast the poly in terms of our generator of interest.
# Also use composite=True with f_num since Poly won't update
# poly as documented in issue 8810.
poly = Poly(f_num, gens[0], composite=True)
# if we aren't on the tsolve-pass, use roots
if not flags.pop('tsolve', False):
soln = None
deg = poly.degree()
flags['tsolve'] = True
solvers = {k: flags.get(k, True) for k in
('cubics', 'quartics', 'quintics')}
soln = roots(poly, **solvers)
if sum(soln.values()) < deg:
# e.g. roots(32*x**5 + 400*x**4 + 2032*x**3 +
# 5000*x**2 + 6250*x + 3189) -> {}
# so all_roots is used and RootOf instances are
# returned *unless* the system is multivariate
# or high-order EX domain.
try:
soln = poly.all_roots()
except NotImplementedError:
if not flags.get('incomplete', True):
raise NotImplementedError(
filldedent('''
Neither high-order multivariate polynomials
nor sorting of EX-domain polynomials is supported.
If you want to see any results, pass keyword incomplete=True to
solve; to see numerical values of roots
for univariate expressions, use nroots.
'''))
else:
pass
else:
soln = list(soln.keys())
if soln is not None:
u = poly.gen
if u != symbol:
try:
t = Dummy('t')
iv = _solve(u - t, symbol, **flags)
soln = list(ordered({i.subs(t, s) for i in iv for s in soln}))
except NotImplementedError:
# perhaps _tsolve can handle f_num
soln = None
else:
check = False # only dens need to be checked
if soln is not None:
if len(soln) > 2:
# if the flag wasn't set then unset it since high-order
# results are quite long. Perhaps one could base this
# decision on a certain critical length of the
# roots. In addition, wester test M2 has an expression
# whose roots can be shown to be real with the
# unsimplified form of the solution whereas only one of
# the simplified forms appears to be real.
flags['simplify'] = flags.get('simplify', False)
result = soln
# fallback if above fails
# -----------------------
if result is False:
# try unrad
if flags.pop('_unrad', True):
try:
u = unrad(f_num, symbol)
except (ValueError, NotImplementedError):
u = False
if u:
eq, cov = u
if cov:
isym, ieq = cov
inv = _solve(ieq, symbol, **flags)[0]
rv = {inv.subs(isym, xi) for xi in _solve(eq, isym, **flags)}
else:
try:
rv = set(_solve(eq, symbol, **flags))
except NotImplementedError:
rv = None
if rv is not None:
result = list(ordered(rv))
# if the flag wasn't set then unset it since unrad results
# can be quite long or of very high order
flags['simplify'] = flags.get('simplify', False)
else:
pass # for coverage
# try _tsolve
if result is False:
flags.pop('tsolve', None) # allow tsolve to be used on next pass
try:
soln = _tsolve(f_num, symbol, **flags)
if soln is not None:
result = soln
except PolynomialError:
pass
# ----------- end of fallback ----------------------------
if result is False:
raise NotImplementedError('\n'.join([msg, not_impl_msg % f]))
if flags.get('simplify', True):
result = list(map(simplify, result))
# we just simplified the solution so we now set the flag to
# False so the simplification doesn't happen again in checksol()
flags['simplify'] = False
if checkdens:
# reject any result that makes any denom. affirmatively 0;
# if in doubt, keep it
dens = _simple_dens(f, symbols)
result = [s for s in result if
all(not checksol(d, {symbol: s}, **flags)
for d in dens)]
if check:
# keep only results if the check is not False
result = [r for r in result if
checksol(f_num, {symbol: r}, **flags) is not False]
return result
def _solve_system(exprs, symbols, **flags):
if not exprs:
return []
if flags.pop('_split', True):
# Split the system into connected components
V = exprs
symsset = set(symbols)
exprsyms = {e: e.free_symbols & symsset for e in exprs}
E = []
sym_indices = {sym: i for i, sym in enumerate(symbols)}
for n, e1 in enumerate(exprs):
for e2 in exprs[:n]:
# Equations are connected if they share a symbol
if exprsyms[e1] & exprsyms[e2]:
E.append((e1, e2))
G = V, E
subexprs = connected_components(G)
if len(subexprs) > 1:
subsols = []
for subexpr in subexprs:
subsyms = set()
for e in subexpr:
subsyms |= exprsyms[e]
subsyms = list(sorted(subsyms, key = lambda x: sym_indices[x]))
# use canonical subset to solve these equations
# since there may be redundant equations in the set:
# take the first equation of several that may have the
# same sub-maximal free symbols of interest; the
# other equations that weren't used should be checked
# to see that they did not fail -- does the solver
# take care of that?
choices = sift(subexpr, lambda x: tuple(ordered(exprsyms[x])))
subexpr = choices.pop(tuple(ordered(subsyms)), [])
for k in choices:
subexpr.append(next(ordered(choices[k])))
flags['_split'] = False # skip split step
subsol = _solve_system(subexpr, subsyms, **flags)
if not isinstance(subsol, list):
subsol = [subsol]
subsols.append(subsol)
# Full solution is cartesion product of subsystems
sols = []
for soldicts in cartes(*subsols):
sols.append(dict(item for sd in soldicts
for item in sd.items()))
# Return a list with one dict as just the dict
if len(sols) == 1:
return sols[0]
return sols
polys = []
dens = set()
failed = []
result = False
linear = False
manual = flags.get('manual', False)
checkdens = check = flags.get('check', True)
for j, g in enumerate(exprs):
dens.update(_simple_dens(g, symbols))
i, d = _invert(g, *symbols)
g = d - i
g = g.as_numer_denom()[0]
if manual:
failed.append(g)
continue
poly = g.as_poly(*symbols, extension=True)
if poly is not None:
polys.append(poly)
else:
failed.append(g)
if not polys:
solved_syms = []
else:
if all(p.is_linear for p in polys):
n, m = len(polys), len(symbols)
matrix = zeros(n, m + 1)
for i, poly in enumerate(polys):
for monom, coeff in poly.terms():
try:
j = monom.index(1)
matrix[i, j] = coeff
except ValueError:
matrix[i, m] = -coeff
# returns a dictionary ({symbols: values}) or None
if flags.pop('particular', False):
result = minsolve_linear_system(matrix, *symbols, **flags)
else:
result = solve_linear_system(matrix, *symbols, **flags)
if failed:
if result:
solved_syms = list(result.keys())
else:
solved_syms = []
else:
linear = True
else:
if len(symbols) > len(polys):
from sympy.utilities.iterables import subsets
free = set().union(*[p.free_symbols for p in polys])
free = list(ordered(free.intersection(symbols)))
got_s = set()
result = []
for syms in subsets(free, len(polys)):
try:
# returns [] or list of tuples of solutions for syms
res = solve_poly_system(polys, *syms)
if res:
for r in res:
skip = False
for r1 in r:
if got_s and any([ss in r1.free_symbols
for ss in got_s]):
# sol depends on previously
# solved symbols: discard it
skip = True
if not skip:
got_s.update(syms)
result.extend([dict(list(zip(syms, r)))])
except NotImplementedError:
pass
if got_s:
solved_syms = list(got_s)
else:
raise NotImplementedError('no valid subset found')
else:
try:
result = solve_poly_system(polys, *symbols)
if result:
solved_syms = symbols
# we don't know here if the symbols provided
# were given or not, so let solve resolve that.
# A list of dictionaries is going to always be
# returned from here.
result = [dict(list(zip(solved_syms, r))) for r in result]
except NotImplementedError:
failed.extend([g.as_expr() for g in polys])
solved_syms = []
result = None
if result:
if isinstance(result, dict):
result = [result]
else:
result = [{}]
if failed:
# For each failed equation, see if we can solve for one of the
# remaining symbols from that equation. If so, we update the
# solution set and continue with the next failed equation,
# repeating until we are done or we get an equation that can't
# be solved.
def _ok_syms(e, sort=False):
rv = (e.free_symbols - solved_syms) & legal
if sort:
rv = list(rv)
rv.sort(key=default_sort_key)
return rv
solved_syms = set(solved_syms) # set of symbols we have solved for
legal = set(symbols) # what we are interested in
# sort so equation with the fewest potential symbols is first
u = Dummy() # used in solution checking
for eq in ordered(failed, lambda _: len(_ok_syms(_))):
newresult = []
bad_results = []
got_s = set()
hit = False
for r in result:
# update eq with everything that is known so far
eq2 = eq.subs(r)
# if check is True then we see if it satisfies this
# equation, otherwise we just accept it
if check and r:
b = checksol(u, u, eq2, minimal=True)
if b is not None:
# this solution is sufficient to know whether
# it is valid or not so we either accept or
# reject it, then continue
if b:
newresult.append(r)
else:
bad_results.append(r)
continue
# search for a symbol amongst those available that
# can be solved for
ok_syms = _ok_syms(eq2, sort=True)
if not ok_syms:
if r:
newresult.append(r)
break # skip as it's independent of desired symbols
for s in ok_syms:
try:
soln = _solve(eq2, s, **flags)
except NotImplementedError:
continue
# put each solution in r and append the now-expanded
# result in the new result list; use copy since the
# solution for s in being added in-place
for sol in soln:
if got_s and any([ss in sol.free_symbols for ss in got_s]):
# sol depends on previously solved symbols: discard it
continue
rnew = r.copy()
for k, v in r.items():
rnew[k] = v.subs(s, sol)
# and add this new solution
rnew[s] = sol
newresult.append(rnew)
hit = True
got_s.add(s)
if not hit:
raise NotImplementedError('could not solve %s' % eq2)
else:
result = newresult
for b in bad_results:
if b in result:
result.remove(b)
default_simplify = bool(failed) # rely on system-solvers to simplify
if flags.get('simplify', default_simplify):
for r in result:
for k in r:
r[k] = simplify(r[k])
flags['simplify'] = False # don't need to do so in checksol now
if checkdens:
result = [r for r in result
if not any(checksol(d, r, **flags) for d in dens)]
if check and not linear:
result = [r for r in result
if not any(checksol(e, r, **flags) is False for e in exprs)]
result = [r for r in result if r]
if linear and result:
result = result[0]
return result
def solve_linear(lhs, rhs=0, symbols=[], exclude=[]):
r"""
Return a tuple derived from ``f = lhs - rhs`` that is one of
the following: ``(0, 1)``, ``(0, 0)``, ``(symbol, solution)``, ``(n, d)``.
Explanation
===========
``(0, 1)`` meaning that ``f`` is independent of the symbols in *symbols*
that are not in *exclude*.
``(0, 0)`` meaning that there is no solution to the equation amongst the
symbols given. If the first element of the tuple is not zero, then the
function is guaranteed to be dependent on a symbol in *symbols*.
``(symbol, solution)`` where symbol appears linearly in the numerator of
``f``, is in *symbols* (if given), and is not in *exclude* (if given). No
simplification is done to ``f`` other than a ``mul=True`` expansion, so the
solution will correspond strictly to a unique solution.
``(n, d)`` where ``n`` and ``d`` are the numerator and denominator of ``f``
when the numerator was not linear in any symbol of interest; ``n`` will
never be a symbol unless a solution for that symbol was found (in which case
the second element is the solution, not the denominator).
Examples
========
>>> from sympy.core.power import Pow
>>> from sympy.polys.polytools import cancel
``f`` is independent of the symbols in *symbols* that are not in
*exclude*:
>>> from sympy.solvers.solvers import solve_linear
>>> from sympy.abc import x, y, z
>>> from sympy import cos, sin
>>> eq = y*cos(x)**2 + y*sin(x)**2 - y # = y*(1 - 1) = 0
>>> solve_linear(eq)
(0, 1)
>>> eq = cos(x)**2 + sin(x)**2 # = 1
>>> solve_linear(eq)
(0, 1)
>>> solve_linear(x, exclude=[x])
(0, 1)
The variable ``x`` appears as a linear variable in each of the
following:
>>> solve_linear(x + y**2)
(x, -y**2)
>>> solve_linear(1/x - y**2)
(x, y**(-2))
When not linear in ``x`` or ``y`` then the numerator and denominator are
returned:
>>> solve_linear(x**2/y**2 - 3)
(x**2 - 3*y**2, y**2)
If the numerator of the expression is a symbol, then ``(0, 0)`` is
returned if the solution for that symbol would have set any
denominator to 0:
>>> eq = 1/(1/x - 2)
>>> eq.as_numer_denom()
(x, 1 - 2*x)
>>> solve_linear(eq)
(0, 0)
But automatic rewriting may cause a symbol in the denominator to
appear in the numerator so a solution will be returned:
>>> (1/x)**-1
x
>>> solve_linear((1/x)**-1)
(x, 0)
Use an unevaluated expression to avoid this:
>>> solve_linear(Pow(1/x, -1, evaluate=False))
(0, 0)
If ``x`` is allowed to cancel in the following expression, then it
appears to be linear in ``x``, but this sort of cancellation is not
done by ``solve_linear`` so the solution will always satisfy the
original expression without causing a division by zero error.
>>> eq = x**2*(1/x - z**2/x)
>>> solve_linear(cancel(eq))
(x, 0)
>>> solve_linear(eq)
(x**2*(1 - z**2), x)
A list of symbols for which a solution is desired may be given:
>>> solve_linear(x + y + z, symbols=[y])
(y, -x - z)
A list of symbols to ignore may also be given:
>>> solve_linear(x + y + z, exclude=[x])
(y, -x - z)
(A solution for ``y`` is obtained because it is the first variable
from the canonically sorted list of symbols that had a linear
solution.)
"""
if isinstance(lhs, Equality):
if rhs:
raise ValueError(filldedent('''
If lhs is an Equality, rhs must be 0 but was %s''' % rhs))
rhs = lhs.rhs
lhs = lhs.lhs
dens = None
eq = lhs - rhs
n, d = eq.as_numer_denom()
if not n:
return S.Zero, S.One
free = n.free_symbols
if not symbols:
symbols = free
else:
bad = [s for s in symbols if not s.is_Symbol]
if bad:
if len(bad) == 1:
bad = bad[0]
if len(symbols) == 1:
eg = 'solve(%s, %s)' % (eq, symbols[0])
else:
eg = 'solve(%s, *%s)' % (eq, list(symbols))
raise ValueError(filldedent('''
solve_linear only handles symbols, not %s. To isolate
non-symbols use solve, e.g. >>> %s <<<.
''' % (bad, eg)))
symbols = free.intersection(symbols)
symbols = symbols.difference(exclude)
if not symbols:
return S.Zero, S.One
# derivatives are easy to do but tricky to analyze to see if they
# are going to disallow a linear solution, so for simplicity we
# just evaluate the ones that have the symbols of interest
derivs = defaultdict(list)
for der in n.atoms(Derivative):
csym = der.free_symbols & symbols
for c in csym:
derivs[c].append(der)
all_zero = True
for xi in sorted(symbols, key=default_sort_key): # canonical order
# if there are derivatives in this var, calculate them now
if isinstance(derivs[xi], list):
derivs[xi] = {der: der.doit() for der in derivs[xi]}
newn = n.subs(derivs[xi])
dnewn_dxi = newn.diff(xi)
# dnewn_dxi can be nonzero if it survives differentation by any
# of its free symbols
free = dnewn_dxi.free_symbols
if dnewn_dxi and (not free or any(dnewn_dxi.diff(s) for s in free)):
all_zero = False
if dnewn_dxi is S.NaN:
break
if xi not in dnewn_dxi.free_symbols:
vi = -1/dnewn_dxi*(newn.subs(xi, 0))
if dens is None:
dens = _simple_dens(eq, symbols)
if not any(checksol(di, {xi: vi}, minimal=True) is True
for di in dens):
# simplify any trivial integral
irep = [(i, i.doit()) for i in vi.atoms(Integral) if
i.function.is_number]
# do a slight bit of simplification
vi = expand_mul(vi.subs(irep))
return xi, vi
if all_zero:
return S.Zero, S.One
if n.is_Symbol: # no solution for this symbol was found
return S.Zero, S.Zero
return n, d
def minsolve_linear_system(system, *symbols, **flags):
r"""
Find a particular solution to a linear system.
Explanation
===========
In particular, try to find a solution with the minimal possible number
of non-zero variables using a naive algorithm with exponential complexity.
If ``quick=True``, a heuristic is used.
"""
quick = flags.get('quick', False)
# Check if there are any non-zero solutions at all
s0 = solve_linear_system(system, *symbols, **flags)
if not s0 or all(v == 0 for v in s0.values()):
return s0
if quick:
# We just solve the system and try to heuristically find a nice
# solution.
s = solve_linear_system(system, *symbols)
def update(determined, solution):
delete = []
for k, v in solution.items():
solution[k] = v.subs(determined)
if not solution[k].free_symbols:
delete.append(k)
determined[k] = solution[k]
for k in delete:
del solution[k]
determined = {}
update(determined, s)
while s:
# NOTE sort by default_sort_key to get deterministic result
k = max((k for k in s.values()),
key=lambda x: (len(x.free_symbols), default_sort_key(x)))
x = max(k.free_symbols, key=default_sort_key)
if len(k.free_symbols) != 1:
determined[x] = S.Zero
else:
val = solve(k)[0]
if val == 0 and all(v.subs(x, val) == 0 for v in s.values()):
determined[x] = S.One
else:
determined[x] = val
update(determined, s)
return determined
else:
# We try to select n variables which we want to be non-zero.
# All others will be assumed zero. We try to solve the modified system.
# If there is a non-trivial solution, just set the free variables to
# one. If we do this for increasing n, trying all combinations of
# variables, we will find an optimal solution.
# We speed up slightly by starting at one less than the number of
# variables the quick method manages.
from itertools import combinations
from sympy.utilities.misc import debug
N = len(symbols)
bestsol = minsolve_linear_system(system, *symbols, quick=True)
n0 = len([x for x in bestsol.values() if x != 0])
for n in range(n0 - 1, 1, -1):
debug('minsolve: %s' % n)
thissol = None
for nonzeros in combinations(list(range(N)), n):
subm = Matrix([system.col(i).T for i in nonzeros] + [system.col(-1).T]).T
s = solve_linear_system(subm, *[symbols[i] for i in nonzeros])
if s and not all(v == 0 for v in s.values()):
subs = [(symbols[v], S.One) for v in nonzeros]
for k, v in s.items():
s[k] = v.subs(subs)
for sym in symbols:
if sym not in s:
if symbols.index(sym) in nonzeros:
s[sym] = S.One
else:
s[sym] = S.Zero
thissol = s
break
if thissol is None:
break
bestsol = thissol
return bestsol
def solve_linear_system(system, *symbols, **flags):
r"""
Solve system of $N$ linear equations with $M$ variables, which means
both under- and overdetermined systems are supported.
Explanation
===========
The possible number of solutions is zero, one, or infinite. Respectively,
this procedure will return None or a dictionary with solutions. In the
case of underdetermined systems, all arbitrary parameters are skipped.
This may cause a situation in which an empty dictionary is returned.
In that case, all symbols can be assigned arbitrary values.
Input to this function is a $N\times M + 1$ matrix, which means it has
to be in augmented form. If you prefer to enter $N$ equations and $M$
unknowns then use ``solve(Neqs, *Msymbols)`` instead. Note: a local
copy of the matrix is made by this routine so the matrix that is
passed will not be modified.
The algorithm used here is fraction-free Gaussian elimination,
which results, after elimination, in an upper-triangular matrix.
Then solutions are found using back-substitution. This approach
is more efficient and compact than the Gauss-Jordan method.
Examples
========
>>> from sympy import Matrix, solve_linear_system
>>> from sympy.abc import x, y
Solve the following system::
x + 4 y == 2
-2 x + y == 14
>>> system = Matrix(( (1, 4, 2), (-2, 1, 14)))
>>> solve_linear_system(system, x, y)
{x: -6, y: 2}
A degenerate system returns an empty dictionary:
>>> system = Matrix(( (0,0,0), (0,0,0) ))
>>> solve_linear_system(system, x, y)
{}
"""
assert system.shape[1] == len(symbols) + 1
# This is just a wrapper for solve_lin_sys
eqs = list(system * Matrix(symbols + (-1,)))
eqs, ring = sympy_eqs_to_ring(eqs, symbols)
sol = solve_lin_sys(eqs, ring, _raw=False)
if sol is not None:
sol = {sym:val for sym, val in sol.items() if sym != val}
return sol
def solve_undetermined_coeffs(equ, coeffs, sym, **flags):
r"""
Solve equation of a type $p(x; a_1, \ldots, a_k) = q(x)$ where both
$p$ and $q$ are univariate polynomials that depend on $k$ parameters.
Explanation
===========
The result of this function is a dictionary with symbolic values of those
parameters with respect to coefficients in $q$.
This function accepts both equations class instances and ordinary
SymPy expressions. Specification of parameters and variables is
obligatory for efficiency and simplicity reasons.
Examples
========
>>> from sympy import Eq
>>> from sympy.abc import a, b, c, x
>>> from sympy.solvers import solve_undetermined_coeffs
>>> solve_undetermined_coeffs(Eq(2*a*x + a+b, x), [a, b], x)
{a: 1/2, b: -1/2}
>>> solve_undetermined_coeffs(Eq(a*c*x + a+b, x), [a, b], x)
{a: 1/c, b: -1/c}
"""
if isinstance(equ, Equality):
# got equation, so move all the
# terms to the left hand side
equ = equ.lhs - equ.rhs
equ = cancel(equ).as_numer_denom()[0]
system = list(collect(equ.expand(), sym, evaluate=False).values())
if not any(equ.has(sym) for equ in system):
# consecutive powers in the input expressions have
# been successfully collected, so solve remaining
# system using Gaussian elimination algorithm
return solve(system, *coeffs, **flags)
else:
return None # no solutions
def solve_linear_system_LU(matrix, syms):
"""
Solves the augmented matrix system using ``LUsolve`` and returns a
dictionary in which solutions are keyed to the symbols of *syms* as ordered.
Explanation
===========
The matrix must be invertible.
Examples
========
>>> from sympy import Matrix
>>> from sympy.abc import x, y, z
>>> from sympy.solvers.solvers import solve_linear_system_LU
>>> solve_linear_system_LU(Matrix([
... [1, 2, 0, 1],
... [3, 2, 2, 1],
... [2, 0, 0, 1]]), [x, y, z])
{x: 1/2, y: 1/4, z: -1/2}
See Also
========
LUsolve
"""
if matrix.rows != matrix.cols - 1:
raise ValueError("Rows should be equal to columns - 1")
A = matrix[:matrix.rows, :matrix.rows]
b = matrix[:, matrix.cols - 1:]
soln = A.LUsolve(b)
solutions = {}
for i in range(soln.rows):
solutions[syms[i]] = soln[i, 0]
return solutions
def det_perm(M):
"""
Return the determinant of *M* by using permutations to select factors.
Explanation
===========
For sizes larger than 8 the number of permutations becomes prohibitively
large, or if there are no symbols in the matrix, it is better to use the
standard determinant routines (e.g., ``M.det()``.)
See Also
========
det_minor
det_quick
"""
args = []
s = True
n = M.rows
list_ = getattr(M, '_mat', None)
if list_ is None:
list_ = flatten(M.tolist())
for perm in generate_bell(n):
fac = []
idx = 0
for j in perm:
fac.append(list_[idx + j])
idx += n
term = Mul(*fac) # disaster with unevaluated Mul -- takes forever for n=7
args.append(term if s else -term)
s = not s
return Add(*args)
def det_minor(M):
"""
Return the ``det(M)`` computed from minors without
introducing new nesting in products.
See Also
========
det_perm
det_quick
"""
n = M.rows
if n == 2:
return M[0, 0]*M[1, 1] - M[1, 0]*M[0, 1]
else:
return sum([(1, -1)[i % 2]*Add(*[M[0, i]*d for d in
Add.make_args(det_minor(M.minor_submatrix(0, i)))])
if M[0, i] else S.Zero for i in range(n)])
def det_quick(M, method=None):
"""
Return ``det(M)`` assuming that either
there are lots of zeros or the size of the matrix
is small. If this assumption is not met, then the normal
Matrix.det function will be used with method = ``method``.
See Also
========
det_minor
det_perm
"""
if any(i.has(Symbol) for i in M):
if M.rows < 8 and all(i.has(Symbol) for i in M):
return det_perm(M)
return det_minor(M)
else:
return M.det(method=method) if method else M.det()
def inv_quick(M):
"""Return the inverse of ``M``, assuming that either
there are lots of zeros or the size of the matrix
is small.
"""
from sympy.matrices import zeros
if not all(i.is_Number for i in M):
if not any(i.is_Number for i in M):
det = lambda _: det_perm(_)
else:
det = lambda _: det_minor(_)
else:
return M.inv()
n = M.rows
d = det(M)
if d == S.Zero:
raise NonInvertibleMatrixError("Matrix det == 0; not invertible")
ret = zeros(n)
s1 = -1
for i in range(n):
s = s1 = -s1
for j in range(n):
di = det(M.minor_submatrix(i, j))
ret[j, i] = s*di/d
s = -s
return ret
# these are functions that have multiple inverse values per period
multi_inverses = {
sin: lambda x: (asin(x), S.Pi - asin(x)),
cos: lambda x: (acos(x), 2*S.Pi - acos(x)),
}
def _tsolve(eq, sym, **flags):
"""
Helper for ``_solve`` that solves a transcendental equation with respect
to the given symbol. Various equations containing powers and logarithms,
can be solved.
There is currently no guarantee that all solutions will be returned or
that a real solution will be favored over a complex one.
Either a list of potential solutions will be returned or None will be
returned (in the case that no method was known to get a solution
for the equation). All other errors (like the inability to cast an
expression as a Poly) are unhandled.
Examples
========
>>> from sympy import log
>>> from sympy.solvers.solvers import _tsolve as tsolve
>>> from sympy.abc import x
>>> tsolve(3**(2*x + 5) - 4, x)
[-5/2 + log(2)/log(3), (-5*log(3)/2 + log(2) + I*pi)/log(3)]
>>> tsolve(log(x) + 2*x, x)
[LambertW(2)/2]
"""
if 'tsolve_saw' not in flags:
flags['tsolve_saw'] = []
if eq in flags['tsolve_saw']:
return None
else:
flags['tsolve_saw'].append(eq)
rhs, lhs = _invert(eq, sym)
if lhs == sym:
return [rhs]
try:
if lhs.is_Add:
# it's time to try factoring; powdenest is used
# to try get powers in standard form for better factoring
f = factor(powdenest(lhs - rhs))
if f.is_Mul:
return _solve(f, sym, **flags)
if rhs:
f = logcombine(lhs, force=flags.get('force', True))
if f.count(log) != lhs.count(log):
if isinstance(f, log):
return _solve(f.args[0] - exp(rhs), sym, **flags)
return _tsolve(f - rhs, sym, **flags)
elif lhs.is_Pow:
if lhs.exp.is_Integer:
if lhs - rhs != eq:
return _solve(lhs - rhs, sym, **flags)
if sym not in lhs.exp.free_symbols:
return _solve(lhs.base - rhs**(1/lhs.exp), sym, **flags)
# _tsolve calls this with Dummy before passing the actual number in.
if any(t.is_Dummy for t in rhs.free_symbols):
raise NotImplementedError # _tsolve will call here again...
# a ** g(x) == 0
if not rhs:
# f(x)**g(x) only has solutions where f(x) == 0 and g(x) != 0 at
# the same place
sol_base = _solve(lhs.base, sym, **flags)
return [s for s in sol_base if lhs.exp.subs(sym, s) != 0]
# a ** g(x) == b
if not lhs.base.has(sym):
if lhs.base == 0:
return _solve(lhs.exp, sym, **flags) if rhs != 0 else []
# Gets most solutions...
if lhs.base == rhs.as_base_exp()[0]:
# handles case when bases are equal
sol = _solve(lhs.exp - rhs.as_base_exp()[1], sym, **flags)
else:
# handles cases when bases are not equal and exp
# may or may not be equal
sol = _solve(exp(log(lhs.base)*lhs.exp)-exp(log(rhs)), sym, **flags)
# Check for duplicate solutions
def equal(expr1, expr2):
_ = Dummy()
eq = checksol(expr1 - _, _, expr2)
if eq is None:
if nsimplify(expr1) != nsimplify(expr2):
return False
# they might be coincidentally the same
# so check more rigorously
eq = expr1.equals(expr2)
return eq
# Guess a rational exponent
e_rat = nsimplify(log(abs(rhs))/log(abs(lhs.base)))
e_rat = simplify(posify(e_rat)[0])
n, d = fraction(e_rat)
if expand(lhs.base**n - rhs**d) == 0:
sol = [s for s in sol if not equal(lhs.exp.subs(sym, s), e_rat)]
sol.extend(_solve(lhs.exp - e_rat, sym, **flags))
return list(ordered(set(sol)))
# f(x) ** g(x) == c
else:
sol = []
logform = lhs.exp*log(lhs.base) - log(rhs)
if logform != lhs - rhs:
try:
sol.extend(_solve(logform, sym, **flags))
except NotImplementedError:
pass
# Collect possible solutions and check with substitution later.
check = []
if rhs == 1:
# f(x) ** g(x) = 1 -- g(x)=0 or f(x)=+-1
check.extend(_solve(lhs.exp, sym, **flags))
check.extend(_solve(lhs.base - 1, sym, **flags))
check.extend(_solve(lhs.base + 1, sym, **flags))
elif rhs.is_Rational:
for d in (i for i in divisors(abs(rhs.p)) if i != 1):
e, t = integer_log(rhs.p, d)
if not t:
continue # rhs.p != d**b
for s in divisors(abs(rhs.q)):
if s**e== rhs.q:
r = Rational(d, s)
check.extend(_solve(lhs.base - r, sym, **flags))
check.extend(_solve(lhs.base + r, sym, **flags))
check.extend(_solve(lhs.exp - e, sym, **flags))
elif rhs.is_irrational:
b_l, e_l = lhs.base.as_base_exp()
n, d = (e_l*lhs.exp).as_numer_denom()
b, e = sqrtdenest(rhs).as_base_exp()
check = [sqrtdenest(i) for i in (_solve(lhs.base - b, sym, **flags))]
check.extend([sqrtdenest(i) for i in (_solve(lhs.exp - e, sym, **flags))])
if e_l*d != 1:
check.extend(_solve(b_l**n - rhs**(e_l*d), sym, **flags))
for s in check:
ok = checksol(eq, sym, s)
if ok is None:
ok = eq.subs(sym, s).equals(0)
if ok:
sol.append(s)
return list(ordered(set(sol)))
elif lhs.is_Function and len(lhs.args) == 1:
if lhs.func in multi_inverses:
# sin(x) = 1/3 -> x - asin(1/3) & x - (pi - asin(1/3))
soln = []
for i in multi_inverses[lhs.func](rhs):
soln.extend(_solve(lhs.args[0] - i, sym, **flags))
return list(ordered(soln))
elif lhs.func == LambertW:
return _solve(lhs.args[0] - rhs*exp(rhs), sym, **flags)
rewrite = lhs.rewrite(exp)
if rewrite != lhs:
return _solve(rewrite - rhs, sym, **flags)
except NotImplementedError:
pass
# maybe it is a lambert pattern
if flags.pop('bivariate', True):
# lambert forms may need some help being recognized, e.g. changing
# 2**(3*x) + x**3*log(2)**3 + 3*x**2*log(2)**2 + 3*x*log(2) + 1
# to 2**(3*x) + (x*log(2) + 1)**3
g = _filtered_gens(eq.as_poly(), sym)
up_or_log = set()
for gi in g:
if isinstance(gi, exp) or (gi.is_Pow and gi.base == S.Exp1) or isinstance(gi, log):
up_or_log.add(gi)
elif gi.is_Pow:
gisimp = powdenest(expand_power_exp(gi))
if gisimp.is_Pow and sym in gisimp.exp.free_symbols:
up_or_log.add(gi)
eq_down = expand_log(expand_power_exp(eq)).subs(
dict(list(zip(up_or_log, [0]*len(up_or_log)))))
eq = expand_power_exp(factor(eq_down, deep=True) + (eq - eq_down))
rhs, lhs = _invert(eq, sym)
if lhs.has(sym):
try:
poly = lhs.as_poly()
g = _filtered_gens(poly, sym)
_eq = lhs - rhs
sols = _solve_lambert(_eq, sym, g)
# use a simplified form if it satisfies eq
# and has fewer operations
for n, s in enumerate(sols):
ns = nsimplify(s)
if ns != s and ns.count_ops() <= s.count_ops():
ok = checksol(_eq, sym, ns)
if ok is None:
ok = _eq.subs(sym, ns).equals(0)
if ok:
sols[n] = ns
return sols
except NotImplementedError:
# maybe it's a convoluted function
if len(g) == 2:
try:
gpu = bivariate_type(lhs - rhs, *g)
if gpu is None:
raise NotImplementedError
g, p, u = gpu
flags['bivariate'] = False
inversion = _tsolve(g - u, sym, **flags)
if inversion:
sol = _solve(p, u, **flags)
return list(ordered({i.subs(u, s)
for i in inversion for s in sol}))
except NotImplementedError:
pass
else:
pass
if flags.pop('force', True):
flags['force'] = False
pos, reps = posify(lhs - rhs)
if rhs == S.ComplexInfinity:
return []
for u, s in reps.items():
if s == sym:
break
else:
u = sym
if pos.has(u):
try:
soln = _solve(pos, u, **flags)
return list(ordered([s.subs(reps) for s in soln]))
except NotImplementedError:
pass
else:
pass # here for coverage
return # here for coverage
# TODO: option for calculating J numerically
@conserve_mpmath_dps
def nsolve(*args, dict=False, **kwargs):
r"""
Solve a nonlinear equation system numerically: ``nsolve(f, [args,] x0,
modules=['mpmath'], **kwargs)``.
Explanation
===========
``f`` is a vector function of symbolic expressions representing the system.
*args* are the variables. If there is only one variable, this argument can
be omitted. ``x0`` is a starting vector close to a solution.
Use the modules keyword to specify which modules should be used to
evaluate the function and the Jacobian matrix. Make sure to use a module
that supports matrices. For more information on the syntax, please see the
docstring of ``lambdify``.
If the keyword arguments contain ``dict=True`` (default is False) ``nsolve``
will return a list (perhaps empty) of solution mappings. This might be
especially useful if you want to use ``nsolve`` as a fallback to solve since
using the dict argument for both methods produces return values of
consistent type structure. Please note: to keep this consistent with
``solve``, the solution will be returned in a list even though ``nsolve``
(currently at least) only finds one solution at a time.
Overdetermined systems are supported.
Examples
========
>>> from sympy import Symbol, nsolve
>>> import mpmath
>>> mpmath.mp.dps = 15
>>> x1 = Symbol('x1')
>>> x2 = Symbol('x2')
>>> f1 = 3 * x1**2 - 2 * x2**2 - 1
>>> f2 = x1**2 - 2 * x1 + x2**2 + 2 * x2 - 8
>>> print(nsolve((f1, f2), (x1, x2), (-1, 1)))
Matrix([[-1.19287309935246], [1.27844411169911]])
For one-dimensional functions the syntax is simplified:
>>> from sympy import sin, nsolve
>>> from sympy.abc import x
>>> nsolve(sin(x), x, 2)
3.14159265358979
>>> nsolve(sin(x), 2)
3.14159265358979
To solve with higher precision than the default, use the prec argument:
>>> from sympy import cos
>>> nsolve(cos(x) - x, 1)
0.739085133215161
>>> nsolve(cos(x) - x, 1, prec=50)
0.73908513321516064165531208767387340401341175890076
>>> cos(_)
0.73908513321516064165531208767387340401341175890076
To solve for complex roots of real functions, a nonreal initial point
must be specified:
>>> from sympy import I
>>> nsolve(x**2 + 2, I)
1.4142135623731*I
``mpmath.findroot`` is used and you can find their more extensive
documentation, especially concerning keyword parameters and
available solvers. Note, however, that functions which are very
steep near the root, the verification of the solution may fail. In
this case you should use the flag ``verify=False`` and
independently verify the solution.
>>> from sympy import cos, cosh
>>> f = cos(x)*cosh(x) - 1
>>> nsolve(f, 3.14*100)
Traceback (most recent call last):
...
ValueError: Could not find root within given tolerance. (1.39267e+230 > 2.1684e-19)
>>> ans = nsolve(f, 3.14*100, verify=False); ans
312.588469032184
>>> f.subs(x, ans).n(2)
2.1e+121
>>> (f/f.diff(x)).subs(x, ans).n(2)
7.4e-15
One might safely skip the verification if bounds of the root are known
and a bisection method is used:
>>> bounds = lambda i: (3.14*i, 3.14*(i + 1))
>>> nsolve(f, bounds(100), solver='bisect', verify=False)
315.730061685774
Alternatively, a function may be better behaved when the
denominator is ignored. Since this is not always the case, however,
the decision of what function to use is left to the discretion of
the user.
>>> eq = x**2/(1 - x)/(1 - 2*x)**2 - 100
>>> nsolve(eq, 0.46)
Traceback (most recent call last):
...
ValueError: Could not find root within given tolerance. (10000 > 2.1684e-19)
Try another starting point or tweak arguments.
>>> nsolve(eq.as_numer_denom()[0], 0.46)
0.46792545969349058
"""
# there are several other SymPy functions that use method= so
# guard against that here
if 'method' in kwargs:
raise ValueError(filldedent('''
Keyword "method" should not be used in this context. When using
some mpmath solvers directly, the keyword "method" is
used, but when using nsolve (and findroot) the keyword to use is
"solver".'''))
if 'prec' in kwargs:
prec = kwargs.pop('prec')
import mpmath
mpmath.mp.dps = prec
else:
prec = None
# keyword argument to return result as a dictionary
as_dict = dict
from builtins import dict # to unhide the builtin
# interpret arguments
if len(args) == 3:
f = args[0]
fargs = args[1]
x0 = args[2]
if iterable(fargs) and iterable(x0):
if len(x0) != len(fargs):
raise TypeError('nsolve expected exactly %i guess vectors, got %i'
% (len(fargs), len(x0)))
elif len(args) == 2:
f = args[0]
fargs = None
x0 = args[1]
if iterable(f):
raise TypeError('nsolve expected 3 arguments, got 2')
elif len(args) < 2:
raise TypeError('nsolve expected at least 2 arguments, got %i'
% len(args))
else:
raise TypeError('nsolve expected at most 3 arguments, got %i'
% len(args))
modules = kwargs.get('modules', ['mpmath'])
if iterable(f):
f = list(f)
for i, fi in enumerate(f):
if isinstance(fi, Equality):
f[i] = fi.lhs - fi.rhs
f = Matrix(f).T
if iterable(x0):
x0 = list(x0)
if not isinstance(f, Matrix):
# assume it's a sympy expression
if isinstance(f, Equality):
f = f.lhs - f.rhs
syms = f.free_symbols
if fargs is None:
fargs = syms.copy().pop()
if not (len(syms) == 1 and (fargs in syms or fargs[0] in syms)):
raise ValueError(filldedent('''
expected a one-dimensional and numerical function'''))
# the function is much better behaved if there is no denominator
# but sending the numerator is left to the user since sometimes
# the function is better behaved when the denominator is present
# e.g., issue 11768
f = lambdify(fargs, f, modules)
x = sympify(findroot(f, x0, **kwargs))
if as_dict:
return [{fargs: x}]
return x
if len(fargs) > f.cols:
raise NotImplementedError(filldedent('''
need at least as many equations as variables'''))
verbose = kwargs.get('verbose', False)
if verbose:
print('f(x):')
print(f)
# derive Jacobian
J = f.jacobian(fargs)
if verbose:
print('J(x):')
print(J)
# create functions
f = lambdify(fargs, f.T, modules)
J = lambdify(fargs, J, modules)
# solve the system numerically
x = findroot(f, x0, J=J, **kwargs)
if as_dict:
return [dict(zip(fargs, [sympify(xi) for xi in x]))]
return Matrix(x)
def _invert(eq, *symbols, **kwargs):
"""
Return tuple (i, d) where ``i`` is independent of *symbols* and ``d``
contains symbols.
Explanation
===========
``i`` and ``d`` are obtained after recursively using algebraic inversion
until an uninvertible ``d`` remains. If there are no free symbols then
``d`` will be zero. Some (but not necessarily all) solutions to the
expression ``i - d`` will be related to the solutions of the original
expression.
Examples
========
>>> from sympy.solvers.solvers import _invert as invert
>>> from sympy import sqrt, cos
>>> from sympy.abc import x, y
>>> invert(x - 3)
(3, x)
>>> invert(3)
(3, 0)
>>> invert(2*cos(x) - 1)
(1/2, cos(x))
>>> invert(sqrt(x) - 3)
(3, sqrt(x))
>>> invert(sqrt(x) + y, x)
(-y, sqrt(x))
>>> invert(sqrt(x) + y, y)
(-sqrt(x), y)
>>> invert(sqrt(x) + y, x, y)
(0, sqrt(x) + y)
If there is more than one symbol in a power's base and the exponent
is not an Integer, then the principal root will be used for the
inversion:
>>> invert(sqrt(x + y) - 2)
(4, x + y)
>>> invert(sqrt(x + y) - 2)
(4, x + y)
If the exponent is an Integer, setting ``integer_power`` to True
will force the principal root to be selected:
>>> invert(x**2 - 4, integer_power=True)
(2, x)
"""
eq = sympify(eq)
if eq.args:
# make sure we are working with flat eq
eq = eq.func(*eq.args)
free = eq.free_symbols
if not symbols:
symbols = free
if not free & set(symbols):
return eq, S.Zero
dointpow = bool(kwargs.get('integer_power', False))
lhs = eq
rhs = S.Zero
while True:
was = lhs
while True:
indep, dep = lhs.as_independent(*symbols)
# dep + indep == rhs
if lhs.is_Add:
# this indicates we have done it all
if indep.is_zero:
break
lhs = dep
rhs -= indep
# dep * indep == rhs
else:
# this indicates we have done it all
if indep is S.One:
break
lhs = dep
rhs /= indep
# collect like-terms in symbols
if lhs.is_Add:
terms = {}
for a in lhs.args:
i, d = a.as_independent(*symbols)
terms.setdefault(d, []).append(i)
if any(len(v) > 1 for v in terms.values()):
args = []
for d, i in terms.items():
if len(i) > 1:
args.append(Add(*i)*d)
else:
args.append(i[0]*d)
lhs = Add(*args)
# if it's a two-term Add with rhs = 0 and two powers we can get the
# dependent terms together, e.g. 3*f(x) + 2*g(x) -> f(x)/g(x) = -2/3
if lhs.is_Add and not rhs and len(lhs.args) == 2 and \
not lhs.is_polynomial(*symbols):
a, b = ordered(lhs.args)
ai, ad = a.as_independent(*symbols)
bi, bd = b.as_independent(*symbols)
if any(_ispow(i) for i in (ad, bd)):
a_base, a_exp = ad.as_base_exp()
b_base, b_exp = bd.as_base_exp()
if a_base == b_base:
# a = -b
lhs = powsimp(powdenest(ad/bd))
rhs = -bi/ai
else:
rat = ad/bd
_lhs = powsimp(ad/bd)
if _lhs != rat:
lhs = _lhs
rhs = -bi/ai
elif ai == -bi:
if isinstance(ad, Function) and ad.func == bd.func:
if len(ad.args) == len(bd.args) == 1:
lhs = ad.args[0] - bd.args[0]
elif len(ad.args) == len(bd.args):
# should be able to solve
# f(x, y) - f(2 - x, 0) == 0 -> x == 1
raise NotImplementedError(
'equal function with more than 1 argument')
else:
raise ValueError(
'function with different numbers of args')
elif lhs.is_Mul and any(_ispow(a) for a in lhs.args):
lhs = powsimp(powdenest(lhs))
if lhs.is_Function:
if hasattr(lhs, 'inverse') and lhs.inverse() is not None and len(lhs.args) == 1:
# -1
# f(x) = g -> x = f (g)
#
# /!\ inverse should not be defined if there are multiple values
# for the function -- these are handled in _tsolve
#
rhs = lhs.inverse()(rhs)
lhs = lhs.args[0]
elif isinstance(lhs, atan2):
y, x = lhs.args
lhs = 2*atan(y/(sqrt(x**2 + y**2) + x))
elif lhs.func == rhs.func:
if len(lhs.args) == len(rhs.args) == 1:
lhs = lhs.args[0]
rhs = rhs.args[0]
elif len(lhs.args) == len(rhs.args):
# should be able to solve
# f(x, y) == f(2, 3) -> x == 2
# f(x, x + y) == f(2, 3) -> x == 2
raise NotImplementedError(
'equal function with more than 1 argument')
else:
raise ValueError(
'function with different numbers of args')
if rhs and lhs.is_Pow and lhs.exp.is_Integer and lhs.exp < 0:
lhs = 1/lhs
rhs = 1/rhs
# base**a = b -> base = b**(1/a) if
# a is an Integer and dointpow=True (this gives real branch of root)
# a is not an Integer and the equation is multivariate and the
# base has more than 1 symbol in it
# The rationale for this is that right now the multi-system solvers
# doesn't try to resolve generators to see, for example, if the whole
# system is written in terms of sqrt(x + y) so it will just fail, so we
# do that step here.
if lhs.is_Pow and (
lhs.exp.is_Integer and dointpow or not lhs.exp.is_Integer and
len(symbols) > 1 and len(lhs.base.free_symbols & set(symbols)) > 1):
rhs = rhs**(1/lhs.exp)
lhs = lhs.base
if lhs == was:
break
return rhs, lhs
def unrad(eq, *syms, **flags):
"""
Remove radicals with symbolic arguments and return (eq, cov),
None, or raise an error.
Explanation
===========
None is returned if there are no radicals to remove.
NotImplementedError is raised if there are radicals and they cannot be
removed or if the relationship between the original symbols and the
change of variable needed to rewrite the system as a polynomial cannot
be solved.
Otherwise the tuple, ``(eq, cov)``, is returned where:
*eq*, ``cov``
*eq* is an equation without radicals (in the symbol(s) of
interest) whose solutions are a superset of the solutions to the
original expression. *eq* might be rewritten in terms of a new
variable; the relationship to the original variables is given by
``cov`` which is a list containing ``v`` and ``v**p - b`` where
``p`` is the power needed to clear the radical and ``b`` is the
radical now expressed as a polynomial in the symbols of interest.
For example, for sqrt(2 - x) the tuple would be
``(c, c**2 - 2 + x)``. The solutions of *eq* will contain
solutions to the original equation (if there are any).
*syms*
An iterable of symbols which, if provided, will limit the focus of
radical removal: only radicals with one or more of the symbols of
interest will be cleared. All free symbols are used if *syms* is not
set.
*flags* are used internally for communication during recursive calls.
Two options are also recognized:
``take``, when defined, is interpreted as a single-argument function
that returns True if a given Pow should be handled.
Radicals can be removed from an expression if:
* All bases of the radicals are the same; a change of variables is
done in this case.
* If all radicals appear in one term of the expression.
* There are only four terms with sqrt() factors or there are less than
four terms having sqrt() factors.
* There are only two terms with radicals.
Examples
========
>>> from sympy.solvers.solvers import unrad
>>> from sympy.abc import x
>>> from sympy import sqrt, Rational, root
>>> unrad(sqrt(x)*x**Rational(1, 3) + 2)
(x**5 - 64, [])
>>> unrad(sqrt(x) + root(x + 1, 3))
(-x**3 + x**2 + 2*x + 1, [])
>>> eq = sqrt(x) + root(x, 3) - 2
>>> unrad(eq)
(_p**3 + _p**2 - 2, [_p, _p**6 - x])
"""
from sympy import Equality as Eq
uflags = dict(check=False, simplify=False)
def _cov(p, e):
if cov:
# XXX - uncovered
oldp, olde = cov
if Poly(e, p).degree(p) in (1, 2):
cov[:] = [p, olde.subs(oldp, _solve(e, p, **uflags)[0])]
else:
raise NotImplementedError
else:
cov[:] = [p, e]
def _canonical(eq, cov):
if cov:
# change symbol to vanilla so no solutions are eliminated
p, e = cov
rep = {p: Dummy(p.name)}
eq = eq.xreplace(rep)
cov = [p.xreplace(rep), e.xreplace(rep)]
# remove constants and powers of factors since these don't change
# the location of the root; XXX should factor or factor_terms be used?
eq = factor_terms(_mexpand(eq.as_numer_denom()[0], recursive=True), clear=True)
if eq.is_Mul:
args = []
for f in eq.args:
if f.is_number:
continue
if f.is_Pow:
args.append(f.base)
else:
args.append(f)
eq = Mul(*args) # leave as Mul for more efficient solving
# make the sign canonical
margs = list(Mul.make_args(eq))
changed = False
for i, m in enumerate(margs):
if m.could_extract_minus_sign():
margs[i] = -m
changed = True
if changed:
eq = Mul(*margs, evaluate=False)
return eq, cov
def _Q(pow):
# return leading Rational of denominator of Pow's exponent
c = pow.as_base_exp()[1].as_coeff_Mul()[0]
if not c.is_Rational:
return S.One
return c.q
# define the _take method that will determine whether a term is of interest
def _take(d):
# return True if coefficient of any factor's exponent's den is not 1
for pow in Mul.make_args(d):
if not pow.is_Pow:
continue
if _Q(pow) == 1:
continue
if pow.free_symbols & syms:
return True
return False
_take = flags.setdefault('_take', _take)
if isinstance(eq, Eq):
eq = eq.lhs - eq.rhs # XXX legacy Eq as Eqn support
elif not isinstance(eq, Expr):
return
cov, nwas, rpt = [flags.setdefault(k, v) for k, v in
sorted(dict(cov=[], n=None, rpt=0).items())]
# preconditioning
eq = powdenest(factor_terms(eq, radical=True, clear=True))
eq = eq.as_numer_denom()[0]
eq = _mexpand(eq, recursive=True)
if eq.is_number:
return
# see if there are radicals in symbols of interest
syms = set(syms) or eq.free_symbols # _take uses this
poly = eq.as_poly()
gens = [g for g in poly.gens if _take(g)]
if not gens:
return
# recast poly in terms of eigen-gens
poly = eq.as_poly(*gens)
# - an exponent has a symbol of interest (don't handle)
if any(g.exp.has(*syms) for g in gens):
return
def _rads_bases_lcm(poly):
# if all the bases are the same or all the radicals are in one
# term, `lcm` will be the lcm of the denominators of the
# exponents of the radicals
lcm = 1
rads = set()
bases = set()
for g in poly.gens:
q = _Q(g)
if q != 1:
rads.add(g)
lcm = ilcm(lcm, q)
bases.add(g.base)
return rads, bases, lcm
rads, bases, lcm = _rads_bases_lcm(poly)
covsym = Dummy('p', nonnegative=True)
# only keep in syms symbols that actually appear in radicals;
# and update gens
newsyms = set()
for r in rads:
newsyms.update(syms & r.free_symbols)
if newsyms != syms:
syms = newsyms
gens = [g for g in gens if g.free_symbols & syms]
# get terms together that have common generators
drad = dict(list(zip(rads, list(range(len(rads))))))
rterms = {(): []}
args = Add.make_args(poly.as_expr())
for t in args:
if _take(t):
common = set(t.as_poly().gens).intersection(rads)
key = tuple(sorted([drad[i] for i in common]))
else:
key = ()
rterms.setdefault(key, []).append(t)
others = Add(*rterms.pop(()))
rterms = [Add(*rterms[k]) for k in rterms.keys()]
# the output will depend on the order terms are processed, so
# make it canonical quickly
rterms = list(reversed(list(ordered(rterms))))
ok = False # we don't have a solution yet
depth = sqrt_depth(eq)
if len(rterms) == 1 and not (rterms[0].is_Add and lcm > 2):
eq = rterms[0]**lcm - ((-others)**lcm)
ok = True
else:
if len(rterms) == 1 and rterms[0].is_Add:
rterms = list(rterms[0].args)
if len(bases) == 1:
b = bases.pop()
if len(syms) > 1:
x = b.free_symbols
else:
x = syms
x = list(ordered(x))[0]
try:
inv = _solve(covsym**lcm - b, x, **uflags)
if not inv:
raise NotImplementedError
eq = poly.as_expr().subs(b, covsym**lcm).subs(x, inv[0])
_cov(covsym, covsym**lcm - b)
return _canonical(eq, cov)
except NotImplementedError:
pass
if len(rterms) == 2:
if not others:
eq = rterms[0]**lcm - (-rterms[1])**lcm
ok = True
elif not log(lcm, 2).is_Integer:
# the lcm-is-power-of-two case is handled below
r0, r1 = rterms
if flags.get('_reverse', False):
r1, r0 = r0, r1
i0 = _rads0, _bases0, lcm0 = _rads_bases_lcm(r0.as_poly())
i1 = _rads1, _bases1, lcm1 = _rads_bases_lcm(r1.as_poly())
for reverse in range(2):
if reverse:
i0, i1 = i1, i0
r0, r1 = r1, r0
_rads1, _, lcm1 = i1
_rads1 = Mul(*_rads1)
t1 = _rads1**lcm1
c = covsym**lcm1 - t1
for x in syms:
try:
sol = _solve(c, x, **uflags)
if not sol:
raise NotImplementedError
neweq = r0.subs(x, sol[0]) + covsym*r1/_rads1 + \
others
tmp = unrad(neweq, covsym)
if tmp:
eq, newcov = tmp
if newcov:
newp, newc = newcov
_cov(newp, c.subs(covsym,
_solve(newc, covsym, **uflags)[0]))
else:
_cov(covsym, c)
else:
eq = neweq
_cov(covsym, c)
ok = True
break
except NotImplementedError:
if reverse:
raise NotImplementedError(
'no successful change of variable found')
else:
pass
if ok:
break
elif len(rterms) == 3:
# two cube roots and another with order less than 5
# (so an analytical solution can be found) or a base
# that matches one of the cube root bases
info = [_rads_bases_lcm(i.as_poly()) for i in rterms]
RAD = 0
BASES = 1
LCM = 2
if info[0][LCM] != 3:
info.append(info.pop(0))
rterms.append(rterms.pop(0))
elif info[1][LCM] != 3:
info.append(info.pop(1))
rterms.append(rterms.pop(1))
if info[0][LCM] == info[1][LCM] == 3:
if info[1][BASES] != info[2][BASES]:
info[0], info[1] = info[1], info[0]
rterms[0], rterms[1] = rterms[1], rterms[0]
if info[1][BASES] == info[2][BASES]:
eq = rterms[0]**3 + (rterms[1] + rterms[2] + others)**3
ok = True
elif info[2][LCM] < 5:
# a*root(A, 3) + b*root(B, 3) + others = c
a, b, c, d, A, B = [Dummy(i) for i in 'abcdAB']
# zz represents the unraded expression into which the
# specifics for this case are substituted
zz = (c - d)*(A**3*a**9 + 3*A**2*B*a**6*b**3 -
3*A**2*a**6*c**3 + 9*A**2*a**6*c**2*d - 9*A**2*a**6*c*d**2 +
3*A**2*a**6*d**3 + 3*A*B**2*a**3*b**6 + 21*A*B*a**3*b**3*c**3 -
63*A*B*a**3*b**3*c**2*d + 63*A*B*a**3*b**3*c*d**2 -
21*A*B*a**3*b**3*d**3 + 3*A*a**3*c**6 - 18*A*a**3*c**5*d +
45*A*a**3*c**4*d**2 - 60*A*a**3*c**3*d**3 + 45*A*a**3*c**2*d**4 -
18*A*a**3*c*d**5 + 3*A*a**3*d**6 + B**3*b**9 - 3*B**2*b**6*c**3 +
9*B**2*b**6*c**2*d - 9*B**2*b**6*c*d**2 + 3*B**2*b**6*d**3 +
3*B*b**3*c**6 - 18*B*b**3*c**5*d + 45*B*b**3*c**4*d**2 -
60*B*b**3*c**3*d**3 + 45*B*b**3*c**2*d**4 - 18*B*b**3*c*d**5 +
3*B*b**3*d**6 - c**9 + 9*c**8*d - 36*c**7*d**2 + 84*c**6*d**3 -
126*c**5*d**4 + 126*c**4*d**5 - 84*c**3*d**6 + 36*c**2*d**7 -
9*c*d**8 + d**9)
def _t(i):
b = Mul(*info[i][RAD])
return cancel(rterms[i]/b), Mul(*info[i][BASES])
aa, AA = _t(0)
bb, BB = _t(1)
cc = -rterms[2]
dd = others
eq = zz.xreplace(dict(zip(
(a, A, b, B, c, d),
(aa, AA, bb, BB, cc, dd))))
ok = True
# handle power-of-2 cases
if not ok:
if log(lcm, 2).is_Integer and (not others and
len(rterms) == 4 or len(rterms) < 4):
def _norm2(a, b):
return a**2 + b**2 + 2*a*b
if len(rterms) == 4:
# (r0+r1)**2 - (r2+r3)**2
r0, r1, r2, r3 = rterms
eq = _norm2(r0, r1) - _norm2(r2, r3)
ok = True
elif len(rterms) == 3:
# (r1+r2)**2 - (r0+others)**2
r0, r1, r2 = rterms
eq = _norm2(r1, r2) - _norm2(r0, others)
ok = True
elif len(rterms) == 2:
# r0**2 - (r1+others)**2
r0, r1 = rterms
eq = r0**2 - _norm2(r1, others)
ok = True
new_depth = sqrt_depth(eq) if ok else depth
rpt += 1 # XXX how many repeats with others unchanging is enough?
if not ok or (
nwas is not None and len(rterms) == nwas and
new_depth is not None and new_depth == depth and
rpt > 3):
raise NotImplementedError('Cannot remove all radicals')
flags.update(dict(cov=cov, n=len(rterms), rpt=rpt))
neq = unrad(eq, *syms, **flags)
if neq:
eq, cov = neq
eq, cov = _canonical(eq, cov)
return eq, cov
from sympy.solvers.bivariate import (
bivariate_type, _solve_lambert, _filtered_gens)
|
01c297c1a0a6450b67f79cca81328b73b98ac3de89fdd55b5c6e896032b9f287 | from sympy import Order, S, log, limit, lcm_list, im, re, Dummy, Piecewise
from sympy.core import Add, Mul, Pow
from sympy.core.basic import Basic
from sympy.core.compatibility import iterable
from sympy.core.expr import AtomicExpr, Expr
from sympy.core.function import expand_mul
from sympy.core.numbers import _sympifyit, oo
from sympy.core.relational import is_le, is_lt, is_ge, is_gt
from sympy.core.sympify import _sympify
from sympy.functions.elementary.miscellaneous import Min, Max
from sympy.logic.boolalg import And
from sympy.sets.sets import (Interval, Intersection, FiniteSet, Union,
Complement, EmptySet)
from sympy.sets.fancysets import ImageSet
from sympy.solvers.inequalities import solve_univariate_inequality
from sympy.utilities import filldedent
from sympy.multipledispatch import dispatch
def continuous_domain(f, symbol, domain):
"""
Returns the intervals in the given domain for which the function
is continuous.
This method is limited by the ability to determine the various
singularities and discontinuities of the given function.
Parameters
==========
f : Expr
The concerned function.
symbol : Symbol
The variable for which the intervals are to be determined.
domain : Interval
The domain over which the continuity of the symbol has to be checked.
Examples
========
>>> from sympy import Symbol, S, tan, log, pi, sqrt
>>> from sympy.sets import Interval
>>> from sympy.calculus.util import continuous_domain
>>> x = Symbol('x')
>>> continuous_domain(1/x, x, S.Reals)
Union(Interval.open(-oo, 0), Interval.open(0, oo))
>>> continuous_domain(tan(x), x, Interval(0, pi))
Union(Interval.Ropen(0, pi/2), Interval.Lopen(pi/2, pi))
>>> continuous_domain(sqrt(x - 2), x, Interval(-5, 5))
Interval(2, 5)
>>> continuous_domain(log(2*x - 1), x, S.Reals)
Interval.open(1/2, oo)
Returns
=======
Interval
Union of all intervals where the function is continuous.
Raises
======
NotImplementedError
If the method to determine continuity of such a function
has not yet been developed.
"""
from sympy.solvers.inequalities import solve_univariate_inequality
from sympy.calculus.singularities import singularities
if domain.is_subset(S.Reals):
constrained_interval = domain
for atom in f.atoms(Pow):
den = atom.exp.as_numer_denom()[1]
if den.is_even and den.is_nonzero:
constraint = solve_univariate_inequality(atom.base >= 0,
symbol).as_set()
constrained_interval = Intersection(constraint,
constrained_interval)
for atom in f.atoms(log):
constraint = solve_univariate_inequality(atom.args[0] > 0,
symbol).as_set()
constrained_interval = Intersection(constraint,
constrained_interval)
return constrained_interval - singularities(f, symbol, domain)
def function_range(f, symbol, domain):
"""
Finds the range of a function in a given domain.
This method is limited by the ability to determine the singularities and
determine limits.
Parameters
==========
f : Expr
The concerned function.
symbol : Symbol
The variable for which the range of function is to be determined.
domain : Interval
The domain under which the range of the function has to be found.
Examples
========
>>> from sympy import Symbol, S, exp, log, pi, sqrt, sin, tan
>>> from sympy.sets import Interval
>>> from sympy.calculus.util import function_range
>>> x = Symbol('x')
>>> function_range(sin(x), x, Interval(0, 2*pi))
Interval(-1, 1)
>>> function_range(tan(x), x, Interval(-pi/2, pi/2))
Interval(-oo, oo)
>>> function_range(1/x, x, S.Reals)
Union(Interval.open(-oo, 0), Interval.open(0, oo))
>>> function_range(exp(x), x, S.Reals)
Interval.open(0, oo)
>>> function_range(log(x), x, S.Reals)
Interval(-oo, oo)
>>> function_range(sqrt(x), x , Interval(-5, 9))
Interval(0, 3)
Returns
=======
Interval
Union of all ranges for all intervals under domain where function is
continuous.
Raises
======
NotImplementedError
If any of the intervals, in the given domain, for which function
is continuous are not finite or real,
OR if the critical points of the function on the domain can't be found.
"""
from sympy.solvers.solveset import solveset
if isinstance(domain, EmptySet):
return S.EmptySet
period = periodicity(f, symbol)
if period == S.Zero:
# the expression is constant wrt symbol
return FiniteSet(f.expand())
if period is not None:
if isinstance(domain, Interval):
if (domain.inf - domain.sup).is_infinite:
domain = Interval(0, period)
elif isinstance(domain, Union):
for sub_dom in domain.args:
if isinstance(sub_dom, Interval) and \
((sub_dom.inf - sub_dom.sup).is_infinite):
domain = Interval(0, period)
intervals = continuous_domain(f, symbol, domain)
range_int = S.EmptySet
if isinstance(intervals,(Interval, FiniteSet)):
interval_iter = (intervals,)
elif isinstance(intervals, Union):
interval_iter = intervals.args
else:
raise NotImplementedError(filldedent('''
Unable to find range for the given domain.
'''))
for interval in interval_iter:
if isinstance(interval, FiniteSet):
for singleton in interval:
if singleton in domain:
range_int += FiniteSet(f.subs(symbol, singleton))
elif isinstance(interval, Interval):
vals = S.EmptySet
critical_points = S.EmptySet
critical_values = S.EmptySet
bounds = ((interval.left_open, interval.inf, '+'),
(interval.right_open, interval.sup, '-'))
for is_open, limit_point, direction in bounds:
if is_open:
critical_values += FiniteSet(limit(f, symbol, limit_point, direction))
vals += critical_values
else:
vals += FiniteSet(f.subs(symbol, limit_point))
solution = solveset(f.diff(symbol), symbol, interval)
if not iterable(solution):
raise NotImplementedError(
'Unable to find critical points for {}'.format(f))
if isinstance(solution, ImageSet):
raise NotImplementedError(
'Infinite number of critical points for {}'.format(f))
critical_points += solution
for critical_point in critical_points:
vals += FiniteSet(f.subs(symbol, critical_point))
left_open, right_open = False, False
if critical_values is not S.EmptySet:
if critical_values.inf == vals.inf:
left_open = True
if critical_values.sup == vals.sup:
right_open = True
range_int += Interval(vals.inf, vals.sup, left_open, right_open)
else:
raise NotImplementedError(filldedent('''
Unable to find range for the given domain.
'''))
return range_int
def not_empty_in(finset_intersection, *syms):
"""
Finds the domain of the functions in `finite_set` in which the
`finite_set` is not-empty
Parameters
==========
finset_intersection : The unevaluated intersection of FiniteSet containing
real-valued functions with Union of Sets
syms : Tuple of symbols
Symbol for which domain is to be found
Raises
======
NotImplementedError
The algorithms to find the non-emptiness of the given FiniteSet are
not yet implemented.
ValueError
The input is not valid.
RuntimeError
It is a bug, please report it to the github issue tracker
(https://github.com/sympy/sympy/issues).
Examples
========
>>> from sympy import FiniteSet, Interval, not_empty_in, oo
>>> from sympy.abc import x
>>> not_empty_in(FiniteSet(x/2).intersect(Interval(0, 1)), x)
Interval(0, 2)
>>> not_empty_in(FiniteSet(x, x**2).intersect(Interval(1, 2)), x)
Union(Interval(1, 2), Interval(-sqrt(2), -1))
>>> not_empty_in(FiniteSet(x**2/(x + 2)).intersect(Interval(1, oo)), x)
Union(Interval.Lopen(-2, -1), Interval(2, oo))
"""
# TODO: handle piecewise defined functions
# TODO: handle transcendental functions
# TODO: handle multivariate functions
if len(syms) == 0:
raise ValueError("One or more symbols must be given in syms.")
if finset_intersection is S.EmptySet:
return S.EmptySet
if isinstance(finset_intersection, Union):
elm_in_sets = finset_intersection.args[0]
return Union(not_empty_in(finset_intersection.args[1], *syms),
elm_in_sets)
if isinstance(finset_intersection, FiniteSet):
finite_set = finset_intersection
_sets = S.Reals
else:
finite_set = finset_intersection.args[1]
_sets = finset_intersection.args[0]
if not isinstance(finite_set, FiniteSet):
raise ValueError('A FiniteSet must be given, not %s: %s' %
(type(finite_set), finite_set))
if len(syms) == 1:
symb = syms[0]
else:
raise NotImplementedError('more than one variables %s not handled' %
(syms,))
def elm_domain(expr, intrvl):
""" Finds the domain of an expression in any given interval """
from sympy.solvers.solveset import solveset
_start = intrvl.start
_end = intrvl.end
_singularities = solveset(expr.as_numer_denom()[1], symb,
domain=S.Reals)
if intrvl.right_open:
if _end is S.Infinity:
_domain1 = S.Reals
else:
_domain1 = solveset(expr < _end, symb, domain=S.Reals)
else:
_domain1 = solveset(expr <= _end, symb, domain=S.Reals)
if intrvl.left_open:
if _start is S.NegativeInfinity:
_domain2 = S.Reals
else:
_domain2 = solveset(expr > _start, symb, domain=S.Reals)
else:
_domain2 = solveset(expr >= _start, symb, domain=S.Reals)
# domain in the interval
expr_with_sing = Intersection(_domain1, _domain2)
expr_domain = Complement(expr_with_sing, _singularities)
return expr_domain
if isinstance(_sets, Interval):
return Union(*[elm_domain(element, _sets) for element in finite_set])
if isinstance(_sets, Union):
_domain = S.EmptySet
for intrvl in _sets.args:
_domain_element = Union(*[elm_domain(element, intrvl)
for element in finite_set])
_domain = Union(_domain, _domain_element)
return _domain
def periodicity(f, symbol, check=False):
"""
Tests the given function for periodicity in the given symbol.
Parameters
==========
f : Expr.
The concerned function.
symbol : Symbol
The variable for which the period is to be determined.
check : Boolean, optional
The flag to verify whether the value being returned is a period or not.
Returns
=======
period
The period of the function is returned.
`None` is returned when the function is aperiodic or has a complex period.
The value of `0` is returned as the period of a constant function.
Raises
======
NotImplementedError
The value of the period computed cannot be verified.
Notes
=====
Currently, we do not support functions with a complex period.
The period of functions having complex periodic values such
as `exp`, `sinh` is evaluated to `None`.
The value returned might not be the "fundamental" period of the given
function i.e. it may not be the smallest periodic value of the function.
The verification of the period through the `check` flag is not reliable
due to internal simplification of the given expression. Hence, it is set
to `False` by default.
Examples
========
>>> from sympy import Symbol, sin, cos, tan, exp
>>> from sympy.calculus.util import periodicity
>>> x = Symbol('x')
>>> f = sin(x) + sin(2*x) + sin(3*x)
>>> periodicity(f, x)
2*pi
>>> periodicity(sin(x)*cos(x), x)
pi
>>> periodicity(exp(tan(2*x) - 1), x)
pi/2
>>> periodicity(sin(4*x)**cos(2*x), x)
pi
>>> periodicity(exp(x), x)
"""
from sympy.core.mod import Mod
from sympy.core.relational import Relational
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.complexes import Abs
from sympy.functions.elementary.trigonometric import (
TrigonometricFunction, sin, cos, csc, sec)
from sympy.simplify.simplify import simplify
from sympy.solvers.decompogen import decompogen
from sympy.polys.polytools import degree
temp = Dummy('x', real=True)
f = f.subs(symbol, temp)
symbol = temp
def _check(orig_f, period):
'''Return the checked period or raise an error.'''
new_f = orig_f.subs(symbol, symbol + period)
if new_f.equals(orig_f):
return period
else:
raise NotImplementedError(filldedent('''
The period of the given function cannot be verified.
When `%s` was replaced with `%s + %s` in `%s`, the result
was `%s` which was not recognized as being the same as
the original function.
So either the period was wrong or the two forms were
not recognized as being equal.
Set check=False to obtain the value.''' %
(symbol, symbol, period, orig_f, new_f)))
orig_f = f
period = None
if isinstance(f, Relational):
f = f.lhs - f.rhs
f = simplify(f)
if symbol not in f.free_symbols:
return S.Zero
if isinstance(f, TrigonometricFunction):
try:
period = f.period(symbol)
except NotImplementedError:
pass
if isinstance(f, Abs):
arg = f.args[0]
if isinstance(arg, (sec, csc, cos)):
# all but tan and cot might have a
# a period that is half as large
# so recast as sin
arg = sin(arg.args[0])
period = periodicity(arg, symbol)
if period is not None and isinstance(arg, sin):
# the argument of Abs was a trigonometric other than
# cot or tan; test to see if the half-period
# is valid. Abs(arg) has behaviour equivalent to
# orig_f, so use that for test:
orig_f = Abs(arg)
try:
return _check(orig_f, period/2)
except NotImplementedError as err:
if check:
raise NotImplementedError(err)
# else let new orig_f and period be
# checked below
if isinstance(f, exp) or (f.is_Pow and f.base == S.Exp1):
f = Pow(S.Exp1, expand_mul(f.exp))
if im(f) != 0:
period_real = periodicity(re(f), symbol)
period_imag = periodicity(im(f), symbol)
if period_real is not None and period_imag is not None:
period = lcim([period_real, period_imag])
if f.is_Pow and f.base != S.Exp1:
base, expo = f.args
base_has_sym = base.has(symbol)
expo_has_sym = expo.has(symbol)
if base_has_sym and not expo_has_sym:
period = periodicity(base, symbol)
elif expo_has_sym and not base_has_sym:
period = periodicity(expo, symbol)
else:
period = _periodicity(f.args, symbol)
elif f.is_Mul:
coeff, g = f.as_independent(symbol, as_Add=False)
if isinstance(g, TrigonometricFunction) or coeff is not S.One:
period = periodicity(g, symbol)
else:
period = _periodicity(g.args, symbol)
elif f.is_Add:
k, g = f.as_independent(symbol)
if k is not S.Zero:
return periodicity(g, symbol)
period = _periodicity(g.args, symbol)
elif isinstance(f, Mod):
a, n = f.args
if a == symbol:
period = n
elif isinstance(a, TrigonometricFunction):
period = periodicity(a, symbol)
#check if 'f' is linear in 'symbol'
elif (a.is_polynomial(symbol) and degree(a, symbol) == 1 and
symbol not in n.free_symbols):
period = Abs(n / a.diff(symbol))
elif isinstance(f, Piecewise):
pass # not handling Piecewise yet as the return type is not favorable
elif period is None:
from sympy.solvers.decompogen import compogen
g_s = decompogen(f, symbol)
num_of_gs = len(g_s)
if num_of_gs > 1:
for index, g in enumerate(reversed(g_s)):
start_index = num_of_gs - 1 - index
g = compogen(g_s[start_index:], symbol)
if g != orig_f and g != f: # Fix for issue 12620
period = periodicity(g, symbol)
if period is not None:
break
if period is not None:
if check:
return _check(orig_f, period)
return period
return None
def _periodicity(args, symbol):
"""
Helper for `periodicity` to find the period of a list of simpler
functions.
It uses the `lcim` method to find the least common period of
all the functions.
Parameters
==========
args : Tuple of Symbol
All the symbols present in a function.
symbol : Symbol
The symbol over which the function is to be evaluated.
Returns
=======
period
The least common period of the function for all the symbols
of the function.
None if for at least one of the symbols the function is aperiodic
"""
periods = []
for f in args:
period = periodicity(f, symbol)
if period is None:
return None
if period is not S.Zero:
periods.append(period)
if len(periods) > 1:
return lcim(periods)
if periods:
return periods[0]
def lcim(numbers):
"""Returns the least common integral multiple of a list of numbers.
The numbers can be rational or irrational or a mixture of both.
`None` is returned for incommensurable numbers.
Parameters
==========
numbers : list
Numbers (rational and/or irrational) for which lcim is to be found.
Returns
=======
number
lcim if it exists, otherwise `None` for incommensurable numbers.
Examples
========
>>> from sympy import S, pi
>>> from sympy.calculus.util import lcim
>>> lcim([S(1)/2, S(3)/4, S(5)/6])
15/2
>>> lcim([2*pi, 3*pi, pi, pi/2])
6*pi
>>> lcim([S(1), 2*pi])
"""
result = None
if all(num.is_irrational for num in numbers):
factorized_nums = list(map(lambda num: num.factor(), numbers))
factors_num = list(
map(lambda num: num.as_coeff_Mul(),
factorized_nums))
term = factors_num[0][1]
if all(factor == term for coeff, factor in factors_num):
common_term = term
coeffs = [coeff for coeff, factor in factors_num]
result = lcm_list(coeffs) * common_term
elif all(num.is_rational for num in numbers):
result = lcm_list(numbers)
else:
pass
return result
def is_convex(f, *syms, domain=S.Reals):
"""Determines the convexity of the function passed in the argument.
Parameters
==========
f : Expr
The concerned function.
syms : Tuple of symbols
The variables with respect to which the convexity is to be determined.
domain : Interval, optional
The domain over which the convexity of the function has to be checked.
If unspecified, S.Reals will be the default domain.
Returns
=======
Boolean
The method returns `True` if the function is convex otherwise it
returns `False`.
Raises
======
NotImplementedError
The check for the convexity of multivariate functions is not implemented yet.
Notes
=====
To determine concavity of a function pass `-f` as the concerned function.
To determine logarithmic convexity of a function pass log(f) as
concerned function.
To determine logartihmic concavity of a function pass -log(f) as
concerned function.
Currently, convexity check of multivariate functions is not handled.
Examples
========
>>> from sympy import symbols, exp, oo, Interval
>>> from sympy.calculus.util import is_convex
>>> x = symbols('x')
>>> is_convex(exp(x), x)
True
>>> is_convex(x**3, x, domain = Interval(-1, oo))
False
References
==========
.. [1] https://en.wikipedia.org/wiki/Convex_function
.. [2] http://www.ifp.illinois.edu/~angelia/L3_convfunc.pdf
.. [3] https://en.wikipedia.org/wiki/Logarithmically_convex_function
.. [4] https://en.wikipedia.org/wiki/Logarithmically_concave_function
.. [5] https://en.wikipedia.org/wiki/Concave_function
"""
if len(syms) > 1:
raise NotImplementedError(
"The check for the convexity of multivariate functions is not implemented yet.")
f = _sympify(f)
var = syms[0]
condition = f.diff(var, 2) < 0
if solve_univariate_inequality(condition, var, False, domain):
return False
return True
def stationary_points(f, symbol, domain=S.Reals):
"""
Returns the stationary points of a function (where derivative of the
function is 0) in the given domain.
Parameters
==========
f : Expr
The concerned function.
symbol : Symbol
The variable for which the stationary points are to be determined.
domain : Interval
The domain over which the stationary points have to be checked.
If unspecified, S.Reals will be the default domain.
Returns
=======
Set
A set of stationary points for the function. If there are no
stationary point, an EmptySet is returned.
Examples
========
>>> from sympy import Symbol, S, sin, pi, pprint, stationary_points
>>> from sympy.sets import Interval
>>> x = Symbol('x')
>>> stationary_points(1/x, x, S.Reals)
EmptySet
>>> pprint(stationary_points(sin(x), x), use_unicode=False)
pi 3*pi
{2*n*pi + -- | n in Integers} U {2*n*pi + ---- | n in Integers}
2 2
>>> stationary_points(sin(x),x, Interval(0, 4*pi))
FiniteSet(pi/2, 3*pi/2, 5*pi/2, 7*pi/2)
"""
from sympy import solveset, diff
if isinstance(domain, EmptySet):
return S.EmptySet
domain = continuous_domain(f, symbol, domain)
set = solveset(diff(f, symbol), symbol, domain)
return set
def maximum(f, symbol, domain=S.Reals):
"""
Returns the maximum value of a function in the given domain.
Parameters
==========
f : Expr
The concerned function.
symbol : Symbol
The variable for maximum value needs to be determined.
domain : Interval
The domain over which the maximum have to be checked.
If unspecified, then Global maximum is returned.
Returns
=======
number
Maximum value of the function in given domain.
Examples
========
>>> from sympy import Symbol, S, sin, cos, pi, maximum
>>> from sympy.sets import Interval
>>> x = Symbol('x')
>>> f = -x**2 + 2*x + 5
>>> maximum(f, x, S.Reals)
6
>>> maximum(sin(x), x, Interval(-pi, pi/4))
sqrt(2)/2
>>> maximum(sin(x)*cos(x), x)
1/2
"""
from sympy import Symbol
if isinstance(symbol, Symbol):
if isinstance(domain, EmptySet):
raise ValueError("Maximum value not defined for empty domain.")
return function_range(f, symbol, domain).sup
else:
raise ValueError("%s is not a valid symbol." % symbol)
def minimum(f, symbol, domain=S.Reals):
"""
Returns the minimum value of a function in the given domain.
Parameters
==========
f : Expr
The concerned function.
symbol : Symbol
The variable for minimum value needs to be determined.
domain : Interval
The domain over which the minimum have to be checked.
If unspecified, then Global minimum is returned.
Returns
=======
number
Minimum value of the function in the given domain.
Examples
========
>>> from sympy import Symbol, S, sin, cos, minimum
>>> from sympy.sets import Interval
>>> x = Symbol('x')
>>> f = x**2 + 2*x + 5
>>> minimum(f, x, S.Reals)
4
>>> minimum(sin(x), x, Interval(2, 3))
sin(3)
>>> minimum(sin(x)*cos(x), x)
-1/2
"""
from sympy import Symbol
if isinstance(symbol, Symbol):
if isinstance(domain, EmptySet):
raise ValueError("Minimum value not defined for empty domain.")
return function_range(f, symbol, domain).inf
else:
raise ValueError("%s is not a valid symbol." % symbol)
class AccumulationBounds(AtomicExpr):
r"""
# Note AccumulationBounds has an alias: AccumBounds
AccumulationBounds represent an interval `[a, b]`, which is always closed
at the ends. Here `a` and `b` can be any value from extended real numbers.
The intended meaning of AccummulationBounds is to give an approximate
location of the accumulation points of a real function at a limit point.
Let `a` and `b` be reals such that a <= b.
`\left\langle a, b\right\rangle = \{x \in \mathbb{R} \mid a \le x \le b\}`
`\left\langle -\infty, b\right\rangle = \{x \in \mathbb{R} \mid x \le b\} \cup \{-\infty, \infty\}`
`\left\langle a, \infty \right\rangle = \{x \in \mathbb{R} \mid a \le x\} \cup \{-\infty, \infty\}`
`\left\langle -\infty, \infty \right\rangle = \mathbb{R} \cup \{-\infty, \infty\}`
`oo` and `-oo` are added to the second and third definition respectively,
since if either `-oo` or `oo` is an argument, then the other one should
be included (though not as an end point). This is forced, since we have,
for example, `1/AccumBounds(0, 1) = AccumBounds(1, oo)`, and the limit at
`0` is not one-sided. As x tends to `0-`, then `1/x -> -oo`, so `-oo`
should be interpreted as belonging to `AccumBounds(1, oo)` though it need
not appear explicitly.
In many cases it suffices to know that the limit set is bounded.
However, in some other cases more exact information could be useful.
For example, all accumulation values of cos(x) + 1 are non-negative.
(AccumBounds(-1, 1) + 1 = AccumBounds(0, 2))
A AccumulationBounds object is defined to be real AccumulationBounds,
if its end points are finite reals.
Let `X`, `Y` be real AccumulationBounds, then their sum, difference,
product are defined to be the following sets:
`X + Y = \{ x+y \mid x \in X \cap y \in Y\}`
`X - Y = \{ x-y \mid x \in X \cap y \in Y\}`
`X * Y = \{ x*y \mid x \in X \cap y \in Y\}`
There is, however, no consensus on Interval division.
`X / Y = \{ z \mid \exists x \in X, y \in Y \mid y \neq 0, z = x/y\}`
Note: According to this definition the quotient of two AccumulationBounds
may not be a AccumulationBounds object but rather a union of
AccumulationBounds.
Note
====
The main focus in the interval arithmetic is on the simplest way to
calculate upper and lower endpoints for the range of values of a
function in one or more variables. These barriers are not necessarily
the supremum or infimum, since the precise calculation of those values
can be difficult or impossible.
Examples
========
>>> from sympy import AccumBounds, sin, exp, log, pi, E, S, oo
>>> from sympy.abc import x
>>> AccumBounds(0, 1) + AccumBounds(1, 2)
AccumBounds(1, 3)
>>> AccumBounds(0, 1) - AccumBounds(0, 2)
AccumBounds(-2, 1)
>>> AccumBounds(-2, 3)*AccumBounds(-1, 1)
AccumBounds(-3, 3)
>>> AccumBounds(1, 2)*AccumBounds(3, 5)
AccumBounds(3, 10)
The exponentiation of AccumulationBounds is defined
as follows:
If 0 does not belong to `X` or `n > 0` then
`X^n = \{ x^n \mid x \in X\}`
otherwise
`X^n = \{ x^n \mid x \neq 0, x \in X\} \cup \{-\infty, \infty\}`
Here for fractional `n`, the part of `X` resulting in a complex
AccumulationBounds object is neglected.
>>> AccumBounds(-1, 4)**(S(1)/2)
AccumBounds(0, 2)
>>> AccumBounds(1, 2)**2
AccumBounds(1, 4)
>>> AccumBounds(-1, oo)**(-1)
AccumBounds(-oo, oo)
Note: `<a, b>^2` is not same as `<a, b>*<a, b>`
>>> AccumBounds(-1, 1)**2
AccumBounds(0, 1)
>>> AccumBounds(1, 3) < 4
True
>>> AccumBounds(1, 3) < -1
False
Some elementary functions can also take AccumulationBounds as input.
A function `f` evaluated for some real AccumulationBounds `<a, b>`
is defined as `f(\left\langle a, b\right\rangle) = \{ f(x) \mid a \le x \le b \}`
>>> sin(AccumBounds(pi/6, pi/3))
AccumBounds(1/2, sqrt(3)/2)
>>> exp(AccumBounds(0, 1))
AccumBounds(1, E)
>>> log(AccumBounds(1, E))
AccumBounds(0, 1)
Some symbol in an expression can be substituted for a AccumulationBounds
object. But it doesn't necessarily evaluate the AccumulationBounds for
that expression.
Same expression can be evaluated to different values depending upon
the form it is used for substitution. For example:
>>> (x**2 + 2*x + 1).subs(x, AccumBounds(-1, 1))
AccumBounds(-1, 4)
>>> ((x + 1)**2).subs(x, AccumBounds(-1, 1))
AccumBounds(0, 4)
References
==========
.. [1] https://en.wikipedia.org/wiki/Interval_arithmetic
.. [2] http://fab.cba.mit.edu/classes/S62.12/docs/Hickey_interval.pdf
Notes
=====
Do not use ``AccumulationBounds`` for floating point interval arithmetic
calculations, use ``mpmath.iv`` instead.
"""
is_extended_real = True
def __new__(cls, min, max):
min = _sympify(min)
max = _sympify(max)
# Only allow real intervals (use symbols with 'is_extended_real=True').
if not min.is_extended_real or not max.is_extended_real:
raise ValueError("Only real AccumulationBounds are supported")
# Make sure that the created AccumBounds object will be valid.
if max.is_comparable and min.is_comparable:
if max < min:
raise ValueError(
"Lower limit should be smaller than upper limit")
if max == min:
return max
return Basic.__new__(cls, min, max)
# setting the operation priority
_op_priority = 11.0
def _eval_is_real(self):
if self.min.is_real and self.max.is_real:
return True
@property
def min(self):
"""
Returns the minimum possible value attained by AccumulationBounds
object.
Examples
========
>>> from sympy import AccumBounds
>>> AccumBounds(1, 3).min
1
"""
return self.args[0]
@property
def max(self):
"""
Returns the maximum possible value attained by AccumulationBounds
object.
Examples
========
>>> from sympy import AccumBounds
>>> AccumBounds(1, 3).max
3
"""
return self.args[1]
@property
def delta(self):
"""
Returns the difference of maximum possible value attained by
AccumulationBounds object and minimum possible value attained
by AccumulationBounds object.
Examples
========
>>> from sympy import AccumBounds
>>> AccumBounds(1, 3).delta
2
"""
return self.max - self.min
@property
def mid(self):
"""
Returns the mean of maximum possible value attained by
AccumulationBounds object and minimum possible value
attained by AccumulationBounds object.
Examples
========
>>> from sympy import AccumBounds
>>> AccumBounds(1, 3).mid
2
"""
return (self.min + self.max) / 2
@_sympifyit('other', NotImplemented)
def _eval_power(self, other):
return self.__pow__(other)
@_sympifyit('other', NotImplemented)
def __add__(self, other):
if isinstance(other, Expr):
if isinstance(other, AccumBounds):
return AccumBounds(
Add(self.min, other.min),
Add(self.max, other.max))
if other is S.Infinity and self.min is S.NegativeInfinity or \
other is S.NegativeInfinity and self.max is S.Infinity:
return AccumBounds(-oo, oo)
elif other.is_extended_real:
if self.min is S.NegativeInfinity and self.max is S.Infinity:
return AccumBounds(-oo, oo)
elif self.min is S.NegativeInfinity:
return AccumBounds(-oo, self.max + other)
elif self.max is S.Infinity:
return AccumBounds(self.min + other, oo)
else:
return AccumBounds(Add(self.min, other), Add(self.max, other))
return Add(self, other, evaluate=False)
return NotImplemented
__radd__ = __add__
def __neg__(self):
return AccumBounds(-self.max, -self.min)
@_sympifyit('other', NotImplemented)
def __sub__(self, other):
if isinstance(other, Expr):
if isinstance(other, AccumBounds):
return AccumBounds(
Add(self.min, -other.max),
Add(self.max, -other.min))
if other is S.NegativeInfinity and self.min is S.NegativeInfinity or \
other is S.Infinity and self.max is S.Infinity:
return AccumBounds(-oo, oo)
elif other.is_extended_real:
if self.min is S.NegativeInfinity and self.max is S.Infinity:
return AccumBounds(-oo, oo)
elif self.min is S.NegativeInfinity:
return AccumBounds(-oo, self.max - other)
elif self.max is S.Infinity:
return AccumBounds(self.min - other, oo)
else:
return AccumBounds(
Add(self.min, -other),
Add(self.max, -other))
return Add(self, -other, evaluate=False)
return NotImplemented
@_sympifyit('other', NotImplemented)
def __rsub__(self, other):
return self.__neg__() + other
@_sympifyit('other', NotImplemented)
def __mul__(self, other):
if isinstance(other, Expr):
if isinstance(other, AccumBounds):
return AccumBounds(Min(Mul(self.min, other.min),
Mul(self.min, other.max),
Mul(self.max, other.min),
Mul(self.max, other.max)),
Max(Mul(self.min, other.min),
Mul(self.min, other.max),
Mul(self.max, other.min),
Mul(self.max, other.max)))
if other is S.Infinity:
if self.min.is_zero:
return AccumBounds(0, oo)
if self.max.is_zero:
return AccumBounds(-oo, 0)
if other is S.NegativeInfinity:
if self.min.is_zero:
return AccumBounds(-oo, 0)
if self.max.is_zero:
return AccumBounds(0, oo)
if other.is_extended_real:
if other.is_zero:
if self == AccumBounds(-oo, oo):
return AccumBounds(-oo, oo)
if self.max is S.Infinity:
return AccumBounds(0, oo)
if self.min is S.NegativeInfinity:
return AccumBounds(-oo, 0)
return S.Zero
if other.is_extended_positive:
return AccumBounds(
Mul(self.min, other),
Mul(self.max, other))
elif other.is_extended_negative:
return AccumBounds(
Mul(self.max, other),
Mul(self.min, other))
if isinstance(other, Order):
return other
return Mul(self, other, evaluate=False)
return NotImplemented
__rmul__ = __mul__
@_sympifyit('other', NotImplemented)
def __truediv__(self, other):
if isinstance(other, Expr):
if isinstance(other, AccumBounds):
if other.min.is_positive or other.max.is_negative:
return self * AccumBounds(1/other.max, 1/other.min)
if (self.min.is_extended_nonpositive and self.max.is_extended_nonnegative and
other.min.is_extended_nonpositive and other.max.is_extended_nonnegative):
if self.min.is_zero and other.min.is_zero:
return AccumBounds(0, oo)
if self.max.is_zero and other.min.is_zero:
return AccumBounds(-oo, 0)
return AccumBounds(-oo, oo)
if self.max.is_extended_negative:
if other.min.is_extended_negative:
if other.max.is_zero:
return AccumBounds(self.max / other.min, oo)
if other.max.is_extended_positive:
# the actual answer is a Union of AccumBounds,
# Union(AccumBounds(-oo, self.max/other.max),
# AccumBounds(self.max/other.min, oo))
return AccumBounds(-oo, oo)
if other.min.is_zero and other.max.is_extended_positive:
return AccumBounds(-oo, self.max / other.max)
if self.min.is_extended_positive:
if other.min.is_extended_negative:
if other.max.is_zero:
return AccumBounds(-oo, self.min / other.min)
if other.max.is_extended_positive:
# the actual answer is a Union of AccumBounds,
# Union(AccumBounds(-oo, self.min/other.min),
# AccumBounds(self.min/other.max, oo))
return AccumBounds(-oo, oo)
if other.min.is_zero and other.max.is_extended_positive:
return AccumBounds(self.min / other.max, oo)
elif other.is_extended_real:
if other is S.Infinity or other is S.NegativeInfinity:
if self == AccumBounds(-oo, oo):
return AccumBounds(-oo, oo)
if self.max is S.Infinity:
return AccumBounds(Min(0, other), Max(0, other))
if self.min is S.NegativeInfinity:
return AccumBounds(Min(0, -other), Max(0, -other))
if other.is_extended_positive:
return AccumBounds(self.min / other, self.max / other)
elif other.is_extended_negative:
return AccumBounds(self.max / other, self.min / other)
if (1 / other) is S.ComplexInfinity:
return Mul(self, 1 / other, evaluate=False)
else:
return Mul(self, 1 / other)
return NotImplemented
@_sympifyit('other', NotImplemented)
def __rtruediv__(self, other):
if isinstance(other, Expr):
if other.is_extended_real:
if other.is_zero:
return S.Zero
if (self.min.is_extended_nonpositive and self.max.is_extended_nonnegative):
if self.min.is_zero:
if other.is_extended_positive:
return AccumBounds(Mul(other, 1 / self.max), oo)
if other.is_extended_negative:
return AccumBounds(-oo, Mul(other, 1 / self.max))
if self.max.is_zero:
if other.is_extended_positive:
return AccumBounds(-oo, Mul(other, 1 / self.min))
if other.is_extended_negative:
return AccumBounds(Mul(other, 1 / self.min), oo)
return AccumBounds(-oo, oo)
else:
return AccumBounds(Min(other / self.min, other / self.max),
Max(other / self.min, other / self.max))
return Mul(other, 1 / self, evaluate=False)
else:
return NotImplemented
@_sympifyit('other', NotImplemented)
def __pow__(self, other):
from sympy.functions.elementary.miscellaneous import real_root
if isinstance(other, Expr):
if other is S.Infinity:
if self.min.is_extended_nonnegative:
if self.max < 1:
return S.Zero
if self.min > 1:
return S.Infinity
return AccumBounds(0, oo)
elif self.max.is_extended_negative:
if self.min > -1:
return S.Zero
if self.max < -1:
return FiniteSet(-oo, oo)
return AccumBounds(-oo, oo)
else:
if self.min > -1:
if self.max < 1:
return S.Zero
return AccumBounds(0, oo)
return AccumBounds(-oo, oo)
if other is S.NegativeInfinity:
return (1 / self)**oo
if other.is_extended_real and other.is_number:
if other.is_zero:
return S.One
if other.is_Integer:
if self.min.is_extended_positive:
return AccumBounds(
Min(self.min ** other, self.max ** other),
Max(self.min ** other, self.max ** other))
elif self.max.is_extended_negative:
return AccumBounds(
Min(self.max ** other, self.min ** other),
Max(self.max ** other, self.min ** other))
if other % 2 == 0:
if other.is_extended_negative:
if self.min.is_zero:
return AccumBounds(self.max**other, oo)
if self.max.is_zero:
return AccumBounds(self.min**other, oo)
return AccumBounds(0, oo)
return AccumBounds(
S.Zero, Max(self.min**other, self.max**other))
else:
if other.is_extended_negative:
if self.min.is_zero:
return AccumBounds(self.max**other, oo)
if self.max.is_zero:
return AccumBounds(-oo, self.min**other)
return AccumBounds(-oo, oo)
return AccumBounds(self.min**other, self.max**other)
num, den = other.as_numer_denom()
if num == S.One:
if den % 2 == 0:
if S.Zero in self:
if self.min.is_extended_negative:
return AccumBounds(0, real_root(self.max, den))
return AccumBounds(real_root(self.min, den),
real_root(self.max, den))
if den!=1:
num_pow = self**num
return num_pow**(1 / den)
return AccumBounds(-oo, oo)
return NotImplemented
def __abs__(self):
if self.max.is_extended_negative:
return self.__neg__()
elif self.min.is_extended_negative:
return AccumBounds(S.Zero, Max(abs(self.min), self.max))
else:
return self
def __contains__(self, other):
"""
Returns True if other is contained in self, where other
belongs to extended real numbers, False if not contained,
otherwise TypeError is raised.
Examples
========
>>> from sympy import AccumBounds, oo
>>> 1 in AccumBounds(-1, 3)
True
-oo and oo go together as limits (in AccumulationBounds).
>>> -oo in AccumBounds(1, oo)
True
>>> oo in AccumBounds(-oo, 0)
True
"""
other = _sympify(other)
if other is S.Infinity or other is S.NegativeInfinity:
if self.min is S.NegativeInfinity or self.max is S.Infinity:
return True
return False
rv = And(self.min <= other, self.max >= other)
if rv not in (True, False):
raise TypeError("input failed to evaluate")
return rv
def intersection(self, other):
"""
Returns the intersection of 'self' and 'other'.
Here other can be an instance of FiniteSet or AccumulationBounds.
Parameters
==========
other: AccumulationBounds
Another AccumulationBounds object with which the intersection
has to be computed.
Returns
=======
AccumulationBounds
Intersection of 'self' and 'other'.
Examples
========
>>> from sympy import AccumBounds, FiniteSet
>>> AccumBounds(1, 3).intersection(AccumBounds(2, 4))
AccumBounds(2, 3)
>>> AccumBounds(1, 3).intersection(AccumBounds(4, 6))
EmptySet
>>> AccumBounds(1, 4).intersection(FiniteSet(1, 2, 5))
FiniteSet(1, 2)
"""
if not isinstance(other, (AccumBounds, FiniteSet)):
raise TypeError(
"Input must be AccumulationBounds or FiniteSet object")
if isinstance(other, FiniteSet):
fin_set = S.EmptySet
for i in other:
if i in self:
fin_set = fin_set + FiniteSet(i)
return fin_set
if self.max < other.min or self.min > other.max:
return S.EmptySet
if self.min <= other.min:
if self.max <= other.max:
return AccumBounds(other.min, self.max)
if self.max > other.max:
return other
if other.min <= self.min:
if other.max < self.max:
return AccumBounds(self.min, other.max)
if other.max > self.max:
return self
def union(self, other):
# TODO : Devise a better method for Union of AccumBounds
# this method is not actually correct and
# can be made better
if not isinstance(other, AccumBounds):
raise TypeError(
"Input must be AccumulationBounds or FiniteSet object")
if self.min <= other.min and self.max >= other.min:
return AccumBounds(self.min, Max(self.max, other.max))
if other.min <= self.min and other.max >= self.min:
return AccumBounds(other.min, Max(self.max, other.max))
@dispatch(AccumulationBounds, AccumulationBounds) # type: ignore # noqa:F811
def _eval_is_le(lhs, rhs): # noqa:F811
if is_le(lhs.max, rhs.min):
return True
if is_gt(lhs.min, rhs.max):
return False
@dispatch(AccumulationBounds, Basic) # type: ignore # noqa:F811
def _eval_is_le(lhs, rhs): # noqa: F811
"""
Returns True if range of values attained by `self` AccumulationBounds
object is greater than the range of values attained by `other`,
where other may be any value of type AccumulationBounds object or
extended real number value, False if `other` satisfies
the same property, else an unevaluated Relational.
Examples
========
>>> from sympy import AccumBounds, oo
>>> AccumBounds(1, 3) > AccumBounds(4, oo)
False
>>> AccumBounds(1, 4) > AccumBounds(3, 4)
AccumBounds(1, 4) > AccumBounds(3, 4)
>>> AccumBounds(1, oo) > -1
True
"""
if not rhs.is_extended_real:
raise TypeError(
"Invalid comparison of %s %s" %
(type(rhs), rhs))
elif rhs.is_comparable:
if is_le(lhs.max, rhs):
return True
if is_gt(lhs.min, rhs):
return False
@dispatch(AccumulationBounds, AccumulationBounds)
def _eval_is_ge(lhs, rhs): # noqa:F811
if is_ge(lhs.min, rhs.max):
return True
if is_lt(lhs.max, rhs.min):
return False
@dispatch(AccumulationBounds, Expr) # type:ignore
def _eval_is_ge(lhs, rhs): # noqa: F811
"""
Returns True if range of values attained by `lhs` AccumulationBounds
object is less that the range of values attained by `rhs`, where
other may be any value of type AccumulationBounds object or extended
real number value, False if `rhs` satisfies the same
property, else an unevaluated Relational.
Examples
========
>>> from sympy import AccumBounds, oo
>>> AccumBounds(1, 3) >= AccumBounds(4, oo)
False
>>> AccumBounds(1, 4) >= AccumBounds(3, 4)
AccumBounds(1, 4) >= AccumBounds(3, 4)
>>> AccumBounds(1, oo) >= 1
True
"""
if not rhs.is_extended_real:
raise TypeError(
"Invalid comparison of %s %s" %
(type(rhs), rhs))
elif rhs.is_comparable:
if is_ge(lhs.min, rhs):
return True
if is_lt(lhs.max, rhs):
return False
@dispatch(Expr, AccumulationBounds) # type:ignore
def _eval_is_ge(lhs, rhs): # noqa:F811
if not lhs.is_extended_real:
raise TypeError(
"Invalid comparison of %s %s" %
(type(lhs), lhs))
elif lhs.is_comparable:
if is_le(rhs.max, lhs):
return True
if is_gt(rhs.min, lhs):
return False
@dispatch(AccumulationBounds, AccumulationBounds) # type:ignore
def _eval_is_ge(lhs, rhs): # noqa:F811
if is_ge(lhs.min, rhs.max):
return True
if is_lt(lhs.max, rhs.min):
return False
# setting an alias for AccumulationBounds
AccumBounds = AccumulationBounds
|
808efb9188d5d6b222c70991881a75a86c603b0ffb0e7f0854fa46483e02b125 | """
module for generating C, C++, Fortran77, Fortran90, Julia, Rust
and Octave/Matlab routines that evaluate sympy expressions.
This module is work in progress.
Only the milestones with a '+' character in the list below have been completed.
--- How is sympy.utilities.codegen different from sympy.printing.ccode? ---
We considered the idea to extend the printing routines for sympy functions in
such a way that it prints complete compilable code, but this leads to a few
unsurmountable issues that can only be tackled with dedicated code generator:
- For C, one needs both a code and a header file, while the printing routines
generate just one string. This code generator can be extended to support
.pyf files for f2py.
- SymPy functions are not concerned with programming-technical issues, such
as input, output and input-output arguments. Other examples are contiguous
or non-contiguous arrays, including headers of other libraries such as gsl
or others.
- It is highly interesting to evaluate several sympy functions in one C
routine, eventually sharing common intermediate results with the help
of the cse routine. This is more than just printing.
- From the programming perspective, expressions with constants should be
evaluated in the code generator as much as possible. This is different
for printing.
--- Basic assumptions ---
* A generic Routine data structure describes the routine that must be
translated into C/Fortran/... code. This data structure covers all
features present in one or more of the supported languages.
* Descendants from the CodeGen class transform multiple Routine instances
into compilable code. Each derived class translates into a specific
language.
* In many cases, one wants a simple workflow. The friendly functions in the
last part are a simple api on top of the Routine/CodeGen stuff. They are
easier to use, but are less powerful.
--- Milestones ---
+ First working version with scalar input arguments, generating C code,
tests
+ Friendly functions that are easier to use than the rigorous
Routine/CodeGen workflow.
+ Integer and Real numbers as input and output
+ Output arguments
+ InputOutput arguments
+ Sort input/output arguments properly
+ Contiguous array arguments (numpy matrices)
+ Also generate .pyf code for f2py (in autowrap module)
+ Isolate constants and evaluate them beforehand in double precision
+ Fortran 90
+ Octave/Matlab
- Common Subexpression Elimination
- User defined comments in the generated code
- Optional extra include lines for libraries/objects that can eval special
functions
- Test other C compilers and libraries: gcc, tcc, libtcc, gcc+gsl, ...
- Contiguous array arguments (sympy matrices)
- Non-contiguous array arguments (sympy matrices)
- ccode must raise an error when it encounters something that can not be
translated into c. ccode(integrate(sin(x)/x, x)) does not make sense.
- Complex numbers as input and output
- A default complex datatype
- Include extra information in the header: date, user, hostname, sha1
hash, ...
- Fortran 77
- C++
- Python
- Julia
- Rust
- ...
"""
import os
import textwrap
from io import StringIO
from sympy import __version__ as sympy_version
from sympy.core import Symbol, S, Tuple, Equality, Function, Basic
from sympy.core.compatibility import is_sequence
from sympy.printing.c import c_code_printers
from sympy.printing.codeprinter import AssignmentError
from sympy.printing.fortran import FCodePrinter
from sympy.printing.julia import JuliaCodePrinter
from sympy.printing.octave import OctaveCodePrinter
from sympy.printing.rust import RustCodePrinter
from sympy.tensor import Idx, Indexed, IndexedBase
from sympy.matrices import (MatrixSymbol, ImmutableMatrix, MatrixBase,
MatrixExpr, MatrixSlice)
__all__ = [
# description of routines
"Routine", "DataType", "default_datatypes", "get_default_datatype",
"Argument", "InputArgument", "OutputArgument", "Result",
# routines -> code
"CodeGen", "CCodeGen", "FCodeGen", "JuliaCodeGen", "OctaveCodeGen",
"RustCodeGen",
# friendly functions
"codegen", "make_routine",
]
#
# Description of routines
#
class Routine:
"""Generic description of evaluation routine for set of expressions.
A CodeGen class can translate instances of this class into code in a
particular language. The routine specification covers all the features
present in these languages. The CodeGen part must raise an exception
when certain features are not present in the target language. For
example, multiple return values are possible in Python, but not in C or
Fortran. Another example: Fortran and Python support complex numbers,
while C does not.
"""
def __init__(self, name, arguments, results, local_vars, global_vars):
"""Initialize a Routine instance.
Parameters
==========
name : string
Name of the routine.
arguments : list of Arguments
These are things that appear in arguments of a routine, often
appearing on the right-hand side of a function call. These are
commonly InputArguments but in some languages, they can also be
OutputArguments or InOutArguments (e.g., pass-by-reference in C
code).
results : list of Results
These are the return values of the routine, often appearing on
the left-hand side of a function call. The difference between
Results and OutputArguments and when you should use each is
language-specific.
local_vars : list of Results
These are variables that will be defined at the beginning of the
function.
global_vars : list of Symbols
Variables which will not be passed into the function.
"""
# extract all input symbols and all symbols appearing in an expression
input_symbols = set()
symbols = set()
for arg in arguments:
if isinstance(arg, OutputArgument):
symbols.update(arg.expr.free_symbols - arg.expr.atoms(Indexed))
elif isinstance(arg, InputArgument):
input_symbols.add(arg.name)
elif isinstance(arg, InOutArgument):
input_symbols.add(arg.name)
symbols.update(arg.expr.free_symbols - arg.expr.atoms(Indexed))
else:
raise ValueError("Unknown Routine argument: %s" % arg)
for r in results:
if not isinstance(r, Result):
raise ValueError("Unknown Routine result: %s" % r)
symbols.update(r.expr.free_symbols - r.expr.atoms(Indexed))
local_symbols = set()
for r in local_vars:
if isinstance(r, Result):
symbols.update(r.expr.free_symbols - r.expr.atoms(Indexed))
local_symbols.add(r.name)
else:
local_symbols.add(r)
symbols = {s.label if isinstance(s, Idx) else s for s in symbols}
# Check that all symbols in the expressions are covered by
# InputArguments/InOutArguments---subset because user could
# specify additional (unused) InputArguments or local_vars.
notcovered = symbols.difference(
input_symbols.union(local_symbols).union(global_vars))
if notcovered != set():
raise ValueError("Symbols needed for output are not in input " +
", ".join([str(x) for x in notcovered]))
self.name = name
self.arguments = arguments
self.results = results
self.local_vars = local_vars
self.global_vars = global_vars
def __str__(self):
return self.__class__.__name__ + "({name!r}, {arguments}, {results}, {local_vars}, {global_vars})".format(**self.__dict__)
__repr__ = __str__
@property
def variables(self):
"""Returns a set of all variables possibly used in the routine.
For routines with unnamed return values, the dummies that may or
may not be used will be included in the set.
"""
v = set(self.local_vars)
for arg in self.arguments:
v.add(arg.name)
for res in self.results:
v.add(res.result_var)
return v
@property
def result_variables(self):
"""Returns a list of OutputArgument, InOutArgument and Result.
If return values are present, they are at the end ot the list.
"""
args = [arg for arg in self.arguments if isinstance(
arg, (OutputArgument, InOutArgument))]
args.extend(self.results)
return args
class DataType:
"""Holds strings for a certain datatype in different languages."""
def __init__(self, cname, fname, pyname, jlname, octname, rsname):
self.cname = cname
self.fname = fname
self.pyname = pyname
self.jlname = jlname
self.octname = octname
self.rsname = rsname
default_datatypes = {
"int": DataType("int", "INTEGER*4", "int", "", "", "i32"),
"float": DataType("double", "REAL*8", "float", "", "", "f64"),
"complex": DataType("double", "COMPLEX*16", "complex", "", "", "float") #FIXME:
# complex is only supported in fortran, python, julia, and octave.
# So to not break c or rust code generation, we stick with double or
# float, respecitvely (but actually should raise an exception for
# explicitly complex variables (x.is_complex==True))
}
COMPLEX_ALLOWED = False
def get_default_datatype(expr, complex_allowed=None):
"""Derives an appropriate datatype based on the expression."""
if complex_allowed is None:
complex_allowed = COMPLEX_ALLOWED
if complex_allowed:
final_dtype = "complex"
else:
final_dtype = "float"
if expr.is_integer:
return default_datatypes["int"]
elif expr.is_real:
return default_datatypes["float"]
elif isinstance(expr, MatrixBase):
#check all entries
dt = "int"
for element in expr:
if dt == "int" and not element.is_integer:
dt = "float"
if dt == "float" and not element.is_real:
return default_datatypes[final_dtype]
return default_datatypes[dt]
else:
return default_datatypes[final_dtype]
class Variable:
"""Represents a typed variable."""
def __init__(self, name, datatype=None, dimensions=None, precision=None):
"""Return a new variable.
Parameters
==========
name : Symbol or MatrixSymbol
datatype : optional
When not given, the data type will be guessed based on the
assumptions on the symbol argument.
dimension : sequence containing tupes, optional
If present, the argument is interpreted as an array, where this
sequence of tuples specifies (lower, upper) bounds for each
index of the array.
precision : int, optional
Controls the precision of floating point constants.
"""
if not isinstance(name, (Symbol, MatrixSymbol)):
raise TypeError("The first argument must be a sympy symbol.")
if datatype is None:
datatype = get_default_datatype(name)
elif not isinstance(datatype, DataType):
raise TypeError("The (optional) `datatype' argument must be an "
"instance of the DataType class.")
if dimensions and not isinstance(dimensions, (tuple, list)):
raise TypeError(
"The dimension argument must be a sequence of tuples")
self._name = name
self._datatype = {
'C': datatype.cname,
'FORTRAN': datatype.fname,
'JULIA': datatype.jlname,
'OCTAVE': datatype.octname,
'PYTHON': datatype.pyname,
'RUST': datatype.rsname,
}
self.dimensions = dimensions
self.precision = precision
def __str__(self):
return "%s(%r)" % (self.__class__.__name__, self.name)
__repr__ = __str__
@property
def name(self):
return self._name
def get_datatype(self, language):
"""Returns the datatype string for the requested language.
Examples
========
>>> from sympy import Symbol
>>> from sympy.utilities.codegen import Variable
>>> x = Variable(Symbol('x'))
>>> x.get_datatype('c')
'double'
>>> x.get_datatype('fortran')
'REAL*8'
"""
try:
return self._datatype[language.upper()]
except KeyError:
raise CodeGenError("Has datatypes for languages: %s" %
", ".join(self._datatype))
class Argument(Variable):
"""An abstract Argument data structure: a name and a data type.
This structure is refined in the descendants below.
"""
pass
class InputArgument(Argument):
pass
class ResultBase:
"""Base class for all "outgoing" information from a routine.
Objects of this class stores a sympy expression, and a sympy object
representing a result variable that will be used in the generated code
only if necessary.
"""
def __init__(self, expr, result_var):
self.expr = expr
self.result_var = result_var
def __str__(self):
return "%s(%r, %r)" % (self.__class__.__name__, self.expr,
self.result_var)
__repr__ = __str__
class OutputArgument(Argument, ResultBase):
"""OutputArgument are always initialized in the routine."""
def __init__(self, name, result_var, expr, datatype=None, dimensions=None, precision=None):
"""Return a new variable.
Parameters
==========
name : Symbol, MatrixSymbol
The name of this variable. When used for code generation, this
might appear, for example, in the prototype of function in the
argument list.
result_var : Symbol, Indexed
Something that can be used to assign a value to this variable.
Typically the same as `name` but for Indexed this should be e.g.,
"y[i]" whereas `name` should be the Symbol "y".
expr : object
The expression that should be output, typically a SymPy
expression.
datatype : optional
When not given, the data type will be guessed based on the
assumptions on the symbol argument.
dimension : sequence containing tupes, optional
If present, the argument is interpreted as an array, where this
sequence of tuples specifies (lower, upper) bounds for each
index of the array.
precision : int, optional
Controls the precision of floating point constants.
"""
Argument.__init__(self, name, datatype, dimensions, precision)
ResultBase.__init__(self, expr, result_var)
def __str__(self):
return "%s(%r, %r, %r)" % (self.__class__.__name__, self.name, self.result_var, self.expr)
__repr__ = __str__
class InOutArgument(Argument, ResultBase):
"""InOutArgument are never initialized in the routine."""
def __init__(self, name, result_var, expr, datatype=None, dimensions=None, precision=None):
if not datatype:
datatype = get_default_datatype(expr)
Argument.__init__(self, name, datatype, dimensions, precision)
ResultBase.__init__(self, expr, result_var)
__init__.__doc__ = OutputArgument.__init__.__doc__
def __str__(self):
return "%s(%r, %r, %r)" % (self.__class__.__name__, self.name, self.expr,
self.result_var)
__repr__ = __str__
class Result(Variable, ResultBase):
"""An expression for a return value.
The name result is used to avoid conflicts with the reserved word
"return" in the python language. It is also shorter than ReturnValue.
These may or may not need a name in the destination (e.g., "return(x*y)"
might return a value without ever naming it).
"""
def __init__(self, expr, name=None, result_var=None, datatype=None,
dimensions=None, precision=None):
"""Initialize a return value.
Parameters
==========
expr : SymPy expression
name : Symbol, MatrixSymbol, optional
The name of this return variable. When used for code generation,
this might appear, for example, in the prototype of function in a
list of return values. A dummy name is generated if omitted.
result_var : Symbol, Indexed, optional
Something that can be used to assign a value to this variable.
Typically the same as `name` but for Indexed this should be e.g.,
"y[i]" whereas `name` should be the Symbol "y". Defaults to
`name` if omitted.
datatype : optional
When not given, the data type will be guessed based on the
assumptions on the expr argument.
dimension : sequence containing tupes, optional
If present, this variable is interpreted as an array,
where this sequence of tuples specifies (lower, upper)
bounds for each index of the array.
precision : int, optional
Controls the precision of floating point constants.
"""
# Basic because it is the base class for all types of expressions
if not isinstance(expr, (Basic, MatrixBase)):
raise TypeError("The first argument must be a sympy expression.")
if name is None:
name = 'result_%d' % abs(hash(expr))
if datatype is None:
#try to infer data type from the expression
datatype = get_default_datatype(expr)
if isinstance(name, str):
if isinstance(expr, (MatrixBase, MatrixExpr)):
name = MatrixSymbol(name, *expr.shape)
else:
name = Symbol(name)
if result_var is None:
result_var = name
Variable.__init__(self, name, datatype=datatype,
dimensions=dimensions, precision=precision)
ResultBase.__init__(self, expr, result_var)
def __str__(self):
return "%s(%r, %r, %r)" % (self.__class__.__name__, self.expr, self.name,
self.result_var)
__repr__ = __str__
#
# Transformation of routine objects into code
#
class CodeGen:
"""Abstract class for the code generators."""
printer = None # will be set to an instance of a CodePrinter subclass
def _indent_code(self, codelines):
return self.printer.indent_code(codelines)
def _printer_method_with_settings(self, method, settings=None, *args, **kwargs):
settings = settings or {}
ori = {k: self.printer._settings[k] for k in settings}
for k, v in settings.items():
self.printer._settings[k] = v
result = getattr(self.printer, method)(*args, **kwargs)
for k, v in ori.items():
self.printer._settings[k] = v
return result
def _get_symbol(self, s):
"""Returns the symbol as fcode prints it."""
if self.printer._settings['human']:
expr_str = self.printer.doprint(s)
else:
constants, not_supported, expr_str = self.printer.doprint(s)
if constants or not_supported:
raise ValueError("Failed to print %s" % str(s))
return expr_str.strip()
def __init__(self, project="project", cse=False):
"""Initialize a code generator.
Derived classes will offer more options that affect the generated
code.
"""
self.project = project
self.cse = cse
def routine(self, name, expr, argument_sequence=None, global_vars=None):
"""Creates an Routine object that is appropriate for this language.
This implementation is appropriate for at least C/Fortran. Subclasses
can override this if necessary.
Here, we assume at most one return value (the l-value) which must be
scalar. Additional outputs are OutputArguments (e.g., pointers on
right-hand-side or pass-by-reference). Matrices are always returned
via OutputArguments. If ``argument_sequence`` is None, arguments will
be ordered alphabetically, but with all InputArguments first, and then
OutputArgument and InOutArguments.
"""
if self.cse:
from sympy.simplify.cse_main import cse
if is_sequence(expr) and not isinstance(expr, (MatrixBase, MatrixExpr)):
if not expr:
raise ValueError("No expression given")
for e in expr:
if not e.is_Equality:
raise CodeGenError("Lists of expressions must all be Equalities. {} is not.".format(e))
# create a list of right hand sides and simplify them
rhs = [e.rhs for e in expr]
common, simplified = cse(rhs)
# pack the simplified expressions back up with their left hand sides
expr = [Equality(e.lhs, rhs) for e, rhs in zip(expr, simplified)]
else:
rhs = [expr]
if isinstance(expr, Equality):
common, simplified = cse(expr.rhs) #, ignore=in_out_args)
expr = Equality(expr.lhs, simplified[0])
else:
common, simplified = cse(expr)
expr = simplified
local_vars = [Result(b,a) for a,b in common]
local_symbols = {a for a,_ in common}
local_expressions = Tuple(*[b for _,b in common])
else:
local_expressions = Tuple()
if is_sequence(expr) and not isinstance(expr, (MatrixBase, MatrixExpr)):
if not expr:
raise ValueError("No expression given")
expressions = Tuple(*expr)
else:
expressions = Tuple(expr)
if self.cse:
if {i.label for i in expressions.atoms(Idx)} != set():
raise CodeGenError("CSE and Indexed expressions do not play well together yet")
else:
# local variables for indexed expressions
local_vars = {i.label for i in expressions.atoms(Idx)}
local_symbols = local_vars
# global variables
global_vars = set() if global_vars is None else set(global_vars)
# symbols that should be arguments
symbols = (expressions.free_symbols | local_expressions.free_symbols) - local_symbols - global_vars
new_symbols = set()
new_symbols.update(symbols)
for symbol in symbols:
if isinstance(symbol, Idx):
new_symbols.remove(symbol)
new_symbols.update(symbol.args[1].free_symbols)
if isinstance(symbol, Indexed):
new_symbols.remove(symbol)
symbols = new_symbols
# Decide whether to use output argument or return value
return_val = []
output_args = []
for expr in expressions:
if isinstance(expr, Equality):
out_arg = expr.lhs
expr = expr.rhs
if isinstance(out_arg, Indexed):
dims = tuple([ (S.Zero, dim - 1) for dim in out_arg.shape])
symbol = out_arg.base.label
elif isinstance(out_arg, Symbol):
dims = []
symbol = out_arg
elif isinstance(out_arg, MatrixSymbol):
dims = tuple([ (S.Zero, dim - 1) for dim in out_arg.shape])
symbol = out_arg
else:
raise CodeGenError("Only Indexed, Symbol, or MatrixSymbol "
"can define output arguments.")
if expr.has(symbol):
output_args.append(
InOutArgument(symbol, out_arg, expr, dimensions=dims))
else:
output_args.append(
OutputArgument(symbol, out_arg, expr, dimensions=dims))
# remove duplicate arguments when they are not local variables
if symbol not in local_vars:
# avoid duplicate arguments
symbols.remove(symbol)
elif isinstance(expr, (ImmutableMatrix, MatrixSlice)):
# Create a "dummy" MatrixSymbol to use as the Output arg
out_arg = MatrixSymbol('out_%s' % abs(hash(expr)), *expr.shape)
dims = tuple([(S.Zero, dim - 1) for dim in out_arg.shape])
output_args.append(
OutputArgument(out_arg, out_arg, expr, dimensions=dims))
else:
return_val.append(Result(expr))
arg_list = []
# setup input argument list
# helper to get dimensions for data for array-like args
def dimensions(s):
return [(S.Zero, dim - 1) for dim in s.shape]
array_symbols = {}
for array in expressions.atoms(Indexed) | local_expressions.atoms(Indexed):
array_symbols[array.base.label] = array
for array in expressions.atoms(MatrixSymbol) | local_expressions.atoms(MatrixSymbol):
array_symbols[array] = array
for symbol in sorted(symbols, key=str):
if symbol in array_symbols:
array = array_symbols[symbol]
metadata = {'dimensions': dimensions(array)}
else:
metadata = {}
arg_list.append(InputArgument(symbol, **metadata))
output_args.sort(key=lambda x: str(x.name))
arg_list.extend(output_args)
if argument_sequence is not None:
# if the user has supplied IndexedBase instances, we'll accept that
new_sequence = []
for arg in argument_sequence:
if isinstance(arg, IndexedBase):
new_sequence.append(arg.label)
else:
new_sequence.append(arg)
argument_sequence = new_sequence
missing = [x for x in arg_list if x.name not in argument_sequence]
if missing:
msg = "Argument list didn't specify: {0} "
msg = msg.format(", ".join([str(m.name) for m in missing]))
raise CodeGenArgumentListError(msg, missing)
# create redundant arguments to produce the requested sequence
name_arg_dict = {x.name: x for x in arg_list}
new_args = []
for symbol in argument_sequence:
try:
new_args.append(name_arg_dict[symbol])
except KeyError:
if isinstance(symbol, (IndexedBase, MatrixSymbol)):
metadata = {'dimensions': dimensions(symbol)}
else:
metadata = {}
new_args.append(InputArgument(symbol, **metadata))
arg_list = new_args
return Routine(name, arg_list, return_val, local_vars, global_vars)
def write(self, routines, prefix, to_files=False, header=True, empty=True):
"""Writes all the source code files for the given routines.
The generated source is returned as a list of (filename, contents)
tuples, or is written to files (see below). Each filename consists
of the given prefix, appended with an appropriate extension.
Parameters
==========
routines : list
A list of Routine instances to be written
prefix : string
The prefix for the output files
to_files : bool, optional
When True, the output is written to files. Otherwise, a list
of (filename, contents) tuples is returned. [default: False]
header : bool, optional
When True, a header comment is included on top of each source
file. [default: True]
empty : bool, optional
When True, empty lines are included to structure the source
files. [default: True]
"""
if to_files:
for dump_fn in self.dump_fns:
filename = "%s.%s" % (prefix, dump_fn.extension)
with open(filename, "w") as f:
dump_fn(self, routines, f, prefix, header, empty)
else:
result = []
for dump_fn in self.dump_fns:
filename = "%s.%s" % (prefix, dump_fn.extension)
contents = StringIO()
dump_fn(self, routines, contents, prefix, header, empty)
result.append((filename, contents.getvalue()))
return result
def dump_code(self, routines, f, prefix, header=True, empty=True):
"""Write the code by calling language specific methods.
The generated file contains all the definitions of the routines in
low-level code and refers to the header file if appropriate.
Parameters
==========
routines : list
A list of Routine instances.
f : file-like
Where to write the file.
prefix : string
The filename prefix, used to refer to the proper header file.
Only the basename of the prefix is used.
header : bool, optional
When True, a header comment is included on top of each source
file. [default : True]
empty : bool, optional
When True, empty lines are included to structure the source
files. [default : True]
"""
code_lines = self._preprocessor_statements(prefix)
for routine in routines:
if empty:
code_lines.append("\n")
code_lines.extend(self._get_routine_opening(routine))
code_lines.extend(self._declare_arguments(routine))
code_lines.extend(self._declare_globals(routine))
code_lines.extend(self._declare_locals(routine))
if empty:
code_lines.append("\n")
code_lines.extend(self._call_printer(routine))
if empty:
code_lines.append("\n")
code_lines.extend(self._get_routine_ending(routine))
code_lines = self._indent_code(''.join(code_lines))
if header:
code_lines = ''.join(self._get_header() + [code_lines])
if code_lines:
f.write(code_lines)
class CodeGenError(Exception):
pass
class CodeGenArgumentListError(Exception):
@property
def missing_args(self):
return self.args[1]
header_comment = """Code generated with sympy %(version)s
See http://www.sympy.org/ for more information.
This file is part of '%(project)s'
"""
class CCodeGen(CodeGen):
"""Generator for C code.
The .write() method inherited from CodeGen will output a code file and
an interface file, <prefix>.c and <prefix>.h respectively.
"""
code_extension = "c"
interface_extension = "h"
standard = 'c99'
def __init__(self, project="project", printer=None,
preprocessor_statements=None, cse=False):
super().__init__(project=project, cse=cse)
self.printer = printer or c_code_printers[self.standard.lower()]()
self.preprocessor_statements = preprocessor_statements
if preprocessor_statements is None:
self.preprocessor_statements = ['#include <math.h>']
def _get_header(self):
"""Writes a common header for the generated files."""
code_lines = []
code_lines.append("/" + "*"*78 + '\n')
tmp = header_comment % {"version": sympy_version,
"project": self.project}
for line in tmp.splitlines():
code_lines.append(" *%s*\n" % line.center(76))
code_lines.append(" " + "*"*78 + "/\n")
return code_lines
def get_prototype(self, routine):
"""Returns a string for the function prototype of the routine.
If the routine has multiple result objects, an CodeGenError is
raised.
See: https://en.wikipedia.org/wiki/Function_prototype
"""
if len(routine.results) > 1:
raise CodeGenError("C only supports a single or no return value.")
elif len(routine.results) == 1:
ctype = routine.results[0].get_datatype('C')
else:
ctype = "void"
type_args = []
for arg in routine.arguments:
name = self.printer.doprint(arg.name)
if arg.dimensions or isinstance(arg, ResultBase):
type_args.append((arg.get_datatype('C'), "*%s" % name))
else:
type_args.append((arg.get_datatype('C'), name))
arguments = ", ".join([ "%s %s" % t for t in type_args])
return "%s %s(%s)" % (ctype, routine.name, arguments)
def _preprocessor_statements(self, prefix):
code_lines = []
code_lines.append('#include "{}.h"'.format(os.path.basename(prefix)))
code_lines.extend(self.preprocessor_statements)
code_lines = ['{}\n'.format(l) for l in code_lines]
return code_lines
def _get_routine_opening(self, routine):
prototype = self.get_prototype(routine)
return ["%s {\n" % prototype]
def _declare_arguments(self, routine):
# arguments are declared in prototype
return []
def _declare_globals(self, routine):
# global variables are not explicitly declared within C functions
return []
def _declare_locals(self, routine):
# Compose a list of symbols to be dereferenced in the function
# body. These are the arguments that were passed by a reference
# pointer, excluding arrays.
dereference = []
for arg in routine.arguments:
if isinstance(arg, ResultBase) and not arg.dimensions:
dereference.append(arg.name)
code_lines = []
for result in routine.local_vars:
# local variables that are simple symbols such as those used as indices into
# for loops are defined declared elsewhere.
if not isinstance(result, Result):
continue
if result.name != result.result_var:
raise CodeGen("Result variable and name should match: {}".format(result))
assign_to = result.name
t = result.get_datatype('c')
if isinstance(result.expr, (MatrixBase, MatrixExpr)):
dims = result.expr.shape
code_lines.append("{} {}[{}];\n".format(t, str(assign_to), dims[0]*dims[1]))
prefix = ""
else:
prefix = "const {} ".format(t)
constants, not_c, c_expr = self._printer_method_with_settings(
'doprint', dict(human=False, dereference=dereference),
result.expr, assign_to=assign_to)
for name, value in sorted(constants, key=str):
code_lines.append("double const %s = %s;\n" % (name, value))
code_lines.append("{}{}\n".format(prefix, c_expr))
return code_lines
def _call_printer(self, routine):
code_lines = []
# Compose a list of symbols to be dereferenced in the function
# body. These are the arguments that were passed by a reference
# pointer, excluding arrays.
dereference = []
for arg in routine.arguments:
if isinstance(arg, ResultBase) and not arg.dimensions:
dereference.append(arg.name)
return_val = None
for result in routine.result_variables:
if isinstance(result, Result):
assign_to = routine.name + "_result"
t = result.get_datatype('c')
code_lines.append("{} {};\n".format(t, str(assign_to)))
return_val = assign_to
else:
assign_to = result.result_var
try:
constants, not_c, c_expr = self._printer_method_with_settings(
'doprint', dict(human=False, dereference=dereference),
result.expr, assign_to=assign_to)
except AssignmentError:
assign_to = result.result_var
code_lines.append(
"%s %s;\n" % (result.get_datatype('c'), str(assign_to)))
constants, not_c, c_expr = self._printer_method_with_settings(
'doprint', dict(human=False, dereference=dereference),
result.expr, assign_to=assign_to)
for name, value in sorted(constants, key=str):
code_lines.append("double const %s = %s;\n" % (name, value))
code_lines.append("%s\n" % c_expr)
if return_val:
code_lines.append(" return %s;\n" % return_val)
return code_lines
def _get_routine_ending(self, routine):
return ["}\n"]
def dump_c(self, routines, f, prefix, header=True, empty=True):
self.dump_code(routines, f, prefix, header, empty)
dump_c.extension = code_extension # type: ignore
dump_c.__doc__ = CodeGen.dump_code.__doc__
def dump_h(self, routines, f, prefix, header=True, empty=True):
"""Writes the C header file.
This file contains all the function declarations.
Parameters
==========
routines : list
A list of Routine instances.
f : file-like
Where to write the file.
prefix : string
The filename prefix, used to construct the include guards.
Only the basename of the prefix is used.
header : bool, optional
When True, a header comment is included on top of each source
file. [default : True]
empty : bool, optional
When True, empty lines are included to structure the source
files. [default : True]
"""
if header:
print(''.join(self._get_header()), file=f)
guard_name = "%s__%s__H" % (self.project.replace(
" ", "_").upper(), prefix.replace("/", "_").upper())
# include guards
if empty:
print(file=f)
print("#ifndef %s" % guard_name, file=f)
print("#define %s" % guard_name, file=f)
if empty:
print(file=f)
# declaration of the function prototypes
for routine in routines:
prototype = self.get_prototype(routine)
print("%s;" % prototype, file=f)
# end if include guards
if empty:
print(file=f)
print("#endif", file=f)
if empty:
print(file=f)
dump_h.extension = interface_extension # type: ignore
# This list of dump functions is used by CodeGen.write to know which dump
# functions it has to call.
dump_fns = [dump_c, dump_h]
class C89CodeGen(CCodeGen):
standard = 'C89'
class C99CodeGen(CCodeGen):
standard = 'C99'
class FCodeGen(CodeGen):
"""Generator for Fortran 95 code
The .write() method inherited from CodeGen will output a code file and
an interface file, <prefix>.f90 and <prefix>.h respectively.
"""
code_extension = "f90"
interface_extension = "h"
def __init__(self, project='project', printer=None):
super().__init__(project)
self.printer = printer or FCodePrinter()
def _get_header(self):
"""Writes a common header for the generated files."""
code_lines = []
code_lines.append("!" + "*"*78 + '\n')
tmp = header_comment % {"version": sympy_version,
"project": self.project}
for line in tmp.splitlines():
code_lines.append("!*%s*\n" % line.center(76))
code_lines.append("!" + "*"*78 + '\n')
return code_lines
def _preprocessor_statements(self, prefix):
return []
def _get_routine_opening(self, routine):
"""Returns the opening statements of the fortran routine."""
code_list = []
if len(routine.results) > 1:
raise CodeGenError(
"Fortran only supports a single or no return value.")
elif len(routine.results) == 1:
result = routine.results[0]
code_list.append(result.get_datatype('fortran'))
code_list.append("function")
else:
code_list.append("subroutine")
args = ", ".join("%s" % self._get_symbol(arg.name)
for arg in routine.arguments)
call_sig = "{}({})\n".format(routine.name, args)
# Fortran 95 requires all lines be less than 132 characters, so wrap
# this line before appending.
call_sig = ' &\n'.join(textwrap.wrap(call_sig,
width=60,
break_long_words=False)) + '\n'
code_list.append(call_sig)
code_list = [' '.join(code_list)]
code_list.append('implicit none\n')
return code_list
def _declare_arguments(self, routine):
# argument type declarations
code_list = []
array_list = []
scalar_list = []
for arg in routine.arguments:
if isinstance(arg, InputArgument):
typeinfo = "%s, intent(in)" % arg.get_datatype('fortran')
elif isinstance(arg, InOutArgument):
typeinfo = "%s, intent(inout)" % arg.get_datatype('fortran')
elif isinstance(arg, OutputArgument):
typeinfo = "%s, intent(out)" % arg.get_datatype('fortran')
else:
raise CodeGenError("Unknown Argument type: %s" % type(arg))
fprint = self._get_symbol
if arg.dimensions:
# fortran arrays start at 1
dimstr = ", ".join(["%s:%s" % (
fprint(dim[0] + 1), fprint(dim[1] + 1))
for dim in arg.dimensions])
typeinfo += ", dimension(%s)" % dimstr
array_list.append("%s :: %s\n" % (typeinfo, fprint(arg.name)))
else:
scalar_list.append("%s :: %s\n" % (typeinfo, fprint(arg.name)))
# scalars first, because they can be used in array declarations
code_list.extend(scalar_list)
code_list.extend(array_list)
return code_list
def _declare_globals(self, routine):
# Global variables not explicitly declared within Fortran 90 functions.
# Note: a future F77 mode may need to generate "common" blocks.
return []
def _declare_locals(self, routine):
code_list = []
for var in sorted(routine.local_vars, key=str):
typeinfo = get_default_datatype(var)
code_list.append("%s :: %s\n" % (
typeinfo.fname, self._get_symbol(var)))
return code_list
def _get_routine_ending(self, routine):
"""Returns the closing statements of the fortran routine."""
if len(routine.results) == 1:
return ["end function\n"]
else:
return ["end subroutine\n"]
def get_interface(self, routine):
"""Returns a string for the function interface.
The routine should have a single result object, which can be None.
If the routine has multiple result objects, a CodeGenError is
raised.
See: https://en.wikipedia.org/wiki/Function_prototype
"""
prototype = [ "interface\n" ]
prototype.extend(self._get_routine_opening(routine))
prototype.extend(self._declare_arguments(routine))
prototype.extend(self._get_routine_ending(routine))
prototype.append("end interface\n")
return "".join(prototype)
def _call_printer(self, routine):
declarations = []
code_lines = []
for result in routine.result_variables:
if isinstance(result, Result):
assign_to = routine.name
elif isinstance(result, (OutputArgument, InOutArgument)):
assign_to = result.result_var
constants, not_fortran, f_expr = self._printer_method_with_settings(
'doprint', dict(human=False, source_format='free', standard=95),
result.expr, assign_to=assign_to)
for obj, v in sorted(constants, key=str):
t = get_default_datatype(obj)
declarations.append(
"%s, parameter :: %s = %s\n" % (t.fname, obj, v))
for obj in sorted(not_fortran, key=str):
t = get_default_datatype(obj)
if isinstance(obj, Function):
name = obj.func
else:
name = obj
declarations.append("%s :: %s\n" % (t.fname, name))
code_lines.append("%s\n" % f_expr)
return declarations + code_lines
def _indent_code(self, codelines):
return self._printer_method_with_settings(
'indent_code', dict(human=False, source_format='free'), codelines)
def dump_f95(self, routines, f, prefix, header=True, empty=True):
# check that symbols are unique with ignorecase
for r in routines:
lowercase = {str(x).lower() for x in r.variables}
orig_case = {str(x) for x in r.variables}
if len(lowercase) < len(orig_case):
raise CodeGenError("Fortran ignores case. Got symbols: %s" %
(", ".join([str(var) for var in r.variables])))
self.dump_code(routines, f, prefix, header, empty)
dump_f95.extension = code_extension # type: ignore
dump_f95.__doc__ = CodeGen.dump_code.__doc__
def dump_h(self, routines, f, prefix, header=True, empty=True):
"""Writes the interface to a header file.
This file contains all the function declarations.
Parameters
==========
routines : list
A list of Routine instances.
f : file-like
Where to write the file.
prefix : string
The filename prefix.
header : bool, optional
When True, a header comment is included on top of each source
file. [default : True]
empty : bool, optional
When True, empty lines are included to structure the source
files. [default : True]
"""
if header:
print(''.join(self._get_header()), file=f)
if empty:
print(file=f)
# declaration of the function prototypes
for routine in routines:
prototype = self.get_interface(routine)
f.write(prototype)
if empty:
print(file=f)
dump_h.extension = interface_extension # type: ignore
# This list of dump functions is used by CodeGen.write to know which dump
# functions it has to call.
dump_fns = [dump_f95, dump_h]
class JuliaCodeGen(CodeGen):
"""Generator for Julia code.
The .write() method inherited from CodeGen will output a code file
<prefix>.jl.
"""
code_extension = "jl"
def __init__(self, project='project', printer=None):
super().__init__(project)
self.printer = printer or JuliaCodePrinter()
def routine(self, name, expr, argument_sequence, global_vars):
"""Specialized Routine creation for Julia."""
if is_sequence(expr) and not isinstance(expr, (MatrixBase, MatrixExpr)):
if not expr:
raise ValueError("No expression given")
expressions = Tuple(*expr)
else:
expressions = Tuple(expr)
# local variables
local_vars = {i.label for i in expressions.atoms(Idx)}
# global variables
global_vars = set() if global_vars is None else set(global_vars)
# symbols that should be arguments
old_symbols = expressions.free_symbols - local_vars - global_vars
symbols = set()
for s in old_symbols:
if isinstance(s, Idx):
symbols.update(s.args[1].free_symbols)
elif not isinstance(s, Indexed):
symbols.add(s)
# Julia supports multiple return values
return_vals = []
output_args = []
for (i, expr) in enumerate(expressions):
if isinstance(expr, Equality):
out_arg = expr.lhs
expr = expr.rhs
symbol = out_arg
if isinstance(out_arg, Indexed):
dims = tuple([ (S.One, dim) for dim in out_arg.shape])
symbol = out_arg.base.label
output_args.append(InOutArgument(symbol, out_arg, expr, dimensions=dims))
if not isinstance(out_arg, (Indexed, Symbol, MatrixSymbol)):
raise CodeGenError("Only Indexed, Symbol, or MatrixSymbol "
"can define output arguments.")
return_vals.append(Result(expr, name=symbol, result_var=out_arg))
if not expr.has(symbol):
# this is a pure output: remove from the symbols list, so
# it doesn't become an input.
symbols.remove(symbol)
else:
# we have no name for this output
return_vals.append(Result(expr, name='out%d' % (i+1)))
# setup input argument list
output_args.sort(key=lambda x: str(x.name))
arg_list = list(output_args)
array_symbols = {}
for array in expressions.atoms(Indexed):
array_symbols[array.base.label] = array
for array in expressions.atoms(MatrixSymbol):
array_symbols[array] = array
for symbol in sorted(symbols, key=str):
arg_list.append(InputArgument(symbol))
if argument_sequence is not None:
# if the user has supplied IndexedBase instances, we'll accept that
new_sequence = []
for arg in argument_sequence:
if isinstance(arg, IndexedBase):
new_sequence.append(arg.label)
else:
new_sequence.append(arg)
argument_sequence = new_sequence
missing = [x for x in arg_list if x.name not in argument_sequence]
if missing:
msg = "Argument list didn't specify: {0} "
msg = msg.format(", ".join([str(m.name) for m in missing]))
raise CodeGenArgumentListError(msg, missing)
# create redundant arguments to produce the requested sequence
name_arg_dict = {x.name: x for x in arg_list}
new_args = []
for symbol in argument_sequence:
try:
new_args.append(name_arg_dict[symbol])
except KeyError:
new_args.append(InputArgument(symbol))
arg_list = new_args
return Routine(name, arg_list, return_vals, local_vars, global_vars)
def _get_header(self):
"""Writes a common header for the generated files."""
code_lines = []
tmp = header_comment % {"version": sympy_version,
"project": self.project}
for line in tmp.splitlines():
if line == '':
code_lines.append("#\n")
else:
code_lines.append("# %s\n" % line)
return code_lines
def _preprocessor_statements(self, prefix):
return []
def _get_routine_opening(self, routine):
"""Returns the opening statements of the routine."""
code_list = []
code_list.append("function ")
# Inputs
args = []
for i, arg in enumerate(routine.arguments):
if isinstance(arg, OutputArgument):
raise CodeGenError("Julia: invalid argument of type %s" %
str(type(arg)))
if isinstance(arg, (InputArgument, InOutArgument)):
args.append("%s" % self._get_symbol(arg.name))
args = ", ".join(args)
code_list.append("%s(%s)\n" % (routine.name, args))
code_list = [ "".join(code_list) ]
return code_list
def _declare_arguments(self, routine):
return []
def _declare_globals(self, routine):
return []
def _declare_locals(self, routine):
return []
def _get_routine_ending(self, routine):
outs = []
for result in routine.results:
if isinstance(result, Result):
# Note: name not result_var; want `y` not `y[i]` for Indexed
s = self._get_symbol(result.name)
else:
raise CodeGenError("unexpected object in Routine results")
outs.append(s)
return ["return " + ", ".join(outs) + "\nend\n"]
def _call_printer(self, routine):
declarations = []
code_lines = []
for i, result in enumerate(routine.results):
if isinstance(result, Result):
assign_to = result.result_var
else:
raise CodeGenError("unexpected object in Routine results")
constants, not_supported, jl_expr = self._printer_method_with_settings(
'doprint', dict(human=False), result.expr, assign_to=assign_to)
for obj, v in sorted(constants, key=str):
declarations.append(
"%s = %s\n" % (obj, v))
for obj in sorted(not_supported, key=str):
if isinstance(obj, Function):
name = obj.func
else:
name = obj
declarations.append(
"# unsupported: %s\n" % (name))
code_lines.append("%s\n" % (jl_expr))
return declarations + code_lines
def _indent_code(self, codelines):
# Note that indenting seems to happen twice, first
# statement-by-statement by JuliaPrinter then again here.
p = JuliaCodePrinter({'human': False})
return p.indent_code(codelines)
def dump_jl(self, routines, f, prefix, header=True, empty=True):
self.dump_code(routines, f, prefix, header, empty)
dump_jl.extension = code_extension # type: ignore
dump_jl.__doc__ = CodeGen.dump_code.__doc__
# This list of dump functions is used by CodeGen.write to know which dump
# functions it has to call.
dump_fns = [dump_jl]
class OctaveCodeGen(CodeGen):
"""Generator for Octave code.
The .write() method inherited from CodeGen will output a code file
<prefix>.m.
Octave .m files usually contain one function. That function name should
match the filename (``prefix``). If you pass multiple ``name_expr`` pairs,
the latter ones are presumed to be private functions accessed by the
primary function.
You should only pass inputs to ``argument_sequence``: outputs are ordered
according to their order in ``name_expr``.
"""
code_extension = "m"
def __init__(self, project='project', printer=None):
super().__init__(project)
self.printer = printer or OctaveCodePrinter()
def routine(self, name, expr, argument_sequence, global_vars):
"""Specialized Routine creation for Octave."""
# FIXME: this is probably general enough for other high-level
# languages, perhaps its the C/Fortran one that is specialized!
if is_sequence(expr) and not isinstance(expr, (MatrixBase, MatrixExpr)):
if not expr:
raise ValueError("No expression given")
expressions = Tuple(*expr)
else:
expressions = Tuple(expr)
# local variables
local_vars = {i.label for i in expressions.atoms(Idx)}
# global variables
global_vars = set() if global_vars is None else set(global_vars)
# symbols that should be arguments
old_symbols = expressions.free_symbols - local_vars - global_vars
symbols = set()
for s in old_symbols:
if isinstance(s, Idx):
symbols.update(s.args[1].free_symbols)
elif not isinstance(s, Indexed):
symbols.add(s)
# Octave supports multiple return values
return_vals = []
for (i, expr) in enumerate(expressions):
if isinstance(expr, Equality):
out_arg = expr.lhs
expr = expr.rhs
symbol = out_arg
if isinstance(out_arg, Indexed):
symbol = out_arg.base.label
if not isinstance(out_arg, (Indexed, Symbol, MatrixSymbol)):
raise CodeGenError("Only Indexed, Symbol, or MatrixSymbol "
"can define output arguments.")
return_vals.append(Result(expr, name=symbol, result_var=out_arg))
if not expr.has(symbol):
# this is a pure output: remove from the symbols list, so
# it doesn't become an input.
symbols.remove(symbol)
else:
# we have no name for this output
return_vals.append(Result(expr, name='out%d' % (i+1)))
# setup input argument list
arg_list = []
array_symbols = {}
for array in expressions.atoms(Indexed):
array_symbols[array.base.label] = array
for array in expressions.atoms(MatrixSymbol):
array_symbols[array] = array
for symbol in sorted(symbols, key=str):
arg_list.append(InputArgument(symbol))
if argument_sequence is not None:
# if the user has supplied IndexedBase instances, we'll accept that
new_sequence = []
for arg in argument_sequence:
if isinstance(arg, IndexedBase):
new_sequence.append(arg.label)
else:
new_sequence.append(arg)
argument_sequence = new_sequence
missing = [x for x in arg_list if x.name not in argument_sequence]
if missing:
msg = "Argument list didn't specify: {0} "
msg = msg.format(", ".join([str(m.name) for m in missing]))
raise CodeGenArgumentListError(msg, missing)
# create redundant arguments to produce the requested sequence
name_arg_dict = {x.name: x for x in arg_list}
new_args = []
for symbol in argument_sequence:
try:
new_args.append(name_arg_dict[symbol])
except KeyError:
new_args.append(InputArgument(symbol))
arg_list = new_args
return Routine(name, arg_list, return_vals, local_vars, global_vars)
def _get_header(self):
"""Writes a common header for the generated files."""
code_lines = []
tmp = header_comment % {"version": sympy_version,
"project": self.project}
for line in tmp.splitlines():
if line == '':
code_lines.append("%\n")
else:
code_lines.append("%% %s\n" % line)
return code_lines
def _preprocessor_statements(self, prefix):
return []
def _get_routine_opening(self, routine):
"""Returns the opening statements of the routine."""
code_list = []
code_list.append("function ")
# Outputs
outs = []
for i, result in enumerate(routine.results):
if isinstance(result, Result):
# Note: name not result_var; want `y` not `y(i)` for Indexed
s = self._get_symbol(result.name)
else:
raise CodeGenError("unexpected object in Routine results")
outs.append(s)
if len(outs) > 1:
code_list.append("[" + (", ".join(outs)) + "]")
else:
code_list.append("".join(outs))
code_list.append(" = ")
# Inputs
args = []
for i, arg in enumerate(routine.arguments):
if isinstance(arg, (OutputArgument, InOutArgument)):
raise CodeGenError("Octave: invalid argument of type %s" %
str(type(arg)))
if isinstance(arg, InputArgument):
args.append("%s" % self._get_symbol(arg.name))
args = ", ".join(args)
code_list.append("%s(%s)\n" % (routine.name, args))
code_list = [ "".join(code_list) ]
return code_list
def _declare_arguments(self, routine):
return []
def _declare_globals(self, routine):
if not routine.global_vars:
return []
s = " ".join(sorted([self._get_symbol(g) for g in routine.global_vars]))
return ["global " + s + "\n"]
def _declare_locals(self, routine):
return []
def _get_routine_ending(self, routine):
return ["end\n"]
def _call_printer(self, routine):
declarations = []
code_lines = []
for i, result in enumerate(routine.results):
if isinstance(result, Result):
assign_to = result.result_var
else:
raise CodeGenError("unexpected object in Routine results")
constants, not_supported, oct_expr = self._printer_method_with_settings(
'doprint', dict(human=False), result.expr, assign_to=assign_to)
for obj, v in sorted(constants, key=str):
declarations.append(
" %s = %s; %% constant\n" % (obj, v))
for obj in sorted(not_supported, key=str):
if isinstance(obj, Function):
name = obj.func
else:
name = obj
declarations.append(
" %% unsupported: %s\n" % (name))
code_lines.append("%s\n" % (oct_expr))
return declarations + code_lines
def _indent_code(self, codelines):
return self._printer_method_with_settings(
'indent_code', dict(human=False), codelines)
def dump_m(self, routines, f, prefix, header=True, empty=True, inline=True):
# Note used to call self.dump_code() but we need more control for header
code_lines = self._preprocessor_statements(prefix)
for i, routine in enumerate(routines):
if i > 0:
if empty:
code_lines.append("\n")
code_lines.extend(self._get_routine_opening(routine))
if i == 0:
if routine.name != prefix:
raise ValueError('Octave function name should match prefix')
if header:
code_lines.append("%" + prefix.upper() +
" Autogenerated by sympy\n")
code_lines.append(''.join(self._get_header()))
code_lines.extend(self._declare_arguments(routine))
code_lines.extend(self._declare_globals(routine))
code_lines.extend(self._declare_locals(routine))
if empty:
code_lines.append("\n")
code_lines.extend(self._call_printer(routine))
if empty:
code_lines.append("\n")
code_lines.extend(self._get_routine_ending(routine))
code_lines = self._indent_code(''.join(code_lines))
if code_lines:
f.write(code_lines)
dump_m.extension = code_extension # type: ignore
dump_m.__doc__ = CodeGen.dump_code.__doc__
# This list of dump functions is used by CodeGen.write to know which dump
# functions it has to call.
dump_fns = [dump_m]
class RustCodeGen(CodeGen):
"""Generator for Rust code.
The .write() method inherited from CodeGen will output a code file
<prefix>.rs
"""
code_extension = "rs"
def __init__(self, project="project", printer=None):
super().__init__(project=project)
self.printer = printer or RustCodePrinter()
def routine(self, name, expr, argument_sequence, global_vars):
"""Specialized Routine creation for Rust."""
if is_sequence(expr) and not isinstance(expr, (MatrixBase, MatrixExpr)):
if not expr:
raise ValueError("No expression given")
expressions = Tuple(*expr)
else:
expressions = Tuple(expr)
# local variables
local_vars = {i.label for i in expressions.atoms(Idx)}
# global variables
global_vars = set() if global_vars is None else set(global_vars)
# symbols that should be arguments
symbols = expressions.free_symbols - local_vars - global_vars - expressions.atoms(Indexed)
# Rust supports multiple return values
return_vals = []
output_args = []
for (i, expr) in enumerate(expressions):
if isinstance(expr, Equality):
out_arg = expr.lhs
expr = expr.rhs
symbol = out_arg
if isinstance(out_arg, Indexed):
dims = tuple([ (S.One, dim) for dim in out_arg.shape])
symbol = out_arg.base.label
output_args.append(InOutArgument(symbol, out_arg, expr, dimensions=dims))
if not isinstance(out_arg, (Indexed, Symbol, MatrixSymbol)):
raise CodeGenError("Only Indexed, Symbol, or MatrixSymbol "
"can define output arguments.")
return_vals.append(Result(expr, name=symbol, result_var=out_arg))
if not expr.has(symbol):
# this is a pure output: remove from the symbols list, so
# it doesn't become an input.
symbols.remove(symbol)
else:
# we have no name for this output
return_vals.append(Result(expr, name='out%d' % (i+1)))
# setup input argument list
output_args.sort(key=lambda x: str(x.name))
arg_list = list(output_args)
array_symbols = {}
for array in expressions.atoms(Indexed):
array_symbols[array.base.label] = array
for array in expressions.atoms(MatrixSymbol):
array_symbols[array] = array
for symbol in sorted(symbols, key=str):
arg_list.append(InputArgument(symbol))
if argument_sequence is not None:
# if the user has supplied IndexedBase instances, we'll accept that
new_sequence = []
for arg in argument_sequence:
if isinstance(arg, IndexedBase):
new_sequence.append(arg.label)
else:
new_sequence.append(arg)
argument_sequence = new_sequence
missing = [x for x in arg_list if x.name not in argument_sequence]
if missing:
msg = "Argument list didn't specify: {0} "
msg = msg.format(", ".join([str(m.name) for m in missing]))
raise CodeGenArgumentListError(msg, missing)
# create redundant arguments to produce the requested sequence
name_arg_dict = {x.name: x for x in arg_list}
new_args = []
for symbol in argument_sequence:
try:
new_args.append(name_arg_dict[symbol])
except KeyError:
new_args.append(InputArgument(symbol))
arg_list = new_args
return Routine(name, arg_list, return_vals, local_vars, global_vars)
def _get_header(self):
"""Writes a common header for the generated files."""
code_lines = []
code_lines.append("/*\n")
tmp = header_comment % {"version": sympy_version,
"project": self.project}
for line in tmp.splitlines():
code_lines.append((" *%s" % line.center(76)).rstrip() + "\n")
code_lines.append(" */\n")
return code_lines
def get_prototype(self, routine):
"""Returns a string for the function prototype of the routine.
If the routine has multiple result objects, an CodeGenError is
raised.
See: https://en.wikipedia.org/wiki/Function_prototype
"""
results = [i.get_datatype('Rust') for i in routine.results]
if len(results) == 1:
rstype = " -> " + results[0]
elif len(routine.results) > 1:
rstype = " -> (" + ", ".join(results) + ")"
else:
rstype = ""
type_args = []
for arg in routine.arguments:
name = self.printer.doprint(arg.name)
if arg.dimensions or isinstance(arg, ResultBase):
type_args.append(("*%s" % name, arg.get_datatype('Rust')))
else:
type_args.append((name, arg.get_datatype('Rust')))
arguments = ", ".join([ "%s: %s" % t for t in type_args])
return "fn %s(%s)%s" % (routine.name, arguments, rstype)
def _preprocessor_statements(self, prefix):
code_lines = []
# code_lines.append("use std::f64::consts::*;\n")
return code_lines
def _get_routine_opening(self, routine):
prototype = self.get_prototype(routine)
return ["%s {\n" % prototype]
def _declare_arguments(self, routine):
# arguments are declared in prototype
return []
def _declare_globals(self, routine):
# global variables are not explicitly declared within C functions
return []
def _declare_locals(self, routine):
# loop variables are declared in loop statement
return []
def _call_printer(self, routine):
code_lines = []
declarations = []
returns = []
# Compose a list of symbols to be dereferenced in the function
# body. These are the arguments that were passed by a reference
# pointer, excluding arrays.
dereference = []
for arg in routine.arguments:
if isinstance(arg, ResultBase) and not arg.dimensions:
dereference.append(arg.name)
for i, result in enumerate(routine.results):
if isinstance(result, Result):
assign_to = result.result_var
returns.append(str(result.result_var))
else:
raise CodeGenError("unexpected object in Routine results")
constants, not_supported, rs_expr = self._printer_method_with_settings(
'doprint', dict(human=False), result.expr, assign_to=assign_to)
for name, value in sorted(constants, key=str):
declarations.append("const %s: f64 = %s;\n" % (name, value))
for obj in sorted(not_supported, key=str):
if isinstance(obj, Function):
name = obj.func
else:
name = obj
declarations.append("// unsupported: %s\n" % (name))
code_lines.append("let %s\n" % rs_expr);
if len(returns) > 1:
returns = ['(' + ', '.join(returns) + ')']
returns.append('\n')
return declarations + code_lines + returns
def _get_routine_ending(self, routine):
return ["}\n"]
def dump_rs(self, routines, f, prefix, header=True, empty=True):
self.dump_code(routines, f, prefix, header, empty)
dump_rs.extension = code_extension # type: ignore
dump_rs.__doc__ = CodeGen.dump_code.__doc__
# This list of dump functions is used by CodeGen.write to know which dump
# functions it has to call.
dump_fns = [dump_rs]
def get_code_generator(language, project=None, standard=None, printer = None):
if language == 'C':
if standard is None:
pass
elif standard.lower() == 'c89':
language = 'C89'
elif standard.lower() == 'c99':
language = 'C99'
CodeGenClass = {"C": CCodeGen, "C89": C89CodeGen, "C99": C99CodeGen,
"F95": FCodeGen, "JULIA": JuliaCodeGen,
"OCTAVE": OctaveCodeGen,
"RUST": RustCodeGen}.get(language.upper())
if CodeGenClass is None:
raise ValueError("Language '%s' is not supported." % language)
return CodeGenClass(project, printer)
#
# Friendly functions
#
def codegen(name_expr, language=None, prefix=None, project="project",
to_files=False, header=True, empty=True, argument_sequence=None,
global_vars=None, standard=None, code_gen=None, printer = None):
"""Generate source code for expressions in a given language.
Parameters
==========
name_expr : tuple, or list of tuples
A single (name, expression) tuple or a list of (name, expression)
tuples. Each tuple corresponds to a routine. If the expression is
an equality (an instance of class Equality) the left hand side is
considered an output argument. If expression is an iterable, then
the routine will have multiple outputs.
language : string,
A string that indicates the source code language. This is case
insensitive. Currently, 'C', 'F95' and 'Octave' are supported.
'Octave' generates code compatible with both Octave and Matlab.
prefix : string, optional
A prefix for the names of the files that contain the source code.
Language-dependent suffixes will be appended. If omitted, the name
of the first name_expr tuple is used.
project : string, optional
A project name, used for making unique preprocessor instructions.
[default: "project"]
to_files : bool, optional
When True, the code will be written to one or more files with the
given prefix, otherwise strings with the names and contents of
these files are returned. [default: False]
header : bool, optional
When True, a header is written on top of each source file.
[default: True]
empty : bool, optional
When True, empty lines are used to structure the code.
[default: True]
argument_sequence : iterable, optional
Sequence of arguments for the routine in a preferred order. A
CodeGenError is raised if required arguments are missing.
Redundant arguments are used without warning. If omitted,
arguments will be ordered alphabetically, but with all input
arguments first, and then output or in-out arguments.
global_vars : iterable, optional
Sequence of global variables used by the routine. Variables
listed here will not show up as function arguments.
standard : string
code_gen : CodeGen instance
An instance of a CodeGen subclass. Overrides ``language``.
Examples
========
>>> from sympy.utilities.codegen import codegen
>>> from sympy.abc import x, y, z
>>> [(c_name, c_code), (h_name, c_header)] = codegen(
... ("f", x+y*z), "C89", "test", header=False, empty=False)
>>> print(c_name)
test.c
>>> print(c_code)
#include "test.h"
#include <math.h>
double f(double x, double y, double z) {
double f_result;
f_result = x + y*z;
return f_result;
}
<BLANKLINE>
>>> print(h_name)
test.h
>>> print(c_header)
#ifndef PROJECT__TEST__H
#define PROJECT__TEST__H
double f(double x, double y, double z);
#endif
<BLANKLINE>
Another example using Equality objects to give named outputs. Here the
filename (prefix) is taken from the first (name, expr) pair.
>>> from sympy.abc import f, g
>>> from sympy import Eq
>>> [(c_name, c_code), (h_name, c_header)] = codegen(
... [("myfcn", x + y), ("fcn2", [Eq(f, 2*x), Eq(g, y)])],
... "C99", header=False, empty=False)
>>> print(c_name)
myfcn.c
>>> print(c_code)
#include "myfcn.h"
#include <math.h>
double myfcn(double x, double y) {
double myfcn_result;
myfcn_result = x + y;
return myfcn_result;
}
void fcn2(double x, double y, double *f, double *g) {
(*f) = 2*x;
(*g) = y;
}
<BLANKLINE>
If the generated function(s) will be part of a larger project where various
global variables have been defined, the 'global_vars' option can be used
to remove the specified variables from the function signature
>>> from sympy.utilities.codegen import codegen
>>> from sympy.abc import x, y, z
>>> [(f_name, f_code), header] = codegen(
... ("f", x+y*z), "F95", header=False, empty=False,
... argument_sequence=(x, y), global_vars=(z,))
>>> print(f_code)
REAL*8 function f(x, y)
implicit none
REAL*8, intent(in) :: x
REAL*8, intent(in) :: y
f = x + y*z
end function
<BLANKLINE>
"""
# Initialize the code generator.
if language is None:
if code_gen is None:
raise ValueError("Need either language or code_gen")
else:
if code_gen is not None:
raise ValueError("You cannot specify both language and code_gen.")
code_gen = get_code_generator(language, project, standard, printer)
if isinstance(name_expr[0], str):
# single tuple is given, turn it into a singleton list with a tuple.
name_expr = [name_expr]
if prefix is None:
prefix = name_expr[0][0]
# Construct Routines appropriate for this code_gen from (name, expr) pairs.
routines = []
for name, expr in name_expr:
routines.append(code_gen.routine(name, expr, argument_sequence,
global_vars))
# Write the code.
return code_gen.write(routines, prefix, to_files, header, empty)
def make_routine(name, expr, argument_sequence=None,
global_vars=None, language="F95"):
"""A factory that makes an appropriate Routine from an expression.
Parameters
==========
name : string
The name of this routine in the generated code.
expr : expression or list/tuple of expressions
A SymPy expression that the Routine instance will represent. If
given a list or tuple of expressions, the routine will be
considered to have multiple return values and/or output arguments.
argument_sequence : list or tuple, optional
List arguments for the routine in a preferred order. If omitted,
the results are language dependent, for example, alphabetical order
or in the same order as the given expressions.
global_vars : iterable, optional
Sequence of global variables used by the routine. Variables
listed here will not show up as function arguments.
language : string, optional
Specify a target language. The Routine itself should be
language-agnostic but the precise way one is created, error
checking, etc depend on the language. [default: "F95"].
A decision about whether to use output arguments or return values is made
depending on both the language and the particular mathematical expressions.
For an expression of type Equality, the left hand side is typically made
into an OutputArgument (or perhaps an InOutArgument if appropriate).
Otherwise, typically, the calculated expression is made a return values of
the routine.
Examples
========
>>> from sympy.utilities.codegen import make_routine
>>> from sympy.abc import x, y, f, g
>>> from sympy import Eq
>>> r = make_routine('test', [Eq(f, 2*x), Eq(g, x + y)])
>>> [arg.result_var for arg in r.results]
[]
>>> [arg.name for arg in r.arguments]
[x, y, f, g]
>>> [arg.name for arg in r.result_variables]
[f, g]
>>> r.local_vars
set()
Another more complicated example with a mixture of specified and
automatically-assigned names. Also has Matrix output.
>>> from sympy import Matrix
>>> r = make_routine('fcn', [x*y, Eq(f, 1), Eq(g, x + g), Matrix([[x, 2]])])
>>> [arg.result_var for arg in r.results] # doctest: +SKIP
[result_5397460570204848505]
>>> [arg.expr for arg in r.results]
[x*y]
>>> [arg.name for arg in r.arguments] # doctest: +SKIP
[x, y, f, g, out_8598435338387848786]
We can examine the various arguments more closely:
>>> from sympy.utilities.codegen import (InputArgument, OutputArgument,
... InOutArgument)
>>> [a.name for a in r.arguments if isinstance(a, InputArgument)]
[x, y]
>>> [a.name for a in r.arguments if isinstance(a, OutputArgument)] # doctest: +SKIP
[f, out_8598435338387848786]
>>> [a.expr for a in r.arguments if isinstance(a, OutputArgument)]
[1, Matrix([[x, 2]])]
>>> [a.name for a in r.arguments if isinstance(a, InOutArgument)]
[g]
>>> [a.expr for a in r.arguments if isinstance(a, InOutArgument)]
[g + x]
"""
# initialize a new code generator
code_gen = get_code_generator(language)
return code_gen.routine(name, expr, argument_sequence, global_vars)
|
69e44ef04e875d5436b9e661301da4e4c37f4654d621340b1ddd974b638fbf04 | """
This module provides convenient functions to transform sympy expressions to
lambda functions which can be used to calculate numerical values very fast.
"""
from typing import Any, Dict, Iterable
import builtins
import inspect
import keyword
import textwrap
import linecache
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.core.compatibility import (is_sequence, iterable,
NotIterable)
from sympy.utilities.misc import filldedent
from sympy.utilities.decorator import doctest_depends_on
__doctest_requires__ = {('lambdify',): ['numpy', 'tensorflow']}
# Default namespaces, letting us define translations that can't be defined
# by simple variable maps, like I => 1j
MATH_DEFAULT = {} # type: Dict[str, Any]
MPMATH_DEFAULT = {} # type: Dict[str, Any]
NUMPY_DEFAULT = {"I": 1j} # type: Dict[str, Any]
SCIPY_DEFAULT = {"I": 1j} # type: Dict[str, Any]
CUPY_DEFAULT = {"I": 1j} # type: Dict[str, Any]
TENSORFLOW_DEFAULT = {} # type: Dict[str, Any]
SYMPY_DEFAULT = {} # type: Dict[str, Any]
NUMEXPR_DEFAULT = {} # type: Dict[str, Any]
# These are the namespaces the lambda functions will use.
# These are separate from the names above because they are modified
# throughout this file, whereas the defaults should remain unmodified.
MATH = MATH_DEFAULT.copy()
MPMATH = MPMATH_DEFAULT.copy()
NUMPY = NUMPY_DEFAULT.copy()
SCIPY = SCIPY_DEFAULT.copy()
CUPY = CUPY_DEFAULT.copy()
TENSORFLOW = TENSORFLOW_DEFAULT.copy()
SYMPY = SYMPY_DEFAULT.copy()
NUMEXPR = NUMEXPR_DEFAULT.copy()
# Mappings between sympy and other modules function names.
MATH_TRANSLATIONS = {
"ceiling": "ceil",
"E": "e",
"ln": "log",
}
# NOTE: This dictionary is reused in Function._eval_evalf to allow subclasses
# of Function to automatically evalf.
MPMATH_TRANSLATIONS = {
"Abs": "fabs",
"elliptic_k": "ellipk",
"elliptic_f": "ellipf",
"elliptic_e": "ellipe",
"elliptic_pi": "ellippi",
"ceiling": "ceil",
"chebyshevt": "chebyt",
"chebyshevu": "chebyu",
"E": "e",
"I": "j",
"ln": "log",
#"lowergamma":"lower_gamma",
"oo": "inf",
#"uppergamma":"upper_gamma",
"LambertW": "lambertw",
"MutableDenseMatrix": "matrix",
"ImmutableDenseMatrix": "matrix",
"conjugate": "conj",
"dirichlet_eta": "altzeta",
"Ei": "ei",
"Shi": "shi",
"Chi": "chi",
"Si": "si",
"Ci": "ci",
"RisingFactorial": "rf",
"FallingFactorial": "ff",
"betainc_regularized": "betainc",
}
NUMPY_TRANSLATIONS = {} # type: Dict[str, str]
SCIPY_TRANSLATIONS = {} # type: Dict[str, str]
CUPY_TRANSLATIONS = {} # type: Dict[str, str]
TENSORFLOW_TRANSLATIONS = {} # type: Dict[str, str]
NUMEXPR_TRANSLATIONS = {} # type: Dict[str, str]
# Available modules:
MODULES = {
"math": (MATH, MATH_DEFAULT, MATH_TRANSLATIONS, ("from math import *",)),
"mpmath": (MPMATH, MPMATH_DEFAULT, MPMATH_TRANSLATIONS, ("from mpmath import *",)),
"numpy": (NUMPY, NUMPY_DEFAULT, NUMPY_TRANSLATIONS, ("import numpy; from numpy import *; from numpy.linalg import *",)),
"scipy": (SCIPY, SCIPY_DEFAULT, SCIPY_TRANSLATIONS, ("import numpy; import scipy; from scipy import *; from scipy.special import *",)),
"cupy": (CUPY, CUPY_DEFAULT, CUPY_TRANSLATIONS, ("import cupy",)),
"tensorflow": (TENSORFLOW, TENSORFLOW_DEFAULT, TENSORFLOW_TRANSLATIONS, ("import tensorflow",)),
"sympy": (SYMPY, SYMPY_DEFAULT, {}, (
"from sympy.functions import *",
"from sympy.matrices import *",
"from sympy import Integral, pi, oo, nan, zoo, E, I",)),
"numexpr" : (NUMEXPR, NUMEXPR_DEFAULT, NUMEXPR_TRANSLATIONS,
("import_module('numexpr')", )),
}
def _import(module, reload=False):
"""
Creates a global translation dictionary for module.
The argument module has to be one of the following strings: "math",
"mpmath", "numpy", "sympy", "tensorflow".
These dictionaries map names of python functions to their equivalent in
other modules.
"""
# Required despite static analysis claiming it is not used
from sympy.external import import_module # noqa:F401
try:
namespace, namespace_default, translations, import_commands = MODULES[
module]
except KeyError:
raise NameError(
"'%s' module can't be used for lambdification" % module)
# Clear namespace or exit
if namespace != namespace_default:
# The namespace was already generated, don't do it again if not forced.
if reload:
namespace.clear()
namespace.update(namespace_default)
else:
return
for import_command in import_commands:
if import_command.startswith('import_module'):
module = eval(import_command)
if module is not None:
namespace.update(module.__dict__)
continue
else:
try:
exec(import_command, {}, namespace)
continue
except ImportError:
pass
raise ImportError(
"can't import '%s' with '%s' command" % (module, import_command))
# Add translated names to namespace
for sympyname, translation in translations.items():
namespace[sympyname] = namespace[translation]
# For computing the modulus of a sympy expression we use the builtin abs
# function, instead of the previously used fabs function for all
# translation modules. This is because the fabs function in the math
# module does not accept complex valued arguments. (see issue 9474). The
# only exception, where we don't use the builtin abs function is the
# mpmath translation module, because mpmath.fabs returns mpf objects in
# contrast to abs().
if 'Abs' not in namespace:
namespace['Abs'] = abs
# Used for dynamically generated filenames that are inserted into the
# linecache.
_lambdify_generated_counter = 1
@doctest_depends_on(modules=('numpy', 'tensorflow', ), python_version=(3,))
def lambdify(args: Iterable, expr, modules=None, printer=None, use_imps=True,
dummify=False):
"""Convert a SymPy expression into a function that allows for fast
numeric evaluation.
.. warning::
This function uses ``exec``, and thus shouldn't be used on
unsanitized input.
.. versionchanged:: 1.7.0
Passing a set for the *args* parameter is deprecated as sets are
unordered. Use an ordered iterable such as a list or tuple.
Explanation
===========
For example, to convert the SymPy expression ``sin(x) + cos(x)`` to an
equivalent NumPy function that numerically evaluates it:
>>> from sympy import sin, cos, symbols, lambdify
>>> import numpy as np
>>> x = symbols('x')
>>> expr = sin(x) + cos(x)
>>> expr
sin(x) + cos(x)
>>> f = lambdify(x, expr, 'numpy')
>>> a = np.array([1, 2])
>>> f(a)
[1.38177329 0.49315059]
The primary purpose of this function is to provide a bridge from SymPy
expressions to numerical libraries such as NumPy, SciPy, NumExpr, mpmath,
and tensorflow. In general, SymPy functions do not work with objects from
other libraries, such as NumPy arrays, and functions from numeric
libraries like NumPy or mpmath do not work on SymPy expressions.
``lambdify`` bridges the two by converting a SymPy expression to an
equivalent numeric function.
The basic workflow with ``lambdify`` is to first create a SymPy expression
representing whatever mathematical function you wish to evaluate. This
should be done using only SymPy functions and expressions. Then, use
``lambdify`` to convert this to an equivalent function for numerical
evaluation. For instance, above we created ``expr`` using the SymPy symbol
``x`` and SymPy functions ``sin`` and ``cos``, then converted it to an
equivalent NumPy function ``f``, and called it on a NumPy array ``a``.
Parameters
==========
args : List[Symbol]
A variable or a list of variables whose nesting represents the
nesting of the arguments that will be passed to the function.
Variables can be symbols, undefined functions, or matrix symbols.
>>> from sympy import Eq
>>> from sympy.abc import x, y, z
The list of variables should match the structure of how the
arguments will be passed to the function. Simply enclose the
parameters as they will be passed in a list.
To call a function like ``f(x)`` then ``[x]``
should be the first argument to ``lambdify``; for this
case a single ``x`` can also be used:
>>> f = lambdify(x, x + 1)
>>> f(1)
2
>>> f = lambdify([x], x + 1)
>>> f(1)
2
To call a function like ``f(x, y)`` then ``[x, y]`` will
be the first argument of the ``lambdify``:
>>> f = lambdify([x, y], x + y)
>>> f(1, 1)
2
To call a function with a single 3-element tuple like
``f((x, y, z))`` then ``[(x, y, z)]`` will be the first
argument of the ``lambdify``:
>>> f = lambdify([(x, y, z)], Eq(z**2, x**2 + y**2))
>>> f((3, 4, 5))
True
If two args will be passed and the first is a scalar but
the second is a tuple with two arguments then the items
in the list should match that structure:
>>> f = lambdify([x, (y, z)], x + y + z)
>>> f(1, (2, 3))
6
expr : Expr
An expression, list of expressions, or matrix to be evaluated.
Lists may be nested.
If the expression is a list, the output will also be a list.
>>> f = lambdify(x, [x, [x + 1, x + 2]])
>>> f(1)
[1, [2, 3]]
If it is a matrix, an array will be returned (for the NumPy module).
>>> from sympy import Matrix
>>> f = lambdify(x, Matrix([x, x + 1]))
>>> f(1)
[[1]
[2]]
Note that the argument order here (variables then expression) is used
to emulate the Python ``lambda`` keyword. ``lambdify(x, expr)`` works
(roughly) like ``lambda x: expr``
(see :ref:`lambdify-how-it-works` below).
modules : str, optional
Specifies the numeric library to use.
If not specified, *modules* defaults to:
- ``["scipy", "numpy"]`` if SciPy is installed
- ``["numpy"]`` if only NumPy is installed
- ``["math", "mpmath", "sympy"]`` if neither is installed.
That is, SymPy functions are replaced as far as possible by
either ``scipy`` or ``numpy`` functions if available, and Python's
standard library ``math``, or ``mpmath`` functions otherwise.
*modules* can be one of the following types:
- The strings ``"math"``, ``"mpmath"``, ``"numpy"``, ``"numexpr"``,
``"scipy"``, ``"sympy"``, or ``"tensorflow"``. This uses the
corresponding printer and namespace mapping for that module.
- A module (e.g., ``math``). This uses the global namespace of the
module. If the module is one of the above known modules, it will
also use the corresponding printer and namespace mapping
(i.e., ``modules=numpy`` is equivalent to ``modules="numpy"``).
- A dictionary that maps names of SymPy functions to arbitrary
functions
(e.g., ``{'sin': custom_sin}``).
- A list that contains a mix of the arguments above, with higher
priority given to entries appearing first
(e.g., to use the NumPy module but override the ``sin`` function
with a custom version, you can use
``[{'sin': custom_sin}, 'numpy']``).
dummify : bool, optional
Whether or not the variables in the provided expression that are not
valid Python identifiers are substituted with dummy symbols.
This allows for undefined functions like ``Function('f')(t)`` to be
supplied as arguments. By default, the variables are only dummified
if they are not valid Python identifiers.
Set ``dummify=True`` to replace all arguments with dummy symbols
(if ``args`` is not a string) - for example, to ensure that the
arguments do not redefine any built-in names.
Examples
========
>>> from sympy.utilities.lambdify import implemented_function
>>> from sympy import sqrt, sin, Matrix
>>> from sympy import Function
>>> from sympy.abc import w, x, y, z
>>> f = lambdify(x, x**2)
>>> f(2)
4
>>> f = lambdify((x, y, z), [z, y, x])
>>> f(1,2,3)
[3, 2, 1]
>>> f = lambdify(x, sqrt(x))
>>> f(4)
2.0
>>> f = lambdify((x, y), sin(x*y)**2)
>>> f(0, 5)
0.0
>>> row = lambdify((x, y), Matrix((x, x + y)).T, modules='sympy')
>>> row(1, 2)
Matrix([[1, 3]])
``lambdify`` can be used to translate SymPy expressions into mpmath
functions. This may be preferable to using ``evalf`` (which uses mpmath on
the backend) in some cases.
>>> f = lambdify(x, sin(x), 'mpmath')
>>> f(1)
0.8414709848078965
Tuple arguments are handled and the lambdified function should
be called with the same type of arguments as were used to create
the function:
>>> f = lambdify((x, (y, z)), x + y)
>>> f(1, (2, 4))
3
The ``flatten`` function can be used to always work with flattened
arguments:
>>> from sympy.utilities.iterables import flatten
>>> args = w, (x, (y, z))
>>> vals = 1, (2, (3, 4))
>>> f = lambdify(flatten(args), w + x + y + z)
>>> f(*flatten(vals))
10
Functions present in ``expr`` can also carry their own numerical
implementations, in a callable attached to the ``_imp_`` attribute. This
can be used with undefined functions using the ``implemented_function``
factory:
>>> f = implemented_function(Function('f'), lambda x: x+1)
>>> func = lambdify(x, f(x))
>>> func(4)
5
``lambdify`` always prefers ``_imp_`` implementations to implementations
in other namespaces, unless the ``use_imps`` input parameter is False.
Usage with Tensorflow:
>>> import tensorflow as tf
>>> from sympy import Max, sin, lambdify
>>> from sympy.abc import x
>>> f = Max(x, sin(x))
>>> func = lambdify(x, f, 'tensorflow')
After tensorflow v2, eager execution is enabled by default.
If you want to get the compatible result across tensorflow v1 and v2
as same as this tutorial, run this line.
>>> tf.compat.v1.enable_eager_execution()
If you have eager execution enabled, you can get the result out
immediately as you can use numpy.
If you pass tensorflow objects, you may get an ``EagerTensor``
object instead of value.
>>> result = func(tf.constant(1.0))
>>> print(result)
tf.Tensor(1.0, shape=(), dtype=float32)
>>> print(result.__class__)
<class 'tensorflow.python.framework.ops.EagerTensor'>
You can use ``.numpy()`` to get the numpy value of the tensor.
>>> result.numpy()
1.0
>>> var = tf.Variable(2.0)
>>> result = func(var) # also works for tf.Variable and tf.Placeholder
>>> result.numpy()
2.0
And it works with any shape array.
>>> tensor = tf.constant([[1.0, 2.0], [3.0, 4.0]])
>>> result = func(tensor)
>>> result.numpy()
[[1. 2.]
[3. 4.]]
Notes
=====
- For functions involving large array calculations, numexpr can provide a
significant speedup over numpy. Please note that the available functions
for numexpr are more limited than numpy but can be expanded with
``implemented_function`` and user defined subclasses of Function. If
specified, numexpr may be the only option in modules. The official list
of numexpr functions can be found at:
https://numexpr.readthedocs.io/en/latest/user_guide.html#supported-functions
- In previous versions of SymPy, ``lambdify`` replaced ``Matrix`` with
``numpy.matrix`` by default. As of SymPy 1.0 ``numpy.array`` is the
default. To get the old default behavior you must pass in
``[{'ImmutableDenseMatrix': numpy.matrix}, 'numpy']`` to the
``modules`` kwarg.
>>> from sympy import lambdify, Matrix
>>> from sympy.abc import x, y
>>> import numpy
>>> array2mat = [{'ImmutableDenseMatrix': numpy.matrix}, 'numpy']
>>> f = lambdify((x, y), Matrix([x, y]), modules=array2mat)
>>> f(1, 2)
[[1]
[2]]
- In the above examples, the generated functions can accept scalar
values or numpy arrays as arguments. However, in some cases
the generated function relies on the input being a numpy array:
>>> from sympy import Piecewise
>>> from sympy.testing.pytest import ignore_warnings
>>> f = lambdify(x, Piecewise((x, x <= 1), (1/x, x > 1)), "numpy")
>>> with ignore_warnings(RuntimeWarning):
... f(numpy.array([-1, 0, 1, 2]))
[-1. 0. 1. 0.5]
>>> f(0)
Traceback (most recent call last):
...
ZeroDivisionError: division by zero
In such cases, the input should be wrapped in a numpy array:
>>> with ignore_warnings(RuntimeWarning):
... float(f(numpy.array([0])))
0.0
Or if numpy functionality is not required another module can be used:
>>> f = lambdify(x, Piecewise((x, x <= 1), (1/x, x > 1)), "math")
>>> f(0)
0
.. _lambdify-how-it-works:
How it works
============
When using this function, it helps a great deal to have an idea of what it
is doing. At its core, lambdify is nothing more than a namespace
translation, on top of a special printer that makes some corner cases work
properly.
To understand lambdify, first we must properly understand how Python
namespaces work. Say we had two files. One called ``sin_cos_sympy.py``,
with
.. code:: python
# sin_cos_sympy.py
from sympy import sin, cos
def sin_cos(x):
return sin(x) + cos(x)
and one called ``sin_cos_numpy.py`` with
.. code:: python
# sin_cos_numpy.py
from numpy import sin, cos
def sin_cos(x):
return sin(x) + cos(x)
The two files define an identical function ``sin_cos``. However, in the
first file, ``sin`` and ``cos`` are defined as the SymPy ``sin`` and
``cos``. In the second, they are defined as the NumPy versions.
If we were to import the first file and use the ``sin_cos`` function, we
would get something like
>>> from sin_cos_sympy import sin_cos # doctest: +SKIP
>>> sin_cos(1) # doctest: +SKIP
cos(1) + sin(1)
On the other hand, if we imported ``sin_cos`` from the second file, we
would get
>>> from sin_cos_numpy import sin_cos # doctest: +SKIP
>>> sin_cos(1) # doctest: +SKIP
1.38177329068
In the first case we got a symbolic output, because it used the symbolic
``sin`` and ``cos`` functions from SymPy. In the second, we got a numeric
result, because ``sin_cos`` used the numeric ``sin`` and ``cos`` functions
from NumPy. But notice that the versions of ``sin`` and ``cos`` that were
used was not inherent to the ``sin_cos`` function definition. Both
``sin_cos`` definitions are exactly the same. Rather, it was based on the
names defined at the module where the ``sin_cos`` function was defined.
The key point here is that when function in Python references a name that
is not defined in the function, that name is looked up in the "global"
namespace of the module where that function is defined.
Now, in Python, we can emulate this behavior without actually writing a
file to disk using the ``exec`` function. ``exec`` takes a string
containing a block of Python code, and a dictionary that should contain
the global variables of the module. It then executes the code "in" that
dictionary, as if it were the module globals. The following is equivalent
to the ``sin_cos`` defined in ``sin_cos_sympy.py``:
>>> import sympy
>>> module_dictionary = {'sin': sympy.sin, 'cos': sympy.cos}
>>> exec('''
... def sin_cos(x):
... return sin(x) + cos(x)
... ''', module_dictionary)
>>> sin_cos = module_dictionary['sin_cos']
>>> sin_cos(1)
cos(1) + sin(1)
and similarly with ``sin_cos_numpy``:
>>> import numpy
>>> module_dictionary = {'sin': numpy.sin, 'cos': numpy.cos}
>>> exec('''
... def sin_cos(x):
... return sin(x) + cos(x)
... ''', module_dictionary)
>>> sin_cos = module_dictionary['sin_cos']
>>> sin_cos(1)
1.38177329068
So now we can get an idea of how ``lambdify`` works. The name "lambdify"
comes from the fact that we can think of something like ``lambdify(x,
sin(x) + cos(x), 'numpy')`` as ``lambda x: sin(x) + cos(x)``, where
``sin`` and ``cos`` come from the ``numpy`` namespace. This is also why
the symbols argument is first in ``lambdify``, as opposed to most SymPy
functions where it comes after the expression: to better mimic the
``lambda`` keyword.
``lambdify`` takes the input expression (like ``sin(x) + cos(x)``) and
1. Converts it to a string
2. Creates a module globals dictionary based on the modules that are
passed in (by default, it uses the NumPy module)
3. Creates the string ``"def func({vars}): return {expr}"``, where ``{vars}`` is the
list of variables separated by commas, and ``{expr}`` is the string
created in step 1., then ``exec``s that string with the module globals
namespace and returns ``func``.
In fact, functions returned by ``lambdify`` support inspection. So you can
see exactly how they are defined by using ``inspect.getsource``, or ``??`` if you
are using IPython or the Jupyter notebook.
>>> f = lambdify(x, sin(x) + cos(x))
>>> import inspect
>>> print(inspect.getsource(f))
def _lambdifygenerated(x):
return (sin(x) + cos(x))
This shows us the source code of the function, but not the namespace it
was defined in. We can inspect that by looking at the ``__globals__``
attribute of ``f``:
>>> f.__globals__['sin']
<ufunc 'sin'>
>>> f.__globals__['cos']
<ufunc 'cos'>
>>> f.__globals__['sin'] is numpy.sin
True
This shows us that ``sin`` and ``cos`` in the namespace of ``f`` will be
``numpy.sin`` and ``numpy.cos``.
Note that there are some convenience layers in each of these steps, but at
the core, this is how ``lambdify`` works. Step 1 is done using the
``LambdaPrinter`` printers defined in the printing module (see
:mod:`sympy.printing.lambdarepr`). This allows different SymPy expressions
to define how they should be converted to a string for different modules.
You can change which printer ``lambdify`` uses by passing a custom printer
in to the ``printer`` argument.
Step 2 is augmented by certain translations. There are default
translations for each module, but you can provide your own by passing a
list to the ``modules`` argument. For instance,
>>> def mysin(x):
... print('taking the sin of', x)
... return numpy.sin(x)
...
>>> f = lambdify(x, sin(x), [{'sin': mysin}, 'numpy'])
>>> f(1)
taking the sin of 1
0.8414709848078965
The globals dictionary is generated from the list by merging the
dictionary ``{'sin': mysin}`` and the module dictionary for NumPy. The
merging is done so that earlier items take precedence, which is why
``mysin`` is used above instead of ``numpy.sin``.
If you want to modify the way ``lambdify`` works for a given function, it
is usually easiest to do so by modifying the globals dictionary as such.
In more complicated cases, it may be necessary to create and pass in a
custom printer.
Finally, step 3 is augmented with certain convenience operations, such as
the addition of a docstring.
Understanding how ``lambdify`` works can make it easier to avoid certain
gotchas when using it. For instance, a common mistake is to create a
lambdified function for one module (say, NumPy), and pass it objects from
another (say, a SymPy expression).
For instance, say we create
>>> from sympy.abc import x
>>> f = lambdify(x, x + 1, 'numpy')
Now if we pass in a NumPy array, we get that array plus 1
>>> import numpy
>>> a = numpy.array([1, 2])
>>> f(a)
[2 3]
But what happens if you make the mistake of passing in a SymPy expression
instead of a NumPy array:
>>> f(x + 1)
x + 2
This worked, but it was only by accident. Now take a different lambdified
function:
>>> from sympy import sin
>>> g = lambdify(x, x + sin(x), 'numpy')
This works as expected on NumPy arrays:
>>> g(a)
[1.84147098 2.90929743]
But if we try to pass in a SymPy expression, it fails
>>> try:
... g(x + 1)
... # NumPy release after 1.17 raises TypeError instead of
... # AttributeError
... except (AttributeError, TypeError):
... raise AttributeError() # doctest: +IGNORE_EXCEPTION_DETAIL
Traceback (most recent call last):
...
AttributeError:
Now, let's look at what happened. The reason this fails is that ``g``
calls ``numpy.sin`` on the input expression, and ``numpy.sin`` does not
know how to operate on a SymPy object. **As a general rule, NumPy
functions do not know how to operate on SymPy expressions, and SymPy
functions do not know how to operate on NumPy arrays. This is why lambdify
exists: to provide a bridge between SymPy and NumPy.**
However, why is it that ``f`` did work? That's because ``f`` doesn't call
any functions, it only adds 1. So the resulting function that is created,
``def _lambdifygenerated(x): return x + 1`` does not depend on the globals
namespace it is defined in. Thus it works, but only by accident. A future
version of ``lambdify`` may remove this behavior.
Be aware that certain implementation details described here may change in
future versions of SymPy. The API of passing in custom modules and
printers will not change, but the details of how a lambda function is
created may change. However, the basic idea will remain the same, and
understanding it will be helpful to understanding the behavior of
lambdify.
**In general: you should create lambdified functions for one module (say,
NumPy), and only pass it input types that are compatible with that module
(say, NumPy arrays).** Remember that by default, if the ``module``
argument is not provided, ``lambdify`` creates functions using the NumPy
and SciPy namespaces.
"""
from sympy.core.symbol import Symbol
# If the user hasn't specified any modules, use what is available.
if modules is None:
try:
_import("scipy")
except ImportError:
try:
_import("numpy")
except ImportError:
# Use either numpy (if available) or python.math where possible.
# XXX: This leads to different behaviour on different systems and
# might be the reason for irreproducible errors.
modules = ["math", "mpmath", "sympy"]
else:
modules = ["numpy"]
else:
modules = ["numpy", "scipy"]
# Get the needed namespaces.
namespaces = []
# First find any function implementations
if use_imps:
namespaces.append(_imp_namespace(expr))
# Check for dict before iterating
if isinstance(modules, (dict, str)) or not hasattr(modules, '__iter__'):
namespaces.append(modules)
else:
# consistency check
if _module_present('numexpr', modules) and len(modules) > 1:
raise TypeError("numexpr must be the only item in 'modules'")
namespaces += list(modules)
# fill namespace with first having highest priority
namespace = {} # type: Dict[str, Any]
for m in namespaces[::-1]:
buf = _get_namespace(m)
namespace.update(buf)
if hasattr(expr, "atoms"):
#Try if you can extract symbols from the expression.
#Move on if expr.atoms in not implemented.
syms = expr.atoms(Symbol)
for term in syms:
namespace.update({str(term): term})
if printer is None:
if _module_present('mpmath', namespaces):
from sympy.printing.pycode import MpmathPrinter as Printer # type: ignore
elif _module_present('scipy', namespaces):
from sympy.printing.numpy import SciPyPrinter as Printer # type: ignore
elif _module_present('numpy', namespaces):
from sympy.printing.numpy import NumPyPrinter as Printer # type: ignore
elif _module_present('cupy', namespaces):
from sympy.printing.numpy import CuPyPrinter as Printer # type: ignore
elif _module_present('numexpr', namespaces):
from sympy.printing.lambdarepr import NumExprPrinter as Printer # type: ignore
elif _module_present('tensorflow', namespaces):
from sympy.printing.tensorflow import TensorflowPrinter as Printer # type: ignore
elif _module_present('sympy', namespaces):
from sympy.printing.pycode import SymPyPrinter as Printer # type: ignore
else:
from sympy.printing.pycode import PythonCodePrinter as Printer # type: ignore
user_functions = {}
for m in namespaces[::-1]:
if isinstance(m, dict):
for k in m:
user_functions[k] = k
printer = Printer({'fully_qualified_modules': False, 'inline': True,
'allow_unknown_functions': True,
'user_functions': user_functions})
if isinstance(args, set):
SymPyDeprecationWarning(
feature="The list of arguments is a `set`. This leads to unpredictable results",
useinstead=": Convert set into list or tuple",
issue=20013,
deprecated_since_version="1.6.3"
).warn()
# Get the names of the args, for creating a docstring
if not iterable(args):
args = (args,)
names = []
# Grab the callers frame, for getting the names by inspection (if needed)
callers_local_vars = inspect.currentframe().f_back.f_locals.items() # type: ignore
for n, var in enumerate(args):
if hasattr(var, 'name'):
names.append(var.name)
else:
# It's an iterable. Try to get name by inspection of calling frame.
name_list = [var_name for var_name, var_val in callers_local_vars
if var_val is var]
if len(name_list) == 1:
names.append(name_list[0])
else:
# Cannot infer name with certainty. arg_# will have to do.
names.append('arg_' + str(n))
# Create the function definition code and execute it
funcname = '_lambdifygenerated'
if _module_present('tensorflow', namespaces):
funcprinter = _TensorflowEvaluatorPrinter(printer, dummify) # type: _EvaluatorPrinter
else:
funcprinter = _EvaluatorPrinter(printer, dummify)
funcstr = funcprinter.doprint(funcname, args, expr)
# Collect the module imports from the code printers.
imp_mod_lines = []
for mod, keys in (getattr(printer, 'module_imports', None) or {}).items():
for k in keys:
if k not in namespace:
ln = "from %s import %s" % (mod, k)
try:
exec(ln, {}, namespace)
except ImportError:
# Tensorflow 2.0 has issues with importing a specific
# function from its submodule.
# https://github.com/tensorflow/tensorflow/issues/33022
ln = "%s = %s.%s" % (k, mod, k)
exec(ln, {}, namespace)
imp_mod_lines.append(ln)
# Provide lambda expression with builtins, and compatible implementation of range
namespace.update({'builtins':builtins, 'range':range})
funclocals = {} # type: Dict[str, Any]
global _lambdify_generated_counter
filename = '<lambdifygenerated-%s>' % _lambdify_generated_counter
_lambdify_generated_counter += 1
c = compile(funcstr, filename, 'exec')
exec(c, namespace, funclocals)
# mtime has to be None or else linecache.checkcache will remove it
linecache.cache[filename] = (len(funcstr), None, funcstr.splitlines(True), filename) # type: ignore
func = funclocals[funcname]
# Apply the docstring
sig = "func({})".format(", ".join(str(i) for i in names))
sig = textwrap.fill(sig, subsequent_indent=' '*8)
expr_str = str(expr)
if len(expr_str) > 78:
expr_str = textwrap.wrap(expr_str, 75)[0] + '...'
func.__doc__ = (
"Created with lambdify. Signature:\n\n"
"{sig}\n\n"
"Expression:\n\n"
"{expr}\n\n"
"Source code:\n\n"
"{src}\n\n"
"Imported modules:\n\n"
"{imp_mods}"
).format(sig=sig, expr=expr_str, src=funcstr, imp_mods='\n'.join(imp_mod_lines))
return func
def _module_present(modname, modlist):
if modname in modlist:
return True
for m in modlist:
if hasattr(m, '__name__') and m.__name__ == modname:
return True
return False
def _get_namespace(m):
"""
This is used by _lambdify to parse its arguments.
"""
if isinstance(m, str):
_import(m)
return MODULES[m][0]
elif isinstance(m, dict):
return m
elif hasattr(m, "__dict__"):
return m.__dict__
else:
raise TypeError("Argument must be either a string, dict or module but it is: %s" % m)
def lambdastr(args, expr, printer=None, dummify=None):
"""
Returns a string that can be evaluated to a lambda function.
Examples
========
>>> from sympy.abc import x, y, z
>>> from sympy.utilities.lambdify import lambdastr
>>> lambdastr(x, x**2)
'lambda x: (x**2)'
>>> lambdastr((x,y,z), [z,y,x])
'lambda x,y,z: ([z, y, x])'
Although tuples may not appear as arguments to lambda in Python 3,
lambdastr will create a lambda function that will unpack the original
arguments so that nested arguments can be handled:
>>> lambdastr((x, (y, z)), x + y)
'lambda _0,_1: (lambda x,y,z: (x + y))(_0,_1[0],_1[1])'
"""
# Transforming everything to strings.
from sympy.matrices import DeferredVector
from sympy import Dummy, sympify, Symbol, Function, flatten, Derivative, Basic
if printer is not None:
if inspect.isfunction(printer):
lambdarepr = printer
else:
if inspect.isclass(printer):
lambdarepr = lambda expr: printer().doprint(expr)
else:
lambdarepr = lambda expr: printer.doprint(expr)
else:
#XXX: This has to be done here because of circular imports
from sympy.printing.lambdarepr import lambdarepr
def sub_args(args, dummies_dict):
if isinstance(args, str):
return args
elif isinstance(args, DeferredVector):
return str(args)
elif iterable(args):
dummies = flatten([sub_args(a, dummies_dict) for a in args])
return ",".join(str(a) for a in dummies)
else:
# replace these with Dummy symbols
if isinstance(args, (Function, Symbol, Derivative)):
dummies = Dummy()
dummies_dict.update({args : dummies})
return str(dummies)
else:
return str(args)
def sub_expr(expr, dummies_dict):
expr = sympify(expr)
# dict/tuple are sympified to Basic
if isinstance(expr, Basic):
expr = expr.xreplace(dummies_dict)
# list is not sympified to Basic
elif isinstance(expr, list):
expr = [sub_expr(a, dummies_dict) for a in expr]
return expr
# Transform args
def isiter(l):
return iterable(l, exclude=(str, DeferredVector, NotIterable))
def flat_indexes(iterable):
n = 0
for el in iterable:
if isiter(el):
for ndeep in flat_indexes(el):
yield (n,) + ndeep
else:
yield (n,)
n += 1
if dummify is None:
dummify = any(isinstance(a, Basic) and
a.atoms(Function, Derivative) for a in (
args if isiter(args) else [args]))
if isiter(args) and any(isiter(i) for i in args):
dum_args = [str(Dummy(str(i))) for i in range(len(args))]
indexed_args = ','.join([
dum_args[ind[0]] + ''.join(["[%s]" % k for k in ind[1:]])
for ind in flat_indexes(args)])
lstr = lambdastr(flatten(args), expr, printer=printer, dummify=dummify)
return 'lambda %s: (%s)(%s)' % (','.join(dum_args), lstr, indexed_args)
dummies_dict = {}
if dummify:
args = sub_args(args, dummies_dict)
else:
if isinstance(args, str):
pass
elif iterable(args, exclude=DeferredVector):
args = ",".join(str(a) for a in args)
# Transform expr
if dummify:
if isinstance(expr, str):
pass
else:
expr = sub_expr(expr, dummies_dict)
expr = lambdarepr(expr)
return "lambda %s: (%s)" % (args, expr)
class _EvaluatorPrinter:
def __init__(self, printer=None, dummify=False):
self._dummify = dummify
#XXX: This has to be done here because of circular imports
from sympy.printing.lambdarepr import LambdaPrinter
if printer is None:
printer = LambdaPrinter()
if inspect.isfunction(printer):
self._exprrepr = printer
else:
if inspect.isclass(printer):
printer = printer()
self._exprrepr = printer.doprint
#if hasattr(printer, '_print_Symbol'):
# symbolrepr = printer._print_Symbol
#if hasattr(printer, '_print_Dummy'):
# dummyrepr = printer._print_Dummy
# Used to print the generated function arguments in a standard way
self._argrepr = LambdaPrinter().doprint
def doprint(self, funcname, args, expr):
"""Returns the function definition code as a string."""
from sympy import Dummy
funcbody = []
if not iterable(args):
args = [args]
argstrs, expr = self._preprocess(args, expr)
# Generate argument unpacking and final argument list
funcargs = []
unpackings = []
for argstr in argstrs:
if iterable(argstr):
funcargs.append(self._argrepr(Dummy()))
unpackings.extend(self._print_unpacking(argstr, funcargs[-1]))
else:
funcargs.append(argstr)
funcsig = 'def {}({}):'.format(funcname, ', '.join(funcargs))
# Wrap input arguments before unpacking
funcbody.extend(self._print_funcargwrapping(funcargs))
funcbody.extend(unpackings)
funcbody.append('return ({})'.format(self._exprrepr(expr)))
funclines = [funcsig]
funclines.extend(' ' + line for line in funcbody)
return '\n'.join(funclines) + '\n'
@classmethod
def _is_safe_ident(cls, ident):
return isinstance(ident, str) and ident.isidentifier() \
and not keyword.iskeyword(ident)
def _preprocess(self, args, expr):
"""Preprocess args, expr to replace arguments that do not map
to valid Python identifiers.
Returns string form of args, and updated expr.
"""
from sympy import Dummy, Function, flatten, Derivative, ordered, Basic
from sympy.matrices import DeferredVector
from sympy.core.symbol import uniquely_named_symbol
from sympy.core.expr import Expr
# Args of type Dummy can cause name collisions with args
# of type Symbol. Force dummify of everything in this
# situation.
dummify = self._dummify or any(
isinstance(arg, Dummy) for arg in flatten(args))
argstrs = [None]*len(args)
for arg, i in reversed(list(ordered(zip(args, range(len(args)))))):
if iterable(arg):
s, expr = self._preprocess(arg, expr)
elif isinstance(arg, DeferredVector):
s = str(arg)
elif isinstance(arg, Basic) and arg.is_symbol:
s = self._argrepr(arg)
if dummify or not self._is_safe_ident(s):
dummy = Dummy()
if isinstance(expr, Expr):
dummy = uniquely_named_symbol(
dummy.name, expr, modify=lambda s: '_' + s)
s = self._argrepr(dummy)
expr = self._subexpr(expr, {arg: dummy})
elif dummify or isinstance(arg, (Function, Derivative)):
dummy = Dummy()
s = self._argrepr(dummy)
expr = self._subexpr(expr, {arg: dummy})
else:
s = str(arg)
argstrs[i] = s
return argstrs, expr
def _subexpr(self, expr, dummies_dict):
from sympy.matrices import DeferredVector
from sympy import sympify
expr = sympify(expr)
xreplace = getattr(expr, 'xreplace', None)
if xreplace is not None:
expr = xreplace(dummies_dict)
else:
if isinstance(expr, DeferredVector):
pass
elif isinstance(expr, dict):
k = [self._subexpr(sympify(a), dummies_dict) for a in expr.keys()]
v = [self._subexpr(sympify(a), dummies_dict) for a in expr.values()]
expr = dict(zip(k, v))
elif isinstance(expr, tuple):
expr = tuple(self._subexpr(sympify(a), dummies_dict) for a in expr)
elif isinstance(expr, list):
expr = [self._subexpr(sympify(a), dummies_dict) for a in expr]
return expr
def _print_funcargwrapping(self, args):
"""Generate argument wrapping code.
args is the argument list of the generated function (strings).
Return value is a list of lines of code that will be inserted at
the beginning of the function definition.
"""
return []
def _print_unpacking(self, unpackto, arg):
"""Generate argument unpacking code.
arg is the function argument to be unpacked (a string), and
unpackto is a list or nested lists of the variable names (strings) to
unpack to.
"""
def unpack_lhs(lvalues):
return '[{}]'.format(', '.join(
unpack_lhs(val) if iterable(val) else val for val in lvalues))
return ['{} = {}'.format(unpack_lhs(unpackto), arg)]
class _TensorflowEvaluatorPrinter(_EvaluatorPrinter):
def _print_unpacking(self, lvalues, rvalue):
"""Generate argument unpacking code.
This method is used when the input value is not interable,
but can be indexed (see issue #14655).
"""
from sympy import flatten
def flat_indexes(elems):
n = 0
for el in elems:
if iterable(el):
for ndeep in flat_indexes(el):
yield (n,) + ndeep
else:
yield (n,)
n += 1
indexed = ', '.join('{}[{}]'.format(rvalue, ']['.join(map(str, ind)))
for ind in flat_indexes(lvalues))
return ['[{}] = [{}]'.format(', '.join(flatten(lvalues)), indexed)]
def _imp_namespace(expr, namespace=None):
""" Return namespace dict with function implementations
We need to search for functions in anything that can be thrown at
us - that is - anything that could be passed as ``expr``. Examples
include sympy expressions, as well as tuples, lists and dicts that may
contain sympy expressions.
Parameters
----------
expr : object
Something passed to lambdify, that will generate valid code from
``str(expr)``.
namespace : None or mapping
Namespace to fill. None results in new empty dict
Returns
-------
namespace : dict
dict with keys of implemented function names within ``expr`` and
corresponding values being the numerical implementation of
function
Examples
========
>>> from sympy.abc import x
>>> from sympy.utilities.lambdify import implemented_function, _imp_namespace
>>> from sympy import Function
>>> f = implemented_function(Function('f'), lambda x: x+1)
>>> g = implemented_function(Function('g'), lambda x: x*10)
>>> namespace = _imp_namespace(f(g(x)))
>>> sorted(namespace.keys())
['f', 'g']
"""
# Delayed import to avoid circular imports
from sympy.core.function import FunctionClass
if namespace is None:
namespace = {}
# tuples, lists, dicts are valid expressions
if is_sequence(expr):
for arg in expr:
_imp_namespace(arg, namespace)
return namespace
elif isinstance(expr, dict):
for key, val in expr.items():
# functions can be in dictionary keys
_imp_namespace(key, namespace)
_imp_namespace(val, namespace)
return namespace
# sympy expressions may be Functions themselves
func = getattr(expr, 'func', None)
if isinstance(func, FunctionClass):
imp = getattr(func, '_imp_', None)
if imp is not None:
name = expr.func.__name__
if name in namespace and namespace[name] != imp:
raise ValueError('We found more than one '
'implementation with name '
'"%s"' % name)
namespace[name] = imp
# and / or they may take Functions as arguments
if hasattr(expr, 'args'):
for arg in expr.args:
_imp_namespace(arg, namespace)
return namespace
def implemented_function(symfunc, implementation):
""" Add numerical ``implementation`` to function ``symfunc``.
``symfunc`` can be an ``UndefinedFunction`` instance, or a name string.
In the latter case we create an ``UndefinedFunction`` instance with that
name.
Be aware that this is a quick workaround, not a general method to create
special symbolic functions. If you want to create a symbolic function to be
used by all the machinery of SymPy you should subclass the ``Function``
class.
Parameters
----------
symfunc : ``str`` or ``UndefinedFunction`` instance
If ``str``, then create new ``UndefinedFunction`` with this as
name. If ``symfunc`` is an Undefined function, create a new function
with the same name and the implemented function attached.
implementation : callable
numerical implementation to be called by ``evalf()`` or ``lambdify``
Returns
-------
afunc : sympy.FunctionClass instance
function with attached implementation
Examples
========
>>> from sympy.abc import x
>>> from sympy.utilities.lambdify import lambdify, implemented_function
>>> f = implemented_function('f', lambda x: x+1)
>>> lam_f = lambdify(x, f(x))
>>> lam_f(4)
5
"""
# Delayed import to avoid circular imports
from sympy.core.function import UndefinedFunction
# if name, create function to hold implementation
kwargs = {}
if isinstance(symfunc, UndefinedFunction):
kwargs = symfunc._kwargs
symfunc = symfunc.__name__
if isinstance(symfunc, str):
# Keyword arguments to UndefinedFunction are added as attributes to
# the created class.
symfunc = UndefinedFunction(
symfunc, _imp_=staticmethod(implementation), **kwargs)
elif not isinstance(symfunc, UndefinedFunction):
raise ValueError(filldedent('''
symfunc should be either a string or
an UndefinedFunction instance.'''))
return symfunc
|
a2b0f5dd41740980b082107d425d52728e5c659a1a80bda5e6b811869916a471 | from sympy.core import S
from .pycode import PythonCodePrinter, _known_functions_math, _print_known_const, _print_known_func, _unpack_integral_limits
from .codeprinter import CodePrinter
_not_in_numpy = 'erf erfc factorial gamma loggamma'.split()
_in_numpy = [(k, v) for k, v in _known_functions_math.items() if k not in _not_in_numpy]
_known_functions_numpy = dict(_in_numpy, **{
'acos': 'arccos',
'acosh': 'arccosh',
'asin': 'arcsin',
'asinh': 'arcsinh',
'atan': 'arctan',
'atan2': 'arctan2',
'atanh': 'arctanh',
'exp2': 'exp2',
'sign': 'sign',
'logaddexp': 'logaddexp',
'logaddexp2': 'logaddexp2',
})
_known_constants_numpy = {
'Exp1': 'e',
'Pi': 'pi',
'EulerGamma': 'euler_gamma',
'NaN': 'nan',
'Infinity': 'PINF',
'NegativeInfinity': 'NINF'
}
_numpy_known_functions = {k: 'numpy.' + v for k, v in _known_functions_numpy.items()}
_numpy_known_constants = {k: 'numpy.' + v for k, v in _known_constants_numpy.items()}
class NumPyPrinter(PythonCodePrinter):
"""
Numpy printer which handles vectorized piecewise functions,
logical operators, etc.
"""
_module = 'numpy'
_kf = _numpy_known_functions
_kc = _numpy_known_constants
def __init__(self, settings=None):
"""
`settings` is passed to CodePrinter.__init__()
`module` specifies the array module to use, currently 'NumPy' or 'CuPy'
"""
self.language = "Python with {}".format(self._module)
self.printmethod = "_{}code".format(self._module)
self._kf = {**PythonCodePrinter._kf, **self._kf}
super().__init__(settings=settings)
def _print_seq(self, seq):
"General sequence printer: converts to tuple"
# Print tuples here instead of lists because numba supports
# tuples in nopython mode.
delimiter=', '
return '({},)'.format(delimiter.join(self._print(item) for item in seq))
def _print_MatMul(self, expr):
"Matrix multiplication printer"
if expr.as_coeff_matrices()[0] is not S.One:
expr_list = expr.as_coeff_matrices()[1]+[(expr.as_coeff_matrices()[0])]
return '({})'.format(').dot('.join(self._print(i) for i in expr_list))
return '({})'.format(').dot('.join(self._print(i) for i in expr.args))
def _print_MatPow(self, expr):
"Matrix power printer"
return '{}({}, {})'.format(self._module_format(self._module + '.linalg.matrix_power'),
self._print(expr.args[0]), self._print(expr.args[1]))
def _print_Inverse(self, expr):
"Matrix inverse printer"
return '{}({})'.format(self._module_format(self._module + '.linalg.inv'),
self._print(expr.args[0]))
def _print_DotProduct(self, expr):
# DotProduct allows any shape order, but numpy.dot does matrix
# multiplication, so we have to make sure it gets 1 x n by n x 1.
arg1, arg2 = expr.args
if arg1.shape[0] != 1:
arg1 = arg1.T
if arg2.shape[1] != 1:
arg2 = arg2.T
return "%s(%s, %s)" % (self._module_format(self._module + '.dot'),
self._print(arg1),
self._print(arg2))
def _print_MatrixSolve(self, expr):
return "%s(%s, %s)" % (self._module_format(self._module + '.linalg.solve'),
self._print(expr.matrix),
self._print(expr.vector))
def _print_ZeroMatrix(self, expr):
return '{}({})'.format(self._module_format(self._module + '.zeros'),
self._print(expr.shape))
def _print_OneMatrix(self, expr):
return '{}({})'.format(self._module_format(self._module + '.ones'),
self._print(expr.shape))
def _print_FunctionMatrix(self, expr):
from sympy.core.function import Lambda
from sympy.abc import i, j
lamda = expr.lamda
if not isinstance(lamda, Lambda):
lamda = Lambda((i, j), lamda(i, j))
return '{}(lambda {}: {}, {})'.format(self._module_format(self._module + '.fromfunction'),
', '.join(self._print(arg) for arg in lamda.args[0]),
self._print(lamda.args[1]), self._print(expr.shape))
def _print_HadamardProduct(self, expr):
func = self._module_format(self._module + '.multiply')
return ''.join('{}({}, '.format(func, self._print(arg)) \
for arg in expr.args[:-1]) + "{}{}".format(self._print(expr.args[-1]),
')' * (len(expr.args) - 1))
def _print_KroneckerProduct(self, expr):
func = self._module_format(self._module + '.kron')
return ''.join('{}({}, '.format(func, self._print(arg)) \
for arg in expr.args[:-1]) + "{}{}".format(self._print(expr.args[-1]),
')' * (len(expr.args) - 1))
def _print_Adjoint(self, expr):
return '{}({}({}))'.format(
self._module_format(self._module + '.conjugate'),
self._module_format(self._module + '.transpose'),
self._print(expr.args[0]))
def _print_DiagonalOf(self, expr):
vect = '{}({})'.format(
self._module_format(self._module + '.diag'),
self._print(expr.arg))
return '{}({}, (-1, 1))'.format(
self._module_format(self._module + '.reshape'), vect)
def _print_DiagMatrix(self, expr):
return '{}({})'.format(self._module_format(self._module + '.diagflat'),
self._print(expr.args[0]))
def _print_DiagonalMatrix(self, expr):
return '{}({}, {}({}, {}))'.format(self._module_format(self._module + '.multiply'),
self._print(expr.arg), self._module_format(self._module + '.eye'),
self._print(expr.shape[0]), self._print(expr.shape[1]))
def _print_Piecewise(self, expr):
"Piecewise function printer"
exprs = '[{}]'.format(','.join(self._print(arg.expr) for arg in expr.args))
conds = '[{}]'.format(','.join(self._print(arg.cond) for arg in expr.args))
# If [default_value, True] is a (expr, cond) sequence in a Piecewise object
# it will behave the same as passing the 'default' kwarg to select()
# *as long as* it is the last element in expr.args.
# If this is not the case, it may be triggered prematurely.
return '{}({}, {}, default={})'.format(
self._module_format(self._module + '.select'), conds, exprs,
self._print(S.NaN))
def _print_Relational(self, expr):
"Relational printer for Equality and Unequality"
op = {
'==' :'equal',
'!=' :'not_equal',
'<' :'less',
'<=' :'less_equal',
'>' :'greater',
'>=' :'greater_equal',
}
if expr.rel_op in op:
lhs = self._print(expr.lhs)
rhs = self._print(expr.rhs)
return '{op}({lhs}, {rhs})'.format(op=self._module_format(self._module + '.'+op[expr.rel_op]),
lhs=lhs, rhs=rhs)
return super()._print_Relational(expr)
def _print_And(self, expr):
"Logical And printer"
# We have to override LambdaPrinter because it uses Python 'and' keyword.
# If LambdaPrinter didn't define it, we could use StrPrinter's
# version of the function and add 'logical_and' to NUMPY_TRANSLATIONS.
return '{}.reduce(({}))'.format(self._module_format(self._module + '.logical_and'), ','.join(self._print(i) for i in expr.args))
def _print_Or(self, expr):
"Logical Or printer"
# We have to override LambdaPrinter because it uses Python 'or' keyword.
# If LambdaPrinter didn't define it, we could use StrPrinter's
# version of the function and add 'logical_or' to NUMPY_TRANSLATIONS.
return '{}.reduce(({}))'.format(self._module_format(self._module + '.logical_or'), ','.join(self._print(i) for i in expr.args))
def _print_Not(self, expr):
"Logical Not printer"
# We have to override LambdaPrinter because it uses Python 'not' keyword.
# If LambdaPrinter didn't define it, we would still have to define our
# own because StrPrinter doesn't define it.
return '{}({})'.format(self._module_format(self._module + '.logical_not'), ','.join(self._print(i) for i in expr.args))
def _print_Pow(self, expr, rational=False):
# XXX Workaround for negative integer power error
from sympy.core.power import Pow
if expr.exp.is_integer and expr.exp.is_negative:
expr = Pow(expr.base, expr.exp.evalf(), evaluate=False)
return self._hprint_Pow(expr, rational=rational, sqrt=self._module + '.sqrt')
def _print_Min(self, expr):
return '{}(({}), axis=0)'.format(self._module_format(self._module + '.amin'), ','.join(self._print(i) for i in expr.args))
def _print_Max(self, expr):
return '{}(({}), axis=0)'.format(self._module_format(self._module + '.amax'), ','.join(self._print(i) for i in expr.args))
def _print_arg(self, expr):
return "%s(%s)" % (self._module_format(self._module + '.angle'), self._print(expr.args[0]))
def _print_im(self, expr):
return "%s(%s)" % (self._module_format(self._module + '.imag'), self._print(expr.args[0]))
def _print_Mod(self, expr):
return "%s(%s)" % (self._module_format(self._module + '.mod'), ', '.join(
map(lambda arg: self._print(arg), expr.args)))
def _print_re(self, expr):
return "%s(%s)" % (self._module_format(self._module + '.real'), self._print(expr.args[0]))
def _print_sinc(self, expr):
return "%s(%s)" % (self._module_format(self._module + '.sinc'), self._print(expr.args[0]/S.Pi))
def _print_MatrixBase(self, expr):
func = self.known_functions.get(expr.__class__.__name__, None)
if func is None:
func = self._module_format(self._module + '.array')
return "%s(%s)" % (func, self._print(expr.tolist()))
def _print_Identity(self, expr):
shape = expr.shape
if all([dim.is_Integer for dim in shape]):
return "%s(%s)" % (self._module_format(self._module + '.eye'), self._print(expr.shape[0]))
else:
raise NotImplementedError("Symbolic matrix dimensions are not yet supported for identity matrices")
def _print_BlockMatrix(self, expr):
return '{}({})'.format(self._module_format(self._module + '.block'),
self._print(expr.args[0].tolist()))
def _print_ArrayTensorProduct(self, expr):
array_list = [j for i, arg in enumerate(expr.args) for j in
(self._print(arg), "[%i, %i]" % (2*i, 2*i+1))]
return "%s(%s)" % (self._module_format(self._module + '.einsum'), ", ".join(array_list))
def _print_ArrayContraction(self, expr):
from ..tensor.array.expressions.array_expressions import ArrayTensorProduct
base = expr.expr
contraction_indices = expr.contraction_indices
if not contraction_indices:
return self._print(base)
if isinstance(base, ArrayTensorProduct):
counter = 0
d = {j: min(i) for i in contraction_indices for j in i}
indices = []
for rank_arg in base.subranks:
lindices = []
for i in range(rank_arg):
if counter in d:
lindices.append(d[counter])
else:
lindices.append(counter)
counter += 1
indices.append(lindices)
elems = ["%s, %s" % (self._print(arg), ind) for arg, ind in zip(base.args, indices)]
return "%s(%s)" % (
self._module_format(self._module + '.einsum'),
", ".join(elems)
)
raise NotImplementedError()
def _print_ArrayDiagonal(self, expr):
diagonal_indices = list(expr.diagonal_indices)
if len(diagonal_indices) > 1:
# TODO: this should be handled in sympy.codegen.array_utils,
# possibly by creating the possibility of unfolding the
# ArrayDiagonal object into nested ones. Same reasoning for
# the array contraction.
raise NotImplementedError
if len(diagonal_indices[0]) != 2:
raise NotImplementedError
return "%s(%s, 0, axis1=%s, axis2=%s)" % (
self._module_format("numpy.diagonal"),
self._print(expr.expr),
diagonal_indices[0][0],
diagonal_indices[0][1],
)
def _print_PermuteDims(self, expr):
return "%s(%s, %s)" % (
self._module_format("numpy.transpose"),
self._print(expr.expr),
self._print(expr.permutation.array_form),
)
def _print_ArrayAdd(self, expr):
return self._expand_fold_binary_op(self._module + '.add', expr.args)
_print_lowergamma = CodePrinter._print_not_supported
_print_uppergamma = CodePrinter._print_not_supported
_print_fresnelc = CodePrinter._print_not_supported
_print_fresnels = CodePrinter._print_not_supported
for func in _numpy_known_functions:
setattr(NumPyPrinter, f'_print_{func}', _print_known_func)
for const in _numpy_known_constants:
setattr(NumPyPrinter, f'_print_{const}', _print_known_const)
_known_functions_scipy_special = {
'erf': 'erf',
'erfc': 'erfc',
'besselj': 'jv',
'bessely': 'yv',
'besseli': 'iv',
'besselk': 'kv',
'cosm1': 'cosm1',
'factorial': 'factorial',
'gamma': 'gamma',
'loggamma': 'gammaln',
'digamma': 'psi',
'RisingFactorial': 'poch',
'jacobi': 'eval_jacobi',
'gegenbauer': 'eval_gegenbauer',
'chebyshevt': 'eval_chebyt',
'chebyshevu': 'eval_chebyu',
'legendre': 'eval_legendre',
'hermite': 'eval_hermite',
'laguerre': 'eval_laguerre',
'assoc_laguerre': 'eval_genlaguerre',
'beta': 'beta',
'LambertW' : 'lambertw',
}
_known_constants_scipy_constants = {
'GoldenRatio': 'golden_ratio',
'Pi': 'pi',
}
_scipy_known_functions = {k : "scipy.special." + v for k, v in _known_functions_scipy_special.items()}
_scipy_known_constants = {k : "scipy.constants." + v for k, v in _known_constants_scipy_constants.items()}
class SciPyPrinter(NumPyPrinter):
_kf = {**NumPyPrinter._kf, **_scipy_known_functions}
_kc = {**NumPyPrinter._kc, **_scipy_known_constants}
def __init__(self, settings=None):
super().__init__(settings=settings)
self.language = "Python with SciPy and NumPy"
def _print_SparseMatrix(self, expr):
i, j, data = [], [], []
for (r, c), v in expr._smat.items():
i.append(r)
j.append(c)
data.append(v)
return "{name}(({data}, ({i}, {j})), shape={shape})".format(
name=self._module_format('scipy.sparse.coo_matrix'),
data=data, i=i, j=j, shape=expr.shape
)
_print_ImmutableSparseMatrix = _print_SparseMatrix
# SciPy's lpmv has a different order of arguments from assoc_legendre
def _print_assoc_legendre(self, expr):
return "{0}({2}, {1}, {3})".format(
self._module_format('scipy.special.lpmv'),
self._print(expr.args[0]),
self._print(expr.args[1]),
self._print(expr.args[2]))
def _print_lowergamma(self, expr):
return "{0}({2})*{1}({2}, {3})".format(
self._module_format('scipy.special.gamma'),
self._module_format('scipy.special.gammainc'),
self._print(expr.args[0]),
self._print(expr.args[1]))
def _print_uppergamma(self, expr):
return "{0}({2})*{1}({2}, {3})".format(
self._module_format('scipy.special.gamma'),
self._module_format('scipy.special.gammaincc'),
self._print(expr.args[0]),
self._print(expr.args[1]))
def _print_betainc(self, expr):
betainc = self._module_format('scipy.special.betainc')
beta = self._module_format('scipy.special.beta')
args = [self._print(arg) for arg in expr.args]
return f"({betainc}({args[0]}, {args[1]}, {args[3]}) - {betainc}({args[0]}, {args[1]}, {args[2]})) \
* {beta}({args[0]}, {args[1]})"
def _print_betainc_regularized(self, expr):
return "{0}({1}, {2}, {4}) - {0}({1}, {2}, {3})".format(
self._module_format('scipy.special.betainc'),
self._print(expr.args[0]),
self._print(expr.args[1]),
self._print(expr.args[2]),
self._print(expr.args[3]))
def _print_fresnels(self, expr):
return "{}({})[0]".format(
self._module_format("scipy.special.fresnel"),
self._print(expr.args[0]))
def _print_fresnelc(self, expr):
return "{}({})[1]".format(
self._module_format("scipy.special.fresnel"),
self._print(expr.args[0]))
def _print_airyai(self, expr):
return "{}({})[0]".format(
self._module_format("scipy.special.airy"),
self._print(expr.args[0]))
def _print_airyaiprime(self, expr):
return "{}({})[1]".format(
self._module_format("scipy.special.airy"),
self._print(expr.args[0]))
def _print_airybi(self, expr):
return "{}({})[2]".format(
self._module_format("scipy.special.airy"),
self._print(expr.args[0]))
def _print_airybiprime(self, expr):
return "{}({})[3]".format(
self._module_format("scipy.special.airy"),
self._print(expr.args[0]))
def _print_Integral(self, e):
integration_vars, limits = _unpack_integral_limits(e)
if len(limits) == 1:
# nicer (but not necessary) to prefer quad over nquad for 1D case
module_str = self._module_format("scipy.integrate.quad")
limit_str = "%s, %s" % tuple(map(self._print, limits[0]))
else:
module_str = self._module_format("scipy.integrate.nquad")
limit_str = "({})".format(", ".join(
"(%s, %s)" % tuple(map(self._print, l)) for l in limits))
return "{}(lambda {}: {}, {})[0]".format(
module_str,
", ".join(map(self._print, integration_vars)),
self._print(e.args[0]),
limit_str)
for func in _scipy_known_functions:
setattr(SciPyPrinter, f'_print_{func}', _print_known_func)
for const in _scipy_known_constants:
setattr(SciPyPrinter, f'_print_{const}', _print_known_const)
_cupy_known_functions = {k : "cupy." + v for k, v in _known_functions_numpy.items()}
_cupy_known_constants = {k : "cupy." + v for k, v in _known_constants_numpy.items()}
class CuPyPrinter(NumPyPrinter):
"""
CuPy printer which handles vectorized piecewise functions,
logical operators, etc.
"""
_module = 'cupy'
_kf = _cupy_known_functions
_kc = _cupy_known_constants
def __init__(self, settings=None):
super().__init__(settings=settings)
for func in _cupy_known_functions:
setattr(CuPyPrinter, f'_print_{func}', _print_known_func)
for const in _cupy_known_constants:
setattr(CuPyPrinter, f'_print_{const}', _print_known_const)
|
f7780de58055842f19daa8247d7217cd1897b03e7445592ba1be03ecf74bafc6 | """
Python code printers
This module contains python code printers for plain python as well as NumPy & SciPy enabled code.
"""
from collections import defaultdict
from itertools import chain
from sympy.core import S
from .precedence import precedence
from .codeprinter import CodePrinter
_kw_py2and3 = {
'and', 'as', 'assert', 'break', 'class', 'continue', 'def', 'del', 'elif',
'else', 'except', 'finally', 'for', 'from', 'global', 'if', 'import', 'in',
'is', 'lambda', 'not', 'or', 'pass', 'raise', 'return', 'try', 'while',
'with', 'yield', 'None' # 'None' is actually not in Python 2's keyword.kwlist
}
_kw_only_py2 = {'exec', 'print'}
_kw_only_py3 = {'False', 'nonlocal', 'True'}
_known_functions = {
'Abs': 'abs',
}
_known_functions_math = {
'acos': 'acos',
'acosh': 'acosh',
'asin': 'asin',
'asinh': 'asinh',
'atan': 'atan',
'atan2': 'atan2',
'atanh': 'atanh',
'ceiling': 'ceil',
'cos': 'cos',
'cosh': 'cosh',
'erf': 'erf',
'erfc': 'erfc',
'exp': 'exp',
'expm1': 'expm1',
'factorial': 'factorial',
'floor': 'floor',
'gamma': 'gamma',
'hypot': 'hypot',
'loggamma': 'lgamma',
'log': 'log',
'ln': 'log',
'log10': 'log10',
'log1p': 'log1p',
'log2': 'log2',
'sin': 'sin',
'sinh': 'sinh',
'Sqrt': 'sqrt',
'tan': 'tan',
'tanh': 'tanh'
} # Not used from ``math``: [copysign isclose isfinite isinf isnan ldexp frexp pow modf
# radians trunc fmod fsum gcd degrees fabs]
_known_constants_math = {
'Exp1': 'e',
'Pi': 'pi',
'E': 'e'
# Only in python >= 3.5:
# 'Infinity': 'inf',
# 'NaN': 'nan'
}
def _print_known_func(self, expr):
known = self.known_functions[expr.__class__.__name__]
return '{name}({args})'.format(name=self._module_format(known),
args=', '.join(map(lambda arg: self._print(arg), expr.args)))
def _print_known_const(self, expr):
known = self.known_constants[expr.__class__.__name__]
return self._module_format(known)
class AbstractPythonCodePrinter(CodePrinter):
printmethod = "_pythoncode"
language = "Python"
reserved_words = _kw_py2and3.union(_kw_only_py3)
modules = None # initialized to a set in __init__
tab = ' '
_kf = dict(chain(
_known_functions.items(),
[(k, 'math.' + v) for k, v in _known_functions_math.items()]
))
_kc = {k: 'math.'+v for k, v in _known_constants_math.items()}
_operators = {'and': 'and', 'or': 'or', 'not': 'not'}
_default_settings = dict(
CodePrinter._default_settings,
user_functions={},
precision=17,
inline=True,
fully_qualified_modules=True,
contract=False,
standard='python3',
)
def __init__(self, settings=None):
super().__init__(settings)
# Python standard handler
std = self._settings['standard']
if std is None:
import sys
std = 'python{}'.format(sys.version_info.major)
if std not in ('python2', 'python3'):
raise ValueError('Unrecognized python standard : {}'.format(std))
self.standard = std
self.module_imports = defaultdict(set)
# Known functions and constants handler
self.known_functions = dict(self._kf, **(settings or {}).get(
'user_functions', {}))
self.known_constants = dict(self._kc, **(settings or {}).get(
'user_constants', {}))
def _declare_number_const(self, name, value):
return "%s = %s" % (name, value)
def _module_format(self, fqn, register=True):
parts = fqn.split('.')
if register and len(parts) > 1:
self.module_imports['.'.join(parts[:-1])].add(parts[-1])
if self._settings['fully_qualified_modules']:
return fqn
else:
return fqn.split('(')[0].split('[')[0].split('.')[-1]
def _format_code(self, lines):
return lines
def _get_statement(self, codestring):
return "{}".format(codestring)
def _get_comment(self, text):
return " # {}".format(text)
def _expand_fold_binary_op(self, op, args):
"""
This method expands a fold on binary operations.
``functools.reduce`` is an example of a folded operation.
For example, the expression
`A + B + C + D`
is folded into
`((A + B) + C) + D`
"""
if len(args) == 1:
return self._print(args[0])
else:
return "%s(%s, %s)" % (
self._module_format(op),
self._expand_fold_binary_op(op, args[:-1]),
self._print(args[-1]),
)
def _expand_reduce_binary_op(self, op, args):
"""
This method expands a reductin on binary operations.
Notice: this is NOT the same as ``functools.reduce``.
For example, the expression
`A + B + C + D`
is reduced into:
`(A + B) + (C + D)`
"""
if len(args) == 1:
return self._print(args[0])
else:
N = len(args)
Nhalf = N // 2
return "%s(%s, %s)" % (
self._module_format(op),
self._expand_reduce_binary_op(args[:Nhalf]),
self._expand_reduce_binary_op(args[Nhalf:]),
)
def _get_einsum_string(self, subranks, contraction_indices):
letters = self._get_letter_generator_for_einsum()
contraction_string = ""
counter = 0
d = {j: min(i) for i in contraction_indices for j in i}
indices = []
for rank_arg in subranks:
lindices = []
for i in range(rank_arg):
if counter in d:
lindices.append(d[counter])
else:
lindices.append(counter)
counter += 1
indices.append(lindices)
mapping = {}
letters_free = []
letters_dum = []
for i in indices:
for j in i:
if j not in mapping:
l = next(letters)
mapping[j] = l
else:
l = mapping[j]
contraction_string += l
if j in d:
if l not in letters_dum:
letters_dum.append(l)
else:
letters_free.append(l)
contraction_string += ","
contraction_string = contraction_string[:-1]
return contraction_string, letters_free, letters_dum
def _print_NaN(self, expr):
return "float('nan')"
def _print_Infinity(self, expr):
return "float('inf')"
def _print_NegativeInfinity(self, expr):
return "float('-inf')"
def _print_ComplexInfinity(self, expr):
return self._print_NaN(expr)
def _print_Mod(self, expr):
PREC = precedence(expr)
return ('{} % {}'.format(*map(lambda x: self.parenthesize(x, PREC), expr.args)))
def _print_Piecewise(self, expr):
result = []
i = 0
for arg in expr.args:
e = arg.expr
c = arg.cond
if i == 0:
result.append('(')
result.append('(')
result.append(self._print(e))
result.append(')')
result.append(' if ')
result.append(self._print(c))
result.append(' else ')
i += 1
result = result[:-1]
if result[-1] == 'True':
result = result[:-2]
result.append(')')
else:
result.append(' else None)')
return ''.join(result)
def _print_Relational(self, expr):
"Relational printer for Equality and Unequality"
op = {
'==' :'equal',
'!=' :'not_equal',
'<' :'less',
'<=' :'less_equal',
'>' :'greater',
'>=' :'greater_equal',
}
if expr.rel_op in op:
lhs = self._print(expr.lhs)
rhs = self._print(expr.rhs)
return '({lhs} {op} {rhs})'.format(op=expr.rel_op, lhs=lhs, rhs=rhs)
return super()._print_Relational(expr)
def _print_ITE(self, expr):
from sympy.functions.elementary.piecewise import Piecewise
return self._print(expr.rewrite(Piecewise))
def _print_Sum(self, expr):
loops = (
'for {i} in range({a}, {b}+1)'.format(
i=self._print(i),
a=self._print(a),
b=self._print(b))
for i, a, b in expr.limits)
return '(builtins.sum({function} {loops}))'.format(
function=self._print(expr.function),
loops=' '.join(loops))
def _print_ImaginaryUnit(self, expr):
return '1j'
def _print_KroneckerDelta(self, expr):
a, b = expr.args
return '(1 if {a} == {b} else 0)'.format(
a = self._print(a),
b = self._print(b)
)
def _print_MatrixBase(self, expr):
name = expr.__class__.__name__
func = self.known_functions.get(name, name)
return "%s(%s)" % (func, self._print(expr.tolist()))
_print_SparseMatrix = \
_print_MutableSparseMatrix = \
_print_ImmutableSparseMatrix = \
_print_Matrix = \
_print_DenseMatrix = \
_print_MutableDenseMatrix = \
_print_ImmutableMatrix = \
_print_ImmutableDenseMatrix = \
lambda self, expr: self._print_MatrixBase(expr)
def _indent_codestring(self, codestring):
return '\n'.join([self.tab + line for line in codestring.split('\n')])
def _print_FunctionDefinition(self, fd):
body = '\n'.join(map(lambda arg: self._print(arg), fd.body))
return "def {name}({parameters}):\n{body}".format(
name=self._print(fd.name),
parameters=', '.join([self._print(var.symbol) for var in fd.parameters]),
body=self._indent_codestring(body)
)
def _print_While(self, whl):
body = '\n'.join(map(lambda arg: self._print(arg), whl.body))
return "while {cond}:\n{body}".format(
cond=self._print(whl.condition),
body=self._indent_codestring(body)
)
def _print_Declaration(self, decl):
return '%s = %s' % (
self._print(decl.variable.symbol),
self._print(decl.variable.value)
)
def _print_Return(self, ret):
arg, = ret.args
return 'return %s' % self._print(arg)
def _print_Print(self, prnt):
print_args = ', '.join(map(lambda arg: self._print(arg), prnt.print_args))
if prnt.format_string != None: # Must be '!= None', cannot be 'is not None'
print_args = '{} % ({})'.format(
self._print(prnt.format_string), print_args)
if prnt.file != None: # Must be '!= None', cannot be 'is not None'
print_args += ', file=%s' % self._print(prnt.file)
if self.standard == 'python2':
return 'print %s' % print_args
return 'print(%s)' % print_args
def _print_Stream(self, strm):
if str(strm.name) == 'stdout':
return self._module_format('sys.stdout')
elif str(strm.name) == 'stderr':
return self._module_format('sys.stderr')
else:
return self._print(strm.name)
def _print_NoneToken(self, arg):
return 'None'
def _hprint_Pow(self, expr, rational=False, sqrt='math.sqrt'):
"""Printing helper function for ``Pow``
Notes
=====
This only preprocesses the ``sqrt`` as math formatter
Examples
========
>>> from sympy.functions import sqrt
>>> from sympy.printing.pycode import PythonCodePrinter
>>> from sympy.abc import x
Python code printer automatically looks up ``math.sqrt``.
>>> printer = PythonCodePrinter({'standard':'python3'})
>>> printer._hprint_Pow(sqrt(x), rational=True)
'x**(1/2)'
>>> printer._hprint_Pow(sqrt(x), rational=False)
'math.sqrt(x)'
>>> printer._hprint_Pow(1/sqrt(x), rational=True)
'x**(-1/2)'
>>> printer._hprint_Pow(1/sqrt(x), rational=False)
'1/math.sqrt(x)'
Using sqrt from numpy or mpmath
>>> printer._hprint_Pow(sqrt(x), sqrt='numpy.sqrt')
'numpy.sqrt(x)'
>>> printer._hprint_Pow(sqrt(x), sqrt='mpmath.sqrt')
'mpmath.sqrt(x)'
See Also
========
sympy.printing.str.StrPrinter._print_Pow
"""
PREC = precedence(expr)
if expr.exp == S.Half and not rational:
func = self._module_format(sqrt)
arg = self._print(expr.base)
return '{func}({arg})'.format(func=func, arg=arg)
if expr.is_commutative:
if -expr.exp is S.Half and not rational:
func = self._module_format(sqrt)
num = self._print(S.One)
arg = self._print(expr.base)
return "{num}/{func}({arg})".format(
num=num, func=func, arg=arg)
base_str = self.parenthesize(expr.base, PREC, strict=False)
exp_str = self.parenthesize(expr.exp, PREC, strict=False)
return "{}**{}".format(base_str, exp_str)
class PythonCodePrinter(AbstractPythonCodePrinter):
def _print_sign(self, e):
return '(0.0 if {e} == 0 else {f}(1, {e}))'.format(
f=self._module_format('math.copysign'), e=self._print(e.args[0]))
def _print_Not(self, expr):
PREC = precedence(expr)
return self._operators['not'] + self.parenthesize(expr.args[0], PREC)
def _print_Indexed(self, expr):
base = expr.args[0]
index = expr.args[1:]
return "{}[{}]".format(str(base), ", ".join([self._print(ind) for ind in index]))
def _print_Pow(self, expr, rational=False):
return self._hprint_Pow(expr, rational=rational)
def _print_Rational(self, expr):
if self.standard == 'python2':
return '{}./{}.'.format(expr.p, expr.q)
return '{}/{}'.format(expr.p, expr.q)
def _print_Half(self, expr):
return self._print_Rational(expr)
def _print_frac(self, expr):
from sympy import Mod
return self._print_Mod(Mod(expr.args[0], 1))
_print_lowergamma = CodePrinter._print_not_supported
_print_uppergamma = CodePrinter._print_not_supported
_print_fresnelc = CodePrinter._print_not_supported
_print_fresnels = CodePrinter._print_not_supported
for k in PythonCodePrinter._kf:
setattr(PythonCodePrinter, '_print_%s' % k, _print_known_func)
for k in _known_constants_math:
setattr(PythonCodePrinter, '_print_%s' % k, _print_known_const)
def pycode(expr, **settings):
""" Converts an expr to a string of Python code
Parameters
==========
expr : Expr
A SymPy expression.
fully_qualified_modules : bool
Whether or not to write out full module names of functions
(``math.sin`` vs. ``sin``). default: ``True``.
standard : str or None, optional
If 'python2', Python 2 sematics will be used.
If 'python3', Python 3 sematics will be used.
If None, the standard will be automatically detected.
Default is 'python3'. And this parameter may be removed in the
future.
Examples
========
>>> from sympy import tan, Symbol
>>> from sympy.printing.pycode import pycode
>>> pycode(tan(Symbol('x')) + 1)
'math.tan(x) + 1'
"""
return PythonCodePrinter(settings).doprint(expr)
_not_in_mpmath = 'log1p log2'.split()
_in_mpmath = [(k, v) for k, v in _known_functions_math.items() if k not in _not_in_mpmath]
_known_functions_mpmath = dict(_in_mpmath, **{
'beta': 'beta',
'frac': 'frac',
'fresnelc': 'fresnelc',
'fresnels': 'fresnels',
'sign': 'sign',
'loggamma': 'loggamma',
})
_known_constants_mpmath = {
'Exp1': 'e',
'Pi': 'pi',
'GoldenRatio': 'phi',
'EulerGamma': 'euler',
'Catalan': 'catalan',
'NaN': 'nan',
'Infinity': 'inf',
'NegativeInfinity': 'ninf'
}
def _unpack_integral_limits(integral_expr):
""" helper function for _print_Integral that
- accepts an Integral expression
- returns a tuple of
- a list variables of integration
- a list of tuples of the upper and lower limits of integration
"""
integration_vars = []
limits = []
for integration_range in integral_expr.limits:
if len(integration_range) == 3:
integration_var, lower_limit, upper_limit = integration_range
else:
raise NotImplementedError("Only definite integrals are supported")
integration_vars.append(integration_var)
limits.append((lower_limit, upper_limit))
return integration_vars, limits
class MpmathPrinter(PythonCodePrinter):
"""
Lambda printer for mpmath which maintains precision for floats
"""
printmethod = "_mpmathcode"
language = "Python with mpmath"
_kf = dict(chain(
_known_functions.items(),
[(k, 'mpmath.' + v) for k, v in _known_functions_mpmath.items()]
))
_kc = {k: 'mpmath.'+v for k, v in _known_constants_mpmath.items()}
def _print_Float(self, e):
# XXX: This does not handle setting mpmath.mp.dps. It is assumed that
# the caller of the lambdified function will have set it to sufficient
# precision to match the Floats in the expression.
# Remove 'mpz' if gmpy is installed.
args = str(tuple(map(int, e._mpf_)))
return '{func}({args})'.format(func=self._module_format('mpmath.mpf'), args=args)
def _print_Rational(self, e):
return "{func}({p})/{func}({q})".format(
func=self._module_format('mpmath.mpf'),
q=self._print(e.q),
p=self._print(e.p)
)
def _print_Half(self, e):
return self._print_Rational(e)
def _print_uppergamma(self, e):
return "{}({}, {}, {})".format(
self._module_format('mpmath.gammainc'),
self._print(e.args[0]),
self._print(e.args[1]),
self._module_format('mpmath.inf'))
def _print_lowergamma(self, e):
return "{}({}, 0, {})".format(
self._module_format('mpmath.gammainc'),
self._print(e.args[0]),
self._print(e.args[1]))
def _print_log2(self, e):
return '{0}({1})/{0}(2)'.format(
self._module_format('mpmath.log'), self._print(e.args[0]))
def _print_log1p(self, e):
return '{}({}+1)'.format(
self._module_format('mpmath.log'), self._print(e.args[0]))
def _print_Pow(self, expr, rational=False):
return self._hprint_Pow(expr, rational=rational, sqrt='mpmath.sqrt')
def _print_Integral(self, e):
integration_vars, limits = _unpack_integral_limits(e)
return "{}(lambda {}: {}, {})".format(
self._module_format("mpmath.quad"),
", ".join(map(self._print, integration_vars)),
self._print(e.args[0]),
", ".join("(%s, %s)" % tuple(map(self._print, l)) for l in limits))
for k in MpmathPrinter._kf:
setattr(MpmathPrinter, '_print_%s' % k, _print_known_func)
for k in _known_constants_mpmath:
setattr(MpmathPrinter, '_print_%s' % k, _print_known_const)
class SymPyPrinter(AbstractPythonCodePrinter):
language = "Python with SymPy"
def _print_Function(self, expr):
mod = expr.func.__module__ or ''
return '%s(%s)' % (self._module_format(mod + ('.' if mod else '') + expr.func.__name__),
', '.join(map(lambda arg: self._print(arg), expr.args)))
def _print_Pow(self, expr, rational=False):
return self._hprint_Pow(expr, rational=rational, sqrt='sympy.sqrt')
|
6805696e719fd0f91ad24e0fd4dd14e159f9afdf267e7101013b3991f1011b16 | """
A Printer for generating readable representation of most sympy classes.
"""
from typing import Any, Dict
from sympy.core import S, Rational, Pow, Basic, Mul, Number
from sympy.core.mul import _keep_coeff
from .printer import Printer, print_function
from sympy.printing.precedence import precedence, PRECEDENCE
from mpmath.libmp import prec_to_dps, to_str as mlib_to_str
from sympy.utilities import default_sort_key
class StrPrinter(Printer):
printmethod = "_sympystr"
_default_settings = {
"order": None,
"full_prec": "auto",
"sympy_integers": False,
"abbrev": False,
"perm_cyclic": True,
"min": None,
"max": None,
} # type: Dict[str, Any]
_relationals = dict() # type: Dict[str, str]
def parenthesize(self, item, level, strict=False):
if (precedence(item) < level) or ((not strict) and precedence(item) <= level):
return "(%s)" % self._print(item)
else:
return self._print(item)
def stringify(self, args, sep, level=0):
return sep.join([self.parenthesize(item, level) for item in args])
def emptyPrinter(self, expr):
if isinstance(expr, str):
return expr
elif isinstance(expr, Basic):
return repr(expr)
else:
return str(expr)
def _print_Add(self, expr, order=None):
terms = self._as_ordered_terms(expr, order=order)
PREC = precedence(expr)
l = []
for term in terms:
t = self._print(term)
if t.startswith('-'):
sign = "-"
t = t[1:]
else:
sign = "+"
if precedence(term) < PREC:
l.extend([sign, "(%s)" % t])
else:
l.extend([sign, t])
sign = l.pop(0)
if sign == '+':
sign = ""
return sign + ' '.join(l)
def _print_BooleanTrue(self, expr):
return "True"
def _print_BooleanFalse(self, expr):
return "False"
def _print_Not(self, expr):
return '~%s' %(self.parenthesize(expr.args[0],PRECEDENCE["Not"]))
def _print_And(self, expr):
return self.stringify(expr.args, " & ", PRECEDENCE["BitwiseAnd"])
def _print_Or(self, expr):
return self.stringify(expr.args, " | ", PRECEDENCE["BitwiseOr"])
def _print_Xor(self, expr):
return self.stringify(expr.args, " ^ ", PRECEDENCE["BitwiseXor"])
def _print_AppliedPredicate(self, expr):
return '%s(%s)' % (
self._print(expr.function), self.stringify(expr.arguments, ", "))
def _print_Basic(self, expr):
l = [self._print(o) for o in expr.args]
return expr.__class__.__name__ + "(%s)" % ", ".join(l)
def _print_BlockMatrix(self, B):
if B.blocks.shape == (1, 1):
self._print(B.blocks[0, 0])
return self._print(B.blocks)
def _print_Catalan(self, expr):
return 'Catalan'
def _print_ComplexInfinity(self, expr):
return 'zoo'
def _print_ConditionSet(self, s):
args = tuple([self._print(i) for i in (s.sym, s.condition)])
if s.base_set is S.UniversalSet:
return 'ConditionSet(%s, %s)' % args
args += (self._print(s.base_set),)
return 'ConditionSet(%s, %s, %s)' % args
def _print_Derivative(self, expr):
dexpr = expr.expr
dvars = [i[0] if i[1] == 1 else i for i in expr.variable_count]
return 'Derivative(%s)' % ", ".join(map(lambda arg: self._print(arg), [dexpr] + dvars))
def _print_dict(self, d):
keys = sorted(d.keys(), key=default_sort_key)
items = []
for key in keys:
item = "%s: %s" % (self._print(key), self._print(d[key]))
items.append(item)
return "{%s}" % ", ".join(items)
def _print_Dict(self, expr):
return self._print_dict(expr)
def _print_RandomDomain(self, d):
if hasattr(d, 'as_boolean'):
return 'Domain: ' + self._print(d.as_boolean())
elif hasattr(d, 'set'):
return ('Domain: ' + self._print(d.symbols) + ' in ' +
self._print(d.set))
else:
return 'Domain on ' + self._print(d.symbols)
def _print_Dummy(self, expr):
return '_' + expr.name
def _print_EulerGamma(self, expr):
return 'EulerGamma'
def _print_Exp1(self, expr):
return 'E'
def _print_ExprCondPair(self, expr):
return '(%s, %s)' % (self._print(expr.expr), self._print(expr.cond))
def _print_Function(self, expr):
return expr.func.__name__ + "(%s)" % self.stringify(expr.args, ", ")
def _print_GoldenRatio(self, expr):
return 'GoldenRatio'
def _print_TribonacciConstant(self, expr):
return 'TribonacciConstant'
def _print_ImaginaryUnit(self, expr):
return 'I'
def _print_Infinity(self, expr):
return 'oo'
def _print_Integral(self, expr):
def _xab_tostr(xab):
if len(xab) == 1:
return self._print(xab[0])
else:
return self._print((xab[0],) + tuple(xab[1:]))
L = ', '.join([_xab_tostr(l) for l in expr.limits])
return 'Integral(%s, %s)' % (self._print(expr.function), L)
def _print_Interval(self, i):
fin = 'Interval{m}({a}, {b})'
a, b, l, r = i.args
if a.is_infinite and b.is_infinite:
m = ''
elif a.is_infinite and not r:
m = ''
elif b.is_infinite and not l:
m = ''
elif not l and not r:
m = ''
elif l and r:
m = '.open'
elif l:
m = '.Lopen'
else:
m = '.Ropen'
return fin.format(**{'a': a, 'b': b, 'm': m})
def _print_AccumulationBounds(self, i):
return "AccumBounds(%s, %s)" % (self._print(i.min),
self._print(i.max))
def _print_Inverse(self, I):
return "%s**(-1)" % self.parenthesize(I.arg, PRECEDENCE["Pow"])
def _print_Lambda(self, obj):
expr = obj.expr
sig = obj.signature
if len(sig) == 1 and sig[0].is_symbol:
sig = sig[0]
return "Lambda(%s, %s)" % (self._print(sig), self._print(expr))
def _print_LatticeOp(self, expr):
args = sorted(expr.args, key=default_sort_key)
return expr.func.__name__ + "(%s)" % ", ".join(self._print(arg) for arg in args)
def _print_Limit(self, expr):
e, z, z0, dir = expr.args
if str(dir) == "+":
return "Limit(%s, %s, %s)" % tuple(map(self._print, (e, z, z0)))
else:
return "Limit(%s, %s, %s, dir='%s')" % tuple(map(self._print,
(e, z, z0, dir)))
def _print_list(self, expr):
return "[%s]" % self.stringify(expr, ", ")
def _print_MatrixBase(self, expr):
return expr._format_str(self)
def _print_MatrixElement(self, expr):
return self.parenthesize(expr.parent, PRECEDENCE["Atom"], strict=True) \
+ '[%s, %s]' % (self._print(expr.i), self._print(expr.j))
def _print_MatrixSlice(self, expr):
def strslice(x, dim):
x = list(x)
if x[2] == 1:
del x[2]
if x[0] == 0:
x[0] = ''
if x[1] == dim:
x[1] = ''
return ':'.join(map(lambda arg: self._print(arg), x))
return (self.parenthesize(expr.parent, PRECEDENCE["Atom"], strict=True) + '[' +
strslice(expr.rowslice, expr.parent.rows) + ', ' +
strslice(expr.colslice, expr.parent.cols) + ']')
def _print_DeferredVector(self, expr):
return expr.name
def _print_Mul(self, expr):
prec = precedence(expr)
# Check for unevaluated Mul. In this case we need to make sure the
# identities are visible, multiple Rational factors are not combined
# etc so we display in a straight-forward form that fully preserves all
# args and their order.
args = expr.args
if args[0] is S.One or any(isinstance(arg, Number) for arg in args[1:]):
factors = [self.parenthesize(a, prec, strict=False) for a in args]
return '*'.join(factors)
c, e = expr.as_coeff_Mul()
if c < 0:
expr = _keep_coeff(-c, e)
sign = "-"
else:
sign = ""
a = [] # items in the numerator
b = [] # items that are in the denominator (if any)
pow_paren = [] # Will collect all pow with more than one base element and exp = -1
if self.order not in ('old', 'none'):
args = expr.as_ordered_factors()
else:
# use make_args in case expr was something like -x -> x
args = Mul.make_args(expr)
# Gather args for numerator/denominator
for item in args:
if item.is_commutative and item.is_Pow and item.exp.is_Rational and item.exp.is_negative:
if item.exp != -1:
b.append(Pow(item.base, -item.exp, evaluate=False))
else:
if len(item.args[0].args) != 1 and isinstance(item.base, Mul): # To avoid situations like #14160
pow_paren.append(item)
b.append(Pow(item.base, -item.exp))
elif item.is_Rational and item is not S.Infinity:
if item.p != 1:
a.append(Rational(item.p))
if item.q != 1:
b.append(Rational(item.q))
else:
a.append(item)
a = a or [S.One]
a_str = [self.parenthesize(x, prec, strict=False) for x in a]
b_str = [self.parenthesize(x, prec, strict=False) for x in b]
# To parenthesize Pow with exp = -1 and having more than one Symbol
for item in pow_paren:
if item.base in b:
b_str[b.index(item.base)] = "(%s)" % b_str[b.index(item.base)]
if not b:
return sign + '*'.join(a_str)
elif len(b) == 1:
return sign + '*'.join(a_str) + "/" + b_str[0]
else:
return sign + '*'.join(a_str) + "/(%s)" % '*'.join(b_str)
def _print_MatMul(self, expr):
c, m = expr.as_coeff_mmul()
sign = ""
if c.is_number:
re, im = c.as_real_imag()
if im.is_zero and re.is_negative:
expr = _keep_coeff(-c, m)
sign = "-"
elif re.is_zero and im.is_negative:
expr = _keep_coeff(-c, m)
sign = "-"
return sign + '*'.join(
[self.parenthesize(arg, precedence(expr)) for arg in expr.args]
)
def _print_ElementwiseApplyFunction(self, expr):
return "{}.({})".format(
expr.function,
self._print(expr.expr),
)
def _print_NaN(self, expr):
return 'nan'
def _print_NegativeInfinity(self, expr):
return '-oo'
def _print_Order(self, expr):
if not expr.variables or all(p is S.Zero for p in expr.point):
if len(expr.variables) <= 1:
return 'O(%s)' % self._print(expr.expr)
else:
return 'O(%s)' % self.stringify((expr.expr,) + expr.variables, ', ', 0)
else:
return 'O(%s)' % self.stringify(expr.args, ', ', 0)
def _print_Ordinal(self, expr):
return expr.__str__()
def _print_Cycle(self, expr):
return expr.__str__()
def _print_Permutation(self, expr):
from sympy.combinatorics.permutations import Permutation, Cycle
from sympy.utilities.exceptions import SymPyDeprecationWarning
perm_cyclic = Permutation.print_cyclic
if perm_cyclic is not None:
SymPyDeprecationWarning(
feature="Permutation.print_cyclic = {}".format(perm_cyclic),
useinstead="init_printing(perm_cyclic={})"
.format(perm_cyclic),
issue=15201,
deprecated_since_version="1.6").warn()
else:
perm_cyclic = self._settings.get("perm_cyclic", True)
if perm_cyclic:
if not expr.size:
return '()'
# before taking Cycle notation, see if the last element is
# a singleton and move it to the head of the string
s = Cycle(expr)(expr.size - 1).__repr__()[len('Cycle'):]
last = s.rfind('(')
if not last == 0 and ',' not in s[last:]:
s = s[last:] + s[:last]
s = s.replace(',', '')
return s
else:
s = expr.support()
if not s:
if expr.size < 5:
return 'Permutation(%s)' % self._print(expr.array_form)
return 'Permutation([], size=%s)' % self._print(expr.size)
trim = self._print(expr.array_form[:s[-1] + 1]) + ', size=%s' % self._print(expr.size)
use = full = self._print(expr.array_form)
if len(trim) < len(full):
use = trim
return 'Permutation(%s)' % use
def _print_Subs(self, obj):
expr, old, new = obj.args
if len(obj.point) == 1:
old = old[0]
new = new[0]
return "Subs(%s, %s, %s)" % (
self._print(expr), self._print(old), self._print(new))
def _print_TensorIndex(self, expr):
return expr._print()
def _print_TensorHead(self, expr):
return expr._print()
def _print_Tensor(self, expr):
return expr._print()
def _print_TensMul(self, expr):
# prints expressions like "A(a)", "3*A(a)", "(1+x)*A(a)"
sign, args = expr._get_args_for_traditional_printer()
return sign + "*".join(
[self.parenthesize(arg, precedence(expr)) for arg in args]
)
def _print_TensAdd(self, expr):
return expr._print()
def _print_ArraySymbol(self, expr):
return self._print(expr.name)
def _print_ArrayElement(self, expr):
return "%s[%s]" % (expr.name, ", ".join([self._print(i) for i in expr.indices]))
def _print_PermutationGroup(self, expr):
p = [' %s' % self._print(a) for a in expr.args]
return 'PermutationGroup([\n%s])' % ',\n'.join(p)
def _print_Pi(self, expr):
return 'pi'
def _print_PolyRing(self, ring):
return "Polynomial ring in %s over %s with %s order" % \
(", ".join(map(lambda rs: self._print(rs), ring.symbols)),
self._print(ring.domain), self._print(ring.order))
def _print_FracField(self, field):
return "Rational function field in %s over %s with %s order" % \
(", ".join(map(lambda fs: self._print(fs), field.symbols)),
self._print(field.domain), self._print(field.order))
def _print_FreeGroupElement(self, elm):
return elm.__str__()
def _print_GaussianElement(self, poly):
return "(%s + %s*I)" % (poly.x, poly.y)
def _print_PolyElement(self, poly):
return poly.str(self, PRECEDENCE, "%s**%s", "*")
def _print_FracElement(self, frac):
if frac.denom == 1:
return self._print(frac.numer)
else:
numer = self.parenthesize(frac.numer, PRECEDENCE["Mul"], strict=True)
denom = self.parenthesize(frac.denom, PRECEDENCE["Atom"], strict=True)
return numer + "/" + denom
def _print_Poly(self, expr):
ATOM_PREC = PRECEDENCE["Atom"] - 1
terms, gens = [], [ self.parenthesize(s, ATOM_PREC) for s in expr.gens ]
for monom, coeff in expr.terms():
s_monom = []
for i, exp in enumerate(monom):
if exp > 0:
if exp == 1:
s_monom.append(gens[i])
else:
s_monom.append(gens[i] + "**%d" % exp)
s_monom = "*".join(s_monom)
if coeff.is_Add:
if s_monom:
s_coeff = "(" + self._print(coeff) + ")"
else:
s_coeff = self._print(coeff)
else:
if s_monom:
if coeff is S.One:
terms.extend(['+', s_monom])
continue
if coeff is S.NegativeOne:
terms.extend(['-', s_monom])
continue
s_coeff = self._print(coeff)
if not s_monom:
s_term = s_coeff
else:
s_term = s_coeff + "*" + s_monom
if s_term.startswith('-'):
terms.extend(['-', s_term[1:]])
else:
terms.extend(['+', s_term])
if terms[0] in ['-', '+']:
modifier = terms.pop(0)
if modifier == '-':
terms[0] = '-' + terms[0]
format = expr.__class__.__name__ + "(%s, %s"
from sympy.polys.polyerrors import PolynomialError
try:
format += ", modulus=%s" % expr.get_modulus()
except PolynomialError:
format += ", domain='%s'" % expr.get_domain()
format += ")"
for index, item in enumerate(gens):
if len(item) > 2 and (item[:1] == "(" and item[len(item) - 1:] == ")"):
gens[index] = item[1:len(item) - 1]
return format % (' '.join(terms), ', '.join(gens))
def _print_UniversalSet(self, p):
return 'UniversalSet'
def _print_AlgebraicNumber(self, expr):
if expr.is_aliased:
return self._print(expr.as_poly().as_expr())
else:
return self._print(expr.as_expr())
def _print_Pow(self, expr, rational=False):
"""Printing helper function for ``Pow``
Parameters
==========
rational : bool, optional
If ``True``, it will not attempt printing ``sqrt(x)`` or
``x**S.Half`` as ``sqrt``, and will use ``x**(1/2)``
instead.
See examples for additional details
Examples
========
>>> from sympy.functions import sqrt
>>> from sympy.printing.str import StrPrinter
>>> from sympy.abc import x
How ``rational`` keyword works with ``sqrt``:
>>> printer = StrPrinter()
>>> printer._print_Pow(sqrt(x), rational=True)
'x**(1/2)'
>>> printer._print_Pow(sqrt(x), rational=False)
'sqrt(x)'
>>> printer._print_Pow(1/sqrt(x), rational=True)
'x**(-1/2)'
>>> printer._print_Pow(1/sqrt(x), rational=False)
'1/sqrt(x)'
Notes
=====
``sqrt(x)`` is canonicalized as ``Pow(x, S.Half)`` in SymPy,
so there is no need of defining a separate printer for ``sqrt``.
Instead, it should be handled here as well.
"""
PREC = precedence(expr)
if expr.exp is S.Half and not rational:
return "sqrt(%s)" % self._print(expr.base)
if expr.is_commutative:
if -expr.exp is S.Half and not rational:
# Note: Don't test "expr.exp == -S.Half" here, because that will
# match -0.5, which we don't want.
return "%s/sqrt(%s)" % tuple(map(lambda arg: self._print(arg), (S.One, expr.base)))
if expr.exp is -S.One:
# Similarly to the S.Half case, don't test with "==" here.
return '%s/%s' % (self._print(S.One),
self.parenthesize(expr.base, PREC, strict=False))
e = self.parenthesize(expr.exp, PREC, strict=False)
if self.printmethod == '_sympyrepr' and expr.exp.is_Rational and expr.exp.q != 1:
# the parenthesized exp should be '(Rational(a, b))' so strip parens,
# but just check to be sure.
if e.startswith('(Rational'):
return '%s**%s' % (self.parenthesize(expr.base, PREC, strict=False), e[1:-1])
return '%s**%s' % (self.parenthesize(expr.base, PREC, strict=False), e)
def _print_UnevaluatedExpr(self, expr):
return self._print(expr.args[0])
def _print_MatPow(self, expr):
PREC = precedence(expr)
return '%s**%s' % (self.parenthesize(expr.base, PREC, strict=False),
self.parenthesize(expr.exp, PREC, strict=False))
def _print_Integer(self, expr):
if self._settings.get("sympy_integers", False):
return "S(%s)" % (expr)
return str(expr.p)
def _print_Integers(self, expr):
return 'Integers'
def _print_Naturals(self, expr):
return 'Naturals'
def _print_Naturals0(self, expr):
return 'Naturals0'
def _print_Rationals(self, expr):
return 'Rationals'
def _print_Reals(self, expr):
return 'Reals'
def _print_Complexes(self, expr):
return 'Complexes'
def _print_EmptySet(self, expr):
return 'EmptySet'
def _print_EmptySequence(self, expr):
return 'EmptySequence'
def _print_int(self, expr):
return str(expr)
def _print_mpz(self, expr):
return str(expr)
def _print_Rational(self, expr):
if expr.q == 1:
return str(expr.p)
else:
if self._settings.get("sympy_integers", False):
return "S(%s)/%s" % (expr.p, expr.q)
return "%s/%s" % (expr.p, expr.q)
def _print_PythonRational(self, expr):
if expr.q == 1:
return str(expr.p)
else:
return "%d/%d" % (expr.p, expr.q)
def _print_Fraction(self, expr):
if expr.denominator == 1:
return str(expr.numerator)
else:
return "%s/%s" % (expr.numerator, expr.denominator)
def _print_mpq(self, expr):
if expr.denominator == 1:
return str(expr.numerator)
else:
return "%s/%s" % (expr.numerator, expr.denominator)
def _print_Float(self, expr):
prec = expr._prec
if prec < 5:
dps = 0
else:
dps = prec_to_dps(expr._prec)
if self._settings["full_prec"] is True:
strip = False
elif self._settings["full_prec"] is False:
strip = True
elif self._settings["full_prec"] == "auto":
strip = self._print_level > 1
low = self._settings["min"] if "min" in self._settings else None
high = self._settings["max"] if "max" in self._settings else None
rv = mlib_to_str(expr._mpf_, dps, strip_zeros=strip, min_fixed=low, max_fixed=high)
if rv.startswith('-.0'):
rv = '-0.' + rv[3:]
elif rv.startswith('.0'):
rv = '0.' + rv[2:]
if rv.startswith('+'):
# e.g., +inf -> inf
rv = rv[1:]
return rv
def _print_Relational(self, expr):
charmap = {
"==": "Eq",
"!=": "Ne",
":=": "Assignment",
'+=': "AddAugmentedAssignment",
"-=": "SubAugmentedAssignment",
"*=": "MulAugmentedAssignment",
"/=": "DivAugmentedAssignment",
"%=": "ModAugmentedAssignment",
}
if expr.rel_op in charmap:
return '%s(%s, %s)' % (charmap[expr.rel_op], self._print(expr.lhs),
self._print(expr.rhs))
return '%s %s %s' % (self.parenthesize(expr.lhs, precedence(expr)),
self._relationals.get(expr.rel_op) or expr.rel_op,
self.parenthesize(expr.rhs, precedence(expr)))
def _print_ComplexRootOf(self, expr):
return "CRootOf(%s, %d)" % (self._print_Add(expr.expr, order='lex'),
expr.index)
def _print_RootSum(self, expr):
args = [self._print_Add(expr.expr, order='lex')]
if expr.fun is not S.IdentityFunction:
args.append(self._print(expr.fun))
return "RootSum(%s)" % ", ".join(args)
def _print_GroebnerBasis(self, basis):
cls = basis.__class__.__name__
exprs = [self._print_Add(arg, order=basis.order) for arg in basis.exprs]
exprs = "[%s]" % ", ".join(exprs)
gens = [ self._print(gen) for gen in basis.gens ]
domain = "domain='%s'" % self._print(basis.domain)
order = "order='%s'" % self._print(basis.order)
args = [exprs] + gens + [domain, order]
return "%s(%s)" % (cls, ", ".join(args))
def _print_set(self, s):
items = sorted(s, key=default_sort_key)
args = ', '.join(self._print(item) for item in items)
if not args:
return "set()"
return '{%s}' % args
def _print_frozenset(self, s):
if not s:
return "frozenset()"
return "frozenset(%s)" % self._print_set(s)
def _print_Sum(self, expr):
def _xab_tostr(xab):
if len(xab) == 1:
return self._print(xab[0])
else:
return self._print((xab[0],) + tuple(xab[1:]))
L = ', '.join([_xab_tostr(l) for l in expr.limits])
return 'Sum(%s, %s)' % (self._print(expr.function), L)
def _print_Symbol(self, expr):
return expr.name
_print_MatrixSymbol = _print_Symbol
_print_RandomSymbol = _print_Symbol
def _print_Identity(self, expr):
return "I"
def _print_ZeroMatrix(self, expr):
return "0"
def _print_OneMatrix(self, expr):
return "1"
def _print_Predicate(self, expr):
return "Q.%s" % expr.name
def _print_str(self, expr):
return str(expr)
def _print_tuple(self, expr):
if len(expr) == 1:
return "(%s,)" % self._print(expr[0])
else:
return "(%s)" % self.stringify(expr, ", ")
def _print_Tuple(self, expr):
return self._print_tuple(expr)
def _print_Transpose(self, T):
return "%s.T" % self.parenthesize(T.arg, PRECEDENCE["Pow"])
def _print_Uniform(self, expr):
return "Uniform(%s, %s)" % (self._print(expr.a), self._print(expr.b))
def _print_Quantity(self, expr):
if self._settings.get("abbrev", False):
return "%s" % expr.abbrev
return "%s" % expr.name
def _print_Quaternion(self, expr):
s = [self.parenthesize(i, PRECEDENCE["Mul"], strict=True) for i in expr.args]
a = [s[0]] + [i+"*"+j for i, j in zip(s[1:], "ijk")]
return " + ".join(a)
def _print_Dimension(self, expr):
return str(expr)
def _print_Wild(self, expr):
return expr.name + '_'
def _print_WildFunction(self, expr):
return expr.name + '_'
def _print_WildDot(self, expr):
return expr.name
def _print_WildPlus(self, expr):
return expr.name
def _print_WildStar(self, expr):
return expr.name
def _print_Zero(self, expr):
if self._settings.get("sympy_integers", False):
return "S(0)"
return "0"
def _print_DMP(self, p):
from sympy.core.sympify import SympifyError
try:
if p.ring is not None:
# TODO incorporate order
return self._print(p.ring.to_sympy(p))
except SympifyError:
pass
cls = p.__class__.__name__
rep = self._print(p.rep)
dom = self._print(p.dom)
ring = self._print(p.ring)
return "%s(%s, %s, %s)" % (cls, rep, dom, ring)
def _print_DMF(self, expr):
return self._print_DMP(expr)
def _print_Object(self, obj):
return 'Object("%s")' % obj.name
def _print_IdentityMorphism(self, morphism):
return 'IdentityMorphism(%s)' % morphism.domain
def _print_NamedMorphism(self, morphism):
return 'NamedMorphism(%s, %s, "%s")' % \
(morphism.domain, morphism.codomain, morphism.name)
def _print_Category(self, category):
return 'Category("%s")' % category.name
def _print_Manifold(self, manifold):
return manifold.name.name
def _print_Patch(self, patch):
return patch.name.name
def _print_CoordSystem(self, coords):
return coords.name.name
def _print_BaseScalarField(self, field):
return field._coord_sys.symbols[field._index].name
def _print_BaseVectorField(self, field):
return 'e_%s' % field._coord_sys.symbols[field._index].name
def _print_Differential(self, diff):
field = diff._form_field
if hasattr(field, '_coord_sys'):
return 'd%s' % field._coord_sys.symbols[field._index].name
else:
return 'd(%s)' % self._print(field)
def _print_Tr(self, expr):
#TODO : Handle indices
return "%s(%s)" % ("Tr", self._print(expr.args[0]))
def _print_Str(self, s):
return self._print(s.name)
def _print_AppliedBinaryRelation(self, expr):
rel, args = expr.function, expr.arguments
lhs, rhs = args
return '%s(%s, %s)' % (self._print(rel),
self._print(expr.lhs),
self._print(expr.rhs))
@print_function(StrPrinter)
def sstr(expr, **settings):
"""Returns the expression as a string.
For large expressions where speed is a concern, use the setting
order='none'. If abbrev=True setting is used then units are printed in
abbreviated form.
Examples
========
>>> from sympy import symbols, Eq, sstr
>>> a, b = symbols('a b')
>>> sstr(Eq(a + b, 0))
'Eq(a + b, 0)'
"""
p = StrPrinter(settings)
s = p.doprint(expr)
return s
class StrReprPrinter(StrPrinter):
"""(internal) -- see sstrrepr"""
def _print_str(self, s):
return repr(s)
def _print_Str(self, s):
# Str does not to be printed same as str here
return "%s(%s)" % (s.__class__.__name__, self._print(s.name))
@print_function(StrReprPrinter)
def sstrrepr(expr, **settings):
"""return expr in mixed str/repr form
i.e. strings are returned in repr form with quotes, and everything else
is returned in str form.
This function could be useful for hooking into sys.displayhook
"""
p = StrReprPrinter(settings)
s = p.doprint(expr)
return s
|
483c5cdfedbb29575aaf07b42cb7363d84b54540233f1f7662b366cec1ed0957 | """
A few practical conventions common to all printers.
"""
import re
from collections.abc import Iterable
from sympy import Derivative
_name_with_digits_p = re.compile(r'^([^\W\d_]+)(\d+)$', re.U)
def split_super_sub(text):
"""Split a symbol name into a name, superscripts and subscripts
The first part of the symbol name is considered to be its actual
'name', followed by super- and subscripts. Each superscript is
preceded with a "^" character or by "__". Each subscript is preceded
by a "_" character. The three return values are the actual name, a
list with superscripts and a list with subscripts.
Examples
========
>>> from sympy.printing.conventions import split_super_sub
>>> split_super_sub('a_x^1')
('a', ['1'], ['x'])
>>> split_super_sub('var_sub1__sup_sub2')
('var', ['sup'], ['sub1', 'sub2'])
"""
if not text:
return text, [], []
pos = 0
name = None
supers = []
subs = []
while pos < len(text):
start = pos + 1
if text[pos:pos + 2] == "__":
start += 1
pos_hat = text.find("^", start)
if pos_hat < 0:
pos_hat = len(text)
pos_usc = text.find("_", start)
if pos_usc < 0:
pos_usc = len(text)
pos_next = min(pos_hat, pos_usc)
part = text[pos:pos_next]
pos = pos_next
if name is None:
name = part
elif part.startswith("^"):
supers.append(part[1:])
elif part.startswith("__"):
supers.append(part[2:])
elif part.startswith("_"):
subs.append(part[1:])
else:
raise RuntimeError("This should never happen.")
# Make a little exception when a name ends with digits, i.e. treat them
# as a subscript too.
m = _name_with_digits_p.match(name)
if m:
name, sub = m.groups()
subs.insert(0, sub)
return name, supers, subs
def requires_partial(expr):
"""Return whether a partial derivative symbol is required for printing
This requires checking how many free variables there are,
filtering out the ones that are integers. Some expressions don't have
free variables. In that case, check its variable list explicitly to
get the context of the expression.
"""
if isinstance(expr, Derivative):
return requires_partial(expr.expr)
if not isinstance(expr.free_symbols, Iterable):
return len(set(expr.variables)) > 1
return sum(not s.is_integer for s in expr.free_symbols) > 1
|
192c370562d77fcaa455f85d58bdd566b7c45f5f39d2207dbd904a3e95909eca | """
A Printer which converts an expression into its LaTeX equivalent.
"""
from typing import Any, Dict
import itertools
from sympy.core import Add, Float, Mod, Mul, Number, S, Symbol
from sympy.core.alphabets import greeks
from sympy.core.containers import Tuple
from sympy.core.function import _coeff_isneg, AppliedUndef, Derivative
from sympy.core.operations import AssocOp
from sympy.core.sympify import SympifyError
from sympy.logic.boolalg import true
# sympy.printing imports
from sympy.printing.precedence import precedence_traditional
from sympy.printing.printer import Printer, print_function
from sympy.printing.conventions import split_super_sub, requires_partial
from sympy.printing.precedence import precedence, PRECEDENCE
import mpmath.libmp as mlib
from mpmath.libmp import prec_to_dps
from sympy.core.compatibility import default_sort_key
from sympy.utilities.iterables import has_variety
import re
# Hand-picked functions which can be used directly in both LaTeX and MathJax
# Complete list at
# https://docs.mathjax.org/en/latest/tex.html#supported-latex-commands
# This variable only contains those functions which sympy uses.
accepted_latex_functions = ['arcsin', 'arccos', 'arctan', 'sin', 'cos', 'tan',
'sinh', 'cosh', 'tanh', 'sqrt', 'ln', 'log', 'sec',
'csc', 'cot', 'coth', 're', 'im', 'frac', 'root',
'arg',
]
tex_greek_dictionary = {
'Alpha': 'A',
'Beta': 'B',
'Gamma': r'\Gamma',
'Delta': r'\Delta',
'Epsilon': 'E',
'Zeta': 'Z',
'Eta': 'H',
'Theta': r'\Theta',
'Iota': 'I',
'Kappa': 'K',
'Lambda': r'\Lambda',
'Mu': 'M',
'Nu': 'N',
'Xi': r'\Xi',
'omicron': 'o',
'Omicron': 'O',
'Pi': r'\Pi',
'Rho': 'P',
'Sigma': r'\Sigma',
'Tau': 'T',
'Upsilon': r'\Upsilon',
'Phi': r'\Phi',
'Chi': 'X',
'Psi': r'\Psi',
'Omega': r'\Omega',
'lamda': r'\lambda',
'Lamda': r'\Lambda',
'khi': r'\chi',
'Khi': r'X',
'varepsilon': r'\varepsilon',
'varkappa': r'\varkappa',
'varphi': r'\varphi',
'varpi': r'\varpi',
'varrho': r'\varrho',
'varsigma': r'\varsigma',
'vartheta': r'\vartheta',
}
other_symbols = {'aleph', 'beth', 'daleth', 'gimel', 'ell', 'eth', 'hbar',
'hslash', 'mho', 'wp'}
# Variable name modifiers
modifier_dict = {
# Accents
'mathring': lambda s: r'\mathring{'+s+r'}',
'ddddot': lambda s: r'\ddddot{'+s+r'}',
'dddot': lambda s: r'\dddot{'+s+r'}',
'ddot': lambda s: r'\ddot{'+s+r'}',
'dot': lambda s: r'\dot{'+s+r'}',
'check': lambda s: r'\check{'+s+r'}',
'breve': lambda s: r'\breve{'+s+r'}',
'acute': lambda s: r'\acute{'+s+r'}',
'grave': lambda s: r'\grave{'+s+r'}',
'tilde': lambda s: r'\tilde{'+s+r'}',
'hat': lambda s: r'\hat{'+s+r'}',
'bar': lambda s: r'\bar{'+s+r'}',
'vec': lambda s: r'\vec{'+s+r'}',
'prime': lambda s: "{"+s+"}'",
'prm': lambda s: "{"+s+"}'",
# Faces
'bold': lambda s: r'\boldsymbol{'+s+r'}',
'bm': lambda s: r'\boldsymbol{'+s+r'}',
'cal': lambda s: r'\mathcal{'+s+r'}',
'scr': lambda s: r'\mathscr{'+s+r'}',
'frak': lambda s: r'\mathfrak{'+s+r'}',
# Brackets
'norm': lambda s: r'\left\|{'+s+r'}\right\|',
'avg': lambda s: r'\left\langle{'+s+r'}\right\rangle',
'abs': lambda s: r'\left|{'+s+r'}\right|',
'mag': lambda s: r'\left|{'+s+r'}\right|',
}
greek_letters_set = frozenset(greeks)
_between_two_numbers_p = (
re.compile(r'[0-9][} ]*$'), # search
re.compile(r'[{ ]*[-+0-9]'), # match
)
def latex_escape(s):
"""
Escape a string such that latex interprets it as plaintext.
We can't use verbatim easily with mathjax, so escaping is easier.
Rules from https://tex.stackexchange.com/a/34586/41112.
"""
s = s.replace('\\', r'\textbackslash')
for c in '&%$#_{}':
s = s.replace(c, '\\' + c)
s = s.replace('~', r'\textasciitilde')
s = s.replace('^', r'\textasciicircum')
return s
class LatexPrinter(Printer):
printmethod = "_latex"
_default_settings = {
"full_prec": False,
"fold_frac_powers": False,
"fold_func_brackets": False,
"fold_short_frac": None,
"inv_trig_style": "abbreviated",
"itex": False,
"ln_notation": False,
"long_frac_ratio": None,
"mat_delim": "[",
"mat_str": None,
"mode": "plain",
"mul_symbol": None,
"order": None,
"symbol_names": {},
"root_notation": True,
"mat_symbol_style": "plain",
"imaginary_unit": "i",
"gothic_re_im": False,
"decimal_separator": "period",
"perm_cyclic": True,
"parenthesize_super": True,
"min": None,
"max": None,
} # type: Dict[str, Any]
def __init__(self, settings=None):
Printer.__init__(self, settings)
if 'mode' in self._settings:
valid_modes = ['inline', 'plain', 'equation',
'equation*']
if self._settings['mode'] not in valid_modes:
raise ValueError("'mode' must be one of 'inline', 'plain', "
"'equation' or 'equation*'")
if self._settings['fold_short_frac'] is None and \
self._settings['mode'] == 'inline':
self._settings['fold_short_frac'] = True
mul_symbol_table = {
None: r" ",
"ldot": r" \,.\, ",
"dot": r" \cdot ",
"times": r" \times "
}
try:
self._settings['mul_symbol_latex'] = \
mul_symbol_table[self._settings['mul_symbol']]
except KeyError:
self._settings['mul_symbol_latex'] = \
self._settings['mul_symbol']
try:
self._settings['mul_symbol_latex_numbers'] = \
mul_symbol_table[self._settings['mul_symbol'] or 'dot']
except KeyError:
if (self._settings['mul_symbol'].strip() in
['', ' ', '\\', '\\,', '\\:', '\\;', '\\quad']):
self._settings['mul_symbol_latex_numbers'] = \
mul_symbol_table['dot']
else:
self._settings['mul_symbol_latex_numbers'] = \
self._settings['mul_symbol']
self._delim_dict = {'(': ')', '[': ']'}
imaginary_unit_table = {
None: r"i",
"i": r"i",
"ri": r"\mathrm{i}",
"ti": r"\text{i}",
"j": r"j",
"rj": r"\mathrm{j}",
"tj": r"\text{j}",
}
try:
self._settings['imaginary_unit_latex'] = \
imaginary_unit_table[self._settings['imaginary_unit']]
except KeyError:
self._settings['imaginary_unit_latex'] = \
self._settings['imaginary_unit']
def _add_parens(self, s):
return r"\left({}\right)".format(s)
# TODO: merge this with the above, which requires a lot of test changes
def _add_parens_lspace(self, s):
return r"\left( {}\right)".format(s)
def parenthesize(self, item, level, is_neg=False, strict=False):
prec_val = precedence_traditional(item)
if is_neg and strict:
return self._add_parens(self._print(item))
if (prec_val < level) or ((not strict) and prec_val <= level):
return self._add_parens(self._print(item))
else:
return self._print(item)
def parenthesize_super(self, s):
"""
Protect superscripts in s
If the parenthesize_super option is set, protect with parentheses, else
wrap in braces.
"""
if "^" in s:
if self._settings['parenthesize_super']:
return self._add_parens(s)
else:
return "{{{}}}".format(s)
return s
def doprint(self, expr):
tex = Printer.doprint(self, expr)
if self._settings['mode'] == 'plain':
return tex
elif self._settings['mode'] == 'inline':
return r"$%s$" % tex
elif self._settings['itex']:
return r"$$%s$$" % tex
else:
env_str = self._settings['mode']
return r"\begin{%s}%s\end{%s}" % (env_str, tex, env_str)
def _needs_brackets(self, expr):
"""
Returns True if the expression needs to be wrapped in brackets when
printed, False otherwise. For example: a + b => True; a => False;
10 => False; -10 => True.
"""
return not ((expr.is_Integer and expr.is_nonnegative)
or (expr.is_Atom and (expr is not S.NegativeOne
and expr.is_Rational is False)))
def _needs_function_brackets(self, expr):
"""
Returns True if the expression needs to be wrapped in brackets when
passed as an argument to a function, False otherwise. This is a more
liberal version of _needs_brackets, in that many expressions which need
to be wrapped in brackets when added/subtracted/raised to a power do
not need them when passed to a function. Such an example is a*b.
"""
if not self._needs_brackets(expr):
return False
else:
# Muls of the form a*b*c... can be folded
if expr.is_Mul and not self._mul_is_clean(expr):
return True
# Pows which don't need brackets can be folded
elif expr.is_Pow and not self._pow_is_clean(expr):
return True
# Add and Function always need brackets
elif expr.is_Add or expr.is_Function:
return True
else:
return False
def _needs_mul_brackets(self, expr, first=False, last=False):
"""
Returns True if the expression needs to be wrapped in brackets when
printed as part of a Mul, False otherwise. This is True for Add,
but also for some container objects that would not need brackets
when appearing last in a Mul, e.g. an Integral. ``last=True``
specifies that this expr is the last to appear in a Mul.
``first=True`` specifies that this expr is the first to appear in
a Mul.
"""
from sympy import Integral, Product, Sum
if expr.is_Mul:
if not first and _coeff_isneg(expr):
return True
elif precedence_traditional(expr) < PRECEDENCE["Mul"]:
return True
elif expr.is_Relational:
return True
if expr.is_Piecewise:
return True
if any([expr.has(x) for x in (Mod,)]):
return True
if (not last and
any([expr.has(x) for x in (Integral, Product, Sum)])):
return True
return False
def _needs_add_brackets(self, expr):
"""
Returns True if the expression needs to be wrapped in brackets when
printed as part of an Add, False otherwise. This is False for most
things.
"""
if expr.is_Relational:
return True
if any([expr.has(x) for x in (Mod,)]):
return True
if expr.is_Add:
return True
return False
def _mul_is_clean(self, expr):
for arg in expr.args:
if arg.is_Function:
return False
return True
def _pow_is_clean(self, expr):
return not self._needs_brackets(expr.base)
def _do_exponent(self, expr, exp):
if exp is not None:
return r"\left(%s\right)^{%s}" % (expr, exp)
else:
return expr
def _print_Basic(self, expr):
ls = [self._print(o) for o in expr.args]
return self._deal_with_super_sub(expr.__class__.__name__) + \
r"\left(%s\right)" % ", ".join(ls)
def _print_bool(self, e):
return r"\text{%s}" % e
_print_BooleanTrue = _print_bool
_print_BooleanFalse = _print_bool
def _print_NoneType(self, e):
return r"\text{%s}" % e
def _print_Add(self, expr, order=None):
terms = self._as_ordered_terms(expr, order=order)
tex = ""
for i, term in enumerate(terms):
if i == 0:
pass
elif _coeff_isneg(term):
tex += " - "
term = -term
else:
tex += " + "
term_tex = self._print(term)
if self._needs_add_brackets(term):
term_tex = r"\left(%s\right)" % term_tex
tex += term_tex
return tex
def _print_Cycle(self, expr):
from sympy.combinatorics.permutations import Permutation
if expr.size == 0:
return r"\left( \right)"
expr = Permutation(expr)
expr_perm = expr.cyclic_form
siz = expr.size
if expr.array_form[-1] == siz - 1:
expr_perm = expr_perm + [[siz - 1]]
term_tex = ''
for i in expr_perm:
term_tex += str(i).replace(',', r"\;")
term_tex = term_tex.replace('[', r"\left( ")
term_tex = term_tex.replace(']', r"\right)")
return term_tex
def _print_Permutation(self, expr):
from sympy.combinatorics.permutations import Permutation
from sympy.utilities.exceptions import SymPyDeprecationWarning
perm_cyclic = Permutation.print_cyclic
if perm_cyclic is not None:
SymPyDeprecationWarning(
feature="Permutation.print_cyclic = {}".format(perm_cyclic),
useinstead="init_printing(perm_cyclic={})"
.format(perm_cyclic),
issue=15201,
deprecated_since_version="1.6").warn()
else:
perm_cyclic = self._settings.get("perm_cyclic", True)
if perm_cyclic:
return self._print_Cycle(expr)
if expr.size == 0:
return r"\left( \right)"
lower = [self._print(arg) for arg in expr.array_form]
upper = [self._print(arg) for arg in range(len(lower))]
row1 = " & ".join(upper)
row2 = " & ".join(lower)
mat = r" \\ ".join((row1, row2))
return r"\begin{pmatrix} %s \end{pmatrix}" % mat
def _print_AppliedPermutation(self, expr):
perm, var = expr.args
return r"\sigma_{%s}(%s)" % (self._print(perm), self._print(var))
def _print_Float(self, expr):
# Based off of that in StrPrinter
dps = prec_to_dps(expr._prec)
strip = False if self._settings['full_prec'] else True
low = self._settings["min"] if "min" in self._settings else None
high = self._settings["max"] if "max" in self._settings else None
str_real = mlib.to_str(expr._mpf_, dps, strip_zeros=strip, min_fixed=low, max_fixed=high)
# Must always have a mul symbol (as 2.5 10^{20} just looks odd)
# thus we use the number separator
separator = self._settings['mul_symbol_latex_numbers']
if 'e' in str_real:
(mant, exp) = str_real.split('e')
if exp[0] == '+':
exp = exp[1:]
if self._settings['decimal_separator'] == 'comma':
mant = mant.replace('.','{,}')
return r"%s%s10^{%s}" % (mant, separator, exp)
elif str_real == "+inf":
return r"\infty"
elif str_real == "-inf":
return r"- \infty"
else:
if self._settings['decimal_separator'] == 'comma':
str_real = str_real.replace('.','{,}')
return str_real
def _print_Cross(self, expr):
vec1 = expr._expr1
vec2 = expr._expr2
return r"%s \times %s" % (self.parenthesize(vec1, PRECEDENCE['Mul']),
self.parenthesize(vec2, PRECEDENCE['Mul']))
def _print_Curl(self, expr):
vec = expr._expr
return r"\nabla\times %s" % self.parenthesize(vec, PRECEDENCE['Mul'])
def _print_Divergence(self, expr):
vec = expr._expr
return r"\nabla\cdot %s" % self.parenthesize(vec, PRECEDENCE['Mul'])
def _print_Dot(self, expr):
vec1 = expr._expr1
vec2 = expr._expr2
return r"%s \cdot %s" % (self.parenthesize(vec1, PRECEDENCE['Mul']),
self.parenthesize(vec2, PRECEDENCE['Mul']))
def _print_Gradient(self, expr):
func = expr._expr
return r"\nabla %s" % self.parenthesize(func, PRECEDENCE['Mul'])
def _print_Laplacian(self, expr):
func = expr._expr
return r"\triangle %s" % self.parenthesize(func, PRECEDENCE['Mul'])
def _print_Mul(self, expr):
from sympy.core.power import Pow
from sympy.physics.units import Quantity
from sympy.simplify import fraction
separator = self._settings['mul_symbol_latex']
numbersep = self._settings['mul_symbol_latex_numbers']
def convert(expr):
if not expr.is_Mul:
return str(self._print(expr))
else:
if self.order not in ('old', 'none'):
args = expr.as_ordered_factors()
else:
args = list(expr.args)
# If quantities are present append them at the back
args = sorted(args, key=lambda x: isinstance(x, Quantity) or
(isinstance(x, Pow) and
isinstance(x.base, Quantity)))
return convert_args(args)
def convert_args(args):
_tex = last_term_tex = ""
for i, term in enumerate(args):
term_tex = self._print(term)
if self._needs_mul_brackets(term, first=(i == 0),
last=(i == len(args) - 1)):
term_tex = r"\left(%s\right)" % term_tex
if _between_two_numbers_p[0].search(last_term_tex) and \
_between_two_numbers_p[1].match(term_tex):
# between two numbers
_tex += numbersep
elif _tex:
_tex += separator
_tex += term_tex
last_term_tex = term_tex
return _tex
# Check for unevaluated Mul. In this case we need to make sure the
# identities are visible, multiple Rational factors are not combined
# etc so we display in a straight-forward form that fully preserves all
# args and their order.
# XXX: _print_Pow calls this routine with instances of Pow...
if isinstance(expr, Mul):
args = expr.args
if args[0] is S.One or any(isinstance(arg, Number) for arg in args[1:]):
return convert_args(args)
include_parens = False
if _coeff_isneg(expr):
expr = -expr
tex = "- "
if expr.is_Add:
tex += "("
include_parens = True
else:
tex = ""
numer, denom = fraction(expr, exact=True)
if denom is S.One and Pow(1, -1, evaluate=False) not in expr.args:
# use the original expression here, since fraction() may have
# altered it when producing numer and denom
tex += convert(expr)
else:
snumer = convert(numer)
sdenom = convert(denom)
ldenom = len(sdenom.split())
ratio = self._settings['long_frac_ratio']
if self._settings['fold_short_frac'] and ldenom <= 2 and \
"^" not in sdenom:
# handle short fractions
if self._needs_mul_brackets(numer, last=False):
tex += r"\left(%s\right) / %s" % (snumer, sdenom)
else:
tex += r"%s / %s" % (snumer, sdenom)
elif ratio is not None and \
len(snumer.split()) > ratio*ldenom:
# handle long fractions
if self._needs_mul_brackets(numer, last=True):
tex += r"\frac{1}{%s}%s\left(%s\right)" \
% (sdenom, separator, snumer)
elif numer.is_Mul:
# split a long numerator
a = S.One
b = S.One
for x in numer.args:
if self._needs_mul_brackets(x, last=False) or \
len(convert(a*x).split()) > ratio*ldenom or \
(b.is_commutative is x.is_commutative is False):
b *= x
else:
a *= x
if self._needs_mul_brackets(b, last=True):
tex += r"\frac{%s}{%s}%s\left(%s\right)" \
% (convert(a), sdenom, separator, convert(b))
else:
tex += r"\frac{%s}{%s}%s%s" \
% (convert(a), sdenom, separator, convert(b))
else:
tex += r"\frac{1}{%s}%s%s" % (sdenom, separator, snumer)
else:
tex += r"\frac{%s}{%s}" % (snumer, sdenom)
if include_parens:
tex += ")"
return tex
def _print_Pow(self, expr):
# Treat x**Rational(1,n) as special case
if expr.exp.is_Rational and abs(expr.exp.p) == 1 and expr.exp.q != 1 \
and self._settings['root_notation']:
base = self._print(expr.base)
expq = expr.exp.q
if expq == 2:
tex = r"\sqrt{%s}" % base
elif self._settings['itex']:
tex = r"\root{%d}{%s}" % (expq, base)
else:
tex = r"\sqrt[%d]{%s}" % (expq, base)
if expr.exp.is_negative:
return r"\frac{1}{%s}" % tex
else:
return tex
elif self._settings['fold_frac_powers'] \
and expr.exp.is_Rational \
and expr.exp.q != 1:
base = self.parenthesize(expr.base, PRECEDENCE['Pow'])
p, q = expr.exp.p, expr.exp.q
# issue #12886: add parentheses for superscripts raised to powers
if expr.base.is_Symbol:
base = self.parenthesize_super(base)
if expr.base.is_Function:
return self._print(expr.base, exp="%s/%s" % (p, q))
return r"%s^{%s/%s}" % (base, p, q)
elif expr.exp.is_Rational and expr.exp.is_negative and \
expr.base.is_commutative:
# special case for 1^(-x), issue 9216
if expr.base == 1:
return r"%s^{%s}" % (expr.base, expr.exp)
# things like 1/x
return self._print_Mul(expr)
else:
if expr.base.is_Function:
return self._print(expr.base, exp=self._print(expr.exp))
else:
tex = r"%s^{%s}"
return self._helper_print_standard_power(expr, tex)
def _helper_print_standard_power(self, expr, template):
exp = self._print(expr.exp)
# issue #12886: add parentheses around superscripts raised
# to powers
base = self.parenthesize(expr.base, PRECEDENCE['Pow'])
if expr.base.is_Symbol:
base = self.parenthesize_super(base)
elif (isinstance(expr.base, Derivative)
and base.startswith(r'\left(')
and re.match(r'\\left\(\\d?d?dot', base)
and base.endswith(r'\right)')):
# don't use parentheses around dotted derivative
base = base[6: -7] # remove outermost added parens
return template % (base, exp)
def _print_UnevaluatedExpr(self, expr):
return self._print(expr.args[0])
def _print_Sum(self, expr):
if len(expr.limits) == 1:
tex = r"\sum_{%s=%s}^{%s} " % \
tuple([self._print(i) for i in expr.limits[0]])
else:
def _format_ineq(l):
return r"%s \leq %s \leq %s" % \
tuple([self._print(s) for s in (l[1], l[0], l[2])])
tex = r"\sum_{\substack{%s}} " % \
str.join('\\\\', [_format_ineq(l) for l in expr.limits])
if isinstance(expr.function, Add):
tex += r"\left(%s\right)" % self._print(expr.function)
else:
tex += self._print(expr.function)
return tex
def _print_Product(self, expr):
if len(expr.limits) == 1:
tex = r"\prod_{%s=%s}^{%s} " % \
tuple([self._print(i) for i in expr.limits[0]])
else:
def _format_ineq(l):
return r"%s \leq %s \leq %s" % \
tuple([self._print(s) for s in (l[1], l[0], l[2])])
tex = r"\prod_{\substack{%s}} " % \
str.join('\\\\', [_format_ineq(l) for l in expr.limits])
if isinstance(expr.function, Add):
tex += r"\left(%s\right)" % self._print(expr.function)
else:
tex += self._print(expr.function)
return tex
def _print_BasisDependent(self, expr):
from sympy.vector import Vector
o1 = []
if expr == expr.zero:
return expr.zero._latex_form
if isinstance(expr, Vector):
items = expr.separate().items()
else:
items = [(0, expr)]
for system, vect in items:
inneritems = list(vect.components.items())
inneritems.sort(key=lambda x: x[0].__str__())
for k, v in inneritems:
if v == 1:
o1.append(' + ' + k._latex_form)
elif v == -1:
o1.append(' - ' + k._latex_form)
else:
arg_str = '(' + self._print(v) + ')'
o1.append(' + ' + arg_str + k._latex_form)
outstr = (''.join(o1))
if outstr[1] != '-':
outstr = outstr[3:]
else:
outstr = outstr[1:]
return outstr
def _print_Indexed(self, expr):
tex_base = self._print(expr.base)
tex = '{'+tex_base+'}'+'_{%s}' % ','.join(
map(self._print, expr.indices))
return tex
def _print_IndexedBase(self, expr):
return self._print(expr.label)
def _print_Derivative(self, expr):
if requires_partial(expr.expr):
diff_symbol = r'\partial'
else:
diff_symbol = r'd'
tex = ""
dim = 0
for x, num in reversed(expr.variable_count):
dim += num
if num == 1:
tex += r"%s %s" % (diff_symbol, self._print(x))
else:
tex += r"%s %s^{%s}" % (diff_symbol,
self.parenthesize_super(self._print(x)),
self._print(num))
if dim == 1:
tex = r"\frac{%s}{%s}" % (diff_symbol, tex)
else:
tex = r"\frac{%s^{%s}}{%s}" % (diff_symbol, self._print(dim), tex)
if any(_coeff_isneg(i) for i in expr.args):
return r"%s %s" % (tex, self.parenthesize(expr.expr,
PRECEDENCE["Mul"],
is_neg=True,
strict=True))
return r"%s %s" % (tex, self.parenthesize(expr.expr,
PRECEDENCE["Mul"],
is_neg=False,
strict=True))
def _print_Subs(self, subs):
expr, old, new = subs.args
latex_expr = self._print(expr)
latex_old = (self._print(e) for e in old)
latex_new = (self._print(e) for e in new)
latex_subs = r'\\ '.join(
e[0] + '=' + e[1] for e in zip(latex_old, latex_new))
return r'\left. %s \right|_{\substack{ %s }}' % (latex_expr,
latex_subs)
def _print_Integral(self, expr):
tex, symbols = "", []
# Only up to \iiiint exists
if len(expr.limits) <= 4 and all(len(lim) == 1 for lim in expr.limits):
# Use len(expr.limits)-1 so that syntax highlighters don't think
# \" is an escaped quote
tex = r"\i" + "i"*(len(expr.limits) - 1) + "nt"
symbols = [r"\, d%s" % self._print(symbol[0])
for symbol in expr.limits]
else:
for lim in reversed(expr.limits):
symbol = lim[0]
tex += r"\int"
if len(lim) > 1:
if self._settings['mode'] != 'inline' \
and not self._settings['itex']:
tex += r"\limits"
if len(lim) == 3:
tex += "_{%s}^{%s}" % (self._print(lim[1]),
self._print(lim[2]))
if len(lim) == 2:
tex += "^{%s}" % (self._print(lim[1]))
symbols.insert(0, r"\, d%s" % self._print(symbol))
return r"%s %s%s" % (tex, self.parenthesize(expr.function,
PRECEDENCE["Mul"],
is_neg=any(_coeff_isneg(i) for i in expr.args),
strict=True),
"".join(symbols))
def _print_Limit(self, expr):
e, z, z0, dir = expr.args
tex = r"\lim_{%s \to " % self._print(z)
if str(dir) == '+-' or z0 in (S.Infinity, S.NegativeInfinity):
tex += r"%s}" % self._print(z0)
else:
tex += r"%s^%s}" % (self._print(z0), self._print(dir))
if isinstance(e, AssocOp):
return r"%s\left(%s\right)" % (tex, self._print(e))
else:
return r"%s %s" % (tex, self._print(e))
def _hprint_Function(self, func):
r'''
Logic to decide how to render a function to latex
- if it is a recognized latex name, use the appropriate latex command
- if it is a single letter, just use that letter
- if it is a longer name, then put \operatorname{} around it and be
mindful of undercores in the name
'''
func = self._deal_with_super_sub(func)
if func in accepted_latex_functions:
name = r"\%s" % func
elif len(func) == 1 or func.startswith('\\'):
name = func
else:
name = r"\operatorname{%s}" % func
return name
def _print_Function(self, expr, exp=None):
r'''
Render functions to LaTeX, handling functions that LaTeX knows about
e.g., sin, cos, ... by using the proper LaTeX command (\sin, \cos, ...).
For single-letter function names, render them as regular LaTeX math
symbols. For multi-letter function names that LaTeX does not know
about, (e.g., Li, sech) use \operatorname{} so that the function name
is rendered in Roman font and LaTeX handles spacing properly.
expr is the expression involving the function
exp is an exponent
'''
func = expr.func.__name__
if hasattr(self, '_print_' + func) and \
not isinstance(expr, AppliedUndef):
return getattr(self, '_print_' + func)(expr, exp)
else:
args = [str(self._print(arg)) for arg in expr.args]
# How inverse trig functions should be displayed, formats are:
# abbreviated: asin, full: arcsin, power: sin^-1
inv_trig_style = self._settings['inv_trig_style']
# If we are dealing with a power-style inverse trig function
inv_trig_power_case = False
# If it is applicable to fold the argument brackets
can_fold_brackets = self._settings['fold_func_brackets'] and \
len(args) == 1 and \
not self._needs_function_brackets(expr.args[0])
inv_trig_table = [
"asin", "acos", "atan",
"acsc", "asec", "acot",
"asinh", "acosh", "atanh",
"acsch", "asech", "acoth",
]
# If the function is an inverse trig function, handle the style
if func in inv_trig_table:
if inv_trig_style == "abbreviated":
pass
elif inv_trig_style == "full":
func = "arc" + func[1:]
elif inv_trig_style == "power":
func = func[1:]
inv_trig_power_case = True
# Can never fold brackets if we're raised to a power
if exp is not None:
can_fold_brackets = False
if inv_trig_power_case:
if func in accepted_latex_functions:
name = r"\%s^{-1}" % func
else:
name = r"\operatorname{%s}^{-1}" % func
elif exp is not None:
func_tex = self._hprint_Function(func)
func_tex = self.parenthesize_super(func_tex)
name = r'%s^{%s}' % (func_tex, exp)
else:
name = self._hprint_Function(func)
if can_fold_brackets:
if func in accepted_latex_functions:
# Wrap argument safely to avoid parse-time conflicts
# with the function name itself
name += r" {%s}"
else:
name += r"%s"
else:
name += r"{\left(%s \right)}"
if inv_trig_power_case and exp is not None:
name += r"^{%s}" % exp
return name % ",".join(args)
def _print_UndefinedFunction(self, expr):
return self._hprint_Function(str(expr))
def _print_ElementwiseApplyFunction(self, expr):
return r"{%s}_{\circ}\left({%s}\right)" % (
self._print(expr.function),
self._print(expr.expr),
)
@property
def _special_function_classes(self):
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.functions.special.gamma_functions import gamma, lowergamma
from sympy.functions.special.beta_functions import beta
from sympy.functions.special.delta_functions import DiracDelta
from sympy.functions.special.error_functions import Chi
return {KroneckerDelta: r'\delta',
gamma: r'\Gamma',
lowergamma: r'\gamma',
beta: r'\operatorname{B}',
DiracDelta: r'\delta',
Chi: r'\operatorname{Chi}'}
def _print_FunctionClass(self, expr):
for cls in self._special_function_classes:
if issubclass(expr, cls) and expr.__name__ == cls.__name__:
return self._special_function_classes[cls]
return self._hprint_Function(str(expr))
def _print_Lambda(self, expr):
symbols, expr = expr.args
if len(symbols) == 1:
symbols = self._print(symbols[0])
else:
symbols = self._print(tuple(symbols))
tex = r"\left( %s \mapsto %s \right)" % (symbols, self._print(expr))
return tex
def _print_IdentityFunction(self, expr):
return r"\left( x \mapsto x \right)"
def _hprint_variadic_function(self, expr, exp=None):
args = sorted(expr.args, key=default_sort_key)
texargs = [r"%s" % self._print(symbol) for symbol in args]
tex = r"\%s\left(%s\right)" % (str(expr.func).lower(),
", ".join(texargs))
if exp is not None:
return r"%s^{%s}" % (tex, exp)
else:
return tex
_print_Min = _print_Max = _hprint_variadic_function
def _print_floor(self, expr, exp=None):
tex = r"\left\lfloor{%s}\right\rfloor" % self._print(expr.args[0])
if exp is not None:
return r"%s^{%s}" % (tex, exp)
else:
return tex
def _print_ceiling(self, expr, exp=None):
tex = r"\left\lceil{%s}\right\rceil" % self._print(expr.args[0])
if exp is not None:
return r"%s^{%s}" % (tex, exp)
else:
return tex
def _print_log(self, expr, exp=None):
if not self._settings["ln_notation"]:
tex = r"\log{\left(%s \right)}" % self._print(expr.args[0])
else:
tex = r"\ln{\left(%s \right)}" % self._print(expr.args[0])
if exp is not None:
return r"%s^{%s}" % (tex, exp)
else:
return tex
def _print_Abs(self, expr, exp=None):
tex = r"\left|{%s}\right|" % self._print(expr.args[0])
if exp is not None:
return r"%s^{%s}" % (tex, exp)
else:
return tex
_print_Determinant = _print_Abs
def _print_re(self, expr, exp=None):
if self._settings['gothic_re_im']:
tex = r"\Re{%s}" % self.parenthesize(expr.args[0], PRECEDENCE['Atom'])
else:
tex = r"\operatorname{{re}}{{{}}}".format(self.parenthesize(expr.args[0], PRECEDENCE['Atom']))
return self._do_exponent(tex, exp)
def _print_im(self, expr, exp=None):
if self._settings['gothic_re_im']:
tex = r"\Im{%s}" % self.parenthesize(expr.args[0], PRECEDENCE['Atom'])
else:
tex = r"\operatorname{{im}}{{{}}}".format(self.parenthesize(expr.args[0], PRECEDENCE['Atom']))
return self._do_exponent(tex, exp)
def _print_Not(self, e):
from sympy import Equivalent, Implies
if isinstance(e.args[0], Equivalent):
return self._print_Equivalent(e.args[0], r"\not\Leftrightarrow")
if isinstance(e.args[0], Implies):
return self._print_Implies(e.args[0], r"\not\Rightarrow")
if (e.args[0].is_Boolean):
return r"\neg \left(%s\right)" % self._print(e.args[0])
else:
return r"\neg %s" % self._print(e.args[0])
def _print_LogOp(self, args, char):
arg = args[0]
if arg.is_Boolean and not arg.is_Not:
tex = r"\left(%s\right)" % self._print(arg)
else:
tex = r"%s" % self._print(arg)
for arg in args[1:]:
if arg.is_Boolean and not arg.is_Not:
tex += r" %s \left(%s\right)" % (char, self._print(arg))
else:
tex += r" %s %s" % (char, self._print(arg))
return tex
def _print_And(self, e):
args = sorted(e.args, key=default_sort_key)
return self._print_LogOp(args, r"\wedge")
def _print_Or(self, e):
args = sorted(e.args, key=default_sort_key)
return self._print_LogOp(args, r"\vee")
def _print_Xor(self, e):
args = sorted(e.args, key=default_sort_key)
return self._print_LogOp(args, r"\veebar")
def _print_Implies(self, e, altchar=None):
return self._print_LogOp(e.args, altchar or r"\Rightarrow")
def _print_Equivalent(self, e, altchar=None):
args = sorted(e.args, key=default_sort_key)
return self._print_LogOp(args, altchar or r"\Leftrightarrow")
def _print_conjugate(self, expr, exp=None):
tex = r"\overline{%s}" % self._print(expr.args[0])
if exp is not None:
return r"%s^{%s}" % (tex, exp)
else:
return tex
def _print_polar_lift(self, expr, exp=None):
func = r"\operatorname{polar\_lift}"
arg = r"{\left(%s \right)}" % self._print(expr.args[0])
if exp is not None:
return r"%s^{%s}%s" % (func, exp, arg)
else:
return r"%s%s" % (func, arg)
def _print_ExpBase(self, expr, exp=None):
# TODO should exp_polar be printed differently?
# what about exp_polar(0), exp_polar(1)?
tex = r"e^{%s}" % self._print(expr.args[0])
return self._do_exponent(tex, exp)
def _print_Exp1(self, expr, exp=None):
return "e"
def _print_elliptic_k(self, expr, exp=None):
tex = r"\left(%s\right)" % self._print(expr.args[0])
if exp is not None:
return r"K^{%s}%s" % (exp, tex)
else:
return r"K%s" % tex
def _print_elliptic_f(self, expr, exp=None):
tex = r"\left(%s\middle| %s\right)" % \
(self._print(expr.args[0]), self._print(expr.args[1]))
if exp is not None:
return r"F^{%s}%s" % (exp, tex)
else:
return r"F%s" % tex
def _print_elliptic_e(self, expr, exp=None):
if len(expr.args) == 2:
tex = r"\left(%s\middle| %s\right)" % \
(self._print(expr.args[0]), self._print(expr.args[1]))
else:
tex = r"\left(%s\right)" % self._print(expr.args[0])
if exp is not None:
return r"E^{%s}%s" % (exp, tex)
else:
return r"E%s" % tex
def _print_elliptic_pi(self, expr, exp=None):
if len(expr.args) == 3:
tex = r"\left(%s; %s\middle| %s\right)" % \
(self._print(expr.args[0]), self._print(expr.args[1]),
self._print(expr.args[2]))
else:
tex = r"\left(%s\middle| %s\right)" % \
(self._print(expr.args[0]), self._print(expr.args[1]))
if exp is not None:
return r"\Pi^{%s}%s" % (exp, tex)
else:
return r"\Pi%s" % tex
def _print_beta(self, expr, exp=None):
tex = r"\left(%s, %s\right)" % (self._print(expr.args[0]),
self._print(expr.args[1]))
if exp is not None:
return r"\operatorname{B}^{%s}%s" % (exp, tex)
else:
return r"\operatorname{B}%s" % tex
def _print_betainc(self, expr, exp=None, operator='B'):
largs = [self._print(arg) for arg in expr.args]
tex = r"\left(%s, %s\right)" % (largs[0], largs[1])
if exp is not None:
return r"\operatorname{%s}_{(%s, %s)}^{%s}%s" % (operator, largs[2], largs[3], exp, tex)
else:
return r"\operatorname{%s}_{(%s, %s)}%s" % (operator, largs[2], largs[3], tex)
def _print_betainc_regularized(self, expr, exp=None):
return self._print_betainc(expr, exp, operator='I')
def _print_uppergamma(self, expr, exp=None):
tex = r"\left(%s, %s\right)" % (self._print(expr.args[0]),
self._print(expr.args[1]))
if exp is not None:
return r"\Gamma^{%s}%s" % (exp, tex)
else:
return r"\Gamma%s" % tex
def _print_lowergamma(self, expr, exp=None):
tex = r"\left(%s, %s\right)" % (self._print(expr.args[0]),
self._print(expr.args[1]))
if exp is not None:
return r"\gamma^{%s}%s" % (exp, tex)
else:
return r"\gamma%s" % tex
def _hprint_one_arg_func(self, expr, exp=None):
tex = r"\left(%s\right)" % self._print(expr.args[0])
if exp is not None:
return r"%s^{%s}%s" % (self._print(expr.func), exp, tex)
else:
return r"%s%s" % (self._print(expr.func), tex)
_print_gamma = _hprint_one_arg_func
def _print_Chi(self, expr, exp=None):
tex = r"\left(%s\right)" % self._print(expr.args[0])
if exp is not None:
return r"\operatorname{Chi}^{%s}%s" % (exp, tex)
else:
return r"\operatorname{Chi}%s" % tex
def _print_expint(self, expr, exp=None):
tex = r"\left(%s\right)" % self._print(expr.args[1])
nu = self._print(expr.args[0])
if exp is not None:
return r"\operatorname{E}_{%s}^{%s}%s" % (nu, exp, tex)
else:
return r"\operatorname{E}_{%s}%s" % (nu, tex)
def _print_fresnels(self, expr, exp=None):
tex = r"\left(%s\right)" % self._print(expr.args[0])
if exp is not None:
return r"S^{%s}%s" % (exp, tex)
else:
return r"S%s" % tex
def _print_fresnelc(self, expr, exp=None):
tex = r"\left(%s\right)" % self._print(expr.args[0])
if exp is not None:
return r"C^{%s}%s" % (exp, tex)
else:
return r"C%s" % tex
def _print_subfactorial(self, expr, exp=None):
tex = r"!%s" % self.parenthesize(expr.args[0], PRECEDENCE["Func"])
if exp is not None:
return r"\left(%s\right)^{%s}" % (tex, exp)
else:
return tex
def _print_factorial(self, expr, exp=None):
tex = r"%s!" % self.parenthesize(expr.args[0], PRECEDENCE["Func"])
if exp is not None:
return r"%s^{%s}" % (tex, exp)
else:
return tex
def _print_factorial2(self, expr, exp=None):
tex = r"%s!!" % self.parenthesize(expr.args[0], PRECEDENCE["Func"])
if exp is not None:
return r"%s^{%s}" % (tex, exp)
else:
return tex
def _print_binomial(self, expr, exp=None):
tex = r"{\binom{%s}{%s}}" % (self._print(expr.args[0]),
self._print(expr.args[1]))
if exp is not None:
return r"%s^{%s}" % (tex, exp)
else:
return tex
def _print_RisingFactorial(self, expr, exp=None):
n, k = expr.args
base = r"%s" % self.parenthesize(n, PRECEDENCE['Func'])
tex = r"{%s}^{\left(%s\right)}" % (base, self._print(k))
return self._do_exponent(tex, exp)
def _print_FallingFactorial(self, expr, exp=None):
n, k = expr.args
sub = r"%s" % self.parenthesize(k, PRECEDENCE['Func'])
tex = r"{\left(%s\right)}_{%s}" % (self._print(n), sub)
return self._do_exponent(tex, exp)
def _hprint_BesselBase(self, expr, exp, sym):
tex = r"%s" % (sym)
need_exp = False
if exp is not None:
if tex.find('^') == -1:
tex = r"%s^{%s}" % (tex, exp)
else:
need_exp = True
tex = r"%s_{%s}\left(%s\right)" % (tex, self._print(expr.order),
self._print(expr.argument))
if need_exp:
tex = self._do_exponent(tex, exp)
return tex
def _hprint_vec(self, vec):
if not vec:
return ""
s = ""
for i in vec[:-1]:
s += "%s, " % self._print(i)
s += self._print(vec[-1])
return s
def _print_besselj(self, expr, exp=None):
return self._hprint_BesselBase(expr, exp, 'J')
def _print_besseli(self, expr, exp=None):
return self._hprint_BesselBase(expr, exp, 'I')
def _print_besselk(self, expr, exp=None):
return self._hprint_BesselBase(expr, exp, 'K')
def _print_bessely(self, expr, exp=None):
return self._hprint_BesselBase(expr, exp, 'Y')
def _print_yn(self, expr, exp=None):
return self._hprint_BesselBase(expr, exp, 'y')
def _print_jn(self, expr, exp=None):
return self._hprint_BesselBase(expr, exp, 'j')
def _print_hankel1(self, expr, exp=None):
return self._hprint_BesselBase(expr, exp, 'H^{(1)}')
def _print_hankel2(self, expr, exp=None):
return self._hprint_BesselBase(expr, exp, 'H^{(2)}')
def _print_hn1(self, expr, exp=None):
return self._hprint_BesselBase(expr, exp, 'h^{(1)}')
def _print_hn2(self, expr, exp=None):
return self._hprint_BesselBase(expr, exp, 'h^{(2)}')
def _hprint_airy(self, expr, exp=None, notation=""):
tex = r"\left(%s\right)" % self._print(expr.args[0])
if exp is not None:
return r"%s^{%s}%s" % (notation, exp, tex)
else:
return r"%s%s" % (notation, tex)
def _hprint_airy_prime(self, expr, exp=None, notation=""):
tex = r"\left(%s\right)" % self._print(expr.args[0])
if exp is not None:
return r"{%s^\prime}^{%s}%s" % (notation, exp, tex)
else:
return r"%s^\prime%s" % (notation, tex)
def _print_airyai(self, expr, exp=None):
return self._hprint_airy(expr, exp, 'Ai')
def _print_airybi(self, expr, exp=None):
return self._hprint_airy(expr, exp, 'Bi')
def _print_airyaiprime(self, expr, exp=None):
return self._hprint_airy_prime(expr, exp, 'Ai')
def _print_airybiprime(self, expr, exp=None):
return self._hprint_airy_prime(expr, exp, 'Bi')
def _print_hyper(self, expr, exp=None):
tex = r"{{}_{%s}F_{%s}\left(\begin{matrix} %s \\ %s \end{matrix}" \
r"\middle| {%s} \right)}" % \
(self._print(len(expr.ap)), self._print(len(expr.bq)),
self._hprint_vec(expr.ap), self._hprint_vec(expr.bq),
self._print(expr.argument))
if exp is not None:
tex = r"{%s}^{%s}" % (tex, exp)
return tex
def _print_meijerg(self, expr, exp=None):
tex = r"{G_{%s, %s}^{%s, %s}\left(\begin{matrix} %s & %s \\" \
r"%s & %s \end{matrix} \middle| {%s} \right)}" % \
(self._print(len(expr.ap)), self._print(len(expr.bq)),
self._print(len(expr.bm)), self._print(len(expr.an)),
self._hprint_vec(expr.an), self._hprint_vec(expr.aother),
self._hprint_vec(expr.bm), self._hprint_vec(expr.bother),
self._print(expr.argument))
if exp is not None:
tex = r"{%s}^{%s}" % (tex, exp)
return tex
def _print_dirichlet_eta(self, expr, exp=None):
tex = r"\left(%s\right)" % self._print(expr.args[0])
if exp is not None:
return r"\eta^{%s}%s" % (exp, tex)
return r"\eta%s" % tex
def _print_zeta(self, expr, exp=None):
if len(expr.args) == 2:
tex = r"\left(%s, %s\right)" % tuple(map(self._print, expr.args))
else:
tex = r"\left(%s\right)" % self._print(expr.args[0])
if exp is not None:
return r"\zeta^{%s}%s" % (exp, tex)
return r"\zeta%s" % tex
def _print_stieltjes(self, expr, exp=None):
if len(expr.args) == 2:
tex = r"_{%s}\left(%s\right)" % tuple(map(self._print, expr.args))
else:
tex = r"_{%s}" % self._print(expr.args[0])
if exp is not None:
return r"\gamma%s^{%s}" % (tex, exp)
return r"\gamma%s" % tex
def _print_lerchphi(self, expr, exp=None):
tex = r"\left(%s, %s, %s\right)" % tuple(map(self._print, expr.args))
if exp is None:
return r"\Phi%s" % tex
return r"\Phi^{%s}%s" % (exp, tex)
def _print_polylog(self, expr, exp=None):
s, z = map(self._print, expr.args)
tex = r"\left(%s\right)" % z
if exp is None:
return r"\operatorname{Li}_{%s}%s" % (s, tex)
return r"\operatorname{Li}_{%s}^{%s}%s" % (s, exp, tex)
def _print_jacobi(self, expr, exp=None):
n, a, b, x = map(self._print, expr.args)
tex = r"P_{%s}^{\left(%s,%s\right)}\left(%s\right)" % (n, a, b, x)
if exp is not None:
tex = r"\left(" + tex + r"\right)^{%s}" % (exp)
return tex
def _print_gegenbauer(self, expr, exp=None):
n, a, x = map(self._print, expr.args)
tex = r"C_{%s}^{\left(%s\right)}\left(%s\right)" % (n, a, x)
if exp is not None:
tex = r"\left(" + tex + r"\right)^{%s}" % (exp)
return tex
def _print_chebyshevt(self, expr, exp=None):
n, x = map(self._print, expr.args)
tex = r"T_{%s}\left(%s\right)" % (n, x)
if exp is not None:
tex = r"\left(" + tex + r"\right)^{%s}" % (exp)
return tex
def _print_chebyshevu(self, expr, exp=None):
n, x = map(self._print, expr.args)
tex = r"U_{%s}\left(%s\right)" % (n, x)
if exp is not None:
tex = r"\left(" + tex + r"\right)^{%s}" % (exp)
return tex
def _print_legendre(self, expr, exp=None):
n, x = map(self._print, expr.args)
tex = r"P_{%s}\left(%s\right)" % (n, x)
if exp is not None:
tex = r"\left(" + tex + r"\right)^{%s}" % (exp)
return tex
def _print_assoc_legendre(self, expr, exp=None):
n, a, x = map(self._print, expr.args)
tex = r"P_{%s}^{\left(%s\right)}\left(%s\right)" % (n, a, x)
if exp is not None:
tex = r"\left(" + tex + r"\right)^{%s}" % (exp)
return tex
def _print_hermite(self, expr, exp=None):
n, x = map(self._print, expr.args)
tex = r"H_{%s}\left(%s\right)" % (n, x)
if exp is not None:
tex = r"\left(" + tex + r"\right)^{%s}" % (exp)
return tex
def _print_laguerre(self, expr, exp=None):
n, x = map(self._print, expr.args)
tex = r"L_{%s}\left(%s\right)" % (n, x)
if exp is not None:
tex = r"\left(" + tex + r"\right)^{%s}" % (exp)
return tex
def _print_assoc_laguerre(self, expr, exp=None):
n, a, x = map(self._print, expr.args)
tex = r"L_{%s}^{\left(%s\right)}\left(%s\right)" % (n, a, x)
if exp is not None:
tex = r"\left(" + tex + r"\right)^{%s}" % (exp)
return tex
def _print_Ynm(self, expr, exp=None):
n, m, theta, phi = map(self._print, expr.args)
tex = r"Y_{%s}^{%s}\left(%s,%s\right)" % (n, m, theta, phi)
if exp is not None:
tex = r"\left(" + tex + r"\right)^{%s}" % (exp)
return tex
def _print_Znm(self, expr, exp=None):
n, m, theta, phi = map(self._print, expr.args)
tex = r"Z_{%s}^{%s}\left(%s,%s\right)" % (n, m, theta, phi)
if exp is not None:
tex = r"\left(" + tex + r"\right)^{%s}" % (exp)
return tex
def __print_mathieu_functions(self, character, args, prime=False, exp=None):
a, q, z = map(self._print, args)
sup = r"^{\prime}" if prime else ""
exp = "" if not exp else "^{%s}" % exp
return r"%s%s\left(%s, %s, %s\right)%s" % (character, sup, a, q, z, exp)
def _print_mathieuc(self, expr, exp=None):
return self.__print_mathieu_functions("C", expr.args, exp=exp)
def _print_mathieus(self, expr, exp=None):
return self.__print_mathieu_functions("S", expr.args, exp=exp)
def _print_mathieucprime(self, expr, exp=None):
return self.__print_mathieu_functions("C", expr.args, prime=True, exp=exp)
def _print_mathieusprime(self, expr, exp=None):
return self.__print_mathieu_functions("S", expr.args, prime=True, exp=exp)
def _print_Rational(self, expr):
if expr.q != 1:
sign = ""
p = expr.p
if expr.p < 0:
sign = "- "
p = -p
if self._settings['fold_short_frac']:
return r"%s%d / %d" % (sign, p, expr.q)
return r"%s\frac{%d}{%d}" % (sign, p, expr.q)
else:
return self._print(expr.p)
def _print_Order(self, expr):
s = self._print(expr.expr)
if expr.point and any(p != S.Zero for p in expr.point) or \
len(expr.variables) > 1:
s += '; '
if len(expr.variables) > 1:
s += self._print(expr.variables)
elif expr.variables:
s += self._print(expr.variables[0])
s += r'\rightarrow '
if len(expr.point) > 1:
s += self._print(expr.point)
else:
s += self._print(expr.point[0])
return r"O\left(%s\right)" % s
def _print_Symbol(self, expr, style='plain'):
if expr in self._settings['symbol_names']:
return self._settings['symbol_names'][expr]
return self._deal_with_super_sub(expr.name, style=style)
_print_RandomSymbol = _print_Symbol
def _deal_with_super_sub(self, string, style='plain'):
if '{' in string:
name, supers, subs = string, [], []
else:
name, supers, subs = split_super_sub(string)
name = translate(name)
supers = [translate(sup) for sup in supers]
subs = [translate(sub) for sub in subs]
# apply the style only to the name
if style == 'bold':
name = "\\mathbf{{{}}}".format(name)
# glue all items together:
if supers:
name += "^{%s}" % " ".join(supers)
if subs:
name += "_{%s}" % " ".join(subs)
return name
def _print_Relational(self, expr):
if self._settings['itex']:
gt = r"\gt"
lt = r"\lt"
else:
gt = ">"
lt = "<"
charmap = {
"==": "=",
">": gt,
"<": lt,
">=": r"\geq",
"<=": r"\leq",
"!=": r"\neq",
}
return "%s %s %s" % (self._print(expr.lhs),
charmap[expr.rel_op], self._print(expr.rhs))
def _print_Piecewise(self, expr):
ecpairs = [r"%s & \text{for}\: %s" % (self._print(e), self._print(c))
for e, c in expr.args[:-1]]
if expr.args[-1].cond == true:
ecpairs.append(r"%s & \text{otherwise}" %
self._print(expr.args[-1].expr))
else:
ecpairs.append(r"%s & \text{for}\: %s" %
(self._print(expr.args[-1].expr),
self._print(expr.args[-1].cond)))
tex = r"\begin{cases} %s \end{cases}"
return tex % r" \\".join(ecpairs)
def _print_MatrixBase(self, expr):
lines = []
for line in range(expr.rows): # horrible, should be 'rows'
lines.append(" & ".join([self._print(i) for i in expr[line, :]]))
mat_str = self._settings['mat_str']
if mat_str is None:
if self._settings['mode'] == 'inline':
mat_str = 'smallmatrix'
else:
if (expr.cols <= 10) is True:
mat_str = 'matrix'
else:
mat_str = 'array'
out_str = r'\begin{%MATSTR%}%s\end{%MATSTR%}'
out_str = out_str.replace('%MATSTR%', mat_str)
if mat_str == 'array':
out_str = out_str.replace('%s', '{' + 'c'*expr.cols + '}%s')
if self._settings['mat_delim']:
left_delim = self._settings['mat_delim']
right_delim = self._delim_dict[left_delim]
out_str = r'\left' + left_delim + out_str + \
r'\right' + right_delim
return out_str % r"\\".join(lines)
def _print_MatrixElement(self, expr):
return self.parenthesize(expr.parent, PRECEDENCE["Atom"], strict=True)\
+ '_{%s, %s}' % (self._print(expr.i), self._print(expr.j))
def _print_MatrixSlice(self, expr):
def latexslice(x, dim):
x = list(x)
if x[2] == 1:
del x[2]
if x[0] == 0:
x[0] = None
if x[1] == dim:
x[1] = None
return ':'.join(self._print(xi) if xi is not None else '' for xi in x)
return (self.parenthesize(expr.parent, PRECEDENCE["Atom"], strict=True) + r'\left[' +
latexslice(expr.rowslice, expr.parent.rows) + ', ' +
latexslice(expr.colslice, expr.parent.cols) + r'\right]')
def _print_BlockMatrix(self, expr):
return self._print(expr.blocks)
def _print_Transpose(self, expr):
mat = expr.arg
from sympy.matrices import MatrixSymbol
if not isinstance(mat, MatrixSymbol):
return r"\left(%s\right)^{T}" % self._print(mat)
else:
return "%s^{T}" % self.parenthesize(mat, precedence_traditional(expr), True)
def _print_Trace(self, expr):
mat = expr.arg
return r"\operatorname{tr}\left(%s \right)" % self._print(mat)
def _print_Adjoint(self, expr):
mat = expr.arg
from sympy.matrices import MatrixSymbol
if not isinstance(mat, MatrixSymbol):
return r"\left(%s\right)^{\dagger}" % self._print(mat)
else:
return r"%s^{\dagger}" % self._print(mat)
def _print_MatMul(self, expr):
from sympy import MatMul, Mul
parens = lambda x: self.parenthesize(x, precedence_traditional(expr),
False)
args = expr.args
if isinstance(args[0], Mul):
args = args[0].as_ordered_factors() + list(args[1:])
else:
args = list(args)
if isinstance(expr, MatMul) and _coeff_isneg(expr):
if args[0] == -1:
args = args[1:]
else:
args[0] = -args[0]
return '- ' + ' '.join(map(parens, args))
else:
return ' '.join(map(parens, args))
def _print_Mod(self, expr, exp=None):
if exp is not None:
return r'\left(%s\bmod{%s}\right)^{%s}' % \
(self.parenthesize(expr.args[0], PRECEDENCE['Mul'],
strict=True), self._print(expr.args[1]),
exp)
return r'%s\bmod{%s}' % (self.parenthesize(expr.args[0],
PRECEDENCE['Mul'], strict=True),
self._print(expr.args[1]))
def _print_HadamardProduct(self, expr):
args = expr.args
prec = PRECEDENCE['Pow']
parens = self.parenthesize
return r' \circ '.join(
map(lambda arg: parens(arg, prec, strict=True), args))
def _print_HadamardPower(self, expr):
if precedence_traditional(expr.exp) < PRECEDENCE["Mul"]:
template = r"%s^{\circ \left({%s}\right)}"
else:
template = r"%s^{\circ {%s}}"
return self._helper_print_standard_power(expr, template)
def _print_KroneckerProduct(self, expr):
args = expr.args
prec = PRECEDENCE['Pow']
parens = self.parenthesize
return r' \otimes '.join(
map(lambda arg: parens(arg, prec, strict=True), args))
def _print_MatPow(self, expr):
base, exp = expr.base, expr.exp
from sympy.matrices import MatrixSymbol
if not isinstance(base, MatrixSymbol):
return "\\left(%s\\right)^{%s}" % (self._print(base),
self._print(exp))
else:
return "%s^{%s}" % (self._print(base), self._print(exp))
def _print_MatrixSymbol(self, expr):
return self._print_Symbol(expr, style=self._settings[
'mat_symbol_style'])
def _print_ZeroMatrix(self, Z):
return r"\mathbb{0}" if self._settings[
'mat_symbol_style'] == 'plain' else r"\mathbf{0}"
def _print_OneMatrix(self, O):
return r"\mathbb{1}" if self._settings[
'mat_symbol_style'] == 'plain' else r"\mathbf{1}"
def _print_Identity(self, I):
return r"\mathbb{I}" if self._settings[
'mat_symbol_style'] == 'plain' else r"\mathbf{I}"
def _print_PermutationMatrix(self, P):
perm_str = self._print(P.args[0])
return "P_{%s}" % perm_str
def _print_NDimArray(self, expr):
if expr.rank() == 0:
return self._print(expr[()])
mat_str = self._settings['mat_str']
if mat_str is None:
if self._settings['mode'] == 'inline':
mat_str = 'smallmatrix'
else:
if (expr.rank() == 0) or (expr.shape[-1] <= 10):
mat_str = 'matrix'
else:
mat_str = 'array'
block_str = r'\begin{%MATSTR%}%s\end{%MATSTR%}'
block_str = block_str.replace('%MATSTR%', mat_str)
if self._settings['mat_delim']:
left_delim = self._settings['mat_delim']
right_delim = self._delim_dict[left_delim]
block_str = r'\left' + left_delim + block_str + \
r'\right' + right_delim
if expr.rank() == 0:
return block_str % ""
level_str = [[]] + [[] for i in range(expr.rank())]
shape_ranges = [list(range(i)) for i in expr.shape]
for outer_i in itertools.product(*shape_ranges):
level_str[-1].append(self._print(expr[outer_i]))
even = True
for back_outer_i in range(expr.rank()-1, -1, -1):
if len(level_str[back_outer_i+1]) < expr.shape[back_outer_i]:
break
if even:
level_str[back_outer_i].append(
r" & ".join(level_str[back_outer_i+1]))
else:
level_str[back_outer_i].append(
block_str % (r"\\".join(level_str[back_outer_i+1])))
if len(level_str[back_outer_i+1]) == 1:
level_str[back_outer_i][-1] = r"\left[" + \
level_str[back_outer_i][-1] + r"\right]"
even = not even
level_str[back_outer_i+1] = []
out_str = level_str[0][0]
if expr.rank() % 2 == 1:
out_str = block_str % out_str
return out_str
def _printer_tensor_indices(self, name, indices, index_map={}):
out_str = self._print(name)
last_valence = None
prev_map = None
for index in indices:
new_valence = index.is_up
if ((index in index_map) or prev_map) and \
last_valence == new_valence:
out_str += ","
if last_valence != new_valence:
if last_valence is not None:
out_str += "}"
if index.is_up:
out_str += "{}^{"
else:
out_str += "{}_{"
out_str += self._print(index.args[0])
if index in index_map:
out_str += "="
out_str += self._print(index_map[index])
prev_map = True
else:
prev_map = False
last_valence = new_valence
if last_valence is not None:
out_str += "}"
return out_str
def _print_Tensor(self, expr):
name = expr.args[0].args[0]
indices = expr.get_indices()
return self._printer_tensor_indices(name, indices)
def _print_TensorElement(self, expr):
name = expr.expr.args[0].args[0]
indices = expr.expr.get_indices()
index_map = expr.index_map
return self._printer_tensor_indices(name, indices, index_map)
def _print_TensMul(self, expr):
# prints expressions like "A(a)", "3*A(a)", "(1+x)*A(a)"
sign, args = expr._get_args_for_traditional_printer()
return sign + "".join(
[self.parenthesize(arg, precedence(expr)) for arg in args]
)
def _print_TensAdd(self, expr):
a = []
args = expr.args
for x in args:
a.append(self.parenthesize(x, precedence(expr)))
a.sort()
s = ' + '.join(a)
s = s.replace('+ -', '- ')
return s
def _print_TensorIndex(self, expr):
return "{}%s{%s}" % (
"^" if expr.is_up else "_",
self._print(expr.args[0])
)
def _print_PartialDerivative(self, expr):
if len(expr.variables) == 1:
return r"\frac{\partial}{\partial {%s}}{%s}" % (
self._print(expr.variables[0]),
self.parenthesize(expr.expr, PRECEDENCE["Mul"], False)
)
else:
return r"\frac{\partial^{%s}}{%s}{%s}" % (
len(expr.variables),
" ".join([r"\partial {%s}" % self._print(i) for i in expr.variables]),
self.parenthesize(expr.expr, PRECEDENCE["Mul"], False)
)
def _print_ArraySymbol(self, expr):
return self._print(expr.name)
def _print_ArrayElement(self, expr):
return "{{%s}_{%s}}" % (expr.name, ", ".join([f"{self._print(i)}" for i in expr.indices]))
def _print_UniversalSet(self, expr):
return r"\mathbb{U}"
def _print_frac(self, expr, exp=None):
if exp is None:
return r"\operatorname{frac}{\left(%s\right)}" % self._print(expr.args[0])
else:
return r"\operatorname{frac}{\left(%s\right)}^{%s}" % (
self._print(expr.args[0]), exp)
def _print_tuple(self, expr):
if self._settings['decimal_separator'] == 'comma':
sep = ";"
elif self._settings['decimal_separator'] == 'period':
sep = ","
else:
raise ValueError('Unknown Decimal Separator')
if len(expr) == 1:
# 1-tuple needs a trailing separator
return self._add_parens_lspace(self._print(expr[0]) + sep)
else:
return self._add_parens_lspace(
(sep + r" \ ").join([self._print(i) for i in expr]))
def _print_TensorProduct(self, expr):
elements = [self._print(a) for a in expr.args]
return r' \otimes '.join(elements)
def _print_WedgeProduct(self, expr):
elements = [self._print(a) for a in expr.args]
return r' \wedge '.join(elements)
def _print_Tuple(self, expr):
return self._print_tuple(expr)
def _print_list(self, expr):
if self._settings['decimal_separator'] == 'comma':
return r"\left[ %s\right]" % \
r"; \ ".join([self._print(i) for i in expr])
elif self._settings['decimal_separator'] == 'period':
return r"\left[ %s\right]" % \
r", \ ".join([self._print(i) for i in expr])
else:
raise ValueError('Unknown Decimal Separator')
def _print_dict(self, d):
keys = sorted(d.keys(), key=default_sort_key)
items = []
for key in keys:
val = d[key]
items.append("%s : %s" % (self._print(key), self._print(val)))
return r"\left\{ %s\right\}" % r", \ ".join(items)
def _print_Dict(self, expr):
return self._print_dict(expr)
def _print_DiracDelta(self, expr, exp=None):
if len(expr.args) == 1 or expr.args[1] == 0:
tex = r"\delta\left(%s\right)" % self._print(expr.args[0])
else:
tex = r"\delta^{\left( %s \right)}\left( %s \right)" % (
self._print(expr.args[1]), self._print(expr.args[0]))
if exp:
tex = r"\left(%s\right)^{%s}" % (tex, exp)
return tex
def _print_SingularityFunction(self, expr, exp=None):
shift = self._print(expr.args[0] - expr.args[1])
power = self._print(expr.args[2])
tex = r"{\left\langle %s \right\rangle}^{%s}" % (shift, power)
if exp is not None:
tex = r"{\left({\langle %s \rangle}^{%s}\right)}^{%s}" % (shift, power, exp)
return tex
def _print_Heaviside(self, expr, exp=None):
tex = r"\theta\left(%s\right)" % self._print(expr.args[0])
if exp:
tex = r"\left(%s\right)^{%s}" % (tex, exp)
return tex
def _print_KroneckerDelta(self, expr, exp=None):
i = self._print(expr.args[0])
j = self._print(expr.args[1])
if expr.args[0].is_Atom and expr.args[1].is_Atom:
tex = r'\delta_{%s %s}' % (i, j)
else:
tex = r'\delta_{%s, %s}' % (i, j)
if exp is not None:
tex = r'\left(%s\right)^{%s}' % (tex, exp)
return tex
def _print_LeviCivita(self, expr, exp=None):
indices = map(self._print, expr.args)
if all(x.is_Atom for x in expr.args):
tex = r'\varepsilon_{%s}' % " ".join(indices)
else:
tex = r'\varepsilon_{%s}' % ", ".join(indices)
if exp:
tex = r'\left(%s\right)^{%s}' % (tex, exp)
return tex
def _print_RandomDomain(self, d):
if hasattr(d, 'as_boolean'):
return '\\text{Domain: }' + self._print(d.as_boolean())
elif hasattr(d, 'set'):
return ('\\text{Domain: }' + self._print(d.symbols) + '\\text{ in }' +
self._print(d.set))
elif hasattr(d, 'symbols'):
return '\\text{Domain on }' + self._print(d.symbols)
else:
return self._print(None)
def _print_FiniteSet(self, s):
items = sorted(s.args, key=default_sort_key)
return self._print_set(items)
def _print_set(self, s):
items = sorted(s, key=default_sort_key)
if self._settings['decimal_separator'] == 'comma':
items = "; ".join(map(self._print, items))
elif self._settings['decimal_separator'] == 'period':
items = ", ".join(map(self._print, items))
else:
raise ValueError('Unknown Decimal Separator')
return r"\left\{%s\right\}" % items
_print_frozenset = _print_set
def _print_Range(self, s):
dots = object()
if s.has(Symbol):
return self._print_Basic(s)
if s.start.is_infinite and s.stop.is_infinite:
if s.step.is_positive:
printset = dots, -1, 0, 1, dots
else:
printset = dots, 1, 0, -1, dots
elif s.start.is_infinite:
printset = dots, s[-1] - s.step, s[-1]
elif s.stop.is_infinite:
it = iter(s)
printset = next(it), next(it), dots
elif len(s) > 4:
it = iter(s)
printset = next(it), next(it), dots, s[-1]
else:
printset = tuple(s)
return (r"\left\{" +
r", ".join(self._print(el) if el is not dots else r'\ldots' for el in printset) +
r"\right\}")
def __print_number_polynomial(self, expr, letter, exp=None):
if len(expr.args) == 2:
if exp is not None:
return r"%s_{%s}^{%s}\left(%s\right)" % (letter,
self._print(expr.args[0]), exp,
self._print(expr.args[1]))
return r"%s_{%s}\left(%s\right)" % (letter,
self._print(expr.args[0]), self._print(expr.args[1]))
tex = r"%s_{%s}" % (letter, self._print(expr.args[0]))
if exp is not None:
tex = r"%s^{%s}" % (tex, exp)
return tex
def _print_bernoulli(self, expr, exp=None):
return self.__print_number_polynomial(expr, "B", exp)
def _print_bell(self, expr, exp=None):
if len(expr.args) == 3:
tex1 = r"B_{%s, %s}" % (self._print(expr.args[0]),
self._print(expr.args[1]))
tex2 = r"\left(%s\right)" % r", ".join(self._print(el) for
el in expr.args[2])
if exp is not None:
tex = r"%s^{%s}%s" % (tex1, exp, tex2)
else:
tex = tex1 + tex2
return tex
return self.__print_number_polynomial(expr, "B", exp)
def _print_fibonacci(self, expr, exp=None):
return self.__print_number_polynomial(expr, "F", exp)
def _print_lucas(self, expr, exp=None):
tex = r"L_{%s}" % self._print(expr.args[0])
if exp is not None:
tex = r"%s^{%s}" % (tex, exp)
return tex
def _print_tribonacci(self, expr, exp=None):
return self.__print_number_polynomial(expr, "T", exp)
def _print_SeqFormula(self, s):
dots = object()
if len(s.start.free_symbols) > 0 or len(s.stop.free_symbols) > 0:
return r"\left\{%s\right\}_{%s=%s}^{%s}" % (
self._print(s.formula),
self._print(s.variables[0]),
self._print(s.start),
self._print(s.stop)
)
if s.start is S.NegativeInfinity:
stop = s.stop
printset = (dots, s.coeff(stop - 3), s.coeff(stop - 2),
s.coeff(stop - 1), s.coeff(stop))
elif s.stop is S.Infinity or s.length > 4:
printset = s[:4]
printset.append(dots)
else:
printset = tuple(s)
return (r"\left[" +
r", ".join(self._print(el) if el is not dots else r'\ldots' for el in printset) +
r"\right]")
_print_SeqPer = _print_SeqFormula
_print_SeqAdd = _print_SeqFormula
_print_SeqMul = _print_SeqFormula
def _print_Interval(self, i):
if i.start == i.end:
return r"\left\{%s\right\}" % self._print(i.start)
else:
if i.left_open:
left = '('
else:
left = '['
if i.right_open:
right = ')'
else:
right = ']'
return r"\left%s%s, %s\right%s" % \
(left, self._print(i.start), self._print(i.end), right)
def _print_AccumulationBounds(self, i):
return r"\left\langle %s, %s\right\rangle" % \
(self._print(i.min), self._print(i.max))
def _print_Union(self, u):
prec = precedence_traditional(u)
args_str = [self.parenthesize(i, prec) for i in u.args]
return r" \cup ".join(args_str)
def _print_Complement(self, u):
prec = precedence_traditional(u)
args_str = [self.parenthesize(i, prec) for i in u.args]
return r" \setminus ".join(args_str)
def _print_Intersection(self, u):
prec = precedence_traditional(u)
args_str = [self.parenthesize(i, prec) for i in u.args]
return r" \cap ".join(args_str)
def _print_SymmetricDifference(self, u):
prec = precedence_traditional(u)
args_str = [self.parenthesize(i, prec) for i in u.args]
return r" \triangle ".join(args_str)
def _print_ProductSet(self, p):
prec = precedence_traditional(p)
if len(p.sets) >= 1 and not has_variety(p.sets):
return self.parenthesize(p.sets[0], prec) + "^{%d}" % len(p.sets)
return r" \times ".join(
self.parenthesize(set, prec) for set in p.sets)
def _print_EmptySet(self, e):
return r"\emptyset"
def _print_Naturals(self, n):
return r"\mathbb{N}"
def _print_Naturals0(self, n):
return r"\mathbb{N}_0"
def _print_Integers(self, i):
return r"\mathbb{Z}"
def _print_Rationals(self, i):
return r"\mathbb{Q}"
def _print_Reals(self, i):
return r"\mathbb{R}"
def _print_Complexes(self, i):
return r"\mathbb{C}"
def _print_ImageSet(self, s):
expr = s.lamda.expr
sig = s.lamda.signature
xys = ((self._print(x), self._print(y)) for x, y in zip(sig, s.base_sets))
xinys = r" , ".join(r"%s \in %s" % xy for xy in xys)
return r"\left\{%s\; \middle|\; %s\right\}" % (self._print(expr), xinys)
def _print_ConditionSet(self, s):
vars_print = ', '.join([self._print(var) for var in Tuple(s.sym)])
if s.base_set is S.UniversalSet:
return r"\left\{%s\; \middle|\; %s \right\}" % \
(vars_print, self._print(s.condition))
return r"\left\{%s\; \middle|\; %s \in %s \wedge %s \right\}" % (
vars_print,
vars_print,
self._print(s.base_set),
self._print(s.condition))
def _print_ComplexRegion(self, s):
vars_print = ', '.join([self._print(var) for var in s.variables])
return r"\left\{%s\; \middle|\; %s \in %s \right\}" % (
self._print(s.expr),
vars_print,
self._print(s.sets))
def _print_Contains(self, e):
return r"%s \in %s" % tuple(self._print(a) for a in e.args)
def _print_FourierSeries(self, s):
return self._print_Add(s.truncate()) + r' + \ldots'
def _print_FormalPowerSeries(self, s):
return self._print_Add(s.infinite)
def _print_FiniteField(self, expr):
return r"\mathbb{F}_{%s}" % expr.mod
def _print_IntegerRing(self, expr):
return r"\mathbb{Z}"
def _print_RationalField(self, expr):
return r"\mathbb{Q}"
def _print_RealField(self, expr):
return r"\mathbb{R}"
def _print_ComplexField(self, expr):
return r"\mathbb{C}"
def _print_PolynomialRing(self, expr):
domain = self._print(expr.domain)
symbols = ", ".join(map(self._print, expr.symbols))
return r"%s\left[%s\right]" % (domain, symbols)
def _print_FractionField(self, expr):
domain = self._print(expr.domain)
symbols = ", ".join(map(self._print, expr.symbols))
return r"%s\left(%s\right)" % (domain, symbols)
def _print_PolynomialRingBase(self, expr):
domain = self._print(expr.domain)
symbols = ", ".join(map(self._print, expr.symbols))
inv = ""
if not expr.is_Poly:
inv = r"S_<^{-1}"
return r"%s%s\left[%s\right]" % (inv, domain, symbols)
def _print_Poly(self, poly):
cls = poly.__class__.__name__
terms = []
for monom, coeff in poly.terms():
s_monom = ''
for i, exp in enumerate(monom):
if exp > 0:
if exp == 1:
s_monom += self._print(poly.gens[i])
else:
s_monom += self._print(pow(poly.gens[i], exp))
if coeff.is_Add:
if s_monom:
s_coeff = r"\left(%s\right)" % self._print(coeff)
else:
s_coeff = self._print(coeff)
else:
if s_monom:
if coeff is S.One:
terms.extend(['+', s_monom])
continue
if coeff is S.NegativeOne:
terms.extend(['-', s_monom])
continue
s_coeff = self._print(coeff)
if not s_monom:
s_term = s_coeff
else:
s_term = s_coeff + " " + s_monom
if s_term.startswith('-'):
terms.extend(['-', s_term[1:]])
else:
terms.extend(['+', s_term])
if terms[0] in ['-', '+']:
modifier = terms.pop(0)
if modifier == '-':
terms[0] = '-' + terms[0]
expr = ' '.join(terms)
gens = list(map(self._print, poly.gens))
domain = "domain=%s" % self._print(poly.get_domain())
args = ", ".join([expr] + gens + [domain])
if cls in accepted_latex_functions:
tex = r"\%s {\left(%s \right)}" % (cls, args)
else:
tex = r"\operatorname{%s}{\left( %s \right)}" % (cls, args)
return tex
def _print_ComplexRootOf(self, root):
cls = root.__class__.__name__
if cls == "ComplexRootOf":
cls = "CRootOf"
expr = self._print(root.expr)
index = root.index
if cls in accepted_latex_functions:
return r"\%s {\left(%s, %d\right)}" % (cls, expr, index)
else:
return r"\operatorname{%s} {\left(%s, %d\right)}" % (cls, expr,
index)
def _print_RootSum(self, expr):
cls = expr.__class__.__name__
args = [self._print(expr.expr)]
if expr.fun is not S.IdentityFunction:
args.append(self._print(expr.fun))
if cls in accepted_latex_functions:
return r"\%s {\left(%s\right)}" % (cls, ", ".join(args))
else:
return r"\operatorname{%s} {\left(%s\right)}" % (cls,
", ".join(args))
def _print_PolyElement(self, poly):
mul_symbol = self._settings['mul_symbol_latex']
return poly.str(self, PRECEDENCE, "{%s}^{%d}", mul_symbol)
def _print_FracElement(self, frac):
if frac.denom == 1:
return self._print(frac.numer)
else:
numer = self._print(frac.numer)
denom = self._print(frac.denom)
return r"\frac{%s}{%s}" % (numer, denom)
def _print_euler(self, expr, exp=None):
m, x = (expr.args[0], None) if len(expr.args) == 1 else expr.args
tex = r"E_{%s}" % self._print(m)
if exp is not None:
tex = r"%s^{%s}" % (tex, exp)
if x is not None:
tex = r"%s\left(%s\right)" % (tex, self._print(x))
return tex
def _print_catalan(self, expr, exp=None):
tex = r"C_{%s}" % self._print(expr.args[0])
if exp is not None:
tex = r"%s^{%s}" % (tex, exp)
return tex
def _print_UnifiedTransform(self, expr, s, inverse=False):
return r"\mathcal{{{}}}{}_{{{}}}\left[{}\right]\left({}\right)".format(s, '^{-1}' if inverse else '', self._print(expr.args[1]), self._print(expr.args[0]), self._print(expr.args[2]))
def _print_MellinTransform(self, expr):
return self._print_UnifiedTransform(expr, 'M')
def _print_InverseMellinTransform(self, expr):
return self._print_UnifiedTransform(expr, 'M', True)
def _print_LaplaceTransform(self, expr):
return self._print_UnifiedTransform(expr, 'L')
def _print_InverseLaplaceTransform(self, expr):
return self._print_UnifiedTransform(expr, 'L', True)
def _print_FourierTransform(self, expr):
return self._print_UnifiedTransform(expr, 'F')
def _print_InverseFourierTransform(self, expr):
return self._print_UnifiedTransform(expr, 'F', True)
def _print_SineTransform(self, expr):
return self._print_UnifiedTransform(expr, 'SIN')
def _print_InverseSineTransform(self, expr):
return self._print_UnifiedTransform(expr, 'SIN', True)
def _print_CosineTransform(self, expr):
return self._print_UnifiedTransform(expr, 'COS')
def _print_InverseCosineTransform(self, expr):
return self._print_UnifiedTransform(expr, 'COS', True)
def _print_DMP(self, p):
try:
if p.ring is not None:
# TODO incorporate order
return self._print(p.ring.to_sympy(p))
except SympifyError:
pass
return self._print(repr(p))
def _print_DMF(self, p):
return self._print_DMP(p)
def _print_Object(self, object):
return self._print(Symbol(object.name))
def _print_LambertW(self, expr):
if len(expr.args) == 1:
return r"W\left(%s\right)" % self._print(expr.args[0])
return r"W_{%s}\left(%s\right)" % \
(self._print(expr.args[1]), self._print(expr.args[0]))
def _print_Morphism(self, morphism):
domain = self._print(morphism.domain)
codomain = self._print(morphism.codomain)
return "%s\\rightarrow %s" % (domain, codomain)
def _print_TransferFunction(self, expr):
from sympy.core import Mul, Pow
num, den = expr.num, expr.den
res = Mul(num, Pow(den, -1, evaluate=False), evaluate=False)
return self._print_Mul(res)
def _print_Series(self, expr):
args = list(expr.args)
parens = lambda x: self.parenthesize(x, precedence_traditional(expr),
False)
return ' '.join(map(parens, args))
def _print_Parallel(self, expr):
args = list(expr.args)
parens = lambda x: self.parenthesize(x, precedence_traditional(expr),
False)
return ' '.join(map(parens, args))
def _print_Feedback(self, expr):
from sympy.physics.control import TransferFunction, Parallel, Series
num, tf = expr.num, TransferFunction(1, 1, expr.num.var)
num_arg_list = list(num.args) if isinstance(num, Series) else [num]
den_arg_list = list(expr.den.args) if isinstance(expr.den, Series) else [expr.den]
if isinstance(num, Series) and isinstance(expr.den, Series):
den = Parallel(tf, Series(*num_arg_list, *den_arg_list))
elif isinstance(num, Series) and isinstance(expr.den, TransferFunction):
if expr.den == tf:
den = Parallel(tf, Series(*num_arg_list))
else:
den = Parallel(tf, Series(*num_arg_list, expr.den))
elif isinstance(num, TransferFunction) and isinstance(expr.den, Series):
if num == tf:
den = Parallel(tf, Series(*den_arg_list))
else:
den = Parallel(tf, Series(num, *den_arg_list))
else:
if num == tf:
den = Parallel(tf, *den_arg_list)
elif expr.den == tf:
den = Parallel(tf, *num_arg_list)
else:
den = Parallel(tf, Series(*num_arg_list, *den_arg_list))
numer = self._print(num)
denom = self._print(den)
return r"\frac{%s}{%s}" % (numer, denom)
def _print_NamedMorphism(self, morphism):
pretty_name = self._print(Symbol(morphism.name))
pretty_morphism = self._print_Morphism(morphism)
return "%s:%s" % (pretty_name, pretty_morphism)
def _print_IdentityMorphism(self, morphism):
from sympy.categories import NamedMorphism
return self._print_NamedMorphism(NamedMorphism(
morphism.domain, morphism.codomain, "id"))
def _print_CompositeMorphism(self, morphism):
# All components of the morphism have names and it is thus
# possible to build the name of the composite.
component_names_list = [self._print(Symbol(component.name)) for
component in morphism.components]
component_names_list.reverse()
component_names = "\\circ ".join(component_names_list) + ":"
pretty_morphism = self._print_Morphism(morphism)
return component_names + pretty_morphism
def _print_Category(self, morphism):
return r"\mathbf{{{}}}".format(self._print(Symbol(morphism.name)))
def _print_Diagram(self, diagram):
if not diagram.premises:
# This is an empty diagram.
return self._print(S.EmptySet)
latex_result = self._print(diagram.premises)
if diagram.conclusions:
latex_result += "\\Longrightarrow %s" % \
self._print(diagram.conclusions)
return latex_result
def _print_DiagramGrid(self, grid):
latex_result = "\\begin{array}{%s}\n" % ("c" * grid.width)
for i in range(grid.height):
for j in range(grid.width):
if grid[i, j]:
latex_result += latex(grid[i, j])
latex_result += " "
if j != grid.width - 1:
latex_result += "& "
if i != grid.height - 1:
latex_result += "\\\\"
latex_result += "\n"
latex_result += "\\end{array}\n"
return latex_result
def _print_FreeModule(self, M):
return '{{{}}}^{{{}}}'.format(self._print(M.ring), self._print(M.rank))
def _print_FreeModuleElement(self, m):
# Print as row vector for convenience, for now.
return r"\left[ {} \right]".format(",".join(
'{' + self._print(x) + '}' for x in m))
def _print_SubModule(self, m):
return r"\left\langle {} \right\rangle".format(",".join(
'{' + self._print(x) + '}' for x in m.gens))
def _print_ModuleImplementedIdeal(self, m):
return r"\left\langle {} \right\rangle".format(",".join(
'{' + self._print(x) + '}' for [x] in m._module.gens))
def _print_Quaternion(self, expr):
# TODO: This expression is potentially confusing,
# shall we print it as `Quaternion( ... )`?
s = [self.parenthesize(i, PRECEDENCE["Mul"], strict=True)
for i in expr.args]
a = [s[0]] + [i+" "+j for i, j in zip(s[1:], "ijk")]
return " + ".join(a)
def _print_QuotientRing(self, R):
# TODO nicer fractions for few generators...
return r"\frac{{{}}}{{{}}}".format(self._print(R.ring),
self._print(R.base_ideal))
def _print_QuotientRingElement(self, x):
return r"{{{}}} + {{{}}}".format(self._print(x.data),
self._print(x.ring.base_ideal))
def _print_QuotientModuleElement(self, m):
return r"{{{}}} + {{{}}}".format(self._print(m.data),
self._print(m.module.killed_module))
def _print_QuotientModule(self, M):
# TODO nicer fractions for few generators...
return r"\frac{{{}}}{{{}}}".format(self._print(M.base),
self._print(M.killed_module))
def _print_MatrixHomomorphism(self, h):
return r"{{{}}} : {{{}}} \to {{{}}}".format(self._print(h._sympy_matrix()),
self._print(h.domain), self._print(h.codomain))
def _print_Manifold(self, manifold):
string = manifold.name.name
if '{' in string:
name, supers, subs = string, [], []
else:
name, supers, subs = split_super_sub(string)
name = translate(name)
supers = [translate(sup) for sup in supers]
subs = [translate(sub) for sub in subs]
name = r'\text{%s}' % name
if supers:
name += "^{%s}" % " ".join(supers)
if subs:
name += "_{%s}" % " ".join(subs)
return name
def _print_Patch(self, patch):
return r'\text{%s}_{%s}' % (self._print(patch.name), self._print(patch.manifold))
def _print_CoordSystem(self, coordsys):
return r'\text{%s}^{\text{%s}}_{%s}' % (
self._print(coordsys.name), self._print(coordsys.patch.name), self._print(coordsys.manifold)
)
def _print_CovarDerivativeOp(self, cvd):
return r'\mathbb{\nabla}_{%s}' % self._print(cvd._wrt)
def _print_BaseScalarField(self, field):
string = field._coord_sys.symbols[field._index].name
return r'\mathbf{{{}}}'.format(self._print(Symbol(string)))
def _print_BaseVectorField(self, field):
string = field._coord_sys.symbols[field._index].name
return r'\partial_{{{}}}'.format(self._print(Symbol(string)))
def _print_Differential(self, diff):
field = diff._form_field
if hasattr(field, '_coord_sys'):
string = field._coord_sys.symbols[field._index].name
return r'\operatorname{{d}}{}'.format(self._print(Symbol(string)))
else:
string = self._print(field)
return r'\operatorname{{d}}\left({}\right)'.format(string)
def _print_Tr(self, p):
# TODO: Handle indices
contents = self._print(p.args[0])
return r'\operatorname{{tr}}\left({}\right)'.format(contents)
def _print_totient(self, expr, exp=None):
if exp is not None:
return r'\left(\phi\left(%s\right)\right)^{%s}' % \
(self._print(expr.args[0]), exp)
return r'\phi\left(%s\right)' % self._print(expr.args[0])
def _print_reduced_totient(self, expr, exp=None):
if exp is not None:
return r'\left(\lambda\left(%s\right)\right)^{%s}' % \
(self._print(expr.args[0]), exp)
return r'\lambda\left(%s\right)' % self._print(expr.args[0])
def _print_divisor_sigma(self, expr, exp=None):
if len(expr.args) == 2:
tex = r"_%s\left(%s\right)" % tuple(map(self._print,
(expr.args[1], expr.args[0])))
else:
tex = r"\left(%s\right)" % self._print(expr.args[0])
if exp is not None:
return r"\sigma^{%s}%s" % (exp, tex)
return r"\sigma%s" % tex
def _print_udivisor_sigma(self, expr, exp=None):
if len(expr.args) == 2:
tex = r"_%s\left(%s\right)" % tuple(map(self._print,
(expr.args[1], expr.args[0])))
else:
tex = r"\left(%s\right)" % self._print(expr.args[0])
if exp is not None:
return r"\sigma^*^{%s}%s" % (exp, tex)
return r"\sigma^*%s" % tex
def _print_primenu(self, expr, exp=None):
if exp is not None:
return r'\left(\nu\left(%s\right)\right)^{%s}' % \
(self._print(expr.args[0]), exp)
return r'\nu\left(%s\right)' % self._print(expr.args[0])
def _print_primeomega(self, expr, exp=None):
if exp is not None:
return r'\left(\Omega\left(%s\right)\right)^{%s}' % \
(self._print(expr.args[0]), exp)
return r'\Omega\left(%s\right)' % self._print(expr.args[0])
def _print_Str(self, s):
return str(s.name)
def _print_float(self, expr):
return self._print(Float(expr))
def _print_int(self, expr):
return str(expr)
def _print_mpz(self, expr):
return str(expr)
def _print_mpq(self, expr):
return str(expr)
def _print_Predicate(self, expr):
return str(expr)
def _print_AppliedPredicate(self, expr):
pred = expr.function
args = expr.arguments
pred_latex = self._print(pred)
args_latex = ', '.join([self._print(a) for a in args])
return '%s(%s)' % (pred_latex, args_latex)
def emptyPrinter(self, expr):
# default to just printing as monospace, like would normally be shown
s = super().emptyPrinter(expr)
return r"\mathtt{\text{%s}}" % latex_escape(s)
def translate(s):
r'''
Check for a modifier ending the string. If present, convert the
modifier to latex and translate the rest recursively.
Given a description of a Greek letter or other special character,
return the appropriate latex.
Let everything else pass as given.
>>> from sympy.printing.latex import translate
>>> translate('alphahatdotprime')
"{\\dot{\\hat{\\alpha}}}'"
'''
# Process the rest
tex = tex_greek_dictionary.get(s)
if tex:
return tex
elif s.lower() in greek_letters_set:
return "\\" + s.lower()
elif s in other_symbols:
return "\\" + s
else:
# Process modifiers, if any, and recurse
for key in sorted(modifier_dict.keys(), key=lambda k:len(k), reverse=True):
if s.lower().endswith(key) and len(s) > len(key):
return modifier_dict[key](translate(s[:-len(key)]))
return s
@print_function(LatexPrinter)
def latex(expr, **settings):
r"""Convert the given expression to LaTeX string representation.
Parameters
==========
full_prec: boolean, optional
If set to True, a floating point number is printed with full precision.
fold_frac_powers : boolean, optional
Emit ``^{p/q}`` instead of ``^{\frac{p}{q}}`` for fractional powers.
fold_func_brackets : boolean, optional
Fold function brackets where applicable.
fold_short_frac : boolean, optional
Emit ``p / q`` instead of ``\frac{p}{q}`` when the denominator is
simple enough (at most two terms and no powers). The default value is
``True`` for inline mode, ``False`` otherwise.
inv_trig_style : string, optional
How inverse trig functions should be displayed. Can be one of
``abbreviated``, ``full``, or ``power``. Defaults to ``abbreviated``.
itex : boolean, optional
Specifies if itex-specific syntax is used, including emitting
``$$...$$``.
ln_notation : boolean, optional
If set to ``True``, ``\ln`` is used instead of default ``\log``.
long_frac_ratio : float or None, optional
The allowed ratio of the width of the numerator to the width of the
denominator before the printer breaks off long fractions. If ``None``
(the default value), long fractions are not broken up.
mat_delim : string, optional
The delimiter to wrap around matrices. Can be one of ``[``, ``(``, or
the empty string. Defaults to ``[``.
mat_str : string, optional
Which matrix environment string to emit. ``smallmatrix``, ``matrix``,
``array``, etc. Defaults to ``smallmatrix`` for inline mode, ``matrix``
for matrices of no more than 10 columns, and ``array`` otherwise.
mode: string, optional
Specifies how the generated code will be delimited. ``mode`` can be one
of ``plain``, ``inline``, ``equation`` or ``equation*``. If ``mode``
is set to ``plain``, then the resulting code will not be delimited at
all (this is the default). If ``mode`` is set to ``inline`` then inline
LaTeX ``$...$`` will be used. If ``mode`` is set to ``equation`` or
``equation*``, the resulting code will be enclosed in the ``equation``
or ``equation*`` environment (remember to import ``amsmath`` for
``equation*``), unless the ``itex`` option is set. In the latter case,
the ``$$...$$`` syntax is used.
mul_symbol : string or None, optional
The symbol to use for multiplication. Can be one of ``None``, ``ldot``,
``dot``, or ``times``.
order: string, optional
Any of the supported monomial orderings (currently ``lex``, ``grlex``,
or ``grevlex``), ``old``, and ``none``. This parameter does nothing for
Mul objects. Setting order to ``old`` uses the compatibility ordering
for Add defined in Printer. For very large expressions, set the
``order`` keyword to ``none`` if speed is a concern.
symbol_names : dictionary of strings mapped to symbols, optional
Dictionary of symbols and the custom strings they should be emitted as.
root_notation : boolean, optional
If set to ``False``, exponents of the form 1/n are printed in fractonal
form. Default is ``True``, to print exponent in root form.
mat_symbol_style : string, optional
Can be either ``plain`` (default) or ``bold``. If set to ``bold``,
a MatrixSymbol A will be printed as ``\mathbf{A}``, otherwise as ``A``.
imaginary_unit : string, optional
String to use for the imaginary unit. Defined options are "i" (default)
and "j". Adding "r" or "t" in front gives ``\mathrm`` or ``\text``, so
"ri" leads to ``\mathrm{i}`` which gives `\mathrm{i}`.
gothic_re_im : boolean, optional
If set to ``True``, `\Re` and `\Im` is used for ``re`` and ``im``, respectively.
The default is ``False`` leading to `\operatorname{re}` and `\operatorname{im}`.
decimal_separator : string, optional
Specifies what separator to use to separate the whole and fractional parts of a
floating point number as in `2.5` for the default, ``period`` or `2{,}5`
when ``comma`` is specified. Lists, sets, and tuple are printed with semicolon
separating the elements when ``comma`` is chosen. For example, [1; 2; 3] when
``comma`` is chosen and [1,2,3] for when ``period`` is chosen.
parenthesize_super : boolean, optional
If set to ``False``, superscripted expressions will not be parenthesized when
powered. Default is ``True``, which parenthesizes the expression when powered.
min: Integer or None, optional
Sets the lower bound for the exponent to print floating point numbers in
fixed-point format.
max: Integer or None, optional
Sets the upper bound for the exponent to print floating point numbers in
fixed-point format.
Notes
=====
Not using a print statement for printing, results in double backslashes for
latex commands since that's the way Python escapes backslashes in strings.
>>> from sympy import latex, Rational
>>> from sympy.abc import tau
>>> latex((2*tau)**Rational(7,2))
'8 \\sqrt{2} \\tau^{\\frac{7}{2}}'
>>> print(latex((2*tau)**Rational(7,2)))
8 \sqrt{2} \tau^{\frac{7}{2}}
Examples
========
>>> from sympy import latex, pi, sin, asin, Integral, Matrix, Rational, log
>>> from sympy.abc import x, y, mu, r, tau
Basic usage:
>>> print(latex((2*tau)**Rational(7,2)))
8 \sqrt{2} \tau^{\frac{7}{2}}
``mode`` and ``itex`` options:
>>> print(latex((2*mu)**Rational(7,2), mode='plain'))
8 \sqrt{2} \mu^{\frac{7}{2}}
>>> print(latex((2*tau)**Rational(7,2), mode='inline'))
$8 \sqrt{2} \tau^{7 / 2}$
>>> print(latex((2*mu)**Rational(7,2), mode='equation*'))
\begin{equation*}8 \sqrt{2} \mu^{\frac{7}{2}}\end{equation*}
>>> print(latex((2*mu)**Rational(7,2), mode='equation'))
\begin{equation}8 \sqrt{2} \mu^{\frac{7}{2}}\end{equation}
>>> print(latex((2*mu)**Rational(7,2), mode='equation', itex=True))
$$8 \sqrt{2} \mu^{\frac{7}{2}}$$
>>> print(latex((2*mu)**Rational(7,2), mode='plain'))
8 \sqrt{2} \mu^{\frac{7}{2}}
>>> print(latex((2*tau)**Rational(7,2), mode='inline'))
$8 \sqrt{2} \tau^{7 / 2}$
>>> print(latex((2*mu)**Rational(7,2), mode='equation*'))
\begin{equation*}8 \sqrt{2} \mu^{\frac{7}{2}}\end{equation*}
>>> print(latex((2*mu)**Rational(7,2), mode='equation'))
\begin{equation}8 \sqrt{2} \mu^{\frac{7}{2}}\end{equation}
>>> print(latex((2*mu)**Rational(7,2), mode='equation', itex=True))
$$8 \sqrt{2} \mu^{\frac{7}{2}}$$
Fraction options:
>>> print(latex((2*tau)**Rational(7,2), fold_frac_powers=True))
8 \sqrt{2} \tau^{7/2}
>>> print(latex((2*tau)**sin(Rational(7,2))))
\left(2 \tau\right)^{\sin{\left(\frac{7}{2} \right)}}
>>> print(latex((2*tau)**sin(Rational(7,2)), fold_func_brackets=True))
\left(2 \tau\right)^{\sin {\frac{7}{2}}}
>>> print(latex(3*x**2/y))
\frac{3 x^{2}}{y}
>>> print(latex(3*x**2/y, fold_short_frac=True))
3 x^{2} / y
>>> print(latex(Integral(r, r)/2/pi, long_frac_ratio=2))
\frac{\int r\, dr}{2 \pi}
>>> print(latex(Integral(r, r)/2/pi, long_frac_ratio=0))
\frac{1}{2 \pi} \int r\, dr
Multiplication options:
>>> print(latex((2*tau)**sin(Rational(7,2)), mul_symbol="times"))
\left(2 \times \tau\right)^{\sin{\left(\frac{7}{2} \right)}}
Trig options:
>>> print(latex(asin(Rational(7,2))))
\operatorname{asin}{\left(\frac{7}{2} \right)}
>>> print(latex(asin(Rational(7,2)), inv_trig_style="full"))
\arcsin{\left(\frac{7}{2} \right)}
>>> print(latex(asin(Rational(7,2)), inv_trig_style="power"))
\sin^{-1}{\left(\frac{7}{2} \right)}
Matrix options:
>>> print(latex(Matrix(2, 1, [x, y])))
\left[\begin{matrix}x\\y\end{matrix}\right]
>>> print(latex(Matrix(2, 1, [x, y]), mat_str = "array"))
\left[\begin{array}{c}x\\y\end{array}\right]
>>> print(latex(Matrix(2, 1, [x, y]), mat_delim="("))
\left(\begin{matrix}x\\y\end{matrix}\right)
Custom printing of symbols:
>>> print(latex(x**2, symbol_names={x: 'x_i'}))
x_i^{2}
Logarithms:
>>> print(latex(log(10)))
\log{\left(10 \right)}
>>> print(latex(log(10), ln_notation=True))
\ln{\left(10 \right)}
``latex()`` also supports the builtin container types :class:`list`,
:class:`tuple`, and :class:`dict`:
>>> print(latex([2/x, y], mode='inline'))
$\left[ 2 / x, \ y\right]$
Unsupported types are rendered as monospaced plaintext:
>>> print(latex(int))
\mathtt{\text{<class 'int'>}}
>>> print(latex("plain % text"))
\mathtt{\text{plain \% text}}
See :ref:`printer_method_example` for an example of how to override
this behavior for your own types by implementing ``_latex``.
.. versionchanged:: 1.7.0
Unsupported types no longer have their ``str`` representation treated as valid latex.
"""
return LatexPrinter(settings).doprint(expr)
def print_latex(expr, **settings):
"""Prints LaTeX representation of the given expression. Takes the same
settings as ``latex()``."""
print(latex(expr, **settings))
def multiline_latex(lhs, rhs, terms_per_line=1, environment="align*", use_dots=False, **settings):
r"""
This function generates a LaTeX equation with a multiline right-hand side
in an ``align*``, ``eqnarray`` or ``IEEEeqnarray`` environment.
Parameters
==========
lhs : Expr
Left-hand side of equation
rhs : Expr
Right-hand side of equation
terms_per_line : integer, optional
Number of terms per line to print. Default is 1.
environment : "string", optional
Which LaTeX wnvironment to use for the output. Options are "align*"
(default), "eqnarray", and "IEEEeqnarray".
use_dots : boolean, optional
If ``True``, ``\\dots`` is added to the end of each line. Default is ``False``.
Examples
========
>>> from sympy import multiline_latex, symbols, sin, cos, exp, log, I
>>> x, y, alpha = symbols('x y alpha')
>>> expr = sin(alpha*y) + exp(I*alpha) - cos(log(y))
>>> print(multiline_latex(x, expr))
\begin{align*}
x = & e^{i \alpha} \\
& + \sin{\left(\alpha y \right)} \\
& - \cos{\left(\log{\left(y \right)} \right)}
\end{align*}
Using at most two terms per line:
>>> print(multiline_latex(x, expr, 2))
\begin{align*}
x = & e^{i \alpha} + \sin{\left(\alpha y \right)} \\
& - \cos{\left(\log{\left(y \right)} \right)}
\end{align*}
Using ``eqnarray`` and dots:
>>> print(multiline_latex(x, expr, terms_per_line=2, environment="eqnarray", use_dots=True))
\begin{eqnarray}
x & = & e^{i \alpha} + \sin{\left(\alpha y \right)} \dots\nonumber\\
& & - \cos{\left(\log{\left(y \right)} \right)}
\end{eqnarray}
Using ``IEEEeqnarray``:
>>> print(multiline_latex(x, expr, environment="IEEEeqnarray"))
\begin{IEEEeqnarray}{rCl}
x & = & e^{i \alpha} \nonumber\\
& & + \sin{\left(\alpha y \right)} \nonumber\\
& & - \cos{\left(\log{\left(y \right)} \right)}
\end{IEEEeqnarray}
Notes
=====
All optional parameters from ``latex`` can also be used.
"""
# Based on code from https://github.com/sympy/sympy/issues/3001
l = LatexPrinter(**settings)
if environment == "eqnarray":
result = r'\begin{eqnarray}' + '\n'
first_term = '& = &'
nonumber = r'\nonumber'
end_term = '\n\\end{eqnarray}'
doubleet = True
elif environment == "IEEEeqnarray":
result = r'\begin{IEEEeqnarray}{rCl}' + '\n'
first_term = '& = &'
nonumber = r'\nonumber'
end_term = '\n\\end{IEEEeqnarray}'
doubleet = True
elif environment == "align*":
result = r'\begin{align*}' + '\n'
first_term = '= &'
nonumber = ''
end_term = '\n\\end{align*}'
doubleet = False
else:
raise ValueError("Unknown environment: {}".format(environment))
dots = ''
if use_dots:
dots=r'\dots'
terms = rhs.as_ordered_terms()
n_terms = len(terms)
term_count = 1
for i in range(n_terms):
term = terms[i]
term_start = ''
term_end = ''
sign = '+'
if term_count > terms_per_line:
if doubleet:
term_start = '& & '
else:
term_start = '& '
term_count = 1
if term_count == terms_per_line:
# End of line
if i < n_terms-1:
# There are terms remaining
term_end = dots + nonumber + r'\\' + '\n'
else:
term_end = ''
if term.as_ordered_factors()[0] == -1:
term = -1*term
sign = r'-'
if i == 0: # beginning
if sign == '+':
sign = ''
result += r'{:s} {:s}{:s} {:s} {:s}'.format(l.doprint(lhs),
first_term, sign, l.doprint(term), term_end)
else:
result += r'{:s}{:s} {:s} {:s}'.format(term_start, sign,
l.doprint(term), term_end)
term_count += 1
result += end_term
return result
|
8ae2f0818a282097c822d59a8d8ff8e0c94088fdcfa0e24d6bace4577260b733 | from .pycode import (
PythonCodePrinter,
MpmathPrinter, # MpmathPrinter is imported for backward compatibility
)
from .numpy import NumPyPrinter # NumPyPrinter is imported for backward compatibility
from sympy.utilities import default_sort_key
__all__ = [
'PythonCodePrinter',
'MpmathPrinter',
'NumPyPrinter',
'LambdaPrinter',
'NumPyPrinter',
'lambdarepr',
]
class LambdaPrinter(PythonCodePrinter):
"""
This printer converts expressions into strings that can be used by
lambdify.
"""
printmethod = "_lambdacode"
def _print_And(self, expr):
result = ['(']
for arg in sorted(expr.args, key=default_sort_key):
result.extend(['(', self._print(arg), ')'])
result.append(' and ')
result = result[:-1]
result.append(')')
return ''.join(result)
def _print_Or(self, expr):
result = ['(']
for arg in sorted(expr.args, key=default_sort_key):
result.extend(['(', self._print(arg), ')'])
result.append(' or ')
result = result[:-1]
result.append(')')
return ''.join(result)
def _print_Not(self, expr):
result = ['(', 'not (', self._print(expr.args[0]), '))']
return ''.join(result)
def _print_BooleanTrue(self, expr):
return "True"
def _print_BooleanFalse(self, expr):
return "False"
def _print_ITE(self, expr):
result = [
'((', self._print(expr.args[1]),
') if (', self._print(expr.args[0]),
') else (', self._print(expr.args[2]), '))'
]
return ''.join(result)
def _print_NumberSymbol(self, expr):
return str(expr)
def _print_Pow(self, expr, **kwargs):
# XXX Temporary workaround. Should python math printer be
# isolated from PythonCodePrinter?
return super(PythonCodePrinter, self)._print_Pow(expr, **kwargs)
# numexpr works by altering the string passed to numexpr.evaluate
# rather than by populating a namespace. Thus a special printer...
class NumExprPrinter(LambdaPrinter):
# key, value pairs correspond to sympy name and numexpr name
# functions not appearing in this dict will raise a TypeError
printmethod = "_numexprcode"
_numexpr_functions = {
'sin' : 'sin',
'cos' : 'cos',
'tan' : 'tan',
'asin': 'arcsin',
'acos': 'arccos',
'atan': 'arctan',
'atan2' : 'arctan2',
'sinh' : 'sinh',
'cosh' : 'cosh',
'tanh' : 'tanh',
'asinh': 'arcsinh',
'acosh': 'arccosh',
'atanh': 'arctanh',
'ln' : 'log',
'log': 'log',
'exp': 'exp',
'sqrt' : 'sqrt',
'Abs' : 'abs',
'conjugate' : 'conj',
'im' : 'imag',
're' : 'real',
'where' : 'where',
'complex' : 'complex',
'contains' : 'contains',
}
def _print_ImaginaryUnit(self, expr):
return '1j'
def _print_seq(self, seq, delimiter=', '):
# simplified _print_seq taken from pretty.py
s = [self._print(item) for item in seq]
if s:
return delimiter.join(s)
else:
return ""
def _print_Function(self, e):
func_name = e.func.__name__
nstr = self._numexpr_functions.get(func_name, None)
if nstr is None:
# check for implemented_function
if hasattr(e, '_imp_'):
return "(%s)" % self._print(e._imp_(*e.args))
else:
raise TypeError("numexpr does not support function '%s'" %
func_name)
return "%s(%s)" % (nstr, self._print_seq(e.args))
def _print_Piecewise(self, expr):
"Piecewise function printer"
exprs = [self._print(arg.expr) for arg in expr.args]
conds = [self._print(arg.cond) for arg in expr.args]
# If [default_value, True] is a (expr, cond) sequence in a Piecewise object
# it will behave the same as passing the 'default' kwarg to select()
# *as long as* it is the last element in expr.args.
# If this is not the case, it may be triggered prematurely.
ans = []
parenthesis_count = 0
is_last_cond_True = False
for cond, expr in zip(conds, exprs):
if cond == 'True':
ans.append(expr)
is_last_cond_True = True
break
else:
ans.append('where(%s, %s, ' % (cond, expr))
parenthesis_count += 1
if not is_last_cond_True:
# simplest way to put a nan but raises
# 'RuntimeWarning: invalid value encountered in log'
ans.append('log(-1)')
return ''.join(ans) + ')' * parenthesis_count
def blacklisted(self, expr):
raise TypeError("numexpr cannot be used with %s" %
expr.__class__.__name__)
# blacklist all Matrix printing
_print_SparseMatrix = \
_print_MutableSparseMatrix = \
_print_ImmutableSparseMatrix = \
_print_Matrix = \
_print_DenseMatrix = \
_print_MutableDenseMatrix = \
_print_ImmutableMatrix = \
_print_ImmutableDenseMatrix = \
blacklisted
# blacklist some python expressions
_print_list = \
_print_tuple = \
_print_Tuple = \
_print_dict = \
_print_Dict = \
blacklisted
def doprint(self, expr):
lstr = super().doprint(expr)
return "evaluate('%s', truediv=True)" % lstr
for k in NumExprPrinter._numexpr_functions:
setattr(NumExprPrinter, '_print_%s' % k, NumExprPrinter._print_Function)
def lambdarepr(expr, **settings):
"""
Returns a string usable for lambdifying.
"""
return LambdaPrinter(settings).doprint(expr)
|
94a2d4cb25e811ec3c2df2fe8e04fe995532b5edc1fff113481b9401f9dab33c | from typing import Any, Dict
from sympy.core.compatibility import is_sequence
from sympy.external import import_module
from sympy.printing.printer import Printer
import sympy
from functools import partial
aesara = import_module('aesara')
if aesara:
aes = aesara.scalar
aet = aesara.tensor
from aesara.tensor import nlinalg
from aesara.tensor.elemwise import Elemwise
from aesara.tensor.elemwise import DimShuffle
mapping = {
sympy.Add: aet.add,
sympy.Mul: aet.mul,
sympy.Abs: aet.abs_,
sympy.sign: aet.sgn,
sympy.ceiling: aet.ceil,
sympy.floor: aet.floor,
sympy.log: aet.log,
sympy.exp: aet.exp,
sympy.sqrt: aet.sqrt,
sympy.cos: aet.cos,
sympy.acos: aet.arccos,
sympy.sin: aet.sin,
sympy.asin: aet.arcsin,
sympy.tan: aet.tan,
sympy.atan: aet.arctan,
sympy.atan2: aet.arctan2,
sympy.cosh: aet.cosh,
sympy.acosh: aet.arccosh,
sympy.sinh: aet.sinh,
sympy.asinh: aet.arcsinh,
sympy.tanh: aet.tanh,
sympy.atanh: aet.arctanh,
sympy.re: aet.real,
sympy.im: aet.imag,
sympy.arg: aet.angle,
sympy.erf: aet.erf,
sympy.gamma: aet.gamma,
sympy.loggamma: aet.gammaln,
sympy.Pow: aet.pow,
sympy.Eq: aet.eq,
sympy.StrictGreaterThan: aet.gt,
sympy.StrictLessThan: aet.lt,
sympy.LessThan: aet.le,
sympy.GreaterThan: aet.ge,
sympy.And: aet.and_,
sympy.Or: aet.or_,
sympy.Max: aet.maximum, # Sympy accept >2 inputs, Aesara only 2
sympy.Min: aet.minimum, # Sympy accept >2 inputs, Aesara only 2
sympy.conjugate: aet.conj,
sympy.core.numbers.ImaginaryUnit: lambda:aet.complex(0,1),
# Matrices
sympy.MatAdd: Elemwise(aes.add),
sympy.HadamardProduct: Elemwise(aes.mul),
sympy.Trace: nlinalg.trace,
sympy.Determinant : nlinalg.det,
sympy.Inverse: nlinalg.matrix_inverse,
sympy.Transpose: DimShuffle((False, False), [1, 0]),
}
class AesaraPrinter(Printer):
""" Code printer which creates Aesara symbolic expression graphs.
Parameters
==========
cache : dict
Cache dictionary to use. If None (default) will use
the global cache. To create a printer which does not depend on or alter
global state pass an empty dictionary. Note: the dictionary is not
copied on initialization of the printer and will be updated in-place,
so using the same dict object when creating multiple printers or making
multiple calls to :func:`.aesara_code` or :func:`.aesara_function` means
the cache is shared between all these applications.
Attributes
==========
cache : dict
A cache of Aesara variables which have been created for Sympy
symbol-like objects (e.g. :class:`sympy.core.symbol.Symbol` or
:class:`sympy.matrices.expressions.MatrixSymbol`). This is used to
ensure that all references to a given symbol in an expression (or
multiple expressions) are printed as the same Aesara variable, which is
created only once. Symbols are differentiated only by name and type. The
format of the cache's contents should be considered opaque to the user.
"""
printmethod = "_aesara"
def __init__(self, *args, **kwargs):
self.cache = kwargs.pop('cache', dict())
super().__init__(*args, **kwargs)
def _get_key(self, s, name=None, dtype=None, broadcastable=None):
""" Get the cache key for a Sympy object.
Parameters
==========
s : sympy.core.basic.Basic
Sympy object to get key for.
name : str
Name of object, if it does not have a ``name`` attribute.
"""
if name is None:
name = s.name
return (name, type(s), s.args, dtype, broadcastable)
def _get_or_create(self, s, name=None, dtype=None, broadcastable=None):
"""
Get the Aesara variable for a Sympy symbol from the cache, or create it
if it does not exist.
"""
# Defaults
if name is None:
name = s.name
if dtype is None:
dtype = 'floatX'
if broadcastable is None:
broadcastable = ()
key = self._get_key(s, name, dtype=dtype, broadcastable=broadcastable)
if key in self.cache:
return self.cache[key]
value = aet.tensor(name=name, dtype=dtype, broadcastable=broadcastable)
self.cache[key] = value
return value
def _print_Symbol(self, s, **kwargs):
dtype = kwargs.get('dtypes', {}).get(s)
bc = kwargs.get('broadcastables', {}).get(s)
return self._get_or_create(s, dtype=dtype, broadcastable=bc)
def _print_AppliedUndef(self, s, **kwargs):
name = str(type(s)) + '_' + str(s.args[0])
dtype = kwargs.get('dtypes', {}).get(s)
bc = kwargs.get('broadcastables', {}).get(s)
return self._get_or_create(s, name=name, dtype=dtype, broadcastable=bc)
def _print_Basic(self, expr, **kwargs):
op = mapping[type(expr)]
children = [self._print(arg, **kwargs) for arg in expr.args]
return op(*children)
def _print_Number(self, n, **kwargs):
# Integers already taken care of below, interpret as float
return float(n.evalf())
def _print_MatrixSymbol(self, X, **kwargs):
dtype = kwargs.get('dtypes', {}).get(X)
return self._get_or_create(X, dtype=dtype, broadcastable=(None, None))
def _print_DenseMatrix(self, X, **kwargs):
if not hasattr(aet, 'stacklists'):
raise NotImplementedError(
"Matrix translation not yet supported in this version of Aesara")
return aet.stacklists([
[self._print(arg, **kwargs) for arg in L]
for L in X.tolist()
])
_print_ImmutableMatrix = _print_ImmutableDenseMatrix = _print_DenseMatrix
def _print_MatMul(self, expr, **kwargs):
children = [self._print(arg, **kwargs) for arg in expr.args]
result = children[0]
for child in children[1:]:
result = aet.dot(result, child)
return result
def _print_MatPow(self, expr, **kwargs):
children = [self._print(arg, **kwargs) for arg in expr.args]
result = 1
if isinstance(children[1], int) and children[1] > 0:
for i in range(children[1]):
result = aet.dot(result, children[0])
else:
raise NotImplementedError('''Only non-negative integer
powers of matrices can be handled by Aesara at the moment''')
return result
def _print_MatrixSlice(self, expr, **kwargs):
parent = self._print(expr.parent, **kwargs)
rowslice = self._print(slice(*expr.rowslice), **kwargs)
colslice = self._print(slice(*expr.colslice), **kwargs)
return parent[rowslice, colslice]
def _print_BlockMatrix(self, expr, **kwargs):
nrows, ncols = expr.blocks.shape
blocks = [[self._print(expr.blocks[r, c], **kwargs)
for c in range(ncols)]
for r in range(nrows)]
return aet.join(0, *[aet.join(1, *row) for row in blocks])
def _print_slice(self, expr, **kwargs):
return slice(*[self._print(i, **kwargs)
if isinstance(i, sympy.Basic) else i
for i in (expr.start, expr.stop, expr.step)])
def _print_Pi(self, expr, **kwargs):
return 3.141592653589793
def _print_Piecewise(self, expr, **kwargs):
import numpy as np
e, cond = expr.args[0].args # First condition and corresponding value
# Print conditional expression and value for first condition
p_cond = self._print(cond, **kwargs)
p_e = self._print(e, **kwargs)
# One condition only
if len(expr.args) == 1:
# Return value if condition else NaN
return aet.switch(p_cond, p_e, np.nan)
# Return value_1 if condition_1 else evaluate remaining conditions
p_remaining = self._print(sympy.Piecewise(*expr.args[1:]), **kwargs)
return aet.switch(p_cond, p_e, p_remaining)
def _print_Rational(self, expr, **kwargs):
return aet.true_div(self._print(expr.p, **kwargs),
self._print(expr.q, **kwargs))
def _print_Integer(self, expr, **kwargs):
return expr.p
def _print_factorial(self, expr, **kwargs):
return self._print(sympy.gamma(expr.args[0] + 1), **kwargs)
def _print_Derivative(self, deriv, **kwargs):
from aesara.gradient import Rop
rv = self._print(deriv.expr, **kwargs)
for var in deriv.variables:
var = self._print(var, **kwargs)
rv = Rop(rv, var, aet.ones_like(var))
return rv
def emptyPrinter(self, expr):
return expr
def doprint(self, expr, dtypes=None, broadcastables=None):
""" Convert a Sympy expression to a Aesara graph variable.
The ``dtypes`` and ``broadcastables`` arguments are used to specify the
data type, dimension, and broadcasting behavior of the Aesara variables
corresponding to the free symbols in ``expr``. Each is a mapping from
Sympy symbols to the value of the corresponding argument to
``aesara.tensor.var.TensorVariable``.
See the corresponding `documentation page`__ for more information on
broadcasting in Aesara.
.. __: https://aesara.readthedocs.io/en/latest/tutorial/broadcasting.html
Parameters
==========
expr : sympy.core.expr.Expr
Sympy expression to print.
dtypes : dict
Mapping from Sympy symbols to Aesara datatypes to use when creating
new Aesara variables for those symbols. Corresponds to the ``dtype``
argument to ``aesara.tensor.var.TensorVariable``. Defaults to ``'floatX'``
for symbols not included in the mapping.
broadcastables : dict
Mapping from Sympy symbols to the value of the ``broadcastable``
argument to ``aesara.tensor.var.TensorVariable`` to use when creating Aesara
variables for those symbols. Defaults to the empty tuple for symbols
not included in the mapping (resulting in a scalar).
Returns
=======
aesara.graph.basic.Variable
A variable corresponding to the expression's value in a Aesara
symbolic expression graph.
"""
if dtypes is None:
dtypes = {}
if broadcastables is None:
broadcastables = {}
return self._print(expr, dtypes=dtypes, broadcastables=broadcastables)
global_cache = {} # type: Dict[Any, Any]
def aesara_code(expr, cache=None, **kwargs):
"""
Convert a Sympy expression into a Aesara graph variable.
Parameters
==========
expr : sympy.core.expr.Expr
Sympy expression object to convert.
cache : dict
Cached Aesara variables (see :class:`AesaraPrinter.cache
<AesaraPrinter>`). Defaults to the module-level global cache.
dtypes : dict
Passed to :meth:`.AesaraPrinter.doprint`.
broadcastables : dict
Passed to :meth:`.AesaraPrinter.doprint`.
Returns
=======
aesara.graph.basic.Variable
A variable corresponding to the expression's value in a Aesara symbolic
expression graph.
"""
if not aesara:
raise ImportError("aesara is required for aesara_code")
if cache is None:
cache = global_cache
return AesaraPrinter(cache=cache, settings={}).doprint(expr, **kwargs)
def dim_handling(inputs, dim=None, dims=None, broadcastables=None):
r"""
Get value of ``broadcastables`` argument to :func:`.aesara_code` from
keyword arguments to :func:`.aesara_function`.
Included for backwards compatibility.
Parameters
==========
inputs
Sequence of input symbols.
dim : int
Common number of dimensions for all inputs. Overrides other arguments
if given.
dims : dict
Mapping from input symbols to number of dimensions. Overrides
``broadcastables`` argument if given.
broadcastables : dict
Explicit value of ``broadcastables`` argument to
:meth:`.AesaraPrinter.doprint`. If not None function will return this value unchanged.
Returns
=======
dict
Dictionary mapping elements of ``inputs`` to their "broadcastable"
values (tuple of ``bool``\ s).
"""
if dim is not None:
return {s: (False,) * dim for s in inputs}
if dims is not None:
maxdim = max(dims.values())
return {
s: (False,) * d + (True,) * (maxdim - d)
for s, d in dims.items()
}
if broadcastables is not None:
return broadcastables
return {}
def aesara_function(inputs, outputs, scalar=False, *,
dim=None, dims=None, broadcastables=None, **kwargs):
"""
Create a Aesara function from SymPy expressions.
The inputs and outputs are converted to Aesara variables using
:func:`.aesara_code` and then passed to ``aesara.function``.
Parameters
==========
inputs
Sequence of symbols which constitute the inputs of the function.
outputs
Sequence of expressions which constitute the outputs(s) of the
function. The free symbols of each expression must be a subset of
``inputs``.
scalar : bool
Convert 0-dimensional arrays in output to scalars. This will return a
Python wrapper function around the Aesara function object.
cache : dict
Cached Aesara variables (see :class:`AesaraPrinter.cache
<AesaraPrinter>`). Defaults to the module-level global cache.
dtypes : dict
Passed to :meth:`.AesaraPrinter.doprint`.
broadcastables : dict
Passed to :meth:`.AesaraPrinter.doprint`.
dims : dict
Alternative to ``broadcastables`` argument. Mapping from elements of
``inputs`` to integers indicating the dimension of their associated
arrays/tensors. Overrides ``broadcastables`` argument if given.
dim : int
Another alternative to the ``broadcastables`` argument. Common number of
dimensions to use for all arrays/tensors.
``aesara_function([x, y], [...], dim=2)`` is equivalent to using
``broadcastables={x: (False, False), y: (False, False)}``.
Returns
=======
callable
A callable object which takes values of ``inputs`` as positional
arguments and returns an output array for each of the expressions
in ``outputs``. If ``outputs`` is a single expression the function will
return a Numpy array, if it is a list of multiple expressions the
function will return a list of arrays. See description of the ``squeeze``
argument above for the behavior when a single output is passed in a list.
The returned object will either be an instance of
``aesara.compile.function.types.Function`` or a Python wrapper
function around one. In both cases, the returned value will have a
``aesara_function`` attribute which points to the return value of
``aesara.function``.
Examples
========
>>> from sympy.abc import x, y, z
>>> from sympy.printing.aesaracode import aesara_function
A simple function with one input and one output:
>>> f1 = aesara_function([x], [x**2 - 1], scalar=True)
>>> f1(3)
8.0
A function with multiple inputs and one output:
>>> f2 = aesara_function([x, y, z], [(x**z + y**z)**(1/z)], scalar=True)
>>> f2(3, 4, 2)
5.0
A function with multiple inputs and multiple outputs:
>>> f3 = aesara_function([x, y], [x**2 + y**2, x**2 - y**2], scalar=True)
>>> f3(2, 3)
[13.0, -5.0]
See also
========
dim_handling
"""
if not aesara:
raise ImportError("Aesara is required for aesara_function")
# Pop off non-aesara keyword args
cache = kwargs.pop('cache', {})
dtypes = kwargs.pop('dtypes', {})
broadcastables = dim_handling(
inputs, dim=dim, dims=dims, broadcastables=broadcastables,
)
# Print inputs/outputs
code = partial(aesara_code, cache=cache, dtypes=dtypes,
broadcastables=broadcastables)
tinputs = list(map(code, inputs))
toutputs = list(map(code, outputs))
#fix constant expressions as variables
toutputs = [output if isinstance(output, aesara.graph.basic.Variable) else aet.as_tensor_variable(output) for output in toutputs]
if len(toutputs) == 1:
toutputs = toutputs[0]
# Compile aesara func
func = aesara.function(tinputs, toutputs, **kwargs)
is_0d = [len(o.variable.broadcastable) == 0 for o in func.outputs]
# No wrapper required
if not scalar or not any(is_0d):
func.aesara_function = func
return func
# Create wrapper to convert 0-dimensional outputs to scalars
def wrapper(*args):
out = func(*args)
# out can be array(1.0) or [array(1.0), array(2.0)]
if is_sequence(out):
return [o[()] if is_0d[i] else o for i, o in enumerate(out)]
else:
return out[()]
wrapper.__wrapped__ = func
wrapper.__doc__ = func.__doc__
wrapper.aesara_function = func
return wrapper
|
60c2c498079363285f01372a88c74489c7a6ac8a8742568ff1327fa2185d9999 | from distutils.version import LooseVersion as V
from collections.abc import Iterable
from sympy import Mul, S
from sympy.codegen.cfunctions import Sqrt
from sympy.external import import_module
from sympy.printing.precedence import PRECEDENCE
from sympy.printing.pycode import AbstractPythonCodePrinter
import sympy
tensorflow = import_module('tensorflow')
class TensorflowPrinter(AbstractPythonCodePrinter):
"""
Tensorflow printer which handles vectorized piecewise functions,
logical operators, max/min, and relational operators.
"""
printmethod = "_tensorflowcode"
mapping = {
sympy.Abs: "tensorflow.math.abs",
sympy.sign: "tensorflow.math.sign",
# XXX May raise error for ints.
sympy.ceiling: "tensorflow.math.ceil",
sympy.floor: "tensorflow.math.floor",
sympy.log: "tensorflow.math.log",
sympy.exp: "tensorflow.math.exp",
Sqrt: "tensorflow.math.sqrt",
sympy.cos: "tensorflow.math.cos",
sympy.acos: "tensorflow.math.acos",
sympy.sin: "tensorflow.math.sin",
sympy.asin: "tensorflow.math.asin",
sympy.tan: "tensorflow.math.tan",
sympy.atan: "tensorflow.math.atan",
sympy.atan2: "tensorflow.math.atan2",
# XXX Also may give NaN for complex results.
sympy.cosh: "tensorflow.math.cosh",
sympy.acosh: "tensorflow.math.acosh",
sympy.sinh: "tensorflow.math.sinh",
sympy.asinh: "tensorflow.math.asinh",
sympy.tanh: "tensorflow.math.tanh",
sympy.atanh: "tensorflow.math.atanh",
sympy.re: "tensorflow.math.real",
sympy.im: "tensorflow.math.imag",
sympy.arg: "tensorflow.math.angle",
# XXX May raise error for ints and complexes
sympy.erf: "tensorflow.math.erf",
sympy.loggamma: "tensorflow.math.lgamma",
sympy.Eq: "tensorflow.math.equal",
sympy.Ne: "tensorflow.math.not_equal",
sympy.StrictGreaterThan: "tensorflow.math.greater",
sympy.StrictLessThan: "tensorflow.math.less",
sympy.LessThan: "tensorflow.math.less_equal",
sympy.GreaterThan: "tensorflow.math.greater_equal",
sympy.And: "tensorflow.math.logical_and",
sympy.Or: "tensorflow.math.logical_or",
sympy.Not: "tensorflow.math.logical_not",
sympy.Max: "tensorflow.math.maximum",
sympy.Min: "tensorflow.math.minimum",
# Matrices
sympy.MatAdd: "tensorflow.math.add",
sympy.HadamardProduct: "tensorflow.math.multiply",
sympy.Trace: "tensorflow.linalg.trace",
# XXX May raise error for integer matrices.
sympy.Determinant : "tensorflow.linalg.det",
}
_default_settings = dict(
AbstractPythonCodePrinter._default_settings,
tensorflow_version=None
)
def __init__(self, settings=None):
super().__init__(settings)
version = self._settings['tensorflow_version']
if version is None and tensorflow:
version = tensorflow.__version__
self.tensorflow_version = version
def _print_Function(self, expr):
op = self.mapping.get(type(expr), None)
if op is None:
return super()._print_Basic(expr)
children = [self._print(arg) for arg in expr.args]
if len(children) == 1:
return "%s(%s)" % (
self._module_format(op),
children[0]
)
else:
return self._expand_fold_binary_op(op, children)
_print_Expr = _print_Function
_print_Application = _print_Function
_print_MatrixExpr = _print_Function
# TODO: a better class structure would avoid this mess:
_print_Relational = _print_Function
_print_Not = _print_Function
_print_And = _print_Function
_print_Or = _print_Function
_print_HadamardProduct = _print_Function
_print_Trace = _print_Function
_print_Determinant = _print_Function
def _print_Inverse(self, expr):
op = self._module_format('tensorflow.linalg.inv')
return "{}({})".format(op, self._print(expr.arg))
def _print_Transpose(self, expr):
version = self.tensorflow_version
if version and V(version) < V('1.14'):
op = self._module_format('tensorflow.matrix_transpose')
else:
op = self._module_format('tensorflow.linalg.matrix_transpose')
return "{}({})".format(op, self._print(expr.arg))
def _print_Derivative(self, expr):
variables = expr.variables
if any(isinstance(i, Iterable) for i in variables):
raise NotImplementedError("derivation by multiple variables is not supported")
def unfold(expr, args):
if not args:
return self._print(expr)
return "%s(%s, %s)[0]" % (
self._module_format("tensorflow.gradients"),
unfold(expr, args[:-1]),
self._print(args[-1]),
)
return unfold(expr.expr, variables)
def _print_Piecewise(self, expr):
version = self.tensorflow_version
if version and V(version) < V('1.0'):
tensorflow_piecewise = "tensorflow.select"
else:
tensorflow_piecewise = "tensorflow.where"
from sympy import Piecewise
e, cond = expr.args[0].args
if len(expr.args) == 1:
return '{}({}, {}, {})'.format(
self._module_format(tensorflow_piecewise),
self._print(cond),
self._print(e),
0)
return '{}({}, {}, {})'.format(
self._module_format(tensorflow_piecewise),
self._print(cond),
self._print(e),
self._print(Piecewise(*expr.args[1:])))
def _print_Pow(self, expr):
# XXX May raise error for
# int**float or int**complex or float**complex
base, exp = expr.args
if expr.exp == S.Half:
return "{}({})".format(
self._module_format("tensorflow.math.sqrt"), self._print(base))
return "{}({}, {})".format(
self._module_format("tensorflow.math.pow"),
self._print(base), self._print(exp))
def _print_MatrixBase(self, expr):
tensorflow_f = "tensorflow.Variable" if expr.free_symbols else "tensorflow.constant"
data = "["+", ".join(["["+", ".join([self._print(j) for j in i])+"]" for i in expr.tolist()])+"]"
return "%s(%s)" % (
self._module_format(tensorflow_f),
data,
)
def _print_MatMul(self, expr):
from sympy.matrices.expressions import MatrixExpr
mat_args = [arg for arg in expr.args if isinstance(arg, MatrixExpr)]
args = [arg for arg in expr.args if arg not in mat_args]
if args:
return "%s*%s" % (
self.parenthesize(Mul.fromiter(args), PRECEDENCE["Mul"]),
self._expand_fold_binary_op(
"tensorflow.linalg.matmul", mat_args)
)
else:
return self._expand_fold_binary_op(
"tensorflow.linalg.matmul", mat_args)
def _print_MatPow(self, expr):
return self._expand_fold_binary_op(
"tensorflow.linalg.matmul", [expr.base]*expr.exp)
def _print_Assignment(self, expr):
# TODO: is this necessary?
return "%s = %s" % (
self._print(expr.lhs),
self._print(expr.rhs),
)
def _print_CodeBlock(self, expr):
# TODO: is this necessary?
ret = []
for subexpr in expr.args:
ret.append(self._print(subexpr))
return "\n".join(ret)
def _get_letter_generator_for_einsum(self):
for i in range(97, 123):
yield chr(i)
for i in range(65, 91):
yield chr(i)
raise ValueError("out of letters")
def _print_ArrayTensorProduct(self, expr):
letters = self._get_letter_generator_for_einsum()
contraction_string = ",".join(["".join([next(letters) for j in range(i)]) for i in expr.subranks])
return '%s("%s", %s)' % (
self._module_format('tensorflow.linalg.einsum'),
contraction_string,
", ".join([self._print(arg) for arg in expr.args])
)
def _print_ArrayContraction(self, expr):
from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct
base = expr.expr
contraction_indices = expr.contraction_indices
contraction_string, letters_free, letters_dum = self._get_einsum_string(base.subranks, contraction_indices)
if not contraction_indices:
return self._print(base)
if isinstance(base, ArrayTensorProduct):
elems = ["%s" % (self._print(arg)) for arg in base.args]
return "%s(\"%s\", %s)" % (
self._module_format("tensorflow.linalg.einsum"),
contraction_string,
", ".join(elems)
)
raise NotImplementedError()
def _print_ArrayDiagonal(self, expr):
from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct
diagonal_indices = list(expr.diagonal_indices)
if len(diagonal_indices) > 1:
# TODO: this should be handled in sympy.codegen.array_utils,
# possibly by creating the possibility of unfolding the
# ArrayDiagonal object into nested ones. Same reasoning for
# the array contraction.
raise NotImplementedError
if len(diagonal_indices[0]) != 2:
raise NotImplementedError
if isinstance(expr.expr, ArrayTensorProduct):
subranks = expr.expr.subranks
elems = expr.expr.args
else:
subranks = expr.subranks
elems = [expr.expr]
diagonal_string, letters_free, letters_dum = self._get_einsum_string(subranks, diagonal_indices)
elems = [self._print(i) for i in elems]
return '%s("%s", %s)' % (
self._module_format("tensorflow.linalg.einsum"),
"{}->{}{}".format(diagonal_string, "".join(letters_free), "".join(letters_dum)),
", ".join(elems)
)
def _print_PermuteDims(self, expr):
return "%s(%s, %s)" % (
self._module_format("tensorflow.transpose"),
self._print(expr.expr),
self._print(expr.permutation.array_form),
)
def _print_ArrayAdd(self, expr):
return self._expand_fold_binary_op('tensorflow.math.add', expr.args)
def tensorflow_code(expr, **settings):
printer = TensorflowPrinter(settings)
return printer.doprint(expr)
|
eabe9ac37575c7b692595916cd67e9d49fd699aaf42826b18c7b2027f101e75e | from typing import Any, Dict
from sympy.core.compatibility import is_sequence
from sympy.external import import_module
from sympy.printing.printer import Printer
import sympy
from functools import partial
from sympy.utilities.decorator import doctest_depends_on
from sympy.utilities.exceptions import SymPyDeprecationWarning
theano = import_module('theano')
if theano:
ts = theano.scalar
tt = theano.tensor
from theano.sandbox import linalg as tlinalg
mapping = {
sympy.Add: tt.add,
sympy.Mul: tt.mul,
sympy.Abs: tt.abs_,
sympy.sign: tt.sgn,
sympy.ceiling: tt.ceil,
sympy.floor: tt.floor,
sympy.log: tt.log,
sympy.exp: tt.exp,
sympy.sqrt: tt.sqrt,
sympy.cos: tt.cos,
sympy.acos: tt.arccos,
sympy.sin: tt.sin,
sympy.asin: tt.arcsin,
sympy.tan: tt.tan,
sympy.atan: tt.arctan,
sympy.atan2: tt.arctan2,
sympy.cosh: tt.cosh,
sympy.acosh: tt.arccosh,
sympy.sinh: tt.sinh,
sympy.asinh: tt.arcsinh,
sympy.tanh: tt.tanh,
sympy.atanh: tt.arctanh,
sympy.re: tt.real,
sympy.im: tt.imag,
sympy.arg: tt.angle,
sympy.erf: tt.erf,
sympy.gamma: tt.gamma,
sympy.loggamma: tt.gammaln,
sympy.Pow: tt.pow,
sympy.Eq: tt.eq,
sympy.StrictGreaterThan: tt.gt,
sympy.StrictLessThan: tt.lt,
sympy.LessThan: tt.le,
sympy.GreaterThan: tt.ge,
sympy.And: tt.and_,
sympy.Or: tt.or_,
sympy.Max: tt.maximum, # Sympy accept >2 inputs, Theano only 2
sympy.Min: tt.minimum, # Sympy accept >2 inputs, Theano only 2
sympy.conjugate: tt.conj,
sympy.core.numbers.ImaginaryUnit: lambda:tt.complex(0,1),
# Matrices
sympy.MatAdd: tt.Elemwise(ts.add),
sympy.HadamardProduct: tt.Elemwise(ts.mul),
sympy.Trace: tlinalg.trace,
sympy.Determinant : tlinalg.det,
sympy.Inverse: tlinalg.matrix_inverse,
sympy.Transpose: tt.DimShuffle((False, False), [1, 0]),
}
class TheanoPrinter(Printer):
""" Code printer which creates Theano symbolic expression graphs.
Parameters
==========
cache : dict
Cache dictionary to use. If None (default) will use
the global cache. To create a printer which does not depend on or alter
global state pass an empty dictionary. Note: the dictionary is not
copied on initialization of the printer and will be updated in-place,
so using the same dict object when creating multiple printers or making
multiple calls to :func:`.theano_code` or :func:`.theano_function` means
the cache is shared between all these applications.
Attributes
==========
cache : dict
A cache of Theano variables which have been created for Sympy
symbol-like objects (e.g. :class:`sympy.core.symbol.Symbol` or
:class:`sympy.matrices.expressions.MatrixSymbol`). This is used to
ensure that all references to a given symbol in an expression (or
multiple expressions) are printed as the same Theano variable, which is
created only once. Symbols are differentiated only by name and type. The
format of the cache's contents should be considered opaque to the user.
"""
printmethod = "_theano"
def __init__(self, *args, **kwargs):
self.cache = kwargs.pop('cache', dict())
super().__init__(*args, **kwargs)
def _get_key(self, s, name=None, dtype=None, broadcastable=None):
""" Get the cache key for a Sympy object.
Parameters
==========
s : sympy.core.basic.Basic
Sympy object to get key for.
name : str
Name of object, if it does not have a ``name`` attribute.
"""
if name is None:
name = s.name
return (name, type(s), s.args, dtype, broadcastable)
def _get_or_create(self, s, name=None, dtype=None, broadcastable=None):
"""
Get the Theano variable for a Sympy symbol from the cache, or create it
if it does not exist.
"""
# Defaults
if name is None:
name = s.name
if dtype is None:
dtype = 'floatX'
if broadcastable is None:
broadcastable = ()
key = self._get_key(s, name, dtype=dtype, broadcastable=broadcastable)
if key in self.cache:
return self.cache[key]
value = tt.tensor(name=name, dtype=dtype, broadcastable=broadcastable)
self.cache[key] = value
return value
def _print_Symbol(self, s, **kwargs):
dtype = kwargs.get('dtypes', {}).get(s)
bc = kwargs.get('broadcastables', {}).get(s)
return self._get_or_create(s, dtype=dtype, broadcastable=bc)
def _print_AppliedUndef(self, s, **kwargs):
name = str(type(s)) + '_' + str(s.args[0])
dtype = kwargs.get('dtypes', {}).get(s)
bc = kwargs.get('broadcastables', {}).get(s)
return self._get_or_create(s, name=name, dtype=dtype, broadcastable=bc)
def _print_Basic(self, expr, **kwargs):
op = mapping[type(expr)]
children = [self._print(arg, **kwargs) for arg in expr.args]
return op(*children)
def _print_Number(self, n, **kwargs):
# Integers already taken care of below, interpret as float
return float(n.evalf())
def _print_MatrixSymbol(self, X, **kwargs):
dtype = kwargs.get('dtypes', {}).get(X)
return self._get_or_create(X, dtype=dtype, broadcastable=(None, None))
def _print_DenseMatrix(self, X, **kwargs):
if not hasattr(tt, 'stacklists'):
raise NotImplementedError(
"Matrix translation not yet supported in this version of Theano")
return tt.stacklists([
[self._print(arg, **kwargs) for arg in L]
for L in X.tolist()
])
_print_ImmutableMatrix = _print_ImmutableDenseMatrix = _print_DenseMatrix
def _print_MatMul(self, expr, **kwargs):
children = [self._print(arg, **kwargs) for arg in expr.args]
result = children[0]
for child in children[1:]:
result = tt.dot(result, child)
return result
def _print_MatPow(self, expr, **kwargs):
children = [self._print(arg, **kwargs) for arg in expr.args]
result = 1
if isinstance(children[1], int) and children[1] > 0:
for i in range(children[1]):
result = tt.dot(result, children[0])
else:
raise NotImplementedError('''Only non-negative integer
powers of matrices can be handled by Theano at the moment''')
return result
def _print_MatrixSlice(self, expr, **kwargs):
parent = self._print(expr.parent, **kwargs)
rowslice = self._print(slice(*expr.rowslice), **kwargs)
colslice = self._print(slice(*expr.colslice), **kwargs)
return parent[rowslice, colslice]
def _print_BlockMatrix(self, expr, **kwargs):
nrows, ncols = expr.blocks.shape
blocks = [[self._print(expr.blocks[r, c], **kwargs)
for c in range(ncols)]
for r in range(nrows)]
return tt.join(0, *[tt.join(1, *row) for row in blocks])
def _print_slice(self, expr, **kwargs):
return slice(*[self._print(i, **kwargs)
if isinstance(i, sympy.Basic) else i
for i in (expr.start, expr.stop, expr.step)])
def _print_Pi(self, expr, **kwargs):
return 3.141592653589793
def _print_Piecewise(self, expr, **kwargs):
import numpy as np
e, cond = expr.args[0].args # First condition and corresponding value
# Print conditional expression and value for first condition
p_cond = self._print(cond, **kwargs)
p_e = self._print(e, **kwargs)
# One condition only
if len(expr.args) == 1:
# Return value if condition else NaN
return tt.switch(p_cond, p_e, np.nan)
# Return value_1 if condition_1 else evaluate remaining conditions
p_remaining = self._print(sympy.Piecewise(*expr.args[1:]), **kwargs)
return tt.switch(p_cond, p_e, p_remaining)
def _print_Rational(self, expr, **kwargs):
return tt.true_div(self._print(expr.p, **kwargs),
self._print(expr.q, **kwargs))
def _print_Integer(self, expr, **kwargs):
return expr.p
def _print_factorial(self, expr, **kwargs):
return self._print(sympy.gamma(expr.args[0] + 1), **kwargs)
def _print_Derivative(self, deriv, **kwargs):
rv = self._print(deriv.expr, **kwargs)
for var in deriv.variables:
var = self._print(var, **kwargs)
rv = tt.Rop(rv, var, tt.ones_like(var))
return rv
def emptyPrinter(self, expr):
return expr
def doprint(self, expr, dtypes=None, broadcastables=None):
""" Convert a Sympy expression to a Theano graph variable.
The ``dtypes`` and ``broadcastables`` arguments are used to specify the
data type, dimension, and broadcasting behavior of the Theano variables
corresponding to the free symbols in ``expr``. Each is a mapping from
Sympy symbols to the value of the corresponding argument to
``theano.tensor.Tensor``.
See the corresponding `documentation page`__ for more information on
broadcasting in Theano.
.. __: http://deeplearning.net/software/theano/tutorial/broadcasting.html
Parameters
==========
expr : sympy.core.expr.Expr
Sympy expression to print.
dtypes : dict
Mapping from Sympy symbols to Theano datatypes to use when creating
new Theano variables for those symbols. Corresponds to the ``dtype``
argument to ``theano.tensor.Tensor``. Defaults to ``'floatX'``
for symbols not included in the mapping.
broadcastables : dict
Mapping from Sympy symbols to the value of the ``broadcastable``
argument to ``theano.tensor.Tensor`` to use when creating Theano
variables for those symbols. Defaults to the empty tuple for symbols
not included in the mapping (resulting in a scalar).
Returns
=======
theano.gof.graph.Variable
A variable corresponding to the expression's value in a Theano
symbolic expression graph.
"""
if dtypes is None:
dtypes = {}
if broadcastables is None:
broadcastables = {}
return self._print(expr, dtypes=dtypes, broadcastables=broadcastables)
global_cache = {} # type: Dict[Any, Any]
def theano_code(expr, cache=None, **kwargs):
"""
Convert a Sympy expression into a Theano graph variable.
Parameters
==========
expr : sympy.core.expr.Expr
Sympy expression object to convert.
cache : dict
Cached Theano variables (see :class:`TheanoPrinter.cache
<TheanoPrinter>`). Defaults to the module-level global cache.
dtypes : dict
Passed to :meth:`.TheanoPrinter.doprint`.
broadcastables : dict
Passed to :meth:`.TheanoPrinter.doprint`.
Returns
=======
theano.gof.graph.Variable
A variable corresponding to the expression's value in a Theano symbolic
expression graph.
"""
SymPyDeprecationWarning(
feature="sympy.printing.theanocode",
useinstead="Theano is deprecated; use Aesara and sympy.printing.aesaracode",
issue=21150,
deprecated_since_version="1.8").warn()
if not theano:
raise ImportError("theano is required for theano_code")
if cache is None:
cache = global_cache
return TheanoPrinter(cache=cache, settings={}).doprint(expr, **kwargs)
def dim_handling(inputs, dim=None, dims=None, broadcastables=None):
r"""
Get value of ``broadcastables`` argument to :func:`.theano_code` from
keyword arguments to :func:`.theano_function`.
Included for backwards compatibility.
Parameters
==========
inputs
Sequence of input symbols.
dim : int
Common number of dimensions for all inputs. Overrides other arguments
if given.
dims : dict
Mapping from input symbols to number of dimensions. Overrides
``broadcastables`` argument if given.
broadcastables : dict
Explicit value of ``broadcastables`` argument to
:meth:`.TheanoPrinter.doprint`. If not None function will return this value unchanged.
Returns
=======
dict
Dictionary mapping elements of ``inputs`` to their "broadcastable"
values (tuple of ``bool``\ s).
"""
if dim is not None:
return {s: (False,) * dim for s in inputs}
if dims is not None:
maxdim = max(dims.values())
return {
s: (False,) * d + (True,) * (maxdim - d)
for s, d in dims.items()
}
if broadcastables is not None:
return broadcastables
return {}
@doctest_depends_on(modules=('theano',))
def theano_function(inputs, outputs, scalar=False, *,
dim=None, dims=None, broadcastables=None, **kwargs):
"""
Create a Theano function from SymPy expressions.
The inputs and outputs are converted to Theano variables using
:func:`.theano_code` and then passed to ``theano.function``.
Parameters
==========
inputs
Sequence of symbols which constitute the inputs of the function.
outputs
Sequence of expressions which constitute the outputs(s) of the
function. The free symbols of each expression must be a subset of
``inputs``.
scalar : bool
Convert 0-dimensional arrays in output to scalars. This will return a
Python wrapper function around the Theano function object.
cache : dict
Cached Theano variables (see :class:`TheanoPrinter.cache
<TheanoPrinter>`). Defaults to the module-level global cache.
dtypes : dict
Passed to :meth:`.TheanoPrinter.doprint`.
broadcastables : dict
Passed to :meth:`.TheanoPrinter.doprint`.
dims : dict
Alternative to ``broadcastables`` argument. Mapping from elements of
``inputs`` to integers indicating the dimension of their associated
arrays/tensors. Overrides ``broadcastables`` argument if given.
dim : int
Another alternative to the ``broadcastables`` argument. Common number of
dimensions to use for all arrays/tensors.
``theano_function([x, y], [...], dim=2)`` is equivalent to using
``broadcastables={x: (False, False), y: (False, False)}``.
Returns
=======
callable
A callable object which takes values of ``inputs`` as positional
arguments and returns an output array for each of the expressions
in ``outputs``. If ``outputs`` is a single expression the function will
return a Numpy array, if it is a list of multiple expressions the
function will return a list of arrays. See description of the ``squeeze``
argument above for the behavior when a single output is passed in a list.
The returned object will either be an instance of
``theano.compile.function_module.Function`` or a Python wrapper
function around one. In both cases, the returned value will have a
``theano_function`` attribute which points to the return value of
``theano.function``.
Examples
========
>>> from sympy.abc import x, y, z
>>> from sympy.printing.theanocode import theano_function
A simple function with one input and one output:
>>> f1 = theano_function([x], [x**2 - 1], scalar=True)
>>> f1(3)
8.0
A function with multiple inputs and one output:
>>> f2 = theano_function([x, y, z], [(x**z + y**z)**(1/z)], scalar=True)
>>> f2(3, 4, 2)
5.0
A function with multiple inputs and multiple outputs:
>>> f3 = theano_function([x, y], [x**2 + y**2, x**2 - y**2], scalar=True)
>>> f3(2, 3)
[13.0, -5.0]
See also
========
dim_handling
"""
SymPyDeprecationWarning(
feature="sympy.printing.theanocode",
useinstead="Theano is deprecated; use Aesara and sympy.printing.aesaracode",
issue=21150,
deprecated_since_version="1.8").warn()
if not theano:
raise ImportError("theano is required for theano_function")
# Pop off non-theano keyword args
cache = kwargs.pop('cache', {})
dtypes = kwargs.pop('dtypes', {})
broadcastables = dim_handling(
inputs, dim=dim, dims=dims, broadcastables=broadcastables,
)
# Print inputs/outputs
code = partial(theano_code, cache=cache, dtypes=dtypes,
broadcastables=broadcastables)
tinputs = list(map(code, inputs))
toutputs = list(map(code, outputs))
#fix constant expressions as variables
toutputs = [output if isinstance(output, theano.Variable) else tt.as_tensor_variable(output) for output in toutputs]
if len(toutputs) == 1:
toutputs = toutputs[0]
# Compile theano func
func = theano.function(tinputs, toutputs, **kwargs)
is_0d = [len(o.variable.broadcastable) == 0 for o in func.outputs]
# No wrapper required
if not scalar or not any(is_0d):
func.theano_function = func
return func
# Create wrapper to convert 0-dimensional outputs to scalars
def wrapper(*args):
out = func(*args)
# out can be array(1.0) or [array(1.0), array(2.0)]
if is_sequence(out):
return [o[()] if is_0d[i] else o for i, o in enumerate(out)]
else:
return out[()]
wrapper.__wrapped__ = func
wrapper.__doc__ = func.__doc__
wrapper.theano_function = func
return wrapper
|
0b8346923de88756208138872f30a428ac5897f5cb190344bf50aa7ed8fe49f2 | """ Integral Transforms """
from functools import reduce
from sympy.core import S
from sympy.core.compatibility import iterable, ordered
from sympy.core.function import Function
from sympy.core.relational import _canonical, Ge, Gt
from sympy.core.numbers import oo
from sympy.core.symbol import Dummy
from sympy.integrals import integrate, Integral
from sympy.integrals.meijerint import _dummy
from sympy.logic.boolalg import to_cnf, conjuncts, disjuncts, Or, And
from sympy.simplify import simplify
from sympy.utilities import default_sort_key
from sympy.matrices.matrices import MatrixBase
##########################################################################
# Helpers / Utilities
##########################################################################
class IntegralTransformError(NotImplementedError):
"""
Exception raised in relation to problems computing transforms.
Explanation
===========
This class is mostly used internally; if integrals cannot be computed
objects representing unevaluated transforms are usually returned.
The hint ``needeval=True`` can be used to disable returning transform
objects, and instead raise this exception if an integral cannot be
computed.
"""
def __init__(self, transform, function, msg):
super().__init__(
"%s Transform could not be computed: %s." % (transform, msg))
self.function = function
class IntegralTransform(Function):
"""
Base class for integral transforms.
Explanation
===========
This class represents unevaluated transforms.
To implement a concrete transform, derive from this class and implement
the ``_compute_transform(f, x, s, **hints)`` and ``_as_integral(f, x, s)``
functions. If the transform cannot be computed, raise :obj:`IntegralTransformError`.
Also set ``cls._name``. For instance,
>>> from sympy.integrals.transforms import LaplaceTransform
>>> LaplaceTransform._name
'Laplace'
Implement ``self._collapse_extra`` if your function returns more than just a
number and possibly a convergence condition.
"""
@property
def function(self):
""" The function to be transformed. """
return self.args[0]
@property
def function_variable(self):
""" The dependent variable of the function to be transformed. """
return self.args[1]
@property
def transform_variable(self):
""" The independent transform variable. """
return self.args[2]
@property
def free_symbols(self):
"""
This method returns the symbols that will exist when the transform
is evaluated.
"""
return self.function.free_symbols.union({self.transform_variable}) \
- {self.function_variable}
def _compute_transform(self, f, x, s, **hints):
raise NotImplementedError
def _as_integral(self, f, x, s):
raise NotImplementedError
def _collapse_extra(self, extra):
cond = And(*extra)
if cond == False:
raise IntegralTransformError(self.__class__.name, None, '')
return cond
def doit(self, **hints):
"""
Try to evaluate the transform in closed form.
Explanation
===========
This general function handles linearity, but apart from that leaves
pretty much everything to _compute_transform.
Standard hints are the following:
- ``simplify``: whether or not to simplify the result
- ``noconds``: if True, don't return convergence conditions
- ``needeval``: if True, raise IntegralTransformError instead of
returning IntegralTransform objects
The default values of these hints depend on the concrete transform,
usually the default is
``(simplify, noconds, needeval) = (True, False, False)``.
"""
from sympy import Add, expand_mul, Mul
from sympy.core.function import AppliedUndef
needeval = hints.pop('needeval', False)
try_directly = not any(func.has(self.function_variable)
for func in self.function.atoms(AppliedUndef))
if try_directly:
try:
return self._compute_transform(self.function,
self.function_variable, self.transform_variable, **hints)
except IntegralTransformError:
pass
fn = self.function
if not fn.is_Add:
fn = expand_mul(fn)
if fn.is_Add:
hints['needeval'] = needeval
res = [self.__class__(*([x] + list(self.args[1:]))).doit(**hints)
for x in fn.args]
extra = []
ress = []
for x in res:
if not isinstance(x, tuple):
x = [x]
ress.append(x[0])
if len(x) == 2:
# only a condition
extra.append(x[1])
elif len(x) > 2:
# some region parameters and a condition (Mellin, Laplace)
extra += [x[1:]]
res = Add(*ress)
if not extra:
return res
try:
extra = self._collapse_extra(extra)
if iterable(extra):
return tuple([res]) + tuple(extra)
else:
return (res, extra)
except IntegralTransformError:
pass
if needeval:
raise IntegralTransformError(
self.__class__._name, self.function, 'needeval')
# TODO handle derivatives etc
# pull out constant coefficients
coeff, rest = fn.as_coeff_mul(self.function_variable)
return coeff*self.__class__(*([Mul(*rest)] + list(self.args[1:])))
@property
def as_integral(self):
return self._as_integral(self.function, self.function_variable,
self.transform_variable)
def _eval_rewrite_as_Integral(self, *args, **kwargs):
return self.as_integral
from sympy.solvers.inequalities import _solve_inequality
def _simplify(expr, doit):
from sympy import powdenest, piecewise_fold
if doit:
return simplify(powdenest(piecewise_fold(expr), polar=True))
return expr
def _noconds_(default):
"""
This is a decorator generator for dropping convergence conditions.
Explanation
===========
Suppose you define a function ``transform(*args)`` which returns a tuple of
the form ``(result, cond1, cond2, ...)``.
Decorating it ``@_noconds_(default)`` will add a new keyword argument
``noconds`` to it. If ``noconds=True``, the return value will be altered to
be only ``result``, whereas if ``noconds=False`` the return value will not
be altered.
The default value of the ``noconds`` keyword will be ``default`` (i.e. the
argument of this function).
"""
def make_wrapper(func):
from sympy.core.decorators import wraps
@wraps(func)
def wrapper(*args, noconds=default, **kwargs):
res = func(*args, **kwargs)
if noconds:
return res[0]
return res
return wrapper
return make_wrapper
_noconds = _noconds_(False)
##########################################################################
# Mellin Transform
##########################################################################
def _default_integrator(f, x):
return integrate(f, (x, 0, oo))
@_noconds
def _mellin_transform(f, x, s_, integrator=_default_integrator, simplify=True):
""" Backend function to compute Mellin transforms. """
from sympy import re, Max, Min, count_ops
# We use a fresh dummy, because assumptions on s might drop conditions on
# convergence of the integral.
s = _dummy('s', 'mellin-transform', f)
F = integrator(x**(s - 1) * f, x)
if not F.has(Integral):
return _simplify(F.subs(s, s_), simplify), (-oo, oo), S.true
if not F.is_Piecewise: # XXX can this work if integration gives continuous result now?
raise IntegralTransformError('Mellin', f, 'could not compute integral')
F, cond = F.args[0]
if F.has(Integral):
raise IntegralTransformError(
'Mellin', f, 'integral in unexpected form')
def process_conds(cond):
"""
Turn ``cond`` into a strip (a, b), and auxiliary conditions.
"""
a = -oo
b = oo
aux = S.true
conds = conjuncts(to_cnf(cond))
t = Dummy('t', real=True)
for c in conds:
a_ = oo
b_ = -oo
aux_ = []
for d in disjuncts(c):
d_ = d.replace(
re, lambda x: x.as_real_imag()[0]).subs(re(s), t)
if not d.is_Relational or \
d.rel_op in ('==', '!=') \
or d_.has(s) or not d_.has(t):
aux_ += [d]
continue
soln = _solve_inequality(d_, t)
if not soln.is_Relational or \
soln.rel_op in ('==', '!='):
aux_ += [d]
continue
if soln.lts == t:
b_ = Max(soln.gts, b_)
else:
a_ = Min(soln.lts, a_)
if a_ != oo and a_ != b:
a = Max(a_, a)
elif b_ != -oo and b_ != a:
b = Min(b_, b)
else:
aux = And(aux, Or(*aux_))
return a, b, aux
conds = [process_conds(c) for c in disjuncts(cond)]
conds = [x for x in conds if x[2] != False]
conds.sort(key=lambda x: (x[0] - x[1], count_ops(x[2])))
if not conds:
raise IntegralTransformError('Mellin', f, 'no convergence found')
a, b, aux = conds[0]
return _simplify(F.subs(s, s_), simplify), (a, b), aux
class MellinTransform(IntegralTransform):
"""
Class representing unevaluated Mellin transforms.
For usage of this class, see the :class:`IntegralTransform` docstring.
For how to compute Mellin transforms, see the :func:`mellin_transform`
docstring.
"""
_name = 'Mellin'
def _compute_transform(self, f, x, s, **hints):
return _mellin_transform(f, x, s, **hints)
def _as_integral(self, f, x, s):
return Integral(f*x**(s - 1), (x, 0, oo))
def _collapse_extra(self, extra):
from sympy import Max, Min
a = []
b = []
cond = []
for (sa, sb), c in extra:
a += [sa]
b += [sb]
cond += [c]
res = (Max(*a), Min(*b)), And(*cond)
if (res[0][0] >= res[0][1]) == True or res[1] == False:
raise IntegralTransformError(
'Mellin', None, 'no combined convergence.')
return res
def mellin_transform(f, x, s, **hints):
r"""
Compute the Mellin transform `F(s)` of `f(x)`,
.. math :: F(s) = \int_0^\infty x^{s-1} f(x) \mathrm{d}x.
For all "sensible" functions, this converges absolutely in a strip
`a < \operatorname{Re}(s) < b`.
Explanation
===========
The Mellin transform is related via change of variables to the Fourier
transform, and also to the (bilateral) Laplace transform.
This function returns ``(F, (a, b), cond)``
where ``F`` is the Mellin transform of ``f``, ``(a, b)`` is the fundamental strip
(as above), and ``cond`` are auxiliary convergence conditions.
If the integral cannot be computed in closed form, this function returns
an unevaluated :class:`MellinTransform` object.
For a description of possible hints, refer to the docstring of
:func:`sympy.integrals.transforms.IntegralTransform.doit`. If ``noconds=False``,
then only `F` will be returned (i.e. not ``cond``, and also not the strip
``(a, b)``).
Examples
========
>>> from sympy.integrals.transforms import mellin_transform
>>> from sympy import exp
>>> from sympy.abc import x, s
>>> mellin_transform(exp(-x), x, s)
(gamma(s), (0, oo), True)
See Also
========
inverse_mellin_transform, laplace_transform, fourier_transform
hankel_transform, inverse_hankel_transform
"""
return MellinTransform(f, x, s).doit(**hints)
def _rewrite_sin(m_n, s, a, b):
"""
Re-write the sine function ``sin(m*s + n)`` as gamma functions, compatible
with the strip (a, b).
Return ``(gamma1, gamma2, fac)`` so that ``f == fac/(gamma1 * gamma2)``.
Examples
========
>>> from sympy.integrals.transforms import _rewrite_sin
>>> from sympy import pi, S
>>> from sympy.abc import s
>>> _rewrite_sin((pi, 0), s, 0, 1)
(gamma(s), gamma(1 - s), pi)
>>> _rewrite_sin((pi, 0), s, 1, 0)
(gamma(s - 1), gamma(2 - s), -pi)
>>> _rewrite_sin((pi, 0), s, -1, 0)
(gamma(s + 1), gamma(-s), -pi)
>>> _rewrite_sin((pi, pi/2), s, S(1)/2, S(3)/2)
(gamma(s - 1/2), gamma(3/2 - s), -pi)
>>> _rewrite_sin((pi, pi), s, 0, 1)
(gamma(s), gamma(1 - s), -pi)
>>> _rewrite_sin((2*pi, 0), s, 0, S(1)/2)
(gamma(2*s), gamma(1 - 2*s), pi)
>>> _rewrite_sin((2*pi, 0), s, S(1)/2, 1)
(gamma(2*s - 1), gamma(2 - 2*s), -pi)
"""
# (This is a separate function because it is moderately complicated,
# and I want to doctest it.)
# We want to use pi/sin(pi*x) = gamma(x)*gamma(1-x).
# But there is one comlication: the gamma functions determine the
# inegration contour in the definition of the G-function. Usually
# it would not matter if this is slightly shifted, unless this way
# we create an undefined function!
# So we try to write this in such a way that the gammas are
# eminently on the right side of the strip.
from sympy import expand_mul, pi, ceiling, gamma
m, n = m_n
m = expand_mul(m/pi)
n = expand_mul(n/pi)
r = ceiling(-m*a - n.as_real_imag()[0]) # Don't use re(n), does not expand
return gamma(m*s + n + r), gamma(1 - n - r - m*s), (-1)**r*pi
class MellinTransformStripError(ValueError):
"""
Exception raised by _rewrite_gamma. Mainly for internal use.
"""
pass
def _rewrite_gamma(f, s, a, b):
"""
Try to rewrite the product f(s) as a product of gamma functions,
so that the inverse Mellin transform of f can be expressed as a meijer
G function.
Explanation
===========
Return (an, ap), (bm, bq), arg, exp, fac such that
G((an, ap), (bm, bq), arg/z**exp)*fac is the inverse Mellin transform of f(s).
Raises IntegralTransformError or MellinTransformStripError on failure.
It is asserted that f has no poles in the fundamental strip designated by
(a, b). One of a and b is allowed to be None. The fundamental strip is
important, because it determines the inversion contour.
This function can handle exponentials, linear factors, trigonometric
functions.
This is a helper function for inverse_mellin_transform that will not
attempt any transformations on f.
Examples
========
>>> from sympy.integrals.transforms import _rewrite_gamma
>>> from sympy.abc import s
>>> from sympy import oo
>>> _rewrite_gamma(s*(s+3)*(s-1), s, -oo, oo)
(([], [-3, 0, 1]), ([-2, 1, 2], []), 1, 1, -1)
>>> _rewrite_gamma((s-1)**2, s, -oo, oo)
(([], [1, 1]), ([2, 2], []), 1, 1, 1)
Importance of the fundamental strip:
>>> _rewrite_gamma(1/s, s, 0, oo)
(([1], []), ([], [0]), 1, 1, 1)
>>> _rewrite_gamma(1/s, s, None, oo)
(([1], []), ([], [0]), 1, 1, 1)
>>> _rewrite_gamma(1/s, s, 0, None)
(([1], []), ([], [0]), 1, 1, 1)
>>> _rewrite_gamma(1/s, s, -oo, 0)
(([], [1]), ([0], []), 1, 1, -1)
>>> _rewrite_gamma(1/s, s, None, 0)
(([], [1]), ([0], []), 1, 1, -1)
>>> _rewrite_gamma(1/s, s, -oo, None)
(([], [1]), ([0], []), 1, 1, -1)
>>> _rewrite_gamma(2**(-s+3), s, -oo, oo)
(([], []), ([], []), 1/2, 1, 8)
"""
from itertools import repeat
from sympy import (Poly, gamma, Mul, re, CRootOf, exp as exp_, expand,
roots, ilcm, pi, sin, cos, tan, cot, igcd, exp_polar)
# Our strategy will be as follows:
# 1) Guess a constant c such that the inversion integral should be
# performed wrt s'=c*s (instead of plain s). Write s for s'.
# 2) Process all factors, rewrite them independently as gamma functions in
# argument s, or exponentials of s.
# 3) Try to transform all gamma functions s.t. they have argument
# a+s or a-s.
# 4) Check that the resulting G function parameters are valid.
# 5) Combine all the exponentials.
a_, b_ = S([a, b])
def left(c, is_numer):
"""
Decide whether pole at c lies to the left of the fundamental strip.
"""
# heuristically, this is the best chance for us to solve the inequalities
c = expand(re(c))
if a_ is None and b_ is oo:
return True
if a_ is None:
return c < b_
if b_ is None:
return c <= a_
if (c >= b_) == True:
return False
if (c <= a_) == True:
return True
if is_numer:
return None
if a_.free_symbols or b_.free_symbols or c.free_symbols:
return None # XXX
#raise IntegralTransformError('Inverse Mellin', f,
# 'Could not determine position of singularity %s'
# ' relative to fundamental strip' % c)
raise MellinTransformStripError('Pole inside critical strip?')
# 1)
s_multipliers = []
for g in f.atoms(gamma):
if not g.has(s):
continue
arg = g.args[0]
if arg.is_Add:
arg = arg.as_independent(s)[1]
coeff, _ = arg.as_coeff_mul(s)
s_multipliers += [coeff]
for g in f.atoms(sin, cos, tan, cot):
if not g.has(s):
continue
arg = g.args[0]
if arg.is_Add:
arg = arg.as_independent(s)[1]
coeff, _ = arg.as_coeff_mul(s)
s_multipliers += [coeff/pi]
s_multipliers = [abs(x) if x.is_extended_real else x for x in s_multipliers]
common_coefficient = S.One
for x in s_multipliers:
if not x.is_Rational:
common_coefficient = x
break
s_multipliers = [x/common_coefficient for x in s_multipliers]
if (any(not x.is_Rational for x in s_multipliers) or
not common_coefficient.is_extended_real):
raise IntegralTransformError("Gamma", None, "Nonrational multiplier")
s_multiplier = common_coefficient/reduce(ilcm, [S(x.q)
for x in s_multipliers], S.One)
if s_multiplier == common_coefficient:
if len(s_multipliers) == 0:
s_multiplier = common_coefficient
else:
s_multiplier = common_coefficient \
*reduce(igcd, [S(x.p) for x in s_multipliers])
f = f.subs(s, s/s_multiplier)
fac = S.One/s_multiplier
exponent = S.One/s_multiplier
if a_ is not None:
a_ *= s_multiplier
if b_ is not None:
b_ *= s_multiplier
# 2)
numer, denom = f.as_numer_denom()
numer = Mul.make_args(numer)
denom = Mul.make_args(denom)
args = list(zip(numer, repeat(True))) + list(zip(denom, repeat(False)))
facs = []
dfacs = []
# *_gammas will contain pairs (a, c) representing Gamma(a*s + c)
numer_gammas = []
denom_gammas = []
# exponentials will contain bases for exponentials of s
exponentials = []
def exception(fact):
return IntegralTransformError("Inverse Mellin", f, "Unrecognised form '%s'." % fact)
while args:
fact, is_numer = args.pop()
if is_numer:
ugammas, lgammas = numer_gammas, denom_gammas
ufacs = facs
else:
ugammas, lgammas = denom_gammas, numer_gammas
ufacs = dfacs
def linear_arg(arg):
""" Test if arg is of form a*s+b, raise exception if not. """
if not arg.is_polynomial(s):
raise exception(fact)
p = Poly(arg, s)
if p.degree() != 1:
raise exception(fact)
return p.all_coeffs()
# constants
if not fact.has(s):
ufacs += [fact]
# exponentials
elif fact.is_Pow or isinstance(fact, exp_):
if fact.is_Pow:
base = fact.base
exp = fact.exp
else:
base = exp_polar(1)
exp = fact.exp
if exp.is_Integer:
cond = is_numer
if exp < 0:
cond = not cond
args += [(base, cond)]*abs(exp)
continue
elif not base.has(s):
a, b = linear_arg(exp)
if not is_numer:
base = 1/base
exponentials += [base**a]
facs += [base**b]
else:
raise exception(fact)
# linear factors
elif fact.is_polynomial(s):
p = Poly(fact, s)
if p.degree() != 1:
# We completely factor the poly. For this we need the roots.
# Now roots() only works in some cases (low degree), and CRootOf
# only works without parameters. So try both...
coeff = p.LT()[1]
rs = roots(p, s)
if len(rs) != p.degree():
rs = CRootOf.all_roots(p)
ufacs += [coeff]
args += [(s - c, is_numer) for c in rs]
continue
a, c = p.all_coeffs()
ufacs += [a]
c /= -a
# Now need to convert s - c
if left(c, is_numer):
ugammas += [(S.One, -c + 1)]
lgammas += [(S.One, -c)]
else:
ufacs += [-1]
ugammas += [(S.NegativeOne, c + 1)]
lgammas += [(S.NegativeOne, c)]
elif isinstance(fact, gamma):
a, b = linear_arg(fact.args[0])
if is_numer:
if (a > 0 and (left(-b/a, is_numer) == False)) or \
(a < 0 and (left(-b/a, is_numer) == True)):
raise NotImplementedError(
'Gammas partially over the strip.')
ugammas += [(a, b)]
elif isinstance(fact, sin):
# We try to re-write all trigs as gammas. This is not in
# general the best strategy, since sometimes this is impossible,
# but rewriting as exponentials would work. However trig functions
# in inverse mellin transforms usually all come from simplifying
# gamma terms, so this should work.
a = fact.args[0]
if is_numer:
# No problem with the poles.
gamma1, gamma2, fac_ = gamma(a/pi), gamma(1 - a/pi), pi
else:
gamma1, gamma2, fac_ = _rewrite_sin(linear_arg(a), s, a_, b_)
args += [(gamma1, not is_numer), (gamma2, not is_numer)]
ufacs += [fac_]
elif isinstance(fact, tan):
a = fact.args[0]
args += [(sin(a, evaluate=False), is_numer),
(sin(pi/2 - a, evaluate=False), not is_numer)]
elif isinstance(fact, cos):
a = fact.args[0]
args += [(sin(pi/2 - a, evaluate=False), is_numer)]
elif isinstance(fact, cot):
a = fact.args[0]
args += [(sin(pi/2 - a, evaluate=False), is_numer),
(sin(a, evaluate=False), not is_numer)]
else:
raise exception(fact)
fac *= Mul(*facs)/Mul(*dfacs)
# 3)
an, ap, bm, bq = [], [], [], []
for gammas, plus, minus, is_numer in [(numer_gammas, an, bm, True),
(denom_gammas, bq, ap, False)]:
while gammas:
a, c = gammas.pop()
if a != -1 and a != +1:
# We use the gamma function multiplication theorem.
p = abs(S(a))
newa = a/p
newc = c/p
if not a.is_Integer:
raise TypeError("a is not an integer")
for k in range(p):
gammas += [(newa, newc + k/p)]
if is_numer:
fac *= (2*pi)**((1 - p)/2) * p**(c - S.Half)
exponentials += [p**a]
else:
fac /= (2*pi)**((1 - p)/2) * p**(c - S.Half)
exponentials += [p**(-a)]
continue
if a == +1:
plus.append(1 - c)
else:
minus.append(c)
# 4)
# TODO
# 5)
arg = Mul(*exponentials)
# for testability, sort the arguments
an.sort(key=default_sort_key)
ap.sort(key=default_sort_key)
bm.sort(key=default_sort_key)
bq.sort(key=default_sort_key)
return (an, ap), (bm, bq), arg, exponent, fac
@_noconds_(True)
def _inverse_mellin_transform(F, s, x_, strip, as_meijerg=False):
""" A helper for the real inverse_mellin_transform function, this one here
assumes x to be real and positive. """
from sympy import (expand, expand_mul, hyperexpand, meijerg,
arg, pi, re, factor, Heaviside, gamma, Add)
x = _dummy('t', 'inverse-mellin-transform', F, positive=True)
# Actually, we won't try integration at all. Instead we use the definition
# of the Meijer G function as a fairly general inverse mellin transform.
F = F.rewrite(gamma)
for g in [factor(F), expand_mul(F), expand(F)]:
if g.is_Add:
# do all terms separately
ress = [_inverse_mellin_transform(G, s, x, strip, as_meijerg,
noconds=False)
for G in g.args]
conds = [p[1] for p in ress]
ress = [p[0] for p in ress]
res = Add(*ress)
if not as_meijerg:
res = factor(res, gens=res.atoms(Heaviside))
return res.subs(x, x_), And(*conds)
try:
a, b, C, e, fac = _rewrite_gamma(g, s, strip[0], strip[1])
except IntegralTransformError:
continue
try:
G = meijerg(a, b, C/x**e)
except ValueError:
continue
if as_meijerg:
h = G
else:
try:
h = hyperexpand(G)
except NotImplementedError:
raise IntegralTransformError(
'Inverse Mellin', F, 'Could not calculate integral')
if h.is_Piecewise and len(h.args) == 3:
# XXX we break modularity here!
h = Heaviside(x - abs(C))*h.args[0].args[0] \
+ Heaviside(abs(C) - x)*h.args[1].args[0]
# We must ensure that the integral along the line we want converges,
# and return that value.
# See [L], 5.2
cond = [abs(arg(G.argument)) < G.delta*pi]
# Note: we allow ">=" here, this corresponds to convergence if we let
# limits go to oo symmetrically. ">" corresponds to absolute convergence.
cond += [And(Or(len(G.ap) != len(G.bq), 0 >= re(G.nu) + 1),
abs(arg(G.argument)) == G.delta*pi)]
cond = Or(*cond)
if cond == False:
raise IntegralTransformError(
'Inverse Mellin', F, 'does not converge')
return (h*fac).subs(x, x_), cond
raise IntegralTransformError('Inverse Mellin', F, '')
_allowed = None
class InverseMellinTransform(IntegralTransform):
"""
Class representing unevaluated inverse Mellin transforms.
For usage of this class, see the :class:`IntegralTransform` docstring.
For how to compute inverse Mellin transforms, see the
:func:`inverse_mellin_transform` docstring.
"""
_name = 'Inverse Mellin'
_none_sentinel = Dummy('None')
_c = Dummy('c')
def __new__(cls, F, s, x, a, b, **opts):
if a is None:
a = InverseMellinTransform._none_sentinel
if b is None:
b = InverseMellinTransform._none_sentinel
return IntegralTransform.__new__(cls, F, s, x, a, b, **opts)
@property
def fundamental_strip(self):
a, b = self.args[3], self.args[4]
if a is InverseMellinTransform._none_sentinel:
a = None
if b is InverseMellinTransform._none_sentinel:
b = None
return a, b
def _compute_transform(self, F, s, x, **hints):
from sympy import postorder_traversal
global _allowed
if _allowed is None:
from sympy import (
exp, gamma, sin, cos, tan, cot, cosh, sinh, tanh,
coth, factorial, rf)
_allowed = {
exp, gamma, sin, cos, tan, cot, cosh, sinh, tanh, coth,
factorial, rf}
for f in postorder_traversal(F):
if f.is_Function and f.has(s) and f.func not in _allowed:
raise IntegralTransformError('Inverse Mellin', F,
'Component %s not recognised.' % f)
strip = self.fundamental_strip
return _inverse_mellin_transform(F, s, x, strip, **hints)
def _as_integral(self, F, s, x):
from sympy import I
c = self.__class__._c
return Integral(F*x**(-s), (s, c - I*oo, c + I*oo))/(2*S.Pi*S.ImaginaryUnit)
def inverse_mellin_transform(F, s, x, strip, **hints):
r"""
Compute the inverse Mellin transform of `F(s)` over the fundamental
strip given by ``strip=(a, b)``.
Explanation
===========
This can be defined as
.. math:: f(x) = \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} x^{-s} F(s) \mathrm{d}s,
for any `c` in the fundamental strip. Under certain regularity
conditions on `F` and/or `f`,
this recovers `f` from its Mellin transform `F`
(and vice versa), for positive real `x`.
One of `a` or `b` may be passed as ``None``; a suitable `c` will be
inferred.
If the integral cannot be computed in closed form, this function returns
an unevaluated :class:`InverseMellinTransform` object.
Note that this function will assume x to be positive and real, regardless
of the sympy assumptions!
For a description of possible hints, refer to the docstring of
:func:`sympy.integrals.transforms.IntegralTransform.doit`.
Examples
========
>>> from sympy.integrals.transforms import inverse_mellin_transform
>>> from sympy import oo, gamma
>>> from sympy.abc import x, s
>>> inverse_mellin_transform(gamma(s), s, x, (0, oo))
exp(-x)
The fundamental strip matters:
>>> f = 1/(s**2 - 1)
>>> inverse_mellin_transform(f, s, x, (-oo, -1))
x*(1 - 1/x**2)*Heaviside(x - 1)/2
>>> inverse_mellin_transform(f, s, x, (-1, 1))
-x*Heaviside(1 - x)/2 - Heaviside(x - 1)/(2*x)
>>> inverse_mellin_transform(f, s, x, (1, oo))
(1/2 - x**2/2)*Heaviside(1 - x)/x
See Also
========
mellin_transform
hankel_transform, inverse_hankel_transform
"""
return InverseMellinTransform(F, s, x, strip[0], strip[1]).doit(**hints)
##########################################################################
# Laplace Transform
##########################################################################
def _simplifyconds(expr, s, a):
r"""
Naively simplify some conditions occurring in ``expr``, given that `\operatorname{Re}(s) > a`.
Examples
========
>>> from sympy.integrals.transforms import _simplifyconds as simp
>>> from sympy.abc import x
>>> from sympy import sympify as S
>>> simp(abs(x**2) < 1, x, 1)
False
>>> simp(abs(x**2) < 1, x, 2)
False
>>> simp(abs(x**2) < 1, x, 0)
Abs(x**2) < 1
>>> simp(abs(1/x**2) < 1, x, 1)
True
>>> simp(S(1) < abs(x), x, 1)
True
>>> simp(S(1) < abs(1/x), x, 1)
False
>>> from sympy import Ne
>>> simp(Ne(1, x**3), x, 1)
True
>>> simp(Ne(1, x**3), x, 2)
True
>>> simp(Ne(1, x**3), x, 0)
Ne(1, x**3)
"""
from sympy.core.relational import ( StrictGreaterThan, StrictLessThan,
Unequality )
from sympy import Abs
def power(ex):
if ex == s:
return 1
if ex.is_Pow and ex.base == s:
return ex.exp
return None
def bigger(ex1, ex2):
""" Return True only if |ex1| > |ex2|, False only if |ex1| < |ex2|.
Else return None. """
if ex1.has(s) and ex2.has(s):
return None
if isinstance(ex1, Abs):
ex1 = ex1.args[0]
if isinstance(ex2, Abs):
ex2 = ex2.args[0]
if ex1.has(s):
return bigger(1/ex2, 1/ex1)
n = power(ex2)
if n is None:
return None
try:
if n > 0 and (abs(ex1) <= abs(a)**n) == True:
return False
if n < 0 and (abs(ex1) >= abs(a)**n) == True:
return True
except TypeError:
pass
def replie(x, y):
""" simplify x < y """
if not (x.is_positive or isinstance(x, Abs)) \
or not (y.is_positive or isinstance(y, Abs)):
return (x < y)
r = bigger(x, y)
if r is not None:
return not r
return (x < y)
def replue(x, y):
b = bigger(x, y)
if b == True or b == False:
return True
return Unequality(x, y)
def repl(ex, *args):
if ex == True or ex == False:
return bool(ex)
return ex.replace(*args)
from sympy.simplify.radsimp import collect_abs
expr = collect_abs(expr)
expr = repl(expr, StrictLessThan, replie)
expr = repl(expr, StrictGreaterThan, lambda x, y: replie(y, x))
expr = repl(expr, Unequality, replue)
return S(expr)
@_noconds
def _laplace_transform(f, t, s_, simplify=True):
""" The backend function for Laplace transforms. """
from sympy import (re, Max, exp, pi, Min, periodic_argument as arg_,
arg, cos, Wild, symbols, polar_lift)
s = Dummy('s')
F = integrate(exp(-s*t) * f, (t, 0, oo))
if not F.has(Integral):
return _simplify(F.subs(s, s_), simplify), -oo, S.true
if not F.is_Piecewise:
raise IntegralTransformError(
'Laplace', f, 'could not compute integral')
F, cond = F.args[0]
if F.has(Integral):
raise IntegralTransformError(
'Laplace', f, 'integral in unexpected form')
def process_conds(conds):
""" Turn ``conds`` into a strip and auxiliary conditions. """
a = -oo
aux = S.true
conds = conjuncts(to_cnf(conds))
p, q, w1, w2, w3, w4, w5 = symbols(
'p q w1 w2 w3 w4 w5', cls=Wild, exclude=[s])
patterns = (
p*abs(arg((s + w3)*q)) < w2,
p*abs(arg((s + w3)*q)) <= w2,
abs(arg_((s + w3)**p*q, w1)) < w2,
abs(arg_((s + w3)**p*q, w1)) <= w2,
abs(arg_((polar_lift(s + w3))**p*q, w1)) < w2,
abs(arg_((polar_lift(s + w3))**p*q, w1)) <= w2)
for c in conds:
a_ = oo
aux_ = []
for d in disjuncts(c):
if d.is_Relational and s in d.rhs.free_symbols:
d = d.reversed
if d.is_Relational and isinstance(d, (Ge, Gt)):
d = d.reversedsign
for pat in patterns:
m = d.match(pat)
if m:
break
if m:
if m[q].is_positive and m[w2]/m[p] == pi/2:
d = -re(s + m[w3]) < 0
m = d.match(p - cos(w1*abs(arg(s*w5))*w2)*abs(s**w3)**w4 < 0)
if not m:
m = d.match(
cos(p - abs(arg_(s**w1*w5, q))*w2)*abs(s**w3)**w4 < 0)
if not m:
m = d.match(
p - cos(abs(arg_(polar_lift(s)**w1*w5, q))*w2
)*abs(s**w3)**w4 < 0)
if m and all(m[wild].is_positive for wild in [w1, w2, w3, w4, w5]):
d = re(s) > m[p]
d_ = d.replace(
re, lambda x: x.expand().as_real_imag()[0]).subs(re(s), t)
if not d.is_Relational or \
d.rel_op in ('==', '!=') \
or d_.has(s) or not d_.has(t):
aux_ += [d]
continue
soln = _solve_inequality(d_, t)
if not soln.is_Relational or \
soln.rel_op in ('==', '!='):
aux_ += [d]
continue
if soln.lts == t:
raise IntegralTransformError('Laplace', f,
'convergence not in half-plane?')
else:
a_ = Min(soln.lts, a_)
if a_ != oo:
a = Max(a_, a)
else:
aux = And(aux, Or(*aux_))
return a, aux.canonical if aux.is_Relational else aux
conds = [process_conds(c) for c in disjuncts(cond)]
conds2 = [x for x in conds if x[1] != False and x[0] != -oo]
if not conds2:
conds2 = [x for x in conds if x[1] != False]
conds = list(ordered(conds2))
def cnt(expr):
if expr == True or expr == False:
return 0
return expr.count_ops()
conds.sort(key=lambda x: (-x[0], cnt(x[1])))
if not conds:
raise IntegralTransformError('Laplace', f, 'no convergence found')
a, aux = conds[0] # XXX is [0] always the right one?
def sbs(expr):
return expr.subs(s, s_)
if simplify:
F = _simplifyconds(F, s, a)
aux = _simplifyconds(aux, s, a)
return _simplify(F.subs(s, s_), simplify), sbs(a), _canonical(sbs(aux))
class LaplaceTransform(IntegralTransform):
"""
Class representing unevaluated Laplace transforms.
For usage of this class, see the :class:`IntegralTransform` docstring.
For how to compute Laplace transforms, see the :func:`laplace_transform`
docstring.
"""
_name = 'Laplace'
def _compute_transform(self, f, t, s, **hints):
return _laplace_transform(f, t, s, **hints)
def _as_integral(self, f, t, s):
from sympy import exp
return Integral(f*exp(-s*t), (t, 0, oo))
def _collapse_extra(self, extra):
from sympy import Max
conds = []
planes = []
for plane, cond in extra:
conds.append(cond)
planes.append(plane)
cond = And(*conds)
plane = Max(*planes)
if cond == False:
raise IntegralTransformError(
'Laplace', None, 'No combined convergence.')
return plane, cond
def laplace_transform(f, t, s, **hints):
r"""
Compute the Laplace Transform `F(s)` of `f(t)`,
.. math :: F(s) = \int_0^\infty e^{-st} f(t) \mathrm{d}t.
Explanation
===========
For all "sensible" functions, this converges absolutely in a
half plane `a < \operatorname{Re}(s)`.
This function returns ``(F, a, cond)``
where ``F`` is the Laplace transform of ``f``, `\operatorname{Re}(s) > a` is the half-plane
of convergence, and ``cond`` are auxiliary convergence conditions.
If the integral cannot be computed in closed form, this function returns
an unevaluated :class:`LaplaceTransform` object.
For a description of possible hints, refer to the docstring of
:func:`sympy.integrals.transforms.IntegralTransform.doit`. If ``noconds=True``,
only `F` will be returned (i.e. not ``cond``, and also not the plane ``a``).
Examples
========
>>> from sympy.integrals import laplace_transform
>>> from sympy.abc import t, s, a
>>> laplace_transform(t**a, t, s)
(s**(-a)*gamma(a + 1)/s, 0, re(a) > -1)
See Also
========
inverse_laplace_transform, mellin_transform, fourier_transform
hankel_transform, inverse_hankel_transform
"""
if isinstance(f, MatrixBase) and hasattr(f, 'applyfunc'):
return f.applyfunc(lambda fij: laplace_transform(fij, t, s, **hints))
return LaplaceTransform(f, t, s).doit(**hints)
@_noconds_(True)
def _inverse_laplace_transform(F, s, t_, plane, simplify=True):
""" The backend function for inverse Laplace transforms. """
from sympy import exp, Heaviside, log, expand_complex, Integral, Piecewise
from sympy.integrals.meijerint import meijerint_inversion, _get_coeff_exp
# There are two strategies we can try:
# 1) Use inverse mellin transforms - related by a simple change of variables.
# 2) Use the inversion integral.
t = Dummy('t', real=True)
def pw_simp(*args):
""" Simplify a piecewise expression from hyperexpand. """
# XXX we break modularity here!
if len(args) != 3:
return Piecewise(*args)
arg = args[2].args[0].argument
coeff, exponent = _get_coeff_exp(arg, t)
e1 = args[0].args[0]
e2 = args[1].args[0]
return Heaviside(1/abs(coeff) - t**exponent)*e1 \
+ Heaviside(t**exponent - 1/abs(coeff))*e2
try:
f, cond = inverse_mellin_transform(F, s, exp(-t), (None, oo),
needeval=True, noconds=False)
except IntegralTransformError:
f = None
if f is None:
f = meijerint_inversion(F, s, t)
if f is None:
raise IntegralTransformError('Inverse Laplace', f, '')
if f.is_Piecewise:
f, cond = f.args[0]
if f.has(Integral):
raise IntegralTransformError('Inverse Laplace', f,
'inversion integral of unrecognised form.')
else:
cond = S.true
f = f.replace(Piecewise, pw_simp)
if f.is_Piecewise:
# many of the functions called below can't work with piecewise
# (b/c it has a bool in args)
return f.subs(t, t_), cond
u = Dummy('u')
def simp_heaviside(arg):
a = arg.subs(exp(-t), u)
if a.has(t):
return Heaviside(arg)
rel = _solve_inequality(a > 0, u)
if rel.lts == u:
k = log(rel.gts)
return Heaviside(t + k)
else:
k = log(rel.lts)
return Heaviside(-(t + k))
f = f.replace(Heaviside, simp_heaviside)
def simp_exp(arg):
return expand_complex(exp(arg))
f = f.replace(exp, simp_exp)
# TODO it would be nice to fix cosh and sinh ... simplify messes these
# exponentials up
return _simplify(f.subs(t, t_), simplify), cond
class InverseLaplaceTransform(IntegralTransform):
"""
Class representing unevaluated inverse Laplace transforms.
For usage of this class, see the :class:`IntegralTransform` docstring.
For how to compute inverse Laplace transforms, see the
:func:`inverse_laplace_transform` docstring.
"""
_name = 'Inverse Laplace'
_none_sentinel = Dummy('None')
_c = Dummy('c')
def __new__(cls, F, s, x, plane, **opts):
if plane is None:
plane = InverseLaplaceTransform._none_sentinel
return IntegralTransform.__new__(cls, F, s, x, plane, **opts)
@property
def fundamental_plane(self):
plane = self.args[3]
if plane is InverseLaplaceTransform._none_sentinel:
plane = None
return plane
def _compute_transform(self, F, s, t, **hints):
return _inverse_laplace_transform(F, s, t, self.fundamental_plane, **hints)
def _as_integral(self, F, s, t):
from sympy import I, exp
c = self.__class__._c
return Integral(exp(s*t)*F, (s, c - I*oo, c + I*oo))/(2*S.Pi*S.ImaginaryUnit)
def inverse_laplace_transform(F, s, t, plane=None, **hints):
r"""
Compute the inverse Laplace transform of `F(s)`, defined as
.. math :: f(t) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} e^{st} F(s) \mathrm{d}s,
for `c` so large that `F(s)` has no singularites in the
half-plane `\operatorname{Re}(s) > c-\epsilon`.
Explanation
===========
The plane can be specified by
argument ``plane``, but will be inferred if passed as None.
Under certain regularity conditions, this recovers `f(t)` from its
Laplace Transform `F(s)`, for non-negative `t`, and vice
versa.
If the integral cannot be computed in closed form, this function returns
an unevaluated :class:`InverseLaplaceTransform` object.
Note that this function will always assume `t` to be real,
regardless of the sympy assumption on `t`.
For a description of possible hints, refer to the docstring of
:func:`sympy.integrals.transforms.IntegralTransform.doit`.
Examples
========
>>> from sympy.integrals.transforms import inverse_laplace_transform
>>> from sympy import exp, Symbol
>>> from sympy.abc import s, t
>>> a = Symbol('a', positive=True)
>>> inverse_laplace_transform(exp(-a*s)/s, s, t)
Heaviside(-a + t)
See Also
========
laplace_transform
hankel_transform, inverse_hankel_transform
"""
if isinstance(F, MatrixBase) and hasattr(F, 'applyfunc'):
return F.applyfunc(lambda Fij: inverse_laplace_transform(Fij, s, t, plane, **hints))
return InverseLaplaceTransform(F, s, t, plane).doit(**hints)
##########################################################################
# Fourier Transform
##########################################################################
@_noconds_(True)
def _fourier_transform(f, x, k, a, b, name, simplify=True):
r"""
Compute a general Fourier-type transform
.. math::
F(k) = a \int_{-\infty}^{\infty} e^{bixk} f(x)\, dx.
For suitable choice of *a* and *b*, this reduces to the standard Fourier
and inverse Fourier transforms.
"""
from sympy import exp, I
F = integrate(a*f*exp(b*I*x*k), (x, -oo, oo))
if not F.has(Integral):
return _simplify(F, simplify), S.true
integral_f = integrate(f, (x, -oo, oo))
if integral_f in (-oo, oo, S.NaN) or integral_f.has(Integral):
raise IntegralTransformError(name, f, 'function not integrable on real axis')
if not F.is_Piecewise:
raise IntegralTransformError(name, f, 'could not compute integral')
F, cond = F.args[0]
if F.has(Integral):
raise IntegralTransformError(name, f, 'integral in unexpected form')
return _simplify(F, simplify), cond
class FourierTypeTransform(IntegralTransform):
""" Base class for Fourier transforms."""
def a(self):
raise NotImplementedError(
"Class %s must implement a(self) but does not" % self.__class__)
def b(self):
raise NotImplementedError(
"Class %s must implement b(self) but does not" % self.__class__)
def _compute_transform(self, f, x, k, **hints):
return _fourier_transform(f, x, k,
self.a(), self.b(),
self.__class__._name, **hints)
def _as_integral(self, f, x, k):
from sympy import exp, I
a = self.a()
b = self.b()
return Integral(a*f*exp(b*I*x*k), (x, -oo, oo))
class FourierTransform(FourierTypeTransform):
"""
Class representing unevaluated Fourier transforms.
For usage of this class, see the :class:`IntegralTransform` docstring.
For how to compute Fourier transforms, see the :func:`fourier_transform`
docstring.
"""
_name = 'Fourier'
def a(self):
return 1
def b(self):
return -2*S.Pi
def fourier_transform(f, x, k, **hints):
r"""
Compute the unitary, ordinary-frequency Fourier transform of ``f``, defined
as
.. math:: F(k) = \int_{-\infty}^\infty f(x) e^{-2\pi i x k} \mathrm{d} x.
Explanation
===========
If the transform cannot be computed in closed form, this
function returns an unevaluated :class:`FourierTransform` object.
For other Fourier transform conventions, see the function
:func:`sympy.integrals.transforms._fourier_transform`.
For a description of possible hints, refer to the docstring of
:func:`sympy.integrals.transforms.IntegralTransform.doit`.
Note that for this transform, by default ``noconds=True``.
Examples
========
>>> from sympy import fourier_transform, exp
>>> from sympy.abc import x, k
>>> fourier_transform(exp(-x**2), x, k)
sqrt(pi)*exp(-pi**2*k**2)
>>> fourier_transform(exp(-x**2), x, k, noconds=False)
(sqrt(pi)*exp(-pi**2*k**2), True)
See Also
========
inverse_fourier_transform
sine_transform, inverse_sine_transform
cosine_transform, inverse_cosine_transform
hankel_transform, inverse_hankel_transform
mellin_transform, laplace_transform
"""
return FourierTransform(f, x, k).doit(**hints)
class InverseFourierTransform(FourierTypeTransform):
"""
Class representing unevaluated inverse Fourier transforms.
For usage of this class, see the :class:`IntegralTransform` docstring.
For how to compute inverse Fourier transforms, see the
:func:`inverse_fourier_transform` docstring.
"""
_name = 'Inverse Fourier'
def a(self):
return 1
def b(self):
return 2*S.Pi
def inverse_fourier_transform(F, k, x, **hints):
r"""
Compute the unitary, ordinary-frequency inverse Fourier transform of `F`,
defined as
.. math:: f(x) = \int_{-\infty}^\infty F(k) e^{2\pi i x k} \mathrm{d} k.
Explanation
===========
If the transform cannot be computed in closed form, this
function returns an unevaluated :class:`InverseFourierTransform` object.
For other Fourier transform conventions, see the function
:func:`sympy.integrals.transforms._fourier_transform`.
For a description of possible hints, refer to the docstring of
:func:`sympy.integrals.transforms.IntegralTransform.doit`.
Note that for this transform, by default ``noconds=True``.
Examples
========
>>> from sympy import inverse_fourier_transform, exp, sqrt, pi
>>> from sympy.abc import x, k
>>> inverse_fourier_transform(sqrt(pi)*exp(-(pi*k)**2), k, x)
exp(-x**2)
>>> inverse_fourier_transform(sqrt(pi)*exp(-(pi*k)**2), k, x, noconds=False)
(exp(-x**2), True)
See Also
========
fourier_transform
sine_transform, inverse_sine_transform
cosine_transform, inverse_cosine_transform
hankel_transform, inverse_hankel_transform
mellin_transform, laplace_transform
"""
return InverseFourierTransform(F, k, x).doit(**hints)
##########################################################################
# Fourier Sine and Cosine Transform
##########################################################################
from sympy import sin, cos, sqrt, pi
@_noconds_(True)
def _sine_cosine_transform(f, x, k, a, b, K, name, simplify=True):
"""
Compute a general sine or cosine-type transform
F(k) = a int_0^oo b*sin(x*k) f(x) dx.
F(k) = a int_0^oo b*cos(x*k) f(x) dx.
For suitable choice of a and b, this reduces to the standard sine/cosine
and inverse sine/cosine transforms.
"""
F = integrate(a*f*K(b*x*k), (x, 0, oo))
if not F.has(Integral):
return _simplify(F, simplify), S.true
if not F.is_Piecewise:
raise IntegralTransformError(name, f, 'could not compute integral')
F, cond = F.args[0]
if F.has(Integral):
raise IntegralTransformError(name, f, 'integral in unexpected form')
return _simplify(F, simplify), cond
class SineCosineTypeTransform(IntegralTransform):
"""
Base class for sine and cosine transforms.
Specify cls._kern.
"""
def a(self):
raise NotImplementedError(
"Class %s must implement a(self) but does not" % self.__class__)
def b(self):
raise NotImplementedError(
"Class %s must implement b(self) but does not" % self.__class__)
def _compute_transform(self, f, x, k, **hints):
return _sine_cosine_transform(f, x, k,
self.a(), self.b(),
self.__class__._kern,
self.__class__._name, **hints)
def _as_integral(self, f, x, k):
a = self.a()
b = self.b()
K = self.__class__._kern
return Integral(a*f*K(b*x*k), (x, 0, oo))
class SineTransform(SineCosineTypeTransform):
"""
Class representing unevaluated sine transforms.
For usage of this class, see the :class:`IntegralTransform` docstring.
For how to compute sine transforms, see the :func:`sine_transform`
docstring.
"""
_name = 'Sine'
_kern = sin
def a(self):
return sqrt(2)/sqrt(pi)
def b(self):
return 1
def sine_transform(f, x, k, **hints):
r"""
Compute the unitary, ordinary-frequency sine transform of `f`, defined
as
.. math:: F(k) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty f(x) \sin(2\pi x k) \mathrm{d} x.
Explanation
===========
If the transform cannot be computed in closed form, this
function returns an unevaluated :class:`SineTransform` object.
For a description of possible hints, refer to the docstring of
:func:`sympy.integrals.transforms.IntegralTransform.doit`.
Note that for this transform, by default ``noconds=True``.
Examples
========
>>> from sympy import sine_transform, exp
>>> from sympy.abc import x, k, a
>>> sine_transform(x*exp(-a*x**2), x, k)
sqrt(2)*k*exp(-k**2/(4*a))/(4*a**(3/2))
>>> sine_transform(x**(-a), x, k)
2**(1/2 - a)*k**(a - 1)*gamma(1 - a/2)/gamma(a/2 + 1/2)
See Also
========
fourier_transform, inverse_fourier_transform
inverse_sine_transform
cosine_transform, inverse_cosine_transform
hankel_transform, inverse_hankel_transform
mellin_transform, laplace_transform
"""
return SineTransform(f, x, k).doit(**hints)
class InverseSineTransform(SineCosineTypeTransform):
"""
Class representing unevaluated inverse sine transforms.
For usage of this class, see the :class:`IntegralTransform` docstring.
For how to compute inverse sine transforms, see the
:func:`inverse_sine_transform` docstring.
"""
_name = 'Inverse Sine'
_kern = sin
def a(self):
return sqrt(2)/sqrt(pi)
def b(self):
return 1
def inverse_sine_transform(F, k, x, **hints):
r"""
Compute the unitary, ordinary-frequency inverse sine transform of `F`,
defined as
.. math:: f(x) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty F(k) \sin(2\pi x k) \mathrm{d} k.
Explanation
===========
If the transform cannot be computed in closed form, this
function returns an unevaluated :class:`InverseSineTransform` object.
For a description of possible hints, refer to the docstring of
:func:`sympy.integrals.transforms.IntegralTransform.doit`.
Note that for this transform, by default ``noconds=True``.
Examples
========
>>> from sympy import inverse_sine_transform, exp, sqrt, gamma
>>> from sympy.abc import x, k, a
>>> inverse_sine_transform(2**((1-2*a)/2)*k**(a - 1)*
... gamma(-a/2 + 1)/gamma((a+1)/2), k, x)
x**(-a)
>>> inverse_sine_transform(sqrt(2)*k*exp(-k**2/(4*a))/(4*sqrt(a)**3), k, x)
x*exp(-a*x**2)
See Also
========
fourier_transform, inverse_fourier_transform
sine_transform
cosine_transform, inverse_cosine_transform
hankel_transform, inverse_hankel_transform
mellin_transform, laplace_transform
"""
return InverseSineTransform(F, k, x).doit(**hints)
class CosineTransform(SineCosineTypeTransform):
"""
Class representing unevaluated cosine transforms.
For usage of this class, see the :class:`IntegralTransform` docstring.
For how to compute cosine transforms, see the :func:`cosine_transform`
docstring.
"""
_name = 'Cosine'
_kern = cos
def a(self):
return sqrt(2)/sqrt(pi)
def b(self):
return 1
def cosine_transform(f, x, k, **hints):
r"""
Compute the unitary, ordinary-frequency cosine transform of `f`, defined
as
.. math:: F(k) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty f(x) \cos(2\pi x k) \mathrm{d} x.
Explanation
===========
If the transform cannot be computed in closed form, this
function returns an unevaluated :class:`CosineTransform` object.
For a description of possible hints, refer to the docstring of
:func:`sympy.integrals.transforms.IntegralTransform.doit`.
Note that for this transform, by default ``noconds=True``.
Examples
========
>>> from sympy import cosine_transform, exp, sqrt, cos
>>> from sympy.abc import x, k, a
>>> cosine_transform(exp(-a*x), x, k)
sqrt(2)*a/(sqrt(pi)*(a**2 + k**2))
>>> cosine_transform(exp(-a*sqrt(x))*cos(a*sqrt(x)), x, k)
a*exp(-a**2/(2*k))/(2*k**(3/2))
See Also
========
fourier_transform, inverse_fourier_transform,
sine_transform, inverse_sine_transform
inverse_cosine_transform
hankel_transform, inverse_hankel_transform
mellin_transform, laplace_transform
"""
return CosineTransform(f, x, k).doit(**hints)
class InverseCosineTransform(SineCosineTypeTransform):
"""
Class representing unevaluated inverse cosine transforms.
For usage of this class, see the :class:`IntegralTransform` docstring.
For how to compute inverse cosine transforms, see the
:func:`inverse_cosine_transform` docstring.
"""
_name = 'Inverse Cosine'
_kern = cos
def a(self):
return sqrt(2)/sqrt(pi)
def b(self):
return 1
def inverse_cosine_transform(F, k, x, **hints):
r"""
Compute the unitary, ordinary-frequency inverse cosine transform of `F`,
defined as
.. math:: f(x) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty F(k) \cos(2\pi x k) \mathrm{d} k.
Explanation
===========
If the transform cannot be computed in closed form, this
function returns an unevaluated :class:`InverseCosineTransform` object.
For a description of possible hints, refer to the docstring of
:func:`sympy.integrals.transforms.IntegralTransform.doit`.
Note that for this transform, by default ``noconds=True``.
Examples
========
>>> from sympy import inverse_cosine_transform, sqrt, pi
>>> from sympy.abc import x, k, a
>>> inverse_cosine_transform(sqrt(2)*a/(sqrt(pi)*(a**2 + k**2)), k, x)
exp(-a*x)
>>> inverse_cosine_transform(1/sqrt(k), k, x)
1/sqrt(x)
See Also
========
fourier_transform, inverse_fourier_transform,
sine_transform, inverse_sine_transform
cosine_transform
hankel_transform, inverse_hankel_transform
mellin_transform, laplace_transform
"""
return InverseCosineTransform(F, k, x).doit(**hints)
##########################################################################
# Hankel Transform
##########################################################################
@_noconds_(True)
def _hankel_transform(f, r, k, nu, name, simplify=True):
r"""
Compute a general Hankel transform
.. math:: F_\nu(k) = \int_{0}^\infty f(r) J_\nu(k r) r \mathrm{d} r.
"""
from sympy import besselj
F = integrate(f*besselj(nu, k*r)*r, (r, 0, oo))
if not F.has(Integral):
return _simplify(F, simplify), S.true
if not F.is_Piecewise:
raise IntegralTransformError(name, f, 'could not compute integral')
F, cond = F.args[0]
if F.has(Integral):
raise IntegralTransformError(name, f, 'integral in unexpected form')
return _simplify(F, simplify), cond
class HankelTypeTransform(IntegralTransform):
"""
Base class for Hankel transforms.
"""
def doit(self, **hints):
return self._compute_transform(self.function,
self.function_variable,
self.transform_variable,
self.args[3],
**hints)
def _compute_transform(self, f, r, k, nu, **hints):
return _hankel_transform(f, r, k, nu, self._name, **hints)
def _as_integral(self, f, r, k, nu):
from sympy import besselj
return Integral(f*besselj(nu, k*r)*r, (r, 0, oo))
@property
def as_integral(self):
return self._as_integral(self.function,
self.function_variable,
self.transform_variable,
self.args[3])
class HankelTransform(HankelTypeTransform):
"""
Class representing unevaluated Hankel transforms.
For usage of this class, see the :class:`IntegralTransform` docstring.
For how to compute Hankel transforms, see the :func:`hankel_transform`
docstring.
"""
_name = 'Hankel'
def hankel_transform(f, r, k, nu, **hints):
r"""
Compute the Hankel transform of `f`, defined as
.. math:: F_\nu(k) = \int_{0}^\infty f(r) J_\nu(k r) r \mathrm{d} r.
Explanation
===========
If the transform cannot be computed in closed form, this
function returns an unevaluated :class:`HankelTransform` object.
For a description of possible hints, refer to the docstring of
:func:`sympy.integrals.transforms.IntegralTransform.doit`.
Note that for this transform, by default ``noconds=True``.
Examples
========
>>> from sympy import hankel_transform, inverse_hankel_transform
>>> from sympy import exp
>>> from sympy.abc import r, k, m, nu, a
>>> ht = hankel_transform(1/r**m, r, k, nu)
>>> ht
2*2**(-m)*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2)
>>> inverse_hankel_transform(ht, k, r, nu)
r**(-m)
>>> ht = hankel_transform(exp(-a*r), r, k, 0)
>>> ht
a/(k**3*(a**2/k**2 + 1)**(3/2))
>>> inverse_hankel_transform(ht, k, r, 0)
exp(-a*r)
See Also
========
fourier_transform, inverse_fourier_transform
sine_transform, inverse_sine_transform
cosine_transform, inverse_cosine_transform
inverse_hankel_transform
mellin_transform, laplace_transform
"""
return HankelTransform(f, r, k, nu).doit(**hints)
class InverseHankelTransform(HankelTypeTransform):
"""
Class representing unevaluated inverse Hankel transforms.
For usage of this class, see the :class:`IntegralTransform` docstring.
For how to compute inverse Hankel transforms, see the
:func:`inverse_hankel_transform` docstring.
"""
_name = 'Inverse Hankel'
def inverse_hankel_transform(F, k, r, nu, **hints):
r"""
Compute the inverse Hankel transform of `F` defined as
.. math:: f(r) = \int_{0}^\infty F_\nu(k) J_\nu(k r) k \mathrm{d} k.
Explanation
===========
If the transform cannot be computed in closed form, this
function returns an unevaluated :class:`InverseHankelTransform` object.
For a description of possible hints, refer to the docstring of
:func:`sympy.integrals.transforms.IntegralTransform.doit`.
Note that for this transform, by default ``noconds=True``.
Examples
========
>>> from sympy import hankel_transform, inverse_hankel_transform
>>> from sympy import exp
>>> from sympy.abc import r, k, m, nu, a
>>> ht = hankel_transform(1/r**m, r, k, nu)
>>> ht
2*2**(-m)*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2)
>>> inverse_hankel_transform(ht, k, r, nu)
r**(-m)
>>> ht = hankel_transform(exp(-a*r), r, k, 0)
>>> ht
a/(k**3*(a**2/k**2 + 1)**(3/2))
>>> inverse_hankel_transform(ht, k, r, 0)
exp(-a*r)
See Also
========
fourier_transform, inverse_fourier_transform
sine_transform, inverse_sine_transform
cosine_transform, inverse_cosine_transform
hankel_transform
mellin_transform, laplace_transform
"""
return InverseHankelTransform(F, k, r, nu).doit(**hints)
|
70433a41fa130b8d0234ee6cf62d733db3d53ee12065a1acb3b842519d49f111 | from typing import Dict, List
from itertools import permutations
from functools import reduce
from sympy.core.add import Add
from sympy.core.basic import Basic
from sympy.core.mul import Mul
from sympy.core.symbol import Wild, Dummy
from sympy.core.basic import sympify
from sympy.core.numbers import Rational, pi, I
from sympy.core.relational import Eq, Ne
from sympy.core.singleton import S
from sympy.functions import exp, sin, cos, tan, cot, asin, atan
from sympy.functions import log, sinh, cosh, tanh, coth, asinh, acosh
from sympy.functions import sqrt, erf, erfi, li, Ei
from sympy.functions import besselj, bessely, besseli, besselk
from sympy.functions import hankel1, hankel2, jn, yn
from sympy.functions.elementary.complexes import Abs, re, im, sign, arg
from sympy.functions.elementary.exponential import LambertW
from sympy.functions.elementary.integers import floor, ceiling
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.special.delta_functions import Heaviside, DiracDelta
from sympy.simplify.radsimp import collect
from sympy.logic.boolalg import And, Or
from sympy.utilities.iterables import uniq
from sympy.polys import quo, gcd, lcm, factor, cancel, PolynomialError
from sympy.polys.monomials import itermonomials
from sympy.polys.polyroots import root_factors
from sympy.polys.rings import PolyRing
from sympy.polys.solvers import solve_lin_sys
from sympy.polys.constructor import construct_domain
from sympy.core.compatibility import ordered
from sympy.integrals.integrals import integrate
def components(f, x):
"""
Returns a set of all functional components of the given expression
which includes symbols, function applications and compositions and
non-integer powers. Fractional powers are collected with
minimal, positive exponents.
Examples
========
>>> from sympy import cos, sin
>>> from sympy.abc import x
>>> from sympy.integrals.heurisch import components
>>> components(sin(x)*cos(x)**2, x)
{x, sin(x), cos(x)}
See Also
========
heurisch
"""
result = set()
if x in f.free_symbols:
if f.is_symbol and f.is_commutative:
result.add(f)
elif f.is_Function or f.is_Derivative:
for g in f.args:
result |= components(g, x)
result.add(f)
elif f.is_Pow:
result |= components(f.base, x)
if not f.exp.is_Integer:
if f.exp.is_Rational:
result.add(f.base**Rational(1, f.exp.q))
else:
result |= components(f.exp, x) | {f}
else:
for g in f.args:
result |= components(g, x)
return result
# name -> [] of symbols
_symbols_cache = {} # type: Dict[str, List[Dummy]]
# NB @cacheit is not convenient here
def _symbols(name, n):
"""get vector of symbols local to this module"""
try:
lsyms = _symbols_cache[name]
except KeyError:
lsyms = []
_symbols_cache[name] = lsyms
while len(lsyms) < n:
lsyms.append( Dummy('%s%i' % (name, len(lsyms))) )
return lsyms[:n]
def heurisch_wrapper(f, x, rewrite=False, hints=None, mappings=None, retries=3,
degree_offset=0, unnecessary_permutations=None,
_try_heurisch=None):
"""
A wrapper around the heurisch integration algorithm.
Explanation
===========
This method takes the result from heurisch and checks for poles in the
denominator. For each of these poles, the integral is reevaluated, and
the final integration result is given in terms of a Piecewise.
Examples
========
>>> from sympy.core import symbols
>>> from sympy.functions import cos
>>> from sympy.integrals.heurisch import heurisch, heurisch_wrapper
>>> n, x = symbols('n x')
>>> heurisch(cos(n*x), x)
sin(n*x)/n
>>> heurisch_wrapper(cos(n*x), x)
Piecewise((sin(n*x)/n, Ne(n, 0)), (x, True))
See Also
========
heurisch
"""
from sympy.solvers.solvers import solve, denoms
f = sympify(f)
if x not in f.free_symbols:
return f*x
res = heurisch(f, x, rewrite, hints, mappings, retries, degree_offset,
unnecessary_permutations, _try_heurisch)
if not isinstance(res, Basic):
return res
# We consider each denominator in the expression, and try to find
# cases where one or more symbolic denominator might be zero. The
# conditions for these cases are stored in the list slns.
slns = []
for d in denoms(res):
try:
slns += solve(d, dict=True, exclude=(x,))
except NotImplementedError:
pass
if not slns:
return res
slns = list(uniq(slns))
# Remove the solutions corresponding to poles in the original expression.
slns0 = []
for d in denoms(f):
try:
slns0 += solve(d, dict=True, exclude=(x,))
except NotImplementedError:
pass
slns = [s for s in slns if s not in slns0]
if not slns:
return res
if len(slns) > 1:
eqs = []
for sub_dict in slns:
eqs.extend([Eq(key, value) for key, value in sub_dict.items()])
slns = solve(eqs, dict=True, exclude=(x,)) + slns
# For each case listed in the list slns, we reevaluate the integral.
pairs = []
for sub_dict in slns:
expr = heurisch(f.subs(sub_dict), x, rewrite, hints, mappings, retries,
degree_offset, unnecessary_permutations,
_try_heurisch)
cond = And(*[Eq(key, value) for key, value in sub_dict.items()])
generic = Or(*[Ne(key, value) for key, value in sub_dict.items()])
if expr is None:
expr = integrate(f.subs(sub_dict),x)
pairs.append((expr, cond))
# If there is one condition, put the generic case first. Otherwise,
# doing so may lead to longer Piecewise formulas
if len(pairs) == 1:
pairs = [(heurisch(f, x, rewrite, hints, mappings, retries,
degree_offset, unnecessary_permutations,
_try_heurisch),
generic),
(pairs[0][0], True)]
else:
pairs.append((heurisch(f, x, rewrite, hints, mappings, retries,
degree_offset, unnecessary_permutations,
_try_heurisch),
True))
return Piecewise(*pairs)
class BesselTable:
"""
Derivatives of Bessel functions of orders n and n-1
in terms of each other.
See the docstring of DiffCache.
"""
def __init__(self):
self.table = {}
self.n = Dummy('n')
self.z = Dummy('z')
self._create_table()
def _create_table(t):
table, n, z = t.table, t.n, t.z
for f in (besselj, bessely, hankel1, hankel2):
table[f] = (f(n-1, z) - n*f(n, z)/z,
(n-1)*f(n-1, z)/z - f(n, z))
f = besseli
table[f] = (f(n-1, z) - n*f(n, z)/z,
(n-1)*f(n-1, z)/z + f(n, z))
f = besselk
table[f] = (-f(n-1, z) - n*f(n, z)/z,
(n-1)*f(n-1, z)/z - f(n, z))
for f in (jn, yn):
table[f] = (f(n-1, z) - (n+1)*f(n, z)/z,
(n-1)*f(n-1, z)/z - f(n, z))
def diffs(t, f, n, z):
if f in t.table:
diff0, diff1 = t.table[f]
repl = [(t.n, n), (t.z, z)]
return (diff0.subs(repl), diff1.subs(repl))
def has(t, f):
return f in t.table
_bessel_table = None
class DiffCache:
"""
Store for derivatives of expressions.
Explanation
===========
The standard form of the derivative of a Bessel function of order n
contains two Bessel functions of orders n-1 and n+1, respectively.
Such forms cannot be used in parallel Risch algorithm, because
there is a linear recurrence relation between the three functions
while the algorithm expects that functions and derivatives are
represented in terms of algebraically independent transcendentals.
The solution is to take two of the functions, e.g., those of orders
n and n-1, and to express the derivatives in terms of the pair.
To guarantee that the proper form is used the two derivatives are
cached as soon as one is encountered.
Derivatives of other functions are also cached at no extra cost.
All derivatives are with respect to the same variable `x`.
"""
def __init__(self, x):
self.cache = {}
self.x = x
global _bessel_table
if not _bessel_table:
_bessel_table = BesselTable()
def get_diff(self, f):
cache = self.cache
if f in cache:
pass
elif (not hasattr(f, 'func') or
not _bessel_table.has(f.func)):
cache[f] = cancel(f.diff(self.x))
else:
n, z = f.args
d0, d1 = _bessel_table.diffs(f.func, n, z)
dz = self.get_diff(z)
cache[f] = d0*dz
cache[f.func(n-1, z)] = d1*dz
return cache[f]
def heurisch(f, x, rewrite=False, hints=None, mappings=None, retries=3,
degree_offset=0, unnecessary_permutations=None,
_try_heurisch=None):
"""
Compute indefinite integral using heuristic Risch algorithm.
Explanation
===========
This is a heuristic approach to indefinite integration in finite
terms using the extended heuristic (parallel) Risch algorithm, based
on Manuel Bronstein's "Poor Man's Integrator".
The algorithm supports various classes of functions including
transcendental elementary or special functions like Airy,
Bessel, Whittaker and Lambert.
Note that this algorithm is not a decision procedure. If it isn't
able to compute the antiderivative for a given function, then this is
not a proof that such a functions does not exist. One should use
recursive Risch algorithm in such case. It's an open question if
this algorithm can be made a full decision procedure.
This is an internal integrator procedure. You should use toplevel
'integrate' function in most cases, as this procedure needs some
preprocessing steps and otherwise may fail.
Specification
=============
heurisch(f, x, rewrite=False, hints=None)
where
f : expression
x : symbol
rewrite -> force rewrite 'f' in terms of 'tan' and 'tanh'
hints -> a list of functions that may appear in anti-derivate
- hints = None --> no suggestions at all
- hints = [ ] --> try to figure out
- hints = [f1, ..., fn] --> we know better
Examples
========
>>> from sympy import tan
>>> from sympy.integrals.heurisch import heurisch
>>> from sympy.abc import x, y
>>> heurisch(y*tan(x), x)
y*log(tan(x)**2 + 1)/2
See Manuel Bronstein's "Poor Man's Integrator":
References
==========
.. [1] http://www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/index.html
For more information on the implemented algorithm refer to:
.. [2] K. Geddes, L. Stefanus, On the Risch-Norman Integration
Method and its Implementation in Maple, Proceedings of
ISSAC'89, ACM Press, 212-217.
.. [3] J. H. Davenport, On the Parallel Risch Algorithm (I),
Proceedings of EUROCAM'82, LNCS 144, Springer, 144-157.
.. [4] J. H. Davenport, On the Parallel Risch Algorithm (III):
Use of Tangents, SIGSAM Bulletin 16 (1982), 3-6.
.. [5] J. H. Davenport, B. M. Trager, On the Parallel Risch
Algorithm (II), ACM Transactions on Mathematical
Software 11 (1985), 356-362.
See Also
========
sympy.integrals.integrals.Integral.doit
sympy.integrals.integrals.Integral
sympy.integrals.heurisch.components
"""
f = sympify(f)
# There are some functions that Heurisch cannot currently handle,
# so do not even try.
# Set _try_heurisch=True to skip this check
if _try_heurisch is not True:
if f.has(Abs, re, im, sign, Heaviside, DiracDelta, floor, ceiling, arg):
return
if x not in f.free_symbols:
return f*x
if not f.is_Add:
indep, f = f.as_independent(x)
else:
indep = S.One
rewritables = {
(sin, cos, cot): tan,
(sinh, cosh, coth): tanh,
}
if rewrite:
for candidates, rule in rewritables.items():
f = f.rewrite(candidates, rule)
else:
for candidates in rewritables.keys():
if f.has(*candidates):
break
else:
rewrite = True
terms = components(f, x)
if hints is not None:
if not hints:
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x])
c = Wild('c', exclude=[x])
for g in set(terms): # using copy of terms
if g.is_Function:
if isinstance(g, li):
M = g.args[0].match(a*x**b)
if M is not None:
terms.add( x*(li(M[a]*x**M[b]) - (M[a]*x**M[b])**(-1/M[b])*Ei((M[b]+1)*log(M[a]*x**M[b])/M[b])) )
#terms.add( x*(li(M[a]*x**M[b]) - (x**M[b])**(-1/M[b])*Ei((M[b]+1)*log(M[a]*x**M[b])/M[b])) )
#terms.add( x*(li(M[a]*x**M[b]) - x*Ei((M[b]+1)*log(M[a]*x**M[b])/M[b])) )
#terms.add( li(M[a]*x**M[b]) - Ei((M[b]+1)*log(M[a]*x**M[b])/M[b]) )
elif isinstance(g, exp):
M = g.args[0].match(a*x**2)
if M is not None:
if M[a].is_positive:
terms.add(erfi(sqrt(M[a])*x))
else: # M[a].is_negative or unknown
terms.add(erf(sqrt(-M[a])*x))
M = g.args[0].match(a*x**2 + b*x + c)
if M is not None:
if M[a].is_positive:
terms.add(sqrt(pi/4*(-M[a]))*exp(M[c] - M[b]**2/(4*M[a]))*
erfi(sqrt(M[a])*x + M[b]/(2*sqrt(M[a]))))
elif M[a].is_negative:
terms.add(sqrt(pi/4*(-M[a]))*exp(M[c] - M[b]**2/(4*M[a]))*
erf(sqrt(-M[a])*x - M[b]/(2*sqrt(-M[a]))))
M = g.args[0].match(a*log(x)**2)
if M is not None:
if M[a].is_positive:
terms.add(erfi(sqrt(M[a])*log(x) + 1/(2*sqrt(M[a]))))
if M[a].is_negative:
terms.add(erf(sqrt(-M[a])*log(x) - 1/(2*sqrt(-M[a]))))
elif g.is_Pow:
if g.exp.is_Rational and g.exp.q == 2:
M = g.base.match(a*x**2 + b)
if M is not None and M[b].is_positive:
if M[a].is_positive:
terms.add(asinh(sqrt(M[a]/M[b])*x))
elif M[a].is_negative:
terms.add(asin(sqrt(-M[a]/M[b])*x))
M = g.base.match(a*x**2 - b)
if M is not None and M[b].is_positive:
if M[a].is_positive:
terms.add(acosh(sqrt(M[a]/M[b])*x))
elif M[a].is_negative:
terms.add(-M[b]/2*sqrt(-M[a])*
atan(sqrt(-M[a])*x/sqrt(M[a]*x**2 - M[b])))
else:
terms |= set(hints)
dcache = DiffCache(x)
for g in set(terms): # using copy of terms
terms |= components(dcache.get_diff(g), x)
# TODO: caching is significant factor for why permutations work at all. Change this.
V = _symbols('x', len(terms))
# sort mapping expressions from largest to smallest (last is always x).
mapping = list(reversed(list(zip(*ordered( #
[(a[0].as_independent(x)[1], a) for a in zip(terms, V)])))[1])) #
rev_mapping = {v: k for k, v in mapping} #
if mappings is None: #
# optimizing the number of permutations of mapping #
assert mapping[-1][0] == x # if not, find it and correct this comment
unnecessary_permutations = [mapping.pop(-1)]
mappings = permutations(mapping)
else:
unnecessary_permutations = unnecessary_permutations or []
def _substitute(expr):
return expr.subs(mapping)
for mapping in mappings:
mapping = list(mapping)
mapping = mapping + unnecessary_permutations
diffs = [ _substitute(dcache.get_diff(g)) for g in terms ]
denoms = [ g.as_numer_denom()[1] for g in diffs ]
if all(h.is_polynomial(*V) for h in denoms) and _substitute(f).is_rational_function(*V):
denom = reduce(lambda p, q: lcm(p, q, *V), denoms)
break
else:
if not rewrite:
result = heurisch(f, x, rewrite=True, hints=hints,
unnecessary_permutations=unnecessary_permutations)
if result is not None:
return indep*result
return None
numers = [ cancel(denom*g) for g in diffs ]
def _derivation(h):
return Add(*[ d * h.diff(v) for d, v in zip(numers, V) ])
def _deflation(p):
for y in V:
if not p.has(y):
continue
if _derivation(p) is not S.Zero:
c, q = p.as_poly(y).primitive()
return _deflation(c)*gcd(q, q.diff(y)).as_expr()
return p
def _splitter(p):
for y in V:
if not p.has(y):
continue
if _derivation(y) is not S.Zero:
c, q = p.as_poly(y).primitive()
q = q.as_expr()
h = gcd(q, _derivation(q), y)
s = quo(h, gcd(q, q.diff(y), y), y)
c_split = _splitter(c)
if s.as_poly(y).degree() == 0:
return (c_split[0], q * c_split[1])
q_split = _splitter(cancel(q / s))
return (c_split[0]*q_split[0]*s, c_split[1]*q_split[1])
return (S.One, p)
special = {}
for term in terms:
if term.is_Function:
if isinstance(term, tan):
special[1 + _substitute(term)**2] = False
elif isinstance(term, tanh):
special[1 + _substitute(term)] = False
special[1 - _substitute(term)] = False
elif isinstance(term, LambertW):
special[_substitute(term)] = True
F = _substitute(f)
P, Q = F.as_numer_denom()
u_split = _splitter(denom)
v_split = _splitter(Q)
polys = set(list(v_split) + [ u_split[0] ] + list(special.keys()))
s = u_split[0] * Mul(*[ k for k, v in special.items() if v ])
polified = [ p.as_poly(*V) for p in [s, P, Q] ]
if None in polified:
return None
#--- definitions for _integrate
a, b, c = [ p.total_degree() for p in polified ]
poly_denom = (s * v_split[0] * _deflation(v_split[1])).as_expr()
def _exponent(g):
if g.is_Pow:
if g.exp.is_Rational and g.exp.q != 1:
if g.exp.p > 0:
return g.exp.p + g.exp.q - 1
else:
return abs(g.exp.p + g.exp.q)
else:
return 1
elif not g.is_Atom and g.args:
return max([ _exponent(h) for h in g.args ])
else:
return 1
A, B = _exponent(f), a + max(b, c)
if A > 1 and B > 1:
monoms = tuple(ordered(itermonomials(V, A + B - 1 + degree_offset)))
else:
monoms = tuple(ordered(itermonomials(V, A + B + degree_offset)))
poly_coeffs = _symbols('A', len(monoms))
poly_part = Add(*[ poly_coeffs[i]*monomial
for i, monomial in enumerate(monoms) ])
reducibles = set()
for poly in polys:
if poly.has(*V):
try:
factorization = factor(poly, greedy=True)
except PolynomialError:
factorization = poly
if factorization.is_Mul:
factors = factorization.args
else:
factors = (factorization, )
for fact in factors:
if fact.is_Pow:
reducibles.add(fact.base)
else:
reducibles.add(fact)
def _integrate(field=None):
irreducibles = set()
atans = set()
pairs = set()
for poly in reducibles:
for z in poly.free_symbols:
if z in V:
break # should this be: `irreducibles |= \
else: # set(root_factors(poly, z, filter=field))`
continue # and the line below deleted?
# |
# V
irreducibles |= set(root_factors(poly, z, filter=field))
log_part, atan_part = [], []
for poly in list(irreducibles):
m = collect(poly, I, evaluate=False)
y = m.get(I, S.Zero)
if y:
x = m.get(S.One, S.Zero)
if x.has(I) or y.has(I):
continue # nontrivial x + I*y
pairs.add((x, y))
irreducibles.remove(poly)
while pairs:
x, y = pairs.pop()
if (x, -y) in pairs:
pairs.remove((x, -y))
# Choosing b with no minus sign
if y.could_extract_minus_sign():
y = -y
irreducibles.add(x*x + y*y)
atans.add(atan(x/y))
else:
irreducibles.add(x + I*y)
B = _symbols('B', len(irreducibles))
C = _symbols('C', len(atans))
# Note: the ordering matters here
for poly, b in reversed(list(zip(ordered(irreducibles), B))):
if poly.has(*V):
poly_coeffs.append(b)
log_part.append(b * log(poly))
for poly, c in reversed(list(zip(ordered(atans), C))):
if poly.has(*V):
poly_coeffs.append(c)
atan_part.append(c * poly)
# TODO: Currently it's better to use symbolic expressions here instead
# of rational functions, because it's simpler and FracElement doesn't
# give big speed improvement yet. This is because cancellation is slow
# due to slow polynomial GCD algorithms. If this gets improved then
# revise this code.
candidate = poly_part/poly_denom + Add(*log_part) + Add(*atan_part)
h = F - _derivation(candidate) / denom
raw_numer = h.as_numer_denom()[0]
# Rewrite raw_numer as a polynomial in K[coeffs][V] where K is a field
# that we have to determine. We can't use simply atoms() because log(3),
# sqrt(y) and similar expressions can appear, leading to non-trivial
# domains.
syms = set(poly_coeffs) | set(V)
non_syms = set()
def find_non_syms(expr):
if expr.is_Integer or expr.is_Rational:
pass # ignore trivial numbers
elif expr in syms:
pass # ignore variables
elif not expr.free_symbols & syms:
non_syms.add(expr)
elif expr.is_Add or expr.is_Mul or expr.is_Pow:
list(map(find_non_syms, expr.args))
else:
# TODO: Non-polynomial expression. This should have been
# filtered out at an earlier stage.
raise PolynomialError
try:
find_non_syms(raw_numer)
except PolynomialError:
return None
else:
ground, _ = construct_domain(non_syms, field=True)
coeff_ring = PolyRing(poly_coeffs, ground)
ring = PolyRing(V, coeff_ring)
try:
numer = ring.from_expr(raw_numer)
except ValueError:
raise PolynomialError
solution = solve_lin_sys(numer.coeffs(), coeff_ring, _raw=False)
if solution is None:
return None
else:
return candidate.xreplace(solution).xreplace(
dict(zip(poly_coeffs, [S.Zero]*len(poly_coeffs))))
if not (F.free_symbols - set(V)):
solution = _integrate('Q')
if solution is None:
solution = _integrate()
else:
solution = _integrate()
if solution is not None:
antideriv = solution.subs(rev_mapping)
antideriv = cancel(antideriv).expand(force=True)
if antideriv.is_Add:
antideriv = antideriv.as_independent(x)[1]
return indep*antideriv
else:
if retries >= 0:
result = heurisch(f, x, mappings=mappings, rewrite=rewrite, hints=hints, retries=retries - 1, unnecessary_permutations=unnecessary_permutations)
if result is not None:
return indep*result
return None
|
9f92b75b10acd07a122c1f248de8be44b35dc608d86323e729fef70a7a08ac1e | """Base class for all the objects in SymPy"""
from collections import defaultdict
from collections.abc import Mapping
from itertools import chain, zip_longest
from .assumptions import BasicMeta, ManagedProperties
from .cache import cacheit
from .sympify import _sympify, sympify, SympifyError
from .compatibility import iterable, ordered
from .kind import UndefinedKind
from ._print_helpers import Printable
from inspect import getmro
def as_Basic(expr):
"""Return expr as a Basic instance using strict sympify
or raise a TypeError; this is just a wrapper to _sympify,
raising a TypeError instead of a SympifyError."""
from sympy.utilities.misc import func_name
try:
return _sympify(expr)
except SympifyError:
raise TypeError(
'Argument must be a Basic object, not `%s`' % func_name(
expr))
class Basic(Printable, metaclass=ManagedProperties):
"""
Base class for all SymPy objects.
Notes and conventions
=====================
1) Always use ``.args``, when accessing parameters of some instance:
>>> from sympy import cot
>>> from sympy.abc import x, y
>>> cot(x).args
(x,)
>>> cot(x).args[0]
x
>>> (x*y).args
(x, y)
>>> (x*y).args[1]
y
2) Never use internal methods or variables (the ones prefixed with ``_``):
>>> cot(x)._args # do not use this, use cot(x).args instead
(x,)
3) By "SymPy object" we mean something that can be returned by
``sympify``. But not all objects one encounters using SymPy are
subclasses of Basic. For example, mutable objects are not:
>>> from sympy import Basic, Matrix, sympify
>>> A = Matrix([[1, 2], [3, 4]]).as_mutable()
>>> isinstance(A, Basic)
False
>>> B = sympify(A)
>>> isinstance(B, Basic)
True
"""
__slots__ = ('_mhash', # hash value
'_args', # arguments
'_assumptions'
)
# To be overridden with True in the appropriate subclasses
is_number = False
is_Atom = False
is_Symbol = False
is_symbol = False
is_Indexed = False
is_Dummy = False
is_Wild = False
is_Function = False
is_Add = False
is_Mul = False
is_Pow = False
is_Number = False
is_Float = False
is_Rational = False
is_Integer = False
is_NumberSymbol = False
is_Order = False
is_Derivative = False
is_Piecewise = False
is_Poly = False
is_AlgebraicNumber = False
is_Relational = False
is_Equality = False
is_Boolean = False
is_Not = False
is_Matrix = False
is_Vector = False
is_Point = False
is_MatAdd = False
is_MatMul = False
kind = UndefinedKind
def __new__(cls, *args):
obj = object.__new__(cls)
obj._assumptions = cls.default_assumptions
obj._mhash = None # will be set by __hash__ method.
obj._args = args # all items in args must be Basic objects
return obj
def copy(self):
return self.func(*self.args)
def __getnewargs__(self):
return self.args
def __getstate__(self):
return None
def __reduce_ex__(self, protocol):
if protocol < 2:
msg = "Only pickle protocol 2 or higher is supported by sympy"
raise NotImplementedError(msg)
return super().__reduce_ex__(protocol)
def __hash__(self):
# hash cannot be cached using cache_it because infinite recurrence
# occurs as hash is needed for setting cache dictionary keys
h = self._mhash
if h is None:
h = hash((type(self).__name__,) + self._hashable_content())
self._mhash = h
return h
def _hashable_content(self):
"""Return a tuple of information about self that can be used to
compute the hash. If a class defines additional attributes,
like ``name`` in Symbol, then this method should be updated
accordingly to return such relevant attributes.
Defining more than _hashable_content is necessary if __eq__ has
been defined by a class. See note about this in Basic.__eq__."""
return self._args
@property
def assumptions0(self):
"""
Return object `type` assumptions.
For example:
Symbol('x', real=True)
Symbol('x', integer=True)
are different objects. In other words, besides Python type (Symbol in
this case), the initial assumptions are also forming their typeinfo.
Examples
========
>>> from sympy import Symbol
>>> from sympy.abc import x
>>> x.assumptions0
{'commutative': True}
>>> x = Symbol("x", positive=True)
>>> x.assumptions0
{'commutative': True, 'complex': True, 'extended_negative': False,
'extended_nonnegative': True, 'extended_nonpositive': False,
'extended_nonzero': True, 'extended_positive': True, 'extended_real':
True, 'finite': True, 'hermitian': True, 'imaginary': False,
'infinite': False, 'negative': False, 'nonnegative': True,
'nonpositive': False, 'nonzero': True, 'positive': True, 'real':
True, 'zero': False}
"""
return {}
def compare(self, other):
"""
Return -1, 0, 1 if the object is smaller, equal, or greater than other.
Not in the mathematical sense. If the object is of a different type
from the "other" then their classes are ordered according to
the sorted_classes list.
Examples
========
>>> from sympy.abc import x, y
>>> x.compare(y)
-1
>>> x.compare(x)
0
>>> y.compare(x)
1
"""
# all redefinitions of __cmp__ method should start with the
# following lines:
if self is other:
return 0
n1 = self.__class__
n2 = other.__class__
c = (n1 > n2) - (n1 < n2)
if c:
return c
#
st = self._hashable_content()
ot = other._hashable_content()
c = (len(st) > len(ot)) - (len(st) < len(ot))
if c:
return c
for l, r in zip(st, ot):
l = Basic(*l) if isinstance(l, frozenset) else l
r = Basic(*r) if isinstance(r, frozenset) else r
if isinstance(l, Basic):
c = l.compare(r)
else:
c = (l > r) - (l < r)
if c:
return c
return 0
@staticmethod
def _compare_pretty(a, b):
from sympy.series.order import Order
if isinstance(a, Order) and not isinstance(b, Order):
return 1
if not isinstance(a, Order) and isinstance(b, Order):
return -1
if a.is_Rational and b.is_Rational:
l = a.p * b.q
r = b.p * a.q
return (l > r) - (l < r)
else:
from sympy.core.symbol import Wild
p1, p2, p3 = Wild("p1"), Wild("p2"), Wild("p3")
r_a = a.match(p1 * p2**p3)
if r_a and p3 in r_a:
a3 = r_a[p3]
r_b = b.match(p1 * p2**p3)
if r_b and p3 in r_b:
b3 = r_b[p3]
c = Basic.compare(a3, b3)
if c != 0:
return c
return Basic.compare(a, b)
@classmethod
def fromiter(cls, args, **assumptions):
"""
Create a new object from an iterable.
This is a convenience function that allows one to create objects from
any iterable, without having to convert to a list or tuple first.
Examples
========
>>> from sympy import Tuple
>>> Tuple.fromiter(i for i in range(5))
(0, 1, 2, 3, 4)
"""
return cls(*tuple(args), **assumptions)
@classmethod
def class_key(cls):
"""Nice order of classes. """
return 5, 0, cls.__name__
@cacheit
def sort_key(self, order=None):
"""
Return a sort key.
Examples
========
>>> from sympy.core import S, I
>>> sorted([S(1)/2, I, -I], key=lambda x: x.sort_key())
[1/2, -I, I]
>>> S("[x, 1/x, 1/x**2, x**2, x**(1/2), x**(1/4), x**(3/2)]")
[x, 1/x, x**(-2), x**2, sqrt(x), x**(1/4), x**(3/2)]
>>> sorted(_, key=lambda x: x.sort_key())
[x**(-2), 1/x, x**(1/4), sqrt(x), x, x**(3/2), x**2]
"""
# XXX: remove this when issue 5169 is fixed
def inner_key(arg):
if isinstance(arg, Basic):
return arg.sort_key(order)
else:
return arg
args = self._sorted_args
args = len(args), tuple([inner_key(arg) for arg in args])
return self.class_key(), args, S.One.sort_key(), S.One
def __eq__(self, other):
"""Return a boolean indicating whether a == b on the basis of
their symbolic trees.
This is the same as a.compare(b) == 0 but faster.
Notes
=====
If a class that overrides __eq__() needs to retain the
implementation of __hash__() from a parent class, the
interpreter must be told this explicitly by setting __hash__ =
<ParentClass>.__hash__. Otherwise the inheritance of __hash__()
will be blocked, just as if __hash__ had been explicitly set to
None.
References
==========
from http://docs.python.org/dev/reference/datamodel.html#object.__hash__
"""
if self is other:
return True
tself = type(self)
tother = type(other)
if tself is not tother:
try:
other = _sympify(other)
tother = type(other)
except SympifyError:
return NotImplemented
# As long as we have the ordering of classes (sympy.core),
# comparing types will be slow in Python 2, because it uses
# __cmp__. Until we can remove it
# (https://github.com/sympy/sympy/issues/4269), we only compare
# types in Python 2 directly if they actually have __ne__.
if type(tself).__ne__ is not type.__ne__:
if tself != tother:
return False
elif tself is not tother:
return False
return self._hashable_content() == other._hashable_content()
def __ne__(self, other):
"""``a != b`` -> Compare two symbolic trees and see whether they are different
this is the same as:
``a.compare(b) != 0``
but faster
"""
return not self == other
def dummy_eq(self, other, symbol=None):
"""
Compare two expressions and handle dummy symbols.
Examples
========
>>> from sympy import Dummy
>>> from sympy.abc import x, y
>>> u = Dummy('u')
>>> (u**2 + 1).dummy_eq(x**2 + 1)
True
>>> (u**2 + 1) == (x**2 + 1)
False
>>> (u**2 + y).dummy_eq(x**2 + y, x)
True
>>> (u**2 + y).dummy_eq(x**2 + y, y)
False
"""
s = self.as_dummy()
o = _sympify(other)
o = o.as_dummy()
dummy_symbols = [i for i in s.free_symbols if i.is_Dummy]
if len(dummy_symbols) == 1:
dummy = dummy_symbols.pop()
else:
return s == o
if symbol is None:
symbols = o.free_symbols
if len(symbols) == 1:
symbol = symbols.pop()
else:
return s == o
tmp = dummy.__class__()
return s.xreplace({dummy: tmp}) == o.xreplace({symbol: tmp})
def atoms(self, *types):
"""Returns the atoms that form the current object.
By default, only objects that are truly atomic and can't
be divided into smaller pieces are returned: symbols, numbers,
and number symbols like I and pi. It is possible to request
atoms of any type, however, as demonstrated below.
Examples
========
>>> from sympy import I, pi, sin
>>> from sympy.abc import x, y
>>> (1 + x + 2*sin(y + I*pi)).atoms()
{1, 2, I, pi, x, y}
If one or more types are given, the results will contain only
those types of atoms.
>>> from sympy import Number, NumberSymbol, Symbol
>>> (1 + x + 2*sin(y + I*pi)).atoms(Symbol)
{x, y}
>>> (1 + x + 2*sin(y + I*pi)).atoms(Number)
{1, 2}
>>> (1 + x + 2*sin(y + I*pi)).atoms(Number, NumberSymbol)
{1, 2, pi}
>>> (1 + x + 2*sin(y + I*pi)).atoms(Number, NumberSymbol, I)
{1, 2, I, pi}
Note that I (imaginary unit) and zoo (complex infinity) are special
types of number symbols and are not part of the NumberSymbol class.
The type can be given implicitly, too:
>>> (1 + x + 2*sin(y + I*pi)).atoms(x) # x is a Symbol
{x, y}
Be careful to check your assumptions when using the implicit option
since ``S(1).is_Integer = True`` but ``type(S(1))`` is ``One``, a special type
of sympy atom, while ``type(S(2))`` is type ``Integer`` and will find all
integers in an expression:
>>> from sympy import S
>>> (1 + x + 2*sin(y + I*pi)).atoms(S(1))
{1}
>>> (1 + x + 2*sin(y + I*pi)).atoms(S(2))
{1, 2}
Finally, arguments to atoms() can select more than atomic atoms: any
sympy type (loaded in core/__init__.py) can be listed as an argument
and those types of "atoms" as found in scanning the arguments of the
expression recursively:
>>> from sympy import Function, Mul
>>> from sympy.core.function import AppliedUndef
>>> f = Function('f')
>>> (1 + f(x) + 2*sin(y + I*pi)).atoms(Function)
{f(x), sin(y + I*pi)}
>>> (1 + f(x) + 2*sin(y + I*pi)).atoms(AppliedUndef)
{f(x)}
>>> (1 + x + 2*sin(y + I*pi)).atoms(Mul)
{I*pi, 2*sin(y + I*pi)}
"""
if types:
types = tuple(
[t if isinstance(t, type) else type(t) for t in types])
nodes = preorder_traversal(self)
if types:
result = {node for node in nodes if isinstance(node, types)}
else:
result = {node for node in nodes if not node.args}
return result
@property
def free_symbols(self):
"""Return from the atoms of self those which are free symbols.
For most expressions, all symbols are free symbols. For some classes
this is not true. e.g. Integrals use Symbols for the dummy variables
which are bound variables, so Integral has a method to return all
symbols except those. Derivative keeps track of symbols with respect
to which it will perform a derivative; those are
bound variables, too, so it has its own free_symbols method.
Any other method that uses bound variables should implement a
free_symbols method."""
return set().union(*[a.free_symbols for a in self.args])
@property
def expr_free_symbols(self):
return set()
def as_dummy(self):
"""Return the expression with any objects having structurally
bound symbols replaced with unique, canonical symbols within
the object in which they appear and having only the default
assumption for commutativity being True. When applied to a
symbol a new symbol having only the same commutativity will be
returned.
Examples
========
>>> from sympy import Integral, Symbol
>>> from sympy.abc import x
>>> r = Symbol('r', real=True)
>>> Integral(r, (r, x)).as_dummy()
Integral(_0, (_0, x))
>>> _.variables[0].is_real is None
True
>>> r.as_dummy()
_r
Notes
=====
Any object that has structurally bound variables should have
a property, `bound_symbols` that returns those symbols
appearing in the object.
"""
from sympy.core.symbol import Dummy, Symbol
def can(x):
# mask free that shadow bound
free = x.free_symbols
bound = set(x.bound_symbols)
d = {i: Dummy() for i in bound & free}
x = x.subs(d)
# replace bound with canonical names
x = x.xreplace(x.canonical_variables)
# return after undoing masking
return x.xreplace({v: k for k, v in d.items()})
if not self.has(Symbol):
return self
return self.replace(
lambda x: hasattr(x, 'bound_symbols'),
lambda x: can(x),
simultaneous=False)
@property
def canonical_variables(self):
"""Return a dictionary mapping any variable defined in
``self.bound_symbols`` to Symbols that do not clash
with any free symbols in the expression.
Examples
========
>>> from sympy import Lambda
>>> from sympy.abc import x
>>> Lambda(x, 2*x).canonical_variables
{x: _0}
"""
from sympy.utilities.iterables import numbered_symbols
if not hasattr(self, 'bound_symbols'):
return {}
dums = numbered_symbols('_')
reps = {}
# watch out for free symbol that are not in bound symbols;
# those that are in bound symbols are about to get changed
bound = self.bound_symbols
names = {i.name for i in self.free_symbols - set(bound)}
for b in bound:
d = next(dums)
if b.is_Symbol:
while d.name in names:
d = next(dums)
reps[b] = d
return reps
def rcall(self, *args):
"""Apply on the argument recursively through the expression tree.
This method is used to simulate a common abuse of notation for
operators. For instance in SymPy the the following will not work:
``(x+Lambda(y, 2*y))(z) == x+2*z``,
however you can use
>>> from sympy import Lambda
>>> from sympy.abc import x, y, z
>>> (x + Lambda(y, 2*y)).rcall(z)
x + 2*z
"""
return Basic._recursive_call(self, args)
@staticmethod
def _recursive_call(expr_to_call, on_args):
"""Helper for rcall method."""
from sympy import Symbol
def the_call_method_is_overridden(expr):
for cls in getmro(type(expr)):
if '__call__' in cls.__dict__:
return cls != Basic
if callable(expr_to_call) and the_call_method_is_overridden(expr_to_call):
if isinstance(expr_to_call, Symbol): # XXX When you call a Symbol it is
return expr_to_call # transformed into an UndefFunction
else:
return expr_to_call(*on_args)
elif expr_to_call.args:
args = [Basic._recursive_call(
sub, on_args) for sub in expr_to_call.args]
return type(expr_to_call)(*args)
else:
return expr_to_call
def is_hypergeometric(self, k):
from sympy.simplify import hypersimp
from sympy.functions import Piecewise
if self.has(Piecewise):
return None
return hypersimp(self, k) is not None
@property
def is_comparable(self):
"""Return True if self can be computed to a real number
(or already is a real number) with precision, else False.
Examples
========
>>> from sympy import exp_polar, pi, I
>>> (I*exp_polar(I*pi/2)).is_comparable
True
>>> (I*exp_polar(I*pi*2)).is_comparable
False
A False result does not mean that `self` cannot be rewritten
into a form that would be comparable. For example, the
difference computed below is zero but without simplification
it does not evaluate to a zero with precision:
>>> e = 2**pi*(1 + 2**pi)
>>> dif = e - e.expand()
>>> dif.is_comparable
False
>>> dif.n(2)._prec
1
"""
is_extended_real = self.is_extended_real
if is_extended_real is False:
return False
if not self.is_number:
return False
# don't re-eval numbers that are already evaluated since
# this will create spurious precision
n, i = [p.evalf(2) if not p.is_Number else p
for p in self.as_real_imag()]
if not (i.is_Number and n.is_Number):
return False
if i:
# if _prec = 1 we can't decide and if not,
# the answer is False because numbers with
# imaginary parts can't be compared
# so return False
return False
else:
return n._prec != 1
@property
def func(self):
"""
The top-level function in an expression.
The following should hold for all objects::
>> x == x.func(*x.args)
Examples
========
>>> from sympy.abc import x
>>> a = 2*x
>>> a.func
<class 'sympy.core.mul.Mul'>
>>> a.args
(2, x)
>>> a.func(*a.args)
2*x
>>> a == a.func(*a.args)
True
"""
return self.__class__
@property
def args(self):
"""Returns a tuple of arguments of 'self'.
Examples
========
>>> from sympy import cot
>>> from sympy.abc import x, y
>>> cot(x).args
(x,)
>>> cot(x).args[0]
x
>>> (x*y).args
(x, y)
>>> (x*y).args[1]
y
Notes
=====
Never use self._args, always use self.args.
Only use _args in __new__ when creating a new function.
Don't override .args() from Basic (so that it's easy to
change the interface in the future if needed).
"""
return self._args
@property
def _sorted_args(self):
"""
The same as ``args``. Derived classes which don't fix an
order on their arguments should override this method to
produce the sorted representation.
"""
return self.args
def as_content_primitive(self, radical=False, clear=True):
"""A stub to allow Basic args (like Tuple) to be skipped when computing
the content and primitive components of an expression.
See Also
========
sympy.core.expr.Expr.as_content_primitive
"""
return S.One, self
def subs(self, *args, **kwargs):
"""
Substitutes old for new in an expression after sympifying args.
`args` is either:
- two arguments, e.g. foo.subs(old, new)
- one iterable argument, e.g. foo.subs(iterable). The iterable may be
o an iterable container with (old, new) pairs. In this case the
replacements are processed in the order given with successive
patterns possibly affecting replacements already made.
o a dict or set whose key/value items correspond to old/new pairs.
In this case the old/new pairs will be sorted by op count and in
case of a tie, by number of args and the default_sort_key. The
resulting sorted list is then processed as an iterable container
(see previous).
If the keyword ``simultaneous`` is True, the subexpressions will not be
evaluated until all the substitutions have been made.
Examples
========
>>> from sympy import pi, exp, limit, oo
>>> from sympy.abc import x, y
>>> (1 + x*y).subs(x, pi)
pi*y + 1
>>> (1 + x*y).subs({x:pi, y:2})
1 + 2*pi
>>> (1 + x*y).subs([(x, pi), (y, 2)])
1 + 2*pi
>>> reps = [(y, x**2), (x, 2)]
>>> (x + y).subs(reps)
6
>>> (x + y).subs(reversed(reps))
x**2 + 2
>>> (x**2 + x**4).subs(x**2, y)
y**2 + y
To replace only the x**2 but not the x**4, use xreplace:
>>> (x**2 + x**4).xreplace({x**2: y})
x**4 + y
To delay evaluation until all substitutions have been made,
set the keyword ``simultaneous`` to True:
>>> (x/y).subs([(x, 0), (y, 0)])
0
>>> (x/y).subs([(x, 0), (y, 0)], simultaneous=True)
nan
This has the added feature of not allowing subsequent substitutions
to affect those already made:
>>> ((x + y)/y).subs({x + y: y, y: x + y})
1
>>> ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True)
y/(x + y)
In order to obtain a canonical result, unordered iterables are
sorted by count_op length, number of arguments and by the
default_sort_key to break any ties. All other iterables are left
unsorted.
>>> from sympy import sqrt, sin, cos
>>> from sympy.abc import a, b, c, d, e
>>> A = (sqrt(sin(2*x)), a)
>>> B = (sin(2*x), b)
>>> C = (cos(2*x), c)
>>> D = (x, d)
>>> E = (exp(x), e)
>>> expr = sqrt(sin(2*x))*sin(exp(x)*x)*cos(2*x) + sin(2*x)
>>> expr.subs(dict([A, B, C, D, E]))
a*c*sin(d*e) + b
The resulting expression represents a literal replacement of the
old arguments with the new arguments. This may not reflect the
limiting behavior of the expression:
>>> (x**3 - 3*x).subs({x: oo})
nan
>>> limit(x**3 - 3*x, x, oo)
oo
If the substitution will be followed by numerical
evaluation, it is better to pass the substitution to
evalf as
>>> (1/x).evalf(subs={x: 3.0}, n=21)
0.333333333333333333333
rather than
>>> (1/x).subs({x: 3.0}).evalf(21)
0.333333333333333314830
as the former will ensure that the desired level of precision is
obtained.
See Also
========
replace: replacement capable of doing wildcard-like matching,
parsing of match, and conditional replacements
xreplace: exact node replacement in expr tree; also capable of
using matching rules
sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision
"""
from sympy.core.compatibility import _nodes, default_sort_key
from sympy.core.containers import Dict
from sympy.core.symbol import Dummy, Symbol
from sympy.utilities.misc import filldedent
unordered = False
if len(args) == 1:
sequence = args[0]
if isinstance(sequence, set):
unordered = True
elif isinstance(sequence, (Dict, Mapping)):
unordered = True
sequence = sequence.items()
elif not iterable(sequence):
raise ValueError(filldedent("""
When a single argument is passed to subs
it should be a dictionary of old: new pairs or an iterable
of (old, new) tuples."""))
elif len(args) == 2:
sequence = [args]
else:
raise ValueError("subs accepts either 1 or 2 arguments")
sequence = list(sequence)
for i, s in enumerate(sequence):
if isinstance(s[0], str):
# when old is a string we prefer Symbol
s = Symbol(s[0]), s[1]
try:
s = [sympify(_, strict=not isinstance(_, (str, type)))
for _ in s]
except SympifyError:
# if it can't be sympified, skip it
sequence[i] = None
continue
# skip if there is no change
sequence[i] = None if _aresame(*s) else tuple(s)
sequence = list(filter(None, sequence))
if unordered:
sequence = dict(sequence)
# order so more complex items are first and items
# of identical complexity are ordered so
# f(x) < f(y) < x < y
# \___ 2 __/ \_1_/ <- number of nodes
#
# For more complex ordering use an unordered sequence.
k = list(ordered(sequence, default=False, keys=(
lambda x: -_nodes(x),
lambda x: default_sort_key(x),
)))
sequence = [(k, sequence[k]) for k in k]
if kwargs.pop('simultaneous', False): # XXX should this be the default for dict subs?
reps = {}
rv = self
kwargs['hack2'] = True
m = Dummy('subs_m')
for old, new in sequence:
com = new.is_commutative
if com is None:
com = True
d = Dummy('subs_d', commutative=com)
# using d*m so Subs will be used on dummy variables
# in things like Derivative(f(x, y), x) in which x
# is both free and bound
rv = rv._subs(old, d*m, **kwargs)
if not isinstance(rv, Basic):
break
reps[d] = new
reps[m] = S.One # get rid of m
return rv.xreplace(reps)
else:
rv = self
for old, new in sequence:
rv = rv._subs(old, new, **kwargs)
if not isinstance(rv, Basic):
break
return rv
@cacheit
def _subs(self, old, new, **hints):
"""Substitutes an expression old -> new.
If self is not equal to old then _eval_subs is called.
If _eval_subs doesn't want to make any special replacement
then a None is received which indicates that the fallback
should be applied wherein a search for replacements is made
amongst the arguments of self.
>>> from sympy import Add
>>> from sympy.abc import x, y, z
Examples
========
Add's _eval_subs knows how to target x + y in the following
so it makes the change:
>>> (x + y + z).subs(x + y, 1)
z + 1
Add's _eval_subs doesn't need to know how to find x + y in
the following:
>>> Add._eval_subs(z*(x + y) + 3, x + y, 1) is None
True
The returned None will cause the fallback routine to traverse the args and
pass the z*(x + y) arg to Mul where the change will take place and the
substitution will succeed:
>>> (z*(x + y) + 3).subs(x + y, 1)
z + 3
** Developers Notes **
An _eval_subs routine for a class should be written if:
1) any arguments are not instances of Basic (e.g. bool, tuple);
2) some arguments should not be targeted (as in integration
variables);
3) if there is something other than a literal replacement
that should be attempted (as in Piecewise where the condition
may be updated without doing a replacement).
If it is overridden, here are some special cases that might arise:
1) If it turns out that no special change was made and all
the original sub-arguments should be checked for
replacements then None should be returned.
2) If it is necessary to do substitutions on a portion of
the expression then _subs should be called. _subs will
handle the case of any sub-expression being equal to old
(which usually would not be the case) while its fallback
will handle the recursion into the sub-arguments. For
example, after Add's _eval_subs removes some matching terms
it must process the remaining terms so it calls _subs
on each of the un-matched terms and then adds them
onto the terms previously obtained.
3) If the initial expression should remain unchanged then
the original expression should be returned. (Whenever an
expression is returned, modified or not, no further
substitution of old -> new is attempted.) Sum's _eval_subs
routine uses this strategy when a substitution is attempted
on any of its summation variables.
"""
def fallback(self, old, new):
"""
Try to replace old with new in any of self's arguments.
"""
hit = False
args = list(self.args)
for i, arg in enumerate(args):
if not hasattr(arg, '_eval_subs'):
continue
arg = arg._subs(old, new, **hints)
if not _aresame(arg, args[i]):
hit = True
args[i] = arg
if hit:
rv = self.func(*args)
hack2 = hints.get('hack2', False)
if hack2 and self.is_Mul and not rv.is_Mul: # 2-arg hack
coeff = S.One
nonnumber = []
for i in args:
if i.is_Number:
coeff *= i
else:
nonnumber.append(i)
nonnumber = self.func(*nonnumber)
if coeff is S.One:
return nonnumber
else:
return self.func(coeff, nonnumber, evaluate=False)
return rv
return self
if _aresame(self, old):
return new
rv = self._eval_subs(old, new)
if rv is None:
rv = fallback(self, old, new)
return rv
def _eval_subs(self, old, new):
"""Override this stub if you want to do anything more than
attempt a replacement of old with new in the arguments of self.
See also
========
_subs
"""
return None
def xreplace(self, rule):
"""
Replace occurrences of objects within the expression.
Parameters
==========
rule : dict-like
Expresses a replacement rule
Returns
=======
xreplace : the result of the replacement
Examples
========
>>> from sympy import symbols, pi, exp
>>> x, y, z = symbols('x y z')
>>> (1 + x*y).xreplace({x: pi})
pi*y + 1
>>> (1 + x*y).xreplace({x: pi, y: 2})
1 + 2*pi
Replacements occur only if an entire node in the expression tree is
matched:
>>> (x*y + z).xreplace({x*y: pi})
z + pi
>>> (x*y*z).xreplace({x*y: pi})
x*y*z
>>> (2*x).xreplace({2*x: y, x: z})
y
>>> (2*2*x).xreplace({2*x: y, x: z})
4*z
>>> (x + y + 2).xreplace({x + y: 2})
x + y + 2
>>> (x + 2 + exp(x + 2)).xreplace({x + 2: y})
x + exp(y) + 2
xreplace doesn't differentiate between free and bound symbols. In the
following, subs(x, y) would not change x since it is a bound symbol,
but xreplace does:
>>> from sympy import Integral
>>> Integral(x, (x, 1, 2*x)).xreplace({x: y})
Integral(y, (y, 1, 2*y))
Trying to replace x with an expression raises an error:
>>> Integral(x, (x, 1, 2*x)).xreplace({x: 2*y}) # doctest: +SKIP
ValueError: Invalid limits given: ((2*y, 1, 4*y),)
See Also
========
replace: replacement capable of doing wildcard-like matching,
parsing of match, and conditional replacements
subs: substitution of subexpressions as defined by the objects
themselves.
"""
value, _ = self._xreplace(rule)
return value
def _xreplace(self, rule):
"""
Helper for xreplace. Tracks whether a replacement actually occurred.
"""
if self in rule:
return rule[self], True
elif rule:
args = []
changed = False
for a in self.args:
_xreplace = getattr(a, '_xreplace', None)
if _xreplace is not None:
a_xr = _xreplace(rule)
args.append(a_xr[0])
changed |= a_xr[1]
else:
args.append(a)
args = tuple(args)
if changed:
return self.func(*args), True
return self, False
@cacheit
def has(self, *patterns):
"""
Test whether any subexpression matches any of the patterns.
Examples
========
>>> from sympy import sin
>>> from sympy.abc import x, y, z
>>> (x**2 + sin(x*y)).has(z)
False
>>> (x**2 + sin(x*y)).has(x, y, z)
True
>>> x.has(x)
True
Note ``has`` is a structural algorithm with no knowledge of
mathematics. Consider the following half-open interval:
>>> from sympy.sets import Interval
>>> i = Interval.Lopen(0, 5); i
Interval.Lopen(0, 5)
>>> i.args
(0, 5, True, False)
>>> i.has(4) # there is no "4" in the arguments
False
>>> i.has(0) # there *is* a "0" in the arguments
True
Instead, use ``contains`` to determine whether a number is in the
interval or not:
>>> i.contains(4)
True
>>> i.contains(0)
False
Note that ``expr.has(*patterns)`` is exactly equivalent to
``any(expr.has(p) for p in patterns)``. In particular, ``False`` is
returned when the list of patterns is empty.
>>> x.has()
False
"""
return any(self._has(pattern) for pattern in patterns)
def _has(self, pattern):
"""Helper for .has()"""
from sympy.core.function import UndefinedFunction, Function
if isinstance(pattern, UndefinedFunction):
return any(f.func == pattern or f == pattern
for f in self.atoms(Function, UndefinedFunction))
if isinstance(pattern, BasicMeta):
subtrees = preorder_traversal(self)
return any(isinstance(arg, pattern) for arg in subtrees)
pattern = _sympify(pattern)
_has_matcher = getattr(pattern, '_has_matcher', None)
if _has_matcher is not None:
match = _has_matcher()
return any(match(arg) for arg in preorder_traversal(self))
else:
return any(arg == pattern for arg in preorder_traversal(self))
def _has_matcher(self):
"""Helper for .has()"""
return lambda other: self == other
def replace(self, query, value, map=False, simultaneous=True, exact=None):
"""
Replace matching subexpressions of ``self`` with ``value``.
If ``map = True`` then also return the mapping {old: new} where ``old``
was a sub-expression found with query and ``new`` is the replacement
value for it. If the expression itself doesn't match the query, then
the returned value will be ``self.xreplace(map)`` otherwise it should
be ``self.subs(ordered(map.items()))``.
Traverses an expression tree and performs replacement of matching
subexpressions from the bottom to the top of the tree. The default
approach is to do the replacement in a simultaneous fashion so
changes made are targeted only once. If this is not desired or causes
problems, ``simultaneous`` can be set to False.
In addition, if an expression containing more than one Wild symbol
is being used to match subexpressions and the ``exact`` flag is None
it will be set to True so the match will only succeed if all non-zero
values are received for each Wild that appears in the match pattern.
Setting this to False accepts a match of 0; while setting it True
accepts all matches that have a 0 in them. See example below for
cautions.
The list of possible combinations of queries and replacement values
is listed below:
Examples
========
Initial setup
>>> from sympy import log, sin, cos, tan, Wild, Mul, Add
>>> from sympy.abc import x, y
>>> f = log(sin(x)) + tan(sin(x**2))
1.1. type -> type
obj.replace(type, newtype)
When object of type ``type`` is found, replace it with the
result of passing its argument(s) to ``newtype``.
>>> f.replace(sin, cos)
log(cos(x)) + tan(cos(x**2))
>>> sin(x).replace(sin, cos, map=True)
(cos(x), {sin(x): cos(x)})
>>> (x*y).replace(Mul, Add)
x + y
1.2. type -> func
obj.replace(type, func)
When object of type ``type`` is found, apply ``func`` to its
argument(s). ``func`` must be written to handle the number
of arguments of ``type``.
>>> f.replace(sin, lambda arg: sin(2*arg))
log(sin(2*x)) + tan(sin(2*x**2))
>>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args)))
sin(2*x*y)
2.1. pattern -> expr
obj.replace(pattern(wild), expr(wild))
Replace subexpressions matching ``pattern`` with the expression
written in terms of the Wild symbols in ``pattern``.
>>> a, b = map(Wild, 'ab')
>>> f.replace(sin(a), tan(a))
log(tan(x)) + tan(tan(x**2))
>>> f.replace(sin(a), tan(a/2))
log(tan(x/2)) + tan(tan(x**2/2))
>>> f.replace(sin(a), a)
log(x) + tan(x**2)
>>> (x*y).replace(a*x, a)
y
Matching is exact by default when more than one Wild symbol
is used: matching fails unless the match gives non-zero
values for all Wild symbols:
>>> (2*x + y).replace(a*x + b, b - a)
y - 2
>>> (2*x).replace(a*x + b, b - a)
2*x
When set to False, the results may be non-intuitive:
>>> (2*x).replace(a*x + b, b - a, exact=False)
2/x
2.2. pattern -> func
obj.replace(pattern(wild), lambda wild: expr(wild))
All behavior is the same as in 2.1 but now a function in terms of
pattern variables is used rather than an expression:
>>> f.replace(sin(a), lambda a: sin(2*a))
log(sin(2*x)) + tan(sin(2*x**2))
3.1. func -> func
obj.replace(filter, func)
Replace subexpression ``e`` with ``func(e)`` if ``filter(e)``
is True.
>>> g = 2*sin(x**3)
>>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2)
4*sin(x**9)
The expression itself is also targeted by the query but is done in
such a fashion that changes are not made twice.
>>> e = x*(x*y + 1)
>>> e.replace(lambda x: x.is_Mul, lambda x: 2*x)
2*x*(2*x*y + 1)
When matching a single symbol, `exact` will default to True, but
this may or may not be the behavior that is desired:
Here, we want `exact=False`:
>>> from sympy import Function
>>> f = Function('f')
>>> e = f(1) + f(0)
>>> q = f(a), lambda a: f(a + 1)
>>> e.replace(*q, exact=False)
f(1) + f(2)
>>> e.replace(*q, exact=True)
f(0) + f(2)
But here, the nature of matching makes selecting
the right setting tricky:
>>> e = x**(1 + y)
>>> (x**(1 + y)).replace(x**(1 + a), lambda a: x**-a, exact=False)
x
>>> (x**(1 + y)).replace(x**(1 + a), lambda a: x**-a, exact=True)
x**(-x - y + 1)
>>> (x**y).replace(x**(1 + a), lambda a: x**-a, exact=False)
x
>>> (x**y).replace(x**(1 + a), lambda a: x**-a, exact=True)
x**(1 - y)
It is probably better to use a different form of the query
that describes the target expression more precisely:
>>> (1 + x**(1 + y)).replace(
... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1,
... lambda x: x.base**(1 - (x.exp - 1)))
...
x**(1 - y) + 1
See Also
========
subs: substitution of subexpressions as defined by the objects
themselves.
xreplace: exact node replacement in expr tree; also capable of
using matching rules
"""
from sympy.core.symbol import Wild
try:
query = _sympify(query)
except SympifyError:
pass
try:
value = _sympify(value)
except SympifyError:
pass
if isinstance(query, type):
_query = lambda expr: isinstance(expr, query)
if isinstance(value, type):
_value = lambda expr, result: value(*expr.args)
elif callable(value):
_value = lambda expr, result: value(*expr.args)
else:
raise TypeError(
"given a type, replace() expects another "
"type or a callable")
elif isinstance(query, Basic):
_query = lambda expr: expr.match(query)
if exact is None:
exact = (len(query.atoms(Wild)) > 1)
if isinstance(value, Basic):
if exact:
_value = lambda expr, result: (value.subs(result)
if all(result.values()) else expr)
else:
_value = lambda expr, result: value.subs(result)
elif callable(value):
# match dictionary keys get the trailing underscore stripped
# from them and are then passed as keywords to the callable;
# if ``exact`` is True, only accept match if there are no null
# values amongst those matched.
if exact:
_value = lambda expr, result: (value(**
{str(k)[:-1]: v for k, v in result.items()})
if all(val for val in result.values()) else expr)
else:
_value = lambda expr, result: value(**
{str(k)[:-1]: v for k, v in result.items()})
else:
raise TypeError(
"given an expression, replace() expects "
"another expression or a callable")
elif callable(query):
_query = query
if callable(value):
_value = lambda expr, result: value(expr)
else:
raise TypeError(
"given a callable, replace() expects "
"another callable")
else:
raise TypeError(
"first argument to replace() must be a "
"type, an expression or a callable")
def walk(rv, F):
"""Apply ``F`` to args and then to result.
"""
args = getattr(rv, 'args', None)
if args is not None:
if args:
newargs = tuple([walk(a, F) for a in args])
if args != newargs:
rv = rv.func(*newargs)
if simultaneous:
# if rv is something that was already
# matched (that was changed) then skip
# applying F again
for i, e in enumerate(args):
if rv == e and e != newargs[i]:
return rv
rv = F(rv)
return rv
mapping = {} # changes that took place
def rec_replace(expr):
result = _query(expr)
if result or result == {}:
v = _value(expr, result)
if v is not None and v != expr:
if map:
mapping[expr] = v
expr = v
return expr
rv = walk(self, rec_replace)
return (rv, mapping) if map else rv
def find(self, query, group=False):
"""Find all subexpressions matching a query. """
query = _make_find_query(query)
results = list(filter(query, preorder_traversal(self)))
if not group:
return set(results)
else:
groups = {}
for result in results:
if result in groups:
groups[result] += 1
else:
groups[result] = 1
return groups
def count(self, query):
"""Count the number of matching subexpressions. """
query = _make_find_query(query)
return sum(bool(query(sub)) for sub in preorder_traversal(self))
def matches(self, expr, repl_dict={}, old=False):
"""
Helper method for match() that looks for a match between Wild symbols
in self and expressions in expr.
Examples
========
>>> from sympy import symbols, Wild, Basic
>>> a, b, c = symbols('a b c')
>>> x = Wild('x')
>>> Basic(a + x, x).matches(Basic(a + b, c)) is None
True
>>> Basic(a + x, x).matches(Basic(a + b + c, b + c))
{x_: b + c}
"""
repl_dict = repl_dict.copy()
expr = sympify(expr)
if not isinstance(expr, self.__class__):
return None
if self == expr:
return repl_dict
if len(self.args) != len(expr.args):
return None
d = repl_dict.copy()
for arg, other_arg in zip(self.args, expr.args):
if arg == other_arg:
continue
d = arg.xreplace(d).matches(other_arg, d, old=old)
if d is None:
return None
return d
def match(self, pattern, old=False):
"""
Pattern matching.
Wild symbols match all.
Return ``None`` when expression (self) does not match
with pattern. Otherwise return a dictionary such that::
pattern.xreplace(self.match(pattern)) == self
Examples
========
>>> from sympy import Wild, Sum
>>> from sympy.abc import x, y
>>> p = Wild("p")
>>> q = Wild("q")
>>> r = Wild("r")
>>> e = (x+y)**(x+y)
>>> e.match(p**p)
{p_: x + y}
>>> e.match(p**q)
{p_: x + y, q_: x + y}
>>> e = (2*x)**2
>>> e.match(p*q**r)
{p_: 4, q_: x, r_: 2}
>>> (p*q**r).xreplace(e.match(p*q**r))
4*x**2
Structurally bound symbols are ignored during matching:
>>> Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p)))
{p_: 2}
But they can be identified if desired:
>>> Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p)))
{p_: 2, q_: x}
The ``old`` flag will give the old-style pattern matching where
expressions and patterns are essentially solved to give the
match. Both of the following give None unless ``old=True``:
>>> (x - 2).match(p - x, old=True)
{p_: 2*x - 2}
>>> (2/x).match(p*x, old=True)
{p_: 2/x**2}
"""
from sympy.core.symbol import Wild
from sympy.core.function import WildFunction
from sympy.utilities.misc import filldedent
pattern = sympify(pattern)
# match non-bound symbols
canonical = lambda x: x if x.is_Symbol else x.as_dummy()
m = canonical(pattern).matches(canonical(self), old=old)
if m is None:
return m
wild = pattern.atoms(Wild, WildFunction)
# sanity check
if set(m) - wild:
raise ValueError(filldedent('''
Some `matches` routine did not use a copy of repl_dict
and injected unexpected symbols. Report this as an
error at https://github.com/sympy/sympy/issues'''))
# now see if bound symbols were requested
bwild = wild - set(m)
if not bwild:
return m
# replace free-Wild symbols in pattern with match result
# so they will match but not be in the next match
wpat = pattern.xreplace(m)
# identify remaining bound wild
w = wpat.matches(self, old=old)
# add them to m
if w:
m.update(w)
# done
return m
def count_ops(self, visual=None):
"""wrapper for count_ops that returns the operation count."""
from sympy import count_ops
return count_ops(self, visual)
def doit(self, **hints):
"""Evaluate objects that are not evaluated by default like limits,
integrals, sums and products. All objects of this kind will be
evaluated recursively, unless some species were excluded via 'hints'
or unless the 'deep' hint was set to 'False'.
>>> from sympy import Integral
>>> from sympy.abc import x
>>> 2*Integral(x, x)
2*Integral(x, x)
>>> (2*Integral(x, x)).doit()
x**2
>>> (2*Integral(x, x)).doit(deep=False)
2*Integral(x, x)
"""
if hints.get('deep', True):
terms = [term.doit(**hints) if isinstance(term, Basic) else term
for term in self.args]
return self.func(*terms)
else:
return self
def simplify(self, **kwargs):
"""See the simplify function in sympy.simplify"""
from sympy.simplify import simplify
return simplify(self, **kwargs)
def refine(self, assumption=True):
"""See the refine function in sympy.assumptions"""
from sympy.assumptions import refine
return refine(self, assumption)
def _eval_rewrite(self, pattern, rule, **hints):
if self.is_Atom:
if hasattr(self, rule):
return getattr(self, rule)()
return self
if hints.get('deep', True):
args = [a._eval_rewrite(pattern, rule, **hints)
if isinstance(a, Basic) else a
for a in self.args]
else:
args = self.args
if pattern is None or isinstance(self, pattern):
if hasattr(self, rule):
rewritten = getattr(self, rule)(*args, **hints)
if rewritten is not None:
return rewritten
return self.func(*args) if hints.get('evaluate', True) else self
def _eval_derivative_n_times(self, s, n):
# This is the default evaluator for derivatives (as called by `diff`
# and `Derivative`), it will attempt a loop to derive the expression
# `n` times by calling the corresponding `_eval_derivative` method,
# while leaving the derivative unevaluated if `n` is symbolic. This
# method should be overridden if the object has a closed form for its
# symbolic n-th derivative.
from sympy import Integer
if isinstance(n, (int, Integer)):
obj = self
for i in range(n):
obj2 = obj._eval_derivative(s)
if obj == obj2 or obj2 is None:
break
obj = obj2
return obj2
else:
return None
def rewrite(self, *args, **hints):
""" Rewrite functions in terms of other functions.
Rewrites expression containing applications of functions
of one kind in terms of functions of different kind. For
example you can rewrite trigonometric functions as complex
exponentials or combinatorial functions as gamma function.
As a pattern this function accepts a list of functions to
to rewrite (instances of DefinedFunction class). As rule
you can use string or a destination function instance (in
this case rewrite() will use the str() function).
There is also the possibility to pass hints on how to rewrite
the given expressions. For now there is only one such hint
defined called 'deep'. When 'deep' is set to False it will
forbid functions to rewrite their contents.
Examples
========
>>> from sympy import sin, exp
>>> from sympy.abc import x
Unspecified pattern:
>>> sin(x).rewrite(exp)
-I*(exp(I*x) - exp(-I*x))/2
Pattern as a single function:
>>> sin(x).rewrite(sin, exp)
-I*(exp(I*x) - exp(-I*x))/2
Pattern as a list of functions:
>>> sin(x).rewrite([sin, ], exp)
-I*(exp(I*x) - exp(-I*x))/2
"""
if not args:
return self
else:
pattern = args[:-1]
if isinstance(args[-1], str):
rule = '_eval_rewrite_as_' + args[-1]
else:
# rewrite arg is usually a class but can also be a
# singleton (e.g. GoldenRatio) so we check
# __name__ or __class__.__name__
clsname = getattr(args[-1], "__name__", None)
if clsname is None:
clsname = args[-1].__class__.__name__
rule = '_eval_rewrite_as_' + clsname
if not pattern:
return self._eval_rewrite(None, rule, **hints)
else:
if iterable(pattern[0]):
pattern = pattern[0]
pattern = [p for p in pattern if self.has(p)]
if pattern:
return self._eval_rewrite(tuple(pattern), rule, **hints)
else:
return self
_constructor_postprocessor_mapping = {} # type: ignore
@classmethod
def _exec_constructor_postprocessors(cls, obj):
# WARNING: This API is experimental.
# This is an experimental API that introduces constructor
# postprosessors for SymPy Core elements. If an argument of a SymPy
# expression has a `_constructor_postprocessor_mapping` attribute, it will
# be interpreted as a dictionary containing lists of postprocessing
# functions for matching expression node names.
clsname = obj.__class__.__name__
postprocessors = defaultdict(list)
for i in obj.args:
try:
postprocessor_mappings = (
Basic._constructor_postprocessor_mapping[cls].items()
for cls in type(i).mro()
if cls in Basic._constructor_postprocessor_mapping
)
for k, v in chain.from_iterable(postprocessor_mappings):
postprocessors[k].extend([j for j in v if j not in postprocessors[k]])
except TypeError:
pass
for f in postprocessors.get(clsname, []):
obj = f(obj)
return obj
class Atom(Basic):
"""
A parent class for atomic things. An atom is an expression with no subexpressions.
Examples
========
Symbol, Number, Rational, Integer, ...
But not: Add, Mul, Pow, ...
"""
is_Atom = True
__slots__ = ()
def matches(self, expr, repl_dict={}, old=False):
if self == expr:
return repl_dict.copy()
def xreplace(self, rule, hack2=False):
return rule.get(self, self)
def doit(self, **hints):
return self
@classmethod
def class_key(cls):
return 2, 0, cls.__name__
@cacheit
def sort_key(self, order=None):
return self.class_key(), (1, (str(self),)), S.One.sort_key(), S.One
def _eval_simplify(self, **kwargs):
return self
@property
def _sorted_args(self):
# this is here as a safeguard against accidentally using _sorted_args
# on Atoms -- they cannot be rebuilt as atom.func(*atom._sorted_args)
# since there are no args. So the calling routine should be checking
# to see that this property is not called for Atoms.
raise AttributeError('Atoms have no args. It might be necessary'
' to make a check for Atoms in the calling code.')
def _aresame(a, b):
"""Return True if a and b are structurally the same, else False.
Examples
========
In SymPy (as in Python) two numbers compare the same if they
have the same underlying base-2 representation even though
they may not be the same type:
>>> from sympy import S
>>> 2.0 == S(2)
True
>>> 0.5 == S.Half
True
This routine was written to provide a query for such cases that
would give false when the types do not match:
>>> from sympy.core.basic import _aresame
>>> _aresame(S(2.0), S(2))
False
"""
from .numbers import Number
from .function import AppliedUndef, UndefinedFunction as UndefFunc
if isinstance(a, Number) and isinstance(b, Number):
return a == b and a.__class__ == b.__class__
for i, j in zip_longest(preorder_traversal(a), preorder_traversal(b)):
if i != j or type(i) != type(j):
if ((isinstance(i, UndefFunc) and isinstance(j, UndefFunc)) or
(isinstance(i, AppliedUndef) and isinstance(j, AppliedUndef))):
if i.class_key() != j.class_key():
return False
else:
return False
return True
def _atomic(e, recursive=False):
"""Return atom-like quantities as far as substitution is
concerned: Derivatives, Functions and Symbols. Don't
return any 'atoms' that are inside such quantities unless
they also appear outside, too, unless `recursive` is True.
Examples
========
>>> from sympy import Derivative, Function, cos
>>> from sympy.abc import x, y
>>> from sympy.core.basic import _atomic
>>> f = Function('f')
>>> _atomic(x + y)
{x, y}
>>> _atomic(x + f(y))
{x, f(y)}
>>> _atomic(Derivative(f(x), x) + cos(x) + y)
{y, cos(x), Derivative(f(x), x)}
"""
from sympy import Derivative, Function, Symbol
pot = preorder_traversal(e)
seen = set()
if isinstance(e, Basic):
free = getattr(e, "free_symbols", None)
if free is None:
return {e}
else:
return set()
atoms = set()
for p in pot:
if p in seen:
pot.skip()
continue
seen.add(p)
if isinstance(p, Symbol) and p in free:
atoms.add(p)
elif isinstance(p, (Derivative, Function)):
if not recursive:
pot.skip()
atoms.add(p)
return atoms
class preorder_traversal:
"""
Do a pre-order traversal of a tree.
This iterator recursively yields nodes that it has visited in a pre-order
fashion. That is, it yields the current node then descends through the
tree breadth-first to yield all of a node's children's pre-order
traversal.
For an expression, the order of the traversal depends on the order of
.args, which in many cases can be arbitrary.
Parameters
==========
node : sympy expression
The expression to traverse.
keys : (default None) sort key(s)
The key(s) used to sort args of Basic objects. When None, args of Basic
objects are processed in arbitrary order. If key is defined, it will
be passed along to ordered() as the only key(s) to use to sort the
arguments; if ``key`` is simply True then the default keys of ordered
will be used.
Yields
======
subtree : sympy expression
All of the subtrees in the tree.
Examples
========
>>> from sympy import symbols
>>> from sympy.core.basic import preorder_traversal
>>> x, y, z = symbols('x y z')
The nodes are returned in the order that they are encountered unless key
is given; simply passing key=True will guarantee that the traversal is
unique.
>>> list(preorder_traversal((x + y)*z, keys=None)) # doctest: +SKIP
[z*(x + y), z, x + y, y, x]
>>> list(preorder_traversal((x + y)*z, keys=True))
[z*(x + y), z, x + y, x, y]
"""
def __init__(self, node, keys=None):
self._skip_flag = False
self._pt = self._preorder_traversal(node, keys)
def _preorder_traversal(self, node, keys):
yield node
if self._skip_flag:
self._skip_flag = False
return
if isinstance(node, Basic):
if not keys and hasattr(node, '_argset'):
# LatticeOp keeps args as a set. We should use this if we
# don't care about the order, to prevent unnecessary sorting.
args = node._argset
else:
args = node.args
if keys:
if keys != True:
args = ordered(args, keys, default=False)
else:
args = ordered(args)
for arg in args:
yield from self._preorder_traversal(arg, keys)
elif iterable(node):
for item in node:
yield from self._preorder_traversal(item, keys)
def skip(self):
"""
Skip yielding current node's (last yielded node's) subtrees.
Examples
========
>>> from sympy.core import symbols
>>> from sympy.core.basic import preorder_traversal
>>> x, y, z = symbols('x y z')
>>> pt = preorder_traversal((x+y*z)*z)
>>> for i in pt:
... print(i)
... if i == x+y*z:
... pt.skip()
z*(x + y*z)
z
x + y*z
"""
self._skip_flag = True
def __next__(self):
return next(self._pt)
def __iter__(self):
return self
def _make_find_query(query):
"""Convert the argument of Basic.find() into a callable"""
try:
query = _sympify(query)
except SympifyError:
pass
if isinstance(query, type):
return lambda expr: isinstance(expr, query)
elif isinstance(query, Basic):
return lambda expr: expr.match(query) is not None
return query
# Delayed to avoid cyclic import
from .singleton import S
|
19b8abfe228bade9505ae3da8aca9bfc09b63d29a5b414679eb12ae3dec50b9a | from typing import Callable
from math import log as _log
from .sympify import _sympify
from .cache import cacheit
from .singleton import S
from .expr import Expr
from .evalf import PrecisionExhausted
from .function import (_coeff_isneg, expand_complex, expand_multinomial,
expand_mul, _mexpand)
from .logic import fuzzy_bool, fuzzy_not, fuzzy_and, fuzzy_or
from .compatibility import as_int, HAS_GMPY, gmpy
from .parameters import global_parameters
from sympy.utilities.iterables import sift
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.multipledispatch import Dispatcher
from mpmath.libmp import sqrtrem as mpmath_sqrtrem
from math import sqrt as _sqrt
def isqrt(n):
"""Return the largest integer less than or equal to sqrt(n)."""
if n < 0:
raise ValueError("n must be nonnegative")
n = int(n)
# Fast path: with IEEE 754 binary64 floats and a correctly-rounded
# math.sqrt, int(math.sqrt(n)) works for any integer n satisfying 0 <= n <
# 4503599761588224 = 2**52 + 2**27. But Python doesn't guarantee either
# IEEE 754 format floats *or* correct rounding of math.sqrt, so check the
# answer and fall back to the slow method if necessary.
if n < 4503599761588224:
s = int(_sqrt(n))
if 0 <= n - s*s <= 2*s:
return s
return integer_nthroot(n, 2)[0]
def integer_nthroot(y, n):
"""
Return a tuple containing x = floor(y**(1/n))
and a boolean indicating whether the result is exact (that is,
whether x**n == y).
Examples
========
>>> from sympy import integer_nthroot
>>> integer_nthroot(16, 2)
(4, True)
>>> integer_nthroot(26, 2)
(5, False)
To simply determine if a number is a perfect square, the is_square
function should be used:
>>> from sympy.ntheory.primetest import is_square
>>> is_square(26)
False
See Also
========
sympy.ntheory.primetest.is_square
integer_log
"""
y, n = as_int(y), as_int(n)
if y < 0:
raise ValueError("y must be nonnegative")
if n < 1:
raise ValueError("n must be positive")
if HAS_GMPY and n < 2**63:
# Currently it works only for n < 2**63, else it produces TypeError
# sympy issue: https://github.com/sympy/sympy/issues/18374
# gmpy2 issue: https://github.com/aleaxit/gmpy/issues/257
if HAS_GMPY >= 2:
x, t = gmpy.iroot(y, n)
else:
x, t = gmpy.root(y, n)
return as_int(x), bool(t)
return _integer_nthroot_python(y, n)
def _integer_nthroot_python(y, n):
if y in (0, 1):
return y, True
if n == 1:
return y, True
if n == 2:
x, rem = mpmath_sqrtrem(y)
return int(x), not rem
if n > y:
return 1, False
# Get initial estimate for Newton's method. Care must be taken to
# avoid overflow
try:
guess = int(y**(1./n) + 0.5)
except OverflowError:
exp = _log(y, 2)/n
if exp > 53:
shift = int(exp - 53)
guess = int(2.0**(exp - shift) + 1) << shift
else:
guess = int(2.0**exp)
if guess > 2**50:
# Newton iteration
xprev, x = -1, guess
while 1:
t = x**(n - 1)
xprev, x = x, ((n - 1)*x + y//t)//n
if abs(x - xprev) < 2:
break
else:
x = guess
# Compensate
t = x**n
while t < y:
x += 1
t = x**n
while t > y:
x -= 1
t = x**n
return int(x), t == y # int converts long to int if possible
def integer_log(y, x):
r"""
Returns ``(e, bool)`` where e is the largest nonnegative integer
such that :math:`|y| \geq |x^e|` and ``bool`` is True if $y = x^e$.
Examples
========
>>> from sympy import integer_log
>>> integer_log(125, 5)
(3, True)
>>> integer_log(17, 9)
(1, False)
>>> integer_log(4, -2)
(2, True)
>>> integer_log(-125,-5)
(3, True)
See Also
========
integer_nthroot
sympy.ntheory.primetest.is_square
sympy.ntheory.factor_.multiplicity
sympy.ntheory.factor_.perfect_power
"""
if x == 1:
raise ValueError('x cannot take value as 1')
if y == 0:
raise ValueError('y cannot take value as 0')
if x in (-2, 2):
x = int(x)
y = as_int(y)
e = y.bit_length() - 1
return e, x**e == y
if x < 0:
n, b = integer_log(y if y > 0 else -y, -x)
return n, b and bool(n % 2 if y < 0 else not n % 2)
x = as_int(x)
y = as_int(y)
r = e = 0
while y >= x:
d = x
m = 1
while y >= d:
y, rem = divmod(y, d)
r = r or rem
e += m
if y > d:
d *= d
m *= 2
return e, r == 0 and y == 1
class Pow(Expr):
"""
Defines the expression x**y as "x raised to a power y"
Singleton definitions involving (0, 1, -1, oo, -oo, I, -I):
+--------------+---------+-----------------------------------------------+
| expr | value | reason |
+==============+=========+===============================================+
| z**0 | 1 | Although arguments over 0**0 exist, see [2]. |
+--------------+---------+-----------------------------------------------+
| z**1 | z | |
+--------------+---------+-----------------------------------------------+
| (-oo)**(-1) | 0 | |
+--------------+---------+-----------------------------------------------+
| (-1)**-1 | -1 | |
+--------------+---------+-----------------------------------------------+
| S.Zero**-1 | zoo | This is not strictly true, as 0**-1 may be |
| | | undefined, but is convenient in some contexts |
| | | where the base is assumed to be positive. |
+--------------+---------+-----------------------------------------------+
| 1**-1 | 1 | |
+--------------+---------+-----------------------------------------------+
| oo**-1 | 0 | |
+--------------+---------+-----------------------------------------------+
| 0**oo | 0 | Because for all complex numbers z near |
| | | 0, z**oo -> 0. |
+--------------+---------+-----------------------------------------------+
| 0**-oo | zoo | This is not strictly true, as 0**oo may be |
| | | oscillating between positive and negative |
| | | values or rotating in the complex plane. |
| | | It is convenient, however, when the base |
| | | is positive. |
+--------------+---------+-----------------------------------------------+
| 1**oo | nan | Because there are various cases where |
| 1**-oo | | lim(x(t),t)=1, lim(y(t),t)=oo (or -oo), |
| | | but lim( x(t)**y(t), t) != 1. See [3]. |
+--------------+---------+-----------------------------------------------+
| b**zoo | nan | Because b**z has no limit as z -> zoo |
+--------------+---------+-----------------------------------------------+
| (-1)**oo | nan | Because of oscillations in the limit. |
| (-1)**(-oo) | | |
+--------------+---------+-----------------------------------------------+
| oo**oo | oo | |
+--------------+---------+-----------------------------------------------+
| oo**-oo | 0 | |
+--------------+---------+-----------------------------------------------+
| (-oo)**oo | nan | |
| (-oo)**-oo | | |
+--------------+---------+-----------------------------------------------+
| oo**I | nan | oo**e could probably be best thought of as |
| (-oo)**I | | the limit of x**e for real x as x tends to |
| | | oo. If e is I, then the limit does not exist |
| | | and nan is used to indicate that. |
+--------------+---------+-----------------------------------------------+
| oo**(1+I) | zoo | If the real part of e is positive, then the |
| (-oo)**(1+I) | | limit of abs(x**e) is oo. So the limit value |
| | | is zoo. |
+--------------+---------+-----------------------------------------------+
| oo**(-1+I) | 0 | If the real part of e is negative, then the |
| -oo**(-1+I) | | limit is 0. |
+--------------+---------+-----------------------------------------------+
Because symbolic computations are more flexible that floating point
calculations and we prefer to never return an incorrect answer,
we choose not to conform to all IEEE 754 conventions. This helps
us avoid extra test-case code in the calculation of limits.
See Also
========
sympy.core.numbers.Infinity
sympy.core.numbers.NegativeInfinity
sympy.core.numbers.NaN
References
==========
.. [1] https://en.wikipedia.org/wiki/Exponentiation
.. [2] https://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_power_of_zero
.. [3] https://en.wikipedia.org/wiki/Indeterminate_forms
"""
is_Pow = True
__slots__ = ('is_commutative',)
@cacheit
def __new__(cls, b, e, evaluate=None):
if evaluate is None:
evaluate = global_parameters.evaluate
from sympy.functions.elementary.exponential import exp_polar
b = _sympify(b)
e = _sympify(e)
# XXX: This can be removed when non-Expr args are disallowed rather
# than deprecated.
from sympy.core.relational import Relational
if isinstance(b, Relational) or isinstance(e, Relational):
raise TypeError('Relational can not be used in Pow')
# XXX: This should raise TypeError once deprecation period is over:
if not (isinstance(b, Expr) and isinstance(e, Expr)):
SymPyDeprecationWarning(
feature="Pow with non-Expr args",
useinstead="Expr args",
issue=19445,
deprecated_since_version="1.7"
).warn()
if evaluate:
if b is S.Zero and e is S.NegativeInfinity:
return S.ComplexInfinity
if e is S.ComplexInfinity:
return S.NaN
if e is S.Zero:
return S.One
elif e is S.One:
return b
elif e == -1 and not b:
return S.ComplexInfinity
elif e.__class__.__name__ == "AccumulationBounds":
if b == S.Exp1:
from sympy import AccumBounds
return AccumBounds(Pow(b, e.min), Pow(b, e.max))
# Only perform autosimplification if exponent or base is a Symbol or number
elif (b.is_Symbol or b.is_number) and (e.is_Symbol or e.is_number) and\
e.is_integer and _coeff_isneg(b):
if e.is_even:
b = -b
elif e.is_odd:
return -Pow(-b, e)
if S.NaN in (b, e): # XXX S.NaN**x -> S.NaN under assumption that x != 0
return S.NaN
elif b is S.One:
if abs(e).is_infinite:
return S.NaN
return S.One
else:
# recognize base as E
if not e.is_Atom and b is not S.Exp1 and not isinstance(b, exp_polar):
from sympy import numer, denom, log, sign, im, factor_terms
c, ex = factor_terms(e, sign=False).as_coeff_Mul()
den = denom(ex)
if isinstance(den, log) and den.args[0] == b:
return S.Exp1**(c*numer(ex))
elif den.is_Add:
s = sign(im(b))
if s.is_Number and s and den == \
log(-factor_terms(b, sign=False)) + s*S.ImaginaryUnit*S.Pi:
return S.Exp1**(c*numer(ex))
obj = b._eval_power(e)
if obj is not None:
return obj
obj = Expr.__new__(cls, b, e)
obj = cls._exec_constructor_postprocessors(obj)
if not isinstance(obj, Pow):
return obj
obj.is_commutative = (b.is_commutative and e.is_commutative)
return obj
def inverse(self, argindex=1):
if self.base == S.Exp1:
from sympy import log
return log
return None
@property
def base(self):
return self._args[0]
@property
def exp(self):
return self._args[1]
@classmethod
def class_key(cls):
return 3, 2, cls.__name__
def _eval_refine(self, assumptions):
from sympy.assumptions.ask import ask, Q
b, e = self.as_base_exp()
if ask(Q.integer(e), assumptions) and _coeff_isneg(b):
if ask(Q.even(e), assumptions):
return Pow(-b, e)
elif ask(Q.odd(e), assumptions):
return -Pow(-b, e)
def _eval_power(self, other):
from sympy import arg, exp, floor, im, log, re, sign
b, e = self.as_base_exp()
if b is S.NaN:
return (b**e)**other # let __new__ handle it
s = None
if other.is_integer:
s = 1
elif b.is_polar: # e.g. exp_polar, besselj, var('p', polar=True)...
s = 1
elif e.is_extended_real is not None:
# helper functions ===========================
def _half(e):
"""Return True if the exponent has a literal 2 as the
denominator, else None."""
if getattr(e, 'q', None) == 2:
return True
n, d = e.as_numer_denom()
if n.is_integer and d == 2:
return True
def _n2(e):
"""Return ``e`` evaluated to a Number with 2 significant
digits, else None."""
try:
rv = e.evalf(2, strict=True)
if rv.is_Number:
return rv
except PrecisionExhausted:
pass
# ===================================================
if e.is_extended_real:
# we need _half(other) with constant floor or
# floor(S.Half - e*arg(b)/2/pi) == 0
# handle -1 as special case
if e == -1:
# floor arg. is 1/2 + arg(b)/2/pi
if _half(other):
if b.is_negative is True:
return S.NegativeOne**other*Pow(-b, e*other)
elif b.is_negative is False:
return Pow(b, -other)
elif e.is_even:
if b.is_extended_real:
b = abs(b)
if b.is_imaginary:
b = abs(im(b))*S.ImaginaryUnit
if (abs(e) < 1) == True or e == 1:
s = 1 # floor = 0
elif b.is_extended_nonnegative:
s = 1 # floor = 0
elif re(b).is_extended_nonnegative and (abs(e) < 2) == True:
s = 1 # floor = 0
elif fuzzy_not(im(b).is_zero) and abs(e) == 2:
s = 1 # floor = 0
elif _half(other):
s = exp(2*S.Pi*S.ImaginaryUnit*other*floor(
S.Half - e*arg(b)/(2*S.Pi)))
if s.is_extended_real and _n2(sign(s) - s) == 0:
s = sign(s)
else:
s = None
else:
# e.is_extended_real is False requires:
# _half(other) with constant floor or
# floor(S.Half - im(e*log(b))/2/pi) == 0
try:
s = exp(2*S.ImaginaryUnit*S.Pi*other*
floor(S.Half - im(e*log(b))/2/S.Pi))
# be careful to test that s is -1 or 1 b/c sign(I) == I:
# so check that s is real
if s.is_extended_real and _n2(sign(s) - s) == 0:
s = sign(s)
else:
s = None
except PrecisionExhausted:
s = None
if s is not None:
return s*Pow(b, e*other)
def _eval_Mod(self, q):
r"""A dispatched function to compute `b^e \bmod q`, dispatched
by ``Mod``.
Notes
=====
Algorithms:
1. For unevaluated integer power, use built-in ``pow`` function
with 3 arguments, if powers are not too large wrt base.
2. For very large powers, use totient reduction if e >= lg(m).
Bound on m, is for safe factorization memory wise ie m^(1/4).
For pollard-rho to be faster than built-in pow lg(e) > m^(1/4)
check is added.
3. For any unevaluated power found in `b` or `e`, the step 2
will be recursed down to the base and the exponent
such that the `b \bmod q` becomes the new base and
``\phi(q) + e \bmod \phi(q)`` becomes the new exponent, and then
the computation for the reduced expression can be done.
"""
from sympy.ntheory import totient
from .mod import Mod
base, exp = self.base, self.exp
if exp.is_integer and exp.is_positive:
if q.is_integer and base % q == 0:
return S.Zero
if base.is_Integer and exp.is_Integer and q.is_Integer:
b, e, m = int(base), int(exp), int(q)
mb = m.bit_length()
if mb <= 80 and e >= mb and e.bit_length()**4 >= m:
phi = totient(m)
return Integer(pow(b, phi + e%phi, m))
return Integer(pow(b, e, m))
if isinstance(base, Pow) and base.is_integer and base.is_number:
base = Mod(base, q)
return Mod(Pow(base, exp, evaluate=False), q)
if isinstance(exp, Pow) and exp.is_integer and exp.is_number:
bit_length = int(q).bit_length()
# XXX Mod-Pow actually attempts to do a hanging evaluation
# if this dispatched function returns None.
# May need some fixes in the dispatcher itself.
if bit_length <= 80:
phi = totient(q)
exp = phi + Mod(exp, phi)
return Mod(Pow(base, exp, evaluate=False), q)
def _eval_is_even(self):
if self.exp.is_integer and self.exp.is_positive:
return self.base.is_even
def _eval_is_negative(self):
ext_neg = Pow._eval_is_extended_negative(self)
if ext_neg is True:
return self.is_finite
return ext_neg
def _eval_is_positive(self):
ext_pos = Pow._eval_is_extended_positive(self)
if ext_pos is True:
return self.is_finite
return ext_pos
def _eval_is_extended_positive(self):
from sympy import log
if self.base == self.exp:
if self.base.is_extended_nonnegative:
return True
elif self.base.is_positive:
if self.exp.is_real:
return True
elif self.base.is_extended_negative:
if self.exp.is_even:
return True
if self.exp.is_odd:
return False
elif self.base.is_zero:
if self.exp.is_extended_real:
return self.exp.is_zero
elif self.base.is_extended_nonpositive:
if self.exp.is_odd:
return False
elif self.base.is_imaginary:
if self.exp.is_integer:
m = self.exp % 4
if m.is_zero:
return True
if m.is_integer and m.is_zero is False:
return False
if self.exp.is_imaginary:
return log(self.base).is_imaginary
def _eval_is_extended_negative(self):
if self.exp is S(1)/2:
if self.base.is_complex or self.base.is_extended_real:
return False
if self.base.is_extended_negative:
if self.exp.is_odd and self.base.is_finite:
return True
if self.exp.is_even:
return False
elif self.base.is_extended_positive:
if self.exp.is_extended_real:
return False
elif self.base.is_zero:
if self.exp.is_extended_real:
return False
elif self.base.is_extended_nonnegative:
if self.exp.is_extended_nonnegative:
return False
elif self.base.is_extended_nonpositive:
if self.exp.is_even:
return False
elif self.base.is_extended_real:
if self.exp.is_even:
return False
def _eval_is_zero(self):
if self.base.is_zero:
if self.exp.is_extended_positive:
return True
elif self.exp.is_extended_nonpositive:
return False
elif self.base == S.Exp1:
return self.exp is S.NegativeInfinity
elif self.base.is_zero is False:
if self.base.is_finite and self.exp.is_finite:
return False
elif self.exp.is_negative:
return self.base.is_infinite
elif self.exp.is_nonnegative:
return False
elif self.exp.is_infinite and self.exp.is_extended_real:
if (1 - abs(self.base)).is_extended_positive:
return self.exp.is_extended_positive
elif (1 - abs(self.base)).is_extended_negative:
return self.exp.is_extended_negative
elif self.base.is_finite and self.exp.is_negative:
# when self.base.is_zero is None
return False
def _eval_is_integer(self):
b, e = self.args
if b.is_rational:
if b.is_integer is False and e.is_positive:
return False # rat**nonneg
if b.is_integer and e.is_integer:
if b is S.NegativeOne:
return True
if e.is_nonnegative or e.is_positive:
return True
if b.is_integer and e.is_negative and (e.is_finite or e.is_integer):
if fuzzy_not((b - 1).is_zero) and fuzzy_not((b + 1).is_zero):
return False
if b.is_Number and e.is_Number:
check = self.func(*self.args)
return check.is_Integer
if e.is_negative and b.is_positive and (b - 1).is_positive:
return False
if e.is_negative and b.is_negative and (b + 1).is_negative:
return False
def _eval_is_extended_real(self):
from ..functions import arg, log, exp
from .mul import Mul
if self.base is S.Exp1:
if self.exp.is_extended_real:
return True
elif self.exp.is_imaginary:
return (2*S.ImaginaryUnit*self.exp/S.Pi).is_even
real_b = self.base.is_extended_real
if real_b is None:
if self.base.func == exp and self.base.exp.is_imaginary:
return self.exp.is_imaginary
if self.base.func == Pow and self.base.base is S.Exp1 and self.base.exp.is_imaginary:
return self.exp.is_imaginary
return
real_e = self.exp.is_extended_real
if real_e is None:
return
if real_b and real_e:
if self.base.is_extended_positive:
return True
elif self.base.is_extended_nonnegative and self.exp.is_extended_nonnegative:
return True
elif self.exp.is_integer and self.base.is_extended_nonzero:
return True
elif self.exp.is_integer and self.exp.is_nonnegative:
return True
elif self.base.is_extended_negative:
if self.exp.is_Rational:
return False
if real_e and self.exp.is_extended_negative and self.base.is_zero is False:
return Pow(self.base, -self.exp).is_extended_real
im_b = self.base.is_imaginary
im_e = self.exp.is_imaginary
if im_b:
if self.exp.is_integer:
if self.exp.is_even:
return True
elif self.exp.is_odd:
return False
elif im_e and log(self.base).is_imaginary:
return True
elif self.exp.is_Add:
c, a = self.exp.as_coeff_Add()
if c and c.is_Integer:
return Mul(
self.base**c, self.base**a, evaluate=False).is_extended_real
elif self.base in (-S.ImaginaryUnit, S.ImaginaryUnit):
if (self.exp/2).is_integer is False:
return False
if real_b and im_e:
if self.base is S.NegativeOne:
return True
c = self.exp.coeff(S.ImaginaryUnit)
if c:
if self.base.is_rational and c.is_rational:
if self.base.is_nonzero and (self.base - 1).is_nonzero and c.is_nonzero:
return False
ok = (c*log(self.base)/S.Pi).is_integer
if ok is not None:
return ok
if real_b is False: # we already know it's not imag
i = arg(self.base)*self.exp/S.Pi
if i.is_complex: # finite
return i.is_integer
def _eval_is_complex(self):
if self.base == S.Exp1:
return fuzzy_or([self.exp.is_complex, self.exp.is_extended_negative])
if all(a.is_complex for a in self.args) and self._eval_is_finite():
return True
def _eval_is_imaginary(self):
from sympy import arg, log
if self.base.is_imaginary:
if self.exp.is_integer:
odd = self.exp.is_odd
if odd is not None:
return odd
return
if self.base == S.Exp1:
f = 2 * self.exp / (S.Pi*S.ImaginaryUnit)
# exp(pi*integer) = 1 or -1, so not imaginary
if f.is_even:
return False
# exp(pi*integer + pi/2) = I or -I, so it is imaginary
if f.is_odd:
return True
return None
if self.exp.is_imaginary:
imlog = log(self.base).is_imaginary
if imlog is not None:
return False # I**i -> real; (2*I)**i -> complex ==> not imaginary
if self.base.is_extended_real and self.exp.is_extended_real:
if self.base.is_positive:
return False
else:
rat = self.exp.is_rational
if not rat:
return rat
if self.exp.is_integer:
return False
else:
half = (2*self.exp).is_integer
if half:
return self.base.is_negative
return half
if self.base.is_extended_real is False: # we already know it's not imag
i = arg(self.base)*self.exp/S.Pi
isodd = (2*i).is_odd
if isodd is not None:
return isodd
def _eval_is_odd(self):
if self.exp.is_integer:
if self.exp.is_positive:
return self.base.is_odd
elif self.exp.is_nonnegative and self.base.is_odd:
return True
elif self.base is S.NegativeOne:
return True
def _eval_is_finite(self):
if self.exp.is_negative:
if self.base.is_zero:
return False
if self.base.is_infinite or self.base.is_nonzero:
return True
c1 = self.base.is_finite
if c1 is None:
return
c2 = self.exp.is_finite
if c2 is None:
return
if c1 and c2:
if self.exp.is_nonnegative or fuzzy_not(self.base.is_zero):
return True
def _eval_is_prime(self):
'''
An integer raised to the n(>=2)-th power cannot be a prime.
'''
if self.base.is_integer and self.exp.is_integer and (self.exp - 1).is_positive:
return False
def _eval_is_composite(self):
"""
A power is composite if both base and exponent are greater than 1
"""
if (self.base.is_integer and self.exp.is_integer and
((self.base - 1).is_positive and (self.exp - 1).is_positive or
(self.base + 1).is_negative and self.exp.is_positive and self.exp.is_even)):
return True
def _eval_is_polar(self):
return self.base.is_polar
def _eval_subs(self, old, new):
from sympy import exp, log, Symbol
def _check(ct1, ct2, old):
"""Return (bool, pow, remainder_pow) where, if bool is True, then the
exponent of Pow `old` will combine with `pow` so the substitution
is valid, otherwise bool will be False.
For noncommutative objects, `pow` will be an integer, and a factor
`Pow(old.base, remainder_pow)` needs to be included. If there is
no such factor, None is returned. For commutative objects,
remainder_pow is always None.
cti are the coefficient and terms of an exponent of self or old
In this _eval_subs routine a change like (b**(2*x)).subs(b**x, y)
will give y**2 since (b**x)**2 == b**(2*x); if that equality does
not hold then the substitution should not occur so `bool` will be
False.
"""
coeff1, terms1 = ct1
coeff2, terms2 = ct2
if terms1 == terms2:
if old.is_commutative:
# Allow fractional powers for commutative objects
pow = coeff1/coeff2
try:
as_int(pow, strict=False)
combines = True
except ValueError:
combines = isinstance(Pow._eval_power(
Pow(*old.as_base_exp(), evaluate=False),
pow), (Pow, exp, Symbol))
return combines, pow, None
else:
# With noncommutative symbols, substitute only integer powers
if not isinstance(terms1, tuple):
terms1 = (terms1,)
if not all(term.is_integer for term in terms1):
return False, None, None
try:
# Round pow toward zero
pow, remainder = divmod(as_int(coeff1), as_int(coeff2))
if pow < 0 and remainder != 0:
pow += 1
remainder -= as_int(coeff2)
if remainder == 0:
remainder_pow = None
else:
remainder_pow = Mul(remainder, *terms1)
return True, pow, remainder_pow
except ValueError:
# Can't substitute
pass
return False, None, None
if old == self.base or (old == exp and self.base == S.Exp1):
if new.is_Function and isinstance(new, Callable):
return new(self.exp._subs(old, new))
else:
return new**self.exp._subs(old, new)
# issue 10829: (4**x - 3*y + 2).subs(2**x, y) -> y**2 - 3*y + 2
if isinstance(old, self.func) and self.exp == old.exp:
l = log(self.base, old.base)
if l.is_Number:
return Pow(new, l)
if isinstance(old, self.func) and self.base == old.base:
if self.exp.is_Add is False:
ct1 = self.exp.as_independent(Symbol, as_Add=False)
ct2 = old.exp.as_independent(Symbol, as_Add=False)
ok, pow, remainder_pow = _check(ct1, ct2, old)
if ok:
# issue 5180: (x**(6*y)).subs(x**(3*y),z)->z**2
result = self.func(new, pow)
if remainder_pow is not None:
result = Mul(result, Pow(old.base, remainder_pow))
return result
else: # b**(6*x + a).subs(b**(3*x), y) -> y**2 * b**a
# exp(exp(x) + exp(x**2)).subs(exp(exp(x)), w) -> w * exp(exp(x**2))
oarg = old.exp
new_l = []
o_al = []
ct2 = oarg.as_coeff_mul()
for a in self.exp.args:
newa = a._subs(old, new)
ct1 = newa.as_coeff_mul()
ok, pow, remainder_pow = _check(ct1, ct2, old)
if ok:
new_l.append(new**pow)
if remainder_pow is not None:
o_al.append(remainder_pow)
continue
elif not old.is_commutative and not newa.is_integer:
# If any term in the exponent is non-integer,
# we do not do any substitutions in the noncommutative case
return
o_al.append(newa)
if new_l:
expo = Add(*o_al)
new_l.append(Pow(self.base, expo, evaluate=False) if expo != 1 else self.base)
return Mul(*new_l)
if (isinstance(old, exp) or (old.is_Pow and old.base is S.Exp1)) and self.exp.is_extended_real and self.base.is_positive:
ct1 = old.exp.as_independent(Symbol, as_Add=False)
ct2 = (self.exp*log(self.base)).as_independent(
Symbol, as_Add=False)
ok, pow, remainder_pow = _check(ct1, ct2, old)
if ok:
result = self.func(new, pow) # (2**x).subs(exp(x*log(2)), z) -> z
if remainder_pow is not None:
result = Mul(result, Pow(old.base, remainder_pow))
return result
def as_base_exp(self):
"""Return base and exp of self.
Explnation
==========
If base is 1/Integer, then return Integer, -exp. If this extra
processing is not needed, the base and exp properties will
give the raw arguments
Examples
========
>>> from sympy import Pow, S
>>> p = Pow(S.Half, 2, evaluate=False)
>>> p.as_base_exp()
(2, -2)
>>> p.args
(1/2, 2)
"""
b, e = self.args
if b.is_Rational and b.p == 1 and b.q != 1:
return Integer(b.q), -e
return b, e
def _eval_adjoint(self):
from sympy.functions.elementary.complexes import adjoint
i, p = self.exp.is_integer, self.base.is_positive
if i:
return adjoint(self.base)**self.exp
if p:
return self.base**adjoint(self.exp)
if i is False and p is False:
expanded = expand_complex(self)
if expanded != self:
return adjoint(expanded)
def _eval_conjugate(self):
from sympy.functions.elementary.complexes import conjugate as c
i, p = self.exp.is_integer, self.base.is_positive
if i:
return c(self.base)**self.exp
if p:
return self.base**c(self.exp)
if i is False and p is False:
expanded = expand_complex(self)
if expanded != self:
return c(expanded)
if self.is_extended_real:
return self
def _eval_transpose(self):
from sympy.functions.elementary.complexes import transpose
if self.base == S.Exp1:
return self.func(S.Exp1, self.exp.transpose())
i, p = self.exp.is_integer, (self.base.is_complex or self.base.is_infinite)
if p:
return self.base**self.exp
if i:
return transpose(self.base)**self.exp
if i is False and p is False:
expanded = expand_complex(self)
if expanded != self:
return transpose(expanded)
def _eval_expand_power_exp(self, **hints):
"""a**(n + m) -> a**n*a**m"""
b = self.base
e = self.exp
if b == S.Exp1:
from sympy import Sum, Product
if isinstance(e, Sum) and e.is_commutative:
return Product(self.func(b, e.function), *e.limits)
if e.is_Add and e.is_commutative:
expr = []
for x in e.args:
expr.append(self.func(b, x))
return Mul(*expr)
return self.func(b, e)
def _eval_expand_power_base(self, **hints):
"""(a*b)**n -> a**n * b**n"""
force = hints.get('force', False)
b = self.base
e = self.exp
if not b.is_Mul:
return self
cargs, nc = b.args_cnc(split_1=False)
# expand each term - this is top-level-only
# expansion but we have to watch out for things
# that don't have an _eval_expand method
if nc:
nc = [i._eval_expand_power_base(**hints)
if hasattr(i, '_eval_expand_power_base') else i
for i in nc]
if e.is_Integer:
if e.is_positive:
rv = Mul(*nc*e)
else:
rv = Mul(*[i**-1 for i in nc[::-1]]*-e)
if cargs:
rv *= Mul(*cargs)**e
return rv
if not cargs:
return self.func(Mul(*nc), e, evaluate=False)
nc = [Mul(*nc)]
# sift the commutative bases
other, maybe_real = sift(cargs, lambda x: x.is_extended_real is False,
binary=True)
def pred(x):
if x is S.ImaginaryUnit:
return S.ImaginaryUnit
polar = x.is_polar
if polar:
return True
if polar is None:
return fuzzy_bool(x.is_extended_nonnegative)
sifted = sift(maybe_real, pred)
nonneg = sifted[True]
other += sifted[None]
neg = sifted[False]
imag = sifted[S.ImaginaryUnit]
if imag:
I = S.ImaginaryUnit
i = len(imag) % 4
if i == 0:
pass
elif i == 1:
other.append(I)
elif i == 2:
if neg:
nonn = -neg.pop()
if nonn is not S.One:
nonneg.append(nonn)
else:
neg.append(S.NegativeOne)
else:
if neg:
nonn = -neg.pop()
if nonn is not S.One:
nonneg.append(nonn)
else:
neg.append(S.NegativeOne)
other.append(I)
del imag
# bring out the bases that can be separated from the base
if force or e.is_integer:
# treat all commutatives the same and put nc in other
cargs = nonneg + neg + other
other = nc
else:
# this is just like what is happening automatically, except
# that now we are doing it for an arbitrary exponent for which
# no automatic expansion is done
assert not e.is_Integer
# handle negatives by making them all positive and putting
# the residual -1 in other
if len(neg) > 1:
o = S.One
if not other and neg[0].is_Number:
o *= neg.pop(0)
if len(neg) % 2:
o = -o
for n in neg:
nonneg.append(-n)
if o is not S.One:
other.append(o)
elif neg and other:
if neg[0].is_Number and neg[0] is not S.NegativeOne:
other.append(S.NegativeOne)
nonneg.append(-neg[0])
else:
other.extend(neg)
else:
other.extend(neg)
del neg
cargs = nonneg
other += nc
rv = S.One
if cargs:
if e.is_Rational:
npow, cargs = sift(cargs, lambda x: x.is_Pow and
x.exp.is_Rational and x.base.is_number,
binary=True)
rv = Mul(*[self.func(b.func(*b.args), e) for b in npow])
rv *= Mul(*[self.func(b, e, evaluate=False) for b in cargs])
if other:
rv *= self.func(Mul(*other), e, evaluate=False)
return rv
def _eval_expand_multinomial(self, **hints):
"""(a + b + ..)**n -> a**n + n*a**(n-1)*b + .., n is nonzero integer"""
base, exp = self.args
result = self
if exp.is_Rational and exp.p > 0 and base.is_Add:
if not exp.is_Integer:
n = Integer(exp.p // exp.q)
if not n:
return result
else:
radical, result = self.func(base, exp - n), []
expanded_base_n = self.func(base, n)
if expanded_base_n.is_Pow:
expanded_base_n = \
expanded_base_n._eval_expand_multinomial()
for term in Add.make_args(expanded_base_n):
result.append(term*radical)
return Add(*result)
n = int(exp)
if base.is_commutative:
order_terms, other_terms = [], []
for b in base.args:
if b.is_Order:
order_terms.append(b)
else:
other_terms.append(b)
if order_terms:
# (f(x) + O(x^n))^m -> f(x)^m + m*f(x)^{m-1} *O(x^n)
f = Add(*other_terms)
o = Add(*order_terms)
if n == 2:
return expand_multinomial(f**n, deep=False) + n*f*o
else:
g = expand_multinomial(f**(n - 1), deep=False)
return expand_mul(f*g, deep=False) + n*g*o
if base.is_number:
# Efficiently expand expressions of the form (a + b*I)**n
# where 'a' and 'b' are real numbers and 'n' is integer.
a, b = base.as_real_imag()
if a.is_Rational and b.is_Rational:
if not a.is_Integer:
if not b.is_Integer:
k = self.func(a.q * b.q, n)
a, b = a.p*b.q, a.q*b.p
else:
k = self.func(a.q, n)
a, b = a.p, a.q*b
elif not b.is_Integer:
k = self.func(b.q, n)
a, b = a*b.q, b.p
else:
k = 1
a, b, c, d = int(a), int(b), 1, 0
while n:
if n & 1:
c, d = a*c - b*d, b*c + a*d
n -= 1
a, b = a*a - b*b, 2*a*b
n //= 2
I = S.ImaginaryUnit
if k == 1:
return c + I*d
else:
return Integer(c)/k + I*d/k
p = other_terms
# (x + y)**3 -> x**3 + 3*x**2*y + 3*x*y**2 + y**3
# in this particular example:
# p = [x,y]; n = 3
# so now it's easy to get the correct result -- we get the
# coefficients first:
from sympy import multinomial_coefficients
from sympy.polys.polyutils import basic_from_dict
expansion_dict = multinomial_coefficients(len(p), n)
# in our example: {(3, 0): 1, (1, 2): 3, (0, 3): 1, (2, 1): 3}
# and now construct the expression.
return basic_from_dict(expansion_dict, *p)
else:
if n == 2:
return Add(*[f*g for f in base.args for g in base.args])
else:
multi = (base**(n - 1))._eval_expand_multinomial()
if multi.is_Add:
return Add(*[f*g for f in base.args
for g in multi.args])
else:
# XXX can this ever happen if base was an Add?
return Add(*[f*multi for f in base.args])
elif (exp.is_Rational and exp.p < 0 and base.is_Add and
abs(exp.p) > exp.q):
return 1 / self.func(base, -exp)._eval_expand_multinomial()
elif exp.is_Add and base.is_Number:
# a + b a b
# n --> n n , where n, a, b are Numbers
coeff, tail = S.One, S.Zero
for term in exp.args:
if term.is_Number:
coeff *= self.func(base, term)
else:
tail += term
return coeff * self.func(base, tail)
else:
return result
def as_real_imag(self, deep=True, **hints):
from sympy import atan2, cos, im, re, sin
from sympy.polys.polytools import poly
if self.exp.is_Integer:
exp = self.exp
re_e, im_e = self.base.as_real_imag(deep=deep)
if not im_e:
return self, S.Zero
a, b = symbols('a b', cls=Dummy)
if exp >= 0:
if re_e.is_Number and im_e.is_Number:
# We can be more efficient in this case
expr = expand_multinomial(self.base**exp)
if expr != self:
return expr.as_real_imag()
expr = poly(
(a + b)**exp) # a = re, b = im; expr = (a + b*I)**exp
else:
mag = re_e**2 + im_e**2
re_e, im_e = re_e/mag, -im_e/mag
if re_e.is_Number and im_e.is_Number:
# We can be more efficient in this case
expr = expand_multinomial((re_e + im_e*S.ImaginaryUnit)**-exp)
if expr != self:
return expr.as_real_imag()
expr = poly((a + b)**-exp)
# Terms with even b powers will be real
r = [i for i in expr.terms() if not i[0][1] % 2]
re_part = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r])
# Terms with odd b powers will be imaginary
r = [i for i in expr.terms() if i[0][1] % 4 == 1]
im_part1 = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r])
r = [i for i in expr.terms() if i[0][1] % 4 == 3]
im_part3 = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r])
return (re_part.subs({a: re_e, b: S.ImaginaryUnit*im_e}),
im_part1.subs({a: re_e, b: im_e}) + im_part3.subs({a: re_e, b: -im_e}))
elif self.exp.is_Rational:
re_e, im_e = self.base.as_real_imag(deep=deep)
if im_e.is_zero and self.exp is S.Half:
if re_e.is_extended_nonnegative:
return self, S.Zero
if re_e.is_extended_nonpositive:
return S.Zero, (-self.base)**self.exp
# XXX: This is not totally correct since for x**(p/q) with
# x being imaginary there are actually q roots, but
# only a single one is returned from here.
r = self.func(self.func(re_e, 2) + self.func(im_e, 2), S.Half)
t = atan2(im_e, re_e)
rp, tp = self.func(r, self.exp), t*self.exp
return rp*cos(tp), rp*sin(tp)
elif self.base is S.Exp1:
from ..functions import exp
re_e, im_e = self.exp.as_real_imag()
if deep:
re_e = re_e.expand(deep, **hints)
im_e = im_e.expand(deep, **hints)
c, s = cos(im_e), sin(im_e)
return exp(re_e)*c, exp(re_e)*s
else:
if deep:
hints['complex'] = False
expanded = self.expand(deep, **hints)
if hints.get('ignore') == expanded:
return None
else:
return (re(expanded), im(expanded))
else:
return re(self), im(self)
def _eval_derivative(self, s):
from sympy import log
dbase = self.base.diff(s)
dexp = self.exp.diff(s)
return self * (dexp * log(self.base) + dbase * self.exp/self.base)
def _eval_evalf(self, prec):
base, exp = self.as_base_exp()
if base == S.Exp1:
# Use mpmath function associated to class "exp":
from sympy import exp as exp_function
return exp_function(self.exp, evaluate=False)._eval_evalf(prec)
base = base._evalf(prec)
if not exp.is_Integer:
exp = exp._evalf(prec)
if exp.is_negative and base.is_number and base.is_extended_real is False:
base = base.conjugate() / (base * base.conjugate())._evalf(prec)
exp = -exp
return self.func(base, exp).expand()
return self.func(base, exp)
def _eval_is_polynomial(self, syms):
if self.exp.has(*syms):
return False
if self.base.has(*syms):
return bool(self.base._eval_is_polynomial(syms) and
self.exp.is_Integer and (self.exp >= 0))
else:
return True
def _eval_is_rational(self):
# The evaluation of self.func below can be very expensive in the case
# of integer**integer if the exponent is large. We should try to exit
# before that if possible:
if (self.exp.is_integer and self.base.is_rational
and fuzzy_not(fuzzy_and([self.exp.is_negative, self.base.is_zero]))):
return True
p = self.func(*self.as_base_exp()) # in case it's unevaluated
if not p.is_Pow:
return p.is_rational
b, e = p.as_base_exp()
if e.is_Rational and b.is_Rational:
# we didn't check that e is not an Integer
# because Rational**Integer autosimplifies
return False
if e.is_integer:
if b.is_rational:
if fuzzy_not(b.is_zero) or e.is_nonnegative:
return True
if b == e: # always rational, even for 0**0
return True
elif b.is_irrational:
return e.is_zero
if b is S.Exp1:
if e.is_rational and e.is_nonzero:
return False
def _eval_is_algebraic(self):
def _is_one(expr):
try:
return (expr - 1).is_zero
except ValueError:
# when the operation is not allowed
return False
if self.base.is_zero or _is_one(self.base):
return True
elif self.base is S.Exp1:
s = self.func(*self.args)
if s.func == self.func:
if self.exp.is_nonzero:
if self.exp.is_algebraic:
return False
elif (self.exp/S.Pi).is_rational:
return False
elif (self.exp/(S.ImaginaryUnit*S.Pi)).is_rational:
return True
else:
return s.is_algebraic
elif self.exp.is_rational:
if self.base.is_algebraic is False:
return self.exp.is_zero
if self.base.is_zero is False:
if self.exp.is_nonzero:
return self.base.is_algebraic
elif self.base.is_algebraic:
return True
if self.exp.is_positive:
return self.base.is_algebraic
elif self.base.is_algebraic and self.exp.is_algebraic:
if ((fuzzy_not(self.base.is_zero)
and fuzzy_not(_is_one(self.base)))
or self.base.is_integer is False
or self.base.is_irrational):
return self.exp.is_rational
def _eval_is_rational_function(self, syms):
if self.exp.has(*syms):
return False
if self.base.has(*syms):
return self.base._eval_is_rational_function(syms) and \
self.exp.is_Integer
else:
return True
def _eval_is_meromorphic(self, x, a):
# f**g is meromorphic if g is an integer and f is meromorphic.
# E**(log(f)*g) is meromorphic if log(f)*g is meromorphic
# and finite.
base_merom = self.base._eval_is_meromorphic(x, a)
exp_integer = self.exp.is_Integer
if exp_integer:
return base_merom
exp_merom = self.exp._eval_is_meromorphic(x, a)
if base_merom is False:
# f**g = E**(log(f)*g) may be meromorphic if the
# singularities of log(f) and g cancel each other,
# for example, if g = 1/log(f). Hence,
return False if exp_merom else None
elif base_merom is None:
return None
b = self.base.subs(x, a)
# b is extended complex as base is meromorphic.
# log(base) is finite and meromorphic when b != 0, zoo.
b_zero = b.is_zero
if b_zero:
log_defined = False
else:
log_defined = fuzzy_and((b.is_finite, fuzzy_not(b_zero)))
if log_defined is False: # zero or pole of base
return exp_integer # False or None
elif log_defined is None:
return None
if not exp_merom:
return exp_merom # False or None
return self.exp.subs(x, a).is_finite
def _eval_is_algebraic_expr(self, syms):
if self.exp.has(*syms):
return False
if self.base.has(*syms):
return self.base._eval_is_algebraic_expr(syms) and \
self.exp.is_Rational
else:
return True
def _eval_rewrite_as_exp(self, base, expo, **kwargs):
from sympy import exp, log, I, arg
if base.is_zero or base.has(exp) or expo.has(exp):
return base**expo
if base.has(Symbol):
# delay evaluation if expo is non symbolic
# (as exp(x*log(5)) automatically reduces to x**5)
if global_parameters.exp_is_pow:
return Pow(S.Exp1, log(base)*expo, evaluate=expo.has(Symbol))
else:
return exp(log(base)*expo, evaluate=expo.has(Symbol))
else:
return exp((log(abs(base)) + I*arg(base))*expo)
def as_numer_denom(self):
if not self.is_commutative:
return self, S.One
base, exp = self.as_base_exp()
n, d = base.as_numer_denom()
# this should be the same as ExpBase.as_numer_denom wrt
# exponent handling
neg_exp = exp.is_negative
if not neg_exp and not (-exp).is_negative:
neg_exp = _coeff_isneg(exp)
int_exp = exp.is_integer
# the denominator cannot be separated from the numerator if
# its sign is unknown unless the exponent is an integer, e.g.
# sqrt(a/b) != sqrt(a)/sqrt(b) when a=1 and b=-1. But if the
# denominator is negative the numerator and denominator can
# be negated and the denominator (now positive) separated.
if not (d.is_extended_real or int_exp):
n = base
d = S.One
dnonpos = d.is_nonpositive
if dnonpos:
n, d = -n, -d
elif dnonpos is None and not int_exp:
n = base
d = S.One
if neg_exp:
n, d = d, n
exp = -exp
if exp.is_infinite:
if n is S.One and d is not S.One:
return n, self.func(d, exp)
if n is not S.One and d is S.One:
return self.func(n, exp), d
return self.func(n, exp), self.func(d, exp)
def matches(self, expr, repl_dict={}, old=False):
expr = _sympify(expr)
repl_dict = repl_dict.copy()
# special case, pattern = 1 and expr.exp can match to 0
if expr is S.One:
d = self.exp.matches(S.Zero, repl_dict)
if d is not None:
return d
# make sure the expression to be matched is an Expr
if not isinstance(expr, Expr):
return None
b, e = expr.as_base_exp()
# special case number
sb, se = self.as_base_exp()
if sb.is_Symbol and se.is_Integer and expr:
if e.is_rational:
return sb.matches(b**(e/se), repl_dict)
return sb.matches(expr**(1/se), repl_dict)
d = repl_dict.copy()
d = self.base.matches(b, d)
if d is None:
return None
d = self.exp.xreplace(d).matches(e, d)
if d is None:
return Expr.matches(self, expr, repl_dict)
return d
def _eval_nseries(self, x, n, logx, cdir=0):
# NOTE! This function is an important part of the gruntz algorithm
# for computing limits. It has to return a generalized power
# series with coefficients in C(log, log(x)). In more detail:
# It has to return an expression
# c_0*x**e_0 + c_1*x**e_1 + ... (finitely many terms)
# where e_i are numbers (not necessarily integers) and c_i are
# expressions involving only numbers, the log function, and log(x).
# The series expansion of b**e is computed as follows:
# 1) We express b as f*(1 + g) where f is the leading term of b.
# g has order O(x**d) where d is strictly positive.
# 2) Then b**e = (f**e)*((1 + g)**e).
# (1 + g)**e is computed using binomial series.
from sympy import im, I, ceiling, polygamma, logcombine, EulerGamma, nan, zoo, factorial, ff, PoleError, O, powdenest, Wild
from itertools import product
from ..functions import exp, log
from ..series import Order, limit
from ..simplify import powsimp
if self.base is S.Exp1:
e_series = self.exp.nseries(x, n=n, logx=logx)
if e_series.is_Order:
return 1 + e_series
e0 = limit(e_series.removeO(), x, 0)
if e0 in (-S.NegativeInfinity, S.Infinity):
return self
t = e_series - e0
exp_series = term = exp(e0)
# series of exp(e0 + t) in t
for i in range(1, n):
term *= t/i
term = term.nseries(x, n=n, logx=logx)
exp_series += term
exp_series += Order(t**n, x)
return powsimp(exp_series, deep=True, combine='exp')
self = powdenest(self, force=True).trigsimp()
b, e = self.as_base_exp()
if e.has(S.Infinity, S.NegativeInfinity, S.ComplexInfinity, S.NaN):
raise PoleError()
if e.has(x):
return exp(e*log(b))._eval_nseries(x, n=n, logx=logx, cdir=cdir)
if logx is not None and b.has(log):
c, ex = symbols('c, ex', cls=Wild, exclude=[x])
b = b.replace(log(c*x**ex), log(c) + ex*logx)
self = b**e
b = b.removeO()
try:
if b.has(polygamma, EulerGamma) and logx is not None:
raise ValueError()
_, m = b.leadterm(x)
except (ValueError, NotImplementedError):
b = b._eval_nseries(x, n=max(2, n), logx=logx, cdir=cdir).removeO()
if b.has(nan, zoo):
raise NotImplementedError()
_, m = b.leadterm(x)
if e.has(log):
e = logcombine(e).cancel()
if not (m.is_zero or e.is_number and e.is_real):
return exp(e*log(b))._eval_nseries(x, n=n, logx=logx, cdir=cdir)
f = b.as_leading_term(x)
g = (b/f - S.One).cancel()
maxpow = n - m*e
if maxpow.is_negative:
return O(x**(m*e), x)
if g.is_zero:
return f**e
def coeff_exp(term, x):
coeff, exp = S.One, S.Zero
for factor in Mul.make_args(term):
if factor.has(x):
base, exp = factor.as_base_exp()
if base != x:
try:
return term.leadterm(x)
except ValueError:
return term, S.Zero
else:
coeff *= factor
return coeff, exp
def mul(d1, d2):
res = {}
for e1, e2 in product(d1, d2):
ex = e1 + e2
if ex < maxpow:
res[ex] = res.get(ex, S.Zero) + d1[e1]*d2[e2]
return res
try:
_, d = g.leadterm(x)
except (ValueError, NotImplementedError):
if limit(g/x**maxpow, x, 0) == 0:
# g has higher order zero
return f**e + e*f**e*g # first term of binomial series
else:
raise NotImplementedError()
if not d.is_positive:
g = (b - f).simplify()/f
_, d = g.leadterm(x)
if not d.is_positive:
raise NotImplementedError()
gpoly = g._eval_nseries(x, n=ceiling(maxpow), logx=logx, cdir=cdir).removeO()
gterms = {}
for term in Add.make_args(gpoly):
co1, e1 = coeff_exp(term, x)
gterms[e1] = gterms.get(e1, S.Zero) + co1
k = S.One
terms = {S.Zero: S.One}
tk = gterms
while (k*d - maxpow).is_negative:
coeff = ff(e, k)/factorial(k)
for ex in tk:
terms[ex] = terms.get(ex, S.Zero) + coeff*tk[ex]
tk = mul(tk, gterms)
k += S.One
if (not e.is_integer and m.is_zero and f.is_real
and f.is_negative and im((b - f).dir(x, cdir)) < 0):
inco, inex = coeff_exp(f**e*exp(-2*e*S.Pi*I), x)
else:
inco, inex = coeff_exp(f**e, x)
res = S.Zero
for e1 in terms:
ex = e1 + inex
res += terms[e1]*inco*x**(ex)
if not (e.is_integer and e.is_positive and (e*d - n).is_nonpositive and
res == _mexpand(self)):
res += O(x**n, x)
return res
def _eval_as_leading_term(self, x, cdir=0):
from ..series import Order
from sympy import exp, I, im, log
e = self.exp
b = self.base
if self.base is S.Exp1:
arg = self.exp
if arg.is_Add:
return Mul(*[(S.Exp1**f).as_leading_term(x) for f in arg.args])
arg_1 = arg.as_leading_term(x)
if Order(x, x).contains(arg_1):
return S.One
if Order(1, x).contains(arg_1):
return S.Exp1**arg_1
####################################################
# The correct result here should be 'None'. #
# Indeed arg in not bounded as x tends to 0. #
# Consequently the series expansion does not admit #
# the leading term. #
# For compatibility reasons, the return value here #
# is the original function, i.e. exp(arg), #
# instead of None. #
####################################################
return S.Exp1**arg
elif e.has(x):
return exp(e * log(b)).as_leading_term(x, cdir=cdir)
else:
f = b.as_leading_term(x, cdir=cdir)
if (not e.is_integer and f.is_constant() and f.is_real
and f.is_negative and im((b - f).dir(x, cdir)) < 0):
return self.func(f, e) * exp(-2 * e * S.Pi * I)
return self.func(f, e)
@cacheit
def _taylor_term(self, n, x, *previous_terms): # of (1 + x)**e
from sympy import binomial
return binomial(self.exp, n) * self.func(x, n)
def taylor_term(self, n, x, *previous_terms):
if self.base is not S.Exp1:
return super().taylor_term(n, x, *previous_terms)
from sympy import sympify, factorial
if n < 0:
return S.Zero
if n == 0:
return S.One
x = sympify(x)
if previous_terms:
p = previous_terms[-1]
if p is not None:
return p * x / n
return x**n/factorial(n)
def _sage_(self):
return self.args[0]._sage_()**self.args[1]._sage_()
def _eval_rewrite_as_sin(self, base, exp):
from ..functions import sin
if self.base is S.Exp1:
return sin(S.ImaginaryUnit*self.exp + S.Pi/2) - S.ImaginaryUnit*sin(S.ImaginaryUnit*self.exp)
def _eval_rewrite_as_cos(self, base, exp):
from ..functions import cos
if self.base is S.Exp1:
return cos(S.ImaginaryUnit*self.exp) + S.ImaginaryUnit*cos(S.ImaginaryUnit*self.exp + S.Pi/2)
def _eval_rewrite_as_tanh(self, base, exp):
from ..functions import tanh
if self.base is S.Exp1:
return (1 + tanh(self.exp/2))/(1 - tanh(self.exp/2))
def _eval_rewrite_as_sqrt(self, base, exp, **kwargs):
from sympy.functions.elementary.trigonometric import sin, cos
if base is not S.Exp1:
return None
if exp.is_Mul:
coeff = exp.coeff(S.Pi * S.ImaginaryUnit)
if coeff and coeff.is_number:
cosine, sine = cos(S.Pi*coeff), sin(S.Pi*coeff)
if not isinstance(cosine, cos) and not isinstance (sine, sin):
return cosine + S.ImaginaryUnit*sine
def as_content_primitive(self, radical=False, clear=True):
"""Return the tuple (R, self/R) where R is the positive Rational
extracted from self.
Examples
========
>>> from sympy import sqrt
>>> sqrt(4 + 4*sqrt(2)).as_content_primitive()
(2, sqrt(1 + sqrt(2)))
>>> sqrt(3 + 3*sqrt(2)).as_content_primitive()
(1, sqrt(3)*sqrt(1 + sqrt(2)))
>>> from sympy import expand_power_base, powsimp, Mul
>>> from sympy.abc import x, y
>>> ((2*x + 2)**2).as_content_primitive()
(4, (x + 1)**2)
>>> (4**((1 + y)/2)).as_content_primitive()
(2, 4**(y/2))
>>> (3**((1 + y)/2)).as_content_primitive()
(1, 3**((y + 1)/2))
>>> (3**((5 + y)/2)).as_content_primitive()
(9, 3**((y + 1)/2))
>>> eq = 3**(2 + 2*x)
>>> powsimp(eq) == eq
True
>>> eq.as_content_primitive()
(9, 3**(2*x))
>>> powsimp(Mul(*_))
3**(2*x + 2)
>>> eq = (2 + 2*x)**y
>>> s = expand_power_base(eq); s.is_Mul, s
(False, (2*x + 2)**y)
>>> eq.as_content_primitive()
(1, (2*(x + 1))**y)
>>> s = expand_power_base(_[1]); s.is_Mul, s
(True, 2**y*(x + 1)**y)
See docstring of Expr.as_content_primitive for more examples.
"""
b, e = self.as_base_exp()
b = _keep_coeff(*b.as_content_primitive(radical=radical, clear=clear))
ce, pe = e.as_content_primitive(radical=radical, clear=clear)
if b.is_Rational:
#e
#= ce*pe
#= ce*(h + t)
#= ce*h + ce*t
#=> self
#= b**(ce*h)*b**(ce*t)
#= b**(cehp/cehq)*b**(ce*t)
#= b**(iceh + r/cehq)*b**(ce*t)
#= b**(iceh)*b**(r/cehq)*b**(ce*t)
#= b**(iceh)*b**(ce*t + r/cehq)
h, t = pe.as_coeff_Add()
if h.is_Rational:
ceh = ce*h
c = self.func(b, ceh)
r = S.Zero
if not c.is_Rational:
iceh, r = divmod(ceh.p, ceh.q)
c = self.func(b, iceh)
return c, self.func(b, _keep_coeff(ce, t + r/ce/ceh.q))
e = _keep_coeff(ce, pe)
# b**e = (h*t)**e = h**e*t**e = c*m*t**e
if e.is_Rational and b.is_Mul:
h, t = b.as_content_primitive(radical=radical, clear=clear) # h is positive
c, m = self.func(h, e).as_coeff_Mul() # so c is positive
m, me = m.as_base_exp()
if m is S.One or me == e: # probably always true
# return the following, not return c, m*Pow(t, e)
# which would change Pow into Mul; we let sympy
# decide what to do by using the unevaluated Mul, e.g
# should it stay as sqrt(2 + 2*sqrt(5)) or become
# sqrt(2)*sqrt(1 + sqrt(5))
return c, self.func(_keep_coeff(m, t), e)
return S.One, self.func(b, e)
def is_constant(self, *wrt, **flags):
expr = self
if flags.get('simplify', True):
expr = expr.simplify()
b, e = expr.as_base_exp()
bz = b.equals(0)
if bz: # recalculate with assumptions in case it's unevaluated
new = b**e
if new != expr:
return new.is_constant()
econ = e.is_constant(*wrt)
bcon = b.is_constant(*wrt)
if bcon:
if econ:
return True
bz = b.equals(0)
if bz is False:
return False
elif bcon is None:
return None
return e.equals(0)
def _eval_difference_delta(self, n, step):
b, e = self.args
if e.has(n) and not b.has(n):
new_e = e.subs(n, n + step)
return (b**(new_e - e) - 1) * self
power = Dispatcher('power')
power.add((object, object), Pow)
from .add import Add
from .numbers import Integer
from .mul import Mul, _keep_coeff
from .symbol import Symbol, Dummy, symbols
|
f7e76954493ea43fd4a6d71cdafcde5edd253b92f83ef93b653977c1a27d5dce | """Singleton mechanism"""
from .core import Registry
from .assumptions import ManagedProperties
from .sympify import sympify
class SingletonRegistry(Registry):
"""
The registry for the singleton classes (accessible as ``S``).
Explanation
===========
This class serves as two separate things.
The first thing it is is the ``SingletonRegistry``. Several classes in
SymPy appear so often that they are singletonized, that is, using some
metaprogramming they are made so that they can only be instantiated once
(see the :class:`sympy.core.singleton.Singleton` class for details). For
instance, every time you create ``Integer(0)``, this will return the same
instance, :class:`sympy.core.numbers.Zero`. All singleton instances are
attributes of the ``S`` object, so ``Integer(0)`` can also be accessed as
``S.Zero``.
Singletonization offers two advantages: it saves memory, and it allows
fast comparison. It saves memory because no matter how many times the
singletonized objects appear in expressions in memory, they all point to
the same single instance in memory. The fast comparison comes from the
fact that you can use ``is`` to compare exact instances in Python
(usually, you need to use ``==`` to compare things). ``is`` compares
objects by memory address, and is very fast.
Examples
========
>>> from sympy import S, Integer
>>> a = Integer(0)
>>> a is S.Zero
True
For the most part, the fact that certain objects are singletonized is an
implementation detail that users shouldn't need to worry about. In SymPy
library code, ``is`` comparison is often used for performance purposes
The primary advantage of ``S`` for end users is the convenient access to
certain instances that are otherwise difficult to type, like ``S.Half``
(instead of ``Rational(1, 2)``).
When using ``is`` comparison, make sure the argument is sympified. For
instance,
>>> x = 0
>>> x is S.Zero
False
This problem is not an issue when using ``==``, which is recommended for
most use-cases:
>>> 0 == S.Zero
True
The second thing ``S`` is is a shortcut for
:func:`sympy.core.sympify.sympify`. :func:`sympy.core.sympify.sympify` is
the function that converts Python objects such as ``int(1)`` into SymPy
objects such as ``Integer(1)``. It also converts the string form of an
expression into a SymPy expression, like ``sympify("x**2")`` ->
``Symbol("x")**2``. ``S(1)`` is the same thing as ``sympify(1)``
(basically, ``S.__call__`` has been defined to call ``sympify``).
This is for convenience, since ``S`` is a single letter. It's mostly
useful for defining rational numbers. Consider an expression like ``x +
1/2``. If you enter this directly in Python, it will evaluate the ``1/2``
and give ``0.5`` (or just ``0`` in Python 2, because of integer division),
because both arguments are ints (see also
:ref:`tutorial-gotchas-final-notes`). However, in SymPy, you usually want
the quotient of two integers to give an exact rational number. The way
Python's evaluation works, at least one side of an operator needs to be a
SymPy object for the SymPy evaluation to take over. You could write this
as ``x + Rational(1, 2)``, but this is a lot more typing. A shorter
version is ``x + S(1)/2``. Since ``S(1)`` returns ``Integer(1)``, the
division will return a ``Rational`` type, since it will call
``Integer.__truediv__``, which knows how to return a ``Rational``.
"""
__slots__ = ()
# Also allow things like S(5)
__call__ = staticmethod(sympify)
def __init__(self):
self._classes_to_install = {}
# Dict of classes that have been registered, but that have not have been
# installed as an attribute of this SingletonRegistry.
# Installation automatically happens at the first attempt to access the
# attribute.
# The purpose of this is to allow registration during class
# initialization during import, but not trigger object creation until
# actual use (which should not happen until after all imports are
# finished).
def register(self, cls):
# Make sure a duplicate class overwrites the old one
if hasattr(self, cls.__name__):
delattr(self, cls.__name__)
self._classes_to_install[cls.__name__] = cls
def __getattr__(self, name):
"""Python calls __getattr__ if no attribute of that name was installed
yet.
Explanation
===========
This __getattr__ checks whether a class with the requested name was
already registered but not installed; if no, raises an AttributeError.
Otherwise, retrieves the class, calculates its singleton value, installs
it as an attribute of the given name, and unregisters the class."""
if name not in self._classes_to_install:
raise AttributeError(
"Attribute '%s' was not installed on SymPy registry %s" % (
name, self))
class_to_install = self._classes_to_install[name]
value_to_install = class_to_install()
self.__setattr__(name, value_to_install)
del self._classes_to_install[name]
return value_to_install
def __repr__(self):
return "S"
S = SingletonRegistry()
class Singleton(ManagedProperties):
"""
Metaclass for singleton classes.
Explanation
===========
A singleton class has only one instance which is returned every time the
class is instantiated. Additionally, this instance can be accessed through
the global registry object ``S`` as ``S.<class_name>``.
Examples
========
>>> from sympy import S, Basic
>>> from sympy.core.singleton import Singleton
>>> class MySingleton(Basic, metaclass=Singleton):
... pass
>>> Basic() is Basic()
False
>>> MySingleton() is MySingleton()
True
>>> S.MySingleton is MySingleton()
True
Notes
=====
Instance creation is delayed until the first time the value is accessed.
(SymPy versions before 1.0 would create the instance during class
creation time, which would be prone to import cycles.)
This metaclass is a subclass of ManagedProperties because that is the
metaclass of many classes that need to be Singletons (Python does not allow
subclasses to have a different metaclass than the superclass, except the
subclass may use a subclassed metaclass).
"""
def __init__(cls, *args, **kwargs):
super().__init__(cls, *args, **kwargs)
cls._instance = obj = Basic.__new__(cls)
cls.__new__ = lambda cls: obj
cls.__getnewargs__ = lambda obj: ()
cls.__getstate__ = lambda obj: None
S.register(cls)
# Delayed to avoid cyclic import
from .basic import Basic
|
735e9a0098e11097bbd8d4838d69a79afd19ca782a97db84794184ee7719c2bd | """
There are three types of functions implemented in SymPy:
1) defined functions (in the sense that they can be evaluated) like
exp or sin; they have a name and a body:
f = exp
2) undefined function which have a name but no body. Undefined
functions can be defined using a Function class as follows:
f = Function('f')
(the result will be a Function instance)
3) anonymous function (or lambda function) which have a body (defined
with dummy variables) but have no name:
f = Lambda(x, exp(x)*x)
f = Lambda((x, y), exp(x)*y)
The fourth type of functions are composites, like (sin + cos)(x); these work in
SymPy core, but are not yet part of SymPy.
Examples
========
>>> import sympy
>>> f = sympy.Function("f")
>>> from sympy.abc import x
>>> f(x)
f(x)
>>> print(sympy.srepr(f(x).func))
Function('f')
>>> f(x).args
(x,)
"""
from typing import Any, Dict as tDict, Optional, Set as tSet, Tuple as tTuple, Union
from .add import Add
from .assumptions import ManagedProperties
from .basic import Basic, _atomic
from .cache import cacheit
from .compatibility import iterable, is_sequence, as_int, ordered, Iterable
from .decorators import _sympifyit
from .expr import Expr, AtomicExpr
from .numbers import Rational, Float
from .operations import LatticeOp
from .rules import Transform
from .singleton import S
from .sympify import sympify
from sympy.core.containers import Tuple, Dict
from sympy.core.parameters import global_parameters
from sympy.core.logic import fuzzy_and, fuzzy_or, fuzzy_not, FuzzyBool
from sympy.utilities import default_sort_key
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.utilities.iterables import has_dups, sift
from sympy.utilities.misc import filldedent
import mpmath
import mpmath.libmp as mlib
import inspect
from collections import Counter
def _coeff_isneg(a):
"""Return True if the leading Number is negative.
Examples
========
>>> from sympy.core.function import _coeff_isneg
>>> from sympy import S, Symbol, oo, pi
>>> _coeff_isneg(-3*pi)
True
>>> _coeff_isneg(S(3))
False
>>> _coeff_isneg(-oo)
True
>>> _coeff_isneg(Symbol('n', negative=True)) # coeff is 1
False
For matrix expressions:
>>> from sympy import MatrixSymbol, sqrt
>>> A = MatrixSymbol("A", 3, 3)
>>> _coeff_isneg(-sqrt(2)*A)
True
>>> _coeff_isneg(sqrt(2)*A)
False
"""
if a.is_MatMul:
a = a.args[0]
if a.is_Mul:
a = a.args[0]
return a.is_Number and a.is_extended_negative
class PoleError(Exception):
pass
class ArgumentIndexError(ValueError):
def __str__(self):
return ("Invalid operation with argument number %s for Function %s" %
(self.args[1], self.args[0]))
class BadSignatureError(TypeError):
'''Raised when a Lambda is created with an invalid signature'''
pass
class BadArgumentsError(TypeError):
'''Raised when a Lambda is called with an incorrect number of arguments'''
pass
# Python 2/3 version that does not raise a Deprecation warning
def arity(cls):
"""Return the arity of the function if it is known, else None.
Explanation
===========
When default values are specified for some arguments, they are
optional and the arity is reported as a tuple of possible values.
Examples
========
>>> from sympy.core.function import arity
>>> from sympy import log
>>> arity(lambda x: x)
1
>>> arity(log)
(1, 2)
>>> arity(lambda *x: sum(x)) is None
True
"""
eval_ = getattr(cls, 'eval', cls)
parameters = inspect.signature(eval_).parameters.items()
if [p for _, p in parameters if p.kind == p.VAR_POSITIONAL]:
return
p_or_k = [p for _, p in parameters if p.kind == p.POSITIONAL_OR_KEYWORD]
# how many have no default and how many have a default value
no, yes = map(len, sift(p_or_k,
lambda p:p.default == p.empty, binary=True))
return no if not yes else tuple(range(no, no + yes + 1))
class FunctionClass(ManagedProperties):
"""
Base class for function classes. FunctionClass is a subclass of type.
Use Function('<function name>' [ , signature ]) to create
undefined function classes.
"""
_new = type.__new__
def __init__(cls, *args, **kwargs):
# honor kwarg value or class-defined value before using
# the number of arguments in the eval function (if present)
nargs = kwargs.pop('nargs', cls.__dict__.get('nargs', arity(cls)))
if nargs is None and 'nargs' not in cls.__dict__:
for supcls in cls.__mro__:
if hasattr(supcls, '_nargs'):
nargs = supcls._nargs
break
else:
continue
# Canonicalize nargs here; change to set in nargs.
if is_sequence(nargs):
if not nargs:
raise ValueError(filldedent('''
Incorrectly specified nargs as %s:
if there are no arguments, it should be
`nargs = 0`;
if there are any number of arguments,
it should be
`nargs = None`''' % str(nargs)))
nargs = tuple(ordered(set(nargs)))
elif nargs is not None:
nargs = (as_int(nargs),)
cls._nargs = nargs
super().__init__(*args, **kwargs)
@property
def __signature__(self):
"""
Allow Python 3's inspect.signature to give a useful signature for
Function subclasses.
"""
# Python 3 only, but backports (like the one in IPython) still might
# call this.
try:
from inspect import signature
except ImportError:
return None
# TODO: Look at nargs
return signature(self.eval)
@property
def free_symbols(self):
return set()
@property
def xreplace(self):
# Function needs args so we define a property that returns
# a function that takes args...and then use that function
# to return the right value
return lambda rule, **_: rule.get(self, self)
@property
def nargs(self):
"""Return a set of the allowed number of arguments for the function.
Examples
========
>>> from sympy.core.function import Function
>>> f = Function('f')
If the function can take any number of arguments, the set of whole
numbers is returned:
>>> Function('f').nargs
Naturals0
If the function was initialized to accept one or more arguments, a
corresponding set will be returned:
>>> Function('f', nargs=1).nargs
FiniteSet(1)
>>> Function('f', nargs=(2, 1)).nargs
FiniteSet(1, 2)
The undefined function, after application, also has the nargs
attribute; the actual number of arguments is always available by
checking the ``args`` attribute:
>>> f = Function('f')
>>> f(1).nargs
Naturals0
>>> len(f(1).args)
1
"""
from sympy.sets.sets import FiniteSet
# XXX it would be nice to handle this in __init__ but there are import
# problems with trying to import FiniteSet there
return FiniteSet(*self._nargs) if self._nargs else S.Naturals0
def __repr__(cls):
return cls.__name__
class Application(Basic, metaclass=FunctionClass):
"""
Base class for applied functions.
Explanation
===========
Instances of Application represent the result of applying an application of
any type to any object.
"""
is_Function = True
@cacheit
def __new__(cls, *args, **options):
from sympy.sets.fancysets import Naturals0
from sympy.sets.sets import FiniteSet
args = list(map(sympify, args))
evaluate = options.pop('evaluate', global_parameters.evaluate)
# WildFunction (and anything else like it) may have nargs defined
# and we throw that value away here
options.pop('nargs', None)
if options:
raise ValueError("Unknown options: %s" % options)
if evaluate:
evaluated = cls.eval(*args)
if evaluated is not None:
return evaluated
obj = super().__new__(cls, *args, **options)
# make nargs uniform here
sentinel = object()
objnargs = getattr(obj, "nargs", sentinel)
if objnargs is not sentinel:
# things passing through here:
# - functions subclassed from Function (e.g. myfunc(1).nargs)
# - functions like cos(1).nargs
# - AppliedUndef with given nargs like Function('f', nargs=1)(1).nargs
# Canonicalize nargs here
if is_sequence(objnargs):
nargs = tuple(ordered(set(objnargs)))
elif objnargs is not None:
nargs = (as_int(objnargs),)
else:
nargs = None
else:
# things passing through here:
# - WildFunction('f').nargs
# - AppliedUndef with no nargs like Function('f')(1).nargs
nargs = obj._nargs # note the underscore here
# convert to FiniteSet
obj.nargs = FiniteSet(*nargs) if nargs else Naturals0()
return obj
@classmethod
def eval(cls, *args):
"""
Returns a canonical form of cls applied to arguments args.
Explanation
===========
The eval() method is called when the class cls is about to be
instantiated and it should return either some simplified instance
(possible of some other class), or if the class cls should be
unmodified, return None.
Examples of eval() for the function "sign"
---------------------------------------------
.. code-block:: python
@classmethod
def eval(cls, arg):
if arg is S.NaN:
return S.NaN
if arg.is_zero: return S.Zero
if arg.is_positive: return S.One
if arg.is_negative: return S.NegativeOne
if isinstance(arg, Mul):
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff is not S.One:
return cls(coeff) * cls(terms)
"""
return
@property
def func(self):
return self.__class__
def _eval_subs(self, old, new):
if (old.is_Function and new.is_Function and
callable(old) and callable(new) and
old == self.func and len(self.args) in new.nargs):
return new(*[i._subs(old, new) for i in self.args])
class Function(Application, Expr):
"""
Base class for applied mathematical functions.
It also serves as a constructor for undefined function classes.
Examples
========
First example shows how to use Function as a constructor for undefined
function classes:
>>> from sympy import Function, Symbol
>>> x = Symbol('x')
>>> f = Function('f')
>>> g = Function('g')(x)
>>> f
f
>>> f(x)
f(x)
>>> g
g(x)
>>> f(x).diff(x)
Derivative(f(x), x)
>>> g.diff(x)
Derivative(g(x), x)
Assumptions can be passed to Function, and if function is initialized with a
Symbol, the function inherits the name and assumptions associated with the Symbol:
>>> f_real = Function('f', real=True)
>>> f_real(x).is_real
True
>>> f_real_inherit = Function(Symbol('f', real=True))
>>> f_real_inherit(x).is_real
True
Note that assumptions on a function are unrelated to the assumptions on
the variable it is called on. If you want to add a relationship, subclass
Function and define the appropriate ``_eval_is_assumption`` methods.
In the following example Function is used as a base class for
``my_func`` that represents a mathematical function *my_func*. Suppose
that it is well known, that *my_func(0)* is *1* and *my_func* at infinity
goes to *0*, so we want those two simplifications to occur automatically.
Suppose also that *my_func(x)* is real exactly when *x* is real. Here is
an implementation that honours those requirements:
>>> from sympy import Function, S, oo, I, sin
>>> class my_func(Function):
...
... @classmethod
... def eval(cls, x):
... if x.is_Number:
... if x.is_zero:
... return S.One
... elif x is S.Infinity:
... return S.Zero
...
... def _eval_is_real(self):
... return self.args[0].is_real
...
>>> x = S('x')
>>> my_func(0) + sin(0)
1
>>> my_func(oo)
0
>>> my_func(3.54).n() # Not yet implemented for my_func.
my_func(3.54)
>>> my_func(I).is_real
False
In order for ``my_func`` to become useful, several other methods would
need to be implemented. See source code of some of the already
implemented functions for more complete examples.
Also, if the function can take more than one argument, then ``nargs``
must be defined, e.g. if ``my_func`` can take one or two arguments
then,
>>> class my_func(Function):
... nargs = (1, 2)
...
>>>
"""
@property
def _diff_wrt(self):
return False
@cacheit
def __new__(cls, *args, **options):
# Handle calls like Function('f')
if cls is Function:
return UndefinedFunction(*args, **options)
n = len(args)
if n not in cls.nargs:
# XXX: exception message must be in exactly this format to
# make it work with NumPy's functions like vectorize(). See,
# for example, https://github.com/numpy/numpy/issues/1697.
# The ideal solution would be just to attach metadata to
# the exception and change NumPy to take advantage of this.
temp = ('%(name)s takes %(qual)s %(args)s '
'argument%(plural)s (%(given)s given)')
raise TypeError(temp % {
'name': cls,
'qual': 'exactly' if len(cls.nargs) == 1 else 'at least',
'args': min(cls.nargs),
'plural': 's'*(min(cls.nargs) != 1),
'given': n})
evaluate = options.get('evaluate', global_parameters.evaluate)
result = super().__new__(cls, *args, **options)
if evaluate and isinstance(result, cls) and result.args:
pr2 = min(cls._should_evalf(a) for a in result.args)
if pr2 > 0:
pr = max(cls._should_evalf(a) for a in result.args)
result = result.evalf(mlib.libmpf.prec_to_dps(pr))
return result
@classmethod
def _should_evalf(cls, arg):
"""
Decide if the function should automatically evalf().
Explanation
===========
By default (in this implementation), this happens if (and only if) the
ARG is a floating point number.
This function is used by __new__.
Returns the precision to evalf to, or -1 if it shouldn't evalf.
"""
from sympy.core.evalf import pure_complex
if arg.is_Float:
return arg._prec
if not arg.is_Add:
return -1
m = pure_complex(arg)
if m is None or not (m[0].is_Float or m[1].is_Float):
return -1
l = [i._prec for i in m if i.is_Float]
l.append(-1)
return max(l)
@classmethod
def class_key(cls):
from sympy.sets.fancysets import Naturals0
funcs = {
'exp': 10,
'log': 11,
'sin': 20,
'cos': 21,
'tan': 22,
'cot': 23,
'sinh': 30,
'cosh': 31,
'tanh': 32,
'coth': 33,
'conjugate': 40,
're': 41,
'im': 42,
'arg': 43,
}
name = cls.__name__
try:
i = funcs[name]
except KeyError:
i = 0 if isinstance(cls.nargs, Naturals0) else 10000
return 4, i, name
def _eval_evalf(self, prec):
def _get_mpmath_func(fname):
"""Lookup mpmath function based on name"""
if isinstance(self, AppliedUndef):
# Shouldn't lookup in mpmath but might have ._imp_
return None
if not hasattr(mpmath, fname):
from sympy.utilities.lambdify import MPMATH_TRANSLATIONS
fname = MPMATH_TRANSLATIONS.get(fname, None)
if fname is None:
return None
return getattr(mpmath, fname)
_eval_mpmath = getattr(self, '_eval_mpmath', None)
if _eval_mpmath is None:
func = _get_mpmath_func(self.func.__name__)
args = self.args
else:
func, args = _eval_mpmath()
# Fall-back evaluation
if func is None:
imp = getattr(self, '_imp_', None)
if imp is None:
return None
try:
return Float(imp(*[i.evalf(prec) for i in self.args]), prec)
except (TypeError, ValueError):
return None
# Convert all args to mpf or mpc
# Convert the arguments to *higher* precision than requested for the
# final result.
# XXX + 5 is a guess, it is similar to what is used in evalf.py. Should
# we be more intelligent about it?
try:
args = [arg._to_mpmath(prec + 5) for arg in args]
def bad(m):
from mpmath import mpf, mpc
# the precision of an mpf value is the last element
# if that is 1 (and m[1] is not 1 which would indicate a
# power of 2), then the eval failed; so check that none of
# the arguments failed to compute to a finite precision.
# Note: An mpc value has two parts, the re and imag tuple;
# check each of those parts, too. Anything else is allowed to
# pass
if isinstance(m, mpf):
m = m._mpf_
return m[1] !=1 and m[-1] == 1
elif isinstance(m, mpc):
m, n = m._mpc_
return m[1] !=1 and m[-1] == 1 and \
n[1] !=1 and n[-1] == 1
else:
return False
if any(bad(a) for a in args):
raise ValueError # one or more args failed to compute with significance
except ValueError:
return
with mpmath.workprec(prec):
v = func(*args)
return Expr._from_mpmath(v, prec)
def _eval_derivative(self, s):
# f(x).diff(s) -> x.diff(s) * f.fdiff(1)(s)
i = 0
l = []
for a in self.args:
i += 1
da = a.diff(s)
if da.is_zero:
continue
try:
df = self.fdiff(i)
except ArgumentIndexError:
df = Function.fdiff(self, i)
l.append(df * da)
return Add(*l)
def _eval_is_commutative(self):
return fuzzy_and(a.is_commutative for a in self.args)
def _eval_is_meromorphic(self, x, a):
if not self.args:
return True
if any(arg.has(x) for arg in self.args[1:]):
return False
arg = self.args[0]
if not arg._eval_is_meromorphic(x, a):
return None
return fuzzy_not(type(self).is_singular(arg.subs(x, a)))
_singularities = None # type: Union[FuzzyBool, tTuple[Expr, ...]]
@classmethod
def is_singular(cls, a):
"""
Tests whether the argument is an essential singularity
or a branch point, or the functions is non-holomorphic.
"""
ss = cls._singularities
if ss in (True, None, False):
return ss
return fuzzy_or(a.is_infinite if s is S.ComplexInfinity
else (a - s).is_zero for s in ss)
def as_base_exp(self):
"""
Returns the method as the 2-tuple (base, exponent).
"""
return self, S.One
def _eval_aseries(self, n, args0, x, logx):
"""
Compute an asymptotic expansion around args0, in terms of self.args.
This function is only used internally by _eval_nseries and should not
be called directly; derived classes can overwrite this to implement
asymptotic expansions.
"""
from sympy.utilities.misc import filldedent
raise PoleError(filldedent('''
Asymptotic expansion of %s around %s is
not implemented.''' % (type(self), args0)))
def _eval_nseries(self, x, n, logx, cdir=0):
"""
This function does compute series for multivariate functions,
but the expansion is always in terms of *one* variable.
Examples
========
>>> from sympy import atan2
>>> from sympy.abc import x, y
>>> atan2(x, y).series(x, n=2)
atan2(0, y) + x/y + O(x**2)
>>> atan2(x, y).series(y, n=2)
-y/x + atan2(x, 0) + O(y**2)
This function also computes asymptotic expansions, if necessary
and possible:
>>> from sympy import loggamma
>>> loggamma(1/x)._eval_nseries(x,0,None)
-1/x - log(x)/x + log(x)/2 + O(1)
"""
from sympy import Order
from sympy.core.symbol import uniquely_named_symbol
from sympy.sets.sets import FiniteSet
args = self.args
args0 = [t.limit(x, 0) for t in args]
if any(t.is_finite is False for t in args0):
from sympy import oo, zoo, nan
# XXX could use t.as_leading_term(x) here but it's a little
# slower
a = [t.compute_leading_term(x, logx=logx) for t in args]
a0 = [t.limit(x, 0) for t in a]
if any([t.has(oo, -oo, zoo, nan) for t in a0]):
return self._eval_aseries(n, args0, x, logx)
# Careful: the argument goes to oo, but only logarithmically so. We
# are supposed to do a power series expansion "around the
# logarithmic term". e.g.
# f(1+x+log(x))
# -> f(1+logx) + x*f'(1+logx) + O(x**2)
# where 'logx' is given in the argument
a = [t._eval_nseries(x, n, logx) for t in args]
z = [r - r0 for (r, r0) in zip(a, a0)]
p = [Dummy() for _ in z]
q = []
v = None
for ai, zi, pi in zip(a0, z, p):
if zi.has(x):
if v is not None:
raise NotImplementedError
q.append(ai + pi)
v = pi
else:
q.append(ai)
e1 = self.func(*q)
if v is None:
return e1
s = e1._eval_nseries(v, n, logx)
o = s.getO()
s = s.removeO()
s = s.subs(v, zi).expand() + Order(o.expr.subs(v, zi), x)
return s
if (self.func.nargs is S.Naturals0
or (self.func.nargs == FiniteSet(1) and args0[0])
or any(c > 1 for c in self.func.nargs)):
e = self
e1 = e.expand()
if e == e1:
#for example when e = sin(x+1) or e = sin(cos(x))
#let's try the general algorithm
if len(e.args) == 1:
# issue 14411
e = e.func(e.args[0].cancel())
term = e.subs(x, S.Zero)
if term.is_finite is False or term is S.NaN:
raise PoleError("Cannot expand %s around 0" % (self))
series = term
fact = S.One
_x = uniquely_named_symbol('xi', self)
e = e.subs(x, _x)
for i in range(n - 1):
i += 1
fact *= Rational(i)
e = e.diff(_x)
subs = e.subs(_x, S.Zero)
if subs is S.NaN:
# try to evaluate a limit if we have to
subs = e.limit(_x, S.Zero)
if subs.is_finite is False:
raise PoleError("Cannot expand %s around 0" % (self))
term = subs*(x**i)/fact
term = term.expand()
series += term
return series + Order(x**n, x)
return e1.nseries(x, n=n, logx=logx)
arg = self.args[0]
l = []
g = None
# try to predict a number of terms needed
nterms = n + 2
cf = Order(arg.as_leading_term(x), x).getn()
if cf != 0:
nterms = (n/cf).ceiling()
for i in range(nterms):
g = self.taylor_term(i, arg, g)
g = g.nseries(x, n=n, logx=logx)
l.append(g)
return Add(*l) + Order(x**n, x)
def fdiff(self, argindex=1):
"""
Returns the first derivative of the function.
"""
if not (1 <= argindex <= len(self.args)):
raise ArgumentIndexError(self, argindex)
ix = argindex - 1
A = self.args[ix]
if A._diff_wrt:
if len(self.args) == 1 or not A.is_Symbol:
return _derivative_dispatch(self, A)
for i, v in enumerate(self.args):
if i != ix and A in v.free_symbols:
# it can't be in any other argument's free symbols
# issue 8510
break
else:
return _derivative_dispatch(self, A)
# See issue 4624 and issue 4719, 5600 and 8510
D = Dummy('xi_%i' % argindex, dummy_index=hash(A))
args = self.args[:ix] + (D,) + self.args[ix + 1:]
return Subs(Derivative(self.func(*args), D), D, A)
def _eval_as_leading_term(self, x, cdir=0):
"""Stub that should be overridden by new Functions to return
the first non-zero term in a series if ever an x-dependent
argument whose leading term vanishes as x -> 0 might be encountered.
See, for example, cos._eval_as_leading_term.
"""
from sympy import Order
args = [a.as_leading_term(x) for a in self.args]
o = Order(1, x)
if any(x in a.free_symbols and o.contains(a) for a in args):
# Whereas x and any finite number are contained in O(1, x),
# expressions like 1/x are not. If any arg simplified to a
# vanishing expression as x -> 0 (like x or x**2, but not
# 3, 1/x, etc...) then the _eval_as_leading_term is needed
# to supply the first non-zero term of the series,
#
# e.g. expression leading term
# ---------- ------------
# cos(1/x) cos(1/x)
# cos(cos(x)) cos(1)
# cos(x) 1 <- _eval_as_leading_term needed
# sin(x) x <- _eval_as_leading_term needed
#
raise NotImplementedError(
'%s has no _eval_as_leading_term routine' % self.func)
else:
return self.func(*args)
def _sage_(self):
import sage.all as sage
fname = self.func.__name__
func = getattr(sage, fname, None)
args = [arg._sage_() for arg in self.args]
# In the case the function is not known in sage:
if func is None:
import sympy
if getattr(sympy, fname, None) is None:
# abstract function
return sage.function(fname)(*args)
else:
# the function defined in sympy is not known in sage
# this exception is caught in sage
raise AttributeError
return func(*args)
class AppliedUndef(Function):
"""
Base class for expressions resulting from the application of an undefined
function.
"""
is_number = False
def __new__(cls, *args, **options):
args = list(map(sympify, args))
u = [a.name for a in args if isinstance(a, UndefinedFunction)]
if u:
raise TypeError('Invalid argument: expecting an expression, not UndefinedFunction%s: %s' % (
's'*(len(u) > 1), ', '.join(u)))
obj = super().__new__(cls, *args, **options)
return obj
def _eval_as_leading_term(self, x, cdir=0):
return self
def _sage_(self):
import sage.all as sage
fname = str(self.func)
args = [arg._sage_() for arg in self.args]
func = sage.function(fname)(*args)
return func
@property
def _diff_wrt(self):
"""
Allow derivatives wrt to undefined functions.
Examples
========
>>> from sympy import Function, Symbol
>>> f = Function('f')
>>> x = Symbol('x')
>>> f(x)._diff_wrt
True
>>> f(x).diff(x)
Derivative(f(x), x)
"""
return True
class UndefSageHelper:
"""
Helper to facilitate Sage conversion.
"""
def __get__(self, ins, typ):
import sage.all as sage
if ins is None:
return lambda: sage.function(typ.__name__)
else:
args = [arg._sage_() for arg in ins.args]
return lambda : sage.function(ins.__class__.__name__)(*args)
_undef_sage_helper = UndefSageHelper()
class UndefinedFunction(FunctionClass):
"""
The (meta)class of undefined functions.
"""
def __new__(mcl, name, bases=(AppliedUndef,), __dict__=None, **kwargs):
from .symbol import _filter_assumptions
# Allow Function('f', real=True)
# and/or Function(Symbol('f', real=True))
assumptions, kwargs = _filter_assumptions(kwargs)
if isinstance(name, Symbol):
assumptions = name._merge(assumptions)
name = name.name
elif not isinstance(name, str):
raise TypeError('expecting string or Symbol for name')
else:
commutative = assumptions.get('commutative', None)
assumptions = Symbol(name, **assumptions).assumptions0
if commutative is None:
assumptions.pop('commutative')
__dict__ = __dict__ or {}
# put the `is_*` for into __dict__
__dict__.update({'is_%s' % k: v for k, v in assumptions.items()})
# You can add other attributes, although they do have to be hashable
# (but seriously, if you want to add anything other than assumptions,
# just subclass Function)
__dict__.update(kwargs)
# add back the sanitized assumptions without the is_ prefix
kwargs.update(assumptions)
# Save these for __eq__
__dict__.update({'_kwargs': kwargs})
# do this for pickling
__dict__['__module__'] = None
obj = super().__new__(mcl, name, bases, __dict__)
obj.name = name
obj._sage_ = _undef_sage_helper
return obj
def __instancecheck__(cls, instance):
return cls in type(instance).__mro__
_kwargs = {} # type: tDict[str, Optional[bool]]
def __hash__(self):
return hash((self.class_key(), frozenset(self._kwargs.items())))
def __eq__(self, other):
return (isinstance(other, self.__class__) and
self.class_key() == other.class_key() and
self._kwargs == other._kwargs)
def __ne__(self, other):
return not self == other
@property
def _diff_wrt(self):
return False
# XXX: The type: ignore on WildFunction is because mypy complains:
#
# sympy/core/function.py:939: error: Cannot determine type of 'sort_key' in
# base class 'Expr'
#
# Somehow this is because of the @cacheit decorator but it is not clear how to
# fix it.
class WildFunction(Function, AtomicExpr): # type: ignore
"""
A WildFunction function matches any function (with its arguments).
Examples
========
>>> from sympy import WildFunction, Function, cos
>>> from sympy.abc import x, y
>>> F = WildFunction('F')
>>> f = Function('f')
>>> F.nargs
Naturals0
>>> x.match(F)
>>> F.match(F)
{F_: F_}
>>> f(x).match(F)
{F_: f(x)}
>>> cos(x).match(F)
{F_: cos(x)}
>>> f(x, y).match(F)
{F_: f(x, y)}
To match functions with a given number of arguments, set ``nargs`` to the
desired value at instantiation:
>>> F = WildFunction('F', nargs=2)
>>> F.nargs
FiniteSet(2)
>>> f(x).match(F)
>>> f(x, y).match(F)
{F_: f(x, y)}
To match functions with a range of arguments, set ``nargs`` to a tuple
containing the desired number of arguments, e.g. if ``nargs = (1, 2)``
then functions with 1 or 2 arguments will be matched.
>>> F = WildFunction('F', nargs=(1, 2))
>>> F.nargs
FiniteSet(1, 2)
>>> f(x).match(F)
{F_: f(x)}
>>> f(x, y).match(F)
{F_: f(x, y)}
>>> f(x, y, 1).match(F)
"""
# XXX: What is this class attribute used for?
include = set() # type: tSet[Any]
def __init__(cls, name, **assumptions):
from sympy.sets.sets import Set, FiniteSet
cls.name = name
nargs = assumptions.pop('nargs', S.Naturals0)
if not isinstance(nargs, Set):
# Canonicalize nargs here. See also FunctionClass.
if is_sequence(nargs):
nargs = tuple(ordered(set(nargs)))
elif nargs is not None:
nargs = (as_int(nargs),)
nargs = FiniteSet(*nargs)
cls.nargs = nargs
def matches(self, expr, repl_dict={}, old=False):
if not isinstance(expr, (AppliedUndef, Function)):
return None
if len(expr.args) not in self.nargs:
return None
repl_dict = repl_dict.copy()
repl_dict[self] = expr
return repl_dict
class Derivative(Expr):
"""
Carries out differentiation of the given expression with respect to symbols.
Examples
========
>>> from sympy import Derivative, Function, symbols, Subs
>>> from sympy.abc import x, y
>>> f, g = symbols('f g', cls=Function)
>>> Derivative(x**2, x, evaluate=True)
2*x
Denesting of derivatives retains the ordering of variables:
>>> Derivative(Derivative(f(x, y), y), x)
Derivative(f(x, y), y, x)
Contiguously identical symbols are merged into a tuple giving
the symbol and the count:
>>> Derivative(f(x), x, x, y, x)
Derivative(f(x), (x, 2), y, x)
If the derivative cannot be performed, and evaluate is True, the
order of the variables of differentiation will be made canonical:
>>> Derivative(f(x, y), y, x, evaluate=True)
Derivative(f(x, y), x, y)
Derivatives with respect to undefined functions can be calculated:
>>> Derivative(f(x)**2, f(x), evaluate=True)
2*f(x)
Such derivatives will show up when the chain rule is used to
evalulate a derivative:
>>> f(g(x)).diff(x)
Derivative(f(g(x)), g(x))*Derivative(g(x), x)
Substitution is used to represent derivatives of functions with
arguments that are not symbols or functions:
>>> f(2*x + 3).diff(x) == 2*Subs(f(y).diff(y), y, 2*x + 3)
True
Notes
=====
Simplification of high-order derivatives:
Because there can be a significant amount of simplification that can be
done when multiple differentiations are performed, results will be
automatically simplified in a fairly conservative fashion unless the
keyword ``simplify`` is set to False.
>>> from sympy import sqrt, diff, Function, symbols
>>> from sympy.abc import x, y, z
>>> f, g = symbols('f,g', cls=Function)
>>> e = sqrt((x + 1)**2 + x)
>>> diff(e, (x, 5), simplify=False).count_ops()
136
>>> diff(e, (x, 5)).count_ops()
30
Ordering of variables:
If evaluate is set to True and the expression cannot be evaluated, the
list of differentiation symbols will be sorted, that is, the expression is
assumed to have continuous derivatives up to the order asked.
Derivative wrt non-Symbols:
For the most part, one may not differentiate wrt non-symbols.
For example, we do not allow differentiation wrt `x*y` because
there are multiple ways of structurally defining where x*y appears
in an expression: a very strict definition would make
(x*y*z).diff(x*y) == 0. Derivatives wrt defined functions (like
cos(x)) are not allowed, either:
>>> (x*y*z).diff(x*y)
Traceback (most recent call last):
...
ValueError: Can't calculate derivative wrt x*y.
To make it easier to work with variational calculus, however,
derivatives wrt AppliedUndef and Derivatives are allowed.
For example, in the Euler-Lagrange method one may write
F(t, u, v) where u = f(t) and v = f'(t). These variables can be
written explicitly as functions of time::
>>> from sympy.abc import t
>>> F = Function('F')
>>> U = f(t)
>>> V = U.diff(t)
The derivative wrt f(t) can be obtained directly:
>>> direct = F(t, U, V).diff(U)
When differentiation wrt a non-Symbol is attempted, the non-Symbol
is temporarily converted to a Symbol while the differentiation
is performed and the same answer is obtained:
>>> indirect = F(t, U, V).subs(U, x).diff(x).subs(x, U)
>>> assert direct == indirect
The implication of this non-symbol replacement is that all
functions are treated as independent of other functions and the
symbols are independent of the functions that contain them::
>>> x.diff(f(x))
0
>>> g(x).diff(f(x))
0
It also means that derivatives are assumed to depend only
on the variables of differentiation, not on anything contained
within the expression being differentiated::
>>> F = f(x)
>>> Fx = F.diff(x)
>>> Fx.diff(F) # derivative depends on x, not F
0
>>> Fxx = Fx.diff(x)
>>> Fxx.diff(Fx) # derivative depends on x, not Fx
0
The last example can be made explicit by showing the replacement
of Fx in Fxx with y:
>>> Fxx.subs(Fx, y)
Derivative(y, x)
Since that in itself will evaluate to zero, differentiating
wrt Fx will also be zero:
>>> _.doit()
0
Replacing undefined functions with concrete expressions
One must be careful to replace undefined functions with expressions
that contain variables consistent with the function definition and
the variables of differentiation or else insconsistent result will
be obtained. Consider the following example:
>>> eq = f(x)*g(y)
>>> eq.subs(f(x), x*y).diff(x, y).doit()
y*Derivative(g(y), y) + g(y)
>>> eq.diff(x, y).subs(f(x), x*y).doit()
y*Derivative(g(y), y)
The results differ because `f(x)` was replaced with an expression
that involved both variables of differentiation. In the abstract
case, differentiation of `f(x)` by `y` is 0; in the concrete case,
the presence of `y` made that derivative nonvanishing and produced
the extra `g(y)` term.
Defining differentiation for an object
An object must define ._eval_derivative(symbol) method that returns
the differentiation result. This function only needs to consider the
non-trivial case where expr contains symbol and it should call the diff()
method internally (not _eval_derivative); Derivative should be the only
one to call _eval_derivative.
Any class can allow derivatives to be taken with respect to
itself (while indicating its scalar nature). See the
docstring of Expr._diff_wrt.
See Also
========
_sort_variable_count
"""
is_Derivative = True
@property
def _diff_wrt(self):
"""An expression may be differentiated wrt a Derivative if
it is in elementary form.
Examples
========
>>> from sympy import Function, Derivative, cos
>>> from sympy.abc import x
>>> f = Function('f')
>>> Derivative(f(x), x)._diff_wrt
True
>>> Derivative(cos(x), x)._diff_wrt
False
>>> Derivative(x + 1, x)._diff_wrt
False
A Derivative might be an unevaluated form of what will not be
a valid variable of differentiation if evaluated. For example,
>>> Derivative(f(f(x)), x).doit()
Derivative(f(x), x)*Derivative(f(f(x)), f(x))
Such an expression will present the same ambiguities as arise
when dealing with any other product, like ``2*x``, so ``_diff_wrt``
is False:
>>> Derivative(f(f(x)), x)._diff_wrt
False
"""
return self.expr._diff_wrt and isinstance(self.doit(), Derivative)
def __new__(cls, expr, *variables, **kwargs):
from sympy.matrices.common import MatrixCommon
from sympy import Integer, MatrixExpr
from sympy.tensor.array import Array, NDimArray
from sympy.utilities.misc import filldedent
expr = sympify(expr)
symbols_or_none = getattr(expr, "free_symbols", None)
has_symbol_set = isinstance(symbols_or_none, set)
if not has_symbol_set:
raise ValueError(filldedent('''
Since there are no variables in the expression %s,
it cannot be differentiated.''' % expr))
# determine value for variables if it wasn't given
if not variables:
variables = expr.free_symbols
if len(variables) != 1:
if expr.is_number:
return S.Zero
if len(variables) == 0:
raise ValueError(filldedent('''
Since there are no variables in the expression,
the variable(s) of differentiation must be supplied
to differentiate %s''' % expr))
else:
raise ValueError(filldedent('''
Since there is more than one variable in the
expression, the variable(s) of differentiation
must be supplied to differentiate %s''' % expr))
# Standardize the variables by sympifying them:
variables = list(sympify(variables))
# Split the list of variables into a list of the variables we are diff
# wrt, where each element of the list has the form (s, count) where
# s is the entity to diff wrt and count is the order of the
# derivative.
variable_count = []
array_likes = (tuple, list, Tuple)
for i, v in enumerate(variables):
if isinstance(v, Integer):
if i == 0:
raise ValueError("First variable cannot be a number: %i" % v)
count = v
prev, prevcount = variable_count[-1]
if prevcount != 1:
raise TypeError("tuple {} followed by number {}".format((prev, prevcount), v))
if count == 0:
variable_count.pop()
else:
variable_count[-1] = Tuple(prev, count)
else:
if isinstance(v, array_likes):
if len(v) == 0:
# Ignore empty tuples: Derivative(expr, ... , (), ... )
continue
if isinstance(v[0], array_likes):
# Derive by array: Derivative(expr, ... , [[x, y, z]], ... )
if len(v) == 1:
v = Array(v[0])
count = 1
else:
v, count = v
v = Array(v)
else:
v, count = v
if count == 0:
continue
elif isinstance(v, UndefinedFunction):
raise TypeError(
"cannot differentiate wrt "
"UndefinedFunction: %s" % v)
else:
count = 1
variable_count.append(Tuple(v, count))
# light evaluation of contiguous, identical
# items: (x, 1), (x, 1) -> (x, 2)
merged = []
for t in variable_count:
v, c = t
if c.is_negative:
raise ValueError(
'order of differentiation must be nonnegative')
if merged and merged[-1][0] == v:
c += merged[-1][1]
if not c:
merged.pop()
else:
merged[-1] = Tuple(v, c)
else:
merged.append(t)
variable_count = merged
# sanity check of variables of differentation; we waited
# until the counts were computed since some variables may
# have been removed because the count was 0
for v, c in variable_count:
# v must have _diff_wrt True
if not v._diff_wrt:
__ = '' # filler to make error message neater
raise ValueError(filldedent('''
Can't calculate derivative wrt %s.%s''' % (v,
__)))
# We make a special case for 0th derivative, because there is no
# good way to unambiguously print this.
if len(variable_count) == 0:
return expr
evaluate = kwargs.get('evaluate', False)
if evaluate:
if isinstance(expr, Derivative):
expr = expr.canonical
variable_count = [
(v.canonical if isinstance(v, Derivative) else v, c)
for v, c in variable_count]
# Look for a quick exit if there are symbols that don't appear in
# expression at all. Note, this cannot check non-symbols like
# Derivatives as those can be created by intermediate
# derivatives.
zero = False
free = expr.free_symbols
for v, c in variable_count:
vfree = v.free_symbols
if c.is_positive and vfree:
if isinstance(v, AppliedUndef):
# these match exactly since
# x.diff(f(x)) == g(x).diff(f(x)) == 0
# and are not created by differentiation
D = Dummy()
if not expr.xreplace({v: D}).has(D):
zero = True
break
elif isinstance(v, MatrixExpr):
zero = False
break
elif isinstance(v, Symbol) and v not in free:
zero = True
break
else:
if not free & vfree:
# e.g. v is IndexedBase or Matrix
zero = True
break
if zero:
return cls._get_zero_with_shape_like(expr)
# make the order of symbols canonical
#TODO: check if assumption of discontinuous derivatives exist
variable_count = cls._sort_variable_count(variable_count)
# denest
if isinstance(expr, Derivative):
variable_count = list(expr.variable_count) + variable_count
expr = expr.expr
return _derivative_dispatch(expr, *variable_count, **kwargs)
# we return here if evaluate is False or if there is no
# _eval_derivative method
if not evaluate or not hasattr(expr, '_eval_derivative'):
# return an unevaluated Derivative
if evaluate and variable_count == [(expr, 1)] and expr.is_scalar:
# special hack providing evaluation for classes
# that have defined is_scalar=True but have no
# _eval_derivative defined
return S.One
return Expr.__new__(cls, expr, *variable_count)
# evaluate the derivative by calling _eval_derivative method
# of expr for each variable
# -------------------------------------------------------------
nderivs = 0 # how many derivatives were performed
unhandled = []
for i, (v, count) in enumerate(variable_count):
old_expr = expr
old_v = None
is_symbol = v.is_symbol or isinstance(v,
(Iterable, Tuple, MatrixCommon, NDimArray))
if not is_symbol:
old_v = v
v = Dummy('xi')
expr = expr.xreplace({old_v: v})
# Derivatives and UndefinedFunctions are independent
# of all others
clashing = not (isinstance(old_v, Derivative) or \
isinstance(old_v, AppliedUndef))
if not v in expr.free_symbols and not clashing:
return expr.diff(v) # expr's version of 0
if not old_v.is_scalar and not hasattr(
old_v, '_eval_derivative'):
# special hack providing evaluation for classes
# that have defined is_scalar=True but have no
# _eval_derivative defined
expr *= old_v.diff(old_v)
obj = cls._dispatch_eval_derivative_n_times(expr, v, count)
if obj is not None and obj.is_zero:
return obj
nderivs += count
if old_v is not None:
if obj is not None:
# remove the dummy that was used
obj = obj.subs(v, old_v)
# restore expr
expr = old_expr
if obj is None:
# we've already checked for quick-exit conditions
# that give 0 so the remaining variables
# are contained in the expression but the expression
# did not compute a derivative so we stop taking
# derivatives
unhandled = variable_count[i:]
break
expr = obj
# what we have so far can be made canonical
expr = expr.replace(
lambda x: isinstance(x, Derivative),
lambda x: x.canonical)
if unhandled:
if isinstance(expr, Derivative):
unhandled = list(expr.variable_count) + unhandled
expr = expr.expr
expr = Expr.__new__(cls, expr, *unhandled)
if (nderivs > 1) == True and kwargs.get('simplify', True):
from sympy.core.exprtools import factor_terms
from sympy.simplify.simplify import signsimp
expr = factor_terms(signsimp(expr))
return expr
@property
def canonical(cls):
return cls.func(cls.expr,
*Derivative._sort_variable_count(cls.variable_count))
@classmethod
def _sort_variable_count(cls, vc):
"""
Sort (variable, count) pairs into canonical order while
retaining order of variables that do not commute during
differentiation:
* symbols and functions commute with each other
* derivatives commute with each other
* a derivative doesn't commute with anything it contains
* any other object is not allowed to commute if it has
free symbols in common with another object
Examples
========
>>> from sympy import Derivative, Function, symbols
>>> vsort = Derivative._sort_variable_count
>>> x, y, z = symbols('x y z')
>>> f, g, h = symbols('f g h', cls=Function)
Contiguous items are collapsed into one pair:
>>> vsort([(x, 1), (x, 1)])
[(x, 2)]
>>> vsort([(y, 1), (f(x), 1), (y, 1), (f(x), 1)])
[(y, 2), (f(x), 2)]
Ordering is canonical.
>>> def vsort0(*v):
... # docstring helper to
... # change vi -> (vi, 0), sort, and return vi vals
... return [i[0] for i in vsort([(i, 0) for i in v])]
>>> vsort0(y, x)
[x, y]
>>> vsort0(g(y), g(x), f(y))
[f(y), g(x), g(y)]
Symbols are sorted as far to the left as possible but never
move to the left of a derivative having the same symbol in
its variables; the same applies to AppliedUndef which are
always sorted after Symbols:
>>> dfx = f(x).diff(x)
>>> assert vsort0(dfx, y) == [y, dfx]
>>> assert vsort0(dfx, x) == [dfx, x]
"""
from sympy.utilities.iterables import uniq, topological_sort
if not vc:
return []
vc = list(vc)
if len(vc) == 1:
return [Tuple(*vc[0])]
V = list(range(len(vc)))
E = []
v = lambda i: vc[i][0]
D = Dummy()
def _block(d, v, wrt=False):
# return True if v should not come before d else False
if d == v:
return wrt
if d.is_Symbol:
return False
if isinstance(d, Derivative):
# a derivative blocks if any of it's variables contain
# v; the wrt flag will return True for an exact match
# and will cause an AppliedUndef to block if v is in
# the arguments
if any(_block(k, v, wrt=True)
for k in d._wrt_variables):
return True
return False
if not wrt and isinstance(d, AppliedUndef):
return False
if v.is_Symbol:
return v in d.free_symbols
if isinstance(v, AppliedUndef):
return _block(d.xreplace({v: D}), D)
return d.free_symbols & v.free_symbols
for i in range(len(vc)):
for j in range(i):
if _block(v(j), v(i)):
E.append((j,i))
# this is the default ordering to use in case of ties
O = dict(zip(ordered(uniq([i for i, c in vc])), range(len(vc))))
ix = topological_sort((V, E), key=lambda i: O[v(i)])
# merge counts of contiguously identical items
merged = []
for v, c in [vc[i] for i in ix]:
if merged and merged[-1][0] == v:
merged[-1][1] += c
else:
merged.append([v, c])
return [Tuple(*i) for i in merged]
def _eval_is_commutative(self):
return self.expr.is_commutative
def _eval_derivative(self, v):
# If v (the variable of differentiation) is not in
# self.variables, we might be able to take the derivative.
if v not in self._wrt_variables:
dedv = self.expr.diff(v)
if isinstance(dedv, Derivative):
return dedv.func(dedv.expr, *(self.variable_count + dedv.variable_count))
# dedv (d(self.expr)/dv) could have simplified things such that the
# derivative wrt things in self.variables can now be done. Thus,
# we set evaluate=True to see if there are any other derivatives
# that can be done. The most common case is when dedv is a simple
# number so that the derivative wrt anything else will vanish.
return self.func(dedv, *self.variables, evaluate=True)
# In this case v was in self.variables so the derivative wrt v has
# already been attempted and was not computed, either because it
# couldn't be or evaluate=False originally.
variable_count = list(self.variable_count)
variable_count.append((v, 1))
return self.func(self.expr, *variable_count, evaluate=False)
def doit(self, **hints):
expr = self.expr
if hints.get('deep', True):
expr = expr.doit(**hints)
hints['evaluate'] = True
rv = self.func(expr, *self.variable_count, **hints)
if rv!= self and rv.has(Derivative):
rv = rv.doit(**hints)
return rv
@_sympifyit('z0', NotImplementedError)
def doit_numerically(self, z0):
"""
Evaluate the derivative at z numerically.
When we can represent derivatives at a point, this should be folded
into the normal evalf. For now, we need a special method.
"""
if len(self.free_symbols) != 1 or len(self.variables) != 1:
raise NotImplementedError('partials and higher order derivatives')
z = list(self.free_symbols)[0]
def eval(x):
f0 = self.expr.subs(z, Expr._from_mpmath(x, prec=mpmath.mp.prec))
f0 = f0.evalf(mlib.libmpf.prec_to_dps(mpmath.mp.prec))
return f0._to_mpmath(mpmath.mp.prec)
return Expr._from_mpmath(mpmath.diff(eval,
z0._to_mpmath(mpmath.mp.prec)),
mpmath.mp.prec)
@property
def expr(self):
return self._args[0]
@property
def _wrt_variables(self):
# return the variables of differentiation without
# respect to the type of count (int or symbolic)
return [i[0] for i in self.variable_count]
@property
def variables(self):
# TODO: deprecate? YES, make this 'enumerated_variables' and
# name _wrt_variables as variables
# TODO: support for `d^n`?
rv = []
for v, count in self.variable_count:
if not count.is_Integer:
raise TypeError(filldedent('''
Cannot give expansion for symbolic count. If you just
want a list of all variables of differentiation, use
_wrt_variables.'''))
rv.extend([v]*count)
return tuple(rv)
@property
def variable_count(self):
return self._args[1:]
@property
def derivative_count(self):
return sum([count for var, count in self.variable_count], 0)
@property
def free_symbols(self):
ret = self.expr.free_symbols
# Add symbolic counts to free_symbols
for var, count in self.variable_count:
ret.update(count.free_symbols)
return ret
def _eval_subs(self, old, new):
# The substitution (old, new) cannot be done inside
# Derivative(expr, vars) for a variety of reasons
# as handled below.
if old in self._wrt_variables:
# first handle the counts
expr = self.func(self.expr, *[(v, c.subs(old, new))
for v, c in self.variable_count])
if expr != self:
return expr._eval_subs(old, new)
# quick exit case
if not getattr(new, '_diff_wrt', False):
# case (0): new is not a valid variable of
# differentiation
if isinstance(old, Symbol):
# don't introduce a new symbol if the old will do
return Subs(self, old, new)
else:
xi = Dummy('xi')
return Subs(self.xreplace({old: xi}), xi, new)
# If both are Derivatives with the same expr, check if old is
# equivalent to self or if old is a subderivative of self.
if old.is_Derivative and old.expr == self.expr:
if self.canonical == old.canonical:
return new
# collections.Counter doesn't have __le__
def _subset(a, b):
return all((a[i] <= b[i]) == True for i in a)
old_vars = Counter(dict(reversed(old.variable_count)))
self_vars = Counter(dict(reversed(self.variable_count)))
if _subset(old_vars, self_vars):
return _derivative_dispatch(new, *(self_vars - old_vars).items()).canonical
args = list(self.args)
newargs = list(x._subs(old, new) for x in args)
if args[0] == old:
# complete replacement of self.expr
# we already checked that the new is valid so we know
# it won't be a problem should it appear in variables
return _derivative_dispatch(*newargs)
if newargs[0] != args[0]:
# case (1) can't change expr by introducing something that is in
# the _wrt_variables if it was already in the expr
# e.g.
# for Derivative(f(x, g(y)), y), x cannot be replaced with
# anything that has y in it; for f(g(x), g(y)).diff(g(y))
# g(x) cannot be replaced with anything that has g(y)
syms = {vi: Dummy() for vi in self._wrt_variables
if not vi.is_Symbol}
wrt = {syms.get(vi, vi) for vi in self._wrt_variables}
forbidden = args[0].xreplace(syms).free_symbols & wrt
nfree = new.xreplace(syms).free_symbols
ofree = old.xreplace(syms).free_symbols
if (nfree - ofree) & forbidden:
return Subs(self, old, new)
viter = ((i, j) for ((i, _), (j, _)) in zip(newargs[1:], args[1:]))
if any(i != j for i, j in viter): # a wrt-variable change
# case (2) can't change vars by introducing a variable
# that is contained in expr, e.g.
# for Derivative(f(z, g(h(x), y)), y), y cannot be changed to
# x, h(x), or g(h(x), y)
for a in _atomic(self.expr, recursive=True):
for i in range(1, len(newargs)):
vi, _ = newargs[i]
if a == vi and vi != args[i][0]:
return Subs(self, old, new)
# more arg-wise checks
vc = newargs[1:]
oldv = self._wrt_variables
newe = self.expr
subs = []
for i, (vi, ci) in enumerate(vc):
if not vi._diff_wrt:
# case (3) invalid differentiation expression so
# create a replacement dummy
xi = Dummy('xi_%i' % i)
# replace the old valid variable with the dummy
# in the expression
newe = newe.xreplace({oldv[i]: xi})
# and replace the bad variable with the dummy
vc[i] = (xi, ci)
# and record the dummy with the new (invalid)
# differentiation expression
subs.append((xi, vi))
if subs:
# handle any residual substitution in the expression
newe = newe._subs(old, new)
# return the Subs-wrapped derivative
return Subs(Derivative(newe, *vc), *zip(*subs))
# everything was ok
return _derivative_dispatch(*newargs)
def _eval_lseries(self, x, logx, cdir=0):
dx = self.variables
for term in self.expr.lseries(x, logx=logx, cdir=cdir):
yield self.func(term, *dx)
def _eval_nseries(self, x, n, logx, cdir=0):
arg = self.expr.nseries(x, n=n, logx=logx)
o = arg.getO()
dx = self.variables
rv = [self.func(a, *dx) for a in Add.make_args(arg.removeO())]
if o:
rv.append(o/x)
return Add(*rv)
def _eval_as_leading_term(self, x, cdir=0):
series_gen = self.expr.lseries(x)
d = S.Zero
for leading_term in series_gen:
d = diff(leading_term, *self.variables)
if d != 0:
break
return d
def _sage_(self):
import sage.all as sage
args = [arg._sage_() for arg in self.args]
return sage.derivative(*args)
def as_finite_difference(self, points=1, x0=None, wrt=None):
""" Expresses a Derivative instance as a finite difference.
Parameters
==========
points : sequence or coefficient, optional
If sequence: discrete values (length >= order+1) of the
independent variable used for generating the finite
difference weights.
If it is a coefficient, it will be used as the step-size
for generating an equidistant sequence of length order+1
centered around ``x0``. Default: 1 (step-size 1)
x0 : number or Symbol, optional
the value of the independent variable (``wrt``) at which the
derivative is to be approximated. Default: same as ``wrt``.
wrt : Symbol, optional
"with respect to" the variable for which the (partial)
derivative is to be approximated for. If not provided it
is required that the derivative is ordinary. Default: ``None``.
Examples
========
>>> from sympy import symbols, Function, exp, sqrt, Symbol
>>> x, h = symbols('x h')
>>> f = Function('f')
>>> f(x).diff(x).as_finite_difference()
-f(x - 1/2) + f(x + 1/2)
The default step size and number of points are 1 and
``order + 1`` respectively. We can change the step size by
passing a symbol as a parameter:
>>> f(x).diff(x).as_finite_difference(h)
-f(-h/2 + x)/h + f(h/2 + x)/h
We can also specify the discretized values to be used in a
sequence:
>>> f(x).diff(x).as_finite_difference([x, x+h, x+2*h])
-3*f(x)/(2*h) + 2*f(h + x)/h - f(2*h + x)/(2*h)
The algorithm is not restricted to use equidistant spacing, nor
do we need to make the approximation around ``x0``, but we can get
an expression estimating the derivative at an offset:
>>> e, sq2 = exp(1), sqrt(2)
>>> xl = [x-h, x+h, x+e*h]
>>> f(x).diff(x, 1).as_finite_difference(xl, x+h*sq2) # doctest: +ELLIPSIS
2*h*((h + sqrt(2)*h)/(2*h) - (-sqrt(2)*h + h)/(2*h))*f(E*h + x)/...
To approximate ``Derivative`` around ``x0`` using a non-equidistant
spacing step, the algorithm supports assignment of undefined
functions to ``points``:
>>> dx = Function('dx')
>>> f(x).diff(x).as_finite_difference(points=dx(x), x0=x-h)
-f(-h + x - dx(-h + x)/2)/dx(-h + x) + f(-h + x + dx(-h + x)/2)/dx(-h + x)
Partial derivatives are also supported:
>>> y = Symbol('y')
>>> d2fdxdy=f(x,y).diff(x,y)
>>> d2fdxdy.as_finite_difference(wrt=x)
-Derivative(f(x - 1/2, y), y) + Derivative(f(x + 1/2, y), y)
We can apply ``as_finite_difference`` to ``Derivative`` instances in
compound expressions using ``replace``:
>>> (1 + 42**f(x).diff(x)).replace(lambda arg: arg.is_Derivative,
... lambda arg: arg.as_finite_difference())
42**(-f(x - 1/2) + f(x + 1/2)) + 1
See also
========
sympy.calculus.finite_diff.apply_finite_diff
sympy.calculus.finite_diff.differentiate_finite
sympy.calculus.finite_diff.finite_diff_weights
"""
from ..calculus.finite_diff import _as_finite_diff
return _as_finite_diff(self, points, x0, wrt)
@classmethod
def _get_zero_with_shape_like(cls, expr):
return S.Zero
@classmethod
def _dispatch_eval_derivative_n_times(cls, expr, v, count):
# Evaluate the derivative `n` times. If
# `_eval_derivative_n_times` is not overridden by the current
# object, the default in `Basic` will call a loop over
# `_eval_derivative`:
return expr._eval_derivative_n_times(v, count)
def _derivative_dispatch(expr, *variables, **kwargs):
from sympy.matrices.common import MatrixCommon
from sympy import MatrixExpr
from sympy import NDimArray
array_types = (MatrixCommon, MatrixExpr, NDimArray, list, tuple, Tuple)
if isinstance(expr, array_types) or any(isinstance(i[0], array_types) if isinstance(i, (tuple, list, Tuple)) else isinstance(i, array_types) for i in variables):
from sympy.tensor.array.array_derivatives import ArrayDerivative
return ArrayDerivative(expr, *variables, **kwargs)
return Derivative(expr, *variables, **kwargs)
class Lambda(Expr):
"""
Lambda(x, expr) represents a lambda function similar to Python's
'lambda x: expr'. A function of several variables is written as
Lambda((x, y, ...), expr).
Examples
========
A simple example:
>>> from sympy import Lambda
>>> from sympy.abc import x
>>> f = Lambda(x, x**2)
>>> f(4)
16
For multivariate functions, use:
>>> from sympy.abc import y, z, t
>>> f2 = Lambda((x, y, z, t), x + y**z + t**z)
>>> f2(1, 2, 3, 4)
73
It is also possible to unpack tuple arguments:
>>> f = Lambda( ((x, y), z) , x + y + z)
>>> f((1, 2), 3)
6
A handy shortcut for lots of arguments:
>>> p = x, y, z
>>> f = Lambda(p, x + y*z)
>>> f(*p)
x + y*z
"""
is_Function = True
def __new__(cls, signature, expr):
if iterable(signature) and not isinstance(signature, (tuple, Tuple)):
SymPyDeprecationWarning(
feature="non tuple iterable of argument symbols to Lambda",
useinstead="tuple of argument symbols",
issue=17474,
deprecated_since_version="1.5").warn()
signature = tuple(signature)
sig = signature if iterable(signature) else (signature,)
sig = sympify(sig)
cls._check_signature(sig)
if len(sig) == 1 and sig[0] == expr:
return S.IdentityFunction
return Expr.__new__(cls, sig, sympify(expr))
@classmethod
def _check_signature(cls, sig):
syms = set()
def rcheck(args):
for a in args:
if a.is_symbol:
if a in syms:
raise BadSignatureError("Duplicate symbol %s" % a)
syms.add(a)
elif isinstance(a, Tuple):
rcheck(a)
else:
raise BadSignatureError("Lambda signature should be only tuples"
" and symbols, not %s" % a)
if not isinstance(sig, Tuple):
raise BadSignatureError("Lambda signature should be a tuple not %s" % sig)
# Recurse through the signature:
rcheck(sig)
@property
def signature(self):
"""The expected form of the arguments to be unpacked into variables"""
return self._args[0]
@property
def expr(self):
"""The return value of the function"""
return self._args[1]
@property
def variables(self):
"""The variables used in the internal representation of the function"""
def _variables(args):
if isinstance(args, Tuple):
for arg in args:
yield from _variables(arg)
else:
yield args
return tuple(_variables(self.signature))
@property
def nargs(self):
from sympy.sets.sets import FiniteSet
return FiniteSet(len(self.signature))
bound_symbols = variables
@property
def free_symbols(self):
return self.expr.free_symbols - set(self.variables)
def __call__(self, *args):
n = len(args)
if n not in self.nargs: # Lambda only ever has 1 value in nargs
# XXX: exception message must be in exactly this format to
# make it work with NumPy's functions like vectorize(). See,
# for example, https://github.com/numpy/numpy/issues/1697.
# The ideal solution would be just to attach metadata to
# the exception and change NumPy to take advantage of this.
## XXX does this apply to Lambda? If not, remove this comment.
temp = ('%(name)s takes exactly %(args)s '
'argument%(plural)s (%(given)s given)')
raise BadArgumentsError(temp % {
'name': self,
'args': list(self.nargs)[0],
'plural': 's'*(list(self.nargs)[0] != 1),
'given': n})
d = self._match_signature(self.signature, args)
return self.expr.xreplace(d)
def _match_signature(self, sig, args):
symargmap = {}
def rmatch(pars, args):
for par, arg in zip(pars, args):
if par.is_symbol:
symargmap[par] = arg
elif isinstance(par, Tuple):
if not isinstance(arg, (tuple, Tuple)) or len(args) != len(pars):
raise BadArgumentsError("Can't match %s and %s" % (args, pars))
rmatch(par, arg)
rmatch(sig, args)
return symargmap
@property
def is_identity(self):
"""Return ``True`` if this ``Lambda`` is an identity function. """
return self.signature == self.expr
def _eval_evalf(self, prec):
from sympy.core.evalf import prec_to_dps
return self.func(self.args[0], self.args[1].evalf(n=prec_to_dps(prec)))
class Subs(Expr):
"""
Represents unevaluated substitutions of an expression.
``Subs(expr, x, x0)`` represents the expression resulting
from substituting x with x0 in expr.
Parameters
==========
expr : Expr
An expression.
x : tuple, variable
A variable or list of distinct variables.
x0 : tuple or list of tuples
A point or list of evaluation points
corresponding to those variables.
Notes
=====
``Subs`` objects are generally useful to represent unevaluated derivatives
calculated at a point.
The variables may be expressions, but they are subjected to the limitations
of subs(), so it is usually a good practice to use only symbols for
variables, since in that case there can be no ambiguity.
There's no automatic expansion - use the method .doit() to effect all
possible substitutions of the object and also of objects inside the
expression.
When evaluating derivatives at a point that is not a symbol, a Subs object
is returned. One is also able to calculate derivatives of Subs objects - in
this case the expression is always expanded (for the unevaluated form, use
Derivative()).
Examples
========
>>> from sympy import Subs, Function, sin, cos
>>> from sympy.abc import x, y, z
>>> f = Function('f')
Subs are created when a particular substitution cannot be made. The
x in the derivative cannot be replaced with 0 because 0 is not a
valid variables of differentiation:
>>> f(x).diff(x).subs(x, 0)
Subs(Derivative(f(x), x), x, 0)
Once f is known, the derivative and evaluation at 0 can be done:
>>> _.subs(f, sin).doit() == sin(x).diff(x).subs(x, 0) == cos(0)
True
Subs can also be created directly with one or more variables:
>>> Subs(f(x)*sin(y) + z, (x, y), (0, 1))
Subs(z + f(x)*sin(y), (x, y), (0, 1))
>>> _.doit()
z + f(0)*sin(1)
Notes
=====
In order to allow expressions to combine before doit is done, a
representation of the Subs expression is used internally to make
expressions that are superficially different compare the same:
>>> a, b = Subs(x, x, 0), Subs(y, y, 0)
>>> a + b
2*Subs(x, x, 0)
This can lead to unexpected consequences when using methods
like `has` that are cached:
>>> s = Subs(x, x, 0)
>>> s.has(x), s.has(y)
(True, False)
>>> ss = s.subs(x, y)
>>> ss.has(x), ss.has(y)
(True, False)
>>> s, ss
(Subs(x, x, 0), Subs(y, y, 0))
"""
def __new__(cls, expr, variables, point, **assumptions):
from sympy import Symbol
if not is_sequence(variables, Tuple):
variables = [variables]
variables = Tuple(*variables)
if has_dups(variables):
repeated = [str(v) for v, i in Counter(variables).items() if i > 1]
__ = ', '.join(repeated)
raise ValueError(filldedent('''
The following expressions appear more than once: %s
''' % __))
point = Tuple(*(point if is_sequence(point, Tuple) else [point]))
if len(point) != len(variables):
raise ValueError('Number of point values must be the same as '
'the number of variables.')
if not point:
return sympify(expr)
# denest
if isinstance(expr, Subs):
variables = expr.variables + variables
point = expr.point + point
expr = expr.expr
else:
expr = sympify(expr)
# use symbols with names equal to the point value (with prepended _)
# to give a variable-independent expression
pre = "_"
pts = sorted(set(point), key=default_sort_key)
from sympy.printing import StrPrinter
class CustomStrPrinter(StrPrinter):
def _print_Dummy(self, expr):
return str(expr) + str(expr.dummy_index)
def mystr(expr, **settings):
p = CustomStrPrinter(settings)
return p.doprint(expr)
while 1:
s_pts = {p: Symbol(pre + mystr(p)) for p in pts}
reps = [(v, s_pts[p])
for v, p in zip(variables, point)]
# if any underscore-prepended symbol is already a free symbol
# and is a variable with a different point value, then there
# is a clash, e.g. _0 clashes in Subs(_0 + _1, (_0, _1), (1, 0))
# because the new symbol that would be created is _1 but _1
# is already mapped to 0 so __0 and __1 are used for the new
# symbols
if any(r in expr.free_symbols and
r in variables and
Symbol(pre + mystr(point[variables.index(r)])) != r
for _, r in reps):
pre += "_"
continue
break
obj = Expr.__new__(cls, expr, Tuple(*variables), point)
obj._expr = expr.xreplace(dict(reps))
return obj
def _eval_is_commutative(self):
return self.expr.is_commutative
def doit(self, **hints):
e, v, p = self.args
# remove self mappings
for i, (vi, pi) in enumerate(zip(v, p)):
if vi == pi:
v = v[:i] + v[i + 1:]
p = p[:i] + p[i + 1:]
if not v:
return self.expr
if isinstance(e, Derivative):
# apply functions first, e.g. f -> cos
undone = []
for i, vi in enumerate(v):
if isinstance(vi, FunctionClass):
e = e.subs(vi, p[i])
else:
undone.append((vi, p[i]))
if not isinstance(e, Derivative):
e = e.doit()
if isinstance(e, Derivative):
# do Subs that aren't related to differentiation
undone2 = []
D = Dummy()
arg = e.args[0]
for vi, pi in undone:
if D not in e.xreplace({vi: D}).free_symbols:
if arg.has(vi):
e = e.subs(vi, pi)
else:
undone2.append((vi, pi))
undone = undone2
# differentiate wrt variables that are present
wrt = []
D = Dummy()
expr = e.expr
free = expr.free_symbols
for vi, ci in e.variable_count:
if isinstance(vi, Symbol) and vi in free:
expr = expr.diff((vi, ci))
elif D in expr.subs(vi, D).free_symbols:
expr = expr.diff((vi, ci))
else:
wrt.append((vi, ci))
# inject remaining subs
rv = expr.subs(undone)
# do remaining differentiation *in order given*
for vc in wrt:
rv = rv.diff(vc)
else:
# inject remaining subs
rv = e.subs(undone)
else:
rv = e.doit(**hints).subs(list(zip(v, p)))
if hints.get('deep', True) and rv != self:
rv = rv.doit(**hints)
return rv
def evalf(self, prec=None, **options):
return self.doit().evalf(prec, **options)
n = evalf
@property
def variables(self):
"""The variables to be evaluated"""
return self._args[1]
bound_symbols = variables
@property
def expr(self):
"""The expression on which the substitution operates"""
return self._args[0]
@property
def point(self):
"""The values for which the variables are to be substituted"""
return self._args[2]
@property
def free_symbols(self):
return (self.expr.free_symbols - set(self.variables) |
set(self.point.free_symbols))
@property
def expr_free_symbols(self):
return (self.expr.expr_free_symbols - set(self.variables) |
set(self.point.expr_free_symbols))
def __eq__(self, other):
if not isinstance(other, Subs):
return False
return self._hashable_content() == other._hashable_content()
def __ne__(self, other):
return not(self == other)
def __hash__(self):
return super().__hash__()
def _hashable_content(self):
return (self._expr.xreplace(self.canonical_variables),
) + tuple(ordered([(v, p) for v, p in
zip(self.variables, self.point) if not self.expr.has(v)]))
def _eval_subs(self, old, new):
# Subs doit will do the variables in order; the semantics
# of subs for Subs is have the following invariant for
# Subs object foo:
# foo.doit().subs(reps) == foo.subs(reps).doit()
pt = list(self.point)
if old in self.variables:
if _atomic(new) == {new} and not any(
i.has(new) for i in self.args):
# the substitution is neutral
return self.xreplace({old: new})
# any occurrence of old before this point will get
# handled by replacements from here on
i = self.variables.index(old)
for j in range(i, len(self.variables)):
pt[j] = pt[j]._subs(old, new)
return self.func(self.expr, self.variables, pt)
v = [i._subs(old, new) for i in self.variables]
if v != list(self.variables):
return self.func(self.expr, self.variables + (old,), pt + [new])
expr = self.expr._subs(old, new)
pt = [i._subs(old, new) for i in self.point]
return self.func(expr, v, pt)
def _eval_derivative(self, s):
# Apply the chain rule of the derivative on the substitution variables:
val = Add.fromiter(p.diff(s) * Subs(self.expr.diff(v), self.variables, self.point).doit() for v, p in zip(self.variables, self.point))
# Check if there are free symbols in `self.expr`:
# First get the `expr_free_symbols`, which returns the free symbols
# that are directly contained in an expression node (i.e. stop
# searching if the node isn't an expression). At this point turn the
# expressions into `free_symbols` and check if there are common free
# symbols in `self.expr` and the deriving factor.
fs1 = {j for i in self.expr_free_symbols for j in i.free_symbols}
if len(fs1 & s.free_symbols) > 0:
val += Subs(self.expr.diff(s), self.variables, self.point).doit()
return val
def _eval_nseries(self, x, n, logx, cdir=0):
if x in self.point:
# x is the variable being substituted into
apos = self.point.index(x)
other = self.variables[apos]
else:
other = x
arg = self.expr.nseries(other, n=n, logx=logx)
o = arg.getO()
terms = Add.make_args(arg.removeO())
rv = Add(*[self.func(a, *self.args[1:]) for a in terms])
if o:
rv += o.subs(other, x)
return rv
def _eval_as_leading_term(self, x, cdir=0):
if x in self.point:
ipos = self.point.index(x)
xvar = self.variables[ipos]
return self.expr.as_leading_term(xvar)
if x in self.variables:
# if `x` is a dummy variable, it means it won't exist after the
# substitution has been performed:
return self
# The variable is independent of the substitution:
return self.expr.as_leading_term(x)
def diff(f, *symbols, **kwargs):
"""
Differentiate f with respect to symbols.
Explanation
===========
This is just a wrapper to unify .diff() and the Derivative class; its
interface is similar to that of integrate(). You can use the same
shortcuts for multiple variables as with Derivative. For example,
diff(f(x), x, x, x) and diff(f(x), x, 3) both return the third derivative
of f(x).
You can pass evaluate=False to get an unevaluated Derivative class. Note
that if there are 0 symbols (such as diff(f(x), x, 0), then the result will
be the function (the zeroth derivative), even if evaluate=False.
Examples
========
>>> from sympy import sin, cos, Function, diff
>>> from sympy.abc import x, y
>>> f = Function('f')
>>> diff(sin(x), x)
cos(x)
>>> diff(f(x), x, x, x)
Derivative(f(x), (x, 3))
>>> diff(f(x), x, 3)
Derivative(f(x), (x, 3))
>>> diff(sin(x)*cos(y), x, 2, y, 2)
sin(x)*cos(y)
>>> type(diff(sin(x), x))
cos
>>> type(diff(sin(x), x, evaluate=False))
<class 'sympy.core.function.Derivative'>
>>> type(diff(sin(x), x, 0))
sin
>>> type(diff(sin(x), x, 0, evaluate=False))
sin
>>> diff(sin(x))
cos(x)
>>> diff(sin(x*y))
Traceback (most recent call last):
...
ValueError: specify differentiation variables to differentiate sin(x*y)
Note that ``diff(sin(x))`` syntax is meant only for convenience
in interactive sessions and should be avoided in library code.
References
==========
http://reference.wolfram.com/legacy/v5_2/Built-inFunctions/AlgebraicComputation/Calculus/D.html
See Also
========
Derivative
idiff: computes the derivative implicitly
"""
if hasattr(f, 'diff'):
return f.diff(*symbols, **kwargs)
kwargs.setdefault('evaluate', True)
return _derivative_dispatch(f, *symbols, **kwargs)
def expand(e, deep=True, modulus=None, power_base=True, power_exp=True,
mul=True, log=True, multinomial=True, basic=True, **hints):
r"""
Expand an expression using methods given as hints.
Explanation
===========
Hints evaluated unless explicitly set to False are: ``basic``, ``log``,
``multinomial``, ``mul``, ``power_base``, and ``power_exp`` The following
hints are supported but not applied unless set to True: ``complex``,
``func``, and ``trig``. In addition, the following meta-hints are
supported by some or all of the other hints: ``frac``, ``numer``,
``denom``, ``modulus``, and ``force``. ``deep`` is supported by all
hints. Additionally, subclasses of Expr may define their own hints or
meta-hints.
The ``basic`` hint is used for any special rewriting of an object that
should be done automatically (along with the other hints like ``mul``)
when expand is called. This is a catch-all hint to handle any sort of
expansion that may not be described by the existing hint names. To use
this hint an object should override the ``_eval_expand_basic`` method.
Objects may also define their own expand methods, which are not run by
default. See the API section below.
If ``deep`` is set to ``True`` (the default), things like arguments of
functions are recursively expanded. Use ``deep=False`` to only expand on
the top level.
If the ``force`` hint is used, assumptions about variables will be ignored
in making the expansion.
Hints
=====
These hints are run by default
mul
---
Distributes multiplication over addition:
>>> from sympy import cos, exp, sin
>>> from sympy.abc import x, y, z
>>> (y*(x + z)).expand(mul=True)
x*y + y*z
multinomial
-----------
Expand (x + y + ...)**n where n is a positive integer.
>>> ((x + y + z)**2).expand(multinomial=True)
x**2 + 2*x*y + 2*x*z + y**2 + 2*y*z + z**2
power_exp
---------
Expand addition in exponents into multiplied bases.
>>> exp(x + y).expand(power_exp=True)
exp(x)*exp(y)
>>> (2**(x + y)).expand(power_exp=True)
2**x*2**y
power_base
----------
Split powers of multiplied bases.
This only happens by default if assumptions allow, or if the
``force`` meta-hint is used:
>>> ((x*y)**z).expand(power_base=True)
(x*y)**z
>>> ((x*y)**z).expand(power_base=True, force=True)
x**z*y**z
>>> ((2*y)**z).expand(power_base=True)
2**z*y**z
Note that in some cases where this expansion always holds, SymPy performs
it automatically:
>>> (x*y)**2
x**2*y**2
log
---
Pull out power of an argument as a coefficient and split logs products
into sums of logs.
Note that these only work if the arguments of the log function have the
proper assumptions--the arguments must be positive and the exponents must
be real--or else the ``force`` hint must be True:
>>> from sympy import log, symbols
>>> log(x**2*y).expand(log=True)
log(x**2*y)
>>> log(x**2*y).expand(log=True, force=True)
2*log(x) + log(y)
>>> x, y = symbols('x,y', positive=True)
>>> log(x**2*y).expand(log=True)
2*log(x) + log(y)
basic
-----
This hint is intended primarily as a way for custom subclasses to enable
expansion by default.
These hints are not run by default:
complex
-------
Split an expression into real and imaginary parts.
>>> x, y = symbols('x,y')
>>> (x + y).expand(complex=True)
re(x) + re(y) + I*im(x) + I*im(y)
>>> cos(x).expand(complex=True)
-I*sin(re(x))*sinh(im(x)) + cos(re(x))*cosh(im(x))
Note that this is just a wrapper around ``as_real_imag()``. Most objects
that wish to redefine ``_eval_expand_complex()`` should consider
redefining ``as_real_imag()`` instead.
func
----
Expand other functions.
>>> from sympy import gamma
>>> gamma(x + 1).expand(func=True)
x*gamma(x)
trig
----
Do trigonometric expansions.
>>> cos(x + y).expand(trig=True)
-sin(x)*sin(y) + cos(x)*cos(y)
>>> sin(2*x).expand(trig=True)
2*sin(x)*cos(x)
Note that the forms of ``sin(n*x)`` and ``cos(n*x)`` in terms of ``sin(x)``
and ``cos(x)`` are not unique, due to the identity `\sin^2(x) + \cos^2(x)
= 1`. The current implementation uses the form obtained from Chebyshev
polynomials, but this may change. See `this MathWorld article
<http://mathworld.wolfram.com/Multiple-AngleFormulas.html>`_ for more
information.
Notes
=====
- You can shut off unwanted methods::
>>> (exp(x + y)*(x + y)).expand()
x*exp(x)*exp(y) + y*exp(x)*exp(y)
>>> (exp(x + y)*(x + y)).expand(power_exp=False)
x*exp(x + y) + y*exp(x + y)
>>> (exp(x + y)*(x + y)).expand(mul=False)
(x + y)*exp(x)*exp(y)
- Use deep=False to only expand on the top level::
>>> exp(x + exp(x + y)).expand()
exp(x)*exp(exp(x)*exp(y))
>>> exp(x + exp(x + y)).expand(deep=False)
exp(x)*exp(exp(x + y))
- Hints are applied in an arbitrary, but consistent order (in the current
implementation, they are applied in alphabetical order, except
multinomial comes before mul, but this may change). Because of this,
some hints may prevent expansion by other hints if they are applied
first. For example, ``mul`` may distribute multiplications and prevent
``log`` and ``power_base`` from expanding them. Also, if ``mul`` is
applied before ``multinomial`, the expression might not be fully
distributed. The solution is to use the various ``expand_hint`` helper
functions or to use ``hint=False`` to this function to finely control
which hints are applied. Here are some examples::
>>> from sympy import expand, expand_mul, expand_power_base
>>> x, y, z = symbols('x,y,z', positive=True)
>>> expand(log(x*(y + z)))
log(x) + log(y + z)
Here, we see that ``log`` was applied before ``mul``. To get the mul
expanded form, either of the following will work::
>>> expand_mul(log(x*(y + z)))
log(x*y + x*z)
>>> expand(log(x*(y + z)), log=False)
log(x*y + x*z)
A similar thing can happen with the ``power_base`` hint::
>>> expand((x*(y + z))**x)
(x*y + x*z)**x
To get the ``power_base`` expanded form, either of the following will
work::
>>> expand((x*(y + z))**x, mul=False)
x**x*(y + z)**x
>>> expand_power_base((x*(y + z))**x)
x**x*(y + z)**x
>>> expand((x + y)*y/x)
y + y**2/x
The parts of a rational expression can be targeted::
>>> expand((x + y)*y/x/(x + 1), frac=True)
(x*y + y**2)/(x**2 + x)
>>> expand((x + y)*y/x/(x + 1), numer=True)
(x*y + y**2)/(x*(x + 1))
>>> expand((x + y)*y/x/(x + 1), denom=True)
y*(x + y)/(x**2 + x)
- The ``modulus`` meta-hint can be used to reduce the coefficients of an
expression post-expansion::
>>> expand((3*x + 1)**2)
9*x**2 + 6*x + 1
>>> expand((3*x + 1)**2, modulus=5)
4*x**2 + x + 1
- Either ``expand()`` the function or ``.expand()`` the method can be
used. Both are equivalent::
>>> expand((x + 1)**2)
x**2 + 2*x + 1
>>> ((x + 1)**2).expand()
x**2 + 2*x + 1
API
===
Objects can define their own expand hints by defining
``_eval_expand_hint()``. The function should take the form::
def _eval_expand_hint(self, **hints):
# Only apply the method to the top-level expression
...
See also the example below. Objects should define ``_eval_expand_hint()``
methods only if ``hint`` applies to that specific object. The generic
``_eval_expand_hint()`` method defined in Expr will handle the no-op case.
Each hint should be responsible for expanding that hint only.
Furthermore, the expansion should be applied to the top-level expression
only. ``expand()`` takes care of the recursion that happens when
``deep=True``.
You should only call ``_eval_expand_hint()`` methods directly if you are
100% sure that the object has the method, as otherwise you are liable to
get unexpected ``AttributeError``s. Note, again, that you do not need to
recursively apply the hint to args of your object: this is handled
automatically by ``expand()``. ``_eval_expand_hint()`` should
generally not be used at all outside of an ``_eval_expand_hint()`` method.
If you want to apply a specific expansion from within another method, use
the public ``expand()`` function, method, or ``expand_hint()`` functions.
In order for expand to work, objects must be rebuildable by their args,
i.e., ``obj.func(*obj.args) == obj`` must hold.
Expand methods are passed ``**hints`` so that expand hints may use
'metahints'--hints that control how different expand methods are applied.
For example, the ``force=True`` hint described above that causes
``expand(log=True)`` to ignore assumptions is such a metahint. The
``deep`` meta-hint is handled exclusively by ``expand()`` and is not
passed to ``_eval_expand_hint()`` methods.
Note that expansion hints should generally be methods that perform some
kind of 'expansion'. For hints that simply rewrite an expression, use the
.rewrite() API.
Examples
========
>>> from sympy import Expr, sympify
>>> class MyClass(Expr):
... def __new__(cls, *args):
... args = sympify(args)
... return Expr.__new__(cls, *args)
...
... def _eval_expand_double(self, *, force=False, **hints):
... '''
... Doubles the args of MyClass.
...
... If there more than four args, doubling is not performed,
... unless force=True is also used (False by default).
... '''
... if not force and len(self.args) > 4:
... return self
... return self.func(*(self.args + self.args))
...
>>> a = MyClass(1, 2, MyClass(3, 4))
>>> a
MyClass(1, 2, MyClass(3, 4))
>>> a.expand(double=True)
MyClass(1, 2, MyClass(3, 4, 3, 4), 1, 2, MyClass(3, 4, 3, 4))
>>> a.expand(double=True, deep=False)
MyClass(1, 2, MyClass(3, 4), 1, 2, MyClass(3, 4))
>>> b = MyClass(1, 2, 3, 4, 5)
>>> b.expand(double=True)
MyClass(1, 2, 3, 4, 5)
>>> b.expand(double=True, force=True)
MyClass(1, 2, 3, 4, 5, 1, 2, 3, 4, 5)
See Also
========
expand_log, expand_mul, expand_multinomial, expand_complex, expand_trig,
expand_power_base, expand_power_exp, expand_func, sympy.simplify.hyperexpand.hyperexpand
"""
# don't modify this; modify the Expr.expand method
hints['power_base'] = power_base
hints['power_exp'] = power_exp
hints['mul'] = mul
hints['log'] = log
hints['multinomial'] = multinomial
hints['basic'] = basic
return sympify(e).expand(deep=deep, modulus=modulus, **hints)
# This is a special application of two hints
def _mexpand(expr, recursive=False):
# expand multinomials and then expand products; this may not always
# be sufficient to give a fully expanded expression (see
# test_issue_8247_8354 in test_arit)
if expr is None:
return
was = None
while was != expr:
was, expr = expr, expand_mul(expand_multinomial(expr))
if not recursive:
break
return expr
# These are simple wrappers around single hints.
def expand_mul(expr, deep=True):
"""
Wrapper around expand that only uses the mul hint. See the expand
docstring for more information.
Examples
========
>>> from sympy import symbols, expand_mul, exp, log
>>> x, y = symbols('x,y', positive=True)
>>> expand_mul(exp(x+y)*(x+y)*log(x*y**2))
x*exp(x + y)*log(x*y**2) + y*exp(x + y)*log(x*y**2)
"""
return sympify(expr).expand(deep=deep, mul=True, power_exp=False,
power_base=False, basic=False, multinomial=False, log=False)
def expand_multinomial(expr, deep=True):
"""
Wrapper around expand that only uses the multinomial hint. See the expand
docstring for more information.
Examples
========
>>> from sympy import symbols, expand_multinomial, exp
>>> x, y = symbols('x y', positive=True)
>>> expand_multinomial((x + exp(x + 1))**2)
x**2 + 2*x*exp(x + 1) + exp(2*x + 2)
"""
return sympify(expr).expand(deep=deep, mul=False, power_exp=False,
power_base=False, basic=False, multinomial=True, log=False)
def expand_log(expr, deep=True, force=False, factor=False):
"""
Wrapper around expand that only uses the log hint. See the expand
docstring for more information.
Examples
========
>>> from sympy import symbols, expand_log, exp, log
>>> x, y = symbols('x,y', positive=True)
>>> expand_log(exp(x+y)*(x+y)*log(x*y**2))
(x + y)*(log(x) + 2*log(y))*exp(x + y)
"""
from sympy import Mul, log
if factor is False:
def _handle(x):
x1 = expand_mul(expand_log(x, deep=deep, force=force, factor=True))
if x1.count(log) <= x.count(log):
return x1
return x
expr = expr.replace(
lambda x: x.is_Mul and all(any(isinstance(i, log) and i.args[0].is_Rational
for i in Mul.make_args(j)) for j in x.as_numer_denom()),
lambda x: _handle(x))
return sympify(expr).expand(deep=deep, log=True, mul=False,
power_exp=False, power_base=False, multinomial=False,
basic=False, force=force, factor=factor)
def expand_func(expr, deep=True):
"""
Wrapper around expand that only uses the func hint. See the expand
docstring for more information.
Examples
========
>>> from sympy import expand_func, gamma
>>> from sympy.abc import x
>>> expand_func(gamma(x + 2))
x*(x + 1)*gamma(x)
"""
return sympify(expr).expand(deep=deep, func=True, basic=False,
log=False, mul=False, power_exp=False, power_base=False, multinomial=False)
def expand_trig(expr, deep=True):
"""
Wrapper around expand that only uses the trig hint. See the expand
docstring for more information.
Examples
========
>>> from sympy import expand_trig, sin
>>> from sympy.abc import x, y
>>> expand_trig(sin(x+y)*(x+y))
(x + y)*(sin(x)*cos(y) + sin(y)*cos(x))
"""
return sympify(expr).expand(deep=deep, trig=True, basic=False,
log=False, mul=False, power_exp=False, power_base=False, multinomial=False)
def expand_complex(expr, deep=True):
"""
Wrapper around expand that only uses the complex hint. See the expand
docstring for more information.
Examples
========
>>> from sympy import expand_complex, exp, sqrt, I
>>> from sympy.abc import z
>>> expand_complex(exp(z))
I*exp(re(z))*sin(im(z)) + exp(re(z))*cos(im(z))
>>> expand_complex(sqrt(I))
sqrt(2)/2 + sqrt(2)*I/2
See Also
========
sympy.core.expr.Expr.as_real_imag
"""
return sympify(expr).expand(deep=deep, complex=True, basic=False,
log=False, mul=False, power_exp=False, power_base=False, multinomial=False)
def expand_power_base(expr, deep=True, force=False):
"""
Wrapper around expand that only uses the power_base hint.
A wrapper to expand(power_base=True) which separates a power with a base
that is a Mul into a product of powers, without performing any other
expansions, provided that assumptions about the power's base and exponent
allow.
deep=False (default is True) will only apply to the top-level expression.
force=True (default is False) will cause the expansion to ignore
assumptions about the base and exponent. When False, the expansion will
only happen if the base is non-negative or the exponent is an integer.
>>> from sympy.abc import x, y, z
>>> from sympy import expand_power_base, sin, cos, exp
>>> (x*y)**2
x**2*y**2
>>> (2*x)**y
(2*x)**y
>>> expand_power_base(_)
2**y*x**y
>>> expand_power_base((x*y)**z)
(x*y)**z
>>> expand_power_base((x*y)**z, force=True)
x**z*y**z
>>> expand_power_base(sin((x*y)**z), deep=False)
sin((x*y)**z)
>>> expand_power_base(sin((x*y)**z), force=True)
sin(x**z*y**z)
>>> expand_power_base((2*sin(x))**y + (2*cos(x))**y)
2**y*sin(x)**y + 2**y*cos(x)**y
>>> expand_power_base((2*exp(y))**x)
2**x*exp(y)**x
>>> expand_power_base((2*cos(x))**y)
2**y*cos(x)**y
Notice that sums are left untouched. If this is not the desired behavior,
apply full ``expand()`` to the expression:
>>> expand_power_base(((x+y)*z)**2)
z**2*(x + y)**2
>>> (((x+y)*z)**2).expand()
x**2*z**2 + 2*x*y*z**2 + y**2*z**2
>>> expand_power_base((2*y)**(1+z))
2**(z + 1)*y**(z + 1)
>>> ((2*y)**(1+z)).expand()
2*2**z*y*y**z
See Also
========
expand
"""
return sympify(expr).expand(deep=deep, log=False, mul=False,
power_exp=False, power_base=True, multinomial=False,
basic=False, force=force)
def expand_power_exp(expr, deep=True):
"""
Wrapper around expand that only uses the power_exp hint.
See the expand docstring for more information.
Examples
========
>>> from sympy import expand_power_exp
>>> from sympy.abc import x, y
>>> expand_power_exp(x**(y + 2))
x**2*x**y
"""
return sympify(expr).expand(deep=deep, complex=False, basic=False,
log=False, mul=False, power_exp=True, power_base=False, multinomial=False)
def count_ops(expr, visual=False):
"""
Return a representation (integer or expression) of the operations in expr.
Parameters
==========
expr : Expr
If expr is an iterable, the sum of the op counts of the
items will be returned.
visual : bool, optional
If ``False`` (default) then the sum of the coefficients of the
visual expression will be returned.
If ``True`` then the number of each type of operation is shown
with the core class types (or their virtual equivalent) multiplied by the
number of times they occur.
Examples
========
>>> from sympy.abc import a, b, x, y
>>> from sympy import sin, count_ops
Although there isn't a SUB object, minus signs are interpreted as
either negations or subtractions:
>>> (x - y).count_ops(visual=True)
SUB
>>> (-x).count_ops(visual=True)
NEG
Here, there are two Adds and a Pow:
>>> (1 + a + b**2).count_ops(visual=True)
2*ADD + POW
In the following, an Add, Mul, Pow and two functions:
>>> (sin(x)*x + sin(x)**2).count_ops(visual=True)
ADD + MUL + POW + 2*SIN
for a total of 5:
>>> (sin(x)*x + sin(x)**2).count_ops(visual=False)
5
Note that "what you type" is not always what you get. The expression
1/x/y is translated by sympy into 1/(x*y) so it gives a DIV and MUL rather
than two DIVs:
>>> (1/x/y).count_ops(visual=True)
DIV + MUL
The visual option can be used to demonstrate the difference in
operations for expressions in different forms. Here, the Horner
representation is compared with the expanded form of a polynomial:
>>> eq=x*(1 + x*(2 + x*(3 + x)))
>>> count_ops(eq.expand(), visual=True) - count_ops(eq, visual=True)
-MUL + 3*POW
The count_ops function also handles iterables:
>>> count_ops([x, sin(x), None, True, x + 2], visual=False)
2
>>> count_ops([x, sin(x), None, True, x + 2], visual=True)
ADD + SIN
>>> count_ops({x: sin(x), x + 2: y + 1}, visual=True)
2*ADD + SIN
"""
from sympy import Integral, Sum, Symbol
from sympy.core.relational import Relational
from sympy.simplify.radsimp import fraction
from sympy.logic.boolalg import BooleanFunction
from sympy.utilities.misc import func_name
expr = sympify(expr)
if isinstance(expr, Expr) and not expr.is_Relational:
ops = []
args = [expr]
NEG = Symbol('NEG')
DIV = Symbol('DIV')
SUB = Symbol('SUB')
ADD = Symbol('ADD')
EXP = Symbol('EXP')
while args:
a = args.pop()
if a.is_Rational:
#-1/3 = NEG + DIV
if a is not S.One:
if a.p < 0:
ops.append(NEG)
if a.q != 1:
ops.append(DIV)
continue
elif a.is_Mul or a.is_MatMul:
if _coeff_isneg(a):
ops.append(NEG)
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.append(DIV)
if n < 0:
ops.append(NEG)
args.append(d)
continue # won't be -Mul but could be Add
elif d is not S.One:
if not d.is_Integer:
args.append(d)
ops.append(DIV)
args.append(n)
continue # could be -Mul
elif a.is_Add or a.is_MatAdd:
aargs = list(a.args)
negs = 0
for i, ai in enumerate(aargs):
if _coeff_isneg(ai):
negs += 1
args.append(-ai)
if i > 0:
ops.append(SUB)
else:
args.append(ai)
if i > 0:
ops.append(ADD)
if negs == len(aargs): # -x - y = NEG + SUB
ops.append(NEG)
elif _coeff_isneg(aargs[0]): # -x + y = SUB, but already recorded ADD
ops.append(SUB - ADD)
continue
if a.is_Pow and a.exp is S.NegativeOne:
ops.append(DIV)
args.append(a.base) # won't be -Mul but could be Add
continue
if a == S.Exp1:
ops.append(EXP)
continue
if a.is_Pow and a.base == S.Exp1:
ops.append(EXP)
args.append(a.exp)
continue
if a.is_Mul or isinstance(a, LatticeOp):
o = Symbol(a.func.__name__.upper())
# count the args
ops.append(o*(len(a.args) - 1))
elif a.args and (
a.is_Pow or
a.is_Function or
isinstance(a, Derivative) or
isinstance(a, Integral) or
isinstance(a, Sum)):
# if it's not in the list above we don't
# consider a.func something to count, e.g.
# Tuple, MatrixSymbol, etc...
o = Symbol(a.func.__name__.upper())
ops.append(o)
if not a.is_Symbol:
args.extend(a.args)
elif isinstance(expr, Dict):
ops = [count_ops(k, visual=visual) +
count_ops(v, visual=visual) for k, v in expr.items()]
elif iterable(expr):
ops = [count_ops(i, visual=visual) for i in expr]
elif isinstance(expr, (Relational, BooleanFunction)):
ops = []
for arg in expr.args:
ops.append(count_ops(arg, visual=True))
o = Symbol(func_name(expr, short=True).upper())
ops.append(o)
elif not isinstance(expr, Basic):
ops = []
else: # it's Basic not isinstance(expr, Expr):
if not isinstance(expr, Basic):
raise TypeError("Invalid type of expr")
else:
ops = []
args = [expr]
while args:
a = args.pop()
if a.args:
o = Symbol(type(a).__name__.upper())
if a.is_Boolean:
ops.append(o*(len(a.args)-1))
else:
ops.append(o)
args.extend(a.args)
if not ops:
if visual:
return S.Zero
return 0
ops = Add(*ops)
if visual:
return ops
if ops.is_Number:
return int(ops)
return sum(int((a.args or [1])[0]) for a in Add.make_args(ops))
def nfloat(expr, n=15, exponent=False, dkeys=False):
"""Make all Rationals in expr Floats except those in exponents
(unless the exponents flag is set to True). When processing
dictionaries, don't modify the keys unless ``dkeys=True``.
Examples
========
>>> from sympy.core.function import nfloat
>>> from sympy.abc import x, y
>>> from sympy import cos, pi, sqrt
>>> nfloat(x**4 + x/2 + cos(pi/3) + 1 + sqrt(y))
x**4 + 0.5*x + sqrt(y) + 1.5
>>> nfloat(x**4 + sqrt(y), exponent=True)
x**4.0 + y**0.5
Container types are not modified:
>>> type(nfloat((1, 2))) is tuple
True
"""
from sympy.core.power import Pow
from sympy.polys.rootoftools import RootOf
from sympy import MatrixBase
kw = dict(n=n, exponent=exponent, dkeys=dkeys)
if isinstance(expr, MatrixBase):
return expr.applyfunc(lambda e: nfloat(e, **kw))
# handling of iterable containers
if iterable(expr, exclude=str):
if isinstance(expr, (dict, Dict)):
if dkeys:
args = [tuple(map(lambda i: nfloat(i, **kw), a))
for a in expr.items()]
else:
args = [(k, nfloat(v, **kw)) for k, v in expr.items()]
if isinstance(expr, dict):
return type(expr)(args)
else:
return expr.func(*args)
elif isinstance(expr, Basic):
return expr.func(*[nfloat(a, **kw) for a in expr.args])
return type(expr)([nfloat(a, **kw) for a in expr])
rv = sympify(expr)
if rv.is_Number:
return Float(rv, n)
elif rv.is_number:
# evalf doesn't always set the precision
rv = rv.n(n)
if rv.is_Number:
rv = Float(rv.n(n), n)
else:
pass # pure_complex(rv) is likely True
return rv
elif rv.is_Atom:
return rv
elif rv.is_Relational:
args_nfloat = (nfloat(arg, **kw) for arg in rv.args)
return rv.func(*args_nfloat)
# watch out for RootOf instances that don't like to have
# their exponents replaced with Dummies and also sometimes have
# problems with evaluating at low precision (issue 6393)
rv = rv.xreplace({ro: ro.n(n) for ro in rv.atoms(RootOf)})
if not exponent:
reps = [(p, Pow(p.base, Dummy())) for p in rv.atoms(Pow)]
rv = rv.xreplace(dict(reps))
rv = rv.n(n)
if not exponent:
rv = rv.xreplace({d.exp: p.exp for p, d in reps})
else:
# Pow._eval_evalf special cases Integer exponents so if
# exponent is suppose to be handled we have to do so here
rv = rv.xreplace(Transform(
lambda x: Pow(x.base, Float(x.exp, n)),
lambda x: x.is_Pow and x.exp.is_Integer))
return rv.xreplace(Transform(
lambda x: x.func(*nfloat(x.args, n, exponent)),
lambda x: isinstance(x, Function)))
from sympy.core.symbol import Dummy, Symbol
|
c44bad27cd40003771e6dd921d5f5a835da95efde1fd0fc7c2dd3da8a2f54c4c | from typing import Tuple as tTuple
from collections.abc import Iterable
from functools import reduce
from .sympify import sympify, _sympify, SympifyError
from .basic import Basic, Atom
from .singleton import S
from .evalf import EvalfMixin, pure_complex
from .decorators import call_highest_priority, sympify_method_args, sympify_return
from .cache import cacheit
from .compatibility import as_int, default_sort_key
from sympy.utilities.misc import func_name
from mpmath.libmp import mpf_log, prec_to_dps
from collections import defaultdict
@sympify_method_args
class Expr(Basic, EvalfMixin):
"""
Base class for algebraic expressions.
Explanation
===========
Everything that requires arithmetic operations to be defined
should subclass this class, instead of Basic (which should be
used only for argument storage and expression manipulation, i.e.
pattern matching, substitutions, etc).
If you want to override the comparisons of expressions:
Should use _eval_is_ge for inequality, or _eval_is_eq, with multiple dispatch.
_eval_is_ge return true if x >= y, false if x < y, and None if the two types
are not comparable or the comparison is indeterminate
See Also
========
sympy.core.basic.Basic
"""
__slots__ = () # type: tTuple[str, ...]
is_scalar = True # self derivative is 1
@property
def _diff_wrt(self):
"""Return True if one can differentiate with respect to this
object, else False.
Explanation
===========
Subclasses such as Symbol, Function and Derivative return True
to enable derivatives wrt them. The implementation in Derivative
separates the Symbol and non-Symbol (_diff_wrt=True) variables and
temporarily converts the non-Symbols into Symbols when performing
the differentiation. By default, any object deriving from Expr
will behave like a scalar with self.diff(self) == 1. If this is
not desired then the object must also set `is_scalar = False` or
else define an _eval_derivative routine.
Note, see the docstring of Derivative for how this should work
mathematically. In particular, note that expr.subs(yourclass, Symbol)
should be well-defined on a structural level, or this will lead to
inconsistent results.
Examples
========
>>> from sympy import Expr
>>> e = Expr()
>>> e._diff_wrt
False
>>> class MyScalar(Expr):
... _diff_wrt = True
...
>>> MyScalar().diff(MyScalar())
1
>>> class MySymbol(Expr):
... _diff_wrt = True
... is_scalar = False
...
>>> MySymbol().diff(MySymbol())
Derivative(MySymbol(), MySymbol())
"""
return False
@cacheit
def sort_key(self, order=None):
coeff, expr = self.as_coeff_Mul()
if expr.is_Pow:
if expr.base is S.Exp1:
# If we remove this, many doctests will go crazy:
# (keeps E**x sorted like the exp(x) function,
# part of exp(x) to E**x transition)
expr, exp = Function("exp")(expr.exp), S.One
else:
expr, exp = expr.args
else:
expr, exp = expr, S.One
if expr.is_Dummy:
args = (expr.sort_key(),)
elif expr.is_Atom:
args = (str(expr),)
else:
if expr.is_Add:
args = expr.as_ordered_terms(order=order)
elif expr.is_Mul:
args = expr.as_ordered_factors(order=order)
else:
args = expr.args
args = tuple(
[ default_sort_key(arg, order=order) for arg in args ])
args = (len(args), tuple(args))
exp = exp.sort_key(order=order)
return expr.class_key(), args, exp, coeff
def __hash__(self) -> int:
# hash cannot be cached using cache_it because infinite recurrence
# occurs as hash is needed for setting cache dictionary keys
h = self._mhash
if h is None:
h = hash((type(self).__name__,) + self._hashable_content())
self._mhash = h
return h
def _hashable_content(self):
"""Return a tuple of information about self that can be used to
compute the hash. If a class defines additional attributes,
like ``name`` in Symbol, then this method should be updated
accordingly to return such relevant attributes.
Defining more than _hashable_content is necessary if __eq__ has
been defined by a class. See note about this in Basic.__eq__."""
return self._args
def __eq__(self, other):
try:
other = _sympify(other)
if not isinstance(other, Expr):
return False
except (SympifyError, SyntaxError):
return False
# check for pure number expr
if not (self.is_Number and other.is_Number) and (
type(self) != type(other)):
return False
a, b = self._hashable_content(), other._hashable_content()
if a != b:
return False
# check number *in* an expression
for a, b in zip(a, b):
if not isinstance(a, Expr):
continue
if a.is_Number and type(a) != type(b):
return False
return True
# ***************
# * Arithmetics *
# ***************
# Expr and its sublcasses use _op_priority to determine which object
# passed to a binary special method (__mul__, etc.) will handle the
# operation. In general, the 'call_highest_priority' decorator will choose
# the object with the highest _op_priority to handle the call.
# Custom subclasses that want to define their own binary special methods
# should set an _op_priority value that is higher than the default.
#
# **NOTE**:
# This is a temporary fix, and will eventually be replaced with
# something better and more powerful. See issue 5510.
_op_priority = 10.0
@property
def _add_handler(self):
return Add
@property
def _mul_handler(self):
return Mul
def __pos__(self):
return self
def __neg__(self):
# Mul has its own __neg__ routine, so we just
# create a 2-args Mul with the -1 in the canonical
# slot 0.
c = self.is_commutative
return Mul._from_args((S.NegativeOne, self), c)
def __abs__(self):
from sympy import Abs
return Abs(self)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__radd__')
def __add__(self, other):
return Add(self, other)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__add__')
def __radd__(self, other):
return Add(other, self)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__rsub__')
def __sub__(self, other):
return Add(self, -other)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__sub__')
def __rsub__(self, other):
return Add(other, -self)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__rmul__')
def __mul__(self, other):
return Mul(self, other)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__mul__')
def __rmul__(self, other):
return Mul(other, self)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__rpow__')
def _pow(self, other):
return Pow(self, other)
def __pow__(self, other, mod=None):
if mod is None:
return self._pow(other)
try:
_self, other, mod = as_int(self), as_int(other), as_int(mod)
if other >= 0:
return pow(_self, other, mod)
else:
from sympy.core.numbers import mod_inverse
return mod_inverse(pow(_self, -other, mod), mod)
except ValueError:
power = self._pow(other)
try:
return power%mod
except TypeError:
return NotImplemented
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__pow__')
def __rpow__(self, other):
return Pow(other, self)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__rtruediv__')
def __truediv__(self, other):
denom = Pow(other, S.NegativeOne)
if self is S.One:
return denom
else:
return Mul(self, denom)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__truediv__')
def __rtruediv__(self, other):
denom = Pow(self, S.NegativeOne)
if other is S.One:
return denom
else:
return Mul(other, denom)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__rmod__')
def __mod__(self, other):
return Mod(self, other)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__mod__')
def __rmod__(self, other):
return Mod(other, self)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__rfloordiv__')
def __floordiv__(self, other):
from sympy.functions.elementary.integers import floor
return floor(self / other)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__floordiv__')
def __rfloordiv__(self, other):
from sympy.functions.elementary.integers import floor
return floor(other / self)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__rdivmod__')
def __divmod__(self, other):
from sympy.functions.elementary.integers import floor
return floor(self / other), Mod(self, other)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__divmod__')
def __rdivmod__(self, other):
from sympy.functions.elementary.integers import floor
return floor(other / self), Mod(other, self)
def __int__(self):
# Although we only need to round to the units position, we'll
# get one more digit so the extra testing below can be avoided
# unless the rounded value rounded to an integer, e.g. if an
# expression were equal to 1.9 and we rounded to the unit position
# we would get a 2 and would not know if this rounded up or not
# without doing a test (as done below). But if we keep an extra
# digit we know that 1.9 is not the same as 1 and there is no
# need for further testing: our int value is correct. If the value
# were 1.99, however, this would round to 2.0 and our int value is
# off by one. So...if our round value is the same as the int value
# (regardless of how much extra work we do to calculate extra decimal
# places) we need to test whether we are off by one.
from sympy import Dummy
if not self.is_number:
raise TypeError("can't convert symbols to int")
r = self.round(2)
if not r.is_Number:
raise TypeError("can't convert complex to int")
if r in (S.NaN, S.Infinity, S.NegativeInfinity):
raise TypeError("can't convert %s to int" % r)
i = int(r)
if not i:
return 0
# off-by-one check
if i == r and not (self - i).equals(0):
isign = 1 if i > 0 else -1
x = Dummy()
# in the following (self - i).evalf(2) will not always work while
# (self - r).evalf(2) and the use of subs does; if the test that
# was added when this comment was added passes, it might be safe
# to simply use sign to compute this rather than doing this by hand:
diff_sign = 1 if (self - x).evalf(2, subs={x: i}) > 0 else -1
if diff_sign != isign:
i -= isign
return i
def __float__(self):
# Don't bother testing if it's a number; if it's not this is going
# to fail, and if it is we still need to check that it evalf'ed to
# a number.
result = self.evalf()
if result.is_Number:
return float(result)
if result.is_number and result.as_real_imag()[1]:
raise TypeError("can't convert complex to float")
raise TypeError("can't convert expression to float")
def __complex__(self):
result = self.evalf()
re, im = result.as_real_imag()
return complex(float(re), float(im))
@sympify_return([('other', 'Expr')], NotImplemented)
def __ge__(self, other):
from .relational import GreaterThan
return GreaterThan(self, other)
@sympify_return([('other', 'Expr')], NotImplemented)
def __le__(self, other):
from .relational import LessThan
return LessThan(self, other)
@sympify_return([('other', 'Expr')], NotImplemented)
def __gt__(self, other):
from .relational import StrictGreaterThan
return StrictGreaterThan(self, other)
@sympify_return([('other', 'Expr')], NotImplemented)
def __lt__(self, other):
from .relational import StrictLessThan
return StrictLessThan(self, other)
def __trunc__(self):
if not self.is_number:
raise TypeError("can't truncate symbols and expressions")
else:
return Integer(self)
@staticmethod
def _from_mpmath(x, prec):
from sympy import Float
if hasattr(x, "_mpf_"):
return Float._new(x._mpf_, prec)
elif hasattr(x, "_mpc_"):
re, im = x._mpc_
re = Float._new(re, prec)
im = Float._new(im, prec)*S.ImaginaryUnit
return re + im
else:
raise TypeError("expected mpmath number (mpf or mpc)")
@property
def is_number(self):
"""Returns True if ``self`` has no free symbols and no
undefined functions (AppliedUndef, to be precise). It will be
faster than ``if not self.free_symbols``, however, since
``is_number`` will fail as soon as it hits a free symbol
or undefined function.
Examples
========
>>> from sympy import Integral, cos, sin, pi
>>> from sympy.core.function import Function
>>> from sympy.abc import x
>>> f = Function('f')
>>> x.is_number
False
>>> f(1).is_number
False
>>> (2*x).is_number
False
>>> (2 + Integral(2, x)).is_number
False
>>> (2 + Integral(2, (x, 1, 2))).is_number
True
Not all numbers are Numbers in the SymPy sense:
>>> pi.is_number, pi.is_Number
(True, False)
If something is a number it should evaluate to a number with
real and imaginary parts that are Numbers; the result may not
be comparable, however, since the real and/or imaginary part
of the result may not have precision.
>>> cos(1).is_number and cos(1).is_comparable
True
>>> z = cos(1)**2 + sin(1)**2 - 1
>>> z.is_number
True
>>> z.is_comparable
False
See Also
========
sympy.core.basic.Basic.is_comparable
"""
return all(obj.is_number for obj in self.args)
def _random(self, n=None, re_min=-1, im_min=-1, re_max=1, im_max=1):
"""Return self evaluated, if possible, replacing free symbols with
random complex values, if necessary.
Explanation
===========
The random complex value for each free symbol is generated
by the random_complex_number routine giving real and imaginary
parts in the range given by the re_min, re_max, im_min, and im_max
values. The returned value is evaluated to a precision of n
(if given) else the maximum of 15 and the precision needed
to get more than 1 digit of precision. If the expression
could not be evaluated to a number, or could not be evaluated
to more than 1 digit of precision, then None is returned.
Examples
========
>>> from sympy import sqrt
>>> from sympy.abc import x, y
>>> x._random() # doctest: +SKIP
0.0392918155679172 + 0.916050214307199*I
>>> x._random(2) # doctest: +SKIP
-0.77 - 0.87*I
>>> (x + y/2)._random(2) # doctest: +SKIP
-0.57 + 0.16*I
>>> sqrt(2)._random(2)
1.4
See Also
========
sympy.testing.randtest.random_complex_number
"""
free = self.free_symbols
prec = 1
if free:
from sympy.testing.randtest import random_complex_number
a, c, b, d = re_min, re_max, im_min, im_max
reps = dict(list(zip(free, [random_complex_number(a, b, c, d, rational=True)
for zi in free])))
try:
nmag = abs(self.evalf(2, subs=reps))
except (ValueError, TypeError):
# if an out of range value resulted in evalf problems
# then return None -- XXX is there a way to know how to
# select a good random number for a given expression?
# e.g. when calculating n! negative values for n should not
# be used
return None
else:
reps = {}
nmag = abs(self.evalf(2))
if not hasattr(nmag, '_prec'):
# e.g. exp_polar(2*I*pi) doesn't evaluate but is_number is True
return None
if nmag._prec == 1:
# increase the precision up to the default maximum
# precision to see if we can get any significance
from mpmath.libmp.libintmath import giant_steps
from sympy.core.evalf import DEFAULT_MAXPREC as target
# evaluate
for prec in giant_steps(2, target):
nmag = abs(self.evalf(prec, subs=reps))
if nmag._prec != 1:
break
if nmag._prec != 1:
if n is None:
n = max(prec, 15)
return self.evalf(n, subs=reps)
# never got any significance
return None
def is_constant(self, *wrt, **flags):
"""Return True if self is constant, False if not, or None if
the constancy could not be determined conclusively.
Explanation
===========
If an expression has no free symbols then it is a constant. If
there are free symbols it is possible that the expression is a
constant, perhaps (but not necessarily) zero. To test such
expressions, a few strategies are tried:
1) numerical evaluation at two random points. If two such evaluations
give two different values and the values have a precision greater than
1 then self is not constant. If the evaluations agree or could not be
obtained with any precision, no decision is made. The numerical testing
is done only if ``wrt`` is different than the free symbols.
2) differentiation with respect to variables in 'wrt' (or all free
symbols if omitted) to see if the expression is constant or not. This
will not always lead to an expression that is zero even though an
expression is constant (see added test in test_expr.py). If
all derivatives are zero then self is constant with respect to the
given symbols.
3) finding out zeros of denominator expression with free_symbols.
It won't be constant if there are zeros. It gives more negative
answers for expression that are not constant.
If neither evaluation nor differentiation can prove the expression is
constant, None is returned unless two numerical values happened to be
the same and the flag ``failing_number`` is True -- in that case the
numerical value will be returned.
If flag simplify=False is passed, self will not be simplified;
the default is True since self should be simplified before testing.
Examples
========
>>> from sympy import cos, sin, Sum, S, pi
>>> from sympy.abc import a, n, x, y
>>> x.is_constant()
False
>>> S(2).is_constant()
True
>>> Sum(x, (x, 1, 10)).is_constant()
True
>>> Sum(x, (x, 1, n)).is_constant()
False
>>> Sum(x, (x, 1, n)).is_constant(y)
True
>>> Sum(x, (x, 1, n)).is_constant(n)
False
>>> Sum(x, (x, 1, n)).is_constant(x)
True
>>> eq = a*cos(x)**2 + a*sin(x)**2 - a
>>> eq.is_constant()
True
>>> eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0
True
>>> (0**x).is_constant()
False
>>> x.is_constant()
False
>>> (x**x).is_constant()
False
>>> one = cos(x)**2 + sin(x)**2
>>> one.is_constant()
True
>>> ((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1
True
"""
def check_denominator_zeros(expression):
from sympy.solvers.solvers import denoms
retNone = False
for den in denoms(expression):
z = den.is_zero
if z is True:
return True
if z is None:
retNone = True
if retNone:
return None
return False
simplify = flags.get('simplify', True)
if self.is_number:
return True
free = self.free_symbols
if not free:
return True # assume f(1) is some constant
# if we are only interested in some symbols and they are not in the
# free symbols then this expression is constant wrt those symbols
wrt = set(wrt)
if wrt and not wrt & free:
return True
wrt = wrt or free
# simplify unless this has already been done
expr = self
if simplify:
expr = expr.simplify()
# is_zero should be a quick assumptions check; it can be wrong for
# numbers (see test_is_not_constant test), giving False when it
# shouldn't, but hopefully it will never give True unless it is sure.
if expr.is_zero:
return True
# try numerical evaluation to see if we get two different values
failing_number = None
if wrt == free:
# try 0 (for a) and 1 (for b)
try:
a = expr.subs(list(zip(free, [0]*len(free))),
simultaneous=True)
if a is S.NaN:
# evaluation may succeed when substitution fails
a = expr._random(None, 0, 0, 0, 0)
except ZeroDivisionError:
a = None
if a is not None and a is not S.NaN:
try:
b = expr.subs(list(zip(free, [1]*len(free))),
simultaneous=True)
if b is S.NaN:
# evaluation may succeed when substitution fails
b = expr._random(None, 1, 0, 1, 0)
except ZeroDivisionError:
b = None
if b is not None and b is not S.NaN and b.equals(a) is False:
return False
# try random real
b = expr._random(None, -1, 0, 1, 0)
if b is not None and b is not S.NaN and b.equals(a) is False:
return False
# try random complex
b = expr._random()
if b is not None and b is not S.NaN:
if b.equals(a) is False:
return False
failing_number = a if a.is_number else b
# now we will test each wrt symbol (or all free symbols) to see if the
# expression depends on them or not using differentiation. This is
# not sufficient for all expressions, however, so we don't return
# False if we get a derivative other than 0 with free symbols.
for w in wrt:
deriv = expr.diff(w)
if simplify:
deriv = deriv.simplify()
if deriv != 0:
if not (pure_complex(deriv, or_real=True)):
if flags.get('failing_number', False):
return failing_number
elif deriv.free_symbols:
# dead line provided _random returns None in such cases
return None
return False
cd = check_denominator_zeros(self)
if cd is True:
return False
elif cd is None:
return None
return True
def equals(self, other, failing_expression=False):
"""Return True if self == other, False if it doesn't, or None. If
failing_expression is True then the expression which did not simplify
to a 0 will be returned instead of None.
Explanation
===========
If ``self`` is a Number (or complex number) that is not zero, then
the result is False.
If ``self`` is a number and has not evaluated to zero, evalf will be
used to test whether the expression evaluates to zero. If it does so
and the result has significance (i.e. the precision is either -1, for
a Rational result, or is greater than 1) then the evalf value will be
used to return True or False.
"""
from sympy.simplify.simplify import nsimplify, simplify
from sympy.solvers.solvers import solve
from sympy.polys.polyerrors import NotAlgebraic
from sympy.polys.numberfields import minimal_polynomial
other = sympify(other)
if self == other:
return True
# they aren't the same so see if we can make the difference 0;
# don't worry about doing simplification steps one at a time
# because if the expression ever goes to 0 then the subsequent
# simplification steps that are done will be very fast.
diff = factor_terms(simplify(self - other), radical=True)
if not diff:
return True
if not diff.has(Add, Mod):
# if there is no expanding to be done after simplifying
# then this can't be a zero
return False
constant = diff.is_constant(simplify=False, failing_number=True)
if constant is False:
return False
if not diff.is_number:
if constant is None:
# e.g. unless the right simplification is done, a symbolic
# zero is possible (see expression of issue 6829: without
# simplification constant will be None).
return
if constant is True:
# this gives a number whether there are free symbols or not
ndiff = diff._random()
# is_comparable will work whether the result is real
# or complex; it could be None, however.
if ndiff and ndiff.is_comparable:
return False
# sometimes we can use a simplified result to give a clue as to
# what the expression should be; if the expression is *not* zero
# then we should have been able to compute that and so now
# we can just consider the cases where the approximation appears
# to be zero -- we try to prove it via minimal_polynomial.
#
# removed
# ns = nsimplify(diff)
# if diff.is_number and (not ns or ns == diff):
#
# The thought was that if it nsimplifies to 0 that's a sure sign
# to try the following to prove it; or if it changed but wasn't
# zero that might be a sign that it's not going to be easy to
# prove. But tests seem to be working without that logic.
#
if diff.is_number:
# try to prove via self-consistency
surds = [s for s in diff.atoms(Pow) if s.args[0].is_Integer]
# it seems to work better to try big ones first
surds.sort(key=lambda x: -x.args[0])
for s in surds:
try:
# simplify is False here -- this expression has already
# been identified as being hard to identify as zero;
# we will handle the checking ourselves using nsimplify
# to see if we are in the right ballpark or not and if so
# *then* the simplification will be attempted.
sol = solve(diff, s, simplify=False)
if sol:
if s in sol:
# the self-consistent result is present
return True
if all(si.is_Integer for si in sol):
# perfect powers are removed at instantiation
# so surd s cannot be an integer
return False
if all(i.is_algebraic is False for i in sol):
# a surd is algebraic
return False
if any(si in surds for si in sol):
# it wasn't equal to s but it is in surds
# and different surds are not equal
return False
if any(nsimplify(s - si) == 0 and
simplify(s - si) == 0 for si in sol):
return True
if s.is_real:
if any(nsimplify(si, [s]) == s and simplify(si) == s
for si in sol):
return True
except NotImplementedError:
pass
# try to prove with minimal_polynomial but know when
# *not* to use this or else it can take a long time. e.g. issue 8354
if True: # change True to condition that assures non-hang
try:
mp = minimal_polynomial(diff)
if mp.is_Symbol:
return True
return False
except (NotAlgebraic, NotImplementedError):
pass
# diff has not simplified to zero; constant is either None, True
# or the number with significance (is_comparable) that was randomly
# calculated twice as the same value.
if constant not in (True, None) and constant != 0:
return False
if failing_expression:
return diff
return None
def _eval_is_positive(self):
finite = self.is_finite
if finite is False:
return False
extended_positive = self.is_extended_positive
if finite is True:
return extended_positive
if extended_positive is False:
return False
def _eval_is_negative(self):
finite = self.is_finite
if finite is False:
return False
extended_negative = self.is_extended_negative
if finite is True:
return extended_negative
if extended_negative is False:
return False
def _eval_is_extended_positive_negative(self, positive):
from sympy.core.numbers import pure_complex
from sympy.polys.numberfields import minimal_polynomial
from sympy.polys.polyerrors import NotAlgebraic
if self.is_number:
if self.is_extended_real is False:
return False
# check to see that we can get a value
try:
n2 = self._eval_evalf(2)
# XXX: This shouldn't be caught here
# Catches ValueError: hypsum() failed to converge to the requested
# 34 bits of accuracy
except ValueError:
return None
if n2 is None:
return None
if getattr(n2, '_prec', 1) == 1: # no significance
return None
if n2 is S.NaN:
return None
f = self.evalf(2)
if f.is_Float:
match = f, S.Zero
else:
match = pure_complex(f)
if match is None:
return False
r, i = match
if not (i.is_Number and r.is_Number):
return False
if r._prec != 1 and i._prec != 1:
return bool(not i and ((r > 0) if positive else (r < 0)))
elif r._prec == 1 and (not i or i._prec == 1) and \
self.is_algebraic and not self.has(Function):
try:
if minimal_polynomial(self).is_Symbol:
return False
except (NotAlgebraic, NotImplementedError):
pass
def _eval_is_extended_positive(self):
return self._eval_is_extended_positive_negative(positive=True)
def _eval_is_extended_negative(self):
return self._eval_is_extended_positive_negative(positive=False)
def _eval_interval(self, x, a, b):
"""
Returns evaluation over an interval. For most functions this is:
self.subs(x, b) - self.subs(x, a),
possibly using limit() if NaN is returned from subs, or if
singularities are found between a and b.
If b or a is None, it only evaluates -self.subs(x, a) or self.subs(b, x),
respectively.
"""
from sympy.series import limit, Limit
from sympy.solvers.solveset import solveset
from sympy.sets.sets import Interval
from sympy.functions.elementary.exponential import log
from sympy.calculus.util import AccumBounds
if (a is None and b is None):
raise ValueError('Both interval ends cannot be None.')
def _eval_endpoint(left):
c = a if left else b
if c is None:
return 0
else:
C = self.subs(x, c)
if C.has(S.NaN, S.Infinity, S.NegativeInfinity,
S.ComplexInfinity, AccumBounds):
if (a < b) != False:
C = limit(self, x, c, "+" if left else "-")
else:
C = limit(self, x, c, "-" if left else "+")
if isinstance(C, Limit):
raise NotImplementedError("Could not compute limit")
return C
if a == b:
return 0
A = _eval_endpoint(left=True)
if A is S.NaN:
return A
B = _eval_endpoint(left=False)
if (a and b) is None:
return B - A
value = B - A
if a.is_comparable and b.is_comparable:
if a < b:
domain = Interval(a, b)
else:
domain = Interval(b, a)
# check the singularities of self within the interval
# if singularities is a ConditionSet (not iterable), catch the exception and pass
singularities = solveset(self.cancel().as_numer_denom()[1], x,
domain=domain)
for logterm in self.atoms(log):
singularities = singularities | solveset(logterm.args[0], x,
domain=domain)
try:
for s in singularities:
if value is S.NaN:
# no need to keep adding, it will stay NaN
break
if not s.is_comparable:
continue
if (a < s) == (s < b) == True:
value += -limit(self, x, s, "+") + limit(self, x, s, "-")
elif (b < s) == (s < a) == True:
value += limit(self, x, s, "+") - limit(self, x, s, "-")
except TypeError:
pass
return value
def _eval_power(self, other):
# subclass to compute self**other for cases when
# other is not NaN, 0, or 1
return None
def _eval_conjugate(self):
if self.is_extended_real:
return self
elif self.is_imaginary:
return -self
def conjugate(self):
"""Returns the complex conjugate of 'self'."""
from sympy.functions.elementary.complexes import conjugate as c
return c(self)
def dir(self, x, cdir):
from sympy import log
minexp = S.Zero
if self.is_zero:
return S.Zero
arg = self
while arg:
minexp += S.One
arg = arg.diff(x)
coeff = arg.subs(x, 0)
if coeff in (S.NaN, S.ComplexInfinity):
try:
coeff, _ = arg.leadterm(x)
if coeff.has(log(x)):
raise ValueError()
except ValueError:
coeff = arg.limit(x, 0)
if coeff != S.Zero:
break
return coeff*cdir**minexp
def _eval_transpose(self):
from sympy.functions.elementary.complexes import conjugate
if (self.is_complex or self.is_infinite):
return self
elif self.is_hermitian:
return conjugate(self)
elif self.is_antihermitian:
return -conjugate(self)
def transpose(self):
from sympy.functions.elementary.complexes import transpose
return transpose(self)
def _eval_adjoint(self):
from sympy.functions.elementary.complexes import conjugate, transpose
if self.is_hermitian:
return self
elif self.is_antihermitian:
return -self
obj = self._eval_conjugate()
if obj is not None:
return transpose(obj)
obj = self._eval_transpose()
if obj is not None:
return conjugate(obj)
def adjoint(self):
from sympy.functions.elementary.complexes import adjoint
return adjoint(self)
@classmethod
def _parse_order(cls, order):
"""Parse and configure the ordering of terms. """
from sympy.polys.orderings import monomial_key
startswith = getattr(order, "startswith", None)
if startswith is None:
reverse = False
else:
reverse = startswith('rev-')
if reverse:
order = order[4:]
monom_key = monomial_key(order)
def neg(monom):
result = []
for m in monom:
if isinstance(m, tuple):
result.append(neg(m))
else:
result.append(-m)
return tuple(result)
def key(term):
_, ((re, im), monom, ncpart) = term
monom = neg(monom_key(monom))
ncpart = tuple([e.sort_key(order=order) for e in ncpart])
coeff = ((bool(im), im), (re, im))
return monom, ncpart, coeff
return key, reverse
def as_ordered_factors(self, order=None):
"""Return list of ordered factors (if Mul) else [self]."""
return [self]
def as_poly(self, *gens, **args):
"""Converts ``self`` to a polynomial or returns ``None``.
Explanation
===========
>>> from sympy import sin
>>> from sympy.abc import x, y
>>> print((x**2 + x*y).as_poly())
Poly(x**2 + x*y, x, y, domain='ZZ')
>>> print((x**2 + x*y).as_poly(x, y))
Poly(x**2 + x*y, x, y, domain='ZZ')
>>> print((x**2 + sin(y)).as_poly(x, y))
None
"""
from sympy.polys import Poly, PolynomialError
try:
poly = Poly(self, *gens, **args)
if not poly.is_Poly:
return None
else:
return poly
except PolynomialError:
return None
def as_ordered_terms(self, order=None, data=False):
"""
Transform an expression to an ordered list of terms.
Examples
========
>>> from sympy import sin, cos
>>> from sympy.abc import x
>>> (sin(x)**2*cos(x) + sin(x)**2 + 1).as_ordered_terms()
[sin(x)**2*cos(x), sin(x)**2, 1]
"""
from .numbers import Number, NumberSymbol
if order is None and self.is_Add:
# Spot the special case of Add(Number, Mul(Number, expr)) with the
# first number positive and thhe second number nagative
key = lambda x:not isinstance(x, (Number, NumberSymbol))
add_args = sorted(Add.make_args(self), key=key)
if (len(add_args) == 2
and isinstance(add_args[0], (Number, NumberSymbol))
and isinstance(add_args[1], Mul)):
mul_args = sorted(Mul.make_args(add_args[1]), key=key)
if (len(mul_args) == 2
and isinstance(mul_args[0], Number)
and add_args[0].is_positive
and mul_args[0].is_negative):
return add_args
key, reverse = self._parse_order(order)
terms, gens = self.as_terms()
if not any(term.is_Order for term, _ in terms):
ordered = sorted(terms, key=key, reverse=reverse)
else:
_terms, _order = [], []
for term, repr in terms:
if not term.is_Order:
_terms.append((term, repr))
else:
_order.append((term, repr))
ordered = sorted(_terms, key=key, reverse=True) \
+ sorted(_order, key=key, reverse=True)
if data:
return ordered, gens
else:
return [term for term, _ in ordered]
def as_terms(self):
"""Transform an expression to a list of terms. """
from .add import Add
from .mul import Mul
from .exprtools import decompose_power
gens, terms = set(), []
for term in Add.make_args(self):
coeff, _term = term.as_coeff_Mul()
coeff = complex(coeff)
cpart, ncpart = {}, []
if _term is not S.One:
for factor in Mul.make_args(_term):
if factor.is_number:
try:
coeff *= complex(factor)
except (TypeError, ValueError):
pass
else:
continue
if factor.is_commutative:
base, exp = decompose_power(factor)
cpart[base] = exp
gens.add(base)
else:
ncpart.append(factor)
coeff = coeff.real, coeff.imag
ncpart = tuple(ncpart)
terms.append((term, (coeff, cpart, ncpart)))
gens = sorted(gens, key=default_sort_key)
k, indices = len(gens), {}
for i, g in enumerate(gens):
indices[g] = i
result = []
for term, (coeff, cpart, ncpart) in terms:
monom = [0]*k
for base, exp in cpart.items():
monom[indices[base]] = exp
result.append((term, (coeff, tuple(monom), ncpart)))
return result, gens
def removeO(self):
"""Removes the additive O(..) symbol if there is one"""
return self
def getO(self):
"""Returns the additive O(..) symbol if there is one, else None."""
return None
def getn(self):
"""
Returns the order of the expression.
Explanation
===========
The order is determined either from the O(...) term. If there
is no O(...) term, it returns None.
Examples
========
>>> from sympy import O
>>> from sympy.abc import x
>>> (1 + x + O(x**2)).getn()
2
>>> (1 + x).getn()
"""
from sympy import Dummy, Symbol
o = self.getO()
if o is None:
return None
elif o.is_Order:
o = o.expr
if o is S.One:
return S.Zero
if o.is_Symbol:
return S.One
if o.is_Pow:
return o.args[1]
if o.is_Mul: # x**n*log(x)**n or x**n/log(x)**n
for oi in o.args:
if oi.is_Symbol:
return S.One
if oi.is_Pow:
syms = oi.atoms(Symbol)
if len(syms) == 1:
x = syms.pop()
oi = oi.subs(x, Dummy('x', positive=True))
if oi.base.is_Symbol and oi.exp.is_Rational:
return abs(oi.exp)
raise NotImplementedError('not sure of order of %s' % o)
def count_ops(self, visual=None):
"""wrapper for count_ops that returns the operation count."""
from .function import count_ops
return count_ops(self, visual)
def args_cnc(self, cset=False, warn=True, split_1=True):
"""Return [commutative factors, non-commutative factors] of self.
Explanation
===========
self is treated as a Mul and the ordering of the factors is maintained.
If ``cset`` is True the commutative factors will be returned in a set.
If there were repeated factors (as may happen with an unevaluated Mul)
then an error will be raised unless it is explicitly suppressed by
setting ``warn`` to False.
Note: -1 is always separated from a Number unless split_1 is False.
Examples
========
>>> from sympy import symbols, oo
>>> A, B = symbols('A B', commutative=0)
>>> x, y = symbols('x y')
>>> (-2*x*y).args_cnc()
[[-1, 2, x, y], []]
>>> (-2.5*x).args_cnc()
[[-1, 2.5, x], []]
>>> (-2*x*A*B*y).args_cnc()
[[-1, 2, x, y], [A, B]]
>>> (-2*x*A*B*y).args_cnc(split_1=False)
[[-2, x, y], [A, B]]
>>> (-2*x*y).args_cnc(cset=True)
[{-1, 2, x, y}, []]
The arg is always treated as a Mul:
>>> (-2 + x + A).args_cnc()
[[], [x - 2 + A]]
>>> (-oo).args_cnc() # -oo is a singleton
[[-1, oo], []]
"""
if self.is_Mul:
args = list(self.args)
else:
args = [self]
for i, mi in enumerate(args):
if not mi.is_commutative:
c = args[:i]
nc = args[i:]
break
else:
c = args
nc = []
if c and split_1 and (
c[0].is_Number and
c[0].is_extended_negative and
c[0] is not S.NegativeOne):
c[:1] = [S.NegativeOne, -c[0]]
if cset:
clen = len(c)
c = set(c)
if clen and warn and len(c) != clen:
raise ValueError('repeated commutative arguments: %s' %
[ci for ci in c if list(self.args).count(ci) > 1])
return [c, nc]
def coeff(self, x, n=1, right=False):
"""
Returns the coefficient from the term(s) containing ``x**n``. If ``n``
is zero then all terms independent of ``x`` will be returned.
Explanation
===========
When ``x`` is noncommutative, the coefficient to the left (default) or
right of ``x`` can be returned. The keyword 'right' is ignored when
``x`` is commutative.
Examples
========
>>> from sympy import symbols
>>> from sympy.abc import x, y, z
You can select terms that have an explicit negative in front of them:
>>> (-x + 2*y).coeff(-1)
x
>>> (x - 2*y).coeff(-1)
2*y
You can select terms with no Rational coefficient:
>>> (x + 2*y).coeff(1)
x
>>> (3 + 2*x + 4*x**2).coeff(1)
0
You can select terms independent of x by making n=0; in this case
expr.as_independent(x)[0] is returned (and 0 will be returned instead
of None):
>>> (3 + 2*x + 4*x**2).coeff(x, 0)
3
>>> eq = ((x + 1)**3).expand() + 1
>>> eq
x**3 + 3*x**2 + 3*x + 2
>>> [eq.coeff(x, i) for i in reversed(range(4))]
[1, 3, 3, 2]
>>> eq -= 2
>>> [eq.coeff(x, i) for i in reversed(range(4))]
[1, 3, 3, 0]
You can select terms that have a numerical term in front of them:
>>> (-x - 2*y).coeff(2)
-y
>>> from sympy import sqrt
>>> (x + sqrt(2)*x).coeff(sqrt(2))
x
The matching is exact:
>>> (3 + 2*x + 4*x**2).coeff(x)
2
>>> (3 + 2*x + 4*x**2).coeff(x**2)
4
>>> (3 + 2*x + 4*x**2).coeff(x**3)
0
>>> (z*(x + y)**2).coeff((x + y)**2)
z
>>> (z*(x + y)**2).coeff(x + y)
0
In addition, no factoring is done, so 1 + z*(1 + y) is not obtained
from the following:
>>> (x + z*(x + x*y)).coeff(x)
1
If such factoring is desired, factor_terms can be used first:
>>> from sympy import factor_terms
>>> factor_terms(x + z*(x + x*y)).coeff(x)
z*(y + 1) + 1
>>> n, m, o = symbols('n m o', commutative=False)
>>> n.coeff(n)
1
>>> (3*n).coeff(n)
3
>>> (n*m + m*n*m).coeff(n) # = (1 + m)*n*m
1 + m
>>> (n*m + m*n*m).coeff(n, right=True) # = (1 + m)*n*m
m
If there is more than one possible coefficient 0 is returned:
>>> (n*m + m*n).coeff(n)
0
If there is only one possible coefficient, it is returned:
>>> (n*m + x*m*n).coeff(m*n)
x
>>> (n*m + x*m*n).coeff(m*n, right=1)
1
See Also
========
as_coefficient: separate the expression into a coefficient and factor
as_coeff_Add: separate the additive constant from an expression
as_coeff_Mul: separate the multiplicative constant from an expression
as_independent: separate x-dependent terms/factors from others
sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly
sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used
"""
x = sympify(x)
if not isinstance(x, Basic):
return S.Zero
n = as_int(n)
if not x:
return S.Zero
if x == self:
if n == 1:
return S.One
return S.Zero
if x is S.One:
co = [a for a in Add.make_args(self)
if a.as_coeff_Mul()[0] is S.One]
if not co:
return S.Zero
return Add(*co)
if n == 0:
if x.is_Add and self.is_Add:
c = self.coeff(x, right=right)
if not c:
return S.Zero
if not right:
return self - Add(*[a*x for a in Add.make_args(c)])
return self - Add(*[x*a for a in Add.make_args(c)])
return self.as_independent(x, as_Add=True)[0]
# continue with the full method, looking for this power of x:
x = x**n
def incommon(l1, l2):
if not l1 or not l2:
return []
n = min(len(l1), len(l2))
for i in range(n):
if l1[i] != l2[i]:
return l1[:i]
return l1[:]
def find(l, sub, first=True):
""" Find where list sub appears in list l. When ``first`` is True
the first occurrence from the left is returned, else the last
occurrence is returned. Return None if sub is not in l.
Examples
========
>> l = range(5)*2
>> find(l, [2, 3])
2
>> find(l, [2, 3], first=0)
7
>> find(l, [2, 4])
None
"""
if not sub or not l or len(sub) > len(l):
return None
n = len(sub)
if not first:
l.reverse()
sub.reverse()
for i in range(0, len(l) - n + 1):
if all(l[i + j] == sub[j] for j in range(n)):
break
else:
i = None
if not first:
l.reverse()
sub.reverse()
if i is not None and not first:
i = len(l) - (i + n)
return i
co = []
args = Add.make_args(self)
self_c = self.is_commutative
x_c = x.is_commutative
if self_c and not x_c:
return S.Zero
one_c = self_c or x_c
xargs, nx = x.args_cnc(cset=True, warn=bool(not x_c))
# find the parts that pass the commutative terms
for a in args:
margs, nc = a.args_cnc(cset=True, warn=bool(not self_c))
if nc is None:
nc = []
if len(xargs) > len(margs):
continue
resid = margs.difference(xargs)
if len(resid) + len(xargs) == len(margs):
if one_c:
co.append(Mul(*(list(resid) + nc)))
else:
co.append((resid, nc))
if one_c:
if co == []:
return S.Zero
elif co:
return Add(*co)
else: # both nc
# now check the non-comm parts
if not co:
return S.Zero
if all(n == co[0][1] for r, n in co):
ii = find(co[0][1], nx, right)
if ii is not None:
if not right:
return Mul(Add(*[Mul(*r) for r, c in co]), Mul(*co[0][1][:ii]))
else:
return Mul(*co[0][1][ii + len(nx):])
beg = reduce(incommon, (n[1] for n in co))
if beg:
ii = find(beg, nx, right)
if ii is not None:
if not right:
gcdc = co[0][0]
for i in range(1, len(co)):
gcdc = gcdc.intersection(co[i][0])
if not gcdc:
break
return Mul(*(list(gcdc) + beg[:ii]))
else:
m = ii + len(nx)
return Add(*[Mul(*(list(r) + n[m:])) for r, n in co])
end = list(reversed(
reduce(incommon, (list(reversed(n[1])) for n in co))))
if end:
ii = find(end, nx, right)
if ii is not None:
if not right:
return Add(*[Mul(*(list(r) + n[:-len(end) + ii])) for r, n in co])
else:
return Mul(*end[ii + len(nx):])
# look for single match
hit = None
for i, (r, n) in enumerate(co):
ii = find(n, nx, right)
if ii is not None:
if not hit:
hit = ii, r, n
else:
break
else:
if hit:
ii, r, n = hit
if not right:
return Mul(*(list(r) + n[:ii]))
else:
return Mul(*n[ii + len(nx):])
return S.Zero
def as_expr(self, *gens):
"""
Convert a polynomial to a SymPy expression.
Examples
========
>>> from sympy import sin
>>> from sympy.abc import x, y
>>> f = (x**2 + x*y).as_poly(x, y)
>>> f.as_expr()
x**2 + x*y
>>> sin(x).as_expr()
sin(x)
"""
return self
def as_coefficient(self, expr):
"""
Extracts symbolic coefficient at the given expression. In
other words, this functions separates 'self' into the product
of 'expr' and 'expr'-free coefficient. If such separation
is not possible it will return None.
Examples
========
>>> from sympy import E, pi, sin, I, Poly
>>> from sympy.abc import x
>>> E.as_coefficient(E)
1
>>> (2*E).as_coefficient(E)
2
>>> (2*sin(E)*E).as_coefficient(E)
Two terms have E in them so a sum is returned. (If one were
desiring the coefficient of the term exactly matching E then
the constant from the returned expression could be selected.
Or, for greater precision, a method of Poly can be used to
indicate the desired term from which the coefficient is
desired.)
>>> (2*E + x*E).as_coefficient(E)
x + 2
>>> _.args[0] # just want the exact match
2
>>> p = Poly(2*E + x*E); p
Poly(x*E + 2*E, x, E, domain='ZZ')
>>> p.coeff_monomial(E)
2
>>> p.nth(0, 1)
2
Since the following cannot be written as a product containing
E as a factor, None is returned. (If the coefficient ``2*x`` is
desired then the ``coeff`` method should be used.)
>>> (2*E*x + x).as_coefficient(E)
>>> (2*E*x + x).coeff(E)
2*x
>>> (E*(x + 1) + x).as_coefficient(E)
>>> (2*pi*I).as_coefficient(pi*I)
2
>>> (2*I).as_coefficient(pi*I)
See Also
========
coeff: return sum of terms have a given factor
as_coeff_Add: separate the additive constant from an expression
as_coeff_Mul: separate the multiplicative constant from an expression
as_independent: separate x-dependent terms/factors from others
sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly
sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used
"""
r = self.extract_multiplicatively(expr)
if r and not r.has(expr):
return r
def as_independent(self, *deps, **hint):
"""
A mostly naive separation of a Mul or Add into arguments that are not
are dependent on deps. To obtain as complete a separation of variables
as possible, use a separation method first, e.g.:
* separatevars() to change Mul, Add and Pow (including exp) into Mul
* .expand(mul=True) to change Add or Mul into Add
* .expand(log=True) to change log expr into an Add
The only non-naive thing that is done here is to respect noncommutative
ordering of variables and to always return (0, 0) for `self` of zero
regardless of hints.
For nonzero `self`, the returned tuple (i, d) has the
following interpretation:
* i will has no variable that appears in deps
* d will either have terms that contain variables that are in deps, or
be equal to 0 (when self is an Add) or 1 (when self is a Mul)
* if self is an Add then self = i + d
* if self is a Mul then self = i*d
* otherwise (self, S.One) or (S.One, self) is returned.
To force the expression to be treated as an Add, use the hint as_Add=True
Examples
========
-- self is an Add
>>> from sympy import sin, cos, exp
>>> from sympy.abc import x, y, z
>>> (x + x*y).as_independent(x)
(0, x*y + x)
>>> (x + x*y).as_independent(y)
(x, x*y)
>>> (2*x*sin(x) + y + x + z).as_independent(x)
(y + z, 2*x*sin(x) + x)
>>> (2*x*sin(x) + y + x + z).as_independent(x, y)
(z, 2*x*sin(x) + x + y)
-- self is a Mul
>>> (x*sin(x)*cos(y)).as_independent(x)
(cos(y), x*sin(x))
non-commutative terms cannot always be separated out when self is a Mul
>>> from sympy import symbols
>>> n1, n2, n3 = symbols('n1 n2 n3', commutative=False)
>>> (n1 + n1*n2).as_independent(n2)
(n1, n1*n2)
>>> (n2*n1 + n1*n2).as_independent(n2)
(0, n1*n2 + n2*n1)
>>> (n1*n2*n3).as_independent(n1)
(1, n1*n2*n3)
>>> (n1*n2*n3).as_independent(n2)
(n1, n2*n3)
>>> ((x-n1)*(x-y)).as_independent(x)
(1, (x - y)*(x - n1))
-- self is anything else:
>>> (sin(x)).as_independent(x)
(1, sin(x))
>>> (sin(x)).as_independent(y)
(sin(x), 1)
>>> exp(x+y).as_independent(x)
(1, exp(x + y))
-- force self to be treated as an Add:
>>> (3*x).as_independent(x, as_Add=True)
(0, 3*x)
-- force self to be treated as a Mul:
>>> (3+x).as_independent(x, as_Add=False)
(1, x + 3)
>>> (-3+x).as_independent(x, as_Add=False)
(1, x - 3)
Note how the below differs from the above in making the
constant on the dep term positive.
>>> (y*(-3+x)).as_independent(x)
(y, x - 3)
-- use .as_independent() for true independence testing instead
of .has(). The former considers only symbols in the free
symbols while the latter considers all symbols
>>> from sympy import Integral
>>> I = Integral(x, (x, 1, 2))
>>> I.has(x)
True
>>> x in I.free_symbols
False
>>> I.as_independent(x) == (I, 1)
True
>>> (I + x).as_independent(x) == (I, x)
True
Note: when trying to get independent terms, a separation method
might need to be used first. In this case, it is important to keep
track of what you send to this routine so you know how to interpret
the returned values
>>> from sympy import separatevars, log
>>> separatevars(exp(x+y)).as_independent(x)
(exp(y), exp(x))
>>> (x + x*y).as_independent(y)
(x, x*y)
>>> separatevars(x + x*y).as_independent(y)
(x, y + 1)
>>> (x*(1 + y)).as_independent(y)
(x, y + 1)
>>> (x*(1 + y)).expand(mul=True).as_independent(y)
(x, x*y)
>>> a, b=symbols('a b', positive=True)
>>> (log(a*b).expand(log=True)).as_independent(b)
(log(a), log(b))
See Also
========
.separatevars(), .expand(log=True), sympy.core.add.Add.as_two_terms(),
sympy.core.mul.Mul.as_two_terms(), .as_coeff_add(), .as_coeff_mul()
"""
from .symbol import Symbol
from .add import _unevaluated_Add
from .mul import _unevaluated_Mul
from sympy.utilities.iterables import sift
if self.is_zero:
return S.Zero, S.Zero
func = self.func
if hint.get('as_Add', isinstance(self, Add) ):
want = Add
else:
want = Mul
# sift out deps into symbolic and other and ignore
# all symbols but those that are in the free symbols
sym = set()
other = []
for d in deps:
if isinstance(d, Symbol): # Symbol.is_Symbol is True
sym.add(d)
else:
other.append(d)
def has(e):
"""return the standard has() if there are no literal symbols, else
check to see that symbol-deps are in the free symbols."""
has_other = e.has(*other)
if not sym:
return has_other
return has_other or e.has(*(e.free_symbols & sym))
if (want is not func or
func is not Add and func is not Mul):
if has(self):
return (want.identity, self)
else:
return (self, want.identity)
else:
if func is Add:
args = list(self.args)
else:
args, nc = self.args_cnc()
d = sift(args, lambda x: has(x))
depend = d[True]
indep = d[False]
if func is Add: # all terms were treated as commutative
return (Add(*indep), _unevaluated_Add(*depend))
else: # handle noncommutative by stopping at first dependent term
for i, n in enumerate(nc):
if has(n):
depend.extend(nc[i:])
break
indep.append(n)
return Mul(*indep), (
Mul(*depend, evaluate=False) if nc else
_unevaluated_Mul(*depend))
def as_real_imag(self, deep=True, **hints):
"""Performs complex expansion on 'self' and returns a tuple
containing collected both real and imaginary parts. This
method can't be confused with re() and im() functions,
which does not perform complex expansion at evaluation.
However it is possible to expand both re() and im()
functions and get exactly the same results as with
a single call to this function.
>>> from sympy import symbols, I
>>> x, y = symbols('x,y', real=True)
>>> (x + y*I).as_real_imag()
(x, y)
>>> from sympy.abc import z, w
>>> (z + w*I).as_real_imag()
(re(z) - im(w), re(w) + im(z))
"""
from sympy import im, re
if hints.get('ignore') == self:
return None
else:
return (re(self), im(self))
def as_powers_dict(self):
"""Return self as a dictionary of factors with each factor being
treated as a power. The keys are the bases of the factors and the
values, the corresponding exponents. The resulting dictionary should
be used with caution if the expression is a Mul and contains non-
commutative factors since the order that they appeared will be lost in
the dictionary.
See Also
========
as_ordered_factors: An alternative for noncommutative applications,
returning an ordered list of factors.
args_cnc: Similar to as_ordered_factors, but guarantees separation
of commutative and noncommutative factors.
"""
d = defaultdict(int)
d.update(dict([self.as_base_exp()]))
return d
def as_coefficients_dict(self):
"""Return a dictionary mapping terms to their Rational coefficient.
Since the dictionary is a defaultdict, inquiries about terms which
were not present will return a coefficient of 0. If an expression is
not an Add it is considered to have a single term.
Examples
========
>>> from sympy.abc import a, x
>>> (3*x + a*x + 4).as_coefficients_dict()
{1: 4, x: 3, a*x: 1}
>>> _[a]
0
>>> (3*a*x).as_coefficients_dict()
{a*x: 3}
"""
c, m = self.as_coeff_Mul()
if not c.is_Rational:
c = S.One
m = self
d = defaultdict(int)
d.update({m: c})
return d
def as_base_exp(self):
# a -> b ** e
return self, S.One
def as_coeff_mul(self, *deps, **kwargs):
"""Return the tuple (c, args) where self is written as a Mul, ``m``.
c should be a Rational multiplied by any factors of the Mul that are
independent of deps.
args should be a tuple of all other factors of m; args is empty
if self is a Number or if self is independent of deps (when given).
This should be used when you don't know if self is a Mul or not but
you want to treat self as a Mul or if you want to process the
individual arguments of the tail of self as a Mul.
- if you know self is a Mul and want only the head, use self.args[0];
- if you don't want to process the arguments of the tail but need the
tail then use self.as_two_terms() which gives the head and tail;
- if you want to split self into an independent and dependent parts
use ``self.as_independent(*deps)``
>>> from sympy import S
>>> from sympy.abc import x, y
>>> (S(3)).as_coeff_mul()
(3, ())
>>> (3*x*y).as_coeff_mul()
(3, (x, y))
>>> (3*x*y).as_coeff_mul(x)
(3*y, (x,))
>>> (3*y).as_coeff_mul(x)
(3*y, ())
"""
if deps:
if not self.has(*deps):
return self, tuple()
return S.One, (self,)
def as_coeff_add(self, *deps):
"""Return the tuple (c, args) where self is written as an Add, ``a``.
c should be a Rational added to any terms of the Add that are
independent of deps.
args should be a tuple of all other terms of ``a``; args is empty
if self is a Number or if self is independent of deps (when given).
This should be used when you don't know if self is an Add or not but
you want to treat self as an Add or if you want to process the
individual arguments of the tail of self as an Add.
- if you know self is an Add and want only the head, use self.args[0];
- if you don't want to process the arguments of the tail but need the
tail then use self.as_two_terms() which gives the head and tail.
- if you want to split self into an independent and dependent parts
use ``self.as_independent(*deps)``
>>> from sympy import S
>>> from sympy.abc import x, y
>>> (S(3)).as_coeff_add()
(3, ())
>>> (3 + x).as_coeff_add()
(3, (x,))
>>> (3 + x + y).as_coeff_add(x)
(y + 3, (x,))
>>> (3 + y).as_coeff_add(x)
(y + 3, ())
"""
if deps:
if not self.has(*deps):
return self, tuple()
return S.Zero, (self,)
def primitive(self):
"""Return the positive Rational that can be extracted non-recursively
from every term of self (i.e., self is treated like an Add). This is
like the as_coeff_Mul() method but primitive always extracts a positive
Rational (never a negative or a Float).
Examples
========
>>> from sympy.abc import x
>>> (3*(x + 1)**2).primitive()
(3, (x + 1)**2)
>>> a = (6*x + 2); a.primitive()
(2, 3*x + 1)
>>> b = (x/2 + 3); b.primitive()
(1/2, x + 6)
>>> (a*b).primitive() == (1, a*b)
True
"""
if not self:
return S.One, S.Zero
c, r = self.as_coeff_Mul(rational=True)
if c.is_negative:
c, r = -c, -r
return c, r
def as_content_primitive(self, radical=False, clear=True):
"""This method should recursively remove a Rational from all arguments
and return that (content) and the new self (primitive). The content
should always be positive and ``Mul(*foo.as_content_primitive()) == foo``.
The primitive need not be in canonical form and should try to preserve
the underlying structure if possible (i.e. expand_mul should not be
applied to self).
Examples
========
>>> from sympy import sqrt
>>> from sympy.abc import x, y, z
>>> eq = 2 + 2*x + 2*y*(3 + 3*y)
The as_content_primitive function is recursive and retains structure:
>>> eq.as_content_primitive()
(2, x + 3*y*(y + 1) + 1)
Integer powers will have Rationals extracted from the base:
>>> ((2 + 6*x)**2).as_content_primitive()
(4, (3*x + 1)**2)
>>> ((2 + 6*x)**(2*y)).as_content_primitive()
(1, (2*(3*x + 1))**(2*y))
Terms may end up joining once their as_content_primitives are added:
>>> ((5*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive()
(11, x*(y + 1))
>>> ((3*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive()
(9, x*(y + 1))
>>> ((3*(z*(1 + y)) + 2.0*x*(3 + 3*y))).as_content_primitive()
(1, 6.0*x*(y + 1) + 3*z*(y + 1))
>>> ((5*(x*(1 + y)) + 2*x*(3 + 3*y))**2).as_content_primitive()
(121, x**2*(y + 1)**2)
>>> ((x*(1 + y) + 0.4*x*(3 + 3*y))**2).as_content_primitive()
(1, 4.84*x**2*(y + 1)**2)
Radical content can also be factored out of the primitive:
>>> (2*sqrt(2) + 4*sqrt(10)).as_content_primitive(radical=True)
(2, sqrt(2)*(1 + 2*sqrt(5)))
If clear=False (default is True) then content will not be removed
from an Add if it can be distributed to leave one or more
terms with integer coefficients.
>>> (x/2 + y).as_content_primitive()
(1/2, x + 2*y)
>>> (x/2 + y).as_content_primitive(clear=False)
(1, x/2 + y)
"""
return S.One, self
def as_numer_denom(self):
""" expression -> a/b -> a, b
This is just a stub that should be defined by
an object's class methods to get anything else.
See Also
========
normal: return ``a/b`` instead of ``(a, b)``
"""
return self, S.One
def normal(self):
""" expression -> a/b
See Also
========
as_numer_denom: return ``(a, b)`` instead of ``a/b``
"""
from .mul import _unevaluated_Mul
n, d = self.as_numer_denom()
if d is S.One:
return n
if d.is_Number:
return _unevaluated_Mul(n, 1/d)
else:
return n/d
def extract_multiplicatively(self, c):
"""Return None if it's not possible to make self in the form
c * something in a nice way, i.e. preserving the properties
of arguments of self.
Examples
========
>>> from sympy import symbols, Rational
>>> x, y = symbols('x,y', real=True)
>>> ((x*y)**3).extract_multiplicatively(x**2 * y)
x*y**2
>>> ((x*y)**3).extract_multiplicatively(x**4 * y)
>>> (2*x).extract_multiplicatively(2)
x
>>> (2*x).extract_multiplicatively(3)
>>> (Rational(1, 2)*x).extract_multiplicatively(3)
x/6
"""
from .add import _unevaluated_Add
c = sympify(c)
if self is S.NaN:
return None
if c is S.One:
return self
elif c == self:
return S.One
if c.is_Add:
cc, pc = c.primitive()
if cc is not S.One:
c = Mul(cc, pc, evaluate=False)
if c.is_Mul:
a, b = c.as_two_terms()
x = self.extract_multiplicatively(a)
if x is not None:
return x.extract_multiplicatively(b)
else:
return x
quotient = self / c
if self.is_Number:
if self is S.Infinity:
if c.is_positive:
return S.Infinity
elif self is S.NegativeInfinity:
if c.is_negative:
return S.Infinity
elif c.is_positive:
return S.NegativeInfinity
elif self is S.ComplexInfinity:
if not c.is_zero:
return S.ComplexInfinity
elif self.is_Integer:
if not quotient.is_Integer:
return None
elif self.is_positive and quotient.is_negative:
return None
else:
return quotient
elif self.is_Rational:
if not quotient.is_Rational:
return None
elif self.is_positive and quotient.is_negative:
return None
else:
return quotient
elif self.is_Float:
if not quotient.is_Float:
return None
elif self.is_positive and quotient.is_negative:
return None
else:
return quotient
elif self.is_NumberSymbol or self.is_Symbol or self is S.ImaginaryUnit:
if quotient.is_Mul and len(quotient.args) == 2:
if quotient.args[0].is_Integer and quotient.args[0].is_positive and quotient.args[1] == self:
return quotient
elif quotient.is_Integer and c.is_Number:
return quotient
elif self.is_Add:
cs, ps = self.primitive()
# assert cs >= 1
if c.is_Number and c is not S.NegativeOne:
# assert c != 1 (handled at top)
if cs is not S.One:
if c.is_negative:
xc = -(cs.extract_multiplicatively(-c))
else:
xc = cs.extract_multiplicatively(c)
if xc is not None:
return xc*ps # rely on 2-arg Mul to restore Add
return # |c| != 1 can only be extracted from cs
if c == ps:
return cs
# check args of ps
newargs = []
for arg in ps.args:
newarg = arg.extract_multiplicatively(c)
if newarg is None:
return # all or nothing
newargs.append(newarg)
if cs is not S.One:
args = [cs*t for t in newargs]
# args may be in different order
return _unevaluated_Add(*args)
else:
return Add._from_args(newargs)
elif self.is_Mul:
args = list(self.args)
for i, arg in enumerate(args):
newarg = arg.extract_multiplicatively(c)
if newarg is not None:
args[i] = newarg
return Mul(*args)
elif self.is_Pow:
if c.is_Pow and c.base == self.base:
new_exp = self.exp.extract_additively(c.exp)
if new_exp is not None:
return self.base ** (new_exp)
elif c == self.base:
new_exp = self.exp.extract_additively(1)
if new_exp is not None:
return self.base ** (new_exp)
def extract_additively(self, c):
"""Return self - c if it's possible to subtract c from self and
make all matching coefficients move towards zero, else return None.
Examples
========
>>> from sympy.abc import x, y
>>> e = 2*x + 3
>>> e.extract_additively(x + 1)
x + 2
>>> e.extract_additively(3*x)
>>> e.extract_additively(4)
>>> (y*(x + 1)).extract_additively(x + 1)
>>> ((x + 1)*(x + 2*y + 1) + 3).extract_additively(x + 1)
(x + 1)*(x + 2*y) + 3
Sometimes auto-expansion will return a less simplified result
than desired; gcd_terms might be used in such cases:
>>> from sympy import gcd_terms
>>> (4*x*(y + 1) + y).extract_additively(x)
4*x*(y + 1) + x*(4*y + 3) - x*(4*y + 4) + y
>>> gcd_terms(_)
x*(4*y + 3) + y
See Also
========
extract_multiplicatively
coeff
as_coefficient
"""
c = sympify(c)
if self is S.NaN:
return None
if c.is_zero:
return self
elif c == self:
return S.Zero
elif self == S.Zero:
return None
if self.is_Number:
if not c.is_Number:
return None
co = self
diff = co - c
# XXX should we match types? i.e should 3 - .1 succeed?
if (co > 0 and diff > 0 and diff < co or
co < 0 and diff < 0 and diff > co):
return diff
return None
if c.is_Number:
co, t = self.as_coeff_Add()
xa = co.extract_additively(c)
if xa is None:
return None
return xa + t
# handle the args[0].is_Number case separately
# since we will have trouble looking for the coeff of
# a number.
if c.is_Add and c.args[0].is_Number:
# whole term as a term factor
co = self.coeff(c)
xa0 = (co.extract_additively(1) or 0)*c
if xa0:
diff = self - co*c
return (xa0 + (diff.extract_additively(c) or diff)) or None
# term-wise
h, t = c.as_coeff_Add()
sh, st = self.as_coeff_Add()
xa = sh.extract_additively(h)
if xa is None:
return None
xa2 = st.extract_additively(t)
if xa2 is None:
return None
return xa + xa2
# whole term as a term factor
co = self.coeff(c)
xa0 = (co.extract_additively(1) or 0)*c
if xa0:
diff = self - co*c
return (xa0 + (diff.extract_additively(c) or diff)) or None
# term-wise
coeffs = []
for a in Add.make_args(c):
ac, at = a.as_coeff_Mul()
co = self.coeff(at)
if not co:
return None
coc, cot = co.as_coeff_Add()
xa = coc.extract_additively(ac)
if xa is None:
return None
self -= co*at
coeffs.append((cot + xa)*at)
coeffs.append(self)
return Add(*coeffs)
@property
def expr_free_symbols(self):
"""
Like ``free_symbols``, but returns the free symbols only if they are contained in an expression node.
Examples
========
>>> from sympy.abc import x, y
>>> (x + y).expr_free_symbols
{x, y}
If the expression is contained in a non-expression object, don't return
the free symbols. Compare:
>>> from sympy import Tuple
>>> t = Tuple(x + y)
>>> t.expr_free_symbols
set()
>>> t.free_symbols
{x, y}
"""
return {j for i in self.args for j in i.expr_free_symbols}
def could_extract_minus_sign(self):
"""Return True if self is not in a canonical form with respect
to its sign.
For most expressions, e, there will be a difference in e and -e.
When there is, True will be returned for one and False for the
other; False will be returned if there is no difference.
Examples
========
>>> from sympy.abc import x, y
>>> e = x - y
>>> {i.could_extract_minus_sign() for i in (e, -e)}
{False, True}
"""
negative_self = -self
if self == negative_self:
return False # e.g. zoo*x == -zoo*x
self_has_minus = (self.extract_multiplicatively(-1) is not None)
negative_self_has_minus = (
(negative_self).extract_multiplicatively(-1) is not None)
if self_has_minus != negative_self_has_minus:
return self_has_minus
else:
if self.is_Add:
# We choose the one with less arguments with minus signs
all_args = len(self.args)
negative_args = len([False for arg in self.args if arg.could_extract_minus_sign()])
positive_args = all_args - negative_args
if positive_args > negative_args:
return False
elif positive_args < negative_args:
return True
elif self.is_Mul:
# We choose the one with an odd number of minus signs
num, den = self.as_numer_denom()
args = Mul.make_args(num) + Mul.make_args(den)
arg_signs = [arg.could_extract_minus_sign() for arg in args]
negative_args = list(filter(None, arg_signs))
return len(negative_args) % 2 == 1
# As a last resort, we choose the one with greater value of .sort_key()
return bool(self.sort_key() < negative_self.sort_key())
def extract_branch_factor(self, allow_half=False):
"""
Try to write self as ``exp_polar(2*pi*I*n)*z`` in a nice way.
Return (z, n).
>>> from sympy import exp_polar, I, pi
>>> from sympy.abc import x, y
>>> exp_polar(I*pi).extract_branch_factor()
(exp_polar(I*pi), 0)
>>> exp_polar(2*I*pi).extract_branch_factor()
(1, 1)
>>> exp_polar(-pi*I).extract_branch_factor()
(exp_polar(I*pi), -1)
>>> exp_polar(3*pi*I + x).extract_branch_factor()
(exp_polar(x + I*pi), 1)
>>> (y*exp_polar(-5*pi*I)*exp_polar(3*pi*I + 2*pi*x)).extract_branch_factor()
(y*exp_polar(2*pi*x), -1)
>>> exp_polar(-I*pi/2).extract_branch_factor()
(exp_polar(-I*pi/2), 0)
If allow_half is True, also extract exp_polar(I*pi):
>>> exp_polar(I*pi).extract_branch_factor(allow_half=True)
(1, 1/2)
>>> exp_polar(2*I*pi).extract_branch_factor(allow_half=True)
(1, 1)
>>> exp_polar(3*I*pi).extract_branch_factor(allow_half=True)
(1, 3/2)
>>> exp_polar(-I*pi).extract_branch_factor(allow_half=True)
(1, -1/2)
"""
from sympy import exp_polar, pi, I, ceiling, Add
n = S.Zero
res = S.One
args = Mul.make_args(self)
exps = []
for arg in args:
if isinstance(arg, exp_polar):
exps += [arg.exp]
else:
res *= arg
piimult = S.Zero
extras = []
while exps:
exp = exps.pop()
if exp.is_Add:
exps += exp.args
continue
if exp.is_Mul:
coeff = exp.as_coefficient(pi*I)
if coeff is not None:
piimult += coeff
continue
extras += [exp]
if piimult.is_number:
coeff = piimult
tail = ()
else:
coeff, tail = piimult.as_coeff_add(*piimult.free_symbols)
# round down to nearest multiple of 2
branchfact = ceiling(coeff/2 - S.Half)*2
n += branchfact/2
c = coeff - branchfact
if allow_half:
nc = c.extract_additively(1)
if nc is not None:
n += S.Half
c = nc
newexp = pi*I*Add(*((c, ) + tail)) + Add(*extras)
if newexp != 0:
res *= exp_polar(newexp)
return res, n
def _eval_is_polynomial(self, syms):
if self.free_symbols.intersection(syms) == set():
return True
return False
def is_polynomial(self, *syms):
r"""
Return True if self is a polynomial in syms and False otherwise.
This checks if self is an exact polynomial in syms. This function
returns False for expressions that are "polynomials" with symbolic
exponents. Thus, you should be able to apply polynomial algorithms to
expressions for which this returns True, and Poly(expr, \*syms) should
work if and only if expr.is_polynomial(\*syms) returns True. The
polynomial does not have to be in expanded form. If no symbols are
given, all free symbols in the expression will be used.
This is not part of the assumptions system. You cannot do
Symbol('z', polynomial=True).
Examples
========
>>> from sympy import Symbol
>>> x = Symbol('x')
>>> ((x**2 + 1)**4).is_polynomial(x)
True
>>> ((x**2 + 1)**4).is_polynomial()
True
>>> (2**x + 1).is_polynomial(x)
False
>>> n = Symbol('n', nonnegative=True, integer=True)
>>> (x**n + 1).is_polynomial(x)
False
This function does not attempt any nontrivial simplifications that may
result in an expression that does not appear to be a polynomial to
become one.
>>> from sympy import sqrt, factor, cancel
>>> y = Symbol('y', positive=True)
>>> a = sqrt(y**2 + 2*y + 1)
>>> a.is_polynomial(y)
False
>>> factor(a)
y + 1
>>> factor(a).is_polynomial(y)
True
>>> b = (y**2 + 2*y + 1)/(y + 1)
>>> b.is_polynomial(y)
False
>>> cancel(b)
y + 1
>>> cancel(b).is_polynomial(y)
True
See also .is_rational_function()
"""
if syms:
syms = set(map(sympify, syms))
else:
syms = self.free_symbols
if syms.intersection(self.free_symbols) == set():
# constant polynomial
return True
else:
return self._eval_is_polynomial(syms)
def _eval_is_rational_function(self, syms):
if self.free_symbols.intersection(syms) == set():
return True
return False
def is_rational_function(self, *syms):
"""
Test whether function is a ratio of two polynomials in the given
symbols, syms. When syms is not given, all free symbols will be used.
The rational function does not have to be in expanded or in any kind of
canonical form.
This function returns False for expressions that are "rational
functions" with symbolic exponents. Thus, you should be able to call
.as_numer_denom() and apply polynomial algorithms to the result for
expressions for which this returns True.
This is not part of the assumptions system. You cannot do
Symbol('z', rational_function=True).
Examples
========
>>> from sympy import Symbol, sin
>>> from sympy.abc import x, y
>>> (x/y).is_rational_function()
True
>>> (x**2).is_rational_function()
True
>>> (x/sin(y)).is_rational_function(y)
False
>>> n = Symbol('n', integer=True)
>>> (x**n + 1).is_rational_function(x)
False
This function does not attempt any nontrivial simplifications that may
result in an expression that does not appear to be a rational function
to become one.
>>> from sympy import sqrt, factor
>>> y = Symbol('y', positive=True)
>>> a = sqrt(y**2 + 2*y + 1)/y
>>> a.is_rational_function(y)
False
>>> factor(a)
(y + 1)/y
>>> factor(a).is_rational_function(y)
True
See also is_algebraic_expr().
"""
if self in [S.NaN, S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:
return False
if syms:
syms = set(map(sympify, syms))
else:
syms = self.free_symbols
if syms.intersection(self.free_symbols) == set():
# constant rational function
return True
else:
return self._eval_is_rational_function(syms)
def _eval_is_meromorphic(self, x, a):
# Default implementation, return True for constants.
return None if self.has(x) else True
def is_meromorphic(self, x, a):
"""
This tests whether an expression is meromorphic as
a function of the given symbol ``x`` at the point ``a``.
This method is intended as a quick test that will return
None if no decision can be made without simplification or
more detailed analysis.
Examples
========
>>> from sympy import zoo, log, sin, sqrt
>>> from sympy.abc import x
>>> f = 1/x**2 + 1 - 2*x**3
>>> f.is_meromorphic(x, 0)
True
>>> f.is_meromorphic(x, 1)
True
>>> f.is_meromorphic(x, zoo)
True
>>> g = x**log(3)
>>> g.is_meromorphic(x, 0)
False
>>> g.is_meromorphic(x, 1)
True
>>> g.is_meromorphic(x, zoo)
False
>>> h = sin(1/x)*x**2
>>> h.is_meromorphic(x, 0)
False
>>> h.is_meromorphic(x, 1)
True
>>> h.is_meromorphic(x, zoo)
True
Multivalued functions are considered meromorphic when their
branches are meromorphic. Thus most functions are meromorphic
everywhere except at essential singularities and branch points.
In particular, they will be meromorphic also on branch cuts
except at their endpoints.
>>> log(x).is_meromorphic(x, -1)
True
>>> log(x).is_meromorphic(x, 0)
False
>>> sqrt(x).is_meromorphic(x, -1)
True
>>> sqrt(x).is_meromorphic(x, 0)
False
"""
if not x.is_symbol:
raise TypeError("{} should be of symbol type".format(x))
a = sympify(a)
return self._eval_is_meromorphic(x, a)
def _eval_is_algebraic_expr(self, syms):
if self.free_symbols.intersection(syms) == set():
return True
return False
def is_algebraic_expr(self, *syms):
"""
This tests whether a given expression is algebraic or not, in the
given symbols, syms. When syms is not given, all free symbols
will be used. The rational function does not have to be in expanded
or in any kind of canonical form.
This function returns False for expressions that are "algebraic
expressions" with symbolic exponents. This is a simple extension to the
is_rational_function, including rational exponentiation.
Examples
========
>>> from sympy import Symbol, sqrt
>>> x = Symbol('x', real=True)
>>> sqrt(1 + x).is_rational_function()
False
>>> sqrt(1 + x).is_algebraic_expr()
True
This function does not attempt any nontrivial simplifications that may
result in an expression that does not appear to be an algebraic
expression to become one.
>>> from sympy import exp, factor
>>> a = sqrt(exp(x)**2 + 2*exp(x) + 1)/(exp(x) + 1)
>>> a.is_algebraic_expr(x)
False
>>> factor(a).is_algebraic_expr()
True
See Also
========
is_rational_function()
References
==========
- https://en.wikipedia.org/wiki/Algebraic_expression
"""
if syms:
syms = set(map(sympify, syms))
else:
syms = self.free_symbols
if syms.intersection(self.free_symbols) == set():
# constant algebraic expression
return True
else:
return self._eval_is_algebraic_expr(syms)
###################################################################################
##################### SERIES, LEADING TERM, LIMIT, ORDER METHODS ##################
###################################################################################
def series(self, x=None, x0=0, n=6, dir="+", logx=None, cdir=0):
"""
Series expansion of "self" around ``x = x0`` yielding either terms of
the series one by one (the lazy series given when n=None), else
all the terms at once when n != None.
Returns the series expansion of "self" around the point ``x = x0``
with respect to ``x`` up to ``O((x - x0)**n, x, x0)`` (default n is 6).
If ``x=None`` and ``self`` is univariate, the univariate symbol will
be supplied, otherwise an error will be raised.
Parameters
==========
expr : Expression
The expression whose series is to be expanded.
x : Symbol
It is the variable of the expression to be calculated.
x0 : Value
The value around which ``x`` is calculated. Can be any value
from ``-oo`` to ``oo``.
n : Value
The number of terms upto which the series is to be expanded.
dir : String, optional
The series-expansion can be bi-directional. If ``dir="+"``,
then (x->x0+). If ``dir="-", then (x->x0-). For infinite
``x0`` (``oo`` or ``-oo``), the ``dir`` argument is determined
from the direction of the infinity (i.e., ``dir="-"`` for
``oo``).
logx : optional
It is used to replace any log(x) in the returned series with a
symbolic value rather than evaluating the actual value.
cdir : optional
It stands for complex direction, and indicates the direction
from which the expansion needs to be evaluated.
Examples
========
>>> from sympy import cos, exp, tan
>>> from sympy.abc import x, y
>>> cos(x).series()
1 - x**2/2 + x**4/24 + O(x**6)
>>> cos(x).series(n=4)
1 - x**2/2 + O(x**4)
>>> cos(x).series(x, x0=1, n=2)
cos(1) - (x - 1)*sin(1) + O((x - 1)**2, (x, 1))
>>> e = cos(x + exp(y))
>>> e.series(y, n=2)
cos(x + 1) - y*sin(x + 1) + O(y**2)
>>> e.series(x, n=2)
cos(exp(y)) - x*sin(exp(y)) + O(x**2)
If ``n=None`` then a generator of the series terms will be returned.
>>> term=cos(x).series(n=None)
>>> [next(term) for i in range(2)]
[1, -x**2/2]
For ``dir=+`` (default) the series is calculated from the right and
for ``dir=-`` the series from the left. For smooth functions this
flag will not alter the results.
>>> abs(x).series(dir="+")
x
>>> abs(x).series(dir="-")
-x
>>> f = tan(x)
>>> f.series(x, 2, 6, "+")
tan(2) + (1 + tan(2)**2)*(x - 2) + (x - 2)**2*(tan(2)**3 + tan(2)) +
(x - 2)**3*(1/3 + 4*tan(2)**2/3 + tan(2)**4) + (x - 2)**4*(tan(2)**5 +
5*tan(2)**3/3 + 2*tan(2)/3) + (x - 2)**5*(2/15 + 17*tan(2)**2/15 +
2*tan(2)**4 + tan(2)**6) + O((x - 2)**6, (x, 2))
>>> f.series(x, 2, 3, "-")
tan(2) + (2 - x)*(-tan(2)**2 - 1) + (2 - x)**2*(tan(2)**3 + tan(2))
+ O((x - 2)**3, (x, 2))
Returns
=======
Expr : Expression
Series expansion of the expression about x0
Raises
======
TypeError
If "n" and "x0" are infinity objects
PoleError
If "x0" is an infinity object
"""
from sympy import collect, Dummy, Order, Rational, Symbol, ceiling
if x is None:
syms = self.free_symbols
if not syms:
return self
elif len(syms) > 1:
raise ValueError('x must be given for multivariate functions.')
x = syms.pop()
if isinstance(x, Symbol):
dep = x in self.free_symbols
else:
d = Dummy()
dep = d in self.xreplace({x: d}).free_symbols
if not dep:
if n is None:
return (s for s in [self])
else:
return self
if len(dir) != 1 or dir not in '+-':
raise ValueError("Dir must be '+' or '-'")
if x0 in [S.Infinity, S.NegativeInfinity]:
sgn = 1 if x0 is S.Infinity else -1
s = self.subs(x, sgn/x).series(x, n=n, dir='+', cdir=cdir)
if n is None:
return (si.subs(x, sgn/x) for si in s)
return s.subs(x, sgn/x)
# use rep to shift origin to x0 and change sign (if dir is negative)
# and undo the process with rep2
if x0 or dir == '-':
if dir == '-':
rep = -x + x0
rep2 = -x
rep2b = x0
else:
rep = x + x0
rep2 = x
rep2b = -x0
s = self.subs(x, rep).series(x, x0=0, n=n, dir='+', logx=logx, cdir=cdir)
if n is None: # lseries...
return (si.subs(x, rep2 + rep2b) for si in s)
return s.subs(x, rep2 + rep2b)
# from here on it's x0=0 and dir='+' handling
if x.is_positive is x.is_negative is None or x.is_Symbol is not True:
# replace x with an x that has a positive assumption
xpos = Dummy('x', positive=True, finite=True)
rv = self.subs(x, xpos).series(xpos, x0, n, dir, logx=logx, cdir=cdir)
if n is None:
return (s.subs(xpos, x) for s in rv)
else:
return rv.subs(xpos, x)
if n is not None: # nseries handling
s1 = self._eval_nseries(x, n=n, logx=logx, cdir=cdir)
o = s1.getO() or S.Zero
if o:
# make sure the requested order is returned
ngot = o.getn()
if ngot > n:
# leave o in its current form (e.g. with x*log(x)) so
# it eats terms properly, then replace it below
if n != 0:
s1 += o.subs(x, x**Rational(n, ngot))
else:
s1 += Order(1, x)
elif ngot < n:
# increase the requested number of terms to get the desired
# number keep increasing (up to 9) until the received order
# is different than the original order and then predict how
# many additional terms are needed
for more in range(1, 9):
s1 = self._eval_nseries(x, n=n + more, logx=logx, cdir=cdir)
newn = s1.getn()
if newn != ngot:
ndo = n + ceiling((n - ngot)*more/(newn - ngot))
s1 = self._eval_nseries(x, n=ndo, logx=logx, cdir=cdir)
while s1.getn() < n:
s1 = self._eval_nseries(x, n=ndo, logx=logx, cdir=cdir)
ndo += 1
break
else:
raise ValueError('Could not calculate %s terms for %s'
% (str(n), self))
s1 += Order(x**n, x)
o = s1.getO()
s1 = s1.removeO()
elif s1.has(Order):
# asymptotic expansion
return s1
else:
o = Order(x**n, x)
s1done = s1.doit()
if (s1done + o).removeO() == s1done:
o = S.Zero
try:
return collect(s1, x) + o
except NotImplementedError:
return s1 + o
else: # lseries handling
def yield_lseries(s):
"""Return terms of lseries one at a time."""
for si in s:
if not si.is_Add:
yield si
continue
# yield terms 1 at a time if possible
# by increasing order until all the
# terms have been returned
yielded = 0
o = Order(si, x)*x
ndid = 0
ndo = len(si.args)
while 1:
do = (si - yielded + o).removeO()
o *= x
if not do or do.is_Order:
continue
if do.is_Add:
ndid += len(do.args)
else:
ndid += 1
yield do
if ndid == ndo:
break
yielded += do
return yield_lseries(self.removeO()._eval_lseries(x, logx=logx, cdir=cdir))
def aseries(self, x=None, n=6, bound=0, hir=False):
"""Asymptotic Series expansion of self.
This is equivalent to ``self.series(x, oo, n)``.
Parameters
==========
self : Expression
The expression whose series is to be expanded.
x : Symbol
It is the variable of the expression to be calculated.
n : Value
The number of terms upto which the series is to be expanded.
hir : Boolean
Set this parameter to be True to produce hierarchical series.
It stops the recursion at an early level and may provide nicer
and more useful results.
bound : Value, Integer
Use the ``bound`` parameter to give limit on rewriting
coefficients in its normalised form.
Examples
========
>>> from sympy import sin, exp
>>> from sympy.abc import x
>>> e = sin(1/x + exp(-x)) - sin(1/x)
>>> e.aseries(x)
(1/(24*x**4) - 1/(2*x**2) + 1 + O(x**(-6), (x, oo)))*exp(-x)
>>> e.aseries(x, n=3, hir=True)
-exp(-2*x)*sin(1/x)/2 + exp(-x)*cos(1/x) + O(exp(-3*x), (x, oo))
>>> e = exp(exp(x)/(1 - 1/x))
>>> e.aseries(x)
exp(exp(x)/(1 - 1/x))
>>> e.aseries(x, bound=3)
exp(exp(x)/x**2)*exp(exp(x)/x)*exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x**2)*exp(exp(x))
Returns
=======
Expr
Asymptotic series expansion of the expression.
Notes
=====
This algorithm is directly induced from the limit computational algorithm provided by Gruntz.
It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first
to look for the most rapidly varying subexpression w of a given expression f and then expands f
in a series in w. Then same thing is recursively done on the leading coefficient
till we get constant coefficients.
If the most rapidly varying subexpression of a given expression f is f itself,
the algorithm tries to find a normalised representation of the mrv set and rewrites f
using this normalised representation.
If the expansion contains an order term, it will be either ``O(x ** (-n))`` or ``O(w ** (-n))``
where ``w`` belongs to the most rapidly varying expression of ``self``.
References
==========
.. [1] A New Algorithm for Computing Asymptotic Series - Dominik Gruntz
.. [2] Gruntz thesis - p90
.. [3] http://en.wikipedia.org/wiki/Asymptotic_expansion
See Also
========
Expr.aseries: See the docstring of this function for complete details of this wrapper.
"""
from sympy import Order, Dummy
from sympy.functions import exp, log
from sympy.series.gruntz import mrv, rewrite
if x.is_positive is x.is_negative is None:
xpos = Dummy('x', positive=True)
return self.subs(x, xpos).aseries(xpos, n, bound, hir).subs(xpos, x)
om, exps = mrv(self, x)
# We move one level up by replacing `x` by `exp(x)`, and then
# computing the asymptotic series for f(exp(x)). Then asymptotic series
# can be obtained by moving one-step back, by replacing x by ln(x).
if x in om:
s = self.subs(x, exp(x)).aseries(x, n, bound, hir).subs(x, log(x))
if s.getO():
return s + Order(1/x**n, (x, S.Infinity))
return s
k = Dummy('k', positive=True)
# f is rewritten in terms of omega
func, logw = rewrite(exps, om, x, k)
if self in om:
if bound <= 0:
return self
s = (self.exp).aseries(x, n, bound=bound)
s = s.func(*[t.removeO() for t in s.args])
res = exp(s.subs(x, 1/x).as_leading_term(x).subs(x, 1/x))
func = exp(self.args[0] - res.args[0]) / k
logw = log(1/res)
s = func.series(k, 0, n)
# Hierarchical series
if hir:
return s.subs(k, exp(logw))
o = s.getO()
terms = sorted(Add.make_args(s.removeO()), key=lambda i: int(i.as_coeff_exponent(k)[1]))
s = S.Zero
has_ord = False
# Then we recursively expand these coefficients one by one into
# their asymptotic series in terms of their most rapidly varying subexpressions.
for t in terms:
coeff, expo = t.as_coeff_exponent(k)
if coeff.has(x):
# Recursive step
snew = coeff.aseries(x, n, bound=bound-1)
if has_ord and snew.getO():
break
elif snew.getO():
has_ord = True
s += (snew * k**expo)
else:
s += t
if not o or has_ord:
return s.subs(k, exp(logw))
return (s + o).subs(k, exp(logw))
def taylor_term(self, n, x, *previous_terms):
"""General method for the taylor term.
This method is slow, because it differentiates n-times. Subclasses can
redefine it to make it faster by using the "previous_terms".
"""
from sympy import Dummy, factorial
x = sympify(x)
_x = Dummy('x')
return self.subs(x, _x).diff(_x, n).subs(_x, x).subs(x, 0) * x**n / factorial(n)
def lseries(self, x=None, x0=0, dir='+', logx=None, cdir=0):
"""
Wrapper for series yielding an iterator of the terms of the series.
Note: an infinite series will yield an infinite iterator. The following,
for exaxmple, will never terminate. It will just keep printing terms
of the sin(x) series::
for term in sin(x).lseries(x):
print term
The advantage of lseries() over nseries() is that many times you are
just interested in the next term in the series (i.e. the first term for
example), but you don't know how many you should ask for in nseries()
using the "n" parameter.
See also nseries().
"""
return self.series(x, x0, n=None, dir=dir, logx=logx, cdir=cdir)
def _eval_lseries(self, x, logx=None, cdir=0):
# default implementation of lseries is using nseries(), and adaptively
# increasing the "n". As you can see, it is not very efficient, because
# we are calculating the series over and over again. Subclasses should
# override this method and implement much more efficient yielding of
# terms.
n = 0
series = self._eval_nseries(x, n=n, logx=logx, cdir=cdir)
while series.is_Order:
n += 1
series = self._eval_nseries(x, n=n, logx=logx, cdir=cdir)
e = series.removeO()
yield e
if e is S.Zero:
return
while 1:
while 1:
n += 1
series = self._eval_nseries(x, n=n, logx=logx, cdir=cdir).removeO()
if e != series:
break
if (series - self).cancel() is S.Zero:
return
yield series - e
e = series
def nseries(self, x=None, x0=0, n=6, dir='+', logx=None, cdir=0):
"""
Wrapper to _eval_nseries if assumptions allow, else to series.
If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is
called. This calculates "n" terms in the innermost expressions and
then builds up the final series just by "cross-multiplying" everything
out.
The optional ``logx`` parameter can be used to replace any log(x) in the
returned series with a symbolic value to avoid evaluating log(x) at 0. A
symbol to use in place of log(x) should be provided.
Advantage -- it's fast, because we don't have to determine how many
terms we need to calculate in advance.
Disadvantage -- you may end up with less terms than you may have
expected, but the O(x**n) term appended will always be correct and
so the result, though perhaps shorter, will also be correct.
If any of those assumptions is not met, this is treated like a
wrapper to series which will try harder to return the correct
number of terms.
See also lseries().
Examples
========
>>> from sympy import sin, log, Symbol
>>> from sympy.abc import x, y
>>> sin(x).nseries(x, 0, 6)
x - x**3/6 + x**5/120 + O(x**6)
>>> log(x+1).nseries(x, 0, 5)
x - x**2/2 + x**3/3 - x**4/4 + O(x**5)
Handling of the ``logx`` parameter --- in the following example the
expansion fails since ``sin`` does not have an asymptotic expansion
at -oo (the limit of log(x) as x approaches 0):
>>> e = sin(log(x))
>>> e.nseries(x, 0, 6)
Traceback (most recent call last):
...
PoleError: ...
...
>>> logx = Symbol('logx')
>>> e.nseries(x, 0, 6, logx=logx)
sin(logx)
In the following example, the expansion works but gives only an Order term
unless the ``logx`` parameter is used:
>>> e = x**y
>>> e.nseries(x, 0, 2)
O(log(x)**2)
>>> e.nseries(x, 0, 2, logx=logx)
exp(logx*y)
"""
if x and not x in self.free_symbols:
return self
if x is None or x0 or dir != '+': # {see XPOS above} or (x.is_positive == x.is_negative == None):
return self.series(x, x0, n, dir, cdir=cdir)
else:
return self._eval_nseries(x, n=n, logx=logx, cdir=cdir)
def _eval_nseries(self, x, n, logx, cdir):
"""
Return terms of series for self up to O(x**n) at x=0
from the positive direction.
This is a method that should be overridden in subclasses. Users should
never call this method directly (use .nseries() instead), so you don't
have to write docstrings for _eval_nseries().
"""
from sympy.utilities.misc import filldedent
raise NotImplementedError(filldedent("""
The _eval_nseries method should be added to
%s to give terms up to O(x**n) at x=0
from the positive direction so it is available when
nseries calls it.""" % self.func)
)
def limit(self, x, xlim, dir='+'):
""" Compute limit x->xlim.
"""
from sympy.series.limits import limit
return limit(self, x, xlim, dir)
def compute_leading_term(self, x, logx=None):
"""
as_leading_term is only allowed for results of .series()
This is a wrapper to compute a series first.
"""
from sympy import Dummy, log, Piecewise, piecewise_fold
from sympy.series.gruntz import calculate_series
if self.has(Piecewise):
expr = piecewise_fold(self)
else:
expr = self
if self.removeO() == 0:
return self
if logx is None:
d = Dummy('logx')
s = calculate_series(expr, x, d).subs(d, log(x))
else:
s = calculate_series(expr, x, logx)
return s.as_leading_term(x)
@cacheit
def as_leading_term(self, *symbols, cdir=0):
"""
Returns the leading (nonzero) term of the series expansion of self.
The _eval_as_leading_term routines are used to do this, and they must
always return a non-zero value.
Examples
========
>>> from sympy.abc import x
>>> (1 + x + x**2).as_leading_term(x)
1
>>> (1/x**2 + x + x**2).as_leading_term(x)
x**(-2)
"""
from sympy import powsimp
if len(symbols) > 1:
c = self
for x in symbols:
c = c.as_leading_term(x, cdir=cdir)
return c
elif not symbols:
return self
x = sympify(symbols[0])
if not x.is_symbol:
raise ValueError('expecting a Symbol but got %s' % x)
if x not in self.free_symbols:
return self
obj = self._eval_as_leading_term(x, cdir=cdir)
if obj is not None:
return powsimp(obj, deep=True, combine='exp')
raise NotImplementedError('as_leading_term(%s, %s)' % (self, x))
def _eval_as_leading_term(self, x, cdir=0):
return self
def as_coeff_exponent(self, x):
""" ``c*x**e -> c,e`` where x can be any symbolic expression.
"""
from sympy import collect
s = collect(self, x)
c, p = s.as_coeff_mul(x)
if len(p) == 1:
b, e = p[0].as_base_exp()
if b == x:
return c, e
return s, S.Zero
def leadterm(self, x, cdir=0):
"""
Returns the leading term a*x**b as a tuple (a, b).
Examples
========
>>> from sympy.abc import x
>>> (1+x+x**2).leadterm(x)
(1, 0)
>>> (1/x**2+x+x**2).leadterm(x)
(1, -2)
"""
from sympy import Dummy, log
l = self.as_leading_term(x, cdir=cdir)
d = Dummy('logx')
if l.has(log(x)):
l = l.subs(log(x), d)
c, e = l.as_coeff_exponent(x)
if x in c.free_symbols:
from sympy.utilities.misc import filldedent
raise ValueError(filldedent("""
cannot compute leadterm(%s, %s). The coefficient
should have been free of %s but got %s""" % (self, x, x, c)))
c = c.subs(d, log(x))
return c, e
def as_coeff_Mul(self, rational=False):
"""Efficiently extract the coefficient of a product. """
return S.One, self
def as_coeff_Add(self, rational=False):
"""Efficiently extract the coefficient of a summation. """
return S.Zero, self
def fps(self, x=None, x0=0, dir=1, hyper=True, order=4, rational=True,
full=False):
"""
Compute formal power power series of self.
See the docstring of the :func:`fps` function in sympy.series.formal for
more information.
"""
from sympy.series.formal import fps
return fps(self, x, x0, dir, hyper, order, rational, full)
def fourier_series(self, limits=None):
"""Compute fourier sine/cosine series of self.
See the docstring of the :func:`fourier_series` in sympy.series.fourier
for more information.
"""
from sympy.series.fourier import fourier_series
return fourier_series(self, limits)
###################################################################################
##################### DERIVATIVE, INTEGRAL, FUNCTIONAL METHODS ####################
###################################################################################
def diff(self, *symbols, **assumptions):
assumptions.setdefault("evaluate", True)
return _derivative_dispatch(self, *symbols, **assumptions)
###########################################################################
###################### EXPRESSION EXPANSION METHODS #######################
###########################################################################
# Relevant subclasses should override _eval_expand_hint() methods. See
# the docstring of expand() for more info.
def _eval_expand_complex(self, **hints):
real, imag = self.as_real_imag(**hints)
return real + S.ImaginaryUnit*imag
@staticmethod
def _expand_hint(expr, hint, deep=True, **hints):
"""
Helper for ``expand()``. Recursively calls ``expr._eval_expand_hint()``.
Returns ``(expr, hit)``, where expr is the (possibly) expanded
``expr`` and ``hit`` is ``True`` if ``expr`` was truly expanded and
``False`` otherwise.
"""
hit = False
# XXX: Hack to support non-Basic args
# |
# V
if deep and getattr(expr, 'args', ()) and not expr.is_Atom:
sargs = []
for arg in expr.args:
arg, arghit = Expr._expand_hint(arg, hint, **hints)
hit |= arghit
sargs.append(arg)
if hit:
expr = expr.func(*sargs)
if hasattr(expr, hint):
newexpr = getattr(expr, hint)(**hints)
if newexpr != expr:
return (newexpr, True)
return (expr, hit)
@cacheit
def expand(self, deep=True, modulus=None, power_base=True, power_exp=True,
mul=True, log=True, multinomial=True, basic=True, **hints):
"""
Expand an expression using hints.
See the docstring of the expand() function in sympy.core.function for
more information.
"""
from sympy.simplify.radsimp import fraction
hints.update(power_base=power_base, power_exp=power_exp, mul=mul,
log=log, multinomial=multinomial, basic=basic)
expr = self
if hints.pop('frac', False):
n, d = [a.expand(deep=deep, modulus=modulus, **hints)
for a in fraction(self)]
return n/d
elif hints.pop('denom', False):
n, d = fraction(self)
return n/d.expand(deep=deep, modulus=modulus, **hints)
elif hints.pop('numer', False):
n, d = fraction(self)
return n.expand(deep=deep, modulus=modulus, **hints)/d
# Although the hints are sorted here, an earlier hint may get applied
# at a given node in the expression tree before another because of how
# the hints are applied. e.g. expand(log(x*(y + z))) -> log(x*y +
# x*z) because while applying log at the top level, log and mul are
# applied at the deeper level in the tree so that when the log at the
# upper level gets applied, the mul has already been applied at the
# lower level.
# Additionally, because hints are only applied once, the expression
# may not be expanded all the way. For example, if mul is applied
# before multinomial, x*(x + 1)**2 won't be expanded all the way. For
# now, we just use a special case to make multinomial run before mul,
# so that at least polynomials will be expanded all the way. In the
# future, smarter heuristics should be applied.
# TODO: Smarter heuristics
def _expand_hint_key(hint):
"""Make multinomial come before mul"""
if hint == 'mul':
return 'mulz'
return hint
for hint in sorted(hints.keys(), key=_expand_hint_key):
use_hint = hints[hint]
if use_hint:
hint = '_eval_expand_' + hint
expr, hit = Expr._expand_hint(expr, hint, deep=deep, **hints)
while True:
was = expr
if hints.get('multinomial', False):
expr, _ = Expr._expand_hint(
expr, '_eval_expand_multinomial', deep=deep, **hints)
if hints.get('mul', False):
expr, _ = Expr._expand_hint(
expr, '_eval_expand_mul', deep=deep, **hints)
if hints.get('log', False):
expr, _ = Expr._expand_hint(
expr, '_eval_expand_log', deep=deep, **hints)
if expr == was:
break
if modulus is not None:
modulus = sympify(modulus)
if not modulus.is_Integer or modulus <= 0:
raise ValueError(
"modulus must be a positive integer, got %s" % modulus)
terms = []
for term in Add.make_args(expr):
coeff, tail = term.as_coeff_Mul(rational=True)
coeff %= modulus
if coeff:
terms.append(coeff*tail)
expr = Add(*terms)
return expr
###########################################################################
################### GLOBAL ACTION VERB WRAPPER METHODS ####################
###########################################################################
def integrate(self, *args, **kwargs):
"""See the integrate function in sympy.integrals"""
from sympy.integrals import integrate
return integrate(self, *args, **kwargs)
def nsimplify(self, constants=[], tolerance=None, full=False):
"""See the nsimplify function in sympy.simplify"""
from sympy.simplify import nsimplify
return nsimplify(self, constants, tolerance, full)
def separate(self, deep=False, force=False):
"""See the separate function in sympy.simplify"""
from sympy.core.function import expand_power_base
return expand_power_base(self, deep=deep, force=force)
def collect(self, syms, func=None, evaluate=True, exact=False, distribute_order_term=True):
"""See the collect function in sympy.simplify"""
from sympy.simplify import collect
return collect(self, syms, func, evaluate, exact, distribute_order_term)
def together(self, *args, **kwargs):
"""See the together function in sympy.polys"""
from sympy.polys import together
return together(self, *args, **kwargs)
def apart(self, x=None, **args):
"""See the apart function in sympy.polys"""
from sympy.polys import apart
return apart(self, x, **args)
def ratsimp(self):
"""See the ratsimp function in sympy.simplify"""
from sympy.simplify import ratsimp
return ratsimp(self)
def trigsimp(self, **args):
"""See the trigsimp function in sympy.simplify"""
from sympy.simplify import trigsimp
return trigsimp(self, **args)
def radsimp(self, **kwargs):
"""See the radsimp function in sympy.simplify"""
from sympy.simplify import radsimp
return radsimp(self, **kwargs)
def powsimp(self, *args, **kwargs):
"""See the powsimp function in sympy.simplify"""
from sympy.simplify import powsimp
return powsimp(self, *args, **kwargs)
def combsimp(self):
"""See the combsimp function in sympy.simplify"""
from sympy.simplify import combsimp
return combsimp(self)
def gammasimp(self):
"""See the gammasimp function in sympy.simplify"""
from sympy.simplify import gammasimp
return gammasimp(self)
def factor(self, *gens, **args):
"""See the factor() function in sympy.polys.polytools"""
from sympy.polys import factor
return factor(self, *gens, **args)
def cancel(self, *gens, **args):
"""See the cancel function in sympy.polys"""
from sympy.polys import cancel
return cancel(self, *gens, **args)
def invert(self, g, *gens, **args):
"""Return the multiplicative inverse of ``self`` mod ``g``
where ``self`` (and ``g``) may be symbolic expressions).
See Also
========
sympy.core.numbers.mod_inverse, sympy.polys.polytools.invert
"""
from sympy.polys.polytools import invert
from sympy.core.numbers import mod_inverse
if self.is_number and getattr(g, 'is_number', True):
return mod_inverse(self, g)
return invert(self, g, *gens, **args)
def round(self, n=None):
"""Return x rounded to the given decimal place.
If a complex number would results, apply round to the real
and imaginary components of the number.
Examples
========
>>> from sympy import pi, E, I, S, Number
>>> pi.round()
3
>>> pi.round(2)
3.14
>>> (2*pi + E*I).round()
6 + 3*I
The round method has a chopping effect:
>>> (2*pi + I/10).round()
6
>>> (pi/10 + 2*I).round()
2*I
>>> (pi/10 + E*I).round(2)
0.31 + 2.72*I
Notes
=====
The Python ``round`` function uses the SymPy ``round`` method so it
will always return a SymPy number (not a Python float or int):
>>> isinstance(round(S(123), -2), Number)
True
"""
from sympy.core.numbers import Float
x = self
if not x.is_number:
raise TypeError("can't round symbolic expression")
if not x.is_Atom:
if not pure_complex(x.n(2), or_real=True):
raise TypeError(
'Expected a number but got %s:' % func_name(x))
elif x in (S.NaN, S.Infinity, S.NegativeInfinity, S.ComplexInfinity):
return x
if x.is_extended_real is False:
r, i = x.as_real_imag()
return r.round(n) + S.ImaginaryUnit*i.round(n)
if not x:
return S.Zero if n is None else x
p = as_int(n or 0)
if x.is_Integer:
return Integer(round(int(x), p))
digits_to_decimal = _mag(x) # _mag(12) = 2, _mag(.012) = -1
allow = digits_to_decimal + p
precs = [f._prec for f in x.atoms(Float)]
dps = prec_to_dps(max(precs)) if precs else None
if dps is None:
# assume everything is exact so use the Python
# float default or whatever was requested
dps = max(15, allow)
else:
allow = min(allow, dps)
# this will shift all digits to right of decimal
# and give us dps to work with as an int
shift = -digits_to_decimal + dps
extra = 1 # how far we look past known digits
# NOTE
# mpmath will calculate the binary representation to
# an arbitrary number of digits but we must base our
# answer on a finite number of those digits, e.g.
# .575 2589569785738035/2**52 in binary.
# mpmath shows us that the first 18 digits are
# >>> Float(.575).n(18)
# 0.574999999999999956
# The default precision is 15 digits and if we ask
# for 15 we get
# >>> Float(.575).n(15)
# 0.575000000000000
# mpmath handles rounding at the 15th digit. But we
# need to be careful since the user might be asking
# for rounding at the last digit and our semantics
# are to round toward the even final digit when there
# is a tie. So the extra digit will be used to make
# that decision. In this case, the value is the same
# to 15 digits:
# >>> Float(.575).n(16)
# 0.5750000000000000
# Now converting this to the 15 known digits gives
# 575000000000000.0
# which rounds to integer
# 5750000000000000
# And now we can round to the desired digt, e.g. at
# the second from the left and we get
# 5800000000000000
# and rescaling that gives
# 0.58
# as the final result.
# If the value is made slightly less than 0.575 we might
# still obtain the same value:
# >>> Float(.575-1e-16).n(16)*10**15
# 574999999999999.8
# What 15 digits best represents the known digits (which are
# to the left of the decimal? 5750000000000000, the same as
# before. The only way we will round down (in this case) is
# if we declared that we had more than 15 digits of precision.
# For example, if we use 16 digits of precision, the integer
# we deal with is
# >>> Float(.575-1e-16).n(17)*10**16
# 5749999999999998.4
# and this now rounds to 5749999999999998 and (if we round to
# the 2nd digit from the left) we get 5700000000000000.
#
xf = x.n(dps + extra)*Pow(10, shift)
xi = Integer(xf)
# use the last digit to select the value of xi
# nearest to x before rounding at the desired digit
sign = 1 if x > 0 else -1
dif2 = sign*(xf - xi).n(extra)
if dif2 < 0:
raise NotImplementedError(
'not expecting int(x) to round away from 0')
if dif2 > .5:
xi += sign # round away from 0
elif dif2 == .5:
xi += sign if xi%2 else -sign # round toward even
# shift p to the new position
ip = p - shift
# let Python handle the int rounding then rescale
xr = round(xi.p, ip)
# restore scale
rv = Rational(xr, Pow(10, shift))
# return Float or Integer
if rv.is_Integer:
if n is None: # the single-arg case
return rv
# use str or else it won't be a float
return Float(str(rv), dps) # keep same precision
else:
if not allow and rv > self:
allow += 1
return Float(rv, allow)
__round__ = round
def _eval_derivative_matrix_lines(self, x):
from sympy.matrices.expressions.matexpr import _LeftRightArgs
return [_LeftRightArgs([S.One, S.One], higher=self._eval_derivative(x))]
class AtomicExpr(Atom, Expr):
"""
A parent class for object which are both atoms and Exprs.
For example: Symbol, Number, Rational, Integer, ...
But not: Add, Mul, Pow, ...
"""
is_number = False
is_Atom = True
__slots__ = ()
def _eval_derivative(self, s):
if self == s:
return S.One
return S.Zero
def _eval_derivative_n_times(self, s, n):
from sympy import Piecewise, Eq
from sympy import Tuple, MatrixExpr
from sympy.matrices.common import MatrixCommon
if isinstance(s, (MatrixCommon, Tuple, Iterable, MatrixExpr)):
return super()._eval_derivative_n_times(s, n)
if self == s:
return Piecewise((self, Eq(n, 0)), (1, Eq(n, 1)), (0, True))
else:
return Piecewise((self, Eq(n, 0)), (0, True))
def _eval_is_polynomial(self, syms):
return True
def _eval_is_rational_function(self, syms):
return True
def _eval_is_meromorphic(self, x, a):
from sympy.calculus.util import AccumBounds
return (not self.is_Number or self.is_finite) and not isinstance(self, AccumBounds)
def _eval_is_algebraic_expr(self, syms):
return True
def _eval_nseries(self, x, n, logx, cdir=0):
return self
@property
def expr_free_symbols(self):
return {self}
def _mag(x):
"""Return integer ``i`` such that .1 <= x/10**i < 1
Examples
========
>>> from sympy.core.expr import _mag
>>> from sympy import Float
>>> _mag(Float(.1))
0
>>> _mag(Float(.01))
-1
>>> _mag(Float(1234))
4
"""
from math import log10, ceil, log
from sympy import Float
xpos = abs(x.n())
if not xpos:
return S.Zero
try:
mag_first_dig = int(ceil(log10(xpos)))
except (ValueError, OverflowError):
mag_first_dig = int(ceil(Float(mpf_log(xpos._mpf_, 53))/log(10)))
# check that we aren't off by 1
if (xpos/10**mag_first_dig) >= 1:
assert 1 <= (xpos/10**mag_first_dig) < 10
mag_first_dig += 1
return mag_first_dig
class UnevaluatedExpr(Expr):
"""
Expression that is not evaluated unless released.
Examples
========
>>> from sympy import UnevaluatedExpr
>>> from sympy.abc import x
>>> x*(1/x)
1
>>> x*UnevaluatedExpr(1/x)
x*1/x
"""
def __new__(cls, arg, **kwargs):
arg = _sympify(arg)
obj = Expr.__new__(cls, arg, **kwargs)
return obj
def doit(self, **kwargs):
if kwargs.get("deep", True):
return self.args[0].doit(**kwargs)
else:
return self.args[0]
def unchanged(func, *args):
"""Return True if `func` applied to the `args` is unchanged.
Can be used instead of `assert foo == foo`.
Examples
========
>>> from sympy import Piecewise, cos, pi
>>> from sympy.core.expr import unchanged
>>> from sympy.abc import x
>>> unchanged(cos, 1) # instead of assert cos(1) == cos(1)
True
>>> unchanged(cos, pi)
False
Comparison of args uses the builtin capabilities of the object's
arguments to test for equality so args can be defined loosely. Here,
the ExprCondPair arguments of Piecewise compare as equal to the
tuples that can be used to create the Piecewise:
>>> unchanged(Piecewise, (x, x > 1), (0, True))
True
"""
f = func(*args)
return f.func == func and f.args == args
class ExprBuilder:
def __init__(self, op, args=[], validator=None, check=True):
if not hasattr(op, "__call__"):
raise TypeError("op {} needs to be callable".format(op))
self.op = op
self.args = args
self.validator = validator
if (validator is not None) and check:
self.validate()
@staticmethod
def _build_args(args):
return [i.build() if isinstance(i, ExprBuilder) else i for i in args]
def validate(self):
if self.validator is None:
return
args = self._build_args(self.args)
self.validator(*args)
def build(self, check=True):
args = self._build_args(self.args)
if self.validator and check:
self.validator(*args)
return self.op(*args)
def append_argument(self, arg, check=True):
self.args.append(arg)
if self.validator and check:
self.validate(*self.args)
def __getitem__(self, item):
if item == 0:
return self.op
else:
return self.args[item-1]
def __repr__(self):
return str(self.build())
def search_element(self, elem):
for i, arg in enumerate(self.args):
if isinstance(arg, ExprBuilder):
ret = arg.search_index(elem)
if ret is not None:
return (i,) + ret
elif id(arg) == id(elem):
return (i,)
return None
from .mul import Mul
from .add import Add
from .power import Pow
from .function import Function, _derivative_dispatch
from .mod import Mod
from .exprtools import factor_terms
from .numbers import Integer, Rational
|
5300b896164394a9fcb14fdbadcc6afbc0b3a23c8bb89ced1672fce7a17811b7 | from typing import Dict, Union, Type
from sympy.utilities.exceptions import SymPyDeprecationWarning
from .basic import S, Atom
from .compatibility import ordered
from .basic import Basic
from .evalf import EvalfMixin
from .function import AppliedUndef
from .sympify import _sympify, SympifyError
from .parameters import global_parameters
from sympy.core.logic import fuzzy_bool, fuzzy_xor, fuzzy_and, fuzzy_not
from sympy.logic.boolalg import Boolean, BooleanAtom
__all__ = (
'Rel', 'Eq', 'Ne', 'Lt', 'Le', 'Gt', 'Ge',
'Relational', 'Equality', 'Unequality', 'StrictLessThan', 'LessThan',
'StrictGreaterThan', 'GreaterThan',
)
from .expr import Expr
from sympy.multipledispatch import dispatch
from .containers import Tuple
from .symbol import Symbol
def _nontrivBool(side):
return isinstance(side, Boolean) and \
not isinstance(side, Atom)
# Note, see issue 4986. Ideally, we wouldn't want to subclass both Boolean
# and Expr.
# from .. import Expr
def _canonical(cond):
# return a condition in which all relationals are canonical
reps = {r: r.canonical for r in cond.atoms(Relational)}
return cond.xreplace(reps)
# XXX: AttributeError was being caught here but it wasn't triggered by any of
# the tests so I've removed it...
class Relational(Boolean, EvalfMixin):
"""Base class for all relation types.
Explanation
===========
Subclasses of Relational should generally be instantiated directly, but
Relational can be instantiated with a valid ``rop`` value to dispatch to
the appropriate subclass.
Parameters
==========
rop : str or None
Indicates what subclass to instantiate. Valid values can be found
in the keys of Relational.ValidRelationOperator.
Examples
========
>>> from sympy import Rel
>>> from sympy.abc import x, y
>>> Rel(y, x + x**2, '==')
Eq(y, x**2 + x)
"""
__slots__ = ()
ValidRelationOperator = {} # type: Dict[Union[str, None], Type[Relational]]
is_Relational = True
# ValidRelationOperator - Defined below, because the necessary classes
# have not yet been defined
def __new__(cls, lhs, rhs, rop=None, **assumptions):
# If called by a subclass, do nothing special and pass on to Basic.
if cls is not Relational:
return Basic.__new__(cls, lhs, rhs, **assumptions)
# XXX: Why do this? There should be a separate function to make a
# particular subclass of Relational from a string.
#
# If called directly with an operator, look up the subclass
# corresponding to that operator and delegate to it
cls = cls.ValidRelationOperator.get(rop, None)
if cls is None:
raise ValueError("Invalid relational operator symbol: %r" % rop)
if not issubclass(cls, (Eq, Ne)):
# validate that Booleans are not being used in a relational
# other than Eq/Ne;
# Note: Symbol is a subclass of Boolean but is considered
# acceptable here.
if any(map(_nontrivBool, (lhs, rhs))):
from sympy.utilities.misc import filldedent
raise TypeError(filldedent('''
A Boolean argument can only be used in
Eq and Ne; all other relationals expect
real expressions.
'''))
return cls(lhs, rhs, **assumptions)
@property
def lhs(self):
"""The left-hand side of the relation."""
return self._args[0]
@property
def rhs(self):
"""The right-hand side of the relation."""
return self._args[1]
@property
def reversed(self):
"""Return the relationship with sides reversed.
Examples
========
>>> from sympy import Eq
>>> from sympy.abc import x
>>> Eq(x, 1)
Eq(x, 1)
>>> _.reversed
Eq(1, x)
>>> x < 1
x < 1
>>> _.reversed
1 > x
"""
ops = {Eq: Eq, Gt: Lt, Ge: Le, Lt: Gt, Le: Ge, Ne: Ne}
a, b = self.args
return Relational.__new__(ops.get(self.func, self.func), b, a)
@property
def reversedsign(self):
"""Return the relationship with signs reversed.
Examples
========
>>> from sympy import Eq
>>> from sympy.abc import x
>>> Eq(x, 1)
Eq(x, 1)
>>> _.reversedsign
Eq(-x, -1)
>>> x < 1
x < 1
>>> _.reversedsign
-x > -1
"""
a, b = self.args
if not (isinstance(a, BooleanAtom) or isinstance(b, BooleanAtom)):
ops = {Eq: Eq, Gt: Lt, Ge: Le, Lt: Gt, Le: Ge, Ne: Ne}
return Relational.__new__(ops.get(self.func, self.func), -a, -b)
else:
return self
@property
def negated(self):
"""Return the negated relationship.
Examples
========
>>> from sympy import Eq
>>> from sympy.abc import x
>>> Eq(x, 1)
Eq(x, 1)
>>> _.negated
Ne(x, 1)
>>> x < 1
x < 1
>>> _.negated
x >= 1
Notes
=====
This works more or less identical to ``~``/``Not``. The difference is
that ``negated`` returns the relationship even if ``evaluate=False``.
Hence, this is useful in code when checking for e.g. negated relations
to existing ones as it will not be affected by the `evaluate` flag.
"""
ops = {Eq: Ne, Ge: Lt, Gt: Le, Le: Gt, Lt: Ge, Ne: Eq}
# If there ever will be new Relational subclasses, the following line
# will work until it is properly sorted out
# return ops.get(self.func, lambda a, b, evaluate=False: ~(self.func(a,
# b, evaluate=evaluate)))(*self.args, evaluate=False)
return Relational.__new__(ops.get(self.func), *self.args)
def _eval_evalf(self, prec):
return self.func(*[s._evalf(prec) for s in self.args])
@property
def canonical(self):
"""Return a canonical form of the relational by putting a
number on the rhs, canonically removing a sign or else
ordering the args canonically. No other simplification is
attempted.
Examples
========
>>> from sympy.abc import x, y
>>> x < 2
x < 2
>>> _.reversed.canonical
x < 2
>>> (-y < x).canonical
x > -y
>>> (-y > x).canonical
x < -y
>>> (-y < -x).canonical
x < y
"""
args = self.args
r = self
if r.rhs.is_number:
if r.rhs.is_Number and r.lhs.is_Number and r.lhs > r.rhs:
r = r.reversed
elif r.lhs.is_number:
r = r.reversed
elif tuple(ordered(args)) != args:
r = r.reversed
LHS_CEMS = getattr(r.lhs, 'could_extract_minus_sign', None)
RHS_CEMS = getattr(r.rhs, 'could_extract_minus_sign', None)
if isinstance(r.lhs, BooleanAtom) or isinstance(r.rhs, BooleanAtom):
return r
# Check if first value has negative sign
if LHS_CEMS and LHS_CEMS():
return r.reversedsign
elif not r.rhs.is_number and RHS_CEMS and RHS_CEMS():
# Right hand side has a minus, but not lhs.
# How does the expression with reversed signs behave?
# This is so that expressions of the type
# Eq(x, -y) and Eq(-x, y)
# have the same canonical representation
expr1, _ = ordered([r.lhs, -r.rhs])
if expr1 != r.lhs:
return r.reversed.reversedsign
return r
def equals(self, other, failing_expression=False):
"""Return True if the sides of the relationship are mathematically
identical and the type of relationship is the same.
If failing_expression is True, return the expression whose truth value
was unknown."""
if isinstance(other, Relational):
if self == other or self.reversed == other:
return True
a, b = self, other
if a.func in (Eq, Ne) or b.func in (Eq, Ne):
if a.func != b.func:
return False
left, right = [i.equals(j,
failing_expression=failing_expression)
for i, j in zip(a.args, b.args)]
if left is True:
return right
if right is True:
return left
lr, rl = [i.equals(j, failing_expression=failing_expression)
for i, j in zip(a.args, b.reversed.args)]
if lr is True:
return rl
if rl is True:
return lr
e = (left, right, lr, rl)
if all(i is False for i in e):
return False
for i in e:
if i not in (True, False):
return i
else:
if b.func != a.func:
b = b.reversed
if a.func != b.func:
return False
left = a.lhs.equals(b.lhs,
failing_expression=failing_expression)
if left is False:
return False
right = a.rhs.equals(b.rhs,
failing_expression=failing_expression)
if right is False:
return False
if left is True:
return right
return left
def _eval_simplify(self, **kwargs):
from .add import Add
from sympy.core.expr import Expr
r = self
r = r.func(*[i.simplify(**kwargs) for i in r.args])
if r.is_Relational:
if not isinstance(r.lhs, Expr) or not isinstance(r.rhs, Expr):
return r
dif = r.lhs - r.rhs
# replace dif with a valid Number that will
# allow a definitive comparison with 0
v = None
if dif.is_comparable:
v = dif.n(2)
elif dif.equals(0): # XXX this is expensive
v = S.Zero
if v is not None:
r = r.func._eval_relation(v, S.Zero)
r = r.canonical
# If there is only one symbol in the expression,
# try to write it on a simplified form
free = list(filter(lambda x: x.is_real is not False, r.free_symbols))
if len(free) == 1:
try:
from sympy.solvers.solveset import linear_coeffs
x = free.pop()
dif = r.lhs - r.rhs
m, b = linear_coeffs(dif, x)
if m.is_zero is False:
if m.is_negative:
# Dividing with a negative number, so change order of arguments
# canonical will put the symbol back on the lhs later
r = r.func(-b / m, x)
else:
r = r.func(x, -b / m)
else:
r = r.func(b, S.zero)
except ValueError:
# maybe not a linear function, try polynomial
from sympy.polys import Poly, poly, PolynomialError, gcd
try:
p = poly(dif, x)
c = p.all_coeffs()
constant = c[-1]
c[-1] = 0
scale = gcd(c)
c = [ctmp / scale for ctmp in c]
r = r.func(Poly.from_list(c, x).as_expr(), -constant / scale)
except PolynomialError:
pass
elif len(free) >= 2:
try:
from sympy.solvers.solveset import linear_coeffs
from sympy.polys import gcd
free = list(ordered(free))
dif = r.lhs - r.rhs
m = linear_coeffs(dif, *free)
constant = m[-1]
del m[-1]
scale = gcd(m)
m = [mtmp / scale for mtmp in m]
nzm = list(filter(lambda f: f[0] != 0, list(zip(m, free))))
if scale.is_zero is False:
if constant != 0:
# lhs: expression, rhs: constant
newexpr = Add(*[i * j for i, j in nzm])
r = r.func(newexpr, -constant / scale)
else:
# keep first term on lhs
lhsterm = nzm[0][0] * nzm[0][1]
del nzm[0]
newexpr = Add(*[i * j for i, j in nzm])
r = r.func(lhsterm, -newexpr)
else:
r = r.func(constant, S.zero)
except ValueError:
pass
# Did we get a simplified result?
r = r.canonical
measure = kwargs['measure']
if measure(r) < kwargs['ratio'] * measure(self):
return r
else:
return self
def _eval_trigsimp(self, **opts):
from sympy.simplify import trigsimp
return self.func(trigsimp(self.lhs, **opts), trigsimp(self.rhs, **opts))
def expand(self, **kwargs):
args = (arg.expand(**kwargs) for arg in self.args)
return self.func(*args)
def __bool__(self):
raise TypeError("cannot determine truth value of Relational")
def _eval_as_set(self):
# self is univariate and periodicity(self, x) in (0, None)
from sympy.solvers.inequalities import solve_univariate_inequality
from sympy.sets.conditionset import ConditionSet
syms = self.free_symbols
assert len(syms) == 1
x = syms.pop()
try:
xset = solve_univariate_inequality(self, x, relational=False)
except NotImplementedError:
# solve_univariate_inequality raises NotImplementedError for
# unsolvable equations/inequalities.
xset = ConditionSet(x, self, S.Reals)
return xset
@property
def binary_symbols(self):
# override where necessary
return set()
Rel = Relational
class Equality(Relational):
"""An equal relation between two objects.
Explanation
===========
Represents that two objects are equal. If they can be easily shown
to be definitively equal (or unequal), this will reduce to True (or
False). Otherwise, the relation is maintained as an unevaluated
Equality object. Use the ``simplify`` function on this object for
more nontrivial evaluation of the equality relation.
As usual, the keyword argument ``evaluate=False`` can be used to
prevent any evaluation.
Examples
========
>>> from sympy import Eq, simplify, exp, cos
>>> from sympy.abc import x, y
>>> Eq(y, x + x**2)
Eq(y, x**2 + x)
>>> Eq(2, 5)
False
>>> Eq(2, 5, evaluate=False)
Eq(2, 5)
>>> _.doit()
False
>>> Eq(exp(x), exp(x).rewrite(cos))
Eq(exp(x), sinh(x) + cosh(x))
>>> simplify(_)
True
See Also
========
sympy.logic.boolalg.Equivalent : for representing equality between two
boolean expressions
Notes
=====
Python treats 1 and True (and 0 and False) as being equal; SymPy
does not. And integer will always compare as unequal to a Boolean:
>>> Eq(True, 1), True == 1
(False, True)
This class is not the same as the == operator. The == operator tests
for exact structural equality between two expressions; this class
compares expressions mathematically.
If either object defines an ``_eval_Eq`` method, it can be used in place of
the default algorithm. If ``lhs._eval_Eq(rhs)`` or ``rhs._eval_Eq(lhs)``
returns anything other than None, that return value will be substituted for
the Equality. If None is returned by ``_eval_Eq``, an Equality object will
be created as usual.
Since this object is already an expression, it does not respond to
the method ``as_expr`` if one tries to create `x - y` from ``Eq(x, y)``.
This can be done with the ``rewrite(Add)`` method.
"""
rel_op = '=='
__slots__ = ()
is_Equality = True
def __new__(cls, lhs, rhs=None, **options):
if rhs is None:
SymPyDeprecationWarning(
feature="Eq(expr) with rhs default to 0",
useinstead="Eq(expr, 0)",
issue=16587,
deprecated_since_version="1.5"
).warn()
rhs = 0
evaluate = options.pop('evaluate', global_parameters.evaluate)
lhs = _sympify(lhs)
rhs = _sympify(rhs)
if evaluate:
val = is_eq(lhs, rhs)
if val is None:
return cls(lhs, rhs, evaluate=False)
else:
return _sympify(val)
return Relational.__new__(cls, lhs, rhs)
@classmethod
def _eval_relation(cls, lhs, rhs):
return _sympify(lhs == rhs)
def _eval_rewrite_as_Add(self, *args, **kwargs):
"""
return Eq(L, R) as L - R. To control the evaluation of
the result set pass `evaluate=True` to give L - R;
if `evaluate=None` then terms in L and R will not cancel
but they will be listed in canonical order; otherwise
non-canonical args will be returned.
Examples
========
>>> from sympy import Eq, Add
>>> from sympy.abc import b, x
>>> eq = Eq(x + b, x - b)
>>> eq.rewrite(Add)
2*b
>>> eq.rewrite(Add, evaluate=None).args
(b, b, x, -x)
>>> eq.rewrite(Add, evaluate=False).args
(b, x, b, -x)
"""
from .add import _unevaluated_Add, Add
L, R = args
evaluate = kwargs.get('evaluate', True)
if evaluate:
# allow cancellation of args
return L - R
args = Add.make_args(L) + Add.make_args(-R)
if evaluate is None:
# no cancellation, but canonical
return _unevaluated_Add(*args)
# no cancellation, not canonical
return Add._from_args(args)
@property
def binary_symbols(self):
if S.true in self.args or S.false in self.args:
if self.lhs.is_Symbol:
return {self.lhs}
elif self.rhs.is_Symbol:
return {self.rhs}
return set()
def _eval_simplify(self, **kwargs):
from .add import Add
from sympy.core.expr import Expr
from sympy.solvers.solveset import linear_coeffs
# standard simplify
e = super()._eval_simplify(**kwargs)
if not isinstance(e, Equality):
return e
if not isinstance(e.lhs, Expr) or not isinstance(e.rhs, Expr):
return e
free = self.free_symbols
if len(free) == 1:
try:
x = free.pop()
m, b = linear_coeffs(
e.rewrite(Add, evaluate=False), x)
if m.is_zero is False:
enew = e.func(x, -b / m)
else:
enew = e.func(m * x, -b)
measure = kwargs['measure']
if measure(enew) <= kwargs['ratio'] * measure(e):
e = enew
except ValueError:
pass
return e.canonical
def integrate(self, *args, **kwargs):
"""See the integrate function in sympy.integrals"""
from sympy.integrals import integrate
return integrate(self, *args, **kwargs)
def as_poly(self, *gens, **kwargs):
'''Returns lhs-rhs as a Poly
Examples
========
>>> from sympy import Eq
>>> from sympy.abc import x
>>> Eq(x**2, 1).as_poly(x)
Poly(x**2 - 1, x, domain='ZZ')
'''
return (self.lhs - self.rhs).as_poly(*gens, **kwargs)
Eq = Equality
class Unequality(Relational):
"""An unequal relation between two objects.
Explanation
===========
Represents that two objects are not equal. If they can be shown to be
definitively equal, this will reduce to False; if definitively unequal,
this will reduce to True. Otherwise, the relation is maintained as an
Unequality object.
Examples
========
>>> from sympy import Ne
>>> from sympy.abc import x, y
>>> Ne(y, x+x**2)
Ne(y, x**2 + x)
See Also
========
Equality
Notes
=====
This class is not the same as the != operator. The != operator tests
for exact structural equality between two expressions; this class
compares expressions mathematically.
This class is effectively the inverse of Equality. As such, it uses the
same algorithms, including any available `_eval_Eq` methods.
"""
rel_op = '!='
__slots__ = ()
def __new__(cls, lhs, rhs, **options):
lhs = _sympify(lhs)
rhs = _sympify(rhs)
evaluate = options.pop('evaluate', global_parameters.evaluate)
if evaluate:
val = is_neq(lhs, rhs)
if val is None:
return cls(lhs, rhs, evaluate=False)
else:
return _sympify(val)
return Relational.__new__(cls, lhs, rhs, **options)
@classmethod
def _eval_relation(cls, lhs, rhs):
return _sympify(lhs != rhs)
@property
def binary_symbols(self):
if S.true in self.args or S.false in self.args:
if self.lhs.is_Symbol:
return {self.lhs}
elif self.rhs.is_Symbol:
return {self.rhs}
return set()
def _eval_simplify(self, **kwargs):
# simplify as an equality
eq = Equality(*self.args)._eval_simplify(**kwargs)
if isinstance(eq, Equality):
# send back Ne with the new args
return self.func(*eq.args)
return eq.negated # result of Ne is the negated Eq
Ne = Unequality
class _Inequality(Relational):
"""Internal base class for all *Than types.
Each subclass must implement _eval_relation to provide the method for
comparing two real numbers.
"""
__slots__ = ()
def __new__(cls, lhs, rhs, **options):
try:
lhs = _sympify(lhs)
rhs = _sympify(rhs)
except SympifyError:
return NotImplemented
evaluate = options.pop('evaluate', global_parameters.evaluate)
if evaluate:
for me in (lhs, rhs):
if me.is_extended_real is False:
raise TypeError("Invalid comparison of non-real %s" % me)
if me is S.NaN:
raise TypeError("Invalid NaN comparison")
# First we invoke the appropriate inequality method of `lhs`
# (e.g., `lhs.__lt__`). That method will try to reduce to
# boolean or raise an exception. It may keep calling
# superclasses until it reaches `Expr` (e.g., `Expr.__lt__`).
# In some cases, `Expr` will just invoke us again (if neither it
# nor a subclass was able to reduce to boolean or raise an
# exception). In that case, it must call us with
# `evaluate=False` to prevent infinite recursion.
return cls._eval_relation(lhs, rhs, **options)
# make a "non-evaluated" Expr for the inequality
return Relational.__new__(cls, lhs, rhs, **options)
@classmethod
def _eval_relation(cls, lhs, rhs, **options):
val = cls._eval_fuzzy_relation(lhs, rhs)
if val is None:
return cls(lhs, rhs, evaluate=False)
else:
return _sympify(val)
class _Greater(_Inequality):
"""Not intended for general use
_Greater is only used so that GreaterThan and StrictGreaterThan may
subclass it for the .gts and .lts properties.
"""
__slots__ = ()
@property
def gts(self):
return self._args[0]
@property
def lts(self):
return self._args[1]
class _Less(_Inequality):
"""Not intended for general use.
_Less is only used so that LessThan and StrictLessThan may subclass it for
the .gts and .lts properties.
"""
__slots__ = ()
@property
def gts(self):
return self._args[1]
@property
def lts(self):
return self._args[0]
class GreaterThan(_Greater):
"""Class representations of inequalities.
Explanation
===========
The ``*Than`` classes represent inequal relationships, where the left-hand
side is generally bigger or smaller than the right-hand side. For example,
the GreaterThan class represents an inequal relationship where the
left-hand side is at least as big as the right side, if not bigger. In
mathematical notation:
lhs >= rhs
In total, there are four ``*Than`` classes, to represent the four
inequalities:
+-----------------+--------+
|Class Name | Symbol |
+=================+========+
|GreaterThan | (>=) |
+-----------------+--------+
|LessThan | (<=) |
+-----------------+--------+
|StrictGreaterThan| (>) |
+-----------------+--------+
|StrictLessThan | (<) |
+-----------------+--------+
All classes take two arguments, lhs and rhs.
+----------------------------+-----------------+
|Signature Example | Math equivalent |
+============================+=================+
|GreaterThan(lhs, rhs) | lhs >= rhs |
+----------------------------+-----------------+
|LessThan(lhs, rhs) | lhs <= rhs |
+----------------------------+-----------------+
|StrictGreaterThan(lhs, rhs) | lhs > rhs |
+----------------------------+-----------------+
|StrictLessThan(lhs, rhs) | lhs < rhs |
+----------------------------+-----------------+
In addition to the normal .lhs and .rhs of Relations, ``*Than`` inequality
objects also have the .lts and .gts properties, which represent the "less
than side" and "greater than side" of the operator. Use of .lts and .gts
in an algorithm rather than .lhs and .rhs as an assumption of inequality
direction will make more explicit the intent of a certain section of code,
and will make it similarly more robust to client code changes:
>>> from sympy import GreaterThan, StrictGreaterThan
>>> from sympy import LessThan, StrictLessThan
>>> from sympy import And, Ge, Gt, Le, Lt, Rel, S
>>> from sympy.abc import x, y, z
>>> from sympy.core.relational import Relational
>>> e = GreaterThan(x, 1)
>>> e
x >= 1
>>> '%s >= %s is the same as %s <= %s' % (e.gts, e.lts, e.lts, e.gts)
'x >= 1 is the same as 1 <= x'
Examples
========
One generally does not instantiate these classes directly, but uses various
convenience methods:
>>> for f in [Ge, Gt, Le, Lt]: # convenience wrappers
... print(f(x, 2))
x >= 2
x > 2
x <= 2
x < 2
Another option is to use the Python inequality operators (>=, >, <=, <)
directly. Their main advantage over the Ge, Gt, Le, and Lt counterparts,
is that one can write a more "mathematical looking" statement rather than
littering the math with oddball function calls. However there are certain
(minor) caveats of which to be aware (search for 'gotcha', below).
>>> x >= 2
x >= 2
>>> _ == Ge(x, 2)
True
However, it is also perfectly valid to instantiate a ``*Than`` class less
succinctly and less conveniently:
>>> Rel(x, 1, ">")
x > 1
>>> Relational(x, 1, ">")
x > 1
>>> StrictGreaterThan(x, 1)
x > 1
>>> GreaterThan(x, 1)
x >= 1
>>> LessThan(x, 1)
x <= 1
>>> StrictLessThan(x, 1)
x < 1
Notes
=====
There are a couple of "gotchas" to be aware of when using Python's
operators.
The first is that what your write is not always what you get:
>>> 1 < x
x > 1
Due to the order that Python parses a statement, it may
not immediately find two objects comparable. When "1 < x"
is evaluated, Python recognizes that the number 1 is a native
number and that x is *not*. Because a native Python number does
not know how to compare itself with a SymPy object
Python will try the reflective operation, "x > 1" and that is the
form that gets evaluated, hence returned.
If the order of the statement is important (for visual output to
the console, perhaps), one can work around this annoyance in a
couple ways:
(1) "sympify" the literal before comparison
>>> S(1) < x
1 < x
(2) use one of the wrappers or less succinct methods described
above
>>> Lt(1, x)
1 < x
>>> Relational(1, x, "<")
1 < x
The second gotcha involves writing equality tests between relationals
when one or both sides of the test involve a literal relational:
>>> e = x < 1; e
x < 1
>>> e == e # neither side is a literal
True
>>> e == x < 1 # expecting True, too
False
>>> e != x < 1 # expecting False
x < 1
>>> x < 1 != x < 1 # expecting False or the same thing as before
Traceback (most recent call last):
...
TypeError: cannot determine truth value of Relational
The solution for this case is to wrap literal relationals in
parentheses:
>>> e == (x < 1)
True
>>> e != (x < 1)
False
>>> (x < 1) != (x < 1)
False
The third gotcha involves chained inequalities not involving
'==' or '!='. Occasionally, one may be tempted to write:
>>> e = x < y < z
Traceback (most recent call last):
...
TypeError: symbolic boolean expression has no truth value.
Due to an implementation detail or decision of Python [1]_,
there is no way for SymPy to create a chained inequality with
that syntax so one must use And:
>>> e = And(x < y, y < z)
>>> type( e )
And
>>> e
(x < y) & (y < z)
Although this can also be done with the '&' operator, it cannot
be done with the 'and' operarator:
>>> (x < y) & (y < z)
(x < y) & (y < z)
>>> (x < y) and (y < z)
Traceback (most recent call last):
...
TypeError: cannot determine truth value of Relational
.. [1] This implementation detail is that Python provides no reliable
method to determine that a chained inequality is being built.
Chained comparison operators are evaluated pairwise, using "and"
logic (see
http://docs.python.org/2/reference/expressions.html#notin). This
is done in an efficient way, so that each object being compared
is only evaluated once and the comparison can short-circuit. For
example, ``1 > 2 > 3`` is evaluated by Python as ``(1 > 2) and (2
> 3)``. The ``and`` operator coerces each side into a bool,
returning the object itself when it short-circuits. The bool of
the --Than operators will raise TypeError on purpose, because
SymPy cannot determine the mathematical ordering of symbolic
expressions. Thus, if we were to compute ``x > y > z``, with
``x``, ``y``, and ``z`` being Symbols, Python converts the
statement (roughly) into these steps:
(1) x > y > z
(2) (x > y) and (y > z)
(3) (GreaterThanObject) and (y > z)
(4) (GreaterThanObject.__bool__()) and (y > z)
(5) TypeError
Because of the "and" added at step 2, the statement gets turned into a
weak ternary statement, and the first object's __bool__ method will
raise TypeError. Thus, creating a chained inequality is not possible.
In Python, there is no way to override the ``and`` operator, or to
control how it short circuits, so it is impossible to make something
like ``x > y > z`` work. There was a PEP to change this,
:pep:`335`, but it was officially closed in March, 2012.
"""
__slots__ = ()
rel_op = '>='
@classmethod
def _eval_fuzzy_relation(cls, lhs, rhs):
return is_ge(lhs, rhs)
Ge = GreaterThan
class LessThan(_Less):
__doc__ = GreaterThan.__doc__
__slots__ = ()
rel_op = '<='
@classmethod
def _eval_fuzzy_relation(cls, lhs, rhs):
return is_le(lhs, rhs)
Le = LessThan
class StrictGreaterThan(_Greater):
__doc__ = GreaterThan.__doc__
__slots__ = ()
rel_op = '>'
@classmethod
def _eval_fuzzy_relation(cls, lhs, rhs):
return is_gt(lhs, rhs)
Gt = StrictGreaterThan
class StrictLessThan(_Less):
__doc__ = GreaterThan.__doc__
__slots__ = ()
rel_op = '<'
@classmethod
def _eval_fuzzy_relation(cls, lhs, rhs):
return is_lt(lhs, rhs)
Lt = StrictLessThan
# A class-specific (not object-specific) data item used for a minor speedup.
# It is defined here, rather than directly in the class, because the classes
# that it references have not been defined until now (e.g. StrictLessThan).
Relational.ValidRelationOperator = {
None: Equality,
'==': Equality,
'eq': Equality,
'!=': Unequality,
'<>': Unequality,
'ne': Unequality,
'>=': GreaterThan,
'ge': GreaterThan,
'<=': LessThan,
'le': LessThan,
'>': StrictGreaterThan,
'gt': StrictGreaterThan,
'<': StrictLessThan,
'lt': StrictLessThan,
}
def _n2(a, b):
"""Return (a - b).evalf(2) if a and b are comparable, else None.
This should only be used when a and b are already sympified.
"""
# /!\ it is very important (see issue 8245) not to
# use a re-evaluated number in the calculation of dif
if a.is_comparable and b.is_comparable:
dif = (a - b).evalf(2)
if dif.is_comparable:
return dif
@dispatch(Expr, Expr)
def _eval_is_ge(lhs, rhs):
return None
@dispatch(Basic, Basic)
def _eval_is_eq(lhs, rhs):
return None
@dispatch(Tuple, Expr) # type: ignore
def _eval_is_eq(lhs, rhs): # noqa:F811
return False
@dispatch(Tuple, AppliedUndef) # type: ignore
def _eval_is_eq(lhs, rhs): # noqa:F811
return None
@dispatch(Tuple, Symbol) # type: ignore
def _eval_is_eq(lhs, rhs): # noqa:F811
return None
@dispatch(Tuple, Tuple) # type: ignore
def _eval_is_eq(lhs, rhs): # noqa:F811
if len(lhs) != len(rhs):
return False
return fuzzy_and(fuzzy_bool(is_eq(s, o)) for s, o in zip(lhs, rhs))
def is_lt(lhs, rhs, assumptions=None):
"""Fuzzy bool for lhs is strictly less than rhs.
See the docstring for :func:`~.is_ge` for more.
"""
return fuzzy_not(is_ge(lhs, rhs, assumptions))
def is_gt(lhs, rhs, assumptions=None):
"""Fuzzy bool for lhs is strictly greater than rhs.
See the docstring for :func:`~.is_ge` for more.
"""
return fuzzy_not(is_le(lhs, rhs, assumptions))
def is_le(lhs, rhs, assumptions=None):
"""Fuzzy bool for lhs is less than or equal to rhs.
See the docstring for :func:`~.is_ge` for more.
"""
return is_ge(rhs, lhs, assumptions)
def is_ge(lhs, rhs, assumptions=None):
"""
Fuzzy bool for *lhs* is greater than or equal to *rhs*.
Parameters
==========
lhs : Expr
The left-hand side of the expression, must be sympified,
and an instance of expression. Throws an exception if
lhs is not an instance of expression.
rhs : Expr
The right-hand side of the expression, must be sympified
and an instance of expression. Throws an exception if
lhs is not an instance of expression.
assumptions: Boolean, optional
Assumptions taken to evaluate the inequality.
Returns
=======
``True`` if *lhs* is greater than or equal to *rhs*, ``False`` if *lhs*
is less than *rhs*, and ``None`` if the comparison between *lhs* and
*rhs* is indeterminate.
Explanation
===========
This function is intended to give a relatively fast determination and
deliberately does not attempt slow calculations that might help in
obtaining a determination of True or False in more difficult cases.
The four comparison functions ``is_le``, ``is_lt``, ``is_ge``, and ``is_gt`` are
each implemented in terms of ``is_ge`` in the following way:
is_ge(x, y) := is_ge(x, y)
is_le(x, y) := is_ge(y, x)
is_lt(x, y) := fuzzy_not(is_ge(x, y))
is_gt(x, y) := fuzzy_not(is_ge(y, x))
Therefore, supporting new type with this function will ensure behavior for
other three functions as well.
To maintain these equivalences in fuzzy logic it is important that in cases where
either x or y is non-real all comparisons will give None.
Examples
========
>>> from sympy import S, Q
>>> from sympy.core.relational import is_ge, is_le, is_gt, is_lt
>>> from sympy.abc import x
>>> is_ge(S(2), S(0))
True
>>> is_ge(S(0), S(2))
False
>>> is_le(S(0), S(2))
True
>>> is_gt(S(0), S(2))
False
>>> is_lt(S(2), S(0))
False
Assumptions can be passed to evaluate the quality which is otherwise
indeterminate.
>>> print(is_ge(x, S(0)))
None
>>> is_ge(x, S(0), assumptions=Q.positive(x))
True
New types can be supported by dispatching to ``_eval_is_ge``.
>>> from sympy import Expr, sympify
>>> from sympy.multipledispatch import dispatch
>>> class MyExpr(Expr):
... def __new__(cls, arg):
... return super().__new__(cls, sympify(arg))
... @property
... def value(self):
... return self.args[0]
>>> @dispatch(MyExpr, MyExpr)
... def _eval_is_ge(a, b):
... return is_ge(a.value, b.value)
>>> a = MyExpr(1)
>>> b = MyExpr(2)
>>> is_ge(b, a)
True
>>> is_le(a, b)
True
"""
from sympy.assumptions.wrapper import AssumptionsWrapper, is_extended_nonnegative
if not (isinstance(lhs, Expr) and isinstance(rhs, Expr)):
raise TypeError("Can only compare inequalities with Expr")
retval = _eval_is_ge(lhs, rhs)
if retval is not None:
return retval
else:
n2 = _n2(lhs, rhs)
if n2 is not None:
# use float comparison for infinity.
# otherwise get stuck in infinite recursion
if n2 in (S.Infinity, S.NegativeInfinity):
n2 = float(n2)
return n2 >= 0
_lhs = AssumptionsWrapper(lhs, assumptions)
_rhs = AssumptionsWrapper(rhs, assumptions)
if _lhs.is_extended_real and _rhs.is_extended_real:
if (_lhs.is_infinite and _lhs.is_extended_positive) or (_rhs.is_infinite and _rhs.is_extended_negative):
return True
diff = lhs - rhs
if diff is not S.NaN:
rv = is_extended_nonnegative(diff, assumptions)
if rv is not None:
return rv
def is_neq(lhs, rhs, assumptions=None):
"""Fuzzy bool for lhs does not equal rhs.
See the docstring for :func:`~.is_eq` for more.
"""
return fuzzy_not(is_eq(lhs, rhs, assumptions))
def is_eq(lhs, rhs, assumptions=None):
"""
Fuzzy bool representing mathematical equality between *lhs* and *rhs*.
Parameters
==========
lhs : Expr
The left-hand side of the expression, must be sympified.
rhs : Expr
The right-hand side of the expression, must be sympified.
assumptions: Boolean, optional
Assumptions taken to evaluate the equality.
Returns
=======
``True`` if *lhs* is equal to *rhs*, ``False`` is *lhs* is not equal to *rhs*,
and ``None`` if the comparison between *lhs* and *rhs* is indeterminate.
Explanation
===========
This function is intended to give a relatively fast determination and
deliberately does not attempt slow calculations that might help in
obtaining a determination of True or False in more difficult cases.
:func:`~.is_neq` calls this function to return its value, so supporting
new type with this function will ensure correct behavior for ``is_neq``
as well.
Examples
========
>>> from sympy import Q, S
>>> from sympy.core.relational import is_eq, is_neq
>>> from sympy.abc import x
>>> is_eq(S(0), S(0))
True
>>> is_neq(S(0), S(0))
False
>>> is_eq(S(0), S(2))
False
>>> is_neq(S(0), S(2))
True
Assumptions can be passed to evaluate the equality which is otherwise
indeterminate.
>>> print(is_eq(x, S(0)))
None
>>> is_eq(x, S(0), assumptions=Q.zero(x))
True
New types can be supported by dispatching to ``_eval_is_eq``.
>>> from sympy import Basic, sympify
>>> from sympy.multipledispatch import dispatch
>>> class MyBasic(Basic):
... def __new__(cls, arg):
... return Basic.__new__(cls, sympify(arg))
... @property
... def value(self):
... return self.args[0]
...
>>> @dispatch(MyBasic, MyBasic)
... def _eval_is_eq(a, b):
... return is_eq(a.value, b.value)
...
>>> a = MyBasic(1)
>>> b = MyBasic(1)
>>> is_eq(a, b)
True
>>> is_neq(a, b)
False
"""
from sympy.assumptions.wrapper import (AssumptionsWrapper,
is_infinite, is_extended_real)
from sympy.core.add import Add
from sympy.functions.elementary.complexes import arg
from sympy.simplify.simplify import clear_coefficients
from sympy.utilities.iterables import sift
# here, _eval_Eq is only called for backwards compatibility
# new code should use is_eq with multiple dispatch as
# outlined in the docstring
for side1, side2 in (lhs, rhs), (rhs, lhs):
eval_func = getattr(side1, '_eval_Eq', None)
if eval_func is not None:
retval = eval_func(side2)
if retval is not None:
return retval
retval = _eval_is_eq(lhs, rhs)
if retval is not None:
return retval
if dispatch(type(lhs), type(rhs)) != dispatch(type(rhs), type(lhs)):
retval = _eval_is_eq(rhs, lhs)
if retval is not None:
return retval
# retval is still None, so go through the equality logic
# If expressions have the same structure, they must be equal.
if lhs == rhs:
return True # e.g. True == True
elif all(isinstance(i, BooleanAtom) for i in (rhs, lhs)):
return False # True != False
elif not (lhs.is_Symbol or rhs.is_Symbol) and (
isinstance(lhs, Boolean) !=
isinstance(rhs, Boolean)):
return False # only Booleans can equal Booleans
_lhs = AssumptionsWrapper(lhs, assumptions)
_rhs = AssumptionsWrapper(rhs, assumptions)
if _lhs.is_infinite or _rhs.is_infinite:
if fuzzy_xor([_lhs.is_infinite, _rhs.is_infinite]):
return False
if fuzzy_xor([_lhs.is_extended_real, _rhs.is_extended_real]):
return False
if fuzzy_and([_lhs.is_extended_real, _rhs.is_extended_real]):
return fuzzy_xor([_lhs.is_extended_positive, fuzzy_not(_rhs.is_extended_positive)])
# Try to split real/imaginary parts and equate them
I = S.ImaginaryUnit
def split_real_imag(expr):
real_imag = lambda t: (
'real' if is_extended_real(t, assumptions) else
'imag' if is_extended_real(I*t, assumptions) else None)
return sift(Add.make_args(expr), real_imag)
lhs_ri = split_real_imag(lhs)
if not lhs_ri[None]:
rhs_ri = split_real_imag(rhs)
if not rhs_ri[None]:
eq_real = is_eq(Add(*lhs_ri['real']), Add(*rhs_ri['real']), assumptions)
eq_imag = is_eq(I * Add(*lhs_ri['imag']), I * Add(*rhs_ri['imag']), assumptions)
return fuzzy_and(map(fuzzy_bool, [eq_real, eq_imag]))
# Compare e.g. zoo with 1+I*oo by comparing args
arglhs = arg(lhs)
argrhs = arg(rhs)
# Guard against Eq(nan, nan) -> Falsesymp
if not (arglhs == S.NaN and argrhs == S.NaN):
return fuzzy_bool(is_eq(arglhs, argrhs, assumptions))
if all(isinstance(i, Expr) for i in (lhs, rhs)):
# see if the difference evaluates
dif = lhs - rhs
_dif = AssumptionsWrapper(dif, assumptions)
z = _dif.is_zero
if z is not None:
if z is False and _dif.is_commutative: # issue 10728
return False
if z:
return True
n2 = _n2(lhs, rhs)
if n2 is not None:
return _sympify(n2 == 0)
# see if the ratio evaluates
n, d = dif.as_numer_denom()
rv = None
_n = AssumptionsWrapper(n, assumptions)
_d = AssumptionsWrapper(d, assumptions)
if _n.is_zero:
rv = _d.is_nonzero
elif _n.is_finite:
if _d.is_infinite:
rv = True
elif _n.is_zero is False:
rv = _d.is_infinite
if rv is None:
# if the condition that makes the denominator
# infinite does not make the original expression
# True then False can be returned
l, r = clear_coefficients(d, S.Infinity)
args = [_.subs(l, r) for _ in (lhs, rhs)]
if args != [lhs, rhs]:
rv = fuzzy_bool(is_eq(*args, assumptions))
if rv is True:
rv = None
elif any(is_infinite(a, assumptions) for a in Add.make_args(n)):
# (inf or nan)/x != 0
rv = False
if rv is not None:
return rv
|
56623c9080748e7884066852581106c058216926c5721a11ebdf8122b379b815 | import numbers
import decimal
import fractions
import math
import re as regex
import sys
from .containers import Tuple
from .sympify import (SympifyError, converter, sympify, _convert_numpy_types, _sympify,
_is_numpy_instance)
from .singleton import S, Singleton
from .expr import Expr, AtomicExpr
from .evalf import pure_complex
from .decorators import _sympifyit
from .cache import cacheit, clear_cache
from .logic import fuzzy_not
from sympy.core.compatibility import (as_int, HAS_GMPY, SYMPY_INTS,
gmpy)
from sympy.core.cache import lru_cache
from .kind import NumberKind
from sympy.multipledispatch import dispatch
import mpmath
import mpmath.libmp as mlib
from mpmath.libmp import bitcount
from mpmath.libmp.backend import MPZ
from mpmath.libmp import mpf_pow, mpf_pi, mpf_e, phi_fixed
from mpmath.ctx_mp import mpnumeric
from mpmath.libmp.libmpf import (
finf as _mpf_inf, fninf as _mpf_ninf,
fnan as _mpf_nan, fzero, _normalize as mpf_normalize,
prec_to_dps)
from sympy.utilities.misc import debug, filldedent
from .parameters import global_parameters
from sympy.utilities.exceptions import SymPyDeprecationWarning
rnd = mlib.round_nearest
_LOG2 = math.log(2)
def comp(z1, z2, tol=None):
"""Return a bool indicating whether the error between z1 and z2
is <= tol.
Examples
========
If ``tol`` is None then True will be returned if
``abs(z1 - z2)*10**p <= 5`` where ``p`` is minimum value of the
decimal precision of each value.
>>> from sympy.core.numbers import comp, pi
>>> pi4 = pi.n(4); pi4
3.142
>>> comp(_, 3.142)
True
>>> comp(pi4, 3.141)
False
>>> comp(pi4, 3.143)
False
A comparison of strings will be made
if ``z1`` is a Number and ``z2`` is a string or ``tol`` is ''.
>>> comp(pi4, 3.1415)
True
>>> comp(pi4, 3.1415, '')
False
When ``tol`` is provided and ``z2`` is non-zero and
``|z1| > 1`` the error is normalized by ``|z1|``:
>>> abs(pi4 - 3.14)/pi4
0.000509791731426756
>>> comp(pi4, 3.14, .001) # difference less than 0.1%
True
>>> comp(pi4, 3.14, .0005) # difference less than 0.1%
False
When ``|z1| <= 1`` the absolute error is used:
>>> 1/pi4
0.3183
>>> abs(1/pi4 - 0.3183)/(1/pi4)
3.07371499106316e-5
>>> abs(1/pi4 - 0.3183)
9.78393554684764e-6
>>> comp(1/pi4, 0.3183, 1e-5)
True
To see if the absolute error between ``z1`` and ``z2`` is less
than or equal to ``tol``, call this as ``comp(z1 - z2, 0, tol)``
or ``comp(z1 - z2, tol=tol)``:
>>> abs(pi4 - 3.14)
0.00160156249999988
>>> comp(pi4 - 3.14, 0, .002)
True
>>> comp(pi4 - 3.14, 0, .001)
False
"""
if type(z2) is str:
if not pure_complex(z1, or_real=True):
raise ValueError('when z2 is a str z1 must be a Number')
return str(z1) == z2
if not z1:
z1, z2 = z2, z1
if not z1:
return True
if not tol:
a, b = z1, z2
if tol == '':
return str(a) == str(b)
if tol is None:
a, b = sympify(a), sympify(b)
if not all(i.is_number for i in (a, b)):
raise ValueError('expecting 2 numbers')
fa = a.atoms(Float)
fb = b.atoms(Float)
if not fa and not fb:
# no floats -- compare exactly
return a == b
# get a to be pure_complex
for do in range(2):
ca = pure_complex(a, or_real=True)
if not ca:
if fa:
a = a.n(prec_to_dps(min([i._prec for i in fa])))
ca = pure_complex(a, or_real=True)
break
else:
fa, fb = fb, fa
a, b = b, a
cb = pure_complex(b)
if not cb and fb:
b = b.n(prec_to_dps(min([i._prec for i in fb])))
cb = pure_complex(b, or_real=True)
if ca and cb and (ca[1] or cb[1]):
return all(comp(i, j) for i, j in zip(ca, cb))
tol = 10**prec_to_dps(min(a._prec, getattr(b, '_prec', a._prec)))
return int(abs(a - b)*tol) <= 5
diff = abs(z1 - z2)
az1 = abs(z1)
if z2 and az1 > 1:
return diff/az1 <= tol
else:
return diff <= tol
def mpf_norm(mpf, prec):
"""Return the mpf tuple normalized appropriately for the indicated
precision after doing a check to see if zero should be returned or
not when the mantissa is 0. ``mpf_normlize`` always assumes that this
is zero, but it may not be since the mantissa for mpf's values "+inf",
"-inf" and "nan" have a mantissa of zero, too.
Note: this is not intended to validate a given mpf tuple, so sending
mpf tuples that were not created by mpmath may produce bad results. This
is only a wrapper to ``mpf_normalize`` which provides the check for non-
zero mpfs that have a 0 for the mantissa.
"""
sign, man, expt, bc = mpf
if not man:
# hack for mpf_normalize which does not do this;
# it assumes that if man is zero the result is 0
# (see issue 6639)
if not bc:
return fzero
else:
# don't change anything; this should already
# be a well formed mpf tuple
return mpf
# Necessary if mpmath is using the gmpy backend
from mpmath.libmp.backend import MPZ
rv = mpf_normalize(sign, MPZ(man), expt, bc, prec, rnd)
return rv
# TODO: we should use the warnings module
_errdict = {"divide": False}
def seterr(divide=False):
"""
Should sympy raise an exception on 0/0 or return a nan?
divide == True .... raise an exception
divide == False ... return nan
"""
if _errdict["divide"] != divide:
clear_cache()
_errdict["divide"] = divide
def _as_integer_ratio(p):
neg_pow, man, expt, bc = getattr(p, '_mpf_', mpmath.mpf(p)._mpf_)
p = [1, -1][neg_pow % 2]*man
if expt < 0:
q = 2**-expt
else:
q = 1
p *= 2**expt
return int(p), int(q)
def _decimal_to_Rational_prec(dec):
"""Convert an ordinary decimal instance to a Rational."""
if not dec.is_finite():
raise TypeError("dec must be finite, got %s." % dec)
s, d, e = dec.as_tuple()
prec = len(d)
if e >= 0: # it's an integer
rv = Integer(int(dec))
else:
s = (-1)**s
d = sum([di*10**i for i, di in enumerate(reversed(d))])
rv = Rational(s*d, 10**-e)
return rv, prec
_floatpat = regex.compile(r"[-+]?((\d*\.\d+)|(\d+\.?))")
def _literal_float(f):
"""Return True if n starts like a floating point number."""
return bool(_floatpat.match(f))
# (a,b) -> gcd(a,b)
# TODO caching with decorator, but not to degrade performance
@lru_cache(1024)
def igcd(*args):
"""Computes nonnegative integer greatest common divisor.
Explanation
===========
The algorithm is based on the well known Euclid's algorithm. To
improve speed, igcd() has its own caching mechanism implemented.
Examples
========
>>> from sympy.core.numbers import igcd
>>> igcd(2, 4)
2
>>> igcd(5, 10, 15)
5
"""
if len(args) < 2:
raise TypeError(
'igcd() takes at least 2 arguments (%s given)' % len(args))
args_temp = [abs(as_int(i)) for i in args]
if 1 in args_temp:
return 1
a = args_temp.pop()
if HAS_GMPY: # Using gmpy if present to speed up.
for b in args_temp:
a = gmpy.gcd(a, b) if b else a
return as_int(a)
for b in args_temp:
a = math.gcd(a, b)
return a
igcd2 = math.gcd
def igcd_lehmer(a, b):
"""Computes greatest common divisor of two integers.
Explanation
===========
Euclid's algorithm for the computation of the greatest
common divisor gcd(a, b) of two (positive) integers
a and b is based on the division identity
a = q*b + r,
where the quotient q and the remainder r are integers
and 0 <= r < b. Then each common divisor of a and b
divides r, and it follows that gcd(a, b) == gcd(b, r).
The algorithm works by constructing the sequence
r0, r1, r2, ..., where r0 = a, r1 = b, and each rn
is the remainder from the division of the two preceding
elements.
In Python, q = a // b and r = a % b are obtained by the
floor division and the remainder operations, respectively.
These are the most expensive arithmetic operations, especially
for large a and b.
Lehmer's algorithm is based on the observation that the quotients
qn = r(n-1) // rn are in general small integers even
when a and b are very large. Hence the quotients can be
usually determined from a relatively small number of most
significant bits.
The efficiency of the algorithm is further enhanced by not
computing each long remainder in Euclid's sequence. The remainders
are linear combinations of a and b with integer coefficients
derived from the quotients. The coefficients can be computed
as far as the quotients can be determined from the chosen
most significant parts of a and b. Only then a new pair of
consecutive remainders is computed and the algorithm starts
anew with this pair.
References
==========
.. [1] https://en.wikipedia.org/wiki/Lehmer%27s_GCD_algorithm
"""
a, b = abs(as_int(a)), abs(as_int(b))
if a < b:
a, b = b, a
# The algorithm works by using one or two digit division
# whenever possible. The outer loop will replace the
# pair (a, b) with a pair of shorter consecutive elements
# of the Euclidean gcd sequence until a and b
# fit into two Python (long) int digits.
nbits = 2*sys.int_info.bits_per_digit
while a.bit_length() > nbits and b != 0:
# Quotients are mostly small integers that can
# be determined from most significant bits.
n = a.bit_length() - nbits
x, y = int(a >> n), int(b >> n) # most significant bits
# Elements of the Euclidean gcd sequence are linear
# combinations of a and b with integer coefficients.
# Compute the coefficients of consecutive pairs
# a' = A*a + B*b, b' = C*a + D*b
# using small integer arithmetic as far as possible.
A, B, C, D = 1, 0, 0, 1 # initial values
while True:
# The coefficients alternate in sign while looping.
# The inner loop combines two steps to keep track
# of the signs.
# At this point we have
# A > 0, B <= 0, C <= 0, D > 0,
# x' = x + B <= x < x" = x + A,
# y' = y + C <= y < y" = y + D,
# and
# x'*N <= a' < x"*N, y'*N <= b' < y"*N,
# where N = 2**n.
# Now, if y' > 0, and x"//y' and x'//y" agree,
# then their common value is equal to q = a'//b'.
# In addition,
# x'%y" = x' - q*y" < x" - q*y' = x"%y',
# and
# (x'%y")*N < a'%b' < (x"%y')*N.
# On the other hand, we also have x//y == q,
# and therefore
# x'%y" = x + B - q*(y + D) = x%y + B',
# x"%y' = x + A - q*(y + C) = x%y + A',
# where
# B' = B - q*D < 0, A' = A - q*C > 0.
if y + C <= 0:
break
q = (x + A) // (y + C)
# Now x'//y" <= q, and equality holds if
# x' - q*y" = (x - q*y) + (B - q*D) >= 0.
# This is a minor optimization to avoid division.
x_qy, B_qD = x - q*y, B - q*D
if x_qy + B_qD < 0:
break
# Next step in the Euclidean sequence.
x, y = y, x_qy
A, B, C, D = C, D, A - q*C, B_qD
# At this point the signs of the coefficients
# change and their roles are interchanged.
# A <= 0, B > 0, C > 0, D < 0,
# x' = x + A <= x < x" = x + B,
# y' = y + D < y < y" = y + C.
if y + D <= 0:
break
q = (x + B) // (y + D)
x_qy, A_qC = x - q*y, A - q*C
if x_qy + A_qC < 0:
break
x, y = y, x_qy
A, B, C, D = C, D, A_qC, B - q*D
# Now the conditions on top of the loop
# are again satisfied.
# A > 0, B < 0, C < 0, D > 0.
if B == 0:
# This can only happen when y == 0 in the beginning
# and the inner loop does nothing.
# Long division is forced.
a, b = b, a % b
continue
# Compute new long arguments using the coefficients.
a, b = A*a + B*b, C*a + D*b
# Small divisors. Finish with the standard algorithm.
while b:
a, b = b, a % b
return a
def ilcm(*args):
"""Computes integer least common multiple.
Examples
========
>>> from sympy.core.numbers import ilcm
>>> ilcm(5, 10)
10
>>> ilcm(7, 3)
21
>>> ilcm(5, 10, 15)
30
"""
if len(args) < 2:
raise TypeError(
'ilcm() takes at least 2 arguments (%s given)' % len(args))
if 0 in args:
return 0
a = args[0]
for b in args[1:]:
a = a // igcd(a, b) * b # since gcd(a,b) | a
return a
def igcdex(a, b):
"""Returns x, y, g such that g = x*a + y*b = gcd(a, b).
Examples
========
>>> from sympy.core.numbers import igcdex
>>> igcdex(2, 3)
(-1, 1, 1)
>>> igcdex(10, 12)
(-1, 1, 2)
>>> x, y, g = igcdex(100, 2004)
>>> x, y, g
(-20, 1, 4)
>>> x*100 + y*2004
4
"""
if (not a) and (not b):
return (0, 1, 0)
if not a:
return (0, b//abs(b), abs(b))
if not b:
return (a//abs(a), 0, abs(a))
if a < 0:
a, x_sign = -a, -1
else:
x_sign = 1
if b < 0:
b, y_sign = -b, -1
else:
y_sign = 1
x, y, r, s = 1, 0, 0, 1
while b:
(c, q) = (a % b, a // b)
(a, b, r, s, x, y) = (b, c, x - q*r, y - q*s, r, s)
return (x*x_sign, y*y_sign, a)
def mod_inverse(a, m):
"""
Return the number c such that, (a * c) = 1 (mod m)
where c has the same sign as m. If no such value exists,
a ValueError is raised.
Examples
========
>>> from sympy import S
>>> from sympy.core.numbers import mod_inverse
Suppose we wish to find multiplicative inverse x of
3 modulo 11. This is the same as finding x such
that 3 * x = 1 (mod 11). One value of x that satisfies
this congruence is 4. Because 3 * 4 = 12 and 12 = 1 (mod 11).
This is the value returned by mod_inverse:
>>> mod_inverse(3, 11)
4
>>> mod_inverse(-3, 11)
7
When there is a common factor between the numerators of
``a`` and ``m`` the inverse does not exist:
>>> mod_inverse(2, 4)
Traceback (most recent call last):
...
ValueError: inverse of 2 mod 4 does not exist
>>> mod_inverse(S(2)/7, S(5)/2)
7/2
References
==========
.. [1] https://en.wikipedia.org/wiki/Modular_multiplicative_inverse
.. [2] https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm
"""
c = None
try:
a, m = as_int(a), as_int(m)
if m != 1 and m != -1:
x, y, g = igcdex(a, m)
if g == 1:
c = x % m
except ValueError:
a, m = sympify(a), sympify(m)
if not (a.is_number and m.is_number):
raise TypeError(filldedent('''
Expected numbers for arguments; symbolic `mod_inverse`
is not implemented
but symbolic expressions can be handled with the
similar function,
sympy.polys.polytools.invert'''))
big = (m > 1)
if not (big is S.true or big is S.false):
raise ValueError('m > 1 did not evaluate; try to simplify %s' % m)
elif big:
c = 1/a
if c is None:
raise ValueError('inverse of %s (mod %s) does not exist' % (a, m))
return c
class Number(AtomicExpr):
"""Represents atomic numbers in SymPy.
Explanation
===========
Floating point numbers are represented by the Float class.
Rational numbers (of any size) are represented by the Rational class.
Integer numbers (of any size) are represented by the Integer class.
Float and Rational are subclasses of Number; Integer is a subclass
of Rational.
For example, ``2/3`` is represented as ``Rational(2, 3)`` which is
a different object from the floating point number obtained with
Python division ``2/3``. Even for numbers that are exactly
represented in binary, there is a difference between how two forms,
such as ``Rational(1, 2)`` and ``Float(0.5)``, are used in SymPy.
The rational form is to be preferred in symbolic computations.
Other kinds of numbers, such as algebraic numbers ``sqrt(2)`` or
complex numbers ``3 + 4*I``, are not instances of Number class as
they are not atomic.
See Also
========
Float, Integer, Rational
"""
is_commutative = True
is_number = True
is_Number = True
__slots__ = ()
# Used to make max(x._prec, y._prec) return x._prec when only x is a float
_prec = -1
kind = NumberKind
def __new__(cls, *obj):
if len(obj) == 1:
obj = obj[0]
if isinstance(obj, Number):
return obj
if isinstance(obj, SYMPY_INTS):
return Integer(obj)
if isinstance(obj, tuple) and len(obj) == 2:
return Rational(*obj)
if isinstance(obj, (float, mpmath.mpf, decimal.Decimal)):
return Float(obj)
if isinstance(obj, str):
_obj = obj.lower() # float('INF') == float('inf')
if _obj == 'nan':
return S.NaN
elif _obj == 'inf':
return S.Infinity
elif _obj == '+inf':
return S.Infinity
elif _obj == '-inf':
return S.NegativeInfinity
val = sympify(obj)
if isinstance(val, Number):
return val
else:
raise ValueError('String "%s" does not denote a Number' % obj)
msg = "expected str|int|long|float|Decimal|Number object but got %r"
raise TypeError(msg % type(obj).__name__)
def invert(self, other, *gens, **args):
from sympy.polys.polytools import invert
if getattr(other, 'is_number', True):
return mod_inverse(self, other)
return invert(self, other, *gens, **args)
def __divmod__(self, other):
from .containers import Tuple
from sympy.functions.elementary.complexes import sign
try:
other = Number(other)
if self.is_infinite or S.NaN in (self, other):
return (S.NaN, S.NaN)
except TypeError:
return NotImplemented
if not other:
raise ZeroDivisionError('modulo by zero')
if self.is_Integer and other.is_Integer:
return Tuple(*divmod(self.p, other.p))
elif isinstance(other, Float):
rat = self/Rational(other)
else:
rat = self/other
if other.is_finite:
w = int(rat) if rat >= 0 else int(rat) - 1
r = self - other*w
else:
w = 0 if not self or (sign(self) == sign(other)) else -1
r = other if w else self
return Tuple(w, r)
def __rdivmod__(self, other):
try:
other = Number(other)
except TypeError:
return NotImplemented
return divmod(other, self)
def _as_mpf_val(self, prec):
"""Evaluation of mpf tuple accurate to at least prec bits."""
raise NotImplementedError('%s needs ._as_mpf_val() method' %
(self.__class__.__name__))
def _eval_evalf(self, prec):
return Float._new(self._as_mpf_val(prec), prec)
def _as_mpf_op(self, prec):
prec = max(prec, self._prec)
return self._as_mpf_val(prec), prec
def __float__(self):
return mlib.to_float(self._as_mpf_val(53))
def floor(self):
raise NotImplementedError('%s needs .floor() method' %
(self.__class__.__name__))
def ceiling(self):
raise NotImplementedError('%s needs .ceiling() method' %
(self.__class__.__name__))
def __floor__(self):
return self.floor()
def __ceil__(self):
return self.ceiling()
def _eval_conjugate(self):
return self
def _eval_order(self, *symbols):
from sympy import Order
# Order(5, x, y) -> Order(1,x,y)
return Order(S.One, *symbols)
def _eval_subs(self, old, new):
if old == -self:
return -new
return self # there is no other possibility
def _eval_is_finite(self):
return True
@classmethod
def class_key(cls):
return 1, 0, 'Number'
@cacheit
def sort_key(self, order=None):
return self.class_key(), (0, ()), (), self
@_sympifyit('other', NotImplemented)
def __add__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
if other is S.NaN:
return S.NaN
elif other is S.Infinity:
return S.Infinity
elif other is S.NegativeInfinity:
return S.NegativeInfinity
return AtomicExpr.__add__(self, other)
@_sympifyit('other', NotImplemented)
def __sub__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
if other is S.NaN:
return S.NaN
elif other is S.Infinity:
return S.NegativeInfinity
elif other is S.NegativeInfinity:
return S.Infinity
return AtomicExpr.__sub__(self, other)
@_sympifyit('other', NotImplemented)
def __mul__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
if other is S.NaN:
return S.NaN
elif other is S.Infinity:
if self.is_zero:
return S.NaN
elif self.is_positive:
return S.Infinity
else:
return S.NegativeInfinity
elif other is S.NegativeInfinity:
if self.is_zero:
return S.NaN
elif self.is_positive:
return S.NegativeInfinity
else:
return S.Infinity
elif isinstance(other, Tuple):
return NotImplemented
return AtomicExpr.__mul__(self, other)
@_sympifyit('other', NotImplemented)
def __truediv__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
if other is S.NaN:
return S.NaN
elif other is S.Infinity or other is S.NegativeInfinity:
return S.Zero
return AtomicExpr.__truediv__(self, other)
def __eq__(self, other):
raise NotImplementedError('%s needs .__eq__() method' %
(self.__class__.__name__))
def __ne__(self, other):
raise NotImplementedError('%s needs .__ne__() method' %
(self.__class__.__name__))
def __lt__(self, other):
try:
other = _sympify(other)
except SympifyError:
raise TypeError("Invalid comparison %s < %s" % (self, other))
raise NotImplementedError('%s needs .__lt__() method' %
(self.__class__.__name__))
def __le__(self, other):
try:
other = _sympify(other)
except SympifyError:
raise TypeError("Invalid comparison %s <= %s" % (self, other))
raise NotImplementedError('%s needs .__le__() method' %
(self.__class__.__name__))
def __gt__(self, other):
try:
other = _sympify(other)
except SympifyError:
raise TypeError("Invalid comparison %s > %s" % (self, other))
return _sympify(other).__lt__(self)
def __ge__(self, other):
try:
other = _sympify(other)
except SympifyError:
raise TypeError("Invalid comparison %s >= %s" % (self, other))
return _sympify(other).__le__(self)
def __hash__(self):
return super().__hash__()
def is_constant(self, *wrt, **flags):
return True
def as_coeff_mul(self, *deps, rational=True, **kwargs):
# a -> c*t
if self.is_Rational or not rational:
return self, tuple()
elif self.is_negative:
return S.NegativeOne, (-self,)
return S.One, (self,)
def as_coeff_add(self, *deps):
# a -> c + t
if self.is_Rational:
return self, tuple()
return S.Zero, (self,)
def as_coeff_Mul(self, rational=False):
"""Efficiently extract the coefficient of a product. """
if rational and not self.is_Rational:
return S.One, self
return (self, S.One) if self else (S.One, self)
def as_coeff_Add(self, rational=False):
"""Efficiently extract the coefficient of a summation. """
if not rational:
return self, S.Zero
return S.Zero, self
def gcd(self, other):
"""Compute GCD of `self` and `other`. """
from sympy.polys import gcd
return gcd(self, other)
def lcm(self, other):
"""Compute LCM of `self` and `other`. """
from sympy.polys import lcm
return lcm(self, other)
def cofactors(self, other):
"""Compute GCD and cofactors of `self` and `other`. """
from sympy.polys import cofactors
return cofactors(self, other)
class Float(Number):
"""Represent a floating-point number of arbitrary precision.
Examples
========
>>> from sympy import Float
>>> Float(3.5)
3.50000000000000
>>> Float(3)
3.00000000000000
Creating Floats from strings (and Python ``int`` and ``long``
types) will give a minimum precision of 15 digits, but the
precision will automatically increase to capture all digits
entered.
>>> Float(1)
1.00000000000000
>>> Float(10**20)
100000000000000000000.
>>> Float('1e20')
100000000000000000000.
However, *floating-point* numbers (Python ``float`` types) retain
only 15 digits of precision:
>>> Float(1e20)
1.00000000000000e+20
>>> Float(1.23456789123456789)
1.23456789123457
It may be preferable to enter high-precision decimal numbers
as strings:
>>> Float('1.23456789123456789')
1.23456789123456789
The desired number of digits can also be specified:
>>> Float('1e-3', 3)
0.00100
>>> Float(100, 4)
100.0
Float can automatically count significant figures if a null string
is sent for the precision; spaces or underscores are also allowed. (Auto-
counting is only allowed for strings, ints and longs).
>>> Float('123 456 789.123_456', '')
123456789.123456
>>> Float('12e-3', '')
0.012
>>> Float(3, '')
3.
If a number is written in scientific notation, only the digits before the
exponent are considered significant if a decimal appears, otherwise the
"e" signifies only how to move the decimal:
>>> Float('60.e2', '') # 2 digits significant
6.0e+3
>>> Float('60e2', '') # 4 digits significant
6000.
>>> Float('600e-2', '') # 3 digits significant
6.00
Notes
=====
Floats are inexact by their nature unless their value is a binary-exact
value.
>>> approx, exact = Float(.1, 1), Float(.125, 1)
For calculation purposes, evalf needs to be able to change the precision
but this will not increase the accuracy of the inexact value. The
following is the most accurate 5-digit approximation of a value of 0.1
that had only 1 digit of precision:
>>> approx.evalf(5)
0.099609
By contrast, 0.125 is exact in binary (as it is in base 10) and so it
can be passed to Float or evalf to obtain an arbitrary precision with
matching accuracy:
>>> Float(exact, 5)
0.12500
>>> exact.evalf(20)
0.12500000000000000000
Trying to make a high-precision Float from a float is not disallowed,
but one must keep in mind that the *underlying float* (not the apparent
decimal value) is being obtained with high precision. For example, 0.3
does not have a finite binary representation. The closest rational is
the fraction 5404319552844595/2**54. So if you try to obtain a Float of
0.3 to 20 digits of precision you will not see the same thing as 0.3
followed by 19 zeros:
>>> Float(0.3, 20)
0.29999999999999998890
If you want a 20-digit value of the decimal 0.3 (not the floating point
approximation of 0.3) you should send the 0.3 as a string. The underlying
representation is still binary but a higher precision than Python's float
is used:
>>> Float('0.3', 20)
0.30000000000000000000
Although you can increase the precision of an existing Float using Float
it will not increase the accuracy -- the underlying value is not changed:
>>> def show(f): # binary rep of Float
... from sympy import Mul, Pow
... s, m, e, b = f._mpf_
... v = Mul(int(m), Pow(2, int(e), evaluate=False), evaluate=False)
... print('%s at prec=%s' % (v, f._prec))
...
>>> t = Float('0.3', 3)
>>> show(t)
4915/2**14 at prec=13
>>> show(Float(t, 20)) # higher prec, not higher accuracy
4915/2**14 at prec=70
>>> show(Float(t, 2)) # lower prec
307/2**10 at prec=10
The same thing happens when evalf is used on a Float:
>>> show(t.evalf(20))
4915/2**14 at prec=70
>>> show(t.evalf(2))
307/2**10 at prec=10
Finally, Floats can be instantiated with an mpf tuple (n, c, p) to
produce the number (-1)**n*c*2**p:
>>> n, c, p = 1, 5, 0
>>> (-1)**n*c*2**p
-5
>>> Float((1, 5, 0))
-5.00000000000000
An actual mpf tuple also contains the number of bits in c as the last
element of the tuple:
>>> _._mpf_
(1, 5, 0, 3)
This is not needed for instantiation and is not the same thing as the
precision. The mpf tuple and the precision are two separate quantities
that Float tracks.
In SymPy, a Float is a number that can be computed with arbitrary
precision. Although floating point 'inf' and 'nan' are not such
numbers, Float can create these numbers:
>>> Float('-inf')
-oo
>>> _.is_Float
False
"""
__slots__ = ('_mpf_', '_prec')
# A Float represents many real numbers,
# both rational and irrational.
is_rational = None
is_irrational = None
is_number = True
is_real = True
is_extended_real = True
is_Float = True
def __new__(cls, num, dps=None, prec=None, precision=None):
if prec is not None:
SymPyDeprecationWarning(
feature="Using 'prec=XX' to denote decimal precision",
useinstead="'dps=XX' for decimal precision and 'precision=XX' "\
"for binary precision",
issue=12820,
deprecated_since_version="1.1").warn()
dps = prec
del prec # avoid using this deprecated kwarg
if dps is not None and precision is not None:
raise ValueError('Both decimal and binary precision supplied. '
'Supply only one. ')
if isinstance(num, str):
# Float accepts spaces as digit separators
num = num.replace(' ', '').lower()
# in Py 3.6
# underscores are allowed. In anticipation of that, we ignore
# legally placed underscores
if '_' in num:
parts = num.split('_')
if not (all(parts) and
all(parts[i][-1].isdigit()
for i in range(0, len(parts), 2)) and
all(parts[i][0].isdigit()
for i in range(1, len(parts), 2))):
# copy Py 3.6 error
raise ValueError("could not convert string to float: '%s'" % num)
num = ''.join(parts)
if num.startswith('.') and len(num) > 1:
num = '0' + num
elif num.startswith('-.') and len(num) > 2:
num = '-0.' + num[2:]
elif num in ('inf', '+inf'):
return S.Infinity
elif num == '-inf':
return S.NegativeInfinity
elif isinstance(num, float) and num == 0:
num = '0'
elif isinstance(num, float) and num == float('inf'):
return S.Infinity
elif isinstance(num, float) and num == float('-inf'):
return S.NegativeInfinity
elif isinstance(num, float) and num == float('nan'):
return S.NaN
elif isinstance(num, (SYMPY_INTS, Integer)):
num = str(num)
elif num is S.Infinity:
return num
elif num is S.NegativeInfinity:
return num
elif num is S.NaN:
return num
elif _is_numpy_instance(num): # support for numpy datatypes
num = _convert_numpy_types(num)
elif isinstance(num, mpmath.mpf):
if precision is None:
if dps is None:
precision = num.context.prec
num = num._mpf_
if dps is None and precision is None:
dps = 15
if isinstance(num, Float):
return num
if isinstance(num, str) and _literal_float(num):
try:
Num = decimal.Decimal(num)
except decimal.InvalidOperation:
pass
else:
isint = '.' not in num
num, dps = _decimal_to_Rational_prec(Num)
if num.is_Integer and isint:
dps = max(dps, len(str(num).lstrip('-')))
dps = max(15, dps)
precision = mlib.libmpf.dps_to_prec(dps)
elif precision == '' and dps is None or precision is None and dps == '':
if not isinstance(num, str):
raise ValueError('The null string can only be used when '
'the number to Float is passed as a string or an integer.')
ok = None
if _literal_float(num):
try:
Num = decimal.Decimal(num)
except decimal.InvalidOperation:
pass
else:
isint = '.' not in num
num, dps = _decimal_to_Rational_prec(Num)
if num.is_Integer and isint:
dps = max(dps, len(str(num).lstrip('-')))
precision = mlib.libmpf.dps_to_prec(dps)
ok = True
if ok is None:
raise ValueError('string-float not recognized: %s' % num)
# decimal precision(dps) is set and maybe binary precision(precision)
# as well.From here on binary precision is used to compute the Float.
# Hence, if supplied use binary precision else translate from decimal
# precision.
if precision is None or precision == '':
precision = mlib.libmpf.dps_to_prec(dps)
precision = int(precision)
if isinstance(num, float):
_mpf_ = mlib.from_float(num, precision, rnd)
elif isinstance(num, str):
_mpf_ = mlib.from_str(num, precision, rnd)
elif isinstance(num, decimal.Decimal):
if num.is_finite():
_mpf_ = mlib.from_str(str(num), precision, rnd)
elif num.is_nan():
return S.NaN
elif num.is_infinite():
if num > 0:
return S.Infinity
return S.NegativeInfinity
else:
raise ValueError("unexpected decimal value %s" % str(num))
elif isinstance(num, tuple) and len(num) in (3, 4):
if type(num[1]) is str:
# it's a hexadecimal (coming from a pickled object)
# assume that it is in standard form
num = list(num)
# If we're loading an object pickled in Python 2 into
# Python 3, we may need to strip a tailing 'L' because
# of a shim for int on Python 3, see issue #13470.
if num[1].endswith('L'):
num[1] = num[1][:-1]
num[1] = MPZ(num[1], 16)
_mpf_ = tuple(num)
else:
if len(num) == 4:
# handle normalization hack
return Float._new(num, precision)
else:
if not all((
num[0] in (0, 1),
num[1] >= 0,
all(type(i) in (int, int) for i in num)
)):
raise ValueError('malformed mpf: %s' % (num,))
# don't compute number or else it may
# over/underflow
return Float._new(
(num[0], num[1], num[2], bitcount(num[1])),
precision)
else:
try:
_mpf_ = num._as_mpf_val(precision)
except (NotImplementedError, AttributeError):
_mpf_ = mpmath.mpf(num, prec=precision)._mpf_
return cls._new(_mpf_, precision, zero=False)
@classmethod
def _new(cls, _mpf_, _prec, zero=True):
# special cases
if zero and _mpf_ == fzero:
return S.Zero # Float(0) -> 0.0; Float._new((0,0,0,0)) -> 0
elif _mpf_ == _mpf_nan:
return S.NaN
elif _mpf_ == _mpf_inf:
return S.Infinity
elif _mpf_ == _mpf_ninf:
return S.NegativeInfinity
obj = Expr.__new__(cls)
obj._mpf_ = mpf_norm(_mpf_, _prec)
obj._prec = _prec
return obj
# mpz can't be pickled
def __getnewargs_ex__(self):
return ((mlib.to_pickable(self._mpf_),), {'precision': self._prec})
def _hashable_content(self):
return (self._mpf_, self._prec)
def floor(self):
return Integer(int(mlib.to_int(
mlib.mpf_floor(self._mpf_, self._prec))))
def ceiling(self):
return Integer(int(mlib.to_int(
mlib.mpf_ceil(self._mpf_, self._prec))))
def __floor__(self):
return self.floor()
def __ceil__(self):
return self.ceiling()
@property
def num(self):
return mpmath.mpf(self._mpf_)
def _as_mpf_val(self, prec):
rv = mpf_norm(self._mpf_, prec)
if rv != self._mpf_ and self._prec == prec:
debug(self._mpf_, rv)
return rv
def _as_mpf_op(self, prec):
return self._mpf_, max(prec, self._prec)
def _eval_is_finite(self):
if self._mpf_ in (_mpf_inf, _mpf_ninf):
return False
return True
def _eval_is_infinite(self):
if self._mpf_ in (_mpf_inf, _mpf_ninf):
return True
return False
def _eval_is_integer(self):
return self._mpf_ == fzero
def _eval_is_negative(self):
if self._mpf_ == _mpf_ninf or self._mpf_ == _mpf_inf:
return False
return self.num < 0
def _eval_is_positive(self):
if self._mpf_ == _mpf_ninf or self._mpf_ == _mpf_inf:
return False
return self.num > 0
def _eval_is_extended_negative(self):
if self._mpf_ == _mpf_ninf:
return True
if self._mpf_ == _mpf_inf:
return False
return self.num < 0
def _eval_is_extended_positive(self):
if self._mpf_ == _mpf_inf:
return True
if self._mpf_ == _mpf_ninf:
return False
return self.num > 0
def _eval_is_zero(self):
return self._mpf_ == fzero
def __bool__(self):
return self._mpf_ != fzero
def __neg__(self):
return Float._new(mlib.mpf_neg(self._mpf_), self._prec)
@_sympifyit('other', NotImplemented)
def __add__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
rhs, prec = other._as_mpf_op(self._prec)
return Float._new(mlib.mpf_add(self._mpf_, rhs, prec, rnd), prec)
return Number.__add__(self, other)
@_sympifyit('other', NotImplemented)
def __sub__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
rhs, prec = other._as_mpf_op(self._prec)
return Float._new(mlib.mpf_sub(self._mpf_, rhs, prec, rnd), prec)
return Number.__sub__(self, other)
@_sympifyit('other', NotImplemented)
def __mul__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
rhs, prec = other._as_mpf_op(self._prec)
return Float._new(mlib.mpf_mul(self._mpf_, rhs, prec, rnd), prec)
return Number.__mul__(self, other)
@_sympifyit('other', NotImplemented)
def __truediv__(self, other):
if isinstance(other, Number) and other != 0 and global_parameters.evaluate:
rhs, prec = other._as_mpf_op(self._prec)
return Float._new(mlib.mpf_div(self._mpf_, rhs, prec, rnd), prec)
return Number.__truediv__(self, other)
@_sympifyit('other', NotImplemented)
def __mod__(self, other):
if isinstance(other, Rational) and other.q != 1 and global_parameters.evaluate:
# calculate mod with Rationals, *then* round the result
return Float(Rational.__mod__(Rational(self), other),
precision=self._prec)
if isinstance(other, Float) and global_parameters.evaluate:
r = self/other
if r == int(r):
return Float(0, precision=max(self._prec, other._prec))
if isinstance(other, Number) and global_parameters.evaluate:
rhs, prec = other._as_mpf_op(self._prec)
return Float._new(mlib.mpf_mod(self._mpf_, rhs, prec, rnd), prec)
return Number.__mod__(self, other)
@_sympifyit('other', NotImplemented)
def __rmod__(self, other):
if isinstance(other, Float) and global_parameters.evaluate:
return other.__mod__(self)
if isinstance(other, Number) and global_parameters.evaluate:
rhs, prec = other._as_mpf_op(self._prec)
return Float._new(mlib.mpf_mod(rhs, self._mpf_, prec, rnd), prec)
return Number.__rmod__(self, other)
def _eval_power(self, expt):
"""
expt is symbolic object but not equal to 0, 1
(-p)**r -> exp(r*log(-p)) -> exp(r*(log(p) + I*Pi)) ->
-> p**r*(sin(Pi*r) + cos(Pi*r)*I)
"""
if self == 0:
if expt.is_positive:
return S.Zero
if expt.is_negative:
return S.Infinity
if isinstance(expt, Number):
if isinstance(expt, Integer):
prec = self._prec
return Float._new(
mlib.mpf_pow_int(self._mpf_, expt.p, prec, rnd), prec)
elif isinstance(expt, Rational) and \
expt.p == 1 and expt.q % 2 and self.is_negative:
return Pow(S.NegativeOne, expt, evaluate=False)*(
-self)._eval_power(expt)
expt, prec = expt._as_mpf_op(self._prec)
mpfself = self._mpf_
try:
y = mpf_pow(mpfself, expt, prec, rnd)
return Float._new(y, prec)
except mlib.ComplexResult:
re, im = mlib.mpc_pow(
(mpfself, fzero), (expt, fzero), prec, rnd)
return Float._new(re, prec) + \
Float._new(im, prec)*S.ImaginaryUnit
def __abs__(self):
return Float._new(mlib.mpf_abs(self._mpf_), self._prec)
def __int__(self):
if self._mpf_ == fzero:
return 0
return int(mlib.to_int(self._mpf_)) # uses round_fast = round_down
def __eq__(self, other):
from sympy.logic.boolalg import Boolean
try:
other = _sympify(other)
except SympifyError:
return NotImplemented
if isinstance(other, Boolean):
return False
if other.is_NumberSymbol:
if other.is_irrational:
return False
return other.__eq__(self)
if other.is_Float:
# comparison is exact
# so Float(.1, 3) != Float(.1, 33)
return self._mpf_ == other._mpf_
if other.is_Rational:
return other.__eq__(self)
if other.is_Number:
# numbers should compare at the same precision;
# all _as_mpf_val routines should be sure to abide
# by the request to change the prec if necessary; if
# they don't, the equality test will fail since it compares
# the mpf tuples
ompf = other._as_mpf_val(self._prec)
return bool(mlib.mpf_eq(self._mpf_, ompf))
if not self:
return not other
return False # Float != non-Number
def __ne__(self, other):
return not self == other
def _Frel(self, other, op):
from sympy.core.numbers import prec_to_dps
try:
other = _sympify(other)
except SympifyError:
return NotImplemented
if other.is_Rational:
# test self*other.q <?> other.p without losing precision
'''
>>> f = Float(.1,2)
>>> i = 1234567890
>>> (f*i)._mpf_
(0, 471, 18, 9)
>>> mlib.mpf_mul(f._mpf_, mlib.from_int(i))
(0, 505555550955, -12, 39)
'''
smpf = mlib.mpf_mul(self._mpf_, mlib.from_int(other.q))
ompf = mlib.from_int(other.p)
return _sympify(bool(op(smpf, ompf)))
elif other.is_Float:
return _sympify(bool(
op(self._mpf_, other._mpf_)))
elif other.is_comparable and other not in (
S.Infinity, S.NegativeInfinity):
other = other.evalf(prec_to_dps(self._prec))
if other._prec > 1:
if other.is_Number:
return _sympify(bool(
op(self._mpf_, other._as_mpf_val(self._prec))))
def __gt__(self, other):
if isinstance(other, NumberSymbol):
return other.__lt__(self)
rv = self._Frel(other, mlib.mpf_gt)
if rv is None:
return Expr.__gt__(self, other)
return rv
def __ge__(self, other):
if isinstance(other, NumberSymbol):
return other.__le__(self)
rv = self._Frel(other, mlib.mpf_ge)
if rv is None:
return Expr.__ge__(self, other)
return rv
def __lt__(self, other):
if isinstance(other, NumberSymbol):
return other.__gt__(self)
rv = self._Frel(other, mlib.mpf_lt)
if rv is None:
return Expr.__lt__(self, other)
return rv
def __le__(self, other):
if isinstance(other, NumberSymbol):
return other.__ge__(self)
rv = self._Frel(other, mlib.mpf_le)
if rv is None:
return Expr.__le__(self, other)
return rv
def __hash__(self):
return super().__hash__()
def epsilon_eq(self, other, epsilon="1e-15"):
return abs(self - other) < Float(epsilon)
def _sage_(self):
import sage.all as sage
return sage.RealNumber(str(self))
def __format__(self, format_spec):
return format(decimal.Decimal(str(self)), format_spec)
# Add sympify converters
converter[float] = converter[decimal.Decimal] = Float
# this is here to work nicely in Sage
RealNumber = Float
class Rational(Number):
"""Represents rational numbers (p/q) of any size.
Examples
========
>>> from sympy import Rational, nsimplify, S, pi
>>> Rational(1, 2)
1/2
Rational is unprejudiced in accepting input. If a float is passed, the
underlying value of the binary representation will be returned:
>>> Rational(.5)
1/2
>>> Rational(.2)
3602879701896397/18014398509481984
If the simpler representation of the float is desired then consider
limiting the denominator to the desired value or convert the float to
a string (which is roughly equivalent to limiting the denominator to
10**12):
>>> Rational(str(.2))
1/5
>>> Rational(.2).limit_denominator(10**12)
1/5
An arbitrarily precise Rational is obtained when a string literal is
passed:
>>> Rational("1.23")
123/100
>>> Rational('1e-2')
1/100
>>> Rational(".1")
1/10
>>> Rational('1e-2/3.2')
1/320
The conversion of other types of strings can be handled by
the sympify() function, and conversion of floats to expressions
or simple fractions can be handled with nsimplify:
>>> S('.[3]') # repeating digits in brackets
1/3
>>> S('3**2/10') # general expressions
9/10
>>> nsimplify(.3) # numbers that have a simple form
3/10
But if the input does not reduce to a literal Rational, an error will
be raised:
>>> Rational(pi)
Traceback (most recent call last):
...
TypeError: invalid input: pi
Low-level
---------
Access numerator and denominator as .p and .q:
>>> r = Rational(3, 4)
>>> r
3/4
>>> r.p
3
>>> r.q
4
Note that p and q return integers (not SymPy Integers) so some care
is needed when using them in expressions:
>>> r.p/r.q
0.75
See Also
========
sympy.core.sympify.sympify, sympy.simplify.simplify.nsimplify
"""
is_real = True
is_integer = False
is_rational = True
is_number = True
__slots__ = ('p', 'q')
is_Rational = True
@cacheit
def __new__(cls, p, q=None, gcd=None):
if q is None:
if isinstance(p, Rational):
return p
if isinstance(p, SYMPY_INTS):
pass
else:
if isinstance(p, (float, Float)):
return Rational(*_as_integer_ratio(p))
if not isinstance(p, str):
try:
p = sympify(p)
except (SympifyError, SyntaxError):
pass # error will raise below
else:
if p.count('/') > 1:
raise TypeError('invalid input: %s' % p)
p = p.replace(' ', '')
pq = p.rsplit('/', 1)
if len(pq) == 2:
p, q = pq
fp = fractions.Fraction(p)
fq = fractions.Fraction(q)
p = fp/fq
try:
p = fractions.Fraction(p)
except ValueError:
pass # error will raise below
else:
return Rational(p.numerator, p.denominator, 1)
if not isinstance(p, Rational):
raise TypeError('invalid input: %s' % p)
q = 1
gcd = 1
else:
p = Rational(p)
q = Rational(q)
if isinstance(q, Rational):
p *= q.q
q = q.p
if isinstance(p, Rational):
q *= p.q
p = p.p
# p and q are now integers
if q == 0:
if p == 0:
if _errdict["divide"]:
raise ValueError("Indeterminate 0/0")
else:
return S.NaN
return S.ComplexInfinity
if q < 0:
q = -q
p = -p
if not gcd:
gcd = igcd(abs(p), q)
if gcd > 1:
p //= gcd
q //= gcd
if q == 1:
return Integer(p)
if p == 1 and q == 2:
return S.Half
obj = Expr.__new__(cls)
obj.p = p
obj.q = q
return obj
def limit_denominator(self, max_denominator=1000000):
"""Closest Rational to self with denominator at most max_denominator.
Examples
========
>>> from sympy import Rational
>>> Rational('3.141592653589793').limit_denominator(10)
22/7
>>> Rational('3.141592653589793').limit_denominator(100)
311/99
"""
f = fractions.Fraction(self.p, self.q)
return Rational(f.limit_denominator(fractions.Fraction(int(max_denominator))))
def __getnewargs__(self):
return (self.p, self.q)
def _hashable_content(self):
return (self.p, self.q)
def _eval_is_positive(self):
return self.p > 0
def _eval_is_zero(self):
return self.p == 0
def __neg__(self):
return Rational(-self.p, self.q)
@_sympifyit('other', NotImplemented)
def __add__(self, other):
if global_parameters.evaluate:
if isinstance(other, Integer):
return Rational(self.p + self.q*other.p, self.q, 1)
elif isinstance(other, Rational):
#TODO: this can probably be optimized more
return Rational(self.p*other.q + self.q*other.p, self.q*other.q)
elif isinstance(other, Float):
return other + self
else:
return Number.__add__(self, other)
return Number.__add__(self, other)
__radd__ = __add__
@_sympifyit('other', NotImplemented)
def __sub__(self, other):
if global_parameters.evaluate:
if isinstance(other, Integer):
return Rational(self.p - self.q*other.p, self.q, 1)
elif isinstance(other, Rational):
return Rational(self.p*other.q - self.q*other.p, self.q*other.q)
elif isinstance(other, Float):
return -other + self
else:
return Number.__sub__(self, other)
return Number.__sub__(self, other)
@_sympifyit('other', NotImplemented)
def __rsub__(self, other):
if global_parameters.evaluate:
if isinstance(other, Integer):
return Rational(self.q*other.p - self.p, self.q, 1)
elif isinstance(other, Rational):
return Rational(self.q*other.p - self.p*other.q, self.q*other.q)
elif isinstance(other, Float):
return -self + other
else:
return Number.__rsub__(self, other)
return Number.__rsub__(self, other)
@_sympifyit('other', NotImplemented)
def __mul__(self, other):
if global_parameters.evaluate:
if isinstance(other, Integer):
return Rational(self.p*other.p, self.q, igcd(other.p, self.q))
elif isinstance(other, Rational):
return Rational(self.p*other.p, self.q*other.q, igcd(self.p, other.q)*igcd(self.q, other.p))
elif isinstance(other, Float):
return other*self
else:
return Number.__mul__(self, other)
return Number.__mul__(self, other)
__rmul__ = __mul__
@_sympifyit('other', NotImplemented)
def __truediv__(self, other):
if global_parameters.evaluate:
if isinstance(other, Integer):
if self.p and other.p == S.Zero:
return S.ComplexInfinity
else:
return Rational(self.p, self.q*other.p, igcd(self.p, other.p))
elif isinstance(other, Rational):
return Rational(self.p*other.q, self.q*other.p, igcd(self.p, other.p)*igcd(self.q, other.q))
elif isinstance(other, Float):
return self*(1/other)
else:
return Number.__truediv__(self, other)
return Number.__truediv__(self, other)
@_sympifyit('other', NotImplemented)
def __rtruediv__(self, other):
if global_parameters.evaluate:
if isinstance(other, Integer):
return Rational(other.p*self.q, self.p, igcd(self.p, other.p))
elif isinstance(other, Rational):
return Rational(other.p*self.q, other.q*self.p, igcd(self.p, other.p)*igcd(self.q, other.q))
elif isinstance(other, Float):
return other*(1/self)
else:
return Number.__rtruediv__(self, other)
return Number.__rtruediv__(self, other)
@_sympifyit('other', NotImplemented)
def __mod__(self, other):
if global_parameters.evaluate:
if isinstance(other, Rational):
n = (self.p*other.q) // (other.p*self.q)
return Rational(self.p*other.q - n*other.p*self.q, self.q*other.q)
if isinstance(other, Float):
# calculate mod with Rationals, *then* round the answer
return Float(self.__mod__(Rational(other)),
precision=other._prec)
return Number.__mod__(self, other)
return Number.__mod__(self, other)
@_sympifyit('other', NotImplemented)
def __rmod__(self, other):
if isinstance(other, Rational):
return Rational.__mod__(other, self)
return Number.__rmod__(self, other)
def _eval_power(self, expt):
if isinstance(expt, Number):
if isinstance(expt, Float):
return self._eval_evalf(expt._prec)**expt
if expt.is_extended_negative:
# (3/4)**-2 -> (4/3)**2
ne = -expt
if (ne is S.One):
return Rational(self.q, self.p)
if self.is_negative:
return S.NegativeOne**expt*Rational(self.q, -self.p)**ne
else:
return Rational(self.q, self.p)**ne
if expt is S.Infinity: # -oo already caught by test for negative
if self.p > self.q:
# (3/2)**oo -> oo
return S.Infinity
if self.p < -self.q:
# (-3/2)**oo -> oo + I*oo
return S.Infinity + S.Infinity*S.ImaginaryUnit
return S.Zero
if isinstance(expt, Integer):
# (4/3)**2 -> 4**2 / 3**2
return Rational(self.p**expt.p, self.q**expt.p, 1)
if isinstance(expt, Rational):
if self.p != 1:
# (4/3)**(5/6) -> 4**(5/6)*3**(-5/6)
return Integer(self.p)**expt*Integer(self.q)**(-expt)
# as the above caught negative self.p, now self is positive
return Integer(self.q)**Rational(
expt.p*(expt.q - 1), expt.q) / \
Integer(self.q)**Integer(expt.p)
if self.is_extended_negative and expt.is_even:
return (-self)**expt
return
def _as_mpf_val(self, prec):
return mlib.from_rational(self.p, self.q, prec, rnd)
def _mpmath_(self, prec, rnd):
return mpmath.make_mpf(mlib.from_rational(self.p, self.q, prec, rnd))
def __abs__(self):
return Rational(abs(self.p), self.q)
def __int__(self):
p, q = self.p, self.q
if p < 0:
return -int(-p//q)
return int(p//q)
def floor(self):
return Integer(self.p // self.q)
def ceiling(self):
return -Integer(-self.p // self.q)
def __floor__(self):
return self.floor()
def __ceil__(self):
return self.ceiling()
def __eq__(self, other):
from sympy.core.power import integer_log
try:
other = _sympify(other)
except SympifyError:
return NotImplemented
if not isinstance(other, Number):
# S(0) == S.false is False
# S(0) == False is True
return False
if not self:
return not other
if other.is_NumberSymbol:
if other.is_irrational:
return False
return other.__eq__(self)
if other.is_Rational:
# a Rational is always in reduced form so will never be 2/4
# so we can just check equivalence of args
return self.p == other.p and self.q == other.q
if other.is_Float:
# all Floats have a denominator that is a power of 2
# so if self doesn't, it can't be equal to other
if self.q & (self.q - 1):
return False
s, m, t = other._mpf_[:3]
if s:
m = -m
if not t:
# other is an odd integer
if not self.is_Integer or self.is_even:
return False
return m == self.p
if t > 0:
# other is an even integer
if not self.is_Integer:
return False
# does m*2**t == self.p
return self.p and not self.p % m and \
integer_log(self.p//m, 2) == (t, True)
# does non-integer s*m/2**-t = p/q?
if self.is_Integer:
return False
return m == self.p and integer_log(self.q, 2) == (-t, True)
return False
def __ne__(self, other):
return not self == other
def _Rrel(self, other, attr):
# if you want self < other, pass self, other, __gt__
try:
other = _sympify(other)
except SympifyError:
return NotImplemented
if other.is_Number:
op = None
s, o = self, other
if other.is_NumberSymbol:
op = getattr(o, attr)
elif other.is_Float:
op = getattr(o, attr)
elif other.is_Rational:
s, o = Integer(s.p*o.q), Integer(s.q*o.p)
op = getattr(o, attr)
if op:
return op(s)
if o.is_number and o.is_extended_real:
return Integer(s.p), s.q*o
def __gt__(self, other):
rv = self._Rrel(other, '__lt__')
if rv is None:
rv = self, other
elif not type(rv) is tuple:
return rv
return Expr.__gt__(*rv)
def __ge__(self, other):
rv = self._Rrel(other, '__le__')
if rv is None:
rv = self, other
elif not type(rv) is tuple:
return rv
return Expr.__ge__(*rv)
def __lt__(self, other):
rv = self._Rrel(other, '__gt__')
if rv is None:
rv = self, other
elif not type(rv) is tuple:
return rv
return Expr.__lt__(*rv)
def __le__(self, other):
rv = self._Rrel(other, '__ge__')
if rv is None:
rv = self, other
elif not type(rv) is tuple:
return rv
return Expr.__le__(*rv)
def __hash__(self):
return super().__hash__()
def factors(self, limit=None, use_trial=True, use_rho=False,
use_pm1=False, verbose=False, visual=False):
"""A wrapper to factorint which return factors of self that are
smaller than limit (or cheap to compute). Special methods of
factoring are disabled by default so that only trial division is used.
"""
from sympy.ntheory import factorrat
return factorrat(self, limit=limit, use_trial=use_trial,
use_rho=use_rho, use_pm1=use_pm1,
verbose=verbose).copy()
@property
def numerator(self):
return self.p
@property
def denominator(self):
return self.q
@_sympifyit('other', NotImplemented)
def gcd(self, other):
if isinstance(other, Rational):
if other == S.Zero:
return other
return Rational(
Integer(igcd(self.p, other.p)),
Integer(ilcm(self.q, other.q)))
return Number.gcd(self, other)
@_sympifyit('other', NotImplemented)
def lcm(self, other):
if isinstance(other, Rational):
return Rational(
self.p // igcd(self.p, other.p) * other.p,
igcd(self.q, other.q))
return Number.lcm(self, other)
def as_numer_denom(self):
return Integer(self.p), Integer(self.q)
def _sage_(self):
import sage.all as sage
return sage.Integer(self.p)/sage.Integer(self.q)
def as_content_primitive(self, radical=False, clear=True):
"""Return the tuple (R, self/R) where R is the positive Rational
extracted from self.
Examples
========
>>> from sympy import S
>>> (S(-3)/2).as_content_primitive()
(3/2, -1)
See docstring of Expr.as_content_primitive for more examples.
"""
if self:
if self.is_positive:
return self, S.One
return -self, S.NegativeOne
return S.One, self
def as_coeff_Mul(self, rational=False):
"""Efficiently extract the coefficient of a product. """
return self, S.One
def as_coeff_Add(self, rational=False):
"""Efficiently extract the coefficient of a summation. """
return self, S.Zero
class Integer(Rational):
"""Represents integer numbers of any size.
Examples
========
>>> from sympy import Integer
>>> Integer(3)
3
If a float or a rational is passed to Integer, the fractional part
will be discarded; the effect is of rounding toward zero.
>>> Integer(3.8)
3
>>> Integer(-3.8)
-3
A string is acceptable input if it can be parsed as an integer:
>>> Integer("9" * 20)
99999999999999999999
It is rarely needed to explicitly instantiate an Integer, because
Python integers are automatically converted to Integer when they
are used in SymPy expressions.
"""
q = 1
is_integer = True
is_number = True
is_Integer = True
__slots__ = ('p',)
def _as_mpf_val(self, prec):
return mlib.from_int(self.p, prec, rnd)
def _mpmath_(self, prec, rnd):
return mpmath.make_mpf(self._as_mpf_val(prec))
@cacheit
def __new__(cls, i):
if isinstance(i, str):
i = i.replace(' ', '')
# whereas we cannot, in general, make a Rational from an
# arbitrary expression, we can make an Integer unambiguously
# (except when a non-integer expression happens to round to
# an integer). So we proceed by taking int() of the input and
# let the int routines determine whether the expression can
# be made into an int or whether an error should be raised.
try:
ival = int(i)
except TypeError:
raise TypeError(
"Argument of Integer should be of numeric type, got %s." % i)
# We only work with well-behaved integer types. This converts, for
# example, numpy.int32 instances.
if ival == 1:
return S.One
if ival == -1:
return S.NegativeOne
if ival == 0:
return S.Zero
obj = Expr.__new__(cls)
obj.p = ival
return obj
def __getnewargs__(self):
return (self.p,)
# Arithmetic operations are here for efficiency
def __int__(self):
return self.p
def floor(self):
return Integer(self.p)
def ceiling(self):
return Integer(self.p)
def __floor__(self):
return self.floor()
def __ceil__(self):
return self.ceiling()
def __neg__(self):
return Integer(-self.p)
def __abs__(self):
if self.p >= 0:
return self
else:
return Integer(-self.p)
def __divmod__(self, other):
from .containers import Tuple
if isinstance(other, Integer) and global_parameters.evaluate:
return Tuple(*(divmod(self.p, other.p)))
else:
return Number.__divmod__(self, other)
def __rdivmod__(self, other):
from .containers import Tuple
if isinstance(other, int) and global_parameters.evaluate:
return Tuple(*(divmod(other, self.p)))
else:
try:
other = Number(other)
except TypeError:
msg = "unsupported operand type(s) for divmod(): '%s' and '%s'"
oname = type(other).__name__
sname = type(self).__name__
raise TypeError(msg % (oname, sname))
return Number.__divmod__(other, self)
# TODO make it decorator + bytecodehacks?
def __add__(self, other):
if global_parameters.evaluate:
if isinstance(other, int):
return Integer(self.p + other)
elif isinstance(other, Integer):
return Integer(self.p + other.p)
elif isinstance(other, Rational):
return Rational(self.p*other.q + other.p, other.q, 1)
return Rational.__add__(self, other)
else:
return Add(self, other)
def __radd__(self, other):
if global_parameters.evaluate:
if isinstance(other, int):
return Integer(other + self.p)
elif isinstance(other, Rational):
return Rational(other.p + self.p*other.q, other.q, 1)
return Rational.__radd__(self, other)
return Rational.__radd__(self, other)
def __sub__(self, other):
if global_parameters.evaluate:
if isinstance(other, int):
return Integer(self.p - other)
elif isinstance(other, Integer):
return Integer(self.p - other.p)
elif isinstance(other, Rational):
return Rational(self.p*other.q - other.p, other.q, 1)
return Rational.__sub__(self, other)
return Rational.__sub__(self, other)
def __rsub__(self, other):
if global_parameters.evaluate:
if isinstance(other, int):
return Integer(other - self.p)
elif isinstance(other, Rational):
return Rational(other.p - self.p*other.q, other.q, 1)
return Rational.__rsub__(self, other)
return Rational.__rsub__(self, other)
def __mul__(self, other):
if global_parameters.evaluate:
if isinstance(other, int):
return Integer(self.p*other)
elif isinstance(other, Integer):
return Integer(self.p*other.p)
elif isinstance(other, Rational):
return Rational(self.p*other.p, other.q, igcd(self.p, other.q))
return Rational.__mul__(self, other)
return Rational.__mul__(self, other)
def __rmul__(self, other):
if global_parameters.evaluate:
if isinstance(other, int):
return Integer(other*self.p)
elif isinstance(other, Rational):
return Rational(other.p*self.p, other.q, igcd(self.p, other.q))
return Rational.__rmul__(self, other)
return Rational.__rmul__(self, other)
def __mod__(self, other):
if global_parameters.evaluate:
if isinstance(other, int):
return Integer(self.p % other)
elif isinstance(other, Integer):
return Integer(self.p % other.p)
return Rational.__mod__(self, other)
return Rational.__mod__(self, other)
def __rmod__(self, other):
if global_parameters.evaluate:
if isinstance(other, int):
return Integer(other % self.p)
elif isinstance(other, Integer):
return Integer(other.p % self.p)
return Rational.__rmod__(self, other)
return Rational.__rmod__(self, other)
def __eq__(self, other):
if isinstance(other, int):
return (self.p == other)
elif isinstance(other, Integer):
return (self.p == other.p)
return Rational.__eq__(self, other)
def __ne__(self, other):
return not self == other
def __gt__(self, other):
try:
other = _sympify(other)
except SympifyError:
return NotImplemented
if other.is_Integer:
return _sympify(self.p > other.p)
return Rational.__gt__(self, other)
def __lt__(self, other):
try:
other = _sympify(other)
except SympifyError:
return NotImplemented
if other.is_Integer:
return _sympify(self.p < other.p)
return Rational.__lt__(self, other)
def __ge__(self, other):
try:
other = _sympify(other)
except SympifyError:
return NotImplemented
if other.is_Integer:
return _sympify(self.p >= other.p)
return Rational.__ge__(self, other)
def __le__(self, other):
try:
other = _sympify(other)
except SympifyError:
return NotImplemented
if other.is_Integer:
return _sympify(self.p <= other.p)
return Rational.__le__(self, other)
def __hash__(self):
return hash(self.p)
def __index__(self):
return self.p
########################################
def _eval_is_odd(self):
return bool(self.p % 2)
def _eval_power(self, expt):
"""
Tries to do some simplifications on self**expt
Returns None if no further simplifications can be done.
Explanation
===========
When exponent is a fraction (so we have for example a square root),
we try to find a simpler representation by factoring the argument
up to factors of 2**15, e.g.
- sqrt(4) becomes 2
- sqrt(-4) becomes 2*I
- (2**(3+7)*3**(6+7))**Rational(1,7) becomes 6*18**(3/7)
Further simplification would require a special call to factorint on
the argument which is not done here for sake of speed.
"""
from sympy.ntheory.factor_ import perfect_power
if expt is S.Infinity:
if self.p > S.One:
return S.Infinity
# cases -1, 0, 1 are done in their respective classes
return S.Infinity + S.ImaginaryUnit*S.Infinity
if expt is S.NegativeInfinity:
return Rational(1, self)**S.Infinity
if not isinstance(expt, Number):
# simplify when expt is even
# (-2)**k --> 2**k
if self.is_negative and expt.is_even:
return (-self)**expt
if isinstance(expt, Float):
# Rational knows how to exponentiate by a Float
return super()._eval_power(expt)
if not isinstance(expt, Rational):
return
if expt is S.Half and self.is_negative:
# we extract I for this special case since everyone is doing so
return S.ImaginaryUnit*Pow(-self, expt)
if expt.is_negative:
# invert base and change sign on exponent
ne = -expt
if self.is_negative:
return S.NegativeOne**expt*Rational(1, -self)**ne
else:
return Rational(1, self.p)**ne
# see if base is a perfect root, sqrt(4) --> 2
x, xexact = integer_nthroot(abs(self.p), expt.q)
if xexact:
# if it's a perfect root we've finished
result = Integer(x**abs(expt.p))
if self.is_negative:
result *= S.NegativeOne**expt
return result
# The following is an algorithm where we collect perfect roots
# from the factors of base.
# if it's not an nth root, it still might be a perfect power
b_pos = int(abs(self.p))
p = perfect_power(b_pos)
if p is not False:
dict = {p[0]: p[1]}
else:
dict = Integer(b_pos).factors(limit=2**15)
# now process the dict of factors
out_int = 1 # integer part
out_rad = 1 # extracted radicals
sqr_int = 1
sqr_gcd = 0
sqr_dict = {}
for prime, exponent in dict.items():
exponent *= expt.p
# remove multiples of expt.q: (2**12)**(1/10) -> 2*(2**2)**(1/10)
div_e, div_m = divmod(exponent, expt.q)
if div_e > 0:
out_int *= prime**div_e
if div_m > 0:
# see if the reduced exponent shares a gcd with e.q
# (2**2)**(1/10) -> 2**(1/5)
g = igcd(div_m, expt.q)
if g != 1:
out_rad *= Pow(prime, Rational(div_m//g, expt.q//g))
else:
sqr_dict[prime] = div_m
# identify gcd of remaining powers
for p, ex in sqr_dict.items():
if sqr_gcd == 0:
sqr_gcd = ex
else:
sqr_gcd = igcd(sqr_gcd, ex)
if sqr_gcd == 1:
break
for k, v in sqr_dict.items():
sqr_int *= k**(v//sqr_gcd)
if sqr_int == b_pos and out_int == 1 and out_rad == 1:
result = None
else:
result = out_int*out_rad*Pow(sqr_int, Rational(sqr_gcd, expt.q))
if self.is_negative:
result *= Pow(S.NegativeOne, expt)
return result
def _eval_is_prime(self):
from sympy.ntheory import isprime
return isprime(self)
def _eval_is_composite(self):
if self > 1:
return fuzzy_not(self.is_prime)
else:
return False
def as_numer_denom(self):
return self, S.One
@_sympifyit('other', NotImplemented)
def __floordiv__(self, other):
if not isinstance(other, Expr):
return NotImplemented
if isinstance(other, Integer):
return Integer(self.p // other)
return Integer(divmod(self, other)[0])
def __rfloordiv__(self, other):
return Integer(Integer(other).p // self.p)
# Add sympify converters
converter[int] = Integer
class AlgebraicNumber(Expr):
"""Class for representing algebraic numbers in SymPy. """
__slots__ = ('rep', 'root', 'alias', 'minpoly')
is_AlgebraicNumber = True
is_algebraic = True
is_number = True
kind = NumberKind
# Optional alias symbol is not free.
# Actually, alias should be a Str, but some methods
# expect that it be an instance of Expr.
free_symbols = set()
def __new__(cls, expr, coeffs=None, alias=None, **args):
"""Construct a new algebraic number. """
from sympy import Poly
from sympy.polys.polyclasses import ANP, DMP
from sympy.polys.numberfields import minimal_polynomial
from sympy.core.symbol import Symbol
expr = sympify(expr)
if isinstance(expr, (tuple, Tuple)):
minpoly, root = expr
if not minpoly.is_Poly:
minpoly = Poly(minpoly)
elif expr.is_AlgebraicNumber:
minpoly, root = expr.minpoly, expr.root
else:
minpoly, root = minimal_polynomial(
expr, args.get('gen'), polys=True), expr
dom = minpoly.get_domain()
if coeffs is not None:
if not isinstance(coeffs, ANP):
rep = DMP.from_sympy_list(sympify(coeffs), 0, dom)
scoeffs = Tuple(*coeffs)
else:
rep = DMP.from_list(coeffs.to_list(), 0, dom)
scoeffs = Tuple(*coeffs.to_list())
if rep.degree() >= minpoly.degree():
rep = rep.rem(minpoly.rep)
else:
rep = DMP.from_list([1, 0], 0, dom)
scoeffs = Tuple(1, 0)
sargs = (root, scoeffs)
if alias is not None:
if not isinstance(alias, Symbol):
alias = Symbol(alias)
sargs = sargs + (alias,)
obj = Expr.__new__(cls, *sargs)
obj.rep = rep
obj.root = root
obj.alias = alias
obj.minpoly = minpoly
return obj
def __hash__(self):
return super().__hash__()
def _eval_evalf(self, prec):
return self.as_expr()._evalf(prec)
@property
def is_aliased(self):
"""Returns ``True`` if ``alias`` was set. """
return self.alias is not None
def as_poly(self, x=None):
"""Create a Poly instance from ``self``. """
from sympy import Dummy, Poly, PurePoly
if x is not None:
return Poly.new(self.rep, x)
else:
if self.alias is not None:
return Poly.new(self.rep, self.alias)
else:
return PurePoly.new(self.rep, Dummy('x'))
def as_expr(self, x=None):
"""Create a Basic expression from ``self``. """
return self.as_poly(x or self.root).as_expr().expand()
def coeffs(self):
"""Returns all SymPy coefficients of an algebraic number. """
return [ self.rep.dom.to_sympy(c) for c in self.rep.all_coeffs() ]
def native_coeffs(self):
"""Returns all native coefficients of an algebraic number. """
return self.rep.all_coeffs()
def to_algebraic_integer(self):
"""Convert ``self`` to an algebraic integer. """
from sympy import Poly
f = self.minpoly
if f.LC() == 1:
return self
coeff = f.LC()**(f.degree() - 1)
poly = f.compose(Poly(f.gen/f.LC()))
minpoly = poly*coeff
root = f.LC()*self.root
return AlgebraicNumber((minpoly, root), self.coeffs())
def _eval_simplify(self, **kwargs):
from sympy.polys import CRootOf, minpoly
measure, ratio = kwargs['measure'], kwargs['ratio']
for r in [r for r in self.minpoly.all_roots() if r.func != CRootOf]:
if minpoly(self.root - r).is_Symbol:
# use the matching root if it's simpler
if measure(r) < ratio*measure(self.root):
return AlgebraicNumber(r)
return self
class RationalConstant(Rational):
"""
Abstract base class for rationals with specific behaviors
Derived classes must define class attributes p and q and should probably all
be singletons.
"""
__slots__ = ()
def __new__(cls):
return AtomicExpr.__new__(cls)
class IntegerConstant(Integer):
__slots__ = ()
def __new__(cls):
return AtomicExpr.__new__(cls)
class Zero(IntegerConstant, metaclass=Singleton):
"""The number zero.
Zero is a singleton, and can be accessed by ``S.Zero``
Examples
========
>>> from sympy import S, Integer
>>> Integer(0) is S.Zero
True
>>> 1/S.Zero
zoo
References
==========
.. [1] https://en.wikipedia.org/wiki/Zero
"""
p = 0
q = 1
is_positive = False
is_negative = False
is_zero = True
is_number = True
is_comparable = True
__slots__ = ()
def __getnewargs__(self):
return ()
@staticmethod
def __abs__():
return S.Zero
@staticmethod
def __neg__():
return S.Zero
def _eval_power(self, expt):
if expt.is_positive:
return self
if expt.is_negative:
return S.ComplexInfinity
if expt.is_extended_real is False:
return S.NaN
# infinities are already handled with pos and neg
# tests above; now throw away leading numbers on Mul
# exponent
coeff, terms = expt.as_coeff_Mul()
if coeff.is_negative:
return S.ComplexInfinity**terms
if coeff is not S.One: # there is a Number to discard
return self**terms
def _eval_order(self, *symbols):
# Order(0,x) -> 0
return self
def __bool__(self):
return False
def as_coeff_Mul(self, rational=False): # XXX this routine should be deleted
"""Efficiently extract the coefficient of a summation. """
return S.One, self
class One(IntegerConstant, metaclass=Singleton):
"""The number one.
One is a singleton, and can be accessed by ``S.One``.
Examples
========
>>> from sympy import S, Integer
>>> Integer(1) is S.One
True
References
==========
.. [1] https://en.wikipedia.org/wiki/1_%28number%29
"""
is_number = True
is_positive = True
p = 1
q = 1
__slots__ = ()
def __getnewargs__(self):
return ()
@staticmethod
def __abs__():
return S.One
@staticmethod
def __neg__():
return S.NegativeOne
def _eval_power(self, expt):
return self
def _eval_order(self, *symbols):
return
@staticmethod
def factors(limit=None, use_trial=True, use_rho=False, use_pm1=False,
verbose=False, visual=False):
if visual:
return S.One
else:
return {}
class NegativeOne(IntegerConstant, metaclass=Singleton):
"""The number negative one.
NegativeOne is a singleton, and can be accessed by ``S.NegativeOne``.
Examples
========
>>> from sympy import S, Integer
>>> Integer(-1) is S.NegativeOne
True
See Also
========
One
References
==========
.. [1] https://en.wikipedia.org/wiki/%E2%88%921_%28number%29
"""
is_number = True
p = -1
q = 1
__slots__ = ()
def __getnewargs__(self):
return ()
@staticmethod
def __abs__():
return S.One
@staticmethod
def __neg__():
return S.One
def _eval_power(self, expt):
if expt.is_odd:
return S.NegativeOne
if expt.is_even:
return S.One
if isinstance(expt, Number):
if isinstance(expt, Float):
return Float(-1.0)**expt
if expt is S.NaN:
return S.NaN
if expt is S.Infinity or expt is S.NegativeInfinity:
return S.NaN
if expt is S.Half:
return S.ImaginaryUnit
if isinstance(expt, Rational):
if expt.q == 2:
return S.ImaginaryUnit**Integer(expt.p)
i, r = divmod(expt.p, expt.q)
if i:
return self**i*self**Rational(r, expt.q)
return
class Half(RationalConstant, metaclass=Singleton):
"""The rational number 1/2.
Half is a singleton, and can be accessed by ``S.Half``.
Examples
========
>>> from sympy import S, Rational
>>> Rational(1, 2) is S.Half
True
References
==========
.. [1] https://en.wikipedia.org/wiki/One_half
"""
is_number = True
p = 1
q = 2
__slots__ = ()
def __getnewargs__(self):
return ()
@staticmethod
def __abs__():
return S.Half
class Infinity(Number, metaclass=Singleton):
r"""Positive infinite quantity.
Explanation
===========
In real analysis the symbol `\infty` denotes an unbounded
limit: `x\to\infty` means that `x` grows without bound.
Infinity is often used not only to define a limit but as a value
in the affinely extended real number system. Points labeled `+\infty`
and `-\infty` can be added to the topological space of the real numbers,
producing the two-point compactification of the real numbers. Adding
algebraic properties to this gives us the extended real numbers.
Infinity is a singleton, and can be accessed by ``S.Infinity``,
or can be imported as ``oo``.
Examples
========
>>> from sympy import oo, exp, limit, Symbol
>>> 1 + oo
oo
>>> 42/oo
0
>>> x = Symbol('x')
>>> limit(exp(x), x, oo)
oo
See Also
========
NegativeInfinity, NaN
References
==========
.. [1] https://en.wikipedia.org/wiki/Infinity
"""
is_commutative = True
is_number = True
is_complex = False
is_extended_real = True
is_infinite = True
is_comparable = True
is_extended_positive = True
is_prime = False
__slots__ = ()
def __new__(cls):
return AtomicExpr.__new__(cls)
def _latex(self, printer):
return r"\infty"
def _eval_subs(self, old, new):
if self == old:
return new
def _eval_evalf(self, prec=None):
return Float('inf')
def evalf(self, prec=None, **options):
return self._eval_evalf(prec)
@_sympifyit('other', NotImplemented)
def __add__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
if other is S.NegativeInfinity or other is S.NaN:
return S.NaN
return self
return Number.__add__(self, other)
__radd__ = __add__
@_sympifyit('other', NotImplemented)
def __sub__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
if other is S.Infinity or other is S.NaN:
return S.NaN
return self
return Number.__sub__(self, other)
@_sympifyit('other', NotImplemented)
def __rsub__(self, other):
return (-self).__add__(other)
@_sympifyit('other', NotImplemented)
def __mul__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
if other.is_zero or other is S.NaN:
return S.NaN
if other.is_extended_positive:
return self
return S.NegativeInfinity
return Number.__mul__(self, other)
__rmul__ = __mul__
@_sympifyit('other', NotImplemented)
def __truediv__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
if other is S.Infinity or \
other is S.NegativeInfinity or \
other is S.NaN:
return S.NaN
if other.is_extended_nonnegative:
return self
return S.NegativeInfinity
return Number.__truediv__(self, other)
def __abs__(self):
return S.Infinity
def __neg__(self):
return S.NegativeInfinity
def _eval_power(self, expt):
"""
``expt`` is symbolic object but not equal to 0 or 1.
================ ======= ==============================
Expression Result Notes
================ ======= ==============================
``oo ** nan`` ``nan``
``oo ** -p`` ``0`` ``p`` is number, ``oo``
================ ======= ==============================
See Also
========
Pow
NaN
NegativeInfinity
"""
from sympy.functions import re
if expt.is_extended_positive:
return S.Infinity
if expt.is_extended_negative:
return S.Zero
if expt is S.NaN:
return S.NaN
if expt is S.ComplexInfinity:
return S.NaN
if expt.is_extended_real is False and expt.is_number:
expt_real = re(expt)
if expt_real.is_positive:
return S.ComplexInfinity
if expt_real.is_negative:
return S.Zero
if expt_real.is_zero:
return S.NaN
return self**expt.evalf()
def _as_mpf_val(self, prec):
return mlib.finf
def _sage_(self):
import sage.all as sage
return sage.oo
def __hash__(self):
return super().__hash__()
def __eq__(self, other):
return other is S.Infinity or other == float('inf')
def __ne__(self, other):
return other is not S.Infinity and other != float('inf')
__gt__ = Expr.__gt__
__ge__ = Expr.__ge__
__lt__ = Expr.__lt__
__le__ = Expr.__le__
@_sympifyit('other', NotImplemented)
def __mod__(self, other):
if not isinstance(other, Expr):
return NotImplemented
return S.NaN
__rmod__ = __mod__
def floor(self):
return self
def ceiling(self):
return self
oo = S.Infinity
class NegativeInfinity(Number, metaclass=Singleton):
"""Negative infinite quantity.
NegativeInfinity is a singleton, and can be accessed
by ``S.NegativeInfinity``.
See Also
========
Infinity
"""
is_extended_real = True
is_complex = False
is_commutative = True
is_infinite = True
is_comparable = True
is_extended_negative = True
is_number = True
is_prime = False
__slots__ = ()
def __new__(cls):
return AtomicExpr.__new__(cls)
def _latex(self, printer):
return r"-\infty"
def _eval_subs(self, old, new):
if self == old:
return new
def _eval_evalf(self, prec=None):
return Float('-inf')
def evalf(self, prec=None, **options):
return self._eval_evalf(prec)
@_sympifyit('other', NotImplemented)
def __add__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
if other is S.Infinity or other is S.NaN:
return S.NaN
return self
return Number.__add__(self, other)
__radd__ = __add__
@_sympifyit('other', NotImplemented)
def __sub__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
if other is S.NegativeInfinity or other is S.NaN:
return S.NaN
return self
return Number.__sub__(self, other)
@_sympifyit('other', NotImplemented)
def __rsub__(self, other):
return (-self).__add__(other)
@_sympifyit('other', NotImplemented)
def __mul__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
if other.is_zero or other is S.NaN:
return S.NaN
if other.is_extended_positive:
return self
return S.Infinity
return Number.__mul__(self, other)
__rmul__ = __mul__
@_sympifyit('other', NotImplemented)
def __truediv__(self, other):
if isinstance(other, Number) and global_parameters.evaluate:
if other is S.Infinity or \
other is S.NegativeInfinity or \
other is S.NaN:
return S.NaN
if other.is_extended_nonnegative:
return self
return S.Infinity
return Number.__truediv__(self, other)
def __abs__(self):
return S.Infinity
def __neg__(self):
return S.Infinity
def _eval_power(self, expt):
"""
``expt`` is symbolic object but not equal to 0 or 1.
================ ======= ==============================
Expression Result Notes
================ ======= ==============================
``(-oo) ** nan`` ``nan``
``(-oo) ** oo`` ``nan``
``(-oo) ** -oo`` ``nan``
``(-oo) ** e`` ``oo`` ``e`` is positive even integer
``(-oo) ** o`` ``-oo`` ``o`` is positive odd integer
================ ======= ==============================
See Also
========
Infinity
Pow
NaN
"""
if expt.is_number:
if expt is S.NaN or \
expt is S.Infinity or \
expt is S.NegativeInfinity:
return S.NaN
if isinstance(expt, Integer) and expt.is_extended_positive:
if expt.is_odd:
return S.NegativeInfinity
else:
return S.Infinity
return S.NegativeOne**expt*S.Infinity**expt
def _as_mpf_val(self, prec):
return mlib.fninf
def _sage_(self):
import sage.all as sage
return -(sage.oo)
def __hash__(self):
return super().__hash__()
def __eq__(self, other):
return other is S.NegativeInfinity or other == float('-inf')
def __ne__(self, other):
return other is not S.NegativeInfinity and other != float('-inf')
__gt__ = Expr.__gt__
__ge__ = Expr.__ge__
__lt__ = Expr.__lt__
__le__ = Expr.__le__
@_sympifyit('other', NotImplemented)
def __mod__(self, other):
if not isinstance(other, Expr):
return NotImplemented
return S.NaN
__rmod__ = __mod__
def floor(self):
return self
def ceiling(self):
return self
def as_powers_dict(self):
return {S.NegativeOne: 1, S.Infinity: 1}
class NaN(Number, metaclass=Singleton):
"""
Not a Number.
Explanation
===========
This serves as a place holder for numeric values that are indeterminate.
Most operations on NaN, produce another NaN. Most indeterminate forms,
such as ``0/0`` or ``oo - oo` produce NaN. Two exceptions are ``0**0``
and ``oo**0``, which all produce ``1`` (this is consistent with Python's
float).
NaN is loosely related to floating point nan, which is defined in the
IEEE 754 floating point standard, and corresponds to the Python
``float('nan')``. Differences are noted below.
NaN is mathematically not equal to anything else, even NaN itself. This
explains the initially counter-intuitive results with ``Eq`` and ``==`` in
the examples below.
NaN is not comparable so inequalities raise a TypeError. This is in
contrast with floating point nan where all inequalities are false.
NaN is a singleton, and can be accessed by ``S.NaN``, or can be imported
as ``nan``.
Examples
========
>>> from sympy import nan, S, oo, Eq
>>> nan is S.NaN
True
>>> oo - oo
nan
>>> nan + 1
nan
>>> Eq(nan, nan) # mathematical equality
False
>>> nan == nan # structural equality
True
References
==========
.. [1] https://en.wikipedia.org/wiki/NaN
"""
is_commutative = True
is_extended_real = None
is_real = None
is_rational = None
is_algebraic = None
is_transcendental = None
is_integer = None
is_comparable = False
is_finite = None
is_zero = None
is_prime = None
is_positive = None
is_negative = None
is_number = True
__slots__ = ()
def __new__(cls):
return AtomicExpr.__new__(cls)
def _latex(self, printer):
return r"\text{NaN}"
def __neg__(self):
return self
@_sympifyit('other', NotImplemented)
def __add__(self, other):
return self
@_sympifyit('other', NotImplemented)
def __sub__(self, other):
return self
@_sympifyit('other', NotImplemented)
def __mul__(self, other):
return self
@_sympifyit('other', NotImplemented)
def __truediv__(self, other):
return self
def floor(self):
return self
def ceiling(self):
return self
def _as_mpf_val(self, prec):
return _mpf_nan
def _sage_(self):
import sage.all as sage
return sage.NaN
def __hash__(self):
return super().__hash__()
def __eq__(self, other):
# NaN is structurally equal to another NaN
return other is S.NaN
def __ne__(self, other):
return other is not S.NaN
# Expr will _sympify and raise TypeError
__gt__ = Expr.__gt__
__ge__ = Expr.__ge__
__lt__ = Expr.__lt__
__le__ = Expr.__le__
nan = S.NaN
@dispatch(NaN, Expr) # type:ignore
def _eval_is_eq(a, b): # noqa:F811
return False
class ComplexInfinity(AtomicExpr, metaclass=Singleton):
r"""Complex infinity.
Explanation
===========
In complex analysis the symbol `\tilde\infty`, called "complex
infinity", represents a quantity with infinite magnitude, but
undetermined complex phase.
ComplexInfinity is a singleton, and can be accessed by
``S.ComplexInfinity``, or can be imported as ``zoo``.
Examples
========
>>> from sympy import zoo
>>> zoo + 42
zoo
>>> 42/zoo
0
>>> zoo + zoo
nan
>>> zoo*zoo
zoo
See Also
========
Infinity
"""
is_commutative = True
is_infinite = True
is_number = True
is_prime = False
is_complex = False
is_extended_real = False
kind = NumberKind
__slots__ = ()
def __new__(cls):
return AtomicExpr.__new__(cls)
def _latex(self, printer):
return r"\tilde{\infty}"
@staticmethod
def __abs__():
return S.Infinity
def floor(self):
return self
def ceiling(self):
return self
@staticmethod
def __neg__():
return S.ComplexInfinity
def _eval_power(self, expt):
if expt is S.ComplexInfinity:
return S.NaN
if isinstance(expt, Number):
if expt.is_zero:
return S.NaN
else:
if expt.is_positive:
return S.ComplexInfinity
else:
return S.Zero
def _sage_(self):
import sage.all as sage
return sage.UnsignedInfinityRing.gen()
zoo = S.ComplexInfinity
class NumberSymbol(AtomicExpr):
is_commutative = True
is_finite = True
is_number = True
__slots__ = ()
is_NumberSymbol = True
kind = NumberKind
def __new__(cls):
return AtomicExpr.__new__(cls)
def approximation(self, number_cls):
""" Return an interval with number_cls endpoints
that contains the value of NumberSymbol.
If not implemented, then return None.
"""
def _eval_evalf(self, prec):
return Float._new(self._as_mpf_val(prec), prec)
def __eq__(self, other):
try:
other = _sympify(other)
except SympifyError:
return NotImplemented
if self is other:
return True
if other.is_Number and self.is_irrational:
return False
return False # NumberSymbol != non-(Number|self)
def __ne__(self, other):
return not self == other
def __le__(self, other):
if self is other:
return S.true
return Expr.__le__(self, other)
def __ge__(self, other):
if self is other:
return S.true
return Expr.__ge__(self, other)
def __int__(self):
# subclass with appropriate return value
raise NotImplementedError
def __hash__(self):
return super().__hash__()
class Exp1(NumberSymbol, metaclass=Singleton):
r"""The `e` constant.
Explanation
===========
The transcendental number `e = 2.718281828\ldots` is the base of the
natural logarithm and of the exponential function, `e = \exp(1)`.
Sometimes called Euler's number or Napier's constant.
Exp1 is a singleton, and can be accessed by ``S.Exp1``,
or can be imported as ``E``.
Examples
========
>>> from sympy import exp, log, E
>>> E is exp(1)
True
>>> log(E)
1
References
==========
.. [1] https://en.wikipedia.org/wiki/E_%28mathematical_constant%29
"""
is_real = True
is_positive = True
is_negative = False # XXX Forces is_negative/is_nonnegative
is_irrational = True
is_number = True
is_algebraic = False
is_transcendental = True
__slots__ = ()
def _latex(self, printer):
return r"e"
@staticmethod
def __abs__():
return S.Exp1
def __int__(self):
return 2
def _as_mpf_val(self, prec):
return mpf_e(prec)
def approximation_interval(self, number_cls):
if issubclass(number_cls, Integer):
return (Integer(2), Integer(3))
elif issubclass(number_cls, Rational):
pass
def _eval_power(self, expt):
from sympy import exp
if global_parameters.exp_is_pow:
return self._eval_power_exp_is_pow(expt)
else:
return exp(expt)
def _eval_power_exp_is_pow(self, arg):
from ..functions.elementary.exponential import log
from . import Add, Mul, Pow
if arg.is_Number:
if arg is oo:
return oo
elif arg == -oo:
return S.Zero
elif isinstance(arg, log):
return arg.args[0]
# don't autoexpand Pow or Mul (see the issue 3351):
elif not arg.is_Add:
Ioo = I*oo
if arg in [Ioo, -Ioo]:
return nan
coeff = arg.coeff(pi*I)
if coeff:
if (2*coeff).is_integer:
if coeff.is_even:
return S.One
elif coeff.is_odd:
return S.NegativeOne
elif (coeff + S.Half).is_even:
return -I
elif (coeff + S.Half).is_odd:
return I
elif coeff.is_Rational:
ncoeff = coeff % 2 # restrict to [0, 2pi)
if ncoeff > 1: # restrict to (-pi, pi]
ncoeff -= 2
if ncoeff != coeff:
return S.Exp1**(ncoeff*S.Pi*S.ImaginaryUnit)
# Warning: code in risch.py will be very sensitive to changes
# in this (see DifferentialExtension).
# look for a single log factor
coeff, terms = arg.as_coeff_Mul()
# but it can't be multiplied by oo
if coeff in (oo, -oo):
return
coeffs, log_term = [coeff], None
for term in Mul.make_args(terms):
if isinstance(term, log):
if log_term is None:
log_term = term.args[0]
else:
return
elif term.is_comparable:
coeffs.append(term)
else:
return
return log_term**Mul(*coeffs) if log_term else None
elif arg.is_Add:
out = []
add = []
argchanged = False
for a in arg.args:
if a is S.One:
add.append(a)
continue
newa = self**a
if isinstance(newa, Pow) and newa.base is self:
if newa.exp != a:
add.append(newa.exp)
argchanged = True
else:
add.append(a)
else:
out.append(newa)
if out or argchanged:
return Mul(*out)*Pow(self, Add(*add), evaluate=False)
elif arg.is_Matrix:
return arg.exp()
def _eval_rewrite_as_sin(self, **kwargs):
from sympy import sin
I = S.ImaginaryUnit
return sin(I + S.Pi/2) - I*sin(I)
def _eval_rewrite_as_cos(self, **kwargs):
from sympy import cos
I = S.ImaginaryUnit
return cos(I) + I*cos(I + S.Pi/2)
def _sage_(self):
import sage.all as sage
return sage.e
E = S.Exp1
class Pi(NumberSymbol, metaclass=Singleton):
r"""The `\pi` constant.
Explanation
===========
The transcendental number `\pi = 3.141592654\ldots` represents the ratio
of a circle's circumference to its diameter, the area of the unit circle,
the half-period of trigonometric functions, and many other things
in mathematics.
Pi is a singleton, and can be accessed by ``S.Pi``, or can
be imported as ``pi``.
Examples
========
>>> from sympy import S, pi, oo, sin, exp, integrate, Symbol
>>> S.Pi
pi
>>> pi > 3
True
>>> pi.is_irrational
True
>>> x = Symbol('x')
>>> sin(x + 2*pi)
sin(x)
>>> integrate(exp(-x**2), (x, -oo, oo))
sqrt(pi)
References
==========
.. [1] https://en.wikipedia.org/wiki/Pi
"""
is_real = True
is_positive = True
is_negative = False
is_irrational = True
is_number = True
is_algebraic = False
is_transcendental = True
__slots__ = ()
def _latex(self, printer):
return r"\pi"
@staticmethod
def __abs__():
return S.Pi
def __int__(self):
return 3
def _as_mpf_val(self, prec):
return mpf_pi(prec)
def approximation_interval(self, number_cls):
if issubclass(number_cls, Integer):
return (Integer(3), Integer(4))
elif issubclass(number_cls, Rational):
return (Rational(223, 71), Rational(22, 7))
def _sage_(self):
import sage.all as sage
return sage.pi
pi = S.Pi
class GoldenRatio(NumberSymbol, metaclass=Singleton):
r"""The golden ratio, `\phi`.
Explanation
===========
`\phi = \frac{1 + \sqrt{5}}{2}` is algebraic number. Two quantities
are in the golden ratio if their ratio is the same as the ratio of
their sum to the larger of the two quantities, i.e. their maximum.
GoldenRatio is a singleton, and can be accessed by ``S.GoldenRatio``.
Examples
========
>>> from sympy import S
>>> S.GoldenRatio > 1
True
>>> S.GoldenRatio.expand(func=True)
1/2 + sqrt(5)/2
>>> S.GoldenRatio.is_irrational
True
References
==========
.. [1] https://en.wikipedia.org/wiki/Golden_ratio
"""
is_real = True
is_positive = True
is_negative = False
is_irrational = True
is_number = True
is_algebraic = True
is_transcendental = False
__slots__ = ()
def _latex(self, printer):
return r"\phi"
def __int__(self):
return 1
def _as_mpf_val(self, prec):
# XXX track down why this has to be increased
rv = mlib.from_man_exp(phi_fixed(prec + 10), -prec - 10)
return mpf_norm(rv, prec)
def _eval_expand_func(self, **hints):
from sympy import sqrt
return S.Half + S.Half*sqrt(5)
def approximation_interval(self, number_cls):
if issubclass(number_cls, Integer):
return (S.One, Rational(2))
elif issubclass(number_cls, Rational):
pass
def _sage_(self):
import sage.all as sage
return sage.golden_ratio
_eval_rewrite_as_sqrt = _eval_expand_func
class TribonacciConstant(NumberSymbol, metaclass=Singleton):
r"""The tribonacci constant.
Explanation
===========
The tribonacci numbers are like the Fibonacci numbers, but instead
of starting with two predetermined terms, the sequence starts with
three predetermined terms and each term afterwards is the sum of the
preceding three terms.
The tribonacci constant is the ratio toward which adjacent tribonacci
numbers tend. It is a root of the polynomial `x^3 - x^2 - x - 1 = 0`,
and also satisfies the equation `x + x^{-3} = 2`.
TribonacciConstant is a singleton, and can be accessed
by ``S.TribonacciConstant``.
Examples
========
>>> from sympy import S
>>> S.TribonacciConstant > 1
True
>>> S.TribonacciConstant.expand(func=True)
1/3 + (19 - 3*sqrt(33))**(1/3)/3 + (3*sqrt(33) + 19)**(1/3)/3
>>> S.TribonacciConstant.is_irrational
True
>>> S.TribonacciConstant.n(20)
1.8392867552141611326
References
==========
.. [1] https://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers#Tribonacci_numbers
"""
is_real = True
is_positive = True
is_negative = False
is_irrational = True
is_number = True
is_algebraic = True
is_transcendental = False
__slots__ = ()
def _latex(self, printer):
return r"\text{TribonacciConstant}"
def __int__(self):
return 2
def _eval_evalf(self, prec):
rv = self._eval_expand_func(function=True)._eval_evalf(prec + 4)
return Float(rv, precision=prec)
def _eval_expand_func(self, **hints):
from sympy import sqrt, cbrt
return (1 + cbrt(19 - 3*sqrt(33)) + cbrt(19 + 3*sqrt(33))) / 3
def approximation_interval(self, number_cls):
if issubclass(number_cls, Integer):
return (S.One, Rational(2))
elif issubclass(number_cls, Rational):
pass
_eval_rewrite_as_sqrt = _eval_expand_func
class EulerGamma(NumberSymbol, metaclass=Singleton):
r"""The Euler-Mascheroni constant.
Explanation
===========
`\gamma = 0.5772157\ldots` (also called Euler's constant) is a mathematical
constant recurring in analysis and number theory. It is defined as the
limiting difference between the harmonic series and the
natural logarithm:
.. math:: \gamma = \lim\limits_{n\to\infty}
\left(\sum\limits_{k=1}^n\frac{1}{k} - \ln n\right)
EulerGamma is a singleton, and can be accessed by ``S.EulerGamma``.
Examples
========
>>> from sympy import S
>>> S.EulerGamma.is_irrational
>>> S.EulerGamma > 0
True
>>> S.EulerGamma > 1
False
References
==========
.. [1] https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant
"""
is_real = True
is_positive = True
is_negative = False
is_irrational = None
is_number = True
__slots__ = ()
def _latex(self, printer):
return r"\gamma"
def __int__(self):
return 0
def _as_mpf_val(self, prec):
# XXX track down why this has to be increased
v = mlib.libhyper.euler_fixed(prec + 10)
rv = mlib.from_man_exp(v, -prec - 10)
return mpf_norm(rv, prec)
def approximation_interval(self, number_cls):
if issubclass(number_cls, Integer):
return (S.Zero, S.One)
elif issubclass(number_cls, Rational):
return (S.Half, Rational(3, 5))
def _sage_(self):
import sage.all as sage
return sage.euler_gamma
class Catalan(NumberSymbol, metaclass=Singleton):
r"""Catalan's constant.
Explanation
===========
`K = 0.91596559\ldots` is given by the infinite series
.. math:: K = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2}
Catalan is a singleton, and can be accessed by ``S.Catalan``.
Examples
========
>>> from sympy import S
>>> S.Catalan.is_irrational
>>> S.Catalan > 0
True
>>> S.Catalan > 1
False
References
==========
.. [1] https://en.wikipedia.org/wiki/Catalan%27s_constant
"""
is_real = True
is_positive = True
is_negative = False
is_irrational = None
is_number = True
__slots__ = ()
def __int__(self):
return 0
def _as_mpf_val(self, prec):
# XXX track down why this has to be increased
v = mlib.catalan_fixed(prec + 10)
rv = mlib.from_man_exp(v, -prec - 10)
return mpf_norm(rv, prec)
def approximation_interval(self, number_cls):
if issubclass(number_cls, Integer):
return (S.Zero, S.One)
elif issubclass(number_cls, Rational):
return (Rational(9, 10), S.One)
def _eval_rewrite_as_Sum(self, k_sym=None, symbols=None):
from sympy import Sum, Dummy
if (k_sym is not None) or (symbols is not None):
return self
k = Dummy('k', integer=True, nonnegative=True)
return Sum((-1)**k / (2*k+1)**2, (k, 0, S.Infinity))
def _sage_(self):
import sage.all as sage
return sage.catalan
class ImaginaryUnit(AtomicExpr, metaclass=Singleton):
r"""The imaginary unit, `i = \sqrt{-1}`.
I is a singleton, and can be accessed by ``S.I``, or can be
imported as ``I``.
Examples
========
>>> from sympy import I, sqrt
>>> sqrt(-1)
I
>>> I*I
-1
>>> 1/I
-I
References
==========
.. [1] https://en.wikipedia.org/wiki/Imaginary_unit
"""
is_commutative = True
is_imaginary = True
is_finite = True
is_number = True
is_algebraic = True
is_transcendental = False
kind = NumberKind
__slots__ = ()
def _latex(self, printer):
return printer._settings['imaginary_unit_latex']
@staticmethod
def __abs__():
return S.One
def _eval_evalf(self, prec):
return self
def _eval_conjugate(self):
return -S.ImaginaryUnit
def _eval_power(self, expt):
"""
b is I = sqrt(-1)
e is symbolic object but not equal to 0, 1
I**r -> (-1)**(r/2) -> exp(r/2*Pi*I) -> sin(Pi*r/2) + cos(Pi*r/2)*I, r is decimal
I**0 mod 4 -> 1
I**1 mod 4 -> I
I**2 mod 4 -> -1
I**3 mod 4 -> -I
"""
if isinstance(expt, Number):
if isinstance(expt, Integer):
expt = expt.p % 4
if expt == 0:
return S.One
if expt == 1:
return S.ImaginaryUnit
if expt == 2:
return -S.One
return -S.ImaginaryUnit
return
def as_base_exp(self):
return S.NegativeOne, S.Half
def _sage_(self):
import sage.all as sage
return sage.I
@property
def _mpc_(self):
return (Float(0)._mpf_, Float(1)._mpf_)
I = S.ImaginaryUnit
@dispatch(Tuple, Number) # type:ignore
def _eval_is_eq(self, other): # noqa: F811
return False
def sympify_fractions(f):
return Rational(f.numerator, f.denominator, 1)
converter[fractions.Fraction] = sympify_fractions
if HAS_GMPY:
def sympify_mpz(x):
return Integer(int(x))
# XXX: The sympify_mpq function here was never used because it is
# overridden by the other sympify_mpq function below. Maybe it should just
# be removed or maybe it should be used for something...
def sympify_mpq(x):
return Rational(int(x.numerator), int(x.denominator))
converter[type(gmpy.mpz(1))] = sympify_mpz
converter[type(gmpy.mpq(1, 2))] = sympify_mpq
def sympify_mpmath_mpq(x):
p, q = x._mpq_
return Rational(p, q, 1)
converter[type(mpmath.rational.mpq(1, 2))] = sympify_mpmath_mpq
def sympify_mpmath(x):
return Expr._from_mpmath(x, x.context.prec)
converter[mpnumeric] = sympify_mpmath
def sympify_complex(a):
real, imag = list(map(sympify, (a.real, a.imag)))
return real + S.ImaginaryUnit*imag
converter[complex] = sympify_complex
from .power import Pow, integer_nthroot
from .mul import Mul
Mul.identity = One()
from .add import Add
Add.identity = Zero()
def _register_classes():
numbers.Number.register(Number)
numbers.Real.register(Float)
numbers.Rational.register(Rational)
numbers.Rational.register(Integer)
_register_classes()
|
fa890e8912f7d3d6e2d32b82a9cd68a601d555212ef2cfe8b21d50f9c4c858af | from operator import attrgetter
from typing import Tuple, Type
from collections import defaultdict
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.core.sympify import _sympify as _sympify_, sympify
from sympy.core.basic import Basic
from sympy.core.cache import cacheit
from sympy.core.compatibility import ordered
from sympy.core.logic import fuzzy_and
from sympy.core.parameters import global_parameters
from sympy.utilities.iterables import sift
from sympy.multipledispatch.dispatcher import (Dispatcher,
ambiguity_register_error_ignore_dup,
str_signature, RaiseNotImplementedError)
class AssocOp(Basic):
""" Associative operations, can separate noncommutative and
commutative parts.
(a op b) op c == a op (b op c) == a op b op c.
Base class for Add and Mul.
This is an abstract base class, concrete derived classes must define
the attribute `identity`.
Parameters
==========
*args :
Arguments which are operated
evaluate : bool, optional
Evaluate the operation. If not passed, refer to ``global_parameters.evaluate``.
"""
# for performance reason, we don't let is_commutative go to assumptions,
# and keep it right here
__slots__ = ('is_commutative',) # type: Tuple[str, ...]
_args_type = None # type: Type[Basic]
@cacheit
def __new__(cls, *args, evaluate=None, _sympify=True):
from sympy import Order
# Allow faster processing by passing ``_sympify=False``, if all arguments
# are already sympified.
if _sympify:
args = list(map(_sympify_, args))
# Disallow non-Expr args in Add/Mul
typ = cls._args_type
if typ is not None:
from sympy.core.relational import Relational
if any(isinstance(arg, Relational) for arg in args):
raise TypeError("Relational can not be used in %s" % cls.__name__)
# This should raise TypeError once deprecation period is over:
if not all(isinstance(arg, typ) for arg in args):
SymPyDeprecationWarning(
feature="Add/Mul with non-Expr args",
useinstead="Expr args",
issue=19445,
deprecated_since_version="1.7"
).warn()
if evaluate is None:
evaluate = global_parameters.evaluate
if not evaluate:
obj = cls._from_args(args)
obj = cls._exec_constructor_postprocessors(obj)
return obj
args = [a for a in args if a is not cls.identity]
if len(args) == 0:
return cls.identity
if len(args) == 1:
return args[0]
c_part, nc_part, order_symbols = cls.flatten(args)
is_commutative = not nc_part
obj = cls._from_args(c_part + nc_part, is_commutative)
obj = cls._exec_constructor_postprocessors(obj)
if order_symbols is not None:
return Order(obj, *order_symbols)
return obj
@classmethod
def _from_args(cls, args, is_commutative=None):
"""Create new instance with already-processed args.
If the args are not in canonical order, then a non-canonical
result will be returned, so use with caution. The order of
args may change if the sign of the args is changed."""
if len(args) == 0:
return cls.identity
elif len(args) == 1:
return args[0]
obj = super().__new__(cls, *args)
if is_commutative is None:
is_commutative = fuzzy_and(a.is_commutative for a in args)
obj.is_commutative = is_commutative
return obj
def _new_rawargs(self, *args, reeval=True, **kwargs):
"""Create new instance of own class with args exactly as provided by
caller but returning the self class identity if args is empty.
Examples
========
This is handy when we want to optimize things, e.g.
>>> from sympy import Mul, S
>>> from sympy.abc import x, y
>>> e = Mul(3, x, y)
>>> e.args
(3, x, y)
>>> Mul(*e.args[1:])
x*y
>>> e._new_rawargs(*e.args[1:]) # the same as above, but faster
x*y
Note: use this with caution. There is no checking of arguments at
all. This is best used when you are rebuilding an Add or Mul after
simply removing one or more args. If, for example, modifications,
result in extra 1s being inserted they will show up in the result:
>>> m = (x*y)._new_rawargs(S.One, x); m
1*x
>>> m == x
False
>>> m.is_Mul
True
Another issue to be aware of is that the commutativity of the result
is based on the commutativity of self. If you are rebuilding the
terms that came from a commutative object then there will be no
problem, but if self was non-commutative then what you are
rebuilding may now be commutative.
Although this routine tries to do as little as possible with the
input, getting the commutativity right is important, so this level
of safety is enforced: commutativity will always be recomputed if
self is non-commutative and kwarg `reeval=False` has not been
passed.
"""
if reeval and self.is_commutative is False:
is_commutative = None
else:
is_commutative = self.is_commutative
return self._from_args(args, is_commutative)
@classmethod
def flatten(cls, seq):
"""Return seq so that none of the elements are of type `cls`. This is
the vanilla routine that will be used if a class derived from AssocOp
does not define its own flatten routine."""
# apply associativity, no commutativity property is used
new_seq = []
while seq:
o = seq.pop()
if o.__class__ is cls: # classes must match exactly
seq.extend(o.args)
else:
new_seq.append(o)
new_seq.reverse()
# c_part, nc_part, order_symbols
return [], new_seq, None
def _matches_commutative(self, expr, repl_dict={}, old=False):
"""
Matches Add/Mul "pattern" to an expression "expr".
repl_dict ... a dictionary of (wild: expression) pairs, that get
returned with the results
This function is the main workhorse for Add/Mul.
Examples
========
>>> from sympy import symbols, Wild, sin
>>> a = Wild("a")
>>> b = Wild("b")
>>> c = Wild("c")
>>> x, y, z = symbols("x y z")
>>> (a+sin(b)*c)._matches_commutative(x+sin(y)*z)
{a_: x, b_: y, c_: z}
In the example above, "a+sin(b)*c" is the pattern, and "x+sin(y)*z" is
the expression.
The repl_dict contains parts that were already matched. For example
here:
>>> (x+sin(b)*c)._matches_commutative(x+sin(y)*z, repl_dict={a: x})
{a_: x, b_: y, c_: z}
the only function of the repl_dict is to return it in the
result, e.g. if you omit it:
>>> (x+sin(b)*c)._matches_commutative(x+sin(y)*z)
{b_: y, c_: z}
the "a: x" is not returned in the result, but otherwise it is
equivalent.
"""
# make sure expr is Expr if pattern is Expr
from .expr import Add, Expr
from sympy import Mul
repl_dict = repl_dict.copy()
if isinstance(self, Expr) and not isinstance(expr, Expr):
return None
# handle simple patterns
if self == expr:
return repl_dict
d = self._matches_simple(expr, repl_dict)
if d is not None:
return d
# eliminate exact part from pattern: (2+a+w1+w2).matches(expr) -> (w1+w2).matches(expr-a-2)
from .function import WildFunction
from .symbol import Wild
wild_part, exact_part = sift(self.args, lambda p:
p.has(Wild, WildFunction) and not expr.has(p),
binary=True)
if not exact_part:
wild_part = list(ordered(wild_part))
if self.is_Add:
# in addition to normal ordered keys, impose
# sorting on Muls with leading Number to put
# them in order
wild_part = sorted(wild_part, key=lambda x:
x.args[0] if x.is_Mul and x.args[0].is_Number else
0)
else:
exact = self._new_rawargs(*exact_part)
free = expr.free_symbols
if free and (exact.free_symbols - free):
# there are symbols in the exact part that are not
# in the expr; but if there are no free symbols, let
# the matching continue
return None
newexpr = self._combine_inverse(expr, exact)
if not old and (expr.is_Add or expr.is_Mul):
if newexpr.count_ops() > expr.count_ops():
return None
newpattern = self._new_rawargs(*wild_part)
return newpattern.matches(newexpr, repl_dict)
# now to real work ;)
i = 0
saw = set()
while expr not in saw:
saw.add(expr)
args = tuple(ordered(self.make_args(expr)))
if self.is_Add and expr.is_Add:
# in addition to normal ordered keys, impose
# sorting on Muls with leading Number to put
# them in order
args = tuple(sorted(args, key=lambda x:
x.args[0] if x.is_Mul and x.args[0].is_Number else
0))
expr_list = (self.identity,) + args
for last_op in reversed(expr_list):
for w in reversed(wild_part):
d1 = w.matches(last_op, repl_dict)
if d1 is not None:
d2 = self.xreplace(d1).matches(expr, d1)
if d2 is not None:
return d2
if i == 0:
if self.is_Mul:
# make e**i look like Mul
if expr.is_Pow and expr.exp.is_Integer:
if expr.exp > 0:
expr = Mul(*[expr.base, expr.base**(expr.exp - 1)], evaluate=False)
else:
expr = Mul(*[1/expr.base, expr.base**(expr.exp + 1)], evaluate=False)
i += 1
continue
elif self.is_Add:
# make i*e look like Add
c, e = expr.as_coeff_Mul()
if abs(c) > 1:
if c > 0:
expr = Add(*[e, (c - 1)*e], evaluate=False)
else:
expr = Add(*[-e, (c + 1)*e], evaluate=False)
i += 1
continue
# try collection on non-Wild symbols
from sympy.simplify.radsimp import collect
was = expr
did = set()
for w in reversed(wild_part):
c, w = w.as_coeff_mul(Wild)
free = c.free_symbols - did
if free:
did.update(free)
expr = collect(expr, free)
if expr != was:
i += 0
continue
break # if we didn't continue, there is nothing more to do
return
def _has_matcher(self):
"""Helper for .has()"""
def _ncsplit(expr):
# this is not the same as args_cnc because here
# we don't assume expr is a Mul -- hence deal with args --
# and always return a set.
cpart, ncpart = sift(expr.args,
lambda arg: arg.is_commutative is True, binary=True)
return set(cpart), ncpart
c, nc = _ncsplit(self)
cls = self.__class__
def is_in(expr):
if expr == self:
return True
elif not isinstance(expr, Basic):
return False
elif isinstance(expr, cls):
_c, _nc = _ncsplit(expr)
if (c & _c) == c:
if not nc:
return True
elif len(nc) <= len(_nc):
for i in range(len(_nc) - len(nc) + 1):
if _nc[i:i + len(nc)] == nc:
return True
return False
return is_in
def _eval_evalf(self, prec):
"""
Evaluate the parts of self that are numbers; if the whole thing
was a number with no functions it would have been evaluated, but
it wasn't so we must judiciously extract the numbers and reconstruct
the object. This is *not* simply replacing numbers with evaluated
numbers. Numbers should be handled in the largest pure-number
expression as possible. So the code below separates ``self`` into
number and non-number parts and evaluates the number parts and
walks the args of the non-number part recursively (doing the same
thing).
"""
from .add import Add
from .mul import Mul
from .symbol import Symbol
from .function import AppliedUndef
if isinstance(self, (Mul, Add)):
x, tail = self.as_independent(Symbol, AppliedUndef)
# if x is an AssocOp Function then the _evalf below will
# call _eval_evalf (here) so we must break the recursion
if not (tail is self.identity or
isinstance(x, AssocOp) and x.is_Function or
x is self.identity and isinstance(tail, AssocOp)):
# here, we have a number so we just call to _evalf with prec;
# prec is not the same as n, it is the binary precision so
# that's why we don't call to evalf.
x = x._evalf(prec) if x is not self.identity else self.identity
args = []
tail_args = tuple(self.func.make_args(tail))
for a in tail_args:
# here we call to _eval_evalf since we don't know what we
# are dealing with and all other _eval_evalf routines should
# be doing the same thing (i.e. taking binary prec and
# finding the evalf-able args)
newa = a._eval_evalf(prec)
if newa is None:
args.append(a)
else:
args.append(newa)
return self.func(x, *args)
# this is the same as above, but there were no pure-number args to
# deal with
args = []
for a in self.args:
newa = a._eval_evalf(prec)
if newa is None:
args.append(a)
else:
args.append(newa)
return self.func(*args)
@classmethod
def make_args(cls, expr):
"""
Return a sequence of elements `args` such that cls(*args) == expr
Examples
========
>>> from sympy import Symbol, Mul, Add
>>> x, y = map(Symbol, 'xy')
>>> Mul.make_args(x*y)
(x, y)
>>> Add.make_args(x*y)
(x*y,)
>>> set(Add.make_args(x*y + y)) == set([y, x*y])
True
"""
if isinstance(expr, cls):
return expr.args
else:
return (sympify(expr),)
def doit(self, **hints):
if hints.get('deep', True):
terms = [term.doit(**hints) for term in self.args]
else:
terms = self.args
return self.func(*terms, evaluate=True)
class ShortCircuit(Exception):
pass
class LatticeOp(AssocOp):
"""
Join/meet operations of an algebraic lattice[1].
Explanation
===========
These binary operations are associative (op(op(a, b), c) = op(a, op(b, c))),
commutative (op(a, b) = op(b, a)) and idempotent (op(a, a) = op(a) = a).
Common examples are AND, OR, Union, Intersection, max or min. They have an
identity element (op(identity, a) = a) and an absorbing element
conventionally called zero (op(zero, a) = zero).
This is an abstract base class, concrete derived classes must declare
attributes zero and identity. All defining properties are then respected.
Examples
========
>>> from sympy import Integer
>>> from sympy.core.operations import LatticeOp
>>> class my_join(LatticeOp):
... zero = Integer(0)
... identity = Integer(1)
>>> my_join(2, 3) == my_join(3, 2)
True
>>> my_join(2, my_join(3, 4)) == my_join(2, 3, 4)
True
>>> my_join(0, 1, 4, 2, 3, 4)
0
>>> my_join(1, 2)
2
References:
.. [1] https://en.wikipedia.org/wiki/Lattice_%28order%29
"""
is_commutative = True
def __new__(cls, *args, **options):
args = (_sympify_(arg) for arg in args)
try:
# /!\ args is a generator and _new_args_filter
# must be careful to handle as such; this
# is done so short-circuiting can be done
# without having to sympify all values
_args = frozenset(cls._new_args_filter(args))
except ShortCircuit:
return sympify(cls.zero)
if not _args:
return sympify(cls.identity)
elif len(_args) == 1:
return set(_args).pop()
else:
# XXX in almost every other case for __new__, *_args is
# passed along, but the expectation here is for _args
obj = super(AssocOp, cls).__new__(cls, *ordered(_args))
obj._argset = _args
return obj
@classmethod
def _new_args_filter(cls, arg_sequence, call_cls=None):
"""Generator filtering args"""
ncls = call_cls or cls
for arg in arg_sequence:
if arg == ncls.zero:
raise ShortCircuit(arg)
elif arg == ncls.identity:
continue
elif arg.func == ncls:
yield from arg.args
else:
yield arg
@classmethod
def make_args(cls, expr):
"""
Return a set of args such that cls(*arg_set) == expr.
"""
if isinstance(expr, cls):
return expr._argset
else:
return frozenset([sympify(expr)])
@staticmethod
def _compare_pretty(a, b):
return (str(a) > str(b)) - (str(a) < str(b))
class AssocOpDispatcher:
"""
Handler dispatcher for associative operators
.. notes::
This approach is experimental, and can be replaced or deleted in the future.
See https://github.com/sympy/sympy/pull/19463.
Explanation
===========
If arguments of different types are passed, the classes which handle the operation for each type
are collected. Then, a class which performs the operation is selected by recursive binary dispatching.
Dispatching relation can be registered by ``register_handlerclass`` method.
Priority registration is unordered. You cannot make ``A*B`` and ``B*A`` refer to
different handler classes. All logic dealing with the order of arguments must be implemented
in the handler class.
Examples
========
>>> from sympy import Add, Expr, Symbol
>>> from sympy.core.add import add
>>> class NewExpr(Expr):
... @property
... def _add_handler(self):
... return NewAdd
>>> class NewAdd(NewExpr, Add):
... pass
>>> add.register_handlerclass((Add, NewAdd), NewAdd)
>>> a, b = Symbol('a'), NewExpr()
>>> add(a, b) == NewAdd(a, b)
True
"""
def __init__(self, name, doc=None):
self.name = name
self.doc = doc
self.handlerattr = "_%s_handler" % name
self._handlergetter = attrgetter(self.handlerattr)
self._dispatcher = Dispatcher(name)
def __repr__(self):
return "<dispatched %s>" % self.name
def register_handlerclass(self, classes, typ, on_ambiguity=ambiguity_register_error_ignore_dup):
"""
Register the handler class for two classes, in both straight and reversed order.
Paramteters
===========
classes : tuple of two types
Classes who are compared with each other.
typ:
Class which is registered to represent *cls1* and *cls2*.
Handler method of *self* must be implemented in this class.
"""
if not len(classes) == 2:
raise RuntimeError(
"Only binary dispatch is supported, but got %s types: <%s>." % (
len(classes), str_signature(classes)
))
if len(set(classes)) == 1:
raise RuntimeError(
"Duplicate types <%s> cannot be dispatched." % str_signature(classes)
)
self._dispatcher.add(tuple(classes), typ, on_ambiguity=on_ambiguity)
self._dispatcher.add(tuple(reversed(classes)), typ, on_ambiguity=on_ambiguity)
@cacheit
def __call__(self, *args, _sympify=True, **kwargs):
"""
Parameters
==========
*args :
Arguments which are operated
"""
if _sympify:
args = tuple(map(_sympify_, args))
handlers = frozenset(map(self._handlergetter, args))
# no need to sympify again
return self.dispatch(handlers)(*args, _sympify=False, **kwargs)
@cacheit
def dispatch(self, handlers):
"""
Select the handler class, and return its handler method.
"""
# Quick exit for the case where all handlers are same
if len(handlers) == 1:
h, = handlers
if not isinstance(h, type):
raise RuntimeError("Handler {!r} is not a type.".format(h))
return h
# Recursively select with registered binary priority
for i, typ in enumerate(handlers):
if not isinstance(typ, type):
raise RuntimeError("Handler {!r} is not a type.".format(typ))
if i == 0:
handler = typ
else:
prev_handler = handler
handler = self._dispatcher.dispatch(prev_handler, typ)
if not isinstance(handler, type):
raise RuntimeError(
"Dispatcher for {!r} and {!r} must return a type, but got {!r}".format(
prev_handler, typ, handler
))
# return handler class
return handler
@property
def __doc__(self):
docs = [
"Multiply dispatched associative operator: %s" % self.name,
"Note that support for this is experimental, see the docs for :class:`AssocOpDispatcher` for details"
]
if self.doc:
docs.append(self.doc)
s = "Registered handler classes\n"
s += '=' * len(s)
docs.append(s)
amb_sigs = []
typ_sigs = defaultdict(list)
for sigs in self._dispatcher.ordering[::-1]:
key = self._dispatcher.funcs[sigs]
typ_sigs[key].append(sigs)
for typ, sigs in typ_sigs.items():
sigs_str = ', '.join('<%s>' % str_signature(sig) for sig in sigs)
if isinstance(typ, RaiseNotImplementedError):
amb_sigs.append(sigs_str)
continue
s = 'Inputs: %s\n' % sigs_str
s += '-' * len(s) + '\n'
s += typ.__name__
docs.append(s)
if amb_sigs:
s = "Ambiguous handler classes\n"
s += '=' * len(s)
docs.append(s)
s = '\n'.join(amb_sigs)
docs.append(s)
return '\n\n'.join(docs)
|
89107cdcfd966fbe9c16ca5a25e9e8c5a8c3be45c4586c5b9554ac3f10a72f3d | from sympy.core.assumptions import StdFactKB, _assume_defined
from sympy.core.compatibility import is_sequence, ordered
from .basic import Basic, Atom
from .sympify import sympify
from .singleton import S
from .expr import Expr, AtomicExpr
from .cache import cacheit
from .function import FunctionClass
from .kind import NumberKind, UndefinedKind
from sympy.core.logic import fuzzy_bool
from sympy.logic.boolalg import Boolean
from sympy.utilities.iterables import cartes, sift
from sympy.core.containers import Tuple
import string
import re as _re
import random
class Str(Atom):
"""
Represents string in SymPy.
Explanation
===========
Previously, ``Symbol`` was used where string is needed in ``args`` of SymPy
objects, e.g. denoting the name of the instance. However, since ``Symbol``
represents mathematical scalar, this class should be used instead.
"""
__slots__ = ('name',)
def __new__(cls, name, **kwargs):
if not isinstance(name, str):
raise TypeError("name should be a string, not %s" % repr(type(name)))
obj = Expr.__new__(cls, **kwargs)
obj.name = name
return obj
def __getnewargs__(self):
return (self.name,)
def _hashable_content(self):
return (self.name,)
def _filter_assumptions(kwargs):
"""Split the given dict into assumptions and non-assumptions.
Keys are taken as assumptions if they correspond to an
entry in ``_assume_defined``.
"""
assumptions, nonassumptions = map(dict, sift(kwargs.items(),
lambda i: i[0] in _assume_defined,
binary=True))
Symbol._sanitize(assumptions)
return assumptions, nonassumptions
def _symbol(s, matching_symbol=None, **assumptions):
"""Return s if s is a Symbol, else if s is a string, return either
the matching_symbol if the names are the same or else a new symbol
with the same assumptions as the matching symbol (or the
assumptions as provided).
Examples
========
>>> from sympy import Symbol
>>> from sympy.core.symbol import _symbol
>>> _symbol('y')
y
>>> _.is_real is None
True
>>> _symbol('y', real=True).is_real
True
>>> x = Symbol('x')
>>> _symbol(x, real=True)
x
>>> _.is_real is None # ignore attribute if s is a Symbol
True
Below, the variable sym has the name 'foo':
>>> sym = Symbol('foo', real=True)
Since 'x' is not the same as sym's name, a new symbol is created:
>>> _symbol('x', sym).name
'x'
It will acquire any assumptions give:
>>> _symbol('x', sym, real=False).is_real
False
Since 'foo' is the same as sym's name, sym is returned
>>> _symbol('foo', sym)
foo
Any assumptions given are ignored:
>>> _symbol('foo', sym, real=False).is_real
True
NB: the symbol here may not be the same as a symbol with the same
name defined elsewhere as a result of different assumptions.
See Also
========
sympy.core.symbol.Symbol
"""
if isinstance(s, str):
if matching_symbol and matching_symbol.name == s:
return matching_symbol
return Symbol(s, **assumptions)
elif isinstance(s, Symbol):
return s
else:
raise ValueError('symbol must be string for symbol name or Symbol')
def uniquely_named_symbol(xname, exprs=(), compare=str, modify=None, **assumptions):
"""Return a symbol which, when printed, will have a name unique
from any other already in the expressions given. The name is made
unique by appending numbers (default) but this can be
customized with the keyword 'modify'.
Parameters
==========
xname : a string or a Symbol (when symbol xname <- str(xname))
compare : a single arg function that takes a symbol and returns
a string to be compared with xname (the default is the str
function which indicates how the name will look when it
is printed, e.g. this includes underscores that appear on
Dummy symbols)
modify : a single arg function that changes its string argument
in some way (the default is to append numbers)
Examples
========
>>> from sympy.core.symbol import uniquely_named_symbol
>>> from sympy.abc import x
>>> uniquely_named_symbol('x', x)
x0
"""
from sympy.core.function import AppliedUndef
def numbered_string_incr(s, start=0):
if not s:
return str(start)
i = len(s) - 1
while i != -1:
if not s[i].isdigit():
break
i -= 1
n = str(int(s[i + 1:] or start - 1) + 1)
return s[:i + 1] + n
default = None
if is_sequence(xname):
xname, default = xname
x = str(xname)
if not exprs:
return _symbol(x, default, **assumptions)
if not is_sequence(exprs):
exprs = [exprs]
names = set().union(
[i.name for e in exprs for i in e.atoms(Symbol)] +
[i.func.name for e in exprs for i in e.atoms(AppliedUndef)])
if modify is None:
modify = numbered_string_incr
while any(x == compare(s) for s in names):
x = modify(x)
return _symbol(x, default, **assumptions)
_uniquely_named_symbol = uniquely_named_symbol
class Symbol(AtomicExpr, Boolean):
"""
Assumptions:
commutative = True
You can override the default assumptions in the constructor.
Examples
========
>>> from sympy import symbols
>>> A,B = symbols('A,B', commutative = False)
>>> bool(A*B != B*A)
True
>>> bool(A*B*2 == 2*A*B) == True # multiplication by scalars is commutative
True
"""
is_comparable = False
__slots__ = ('name',)
is_Symbol = True
is_symbol = True
@property
def kind(self):
if self.is_commutative:
return NumberKind
return UndefinedKind
@property
def _diff_wrt(self):
"""Allow derivatives wrt Symbols.
Examples
========
>>> from sympy import Symbol
>>> x = Symbol('x')
>>> x._diff_wrt
True
"""
return True
@staticmethod
def _sanitize(assumptions, obj=None):
"""Remove None, covert values to bool, check commutativity *in place*.
"""
# be strict about commutativity: cannot be None
is_commutative = fuzzy_bool(assumptions.get('commutative', True))
if is_commutative is None:
whose = '%s ' % obj.__name__ if obj else ''
raise ValueError(
'%scommutativity must be True or False.' % whose)
# sanitize other assumptions so 1 -> True and 0 -> False
for key in list(assumptions.keys()):
v = assumptions[key]
if v is None:
assumptions.pop(key)
continue
assumptions[key] = bool(v)
def _merge(self, assumptions):
base = self.assumptions0
for k in set(assumptions) & set(base):
if assumptions[k] != base[k]:
from sympy.utilities.misc import filldedent
raise ValueError(filldedent('''
non-matching assumptions for %s: existing value
is %s and new value is %s''' % (
k, base[k], assumptions[k])))
base.update(assumptions)
return base
def __new__(cls, name, **assumptions):
"""Symbols are identified by name and assumptions::
>>> from sympy import Symbol
>>> Symbol("x") == Symbol("x")
True
>>> Symbol("x", real=True) == Symbol("x", real=False)
False
"""
cls._sanitize(assumptions, cls)
return Symbol.__xnew_cached_(cls, name, **assumptions)
def __new_stage2__(cls, name, **assumptions):
if not isinstance(name, str):
raise TypeError("name should be a string, not %s" % repr(type(name)))
obj = Expr.__new__(cls)
obj.name = name
# TODO: Issue #8873: Forcing the commutative assumption here means
# later code such as ``srepr()`` cannot tell whether the user
# specified ``commutative=True`` or omitted it. To workaround this,
# we keep a copy of the assumptions dict, then create the StdFactKB,
# and finally overwrite its ``._generator`` with the dict copy. This
# is a bit of a hack because we assume StdFactKB merely copies the
# given dict as ``._generator``, but future modification might, e.g.,
# compute a minimal equivalent assumption set.
tmp_asm_copy = assumptions.copy()
# be strict about commutativity
is_commutative = fuzzy_bool(assumptions.get('commutative', True))
assumptions['commutative'] = is_commutative
obj._assumptions = StdFactKB(assumptions)
obj._assumptions._generator = tmp_asm_copy # Issue #8873
return obj
__xnew__ = staticmethod(
__new_stage2__) # never cached (e.g. dummy)
__xnew_cached_ = staticmethod(
cacheit(__new_stage2__)) # symbols are always cached
def __getnewargs_ex__(self):
return ((self.name,), self.assumptions0)
def _hashable_content(self):
# Note: user-specified assumptions not hashed, just derived ones
return (self.name,) + tuple(sorted(self.assumptions0.items()))
def _eval_subs(self, old, new):
from sympy.core.power import Pow
if old.is_Pow:
return Pow(self, S.One, evaluate=False)._eval_subs(old, new)
def _eval_refine(self, assumptions):
return self
@property
def assumptions0(self):
return {key: value for key, value
in self._assumptions.items() if value is not None}
@cacheit
def sort_key(self, order=None):
return self.class_key(), (1, (self.name,)), S.One.sort_key(), S.One
def as_dummy(self):
# only put commutativity in explicitly if it is False
return Dummy(self.name) if self.is_commutative is not False \
else Dummy(self.name, commutative=self.is_commutative)
def as_real_imag(self, deep=True, **hints):
from sympy import im, re
if hints.get('ignore') == self:
return None
else:
return (re(self), im(self))
def _sage_(self):
import sage.all as sage
return sage.var(self.name)
def is_constant(self, *wrt, **flags):
if not wrt:
return False
return not self in wrt
@property
def free_symbols(self):
return {self}
binary_symbols = free_symbols # in this case, not always
def as_set(self):
return S.UniversalSet
class Dummy(Symbol):
"""Dummy symbols are each unique, even if they have the same name:
Examples
========
>>> from sympy import Dummy
>>> Dummy("x") == Dummy("x")
False
If a name is not supplied then a string value of an internal count will be
used. This is useful when a temporary variable is needed and the name
of the variable used in the expression is not important.
>>> Dummy() #doctest: +SKIP
_Dummy_10
"""
# In the rare event that a Dummy object needs to be recreated, both the
# `name` and `dummy_index` should be passed. This is used by `srepr` for
# example:
# >>> d1 = Dummy()
# >>> d2 = eval(srepr(d1))
# >>> d2 == d1
# True
#
# If a new session is started between `srepr` and `eval`, there is a very
# small chance that `d2` will be equal to a previously-created Dummy.
_count = 0
_prng = random.Random()
_base_dummy_index = _prng.randint(10**6, 9*10**6)
__slots__ = ('dummy_index',)
is_Dummy = True
def __new__(cls, name=None, dummy_index=None, **assumptions):
if dummy_index is not None:
assert name is not None, "If you specify a dummy_index, you must also provide a name"
if name is None:
name = "Dummy_" + str(Dummy._count)
if dummy_index is None:
dummy_index = Dummy._base_dummy_index + Dummy._count
Dummy._count += 1
cls._sanitize(assumptions, cls)
obj = Symbol.__xnew__(cls, name, **assumptions)
obj.dummy_index = dummy_index
return obj
def __getnewargs_ex__(self):
return ((self.name, self.dummy_index), self.assumptions0)
@cacheit
def sort_key(self, order=None):
return self.class_key(), (
2, (self.name, self.dummy_index)), S.One.sort_key(), S.One
def _hashable_content(self):
return Symbol._hashable_content(self) + (self.dummy_index,)
class Wild(Symbol):
"""
A Wild symbol matches anything, or anything
without whatever is explicitly excluded.
Parameters
==========
name : str
Name of the Wild instance.
exclude : iterable, optional
Instances in ``exclude`` will not be matched.
properties : iterable of functions, optional
Functions, each taking an expressions as input
and returns a ``bool``. All functions in ``properties``
need to return ``True`` in order for the Wild instance
to match the expression.
Examples
========
>>> from sympy import Wild, WildFunction, cos, pi
>>> from sympy.abc import x, y, z
>>> a = Wild('a')
>>> x.match(a)
{a_: x}
>>> pi.match(a)
{a_: pi}
>>> (3*x**2).match(a*x)
{a_: 3*x}
>>> cos(x).match(a)
{a_: cos(x)}
>>> b = Wild('b', exclude=[x])
>>> (3*x**2).match(b*x)
>>> b.match(a)
{a_: b_}
>>> A = WildFunction('A')
>>> A.match(a)
{a_: A_}
Tips
====
When using Wild, be sure to use the exclude
keyword to make the pattern more precise.
Without the exclude pattern, you may get matches
that are technically correct, but not what you
wanted. For example, using the above without
exclude:
>>> from sympy import symbols
>>> a, b = symbols('a b', cls=Wild)
>>> (2 + 3*y).match(a*x + b*y)
{a_: 2/x, b_: 3}
This is technically correct, because
(2/x)*x + 3*y == 2 + 3*y, but you probably
wanted it to not match at all. The issue is that
you really didn't want a and b to include x and y,
and the exclude parameter lets you specify exactly
this. With the exclude parameter, the pattern will
not match.
>>> a = Wild('a', exclude=[x, y])
>>> b = Wild('b', exclude=[x, y])
>>> (2 + 3*y).match(a*x + b*y)
Exclude also helps remove ambiguity from matches.
>>> E = 2*x**3*y*z
>>> a, b = symbols('a b', cls=Wild)
>>> E.match(a*b)
{a_: 2*y*z, b_: x**3}
>>> a = Wild('a', exclude=[x, y])
>>> E.match(a*b)
{a_: z, b_: 2*x**3*y}
>>> a = Wild('a', exclude=[x, y, z])
>>> E.match(a*b)
{a_: 2, b_: x**3*y*z}
Wild also accepts a ``properties`` parameter:
>>> a = Wild('a', properties=[lambda k: k.is_Integer])
>>> E.match(a*b)
{a_: 2, b_: x**3*y*z}
"""
is_Wild = True
__slots__ = ('exclude', 'properties')
def __new__(cls, name, exclude=(), properties=(), **assumptions):
exclude = tuple([sympify(x) for x in exclude])
properties = tuple(properties)
cls._sanitize(assumptions, cls)
return Wild.__xnew__(cls, name, exclude, properties, **assumptions)
def __getnewargs__(self):
return (self.name, self.exclude, self.properties)
@staticmethod
@cacheit
def __xnew__(cls, name, exclude, properties, **assumptions):
obj = Symbol.__xnew__(cls, name, **assumptions)
obj.exclude = exclude
obj.properties = properties
return obj
def _hashable_content(self):
return super()._hashable_content() + (self.exclude, self.properties)
# TODO add check against another Wild
def matches(self, expr, repl_dict={}, old=False):
if any(expr.has(x) for x in self.exclude):
return None
if any(not f(expr) for f in self.properties):
return None
repl_dict = repl_dict.copy()
repl_dict[self] = expr
return repl_dict
_range = _re.compile('([0-9]*:[0-9]+|[a-zA-Z]?:[a-zA-Z])')
def symbols(names, *, cls=Symbol, **args):
r"""
Transform strings into instances of :class:`Symbol` class.
:func:`symbols` function returns a sequence of symbols with names taken
from ``names`` argument, which can be a comma or whitespace delimited
string, or a sequence of strings::
>>> from sympy import symbols, Function
>>> x, y, z = symbols('x,y,z')
>>> a, b, c = symbols('a b c')
The type of output is dependent on the properties of input arguments::
>>> symbols('x')
x
>>> symbols('x,')
(x,)
>>> symbols('x,y')
(x, y)
>>> symbols(('a', 'b', 'c'))
(a, b, c)
>>> symbols(['a', 'b', 'c'])
[a, b, c]
>>> symbols({'a', 'b', 'c'})
{a, b, c}
If an iterable container is needed for a single symbol, set the ``seq``
argument to ``True`` or terminate the symbol name with a comma::
>>> symbols('x', seq=True)
(x,)
To reduce typing, range syntax is supported to create indexed symbols.
Ranges are indicated by a colon and the type of range is determined by
the character to the right of the colon. If the character is a digit
then all contiguous digits to the left are taken as the nonnegative
starting value (or 0 if there is no digit left of the colon) and all
contiguous digits to the right are taken as 1 greater than the ending
value::
>>> symbols('x:10')
(x0, x1, x2, x3, x4, x5, x6, x7, x8, x9)
>>> symbols('x5:10')
(x5, x6, x7, x8, x9)
>>> symbols('x5(:2)')
(x50, x51)
>>> symbols('x5:10,y:5')
(x5, x6, x7, x8, x9, y0, y1, y2, y3, y4)
>>> symbols(('x5:10', 'y:5'))
((x5, x6, x7, x8, x9), (y0, y1, y2, y3, y4))
If the character to the right of the colon is a letter, then the single
letter to the left (or 'a' if there is none) is taken as the start
and all characters in the lexicographic range *through* the letter to
the right are used as the range::
>>> symbols('x:z')
(x, y, z)
>>> symbols('x:c') # null range
()
>>> symbols('x(:c)')
(xa, xb, xc)
>>> symbols(':c')
(a, b, c)
>>> symbols('a:d, x:z')
(a, b, c, d, x, y, z)
>>> symbols(('a:d', 'x:z'))
((a, b, c, d), (x, y, z))
Multiple ranges are supported; contiguous numerical ranges should be
separated by parentheses to disambiguate the ending number of one
range from the starting number of the next::
>>> symbols('x:2(1:3)')
(x01, x02, x11, x12)
>>> symbols(':3:2') # parsing is from left to right
(00, 01, 10, 11, 20, 21)
Only one pair of parentheses surrounding ranges are removed, so to
include parentheses around ranges, double them. And to include spaces,
commas, or colons, escape them with a backslash::
>>> symbols('x((a:b))')
(x(a), x(b))
>>> symbols(r'x(:1\,:2)') # or r'x((:1)\,(:2))'
(x(0,0), x(0,1))
All newly created symbols have assumptions set according to ``args``::
>>> a = symbols('a', integer=True)
>>> a.is_integer
True
>>> x, y, z = symbols('x,y,z', real=True)
>>> x.is_real and y.is_real and z.is_real
True
Despite its name, :func:`symbols` can create symbol-like objects like
instances of Function or Wild classes. To achieve this, set ``cls``
keyword argument to the desired type::
>>> symbols('f,g,h', cls=Function)
(f, g, h)
>>> type(_[0])
<class 'sympy.core.function.UndefinedFunction'>
"""
result = []
if isinstance(names, str):
marker = 0
literals = [r'\,', r'\:', r'\ ']
for i in range(len(literals)):
lit = literals.pop(0)
if lit in names:
while chr(marker) in names:
marker += 1
lit_char = chr(marker)
marker += 1
names = names.replace(lit, lit_char)
literals.append((lit_char, lit[1:]))
def literal(s):
if literals:
for c, l in literals:
s = s.replace(c, l)
return s
names = names.strip()
as_seq = names.endswith(',')
if as_seq:
names = names[:-1].rstrip()
if not names:
raise ValueError('no symbols given')
# split on commas
names = [n.strip() for n in names.split(',')]
if not all(n for n in names):
raise ValueError('missing symbol between commas')
# split on spaces
for i in range(len(names) - 1, -1, -1):
names[i: i + 1] = names[i].split()
seq = args.pop('seq', as_seq)
for name in names:
if not name:
raise ValueError('missing symbol')
if ':' not in name:
symbol = cls(literal(name), **args)
result.append(symbol)
continue
split = _range.split(name)
# remove 1 layer of bounding parentheses around ranges
for i in range(len(split) - 1):
if i and ':' in split[i] and split[i] != ':' and \
split[i - 1].endswith('(') and \
split[i + 1].startswith(')'):
split[i - 1] = split[i - 1][:-1]
split[i + 1] = split[i + 1][1:]
for i, s in enumerate(split):
if ':' in s:
if s[-1].endswith(':'):
raise ValueError('missing end range')
a, b = s.split(':')
if b[-1] in string.digits:
a = 0 if not a else int(a)
b = int(b)
split[i] = [str(c) for c in range(a, b)]
else:
a = a or 'a'
split[i] = [string.ascii_letters[c] for c in range(
string.ascii_letters.index(a),
string.ascii_letters.index(b) + 1)] # inclusive
if not split[i]:
break
else:
split[i] = [s]
else:
seq = True
if len(split) == 1:
names = split[0]
else:
names = [''.join(s) for s in cartes(*split)]
if literals:
result.extend([cls(literal(s), **args) for s in names])
else:
result.extend([cls(s, **args) for s in names])
if not seq and len(result) <= 1:
if not result:
return ()
return result[0]
return tuple(result)
else:
for name in names:
result.append(symbols(name, **args))
return type(names)(result)
def var(names, **args):
"""
Create symbols and inject them into the global namespace.
Explanation
===========
This calls :func:`symbols` with the same arguments and puts the results
into the *global* namespace. It's recommended not to use :func:`var` in
library code, where :func:`symbols` has to be used::
Examples
========
>>> from sympy import var
>>> var('x')
x
>>> x # noqa: F821
x
>>> var('a,ab,abc')
(a, ab, abc)
>>> abc # noqa: F821
abc
>>> var('x,y', real=True)
(x, y)
>>> x.is_real and y.is_real # noqa: F821
True
See :func:`symbols` documentation for more details on what kinds of
arguments can be passed to :func:`var`.
"""
def traverse(symbols, frame):
"""Recursively inject symbols to the global namespace. """
for symbol in symbols:
if isinstance(symbol, Basic):
frame.f_globals[symbol.name] = symbol
elif isinstance(symbol, FunctionClass):
frame.f_globals[symbol.__name__] = symbol
else:
traverse(symbol, frame)
from inspect import currentframe
frame = currentframe().f_back
try:
syms = symbols(names, **args)
if syms is not None:
if isinstance(syms, Basic):
frame.f_globals[syms.name] = syms
elif isinstance(syms, FunctionClass):
frame.f_globals[syms.__name__] = syms
else:
traverse(syms, frame)
finally:
del frame # break cyclic dependencies as stated in inspect docs
return syms
def disambiguate(*iter):
"""
Return a Tuple containing the passed expressions with symbols
that appear the same when printed replaced with numerically
subscripted symbols, and all Dummy symbols replaced with Symbols.
Parameters
==========
iter: list of symbols or expressions.
Examples
========
>>> from sympy.core.symbol import disambiguate
>>> from sympy import Dummy, Symbol, Tuple
>>> from sympy.abc import y
>>> tup = Symbol('_x'), Dummy('x'), Dummy('x')
>>> disambiguate(*tup)
(x_2, x, x_1)
>>> eqs = Tuple(Symbol('x')/y, Dummy('x')/y)
>>> disambiguate(*eqs)
(x_1/y, x/y)
>>> ix = Symbol('x', integer=True)
>>> vx = Symbol('x')
>>> disambiguate(vx + ix)
(x + x_1,)
To make your own mapping of symbols to use, pass only the free symbols
of the expressions and create a dictionary:
>>> free = eqs.free_symbols
>>> mapping = dict(zip(free, disambiguate(*free)))
>>> eqs.xreplace(mapping)
(x_1/y, x/y)
"""
new_iter = Tuple(*iter)
key = lambda x:tuple(sorted(x.assumptions0.items()))
syms = ordered(new_iter.free_symbols, keys=key)
mapping = {}
for s in syms:
mapping.setdefault(str(s).lstrip('_'), []).append(s)
reps = {}
for k in mapping:
# the first or only symbol doesn't get subscripted but make
# sure that it's a Symbol, not a Dummy
mapk0 = Symbol("%s" % (k), **mapping[k][0].assumptions0)
if mapping[k][0] != mapk0:
reps[mapping[k][0]] = mapk0
# the others get subscripts (and are made into Symbols)
skip = 0
for i in range(1, len(mapping[k])):
while True:
name = "%s_%i" % (k, i + skip)
if name not in mapping:
break
skip += 1
ki = mapping[k][i]
reps[ki] = Symbol(name, **ki.assumptions0)
return new_iter.xreplace(reps)
|
37123fc861641d0f6eae6c8722d1249bc84e45aa93e39aa6d2767dfc8ac7f460 | """
Reimplementations of constructs introduced in later versions of Python than
we support. Also some functions that are needed SymPy-wide and are located
here for easy import.
"""
import operator
from collections import defaultdict
from time import perf_counter as clock
from sympy.external.gmpy import SYMPY_INTS, HAS_GMPY, GROUND_TYPES, gmpy
"""
Python 2 and Python 3 compatible imports
String and Unicode compatible changes:
* `unicode()` removed in Python 3, import `unicode` for Python 2/3
compatible function
* Use `u()` for escaped unicode sequences (e.g. u'\u2020' -> u('\u2020'))
* Use `u_decode()` to decode utf-8 formatted unicode strings
Renamed function attributes:
* Python 2 `.func_code`, Python 3 `.__func__`, access with
`get_function_code()`
* Python 2 `.func_globals`, Python 3 `.__globals__`, access with
`get_function_globals()`
* Python 2 `.func_name`, Python 3 `.__name__`, access with
`get_function_name()`
Moved modules:
* `reduce()`
* `StringIO()`
* `cStringIO()` (same as `StingIO()` in Python 3)
* Python 2 `__builtin__`, access with Python 3 name, `builtins`
exec:
* Use `exec_()`, with parameters `exec_(code, globs=None, locs=None)`
Metaclasses:
* Use `with_metaclass()`, examples below
* Define class `Foo` with metaclass `Meta`, and no parent:
class Foo(with_metaclass(Meta)):
pass
* Define class `Foo` with metaclass `Meta` and parent class `Bar`:
class Foo(with_metaclass(Meta, Bar)):
pass
"""
__all__ = [
'PY3', 'int_info', 'SYMPY_INTS', 'clock',
'unicode', 'u_decode', 'get_function_code', 'gmpy',
'get_function_globals', 'get_function_name', 'builtins', 'reduce',
'StringIO', 'cStringIO', 'exec_', 'Mapping', 'Callable',
'MutableMapping', 'MutableSet', 'Iterable', 'Hashable', 'unwrap',
'accumulate', 'with_metaclass', 'NotIterable', 'iterable', 'is_sequence',
'as_int', 'default_sort_key', 'ordered', 'GROUND_TYPES', 'HAS_GMPY',
]
import sys
PY3 = True
int_info = sys.int_info
# String / unicode compatibility
unicode = str
def u_decode(x):
return x
# Moved definitions
get_function_code = operator.attrgetter("__code__")
get_function_globals = operator.attrgetter("__globals__")
get_function_name = operator.attrgetter("__name__")
import builtins
from functools import reduce
from io import StringIO
cStringIO = StringIO
exec_ = getattr(builtins, "exec")
from collections.abc import (Mapping, Callable, MutableMapping,
MutableSet, Iterable, Hashable)
from inspect import unwrap
from itertools import accumulate
def with_metaclass(meta, *bases):
"""
Create a base class with a metaclass.
For example, if you have the metaclass
>>> class Meta(type):
... pass
Use this as the metaclass by doing
>>> from sympy.core.compatibility import with_metaclass
>>> class MyClass(with_metaclass(Meta, object)):
... pass
This is equivalent to the Python 2::
class MyClass(object):
__metaclass__ = Meta
or Python 3::
class MyClass(object, metaclass=Meta):
pass
That is, the first argument is the metaclass, and the remaining arguments
are the base classes. Note that if the base class is just ``object``, you
may omit it.
>>> MyClass.__mro__
(<class '...MyClass'>, <... 'object'>)
>>> type(MyClass)
<class '...Meta'>
"""
# This requires a bit of explanation: the basic idea is to make a dummy
# metaclass for one level of class instantiation that replaces itself with
# the actual metaclass.
# Code copied from the 'six' library.
class metaclass(meta):
def __new__(cls, name, this_bases, d):
return meta(name, bases, d)
return type.__new__(metaclass, "NewBase", (), {})
# These are in here because telling if something is an iterable just by calling
# hasattr(obj, "__iter__") behaves differently in Python 2 and Python 3. In
# particular, hasattr(str, "__iter__") is False in Python 2 and True in Python 3.
# I think putting them here also makes it easier to use them in the core.
class NotIterable:
"""
Use this as mixin when creating a class which is not supposed to
return true when iterable() is called on its instances because
calling list() on the instance, for example, would result in
an infinite loop.
"""
pass
def iterable(i, exclude=(str, dict, NotIterable)):
"""
Return a boolean indicating whether ``i`` is SymPy iterable.
True also indicates that the iterator is finite, e.g. you can
call list(...) on the instance.
When SymPy is working with iterables, it is almost always assuming
that the iterable is not a string or a mapping, so those are excluded
by default. If you want a pure Python definition, make exclude=None. To
exclude multiple items, pass them as a tuple.
You can also set the _iterable attribute to True or False on your class,
which will override the checks here, including the exclude test.
As a rule of thumb, some SymPy functions use this to check if they should
recursively map over an object. If an object is technically iterable in
the Python sense but does not desire this behavior (e.g., because its
iteration is not finite, or because iteration might induce an unwanted
computation), it should disable it by setting the _iterable attribute to False.
See also: is_sequence
Examples
========
>>> from sympy.utilities.iterables import iterable
>>> from sympy import Tuple
>>> things = [[1], (1,), set([1]), Tuple(1), (j for j in [1, 2]), {1:2}, '1', 1]
>>> for i in things:
... print('%s %s' % (iterable(i), type(i)))
True <... 'list'>
True <... 'tuple'>
True <... 'set'>
True <class 'sympy.core.containers.Tuple'>
True <... 'generator'>
False <... 'dict'>
False <... 'str'>
False <... 'int'>
>>> iterable({}, exclude=None)
True
>>> iterable({}, exclude=str)
True
>>> iterable("no", exclude=str)
False
"""
if hasattr(i, '_iterable'):
return i._iterable
try:
iter(i)
except TypeError:
return False
if exclude:
return not isinstance(i, exclude)
return True
def is_sequence(i, include=None):
"""
Return a boolean indicating whether ``i`` is a sequence in the SymPy
sense. If anything that fails the test below should be included as
being a sequence for your application, set 'include' to that object's
type; multiple types should be passed as a tuple of types.
Note: although generators can generate a sequence, they often need special
handling to make sure their elements are captured before the generator is
exhausted, so these are not included by default in the definition of a
sequence.
See also: iterable
Examples
========
>>> from sympy.utilities.iterables import is_sequence
>>> from types import GeneratorType
>>> is_sequence([])
True
>>> is_sequence(set())
False
>>> is_sequence('abc')
False
>>> is_sequence('abc', include=str)
True
>>> generator = (c for c in 'abc')
>>> is_sequence(generator)
False
>>> is_sequence(generator, include=(str, GeneratorType))
True
"""
return (hasattr(i, '__getitem__') and
iterable(i) or
bool(include) and
isinstance(i, include))
def as_int(n, strict=True):
"""
Convert the argument to a builtin integer.
The return value is guaranteed to be equal to the input. ValueError is
raised if the input has a non-integral value. When ``strict`` is True, this
uses `__index__ <https://docs.python.org/3/reference/datamodel.html#object.__index__>`_
and when it is False it uses ``int``.
Examples
========
>>> from sympy.core.compatibility import as_int
>>> from sympy import sqrt, S
The function is primarily concerned with sanitizing input for
functions that need to work with builtin integers, so anything that
is unambiguously an integer should be returned as an int:
>>> as_int(S(3))
3
Floats, being of limited precision, are not assumed to be exact and
will raise an error unless the ``strict`` flag is False. This
precision issue becomes apparent for large floating point numbers:
>>> big = 1e23
>>> type(big) is float
True
>>> big == int(big)
True
>>> as_int(big)
Traceback (most recent call last):
...
ValueError: ... is not an integer
>>> as_int(big, strict=False)
99999999999999991611392
Input that might be a complex representation of an integer value is
also rejected by default:
>>> one = sqrt(3 + 2*sqrt(2)) - sqrt(2)
>>> int(one) == 1
True
>>> as_int(one)
Traceback (most recent call last):
...
ValueError: ... is not an integer
"""
if strict:
try:
if type(n) is bool:
raise TypeError
return operator.index(n)
except TypeError:
raise ValueError('%s is not an integer' % (n,))
else:
try:
result = int(n)
except TypeError:
raise ValueError('%s is not an integer' % (n,))
if n != result:
raise ValueError('%s is not an integer' % (n,))
return result
def default_sort_key(item, order=None):
"""Return a key that can be used for sorting.
The key has the structure:
(class_key, (len(args), args), exponent.sort_key(), coefficient)
This key is supplied by the sort_key routine of Basic objects when
``item`` is a Basic object or an object (other than a string) that
sympifies to a Basic object. Otherwise, this function produces the
key.
The ``order`` argument is passed along to the sort_key routine and is
used to determine how the terms *within* an expression are ordered.
(See examples below) ``order`` options are: 'lex', 'grlex', 'grevlex',
and reversed values of the same (e.g. 'rev-lex'). The default order
value is None (which translates to 'lex').
Examples
========
>>> from sympy import S, I, default_sort_key, sin, cos, sqrt
>>> from sympy.core.function import UndefinedFunction
>>> from sympy.abc import x
The following are equivalent ways of getting the key for an object:
>>> x.sort_key() == default_sort_key(x)
True
Here are some examples of the key that is produced:
>>> default_sort_key(UndefinedFunction('f'))
((0, 0, 'UndefinedFunction'), (1, ('f',)), ((1, 0, 'Number'),
(0, ()), (), 1), 1)
>>> default_sort_key('1')
((0, 0, 'str'), (1, ('1',)), ((1, 0, 'Number'), (0, ()), (), 1), 1)
>>> default_sort_key(S.One)
((1, 0, 'Number'), (0, ()), (), 1)
>>> default_sort_key(2)
((1, 0, 'Number'), (0, ()), (), 2)
While sort_key is a method only defined for SymPy objects,
default_sort_key will accept anything as an argument so it is
more robust as a sorting key. For the following, using key=
lambda i: i.sort_key() would fail because 2 doesn't have a sort_key
method; that's why default_sort_key is used. Note, that it also
handles sympification of non-string items likes ints:
>>> a = [2, I, -I]
>>> sorted(a, key=default_sort_key)
[2, -I, I]
The returned key can be used anywhere that a key can be specified for
a function, e.g. sort, min, max, etc...:
>>> a.sort(key=default_sort_key); a[0]
2
>>> min(a, key=default_sort_key)
2
Note
----
The key returned is useful for getting items into a canonical order
that will be the same across platforms. It is not directly useful for
sorting lists of expressions:
>>> a, b = x, 1/x
Since ``a`` has only 1 term, its value of sort_key is unaffected by
``order``:
>>> a.sort_key() == a.sort_key('rev-lex')
True
If ``a`` and ``b`` are combined then the key will differ because there
are terms that can be ordered:
>>> eq = a + b
>>> eq.sort_key() == eq.sort_key('rev-lex')
False
>>> eq.as_ordered_terms()
[x, 1/x]
>>> eq.as_ordered_terms('rev-lex')
[1/x, x]
But since the keys for each of these terms are independent of ``order``'s
value, they don't sort differently when they appear separately in a list:
>>> sorted(eq.args, key=default_sort_key)
[1/x, x]
>>> sorted(eq.args, key=lambda i: default_sort_key(i, order='rev-lex'))
[1/x, x]
The order of terms obtained when using these keys is the order that would
be obtained if those terms were *factors* in a product.
Although it is useful for quickly putting expressions in canonical order,
it does not sort expressions based on their complexity defined by the
number of operations, power of variables and others:
>>> sorted([sin(x)*cos(x), sin(x)], key=default_sort_key)
[sin(x)*cos(x), sin(x)]
>>> sorted([x, x**2, sqrt(x), x**3], key=default_sort_key)
[sqrt(x), x, x**2, x**3]
See Also
========
ordered, sympy.core.expr.as_ordered_factors, sympy.core.expr.as_ordered_terms
"""
from .singleton import S
from .basic import Basic
from .sympify import sympify, SympifyError
from .compatibility import iterable
if isinstance(item, Basic):
return item.sort_key(order=order)
if iterable(item, exclude=str):
if isinstance(item, dict):
args = item.items()
unordered = True
elif isinstance(item, set):
args = item
unordered = True
else:
# e.g. tuple, list
args = list(item)
unordered = False
args = [default_sort_key(arg, order=order) for arg in args]
if unordered:
# e.g. dict, set
args = sorted(args)
cls_index, args = 10, (len(args), tuple(args))
else:
if not isinstance(item, str):
try:
item = sympify(item, strict=True)
except SympifyError:
# e.g. lambda x: x
pass
else:
if isinstance(item, Basic):
# e.g int -> Integer
return default_sort_key(item)
# e.g. UndefinedFunction
# e.g. str
cls_index, args = 0, (1, (str(item),))
return (cls_index, 0, item.__class__.__name__
), args, S.One.sort_key(), S.One
def _node_count(e):
return 1 + sum(map(_node_count, e.args))
def _nodes(e):
"""
A helper for ordered() which returns the node count of ``e`` which
for Basic objects is the number of Basic nodes in the expression tree
but for other objects is 1 (unless the object is an iterable or dict
for which the sum of nodes is returned).
"""
from .basic import Basic
from .function import Derivative
if isinstance(e, Basic):
if isinstance(e, Derivative):
return _nodes(e.expr) + len(e.variables)
return _node_count(e)
elif iterable(e):
return 1 + sum(_nodes(ei) for ei in e)
elif isinstance(e, dict):
return 1 + sum(_nodes(k) + _nodes(v) for k, v in e.items())
else:
return 1
def ordered(seq, keys=None, default=True, warn=False):
"""Return an iterator of the seq where keys are used to break ties in
a conservative fashion: if, after applying a key, there are no ties
then no other keys will be computed.
Two default keys will be applied if 1) keys are not provided or 2) the
given keys don't resolve all ties (but only if ``default`` is True). The
two keys are ``_nodes`` (which places smaller expressions before large) and
``default_sort_key`` which (if the ``sort_key`` for an object is defined
properly) should resolve any ties.
If ``warn`` is True then an error will be raised if there were no
keys remaining to break ties. This can be used if it was expected that
there should be no ties between items that are not identical.
Examples
========
>>> from sympy.utilities.iterables import ordered
>>> from sympy import count_ops
>>> from sympy.abc import x, y
The count_ops is not sufficient to break ties in this list and the first
two items appear in their original order (i.e. the sorting is stable):
>>> list(ordered([y + 2, x + 2, x**2 + y + 3],
... count_ops, default=False, warn=False))
...
[y + 2, x + 2, x**2 + y + 3]
The default_sort_key allows the tie to be broken:
>>> list(ordered([y + 2, x + 2, x**2 + y + 3]))
...
[x + 2, y + 2, x**2 + y + 3]
Here, sequences are sorted by length, then sum:
>>> seq, keys = [[[1, 2, 1], [0, 3, 1], [1, 1, 3], [2], [1]], [
... lambda x: len(x),
... lambda x: sum(x)]]
...
>>> list(ordered(seq, keys, default=False, warn=False))
[[1], [2], [1, 2, 1], [0, 3, 1], [1, 1, 3]]
If ``warn`` is True, an error will be raised if there were not
enough keys to break ties:
>>> list(ordered(seq, keys, default=False, warn=True))
Traceback (most recent call last):
...
ValueError: not enough keys to break ties
Notes
=====
The decorated sort is one of the fastest ways to sort a sequence for
which special item comparison is desired: the sequence is decorated,
sorted on the basis of the decoration (e.g. making all letters lower
case) and then undecorated. If one wants to break ties for items that
have the same decorated value, a second key can be used. But if the
second key is expensive to compute then it is inefficient to decorate
all items with both keys: only those items having identical first key
values need to be decorated. This function applies keys successively
only when needed to break ties. By yielding an iterator, use of the
tie-breaker is delayed as long as possible.
This function is best used in cases when use of the first key is
expected to be a good hashing function; if there are no unique hashes
from application of a key, then that key should not have been used. The
exception, however, is that even if there are many collisions, if the
first group is small and one does not need to process all items in the
list then time will not be wasted sorting what one was not interested
in. For example, if one were looking for the minimum in a list and
there were several criteria used to define the sort order, then this
function would be good at returning that quickly if the first group
of candidates is small relative to the number of items being processed.
"""
d = defaultdict(list)
if keys:
if not isinstance(keys, (list, tuple)):
keys = [keys]
keys = list(keys)
f = keys.pop(0)
for a in seq:
d[f(a)].append(a)
else:
if not default:
raise ValueError('if default=False then keys must be provided')
d[None].extend(seq)
for k in sorted(d.keys()):
if len(d[k]) > 1:
if keys:
d[k] = ordered(d[k], keys, default, warn)
elif default:
d[k] = ordered(d[k], (_nodes, default_sort_key,),
default=False, warn=warn)
elif warn:
from sympy.utilities.iterables import uniq
u = list(uniq(d[k]))
if len(u) > 1:
raise ValueError(
'not enough keys to break ties: %s' % u)
yield from d[k]
d.pop(k)
|
bd6ebbf5969480f05d252a451fef7ef5d00d045dfe6073054435b167ce609896 | """sympify -- convert objects SymPy internal format"""
import typing
if typing.TYPE_CHECKING:
from typing import Any, Callable, Dict, Type
from inspect import getmro
from .compatibility import iterable
from .parameters import global_parameters
class SympifyError(ValueError):
def __init__(self, expr, base_exc=None):
self.expr = expr
self.base_exc = base_exc
def __str__(self):
if self.base_exc is None:
return "SympifyError: %r" % (self.expr,)
return ("Sympify of expression '%s' failed, because of exception being "
"raised:\n%s: %s" % (self.expr, self.base_exc.__class__.__name__,
str(self.base_exc)))
# See sympify docstring.
converter = {} # type: Dict[Type[Any], Callable[[Any], Basic]]
class CantSympify:
"""
Mix in this trait to a class to disallow sympification of its instances.
Examples
========
>>> from sympy.core.sympify import sympify, CantSympify
>>> class Something(dict):
... pass
...
>>> sympify(Something())
{}
>>> class Something(dict, CantSympify):
... pass
...
>>> sympify(Something())
Traceback (most recent call last):
...
SympifyError: SympifyError: {}
"""
pass
def _is_numpy_instance(a):
"""
Checks if an object is an instance of a type from the numpy module.
"""
# This check avoids unnecessarily importing NumPy. We check the whole
# __mro__ in case any base type is a numpy type.
return any(type_.__module__ == 'numpy'
for type_ in type(a).__mro__)
def _convert_numpy_types(a, **sympify_args):
"""
Converts a numpy datatype input to an appropriate SymPy type.
"""
import numpy as np
if not isinstance(a, np.floating):
if np.iscomplex(a):
return converter[complex](a.item())
else:
return sympify(a.item(), **sympify_args)
else:
try:
from sympy.core.numbers import Float
prec = np.finfo(a).nmant + 1
# E.g. double precision means prec=53 but nmant=52
# Leading bit of mantissa is always 1, so is not stored
a = str(list(np.reshape(np.asarray(a),
(1, np.size(a)))[0]))[1:-1]
return Float(a, precision=prec)
except NotImplementedError:
raise SympifyError('Translation for numpy float : %s '
'is not implemented' % a)
def sympify(a, locals=None, convert_xor=True, strict=False, rational=False,
evaluate=None):
"""
Converts an arbitrary expression to a type that can be used inside SymPy.
Explanation
===========
It will convert Python ints into instances of sympy.Integer,
floats into instances of sympy.Float, etc. It is also able to coerce symbolic
expressions which inherit from Basic. This can be useful in cooperation
with SAGE.
.. warning::
Note that this function uses ``eval``, and thus shouldn't be used on
unsanitized input.
If the argument is already a type that SymPy understands, it will do
nothing but return that value. This can be used at the beginning of a
function to ensure you are working with the correct type.
Examples
========
>>> from sympy import sympify
>>> sympify(2).is_integer
True
>>> sympify(2).is_real
True
>>> sympify(2.0).is_real
True
>>> sympify("2.0").is_real
True
>>> sympify("2e-45").is_real
True
If the expression could not be converted, a SympifyError is raised.
>>> sympify("x***2")
Traceback (most recent call last):
...
SympifyError: SympifyError: "could not parse 'x***2'"
Locals
------
The sympification happens with access to everything that is loaded
by ``from sympy import *``; anything used in a string that is not
defined by that import will be converted to a symbol. In the following,
the ``bitcount`` function is treated as a symbol and the ``O`` is
interpreted as the Order object (used with series) and it raises
an error when used improperly:
>>> s = 'bitcount(42)'
>>> sympify(s)
bitcount(42)
>>> sympify("O(x)")
O(x)
>>> sympify("O + 1")
Traceback (most recent call last):
...
TypeError: unbound method...
In order to have ``bitcount`` be recognized it can be imported into a
namespace dictionary and passed as locals:
>>> ns = {}
>>> exec('from sympy.core.evalf import bitcount', ns)
>>> sympify(s, locals=ns)
6
In order to have the ``O`` interpreted as a Symbol, identify it as such
in the namespace dictionary. This can be done in a variety of ways; all
three of the following are possibilities:
>>> from sympy import Symbol
>>> ns["O"] = Symbol("O") # method 1
>>> exec('from sympy.abc import O', ns) # method 2
>>> ns.update(dict(O=Symbol("O"))) # method 3
>>> sympify("O + 1", locals=ns)
O + 1
If you want *all* single-letter and Greek-letter variables to be symbols
then you can use the clashing-symbols dictionaries that have been defined
there as private variables: _clash1 (single-letter variables), _clash2
(the multi-letter Greek names) or _clash (both single and multi-letter
names that are defined in abc).
>>> from sympy.abc import _clash1
>>> _clash1
{'E': E, 'I': I, 'N': N, 'O': O, 'Q': Q, 'S': S}
>>> sympify('I & Q', _clash1)
I & Q
Strict
------
If the option ``strict`` is set to ``True``, only the types for which an
explicit conversion has been defined are converted. In the other
cases, a SympifyError is raised.
>>> print(sympify(None))
None
>>> sympify(None, strict=True)
Traceback (most recent call last):
...
SympifyError: SympifyError: None
Evaluation
----------
If the option ``evaluate`` is set to ``False``, then arithmetic and
operators will be converted into their SymPy equivalents and the
``evaluate=False`` option will be added. Nested ``Add`` or ``Mul`` will
be denested first. This is done via an AST transformation that replaces
operators with their SymPy equivalents, so if an operand redefines any
of those operations, the redefined operators will not be used. If
argument a is not a string, the mathematical expression is evaluated
before being passed to sympify, so adding evaluate=False will still
return the evaluated result of expression.
>>> sympify('2**2 / 3 + 5')
19/3
>>> sympify('2**2 / 3 + 5', evaluate=False)
2**2/3 + 5
>>> sympify('4/2+7', evaluate=True)
9
>>> sympify('4/2+7', evaluate=False)
4/2 + 7
>>> sympify(4/2+7, evaluate=False)
9.00000000000000
Extending
---------
To extend ``sympify`` to convert custom objects (not derived from ``Basic``),
just define a ``_sympy_`` method to your class. You can do that even to
classes that you do not own by subclassing or adding the method at runtime.
>>> from sympy import Matrix
>>> class MyList1(object):
... def __iter__(self):
... yield 1
... yield 2
... return
... def __getitem__(self, i): return list(self)[i]
... def _sympy_(self): return Matrix(self)
>>> sympify(MyList1())
Matrix([
[1],
[2]])
If you do not have control over the class definition you could also use the
``converter`` global dictionary. The key is the class and the value is a
function that takes a single argument and returns the desired SymPy
object, e.g. ``converter[MyList] = lambda x: Matrix(x)``.
>>> class MyList2(object): # XXX Do not do this if you control the class!
... def __iter__(self): # Use _sympy_!
... yield 1
... yield 2
... return
... def __getitem__(self, i): return list(self)[i]
>>> from sympy.core.sympify import converter
>>> converter[MyList2] = lambda x: Matrix(x)
>>> sympify(MyList2())
Matrix([
[1],
[2]])
Notes
=====
The keywords ``rational`` and ``convert_xor`` are only used
when the input is a string.
convert_xor
-----------
>>> sympify('x^y',convert_xor=True)
x**y
>>> sympify('x^y',convert_xor=False)
x ^ y
rational
--------
>>> sympify('0.1',rational=False)
0.1
>>> sympify('0.1',rational=True)
1/10
Sometimes autosimplification during sympification results in expressions
that are very different in structure than what was entered. Until such
autosimplification is no longer done, the ``kernS`` function might be of
some use. In the example below you can see how an expression reduces to
-1 by autosimplification, but does not do so when ``kernS`` is used.
>>> from sympy.core.sympify import kernS
>>> from sympy.abc import x
>>> -2*(-(-x + 1/x)/(x*(x - 1/x)**2) - 1/(x*(x - 1/x))) - 1
-1
>>> s = '-2*(-(-x + 1/x)/(x*(x - 1/x)**2) - 1/(x*(x - 1/x))) - 1'
>>> sympify(s)
-1
>>> kernS(s)
-2*(-(-x + 1/x)/(x*(x - 1/x)**2) - 1/(x*(x - 1/x))) - 1
Parameters
==========
a :
- any object defined in SymPy
- standard numeric python types: int, long, float, Decimal
- strings (like "0.09", "2e-19" or 'sin(x)')
- booleans, including ``None`` (will leave ``None`` unchanged)
- dict, lists, sets or tuples containing any of the above
convert_xor : boolean, optional
If true, treats XOR as exponentiation.
If False, treats XOR as XOR itself.
Used only when input is a string.
locals : any object defined in SymPy, optional
In order to have strings be recognized it can be imported
into a namespace dictionary and passed as locals.
strict : boolean, optional
If the option strict is set to True, only the types for which
an explicit conversion has been defined are converted. In the
other cases, a SympifyError is raised.
rational : boolean, optional
If true, converts floats into Rational.
If false, it lets floats remain as it is.
Used only when input is a string.
evaluate : boolean, optional
If False, then arithmetic and operators will be converted into
their SymPy equivalents. If True the expression will be evaluated
and the result will be returned.
"""
# XXX: If a is a Basic subclass rather than instance (e.g. sin rather than
# sin(x)) then a.__sympy__ will be the property. Only on the instance will
# a.__sympy__ give the *value* of the property (True). Since sympify(sin)
# was used for a long time we allow it to pass. However if strict=True as
# is the case in internal calls to _sympify then we only allow
# is_sympy=True.
#
# https://github.com/sympy/sympy/issues/20124
is_sympy = getattr(a, '__sympy__', None)
if is_sympy is True:
return a
elif is_sympy is not None:
if not strict:
return a
else:
raise SympifyError(a)
if isinstance(a, CantSympify):
raise SympifyError(a)
cls = getattr(a, "__class__", None)
if cls is None:
cls = type(a) # Probably an old-style class
conv = converter.get(cls, None)
if conv is not None:
return conv(a)
for superclass in getmro(cls):
try:
return converter[superclass](a)
except KeyError:
continue
if cls is type(None):
if strict:
raise SympifyError(a)
else:
return a
if evaluate is None:
evaluate = global_parameters.evaluate
# Support for basic numpy datatypes
if _is_numpy_instance(a):
import numpy as np
if np.isscalar(a):
return _convert_numpy_types(a, locals=locals,
convert_xor=convert_xor, strict=strict, rational=rational,
evaluate=evaluate)
_sympy_ = getattr(a, "_sympy_", None)
if _sympy_ is not None:
try:
return a._sympy_()
# XXX: Catches AttributeError: 'SympyConverter' object has no
# attribute 'tuple'
# This is probably a bug somewhere but for now we catch it here.
except AttributeError:
pass
if not strict:
# Put numpy array conversion _before_ float/int, see
# <https://github.com/sympy/sympy/issues/13924>.
flat = getattr(a, "flat", None)
if flat is not None:
shape = getattr(a, "shape", None)
if shape is not None:
from ..tensor.array import Array
return Array(a.flat, a.shape) # works with e.g. NumPy arrays
if not isinstance(a, str):
if _is_numpy_instance(a):
import numpy as np
assert not isinstance(a, np.number)
if isinstance(a, np.ndarray):
# Scalar arrays (those with zero dimensions) have sympify
# called on the scalar element.
if a.ndim == 0:
try:
return sympify(a.item(),
locals=locals,
convert_xor=convert_xor,
strict=strict,
rational=rational,
evaluate=evaluate)
except SympifyError:
pass
else:
# float and int can coerce size-one numpy arrays to their lone
# element. See issue https://github.com/numpy/numpy/issues/10404.
for coerce in (float, int):
try:
return sympify(coerce(a))
except (TypeError, ValueError, AttributeError, SympifyError):
continue
if strict:
raise SympifyError(a)
if iterable(a):
try:
return type(a)([sympify(x, locals=locals, convert_xor=convert_xor,
rational=rational) for x in a])
except TypeError:
# Not all iterables are rebuildable with their type.
pass
if isinstance(a, dict):
try:
return type(a)([sympify(x, locals=locals, convert_xor=convert_xor,
rational=rational) for x in a.items()])
except TypeError:
# Not all iterables are rebuildable with their type.
pass
if not isinstance(a, str):
try:
a = str(a)
except Exception as exc:
raise SympifyError(a, exc)
from sympy.utilities.exceptions import SymPyDeprecationWarning
SymPyDeprecationWarning(
feature="String fallback in sympify",
useinstead= \
'sympify(str(obj)) or ' + \
'sympy.core.sympify.converter or obj._sympy_',
issue=18066,
deprecated_since_version='1.6'
).warn()
from sympy.parsing.sympy_parser import (parse_expr, TokenError,
standard_transformations)
from sympy.parsing.sympy_parser import convert_xor as t_convert_xor
from sympy.parsing.sympy_parser import rationalize as t_rationalize
transformations = standard_transformations
if rational:
transformations += (t_rationalize,)
if convert_xor:
transformations += (t_convert_xor,)
try:
a = a.replace('\n', '')
expr = parse_expr(a, local_dict=locals, transformations=transformations, evaluate=evaluate)
except (TokenError, SyntaxError) as exc:
raise SympifyError('could not parse %r' % a, exc)
return expr
def _sympify(a):
"""
Short version of sympify for internal usage for __add__ and __eq__ methods
where it is ok to allow some things (like Python integers and floats) in
the expression. This excludes things (like strings) that are unwise to
allow into such an expression.
>>> from sympy import Integer
>>> Integer(1) == 1
True
>>> Integer(1) == '1'
False
>>> from sympy.abc import x
>>> x + 1
x + 1
>>> x + '1'
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for +: 'Symbol' and 'str'
see: sympify
"""
return sympify(a, strict=True)
def kernS(s):
"""Use a hack to try keep autosimplification from distributing a
a number into an Add; this modification doesn't
prevent the 2-arg Mul from becoming an Add, however.
Examples
========
>>> from sympy.core.sympify import kernS
>>> from sympy.abc import x, y
The 2-arg Mul distributes a number (or minus sign) across the terms
of an expression, but kernS will prevent that:
>>> 2*(x + y), -(x + 1)
(2*x + 2*y, -x - 1)
>>> kernS('2*(x + y)')
2*(x + y)
>>> kernS('-(x + 1)')
-(x + 1)
If use of the hack fails, the un-hacked string will be passed to sympify...
and you get what you get.
XXX This hack should not be necessary once issue 4596 has been resolved.
"""
import string
from random import choice
from sympy.core.symbol import Symbol
hit = False
quoted = '"' in s or "'" in s
if '(' in s and not quoted:
if s.count('(') != s.count(")"):
raise SympifyError('unmatched left parenthesis')
# strip all space from s
s = ''.join(s.split())
olds = s
# now use space to represent a symbol that
# will
# step 1. turn potential 2-arg Muls into 3-arg versions
# 1a. *( -> * *(
s = s.replace('*(', '* *(')
# 1b. close up exponentials
s = s.replace('** *', '**')
# 2. handle the implied multiplication of a negated
# parenthesized expression in two steps
# 2a: -(...) --> -( *(...)
target = '-( *('
s = s.replace('-(', target)
# 2b: double the matching closing parenthesis
# -( *(...) --> -( *(...))
i = nest = 0
assert target.endswith('(') # assumption below
while True:
j = s.find(target, i)
if j == -1:
break
j += len(target) - 1
for j in range(j, len(s)):
if s[j] == "(":
nest += 1
elif s[j] == ")":
nest -= 1
if nest == 0:
break
s = s[:j] + ")" + s[j:]
i = j + 2 # the first char after 2nd )
if ' ' in s:
# get a unique kern
kern = '_'
while kern in s:
kern += choice(string.ascii_letters + string.digits)
s = s.replace(' ', kern)
hit = kern in s
else:
hit = False
for i in range(2):
try:
expr = sympify(s)
break
except TypeError: # the kern might cause unknown errors...
if hit:
s = olds # maybe it didn't like the kern; use un-kerned s
hit = False
continue
expr = sympify(s) # let original error raise
if not hit:
return expr
rep = {Symbol(kern): 1}
def _clear(expr):
if isinstance(expr, (list, tuple, set)):
return type(expr)([_clear(e) for e in expr])
if hasattr(expr, 'subs'):
return expr.subs(rep, hack2=True)
return expr
expr = _clear(expr)
# hope that kern is not there anymore
return expr
# Avoid circular import
from .basic import Basic
|
892594c52a5393554fc8621174a1c06438191789508358decd88d0200bb69833 | """
Adaptive numerical evaluation of SymPy expressions, using mpmath
for mathematical functions.
"""
from typing import Tuple
import math
import mpmath.libmp as libmp
from mpmath import (
make_mpc, make_mpf, mp, mpc, mpf, nsum, quadts, quadosc, workprec)
from mpmath import inf as mpmath_inf
from mpmath.libmp import (from_int, from_man_exp, from_rational, fhalf,
fnan, fnone, fone, fzero, mpf_abs, mpf_add,
mpf_atan, mpf_atan2, mpf_cmp, mpf_cos, mpf_e, mpf_exp, mpf_log, mpf_lt,
mpf_mul, mpf_neg, mpf_pi, mpf_pow, mpf_pow_int, mpf_shift, mpf_sin,
mpf_sqrt, normalize, round_nearest, to_int, to_str)
from mpmath.libmp import bitcount as mpmath_bitcount
from mpmath.libmp.backend import MPZ
from mpmath.libmp.libmpc import _infs_nan
from mpmath.libmp.libmpf import dps_to_prec, prec_to_dps
from mpmath.libmp.gammazeta import mpf_bernoulli
from .compatibility import SYMPY_INTS
from .sympify import sympify
from .singleton import S
from sympy.utilities.iterables import is_sequence
LG10 = math.log(10, 2)
rnd = round_nearest
def bitcount(n):
"""Return smallest integer, b, such that |n|/2**b < 1.
"""
return mpmath_bitcount(abs(int(n)))
# Used in a few places as placeholder values to denote exponents and
# precision levels, e.g. of exact numbers. Must be careful to avoid
# passing these to mpmath functions or returning them in final results.
INF = float(mpmath_inf)
MINUS_INF = float(-mpmath_inf)
# ~= 100 digits. Real men set this to INF.
DEFAULT_MAXPREC = 333
class PrecisionExhausted(ArithmeticError):
pass
#----------------------------------------------------------------------------#
# #
# Helper functions for arithmetic and complex parts #
# #
#----------------------------------------------------------------------------#
"""
An mpf value tuple is a tuple of integers (sign, man, exp, bc)
representing a floating-point number: [1, -1][sign]*man*2**exp where
sign is 0 or 1 and bc should correspond to the number of bits used to
represent the mantissa (man) in binary notation, e.g.
Explanation
===========
>>> from sympy.core.evalf import bitcount
>>> sign, man, exp, bc = 0, 5, 1, 3
>>> n = [1, -1][sign]*man*2**exp
>>> n, bitcount(man)
(10, 3)
A temporary result is a tuple (re, im, re_acc, im_acc) where
re and im are nonzero mpf value tuples representing approximate
numbers, or None to denote exact zeros.
re_acc, im_acc are integers denoting log2(e) where e is the estimated
relative accuracy of the respective complex part, but may be anything
if the corresponding complex part is None.
"""
def fastlog(x):
"""Fast approximation of log2(x) for an mpf value tuple x.
Explanation
===========
Calculated as exponent + width of mantissa. This is an
approximation for two reasons: 1) it gives the ceil(log2(abs(x)))
value and 2) it is too high by 1 in the case that x is an exact
power of 2. Although this is easy to remedy by testing to see if
the odd mpf mantissa is 1 (indicating that one was dealing with
an exact power of 2) that would decrease the speed and is not
necessary as this is only being used as an approximation for the
number of bits in x. The correct return value could be written as
"x[2] + (x[3] if x[1] != 1 else 0)".
Since mpf tuples always have an odd mantissa, no check is done
to see if the mantissa is a multiple of 2 (in which case the
result would be too large by 1).
Examples
========
>>> from sympy import log
>>> from sympy.core.evalf import fastlog, bitcount
>>> s, m, e = 0, 5, 1
>>> bc = bitcount(m)
>>> n = [1, -1][s]*m*2**e
>>> n, (log(n)/log(2)).evalf(2), fastlog((s, m, e, bc))
(10, 3.3, 4)
"""
if not x or x == fzero:
return MINUS_INF
return x[2] + x[3]
def pure_complex(v, or_real=False):
"""Return a and b if v matches a + I*b where b is not zero and
a and b are Numbers, else None. If `or_real` is True then 0 will
be returned for `b` if `v` is a real number.
Examples
========
>>> from sympy.core.evalf import pure_complex
>>> from sympy import sqrt, I, S
>>> a, b, surd = S(2), S(3), sqrt(2)
>>> pure_complex(a)
>>> pure_complex(a, or_real=True)
(2, 0)
>>> pure_complex(surd)
>>> pure_complex(a + b*I)
(2, 3)
>>> pure_complex(I)
(0, 1)
"""
h, t = v.as_coeff_Add()
if not t:
if or_real:
return h, t
return
c, i = t.as_coeff_Mul()
if i is S.ImaginaryUnit:
return h, c
def scaled_zero(mag, sign=1):
"""Return an mpf representing a power of two with magnitude ``mag``
and -1 for precision. Or, if ``mag`` is a scaled_zero tuple, then just
remove the sign from within the list that it was initially wrapped
in.
Examples
========
>>> from sympy.core.evalf import scaled_zero
>>> from sympy import Float
>>> z, p = scaled_zero(100)
>>> z, p
(([0], 1, 100, 1), -1)
>>> ok = scaled_zero(z)
>>> ok
(0, 1, 100, 1)
>>> Float(ok)
1.26765060022823e+30
>>> Float(ok, p)
0.e+30
>>> ok, p = scaled_zero(100, -1)
>>> Float(scaled_zero(ok), p)
-0.e+30
"""
if type(mag) is tuple and len(mag) == 4 and iszero(mag, scaled=True):
return (mag[0][0],) + mag[1:]
elif isinstance(mag, SYMPY_INTS):
if sign not in [-1, 1]:
raise ValueError('sign must be +/-1')
rv, p = mpf_shift(fone, mag), -1
s = 0 if sign == 1 else 1
rv = ([s],) + rv[1:]
return rv, p
else:
raise ValueError('scaled zero expects int or scaled_zero tuple.')
def iszero(mpf, scaled=False):
if not scaled:
return not mpf or not mpf[1] and not mpf[-1]
return mpf and type(mpf[0]) is list and mpf[1] == mpf[-1] == 1
def complex_accuracy(result):
"""
Returns relative accuracy of a complex number with given accuracies
for the real and imaginary parts. The relative accuracy is defined
in the complex norm sense as ||z|+|error|| / |z| where error
is equal to (real absolute error) + (imag absolute error)*i.
The full expression for the (logarithmic) error can be approximated
easily by using the max norm to approximate the complex norm.
In the worst case (re and im equal), this is wrong by a factor
sqrt(2), or by log2(sqrt(2)) = 0.5 bit.
"""
re, im, re_acc, im_acc = result
if not im:
if not re:
return INF
return re_acc
if not re:
return im_acc
re_size = fastlog(re)
im_size = fastlog(im)
absolute_error = max(re_size - re_acc, im_size - im_acc)
relative_error = absolute_error - max(re_size, im_size)
return -relative_error
def get_abs(expr, prec, options):
re, im, re_acc, im_acc = evalf(expr, prec + 2, options)
if not re:
re, re_acc, im, im_acc = im, im_acc, re, re_acc
if im:
if expr.is_number:
abs_expr, _, acc, _ = evalf(abs(N(expr, prec + 2)),
prec + 2, options)
return abs_expr, None, acc, None
else:
if 'subs' in options:
return libmp.mpc_abs((re, im), prec), None, re_acc, None
return abs(expr), None, prec, None
elif re:
return mpf_abs(re), None, re_acc, None
else:
return None, None, None, None
def get_complex_part(expr, no, prec, options):
"""no = 0 for real part, no = 1 for imaginary part"""
workprec = prec
i = 0
while 1:
res = evalf(expr, workprec, options)
value, accuracy = res[no::2]
# XXX is the last one correct? Consider re((1+I)**2).n()
if (not value) or accuracy >= prec or -value[2] > prec:
return value, None, accuracy, None
workprec += max(30, 2**i)
i += 1
def evalf_abs(expr, prec, options):
return get_abs(expr.args[0], prec, options)
def evalf_re(expr, prec, options):
return get_complex_part(expr.args[0], 0, prec, options)
def evalf_im(expr, prec, options):
return get_complex_part(expr.args[0], 1, prec, options)
def finalize_complex(re, im, prec):
if re == fzero and im == fzero:
raise ValueError("got complex zero with unknown accuracy")
elif re == fzero:
return None, im, None, prec
elif im == fzero:
return re, None, prec, None
size_re = fastlog(re)
size_im = fastlog(im)
if size_re > size_im:
re_acc = prec
im_acc = prec + min(-(size_re - size_im), 0)
else:
im_acc = prec
re_acc = prec + min(-(size_im - size_re), 0)
return re, im, re_acc, im_acc
def chop_parts(value, prec):
"""
Chop off tiny real or complex parts.
"""
re, im, re_acc, im_acc = value
# Method 1: chop based on absolute value
if re and re not in _infs_nan and (fastlog(re) < -prec + 4):
re, re_acc = None, None
if im and im not in _infs_nan and (fastlog(im) < -prec + 4):
im, im_acc = None, None
# Method 2: chop if inaccurate and relatively small
if re and im:
delta = fastlog(re) - fastlog(im)
if re_acc < 2 and (delta - re_acc <= -prec + 4):
re, re_acc = None, None
if im_acc < 2 and (delta - im_acc >= prec - 4):
im, im_acc = None, None
return re, im, re_acc, im_acc
def check_target(expr, result, prec):
a = complex_accuracy(result)
if a < prec:
raise PrecisionExhausted("Failed to distinguish the expression: \n\n%s\n\n"
"from zero. Try simplifying the input, using chop=True, or providing "
"a higher maxn for evalf" % (expr))
def get_integer_part(expr, no, options, return_ints=False):
"""
With no = 1, computes ceiling(expr)
With no = -1, computes floor(expr)
Note: this function either gives the exact result or signals failure.
"""
from sympy.functions.elementary.complexes import re, im
# The expression is likely less than 2^30 or so
assumed_size = 30
ire, iim, ire_acc, iim_acc = evalf(expr, assumed_size, options)
# We now know the size, so we can calculate how much extra precision
# (if any) is needed to get within the nearest integer
if ire and iim:
gap = max(fastlog(ire) - ire_acc, fastlog(iim) - iim_acc)
elif ire:
gap = fastlog(ire) - ire_acc
elif iim:
gap = fastlog(iim) - iim_acc
else:
# ... or maybe the expression was exactly zero
if return_ints:
return 0, 0
else:
return None, None, None, None
margin = 10
if gap >= -margin:
prec = margin + assumed_size + gap
ire, iim, ire_acc, iim_acc = evalf(
expr, prec, options)
else:
prec = assumed_size
# We can now easily find the nearest integer, but to find floor/ceil, we
# must also calculate whether the difference to the nearest integer is
# positive or negative (which may fail if very close).
def calc_part(re_im, nexpr):
from sympy.core.add import Add
_, _, exponent, _ = nexpr
is_int = exponent == 0
nint = int(to_int(nexpr, rnd))
if is_int:
# make sure that we had enough precision to distinguish
# between nint and the re or im part (re_im) of expr that
# was passed to calc_part
ire, iim, ire_acc, iim_acc = evalf(
re_im - nint, 10, options) # don't need much precision
assert not iim
size = -fastlog(ire) + 2 # -ve b/c ire is less than 1
if size > prec:
ire, iim, ire_acc, iim_acc = evalf(
re_im, size, options)
assert not iim
nexpr = ire
nint = int(to_int(nexpr, rnd))
_, _, new_exp, _ = ire
is_int = new_exp == 0
if not is_int:
# if there are subs and they all contain integer re/im parts
# then we can (hopefully) safely substitute them into the
# expression
s = options.get('subs', False)
if s:
doit = True
from sympy.core.compatibility import as_int
# use strict=False with as_int because we take
# 2.0 == 2
for v in s.values():
try:
as_int(v, strict=False)
except ValueError:
try:
[as_int(i, strict=False) for i in v.as_real_imag()]
continue
except (ValueError, AttributeError):
doit = False
break
if doit:
re_im = re_im.subs(s)
re_im = Add(re_im, -nint, evaluate=False)
x, _, x_acc, _ = evalf(re_im, 10, options)
try:
check_target(re_im, (x, None, x_acc, None), 3)
except PrecisionExhausted:
if not re_im.equals(0):
raise PrecisionExhausted
x = fzero
nint += int(no*(mpf_cmp(x or fzero, fzero) == no))
nint = from_int(nint)
return nint, INF
re_, im_, re_acc, im_acc = None, None, None, None
if ire:
re_, re_acc = calc_part(re(expr, evaluate=False), ire)
if iim:
im_, im_acc = calc_part(im(expr, evaluate=False), iim)
if return_ints:
return int(to_int(re_ or fzero)), int(to_int(im_ or fzero))
return re_, im_, re_acc, im_acc
def evalf_ceiling(expr, prec, options):
return get_integer_part(expr.args[0], 1, options)
def evalf_floor(expr, prec, options):
return get_integer_part(expr.args[0], -1, options)
#----------------------------------------------------------------------------#
# #
# Arithmetic operations #
# #
#----------------------------------------------------------------------------#
def add_terms(terms, prec, target_prec):
"""
Helper for evalf_add. Adds a list of (mpfval, accuracy) terms.
Returns
=======
- None, None if there are no non-zero terms;
- terms[0] if there is only 1 term;
- scaled_zero if the sum of the terms produces a zero by cancellation
e.g. mpfs representing 1 and -1 would produce a scaled zero which need
special handling since they are not actually zero and they are purposely
malformed to ensure that they can't be used in anything but accuracy
calculations;
- a tuple that is scaled to target_prec that corresponds to the
sum of the terms.
The returned mpf tuple will be normalized to target_prec; the input
prec is used to define the working precision.
XXX explain why this is needed and why one can't just loop using mpf_add
"""
terms = [t for t in terms if not iszero(t[0])]
if not terms:
return None, None
elif len(terms) == 1:
return terms[0]
# see if any argument is NaN or oo and thus warrants a special return
special = []
from sympy.core.numbers import Float
for t in terms:
arg = Float._new(t[0], 1)
if arg is S.NaN or arg.is_infinite:
special.append(arg)
if special:
from sympy.core.add import Add
rv = evalf(Add(*special), prec + 4, {})
return rv[0], rv[2]
working_prec = 2*prec
sum_man, sum_exp, absolute_error = 0, 0, MINUS_INF
for x, accuracy in terms:
sign, man, exp, bc = x
if sign:
man = -man
absolute_error = max(absolute_error, bc + exp - accuracy)
delta = exp - sum_exp
if exp >= sum_exp:
# x much larger than existing sum?
# first: quick test
if ((delta > working_prec) and
((not sum_man) or
delta - bitcount(abs(sum_man)) > working_prec)):
sum_man = man
sum_exp = exp
else:
sum_man += (man << delta)
else:
delta = -delta
# x much smaller than existing sum?
if delta - bc > working_prec:
if not sum_man:
sum_man, sum_exp = man, exp
else:
sum_man = (sum_man << delta) + man
sum_exp = exp
if not sum_man:
return scaled_zero(absolute_error)
if sum_man < 0:
sum_sign = 1
sum_man = -sum_man
else:
sum_sign = 0
sum_bc = bitcount(sum_man)
sum_accuracy = sum_exp + sum_bc - absolute_error
r = normalize(sum_sign, sum_man, sum_exp, sum_bc, target_prec,
rnd), sum_accuracy
return r
def evalf_add(v, prec, options):
res = pure_complex(v)
if res:
h, c = res
re, _, re_acc, _ = evalf(h, prec, options)
im, _, im_acc, _ = evalf(c, prec, options)
return re, im, re_acc, im_acc
oldmaxprec = options.get('maxprec', DEFAULT_MAXPREC)
i = 0
target_prec = prec
while 1:
options['maxprec'] = min(oldmaxprec, 2*prec)
terms = [evalf(arg, prec + 10, options) for arg in v.args]
re, re_acc = add_terms(
[a[0::2] for a in terms if a[0]], prec, target_prec)
im, im_acc = add_terms(
[a[1::2] for a in terms if a[1]], prec, target_prec)
acc = complex_accuracy((re, im, re_acc, im_acc))
if acc >= target_prec:
if options.get('verbose'):
print("ADD: wanted", target_prec, "accurate bits, got", re_acc, im_acc)
break
else:
if (prec - target_prec) > options['maxprec']:
break
prec = prec + max(10 + 2**i, target_prec - acc)
i += 1
if options.get('verbose'):
print("ADD: restarting with prec", prec)
options['maxprec'] = oldmaxprec
if iszero(re, scaled=True):
re = scaled_zero(re)
if iszero(im, scaled=True):
im = scaled_zero(im)
return re, im, re_acc, im_acc
def evalf_mul(v, prec, options):
res = pure_complex(v)
if res:
# the only pure complex that is a mul is h*I
_, h = res
im, _, im_acc, _ = evalf(h, prec, options)
return None, im, None, im_acc
args = list(v.args)
# see if any argument is NaN or oo and thus warrants a special return
special = []
from sympy.core.numbers import Float
for arg in args:
arg = evalf(arg, prec, options)
if arg[0] is None:
continue
arg = Float._new(arg[0], 1)
if arg is S.NaN or arg.is_infinite:
special.append(arg)
if special:
from sympy.core.mul import Mul
special = Mul(*special)
return evalf(special, prec + 4, {})
# With guard digits, multiplication in the real case does not destroy
# accuracy. This is also true in the complex case when considering the
# total accuracy; however accuracy for the real or imaginary parts
# separately may be lower.
acc = prec
# XXX: big overestimate
working_prec = prec + len(args) + 5
# Empty product is 1
start = man, exp, bc = MPZ(1), 0, 1
# First, we multiply all pure real or pure imaginary numbers.
# direction tells us that the result should be multiplied by
# I**direction; all other numbers get put into complex_factors
# to be multiplied out after the first phase.
last = len(args)
direction = 0
args.append(S.One)
complex_factors = []
for i, arg in enumerate(args):
if i != last and pure_complex(arg):
args[-1] = (args[-1]*arg).expand()
continue
elif i == last and arg is S.One:
continue
re, im, re_acc, im_acc = evalf(arg, working_prec, options)
if re and im:
complex_factors.append((re, im, re_acc, im_acc))
continue
elif re:
(s, m, e, b), w_acc = re, re_acc
elif im:
(s, m, e, b), w_acc = im, im_acc
direction += 1
else:
return None, None, None, None
direction += 2*s
man *= m
exp += e
bc += b
if bc > 3*working_prec:
man >>= working_prec
exp += working_prec
acc = min(acc, w_acc)
sign = (direction & 2) >> 1
if not complex_factors:
v = normalize(sign, man, exp, bitcount(man), prec, rnd)
# multiply by i
if direction & 1:
return None, v, None, acc
else:
return v, None, acc, None
else:
# initialize with the first term
if (man, exp, bc) != start:
# there was a real part; give it an imaginary part
re, im = (sign, man, exp, bitcount(man)), (0, MPZ(0), 0, 0)
i0 = 0
else:
# there is no real part to start (other than the starting 1)
wre, wim, wre_acc, wim_acc = complex_factors[0]
acc = min(acc,
complex_accuracy((wre, wim, wre_acc, wim_acc)))
re = wre
im = wim
i0 = 1
for wre, wim, wre_acc, wim_acc in complex_factors[i0:]:
# acc is the overall accuracy of the product; we aren't
# computing exact accuracies of the product.
acc = min(acc,
complex_accuracy((wre, wim, wre_acc, wim_acc)))
use_prec = working_prec
A = mpf_mul(re, wre, use_prec)
B = mpf_mul(mpf_neg(im), wim, use_prec)
C = mpf_mul(re, wim, use_prec)
D = mpf_mul(im, wre, use_prec)
re = mpf_add(A, B, use_prec)
im = mpf_add(C, D, use_prec)
if options.get('verbose'):
print("MUL: wanted", prec, "accurate bits, got", acc)
# multiply by I
if direction & 1:
re, im = mpf_neg(im), re
return re, im, acc, acc
def evalf_pow(v, prec, options):
target_prec = prec
base, exp = v.args
# We handle x**n separately. This has two purposes: 1) it is much
# faster, because we avoid calling evalf on the exponent, and 2) it
# allows better handling of real/imaginary parts that are exactly zero
if exp.is_Integer:
p = exp.p
# Exact
if not p:
return fone, None, prec, None
# Exponentiation by p magnifies relative error by |p|, so the
# base must be evaluated with increased precision if p is large
prec += int(math.log(abs(p), 2))
re, im, re_acc, im_acc = evalf(base, prec + 5, options)
# Real to integer power
if re and not im:
return mpf_pow_int(re, p, target_prec), None, target_prec, None
# (x*I)**n = I**n * x**n
if im and not re:
z = mpf_pow_int(im, p, target_prec)
case = p % 4
if case == 0:
return z, None, target_prec, None
if case == 1:
return None, z, None, target_prec
if case == 2:
return mpf_neg(z), None, target_prec, None
if case == 3:
return None, mpf_neg(z), None, target_prec
# Zero raised to an integer power
if not re:
return None, None, None, None
# General complex number to arbitrary integer power
re, im = libmp.mpc_pow_int((re, im), p, prec)
# Assumes full accuracy in input
return finalize_complex(re, im, target_prec)
# Pure square root
if exp is S.Half:
xre, xim, _, _ = evalf(base, prec + 5, options)
# General complex square root
if xim:
re, im = libmp.mpc_sqrt((xre or fzero, xim), prec)
return finalize_complex(re, im, prec)
if not xre:
return None, None, None, None
# Square root of a negative real number
if mpf_lt(xre, fzero):
return None, mpf_sqrt(mpf_neg(xre), prec), None, prec
# Positive square root
return mpf_sqrt(xre, prec), None, prec, None
# We first evaluate the exponent to find its magnitude
# This determines the working precision that must be used
prec += 10
yre, yim, _, _ = evalf(exp, prec, options)
# Special cases: x**0
if not (yre or yim):
return fone, None, prec, None
ysize = fastlog(yre)
# Restart if too big
# XXX: prec + ysize might exceed maxprec
if ysize > 5:
prec += ysize
yre, yim, _, _ = evalf(exp, prec, options)
# Pure exponential function; no need to evalf the base
if base is S.Exp1:
if yim:
re, im = libmp.mpc_exp((yre or fzero, yim), prec)
return finalize_complex(re, im, target_prec)
return mpf_exp(yre, target_prec), None, target_prec, None
xre, xim, _, _ = evalf(base, prec + 5, options)
# 0**y
if not (xre or xim):
return None, None, None, None
# (real ** complex) or (complex ** complex)
if yim:
re, im = libmp.mpc_pow(
(xre or fzero, xim or fzero), (yre or fzero, yim),
target_prec)
return finalize_complex(re, im, target_prec)
# complex ** real
if xim:
re, im = libmp.mpc_pow_mpf((xre or fzero, xim), yre, target_prec)
return finalize_complex(re, im, target_prec)
# negative ** real
elif mpf_lt(xre, fzero):
re, im = libmp.mpc_pow_mpf((xre, fzero), yre, target_prec)
return finalize_complex(re, im, target_prec)
# positive ** real
else:
return mpf_pow(xre, yre, target_prec), None, target_prec, None
#----------------------------------------------------------------------------#
# #
# Special functions #
# #
#----------------------------------------------------------------------------#
def evalf_trig(v, prec, options):
"""
This function handles sin and cos of complex arguments.
TODO: should also handle tan of complex arguments.
"""
from sympy import cos, sin
if isinstance(v, cos):
func = mpf_cos
elif isinstance(v, sin):
func = mpf_sin
else:
raise NotImplementedError
arg = v.args[0]
# 20 extra bits is possibly overkill. It does make the need
# to restart very unlikely
xprec = prec + 20
re, im, re_acc, im_acc = evalf(arg, xprec, options)
if im:
if 'subs' in options:
v = v.subs(options['subs'])
return evalf(v._eval_evalf(prec), prec, options)
if not re:
if isinstance(v, cos):
return fone, None, prec, None
elif isinstance(v, sin):
return None, None, None, None
else:
raise NotImplementedError
# For trigonometric functions, we are interested in the
# fixed-point (absolute) accuracy of the argument.
xsize = fastlog(re)
# Magnitude <= 1.0. OK to compute directly, because there is no
# danger of hitting the first root of cos (with sin, magnitude
# <= 2.0 would actually be ok)
if xsize < 1:
return func(re, prec, rnd), None, prec, None
# Very large
if xsize >= 10:
xprec = prec + xsize
re, im, re_acc, im_acc = evalf(arg, xprec, options)
# Need to repeat in case the argument is very close to a
# multiple of pi (or pi/2), hitting close to a root
while 1:
y = func(re, prec, rnd)
ysize = fastlog(y)
gap = -ysize
accuracy = (xprec - xsize) - gap
if accuracy < prec:
if options.get('verbose'):
print("SIN/COS", accuracy, "wanted", prec, "gap", gap)
print(to_str(y, 10))
if xprec > options.get('maxprec', DEFAULT_MAXPREC):
return y, None, accuracy, None
xprec += gap
re, im, re_acc, im_acc = evalf(arg, xprec, options)
continue
else:
return y, None, prec, None
def evalf_log(expr, prec, options):
from sympy import Abs, Add, log
if len(expr.args)>1:
expr = expr.doit()
return evalf(expr, prec, options)
arg = expr.args[0]
workprec = prec + 10
xre, xim, xacc, _ = evalf(arg, workprec, options)
# evalf can return NoneTypes if chop=True
# issue 18516, 19623
if xre is xim is None:
# Dear reviewer, I do not know what -inf is;
# it looks to be (1, 0, -789, -3)
# but I'm not sure in general,
# so we just let mpmath figure
# it out by taking log of 0 directly.
# It would be better to return -inf instead.
xre = fzero
if xim:
# XXX: use get_abs etc instead
re = evalf_log(
log(Abs(arg, evaluate=False), evaluate=False), prec, options)
im = mpf_atan2(xim, xre or fzero, prec)
return re[0], im, re[2], prec
imaginary_term = (mpf_cmp(xre, fzero) < 0)
re = mpf_log(mpf_abs(xre), prec, rnd)
size = fastlog(re)
if prec - size > workprec and re != fzero:
# We actually need to compute 1+x accurately, not x
arg = Add(S.NegativeOne, arg, evaluate=False)
xre, xim, _, _ = evalf_add(arg, prec, options)
prec2 = workprec - fastlog(xre)
# xre is now x - 1 so we add 1 back here to calculate x
re = mpf_log(mpf_abs(mpf_add(xre, fone, prec2)), prec, rnd)
re_acc = prec
if imaginary_term:
return re, mpf_pi(prec), re_acc, prec
else:
return re, None, re_acc, None
def evalf_atan(v, prec, options):
arg = v.args[0]
xre, xim, reacc, imacc = evalf(arg, prec + 5, options)
if xre is xim is None:
return (None,)*4
if xim:
raise NotImplementedError
return mpf_atan(xre, prec, rnd), None, prec, None
def evalf_subs(prec, subs):
""" Change all Float entries in `subs` to have precision prec. """
newsubs = {}
for a, b in subs.items():
b = S(b)
if b.is_Float:
b = b._eval_evalf(prec)
newsubs[a] = b
return newsubs
def evalf_piecewise(expr, prec, options):
from sympy import Float, Integer
if 'subs' in options:
expr = expr.subs(evalf_subs(prec, options['subs']))
newopts = options.copy()
del newopts['subs']
if hasattr(expr, 'func'):
return evalf(expr, prec, newopts)
if type(expr) == float:
return evalf(Float(expr), prec, newopts)
if type(expr) == int:
return evalf(Integer(expr), prec, newopts)
# We still have undefined symbols
raise NotImplementedError
def evalf_bernoulli(expr, prec, options):
arg = expr.args[0]
if not arg.is_Integer:
raise ValueError("Bernoulli number index must be an integer")
n = int(arg)
b = mpf_bernoulli(n, prec, rnd)
if b == fzero:
return None, None, None, None
return b, None, prec, None
#----------------------------------------------------------------------------#
# #
# High-level operations #
# #
#----------------------------------------------------------------------------#
def as_mpmath(x, prec, options):
from sympy.core.numbers import Infinity, NegativeInfinity, Zero
x = sympify(x)
if isinstance(x, Zero) or x == 0:
return mpf(0)
if isinstance(x, Infinity):
return mpf('inf')
if isinstance(x, NegativeInfinity):
return mpf('-inf')
# XXX
re, im, _, _ = evalf(x, prec, options)
if im:
return mpc(re or fzero, im)
return mpf(re)
def do_integral(expr, prec, options):
func = expr.args[0]
x, xlow, xhigh = expr.args[1]
if xlow == xhigh:
xlow = xhigh = 0
elif x not in func.free_symbols:
# only the difference in limits matters in this case
# so if there is a symbol in common that will cancel
# out when taking the difference, then use that
# difference
if xhigh.free_symbols & xlow.free_symbols:
diff = xhigh - xlow
if diff.is_number:
xlow, xhigh = 0, diff
oldmaxprec = options.get('maxprec', DEFAULT_MAXPREC)
options['maxprec'] = min(oldmaxprec, 2*prec)
with workprec(prec + 5):
xlow = as_mpmath(xlow, prec + 15, options)
xhigh = as_mpmath(xhigh, prec + 15, options)
# Integration is like summation, and we can phone home from
# the integrand function to update accuracy summation style
# Note that this accuracy is inaccurate, since it fails
# to account for the variable quadrature weights,
# but it is better than nothing
from sympy import cos, sin, Wild
have_part = [False, False]
max_real_term = [MINUS_INF]
max_imag_term = [MINUS_INF]
def f(t):
re, im, re_acc, im_acc = evalf(func, mp.prec, {'subs': {x: t}})
have_part[0] = re or have_part[0]
have_part[1] = im or have_part[1]
max_real_term[0] = max(max_real_term[0], fastlog(re))
max_imag_term[0] = max(max_imag_term[0], fastlog(im))
if im:
return mpc(re or fzero, im)
return mpf(re or fzero)
if options.get('quad') == 'osc':
A = Wild('A', exclude=[x])
B = Wild('B', exclude=[x])
D = Wild('D')
m = func.match(cos(A*x + B)*D)
if not m:
m = func.match(sin(A*x + B)*D)
if not m:
raise ValueError("An integrand of the form sin(A*x+B)*f(x) "
"or cos(A*x+B)*f(x) is required for oscillatory quadrature")
period = as_mpmath(2*S.Pi/m[A], prec + 15, options)
result = quadosc(f, [xlow, xhigh], period=period)
# XXX: quadosc does not do error detection yet
quadrature_error = MINUS_INF
else:
result, quadrature_error = quadts(f, [xlow, xhigh], error=1)
quadrature_error = fastlog(quadrature_error._mpf_)
options['maxprec'] = oldmaxprec
if have_part[0]:
re = result.real._mpf_
if re == fzero:
re, re_acc = scaled_zero(
min(-prec, -max_real_term[0], -quadrature_error))
re = scaled_zero(re) # handled ok in evalf_integral
else:
re_acc = -max(max_real_term[0] - fastlog(re) -
prec, quadrature_error)
else:
re, re_acc = None, None
if have_part[1]:
im = result.imag._mpf_
if im == fzero:
im, im_acc = scaled_zero(
min(-prec, -max_imag_term[0], -quadrature_error))
im = scaled_zero(im) # handled ok in evalf_integral
else:
im_acc = -max(max_imag_term[0] - fastlog(im) -
prec, quadrature_error)
else:
im, im_acc = None, None
result = re, im, re_acc, im_acc
return result
def evalf_integral(expr, prec, options):
limits = expr.limits
if len(limits) != 1 or len(limits[0]) != 3:
raise NotImplementedError
workprec = prec
i = 0
maxprec = options.get('maxprec', INF)
while 1:
result = do_integral(expr, workprec, options)
accuracy = complex_accuracy(result)
if accuracy >= prec: # achieved desired precision
break
if workprec >= maxprec: # can't increase accuracy any more
break
if accuracy == -1:
# maybe the answer really is zero and maybe we just haven't increased
# the precision enough. So increase by doubling to not take too long
# to get to maxprec.
workprec *= 2
else:
workprec += max(prec, 2**i)
workprec = min(workprec, maxprec)
i += 1
return result
def check_convergence(numer, denom, n):
"""
Returns
=======
(h, g, p) where
-- h is:
> 0 for convergence of rate 1/factorial(n)**h
< 0 for divergence of rate factorial(n)**(-h)
= 0 for geometric or polynomial convergence or divergence
-- abs(g) is:
> 1 for geometric convergence of rate 1/h**n
< 1 for geometric divergence of rate h**n
= 1 for polynomial convergence or divergence
(g < 0 indicates an alternating series)
-- p is:
> 1 for polynomial convergence of rate 1/n**h
<= 1 for polynomial divergence of rate n**(-h)
"""
from sympy import Poly
npol = Poly(numer, n)
dpol = Poly(denom, n)
p = npol.degree()
q = dpol.degree()
rate = q - p
if rate:
return rate, None, None
constant = dpol.LC() / npol.LC()
if abs(constant) != 1:
return rate, constant, None
if npol.degree() == dpol.degree() == 0:
return rate, constant, 0
pc = npol.all_coeffs()[1]
qc = dpol.all_coeffs()[1]
return rate, constant, (qc - pc)/dpol.LC()
def hypsum(expr, n, start, prec):
"""
Sum a rapidly convergent infinite hypergeometric series with
given general term, e.g. e = hypsum(1/factorial(n), n). The
quotient between successive terms must be a quotient of integer
polynomials.
"""
from sympy import Float, hypersimp, lambdify
if prec == float('inf'):
raise NotImplementedError('does not support inf prec')
if start:
expr = expr.subs(n, n + start)
hs = hypersimp(expr, n)
if hs is None:
raise NotImplementedError("a hypergeometric series is required")
num, den = hs.as_numer_denom()
func1 = lambdify(n, num)
func2 = lambdify(n, den)
h, g, p = check_convergence(num, den, n)
if h < 0:
raise ValueError("Sum diverges like (n!)^%i" % (-h))
term = expr.subs(n, 0)
if not term.is_Rational:
raise NotImplementedError("Non rational term functionality is not implemented.")
# Direct summation if geometric or faster
if h > 0 or (h == 0 and abs(g) > 1):
term = (MPZ(term.p) << prec) // term.q
s = term
k = 1
while abs(term) > 5:
term *= MPZ(func1(k - 1))
term //= MPZ(func2(k - 1))
s += term
k += 1
return from_man_exp(s, -prec)
else:
alt = g < 0
if abs(g) < 1:
raise ValueError("Sum diverges like (%i)^n" % abs(1/g))
if p < 1 or (p == 1 and not alt):
raise ValueError("Sum diverges like n^%i" % (-p))
# We have polynomial convergence: use Richardson extrapolation
vold = None
ndig = prec_to_dps(prec)
while True:
# Need to use at least quad precision because a lot of cancellation
# might occur in the extrapolation process; we check the answer to
# make sure that the desired precision has been reached, too.
prec2 = 4*prec
term0 = (MPZ(term.p) << prec2) // term.q
def summand(k, _term=[term0]):
if k:
k = int(k)
_term[0] *= MPZ(func1(k - 1))
_term[0] //= MPZ(func2(k - 1))
return make_mpf(from_man_exp(_term[0], -prec2))
with workprec(prec):
v = nsum(summand, [0, mpmath_inf], method='richardson')
vf = Float(v, ndig)
if vold is not None and vold == vf:
break
prec += prec # double precision each time
vold = vf
return v._mpf_
def evalf_prod(expr, prec, options):
from sympy import Sum
if all((l[1] - l[2]).is_Integer for l in expr.limits):
re, im, re_acc, im_acc = evalf(expr.doit(), prec=prec, options=options)
else:
re, im, re_acc, im_acc = evalf(expr.rewrite(Sum), prec=prec, options=options)
return re, im, re_acc, im_acc
def evalf_sum(expr, prec, options):
from sympy import Float
if 'subs' in options:
expr = expr.subs(options['subs'])
func = expr.function
limits = expr.limits
if len(limits) != 1 or len(limits[0]) != 3:
raise NotImplementedError
if func.is_zero:
return None, None, prec, None
prec2 = prec + 10
try:
n, a, b = limits[0]
if b != S.Infinity or a != int(a):
raise NotImplementedError
# Use fast hypergeometric summation if possible
v = hypsum(func, n, int(a), prec2)
delta = prec - fastlog(v)
if fastlog(v) < -10:
v = hypsum(func, n, int(a), delta)
return v, None, min(prec, delta), None
except NotImplementedError:
# Euler-Maclaurin summation for general series
eps = Float(2.0)**(-prec)
for i in range(1, 5):
m = n = 2**i * prec
s, err = expr.euler_maclaurin(m=m, n=n, eps=eps,
eval_integral=False)
err = err.evalf()
if err <= eps:
break
err = fastlog(evalf(abs(err), 20, options)[0])
re, im, re_acc, im_acc = evalf(s, prec2, options)
if re_acc is None:
re_acc = -err
if im_acc is None:
im_acc = -err
return re, im, re_acc, im_acc
#----------------------------------------------------------------------------#
# #
# Symbolic interface #
# #
#----------------------------------------------------------------------------#
def evalf_symbol(x, prec, options):
val = options['subs'][x]
if isinstance(val, mpf):
if not val:
return None, None, None, None
return val._mpf_, None, prec, None
else:
if not '_cache' in options:
options['_cache'] = {}
cache = options['_cache']
cached, cached_prec = cache.get(x, (None, MINUS_INF))
if cached_prec >= prec:
return cached
v = evalf(sympify(val), prec, options)
cache[x] = (v, prec)
return v
evalf_table = None
def _create_evalf_table():
global evalf_table
from sympy.functions.combinatorial.numbers import bernoulli
from sympy.concrete.products import Product
from sympy.concrete.summations import Sum
from sympy.core.add import Add
from sympy.core.mul import Mul
from sympy.core.numbers import Exp1, Float, Half, ImaginaryUnit, Integer, NaN, NegativeOne, One, Pi, Rational, Zero
from sympy.core.power import Pow
from sympy.core.symbol import Dummy, Symbol
from sympy.functions.elementary.complexes import Abs, im, re
from sympy.functions.elementary.exponential import exp, log
from sympy.functions.elementary.integers import ceiling, floor
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import atan, cos, sin
from sympy.integrals.integrals import Integral
evalf_table = {
Symbol: evalf_symbol,
Dummy: evalf_symbol,
Float: lambda x, prec, options: (x._mpf_, None, prec, None),
Rational: lambda x, prec, options: (from_rational(x.p, x.q, prec), None, prec, None),
Integer: lambda x, prec, options: (from_int(x.p, prec), None, prec, None),
Zero: lambda x, prec, options: (None, None, prec, None),
One: lambda x, prec, options: (fone, None, prec, None),
Half: lambda x, prec, options: (fhalf, None, prec, None),
Pi: lambda x, prec, options: (mpf_pi(prec), None, prec, None),
Exp1: lambda x, prec, options: (mpf_e(prec), None, prec, None),
ImaginaryUnit: lambda x, prec, options: (None, fone, None, prec),
NegativeOne: lambda x, prec, options: (fnone, None, prec, None),
NaN: lambda x, prec, options: (fnan, None, prec, None),
exp: lambda x, prec, options: evalf_pow(
Pow(S.Exp1, x.exp, evaluate=False), prec, options),
cos: evalf_trig,
sin: evalf_trig,
Add: evalf_add,
Mul: evalf_mul,
Pow: evalf_pow,
log: evalf_log,
atan: evalf_atan,
Abs: evalf_abs,
re: evalf_re,
im: evalf_im,
floor: evalf_floor,
ceiling: evalf_ceiling,
Integral: evalf_integral,
Sum: evalf_sum,
Product: evalf_prod,
Piecewise: evalf_piecewise,
bernoulli: evalf_bernoulli,
}
def evalf(x, prec, options):
"""
Evaluate the ``Basic`` instance, ``x``
to a binary precision of ``prec``. This
function is supposed to be used internally.
Parameters
==========
x : Basic
The formula to evaluate to a float.
prec : int
The binary precision that the output should have.
options : dict
A dictionary with the same entries as
``EvalfMixin.evalf`` and in addition,
``maxprec`` which is the maximum working precision.
Returns
=======
An optional tuple, ``(re, im, re_acc, im_acc)``
which are the real, imaginary, real accuracy
and imaginary accuracy respectively. ``re`` is
an mpf value tuple and so is ``im``. ``re_acc``
and ``im_acc`` are ints.
NB: all these return values can be ``None``.
If all values are ``None``, then that represents 0.
Note that 0 is also represented as ``fzero = (0, 0, 0, 0)``.
"""
from sympy import re as re_, im as im_
try:
rf = evalf_table[x.func]
r = rf(x, prec, options)
except KeyError:
# Fall back to ordinary evalf if possible
if 'subs' in options:
x = x.subs(evalf_subs(prec, options['subs']))
xe = x._eval_evalf(prec)
if xe is None:
raise NotImplementedError
as_real_imag = getattr(xe, "as_real_imag", None)
if as_real_imag is None:
raise NotImplementedError # e.g. FiniteSet(-1.0, 1.0).evalf()
re, im = as_real_imag()
if re.has(re_) or im.has(im_):
raise NotImplementedError
if re == 0:
re = None
reprec = None
elif re.is_number:
re = re._to_mpmath(prec, allow_ints=False)._mpf_
reprec = prec
else:
raise NotImplementedError
if im == 0:
im = None
imprec = None
elif im.is_number:
im = im._to_mpmath(prec, allow_ints=False)._mpf_
imprec = prec
else:
raise NotImplementedError
r = re, im, reprec, imprec
if options.get("verbose"):
print("### input", x)
print("### output", to_str(r[0] or fzero, 50))
print("### raw", r) # r[0], r[2]
print()
chop = options.get('chop', False)
if chop:
if chop is True:
chop_prec = prec
else:
# convert (approximately) from given tolerance;
# the formula here will will make 1e-i rounds to 0 for
# i in the range +/-27 while 2e-i will not be chopped
chop_prec = int(round(-3.321*math.log10(chop) + 2.5))
if chop_prec == 3:
chop_prec -= 1
r = chop_parts(r, chop_prec)
if options.get("strict"):
check_target(x, r, prec)
return r
class EvalfMixin:
"""Mixin class adding evalf capabililty."""
__slots__ = () # type: Tuple[str, ...]
def evalf(self, n=15, subs=None, maxn=100, chop=False, strict=False, quad=None, verbose=False):
"""
Evaluate the given formula to an accuracy of *n* digits.
Parameters
==========
subs : dict, optional
Substitute numerical values for symbols, e.g.
``subs={x:3, y:1+pi}``. The substitutions must be given as a
dictionary.
maxn : int, optional
Allow a maximum temporary working precision of maxn digits.
chop : bool or number, optional
Specifies how to replace tiny real or imaginary parts in
subresults by exact zeros.
When ``True`` the chop value defaults to standard precision.
Otherwise the chop value is used to determine the
magnitude of "small" for purposes of chopping.
>>> from sympy import N
>>> x = 1e-4
>>> N(x, chop=True)
0.000100000000000000
>>> N(x, chop=1e-5)
0.000100000000000000
>>> N(x, chop=1e-4)
0
strict : bool, optional
Raise ``PrecisionExhausted`` if any subresult fails to
evaluate to full accuracy, given the available maxprec.
quad : str, optional
Choose algorithm for numerical quadrature. By default,
tanh-sinh quadrature is used. For oscillatory
integrals on an infinite interval, try ``quad='osc'``.
verbose : bool, optional
Print debug information.
Notes
=====
When Floats are naively substituted into an expression,
precision errors may adversely affect the result. For example,
adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is
then subtracted, the result will be 0.
That is exactly what happens in the following:
>>> from sympy.abc import x, y, z
>>> values = {x: 1e16, y: 1, z: 1e16}
>>> (x + y - z).subs(values)
0
Using the subs argument for evalf is the accurate way to
evaluate such an expression:
>>> (x + y - z).evalf(subs=values)
1.00000000000000
"""
from sympy import Float, Number
n = n if n is not None else 15
if subs and is_sequence(subs):
raise TypeError('subs must be given as a dictionary')
# for sake of sage that doesn't like evalf(1)
if n == 1 and isinstance(self, Number):
from sympy.core.expr import _mag
rv = self.evalf(2, subs, maxn, chop, strict, quad, verbose)
m = _mag(rv)
rv = rv.round(1 - m)
return rv
if not evalf_table:
_create_evalf_table()
prec = dps_to_prec(n)
options = {'maxprec': max(prec, int(maxn*LG10)), 'chop': chop,
'strict': strict, 'verbose': verbose}
if subs is not None:
options['subs'] = subs
if quad is not None:
options['quad'] = quad
try:
result = evalf(self, prec + 4, options)
except NotImplementedError:
# Fall back to the ordinary evalf
if hasattr(self, 'subs') and subs is not None: # issue 20291
v = self.subs(subs)._eval_evalf(prec)
else:
v = self._eval_evalf(prec)
if v is None:
return self
elif not v.is_number:
return v
try:
# If the result is numerical, normalize it
result = evalf(v, prec, options)
except NotImplementedError:
# Probably contains symbols or unknown functions
return v
re, im, re_acc, im_acc = result
if re:
p = max(min(prec, re_acc), 1)
re = Float._new(re, p)
else:
re = S.Zero
if im:
p = max(min(prec, im_acc), 1)
im = Float._new(im, p)
return re + im*S.ImaginaryUnit
else:
return re
n = evalf
def _evalf(self, prec):
"""Helper for evalf. Does the same thing but takes binary precision"""
r = self._eval_evalf(prec)
if r is None:
r = self
return r
def _eval_evalf(self, prec):
return
def _to_mpmath(self, prec, allow_ints=True):
# mpmath functions accept ints as input
errmsg = "cannot convert to mpmath number"
if allow_ints and self.is_Integer:
return self.p
if hasattr(self, '_as_mpf_val'):
return make_mpf(self._as_mpf_val(prec))
try:
re, im, _, _ = evalf(self, prec, {})
if im:
if not re:
re = fzero
return make_mpc((re, im))
elif re:
return make_mpf(re)
else:
return make_mpf(fzero)
except NotImplementedError:
v = self._eval_evalf(prec)
if v is None:
raise ValueError(errmsg)
if v.is_Float:
return make_mpf(v._mpf_)
# Number + Number*I is also fine
re, im = v.as_real_imag()
if allow_ints and re.is_Integer:
re = from_int(re.p)
elif re.is_Float:
re = re._mpf_
else:
raise ValueError(errmsg)
if allow_ints and im.is_Integer:
im = from_int(im.p)
elif im.is_Float:
im = im._mpf_
else:
raise ValueError(errmsg)
return make_mpc((re, im))
def N(x, n=15, **options):
r"""
Calls x.evalf(n, \*\*options).
Explanations
============
Both .n() and N() are equivalent to .evalf(); use the one that you like better.
See also the docstring of .evalf() for information on the options.
Examples
========
>>> from sympy import Sum, oo, N
>>> from sympy.abc import k
>>> Sum(1/k**k, (k, 1, oo))
Sum(k**(-k), (k, 1, oo))
>>> N(_, 4)
1.291
"""
# by using rational=True, any evaluation of a string
# will be done using exact values for the Floats
return sympify(x, rational=True).evalf(n, **options)
|
4a69209f5a1ebdb8d34f711d50ebef26b5dd9310ccadc50a9eb0a19af9623d34 | from collections import defaultdict
from functools import cmp_to_key, reduce
import operator
from .sympify import sympify
from .basic import Basic
from .singleton import S
from .operations import AssocOp, AssocOpDispatcher
from .cache import cacheit
from .logic import fuzzy_not, _fuzzy_group
from .expr import Expr
from .parameters import global_parameters
from .kind import KindDispatcher
# internal marker to indicate:
# "there are still non-commutative objects -- don't forget to process them"
class NC_Marker:
is_Order = False
is_Mul = False
is_Number = False
is_Poly = False
is_commutative = False
# Key for sorting commutative args in canonical order
_args_sortkey = cmp_to_key(Basic.compare)
def _mulsort(args):
# in-place sorting of args
args.sort(key=_args_sortkey)
def _unevaluated_Mul(*args):
"""Return a well-formed unevaluated Mul: Numbers are collected and
put in slot 0, any arguments that are Muls will be flattened, and args
are sorted. Use this when args have changed but you still want to return
an unevaluated Mul.
Examples
========
>>> from sympy.core.mul import _unevaluated_Mul as uMul
>>> from sympy import S, sqrt, Mul
>>> from sympy.abc import x
>>> a = uMul(*[S(3.0), x, S(2)])
>>> a.args[0]
6.00000000000000
>>> a.args[1]
x
Two unevaluated Muls with the same arguments will
always compare as equal during testing:
>>> m = uMul(sqrt(2), sqrt(3))
>>> m == uMul(sqrt(3), sqrt(2))
True
>>> u = Mul(sqrt(3), sqrt(2), evaluate=False)
>>> m == uMul(u)
True
>>> m == Mul(*m.args)
False
"""
args = list(args)
newargs = []
ncargs = []
co = S.One
while args:
a = args.pop()
if a.is_Mul:
c, nc = a.args_cnc()
args.extend(c)
if nc:
ncargs.append(Mul._from_args(nc))
elif a.is_Number:
co *= a
else:
newargs.append(a)
_mulsort(newargs)
if co is not S.One:
newargs.insert(0, co)
if ncargs:
newargs.append(Mul._from_args(ncargs))
return Mul._from_args(newargs)
class Mul(Expr, AssocOp):
"""
Expression representing multiplication operation for algebraic field.
Every argument of ``Mul()`` must be ``Expr``. Infix operator ``*``
on most scalar objects in SymPy calls this class.
Another use of ``Mul()`` is to represent the structure of abstract
multiplication so that its arguments can be substituted to return
different class. Refer to examples section for this.
``Mul()`` evaluates the argument unless ``evaluate=False`` is passed.
The evaluation logic includes:
1. Flattening
``Mul(x, Mul(y, z))`` -> ``Mul(x, y, z)``
2. Identity removing
``Mul(x, 1, y)`` -> ``Mul(x, y)``
3. Exponent collecting by ``.as_base_exp()``
``Mul(x, x**2)`` -> ``Pow(x, 3)``
4. Term sorting
``Mul(y, x, 2)`` -> ``Mul(2, x, y)``
Since multiplication can be vector space operation, arguments may
have the different :obj:`sympy.core.kind.Kind()`. Kind of the
resulting object is automatically inferred.
Examples
========
>>> from sympy import Mul
>>> from sympy.abc import x, y
>>> Mul(x, 1)
x
>>> Mul(x, x)
x**2
If ``evaluate=False`` is passed, result is not evaluated.
>>> Mul(1, 2, evaluate=False)
1*2
>>> Mul(x, x, evaluate=False)
x*x
``Mul()`` also represents the general structure of multiplication
operation.
>>> from sympy import MatrixSymbol
>>> A = MatrixSymbol('A', 2,2)
>>> expr = Mul(x,y).subs({y:A})
>>> expr
x*A
>>> type(expr)
<class 'sympy.matrices.expressions.matmul.MatMul'>
See Also
========
MatMul
"""
__slots__ = ()
is_Mul = True
_args_type = Expr
_kind_dispatcher = KindDispatcher("Mul_kind_dispatcher", commutative=True)
@property
def kind(self):
arg_kinds = (a.kind for a in self.args)
return self._kind_dispatcher(*arg_kinds)
def __neg__(self):
c, args = self.as_coeff_mul()
c = -c
if c is not S.One:
if args[0].is_Number:
args = list(args)
if c is S.NegativeOne:
args[0] = -args[0]
else:
args[0] *= c
else:
args = (c,) + args
return self._from_args(args, self.is_commutative)
@classmethod
def flatten(cls, seq):
"""Return commutative, noncommutative and order arguments by
combining related terms.
Notes
=====
* In an expression like ``a*b*c``, python process this through sympy
as ``Mul(Mul(a, b), c)``. This can have undesirable consequences.
- Sometimes terms are not combined as one would like:
{c.f. https://github.com/sympy/sympy/issues/4596}
>>> from sympy import Mul, sqrt
>>> from sympy.abc import x, y, z
>>> 2*(x + 1) # this is the 2-arg Mul behavior
2*x + 2
>>> y*(x + 1)*2
2*y*(x + 1)
>>> 2*(x + 1)*y # 2-arg result will be obtained first
y*(2*x + 2)
>>> Mul(2, x + 1, y) # all 3 args simultaneously processed
2*y*(x + 1)
>>> 2*((x + 1)*y) # parentheses can control this behavior
2*y*(x + 1)
Powers with compound bases may not find a single base to
combine with unless all arguments are processed at once.
Post-processing may be necessary in such cases.
{c.f. https://github.com/sympy/sympy/issues/5728}
>>> a = sqrt(x*sqrt(y))
>>> a**3
(x*sqrt(y))**(3/2)
>>> Mul(a,a,a)
(x*sqrt(y))**(3/2)
>>> a*a*a
x*sqrt(y)*sqrt(x*sqrt(y))
>>> _.subs(a.base, z).subs(z, a.base)
(x*sqrt(y))**(3/2)
- If more than two terms are being multiplied then all the
previous terms will be re-processed for each new argument.
So if each of ``a``, ``b`` and ``c`` were :class:`Mul`
expression, then ``a*b*c`` (or building up the product
with ``*=``) will process all the arguments of ``a`` and
``b`` twice: once when ``a*b`` is computed and again when
``c`` is multiplied.
Using ``Mul(a, b, c)`` will process all arguments once.
* The results of Mul are cached according to arguments, so flatten
will only be called once for ``Mul(a, b, c)``. If you can
structure a calculation so the arguments are most likely to be
repeats then this can save time in computing the answer. For
example, say you had a Mul, M, that you wished to divide by ``d[i]``
and multiply by ``n[i]`` and you suspect there are many repeats
in ``n``. It would be better to compute ``M*n[i]/d[i]`` rather
than ``M/d[i]*n[i]`` since every time n[i] is a repeat, the
product, ``M*n[i]`` will be returned without flattening -- the
cached value will be returned. If you divide by the ``d[i]``
first (and those are more unique than the ``n[i]``) then that will
create a new Mul, ``M/d[i]`` the args of which will be traversed
again when it is multiplied by ``n[i]``.
{c.f. https://github.com/sympy/sympy/issues/5706}
This consideration is moot if the cache is turned off.
NB
--
The validity of the above notes depends on the implementation
details of Mul and flatten which may change at any time. Therefore,
you should only consider them when your code is highly performance
sensitive.
Removal of 1 from the sequence is already handled by AssocOp.__new__.
"""
from sympy.calculus.util import AccumBounds
from sympy.matrices.expressions import MatrixExpr
rv = None
if len(seq) == 2:
a, b = seq
if b.is_Rational:
a, b = b, a
seq = [a, b]
assert not a is S.One
if not a.is_zero and a.is_Rational:
r, b = b.as_coeff_Mul()
if b.is_Add:
if r is not S.One: # 2-arg hack
# leave the Mul as a Mul?
ar = a*r
if ar is S.One:
arb = b
else:
arb = cls(a*r, b, evaluate=False)
rv = [arb], [], None
elif global_parameters.distribute and b.is_commutative:
r, b = b.as_coeff_Add()
bargs = [_keep_coeff(a, bi) for bi in Add.make_args(b)]
_addsort(bargs)
ar = a*r
if ar:
bargs.insert(0, ar)
bargs = [Add._from_args(bargs)]
rv = bargs, [], None
if rv:
return rv
# apply associativity, separate commutative part of seq
c_part = [] # out: commutative factors
nc_part = [] # out: non-commutative factors
nc_seq = []
coeff = S.One # standalone term
# e.g. 3 * ...
c_powers = [] # (base,exp) n
# e.g. (x,n) for x
num_exp = [] # (num-base, exp) y
# e.g. (3, y) for ... * 3 * ...
neg1e = S.Zero # exponent on -1 extracted from Number-based Pow and I
pnum_rat = {} # (num-base, Rat-exp) 1/2
# e.g. (3, 1/2) for ... * 3 * ...
order_symbols = None
# --- PART 1 ---
#
# "collect powers and coeff":
#
# o coeff
# o c_powers
# o num_exp
# o neg1e
# o pnum_rat
#
# NOTE: this is optimized for all-objects-are-commutative case
for o in seq:
# O(x)
if o.is_Order:
o, order_symbols = o.as_expr_variables(order_symbols)
# Mul([...])
if o.is_Mul:
if o.is_commutative:
seq.extend(o.args) # XXX zerocopy?
else:
# NCMul can have commutative parts as well
for q in o.args:
if q.is_commutative:
seq.append(q)
else:
nc_seq.append(q)
# append non-commutative marker, so we don't forget to
# process scheduled non-commutative objects
seq.append(NC_Marker)
continue
# 3
elif o.is_Number:
if o is S.NaN or coeff is S.ComplexInfinity and o.is_zero:
# we know for sure the result will be nan
return [S.NaN], [], None
elif coeff.is_Number or isinstance(coeff, AccumBounds): # it could be zoo
coeff *= o
if coeff is S.NaN:
# we know for sure the result will be nan
return [S.NaN], [], None
continue
elif isinstance(o, AccumBounds):
coeff = o.__mul__(coeff)
continue
elif o is S.ComplexInfinity:
if not coeff:
# 0 * zoo = NaN
return [S.NaN], [], None
coeff = S.ComplexInfinity
continue
elif o is S.ImaginaryUnit:
neg1e += S.Half
continue
elif o.is_commutative:
# e
# o = b
b, e = o.as_base_exp()
# y
# 3
if o.is_Pow:
if b.is_Number:
# get all the factors with numeric base so they can be
# combined below, but don't combine negatives unless
# the exponent is an integer
if e.is_Rational:
if e.is_Integer:
coeff *= Pow(b, e) # it is an unevaluated power
continue
elif e.is_negative: # also a sign of an unevaluated power
seq.append(Pow(b, e))
continue
elif b.is_negative:
neg1e += e
b = -b
if b is not S.One:
pnum_rat.setdefault(b, []).append(e)
continue
elif b.is_positive or e.is_integer:
num_exp.append((b, e))
continue
c_powers.append((b, e))
# NON-COMMUTATIVE
# TODO: Make non-commutative exponents not combine automatically
else:
if o is not NC_Marker:
nc_seq.append(o)
# process nc_seq (if any)
while nc_seq:
o = nc_seq.pop(0)
if not nc_part:
nc_part.append(o)
continue
# b c b+c
# try to combine last terms: a * a -> a
o1 = nc_part.pop()
b1, e1 = o1.as_base_exp()
b2, e2 = o.as_base_exp()
new_exp = e1 + e2
# Only allow powers to combine if the new exponent is
# not an Add. This allow things like a**2*b**3 == a**5
# if a.is_commutative == False, but prohibits
# a**x*a**y and x**a*x**b from combining (x,y commute).
if b1 == b2 and (not new_exp.is_Add):
o12 = b1 ** new_exp
# now o12 could be a commutative object
if o12.is_commutative:
seq.append(o12)
continue
else:
nc_seq.insert(0, o12)
else:
nc_part.append(o1)
nc_part.append(o)
# We do want a combined exponent if it would not be an Add, such as
# y 2y 3y
# x * x -> x
# We determine if two exponents have the same term by using
# as_coeff_Mul.
#
# Unfortunately, this isn't smart enough to consider combining into
# exponents that might already be adds, so things like:
# z - y y
# x * x will be left alone. This is because checking every possible
# combination can slow things down.
# gather exponents of common bases...
def _gather(c_powers):
common_b = {} # b:e
for b, e in c_powers:
co = e.as_coeff_Mul()
common_b.setdefault(b, {}).setdefault(
co[1], []).append(co[0])
for b, d in common_b.items():
for di, li in d.items():
d[di] = Add(*li)
new_c_powers = []
for b, e in common_b.items():
new_c_powers.extend([(b, c*t) for t, c in e.items()])
return new_c_powers
# in c_powers
c_powers = _gather(c_powers)
# and in num_exp
num_exp = _gather(num_exp)
# --- PART 2 ---
#
# o process collected powers (x**0 -> 1; x**1 -> x; otherwise Pow)
# o combine collected powers (2**x * 3**x -> 6**x)
# with numeric base
# ................................
# now we have:
# - coeff:
# - c_powers: (b, e)
# - num_exp: (2, e)
# - pnum_rat: {(1/3, [1/3, 2/3, 1/4])}
# 0 1
# x -> 1 x -> x
# this should only need to run twice; if it fails because
# it needs to be run more times, perhaps this should be
# changed to a "while True" loop -- the only reason it
# isn't such now is to allow a less-than-perfect result to
# be obtained rather than raising an error or entering an
# infinite loop
for i in range(2):
new_c_powers = []
changed = False
for b, e in c_powers:
if e.is_zero:
# canceling out infinities yields NaN
if (b.is_Add or b.is_Mul) and any(infty in b.args
for infty in (S.ComplexInfinity, S.Infinity,
S.NegativeInfinity)):
return [S.NaN], [], None
continue
if e is S.One:
if b.is_Number:
coeff *= b
continue
p = b
if e is not S.One:
p = Pow(b, e)
# check to make sure that the base doesn't change
# after exponentiation; to allow for unevaluated
# Pow, we only do so if b is not already a Pow
if p.is_Pow and not b.is_Pow:
bi = b
b, e = p.as_base_exp()
if b != bi:
changed = True
c_part.append(p)
new_c_powers.append((b, e))
# there might have been a change, but unless the base
# matches some other base, there is nothing to do
if changed and len({
b for b, e in new_c_powers}) != len(new_c_powers):
# start over again
c_part = []
c_powers = _gather(new_c_powers)
else:
break
# x x x
# 2 * 3 -> 6
inv_exp_dict = {} # exp:Mul(num-bases) x x
# e.g. x:6 for ... * 2 * 3 * ...
for b, e in num_exp:
inv_exp_dict.setdefault(e, []).append(b)
for e, b in inv_exp_dict.items():
inv_exp_dict[e] = cls(*b)
c_part.extend([Pow(b, e) for e, b in inv_exp_dict.items() if e])
# b, e -> e' = sum(e), b
# {(1/5, [1/3]), (1/2, [1/12, 1/4]} -> {(1/3, [1/5, 1/2])}
comb_e = {}
for b, e in pnum_rat.items():
comb_e.setdefault(Add(*e), []).append(b)
del pnum_rat
# process them, reducing exponents to values less than 1
# and updating coeff if necessary else adding them to
# num_rat for further processing
num_rat = []
for e, b in comb_e.items():
b = cls(*b)
if e.q == 1:
coeff *= Pow(b, e)
continue
if e.p > e.q:
e_i, ep = divmod(e.p, e.q)
coeff *= Pow(b, e_i)
e = Rational(ep, e.q)
num_rat.append((b, e))
del comb_e
# extract gcd of bases in num_rat
# 2**(1/3)*6**(1/4) -> 2**(1/3+1/4)*3**(1/4)
pnew = defaultdict(list)
i = 0 # steps through num_rat which may grow
while i < len(num_rat):
bi, ei = num_rat[i]
grow = []
for j in range(i + 1, len(num_rat)):
bj, ej = num_rat[j]
g = bi.gcd(bj)
if g is not S.One:
# 4**r1*6**r2 -> 2**(r1+r2) * 2**r1 * 3**r2
# this might have a gcd with something else
e = ei + ej
if e.q == 1:
coeff *= Pow(g, e)
else:
if e.p > e.q:
e_i, ep = divmod(e.p, e.q) # change e in place
coeff *= Pow(g, e_i)
e = Rational(ep, e.q)
grow.append((g, e))
# update the jth item
num_rat[j] = (bj/g, ej)
# update bi that we are checking with
bi = bi/g
if bi is S.One:
break
if bi is not S.One:
obj = Pow(bi, ei)
if obj.is_Number:
coeff *= obj
else:
# changes like sqrt(12) -> 2*sqrt(3)
for obj in Mul.make_args(obj):
if obj.is_Number:
coeff *= obj
else:
assert obj.is_Pow
bi, ei = obj.args
pnew[ei].append(bi)
num_rat.extend(grow)
i += 1
# combine bases of the new powers
for e, b in pnew.items():
pnew[e] = cls(*b)
# handle -1 and I
if neg1e:
# treat I as (-1)**(1/2) and compute -1's total exponent
p, q = neg1e.as_numer_denom()
# if the integer part is odd, extract -1
n, p = divmod(p, q)
if n % 2:
coeff = -coeff
# if it's a multiple of 1/2 extract I
if q == 2:
c_part.append(S.ImaginaryUnit)
elif p:
# see if there is any positive base this power of
# -1 can join
neg1e = Rational(p, q)
for e, b in pnew.items():
if e == neg1e and b.is_positive:
pnew[e] = -b
break
else:
# keep it separate; we've already evaluated it as
# much as possible so evaluate=False
c_part.append(Pow(S.NegativeOne, neg1e, evaluate=False))
# add all the pnew powers
c_part.extend([Pow(b, e) for e, b in pnew.items()])
# oo, -oo
if (coeff is S.Infinity) or (coeff is S.NegativeInfinity):
def _handle_for_oo(c_part, coeff_sign):
new_c_part = []
for t in c_part:
if t.is_extended_positive:
continue
if t.is_extended_negative:
coeff_sign *= -1
continue
new_c_part.append(t)
return new_c_part, coeff_sign
c_part, coeff_sign = _handle_for_oo(c_part, 1)
nc_part, coeff_sign = _handle_for_oo(nc_part, coeff_sign)
coeff *= coeff_sign
# zoo
if coeff is S.ComplexInfinity:
# zoo might be
# infinite_real + bounded_im
# bounded_real + infinite_im
# infinite_real + infinite_im
# and non-zero real or imaginary will not change that status.
c_part = [c for c in c_part if not (fuzzy_not(c.is_zero) and
c.is_extended_real is not None)]
nc_part = [c for c in nc_part if not (fuzzy_not(c.is_zero) and
c.is_extended_real is not None)]
# 0
elif coeff.is_zero:
# we know for sure the result will be 0 except the multiplicand
# is infinity or a matrix
if any(isinstance(c, MatrixExpr) for c in nc_part):
return [coeff], nc_part, order_symbols
if any(c.is_finite == False for c in c_part):
return [S.NaN], [], order_symbols
return [coeff], [], order_symbols
# check for straggling Numbers that were produced
_new = []
for i in c_part:
if i.is_Number:
coeff *= i
else:
_new.append(i)
c_part = _new
# order commutative part canonically
_mulsort(c_part)
# current code expects coeff to be always in slot-0
if coeff is not S.One:
c_part.insert(0, coeff)
# we are done
if (global_parameters.distribute and not nc_part and len(c_part) == 2 and
c_part[0].is_Number and c_part[0].is_finite and c_part[1].is_Add):
# 2*(1+a) -> 2 + 2 * a
coeff = c_part[0]
c_part = [Add(*[coeff*f for f in c_part[1].args])]
return c_part, nc_part, order_symbols
def _eval_power(self, e):
# don't break up NC terms: (A*B)**3 != A**3*B**3, it is A*B*A*B*A*B
cargs, nc = self.args_cnc(split_1=False)
if e.is_Integer:
return Mul(*[Pow(b, e, evaluate=False) for b in cargs]) * \
Pow(Mul._from_args(nc), e, evaluate=False)
if e.is_Rational and e.q == 2:
from sympy.core.power import integer_nthroot
from sympy.functions.elementary.complexes import sign
if self.is_imaginary:
a = self.as_real_imag()[1]
if a.is_Rational:
n, d = abs(a/2).as_numer_denom()
n, t = integer_nthroot(n, 2)
if t:
d, t = integer_nthroot(d, 2)
if t:
r = sympify(n)/d
return _unevaluated_Mul(r**e.p, (1 + sign(a)*S.ImaginaryUnit)**e.p)
p = Pow(self, e, evaluate=False)
if e.is_Rational or e.is_Float:
return p._eval_expand_power_base()
return p
@classmethod
def class_key(cls):
return 3, 0, cls.__name__
def _eval_evalf(self, prec):
c, m = self.as_coeff_Mul()
if c is S.NegativeOne:
if m.is_Mul:
rv = -AssocOp._eval_evalf(m, prec)
else:
mnew = m._eval_evalf(prec)
if mnew is not None:
m = mnew
rv = -m
else:
rv = AssocOp._eval_evalf(self, prec)
if rv.is_number:
return rv.expand()
return rv
@property
def _mpc_(self):
"""
Convert self to an mpmath mpc if possible
"""
from sympy.core.numbers import I, Float
im_part, imag_unit = self.as_coeff_Mul()
if not imag_unit == I:
# ValueError may seem more reasonable but since it's a @property,
# we need to use AttributeError to keep from confusing things like
# hasattr.
raise AttributeError("Cannot convert Mul to mpc. Must be of the form Number*I")
return (Float(0)._mpf_, Float(im_part)._mpf_)
@cacheit
def as_two_terms(self):
"""Return head and tail of self.
This is the most efficient way to get the head and tail of an
expression.
- if you want only the head, use self.args[0];
- if you want to process the arguments of the tail then use
self.as_coef_mul() which gives the head and a tuple containing
the arguments of the tail when treated as a Mul.
- if you want the coefficient when self is treated as an Add
then use self.as_coeff_add()[0]
Examples
========
>>> from sympy.abc import x, y
>>> (3*x*y).as_two_terms()
(3, x*y)
"""
args = self.args
if len(args) == 1:
return S.One, self
elif len(args) == 2:
return args
else:
return args[0], self._new_rawargs(*args[1:])
@cacheit
def as_coefficients_dict(self):
"""Return a dictionary mapping terms to their coefficient.
Since the dictionary is a defaultdict, inquiries about terms which
were not present will return a coefficient of 0. The dictionary
is considered to have a single term.
Examples
========
>>> from sympy.abc import a, x
>>> (3*a*x).as_coefficients_dict()
{a*x: 3}
>>> _[a]
0
"""
d = defaultdict(int)
args = self.args
if len(args) == 1 or not args[0].is_Number:
d[self] = S.One
else:
d[self._new_rawargs(*args[1:])] = args[0]
return d
@cacheit
def as_coeff_mul(self, *deps, rational=True, **kwargs):
if deps:
from sympy.utilities.iterables import sift
l1, l2 = sift(self.args, lambda x: x.has(*deps), binary=True)
return self._new_rawargs(*l2), tuple(l1)
args = self.args
if args[0].is_Number:
if not rational or args[0].is_Rational:
return args[0], args[1:]
elif args[0].is_extended_negative:
return S.NegativeOne, (-args[0],) + args[1:]
return S.One, args
def as_coeff_Mul(self, rational=False):
"""
Efficiently extract the coefficient of a product.
"""
coeff, args = self.args[0], self.args[1:]
if coeff.is_Number:
if not rational or coeff.is_Rational:
if len(args) == 1:
return coeff, args[0]
else:
return coeff, self._new_rawargs(*args)
elif coeff.is_extended_negative:
return S.NegativeOne, self._new_rawargs(*((-coeff,) + args))
return S.One, self
def as_real_imag(self, deep=True, **hints):
from sympy import Abs, expand_mul, im, re
other = []
coeffr = []
coeffi = []
addterms = S.One
for a in self.args:
r, i = a.as_real_imag()
if i.is_zero:
coeffr.append(r)
elif r.is_zero:
coeffi.append(i*S.ImaginaryUnit)
elif a.is_commutative:
# search for complex conjugate pairs:
for i, x in enumerate(other):
if x == a.conjugate():
coeffr.append(Abs(x)**2)
del other[i]
break
else:
if a.is_Add:
addterms *= a
else:
other.append(a)
else:
other.append(a)
m = self.func(*other)
if hints.get('ignore') == m:
return
if len(coeffi) % 2:
imco = im(coeffi.pop(0))
# all other pairs make a real factor; they will be
# put into reco below
else:
imco = S.Zero
reco = self.func(*(coeffr + coeffi))
r, i = (reco*re(m), reco*im(m))
if addterms == 1:
if m == 1:
if imco.is_zero:
return (reco, S.Zero)
else:
return (S.Zero, reco*imco)
if imco is S.Zero:
return (r, i)
return (-imco*i, imco*r)
addre, addim = expand_mul(addterms, deep=False).as_real_imag()
if imco is S.Zero:
return (r*addre - i*addim, i*addre + r*addim)
else:
r, i = -imco*i, imco*r
return (r*addre - i*addim, r*addim + i*addre)
@staticmethod
def _expandsums(sums):
"""
Helper function for _eval_expand_mul.
sums must be a list of instances of Basic.
"""
L = len(sums)
if L == 1:
return sums[0].args
terms = []
left = Mul._expandsums(sums[:L//2])
right = Mul._expandsums(sums[L//2:])
terms = [Mul(a, b) for a in left for b in right]
added = Add(*terms)
return Add.make_args(added) # it may have collapsed down to one term
def _eval_expand_mul(self, **hints):
from sympy import fraction
# Handle things like 1/(x*(x + 1)), which are automatically converted
# to 1/x*1/(x + 1)
expr = self
n, d = fraction(expr)
if d.is_Mul:
n, d = [i._eval_expand_mul(**hints) if i.is_Mul else i
for i in (n, d)]
expr = n/d
if not expr.is_Mul:
return expr
plain, sums, rewrite = [], [], False
for factor in expr.args:
if factor.is_Add:
sums.append(factor)
rewrite = True
else:
if factor.is_commutative:
plain.append(factor)
else:
sums.append(Basic(factor)) # Wrapper
if not rewrite:
return expr
else:
plain = self.func(*plain)
if sums:
deep = hints.get("deep", False)
terms = self.func._expandsums(sums)
args = []
for term in terms:
t = self.func(plain, term)
if t.is_Mul and any(a.is_Add for a in t.args) and deep:
t = t._eval_expand_mul()
args.append(t)
return Add(*args)
else:
return plain
@cacheit
def _eval_derivative(self, s):
args = list(self.args)
terms = []
for i in range(len(args)):
d = args[i].diff(s)
if d:
# Note: reduce is used in step of Mul as Mul is unable to
# handle subtypes and operation priority:
terms.append(reduce(lambda x, y: x*y, (args[:i] + [d] + args[i + 1:]), S.One))
return Add.fromiter(terms)
@cacheit
def _eval_derivative_n_times(self, s, n):
from sympy import Integer, factorial, prod, Sum, Max
from sympy.ntheory.multinomial import multinomial_coefficients_iterator
from .function import AppliedUndef
from .symbol import Symbol, symbols, Dummy
if not isinstance(s, AppliedUndef) and not isinstance(s, Symbol):
# other types of s may not be well behaved, e.g.
# (cos(x)*sin(y)).diff([[x, y, z]])
return super()._eval_derivative_n_times(s, n)
args = self.args
m = len(args)
if isinstance(n, (int, Integer)):
# https://en.wikipedia.org/wiki/General_Leibniz_rule#More_than_two_factors
terms = []
for kvals, c in multinomial_coefficients_iterator(m, n):
p = prod([arg.diff((s, k)) for k, arg in zip(kvals, args)])
terms.append(c * p)
return Add(*terms)
kvals = symbols("k1:%i" % m, cls=Dummy)
klast = n - sum(kvals)
nfact = factorial(n)
e, l = (# better to use the multinomial?
nfact/prod(map(factorial, kvals))/factorial(klast)*\
prod([args[t].diff((s, kvals[t])) for t in range(m-1)])*\
args[-1].diff((s, Max(0, klast))),
[(k, 0, n) for k in kvals])
return Sum(e, *l)
def _eval_difference_delta(self, n, step):
from sympy.series.limitseq import difference_delta as dd
arg0 = self.args[0]
rest = Mul(*self.args[1:])
return (arg0.subs(n, n + step) * dd(rest, n, step) + dd(arg0, n, step) *
rest)
def _matches_simple(self, expr, repl_dict):
# handle (w*3).matches('x*5') -> {w: x*5/3}
coeff, terms = self.as_coeff_Mul()
terms = Mul.make_args(terms)
if len(terms) == 1:
newexpr = self.__class__._combine_inverse(expr, coeff)
return terms[0].matches(newexpr, repl_dict)
return
def matches(self, expr, repl_dict={}, old=False):
expr = sympify(expr)
repl_dict = repl_dict.copy()
if self.is_commutative and expr.is_commutative:
return self._matches_commutative(expr, repl_dict, old)
elif self.is_commutative is not expr.is_commutative:
return None
# Proceed only if both both expressions are non-commutative
c1, nc1 = self.args_cnc()
c2, nc2 = expr.args_cnc()
c1, c2 = [c or [1] for c in [c1, c2]]
# TODO: Should these be self.func?
comm_mul_self = Mul(*c1)
comm_mul_expr = Mul(*c2)
repl_dict = comm_mul_self.matches(comm_mul_expr, repl_dict, old)
# If the commutative arguments didn't match and aren't equal, then
# then the expression as a whole doesn't match
if repl_dict is None and c1 != c2:
return None
# Now match the non-commutative arguments, expanding powers to
# multiplications
nc1 = Mul._matches_expand_pows(nc1)
nc2 = Mul._matches_expand_pows(nc2)
repl_dict = Mul._matches_noncomm(nc1, nc2, repl_dict)
return repl_dict or None
@staticmethod
def _matches_expand_pows(arg_list):
new_args = []
for arg in arg_list:
if arg.is_Pow and arg.exp > 0:
new_args.extend([arg.base] * arg.exp)
else:
new_args.append(arg)
return new_args
@staticmethod
def _matches_noncomm(nodes, targets, repl_dict={}):
"""Non-commutative multiplication matcher.
`nodes` is a list of symbols within the matcher multiplication
expression, while `targets` is a list of arguments in the
multiplication expression being matched against.
"""
repl_dict = repl_dict.copy()
# List of possible future states to be considered
agenda = []
# The current matching state, storing index in nodes and targets
state = (0, 0)
node_ind, target_ind = state
# Mapping between wildcard indices and the index ranges they match
wildcard_dict = {}
repl_dict = repl_dict.copy()
while target_ind < len(targets) and node_ind < len(nodes):
node = nodes[node_ind]
if node.is_Wild:
Mul._matches_add_wildcard(wildcard_dict, state)
states_matches = Mul._matches_new_states(wildcard_dict, state,
nodes, targets)
if states_matches:
new_states, new_matches = states_matches
agenda.extend(new_states)
if new_matches:
for match in new_matches:
repl_dict[match] = new_matches[match]
if not agenda:
return None
else:
state = agenda.pop()
node_ind, target_ind = state
return repl_dict
@staticmethod
def _matches_add_wildcard(dictionary, state):
node_ind, target_ind = state
if node_ind in dictionary:
begin, end = dictionary[node_ind]
dictionary[node_ind] = (begin, target_ind)
else:
dictionary[node_ind] = (target_ind, target_ind)
@staticmethod
def _matches_new_states(dictionary, state, nodes, targets):
node_ind, target_ind = state
node = nodes[node_ind]
target = targets[target_ind]
# Don't advance at all if we've exhausted the targets but not the nodes
if target_ind >= len(targets) - 1 and node_ind < len(nodes) - 1:
return None
if node.is_Wild:
match_attempt = Mul._matches_match_wilds(dictionary, node_ind,
nodes, targets)
if match_attempt:
# If the same node has been matched before, don't return
# anything if the current match is diverging from the previous
# match
other_node_inds = Mul._matches_get_other_nodes(dictionary,
nodes, node_ind)
for ind in other_node_inds:
other_begin, other_end = dictionary[ind]
curr_begin, curr_end = dictionary[node_ind]
other_targets = targets[other_begin:other_end + 1]
current_targets = targets[curr_begin:curr_end + 1]
for curr, other in zip(current_targets, other_targets):
if curr != other:
return None
# A wildcard node can match more than one target, so only the
# target index is advanced
new_state = [(node_ind, target_ind + 1)]
# Only move on to the next node if there is one
if node_ind < len(nodes) - 1:
new_state.append((node_ind + 1, target_ind + 1))
return new_state, match_attempt
else:
# If we're not at a wildcard, then make sure we haven't exhausted
# nodes but not targets, since in this case one node can only match
# one target
if node_ind >= len(nodes) - 1 and target_ind < len(targets) - 1:
return None
match_attempt = node.matches(target)
if match_attempt:
return [(node_ind + 1, target_ind + 1)], match_attempt
elif node == target:
return [(node_ind + 1, target_ind + 1)], None
else:
return None
@staticmethod
def _matches_match_wilds(dictionary, wildcard_ind, nodes, targets):
"""Determine matches of a wildcard with sub-expression in `target`."""
wildcard = nodes[wildcard_ind]
begin, end = dictionary[wildcard_ind]
terms = targets[begin:end + 1]
# TODO: Should this be self.func?
mul = Mul(*terms) if len(terms) > 1 else terms[0]
return wildcard.matches(mul)
@staticmethod
def _matches_get_other_nodes(dictionary, nodes, node_ind):
"""Find other wildcards that may have already been matched."""
other_node_inds = []
for ind in dictionary:
if nodes[ind] == nodes[node_ind]:
other_node_inds.append(ind)
return other_node_inds
@staticmethod
def _combine_inverse(lhs, rhs):
"""
Returns lhs/rhs, but treats arguments like symbols, so things
like oo/oo return 1 (instead of a nan) and ``I`` behaves like
a symbol instead of sqrt(-1).
"""
from sympy.simplify.simplify import signsimp
from .symbol import Dummy
if lhs == rhs:
return S.One
def check(l, r):
if l.is_Float and r.is_comparable:
# if both objects are added to 0 they will share the same "normalization"
# and are more likely to compare the same. Since Add(foo, 0) will not allow
# the 0 to pass, we use __add__ directly.
return l.__add__(0) == r.evalf().__add__(0)
return False
if check(lhs, rhs) or check(rhs, lhs):
return S.One
if any(i.is_Pow or i.is_Mul for i in (lhs, rhs)):
# gruntz and limit wants a literal I to not combine
# with a power of -1
d = Dummy('I')
_i = {S.ImaginaryUnit: d}
i_ = {d: S.ImaginaryUnit}
a = lhs.xreplace(_i).as_powers_dict()
b = rhs.xreplace(_i).as_powers_dict()
blen = len(b)
for bi in tuple(b.keys()):
if bi in a:
a[bi] -= b.pop(bi)
if not a[bi]:
a.pop(bi)
if len(b) != blen:
lhs = Mul(*[k**v for k, v in a.items()]).xreplace(i_)
rhs = Mul(*[k**v for k, v in b.items()]).xreplace(i_)
return signsimp(lhs/rhs)
def as_powers_dict(self):
d = defaultdict(int)
for term in self.args:
for b, e in term.as_powers_dict().items():
d[b] += e
return d
def as_numer_denom(self):
# don't use _from_args to rebuild the numerators and denominators
# as the order is not guaranteed to be the same once they have
# been separated from each other
numers, denoms = list(zip(*[f.as_numer_denom() for f in self.args]))
return self.func(*numers), self.func(*denoms)
def as_base_exp(self):
e1 = None
bases = []
nc = 0
for m in self.args:
b, e = m.as_base_exp()
if not b.is_commutative:
nc += 1
if e1 is None:
e1 = e
elif e != e1 or nc > 1:
return self, S.One
bases.append(b)
return self.func(*bases), e1
def _eval_is_polynomial(self, syms):
return all(term._eval_is_polynomial(syms) for term in self.args)
def _eval_is_rational_function(self, syms):
return all(term._eval_is_rational_function(syms) for term in self.args)
def _eval_is_meromorphic(self, x, a):
return _fuzzy_group((arg.is_meromorphic(x, a) for arg in self.args),
quick_exit=True)
def _eval_is_algebraic_expr(self, syms):
return all(term._eval_is_algebraic_expr(syms) for term in self.args)
_eval_is_commutative = lambda self: _fuzzy_group(
a.is_commutative for a in self.args)
def _eval_is_complex(self):
comp = _fuzzy_group(a.is_complex for a in self.args)
if comp is False:
if any(a.is_infinite for a in self.args):
if any(a.is_zero is not False for a in self.args):
return None
return False
return comp
def _eval_is_finite(self):
if all(a.is_finite for a in self.args):
return True
if any(a.is_infinite for a in self.args):
if all(a.is_zero is False for a in self.args):
return False
def _eval_is_infinite(self):
if any(a.is_infinite for a in self.args):
if any(a.is_zero for a in self.args):
return S.NaN.is_infinite
if any(a.is_zero is None for a in self.args):
return None
return True
def _eval_is_rational(self):
r = _fuzzy_group((a.is_rational for a in self.args), quick_exit=True)
if r:
return r
elif r is False:
return self.is_zero
def _eval_is_algebraic(self):
r = _fuzzy_group((a.is_algebraic for a in self.args), quick_exit=True)
if r:
return r
elif r is False:
return self.is_zero
def _eval_is_zero(self):
zero = infinite = False
for a in self.args:
z = a.is_zero
if z:
if infinite:
return # 0*oo is nan and nan.is_zero is None
zero = True
else:
if not a.is_finite:
if zero:
return # 0*oo is nan and nan.is_zero is None
infinite = True
if zero is False and z is None: # trap None
zero = None
return zero
# without involving odd/even checks this code would suffice:
#_eval_is_integer = lambda self: _fuzzy_group(
# (a.is_integer for a in self.args), quick_exit=True)
def _eval_is_integer(self):
is_rational = self._eval_is_rational()
if is_rational is False:
return False
numerators = []
denominators = []
for a in self.args:
if a.is_integer:
numerators.append(a)
elif a.is_Rational:
n, d = a.as_numer_denom()
numerators.append(n)
denominators.append(d)
elif a.is_Pow:
b, e = a.as_base_exp()
if not b.is_integer or not e.is_integer: return
if e.is_negative:
denominators.append(b)
else:
# for integer b and positive integer e: a = b**e would be integer
assert not e.is_positive
# for self being rational and e equal to zero: a = b**e would be 1
assert not e.is_zero
return # sign of e unknown -> self.is_integer cannot be decided
else:
return
if not denominators:
return True
odd = lambda ints: all(i.is_odd for i in ints)
even = lambda ints: any(i.is_even for i in ints)
if odd(numerators) and even(denominators):
return False
elif even(numerators) and denominators == [2]:
return True
def _eval_is_polar(self):
has_polar = any(arg.is_polar for arg in self.args)
return has_polar and \
all(arg.is_polar or arg.is_positive for arg in self.args)
def _eval_is_extended_real(self):
return self._eval_real_imag(True)
def _eval_real_imag(self, real):
zero = False
t_not_re_im = None
for t in self.args:
if (t.is_complex or t.is_infinite) is False and t.is_extended_real is False:
return False
elif t.is_imaginary: # I
real = not real
elif t.is_extended_real: # 2
if not zero:
z = t.is_zero
if not z and zero is False:
zero = z
elif z:
if all(a.is_finite for a in self.args):
return True
return
elif t.is_extended_real is False:
# symbolic or literal like `2 + I` or symbolic imaginary
if t_not_re_im:
return # complex terms might cancel
t_not_re_im = t
elif t.is_imaginary is False: # symbolic like `2` or `2 + I`
if t_not_re_im:
return # complex terms might cancel
t_not_re_im = t
else:
return
if t_not_re_im:
if t_not_re_im.is_extended_real is False:
if real: # like 3
return zero # 3*(smthng like 2 + I or i) is not real
if t_not_re_im.is_imaginary is False: # symbolic 2 or 2 + I
if not real: # like I
return zero # I*(smthng like 2 or 2 + I) is not real
elif zero is False:
return real # can't be trumped by 0
elif real:
return real # doesn't matter what zero is
def _eval_is_imaginary(self):
z = self.is_zero
if z:
return False
if self.is_finite is False:
return False
elif z is False and self.is_finite is True:
return self._eval_real_imag(False)
def _eval_is_hermitian(self):
return self._eval_herm_antiherm(True)
def _eval_herm_antiherm(self, real):
one_nc = zero = one_neither = False
for t in self.args:
if not t.is_commutative:
if one_nc:
return
one_nc = True
if t.is_antihermitian:
real = not real
elif t.is_hermitian:
if not zero:
z = t.is_zero
if not z and zero is False:
zero = z
elif z:
if all(a.is_finite for a in self.args):
return True
return
elif t.is_hermitian is False:
if one_neither:
return
one_neither = True
else:
return
if one_neither:
if real:
return zero
elif zero is False or real:
return real
def _eval_is_antihermitian(self):
z = self.is_zero
if z:
return False
elif z is False:
return self._eval_herm_antiherm(False)
def _eval_is_irrational(self):
for t in self.args:
a = t.is_irrational
if a:
others = list(self.args)
others.remove(t)
if all((x.is_rational and fuzzy_not(x.is_zero)) is True for x in others):
return True
return
if a is None:
return
if all(x.is_real for x in self.args):
return False
def _eval_is_extended_positive(self):
"""Return True if self is positive, False if not, and None if it
cannot be determined.
Explanation
===========
This algorithm is non-recursive and works by keeping track of the
sign which changes when a negative or nonpositive is encountered.
Whether a nonpositive or nonnegative is seen is also tracked since
the presence of these makes it impossible to return True, but
possible to return False if the end result is nonpositive. e.g.
pos * neg * nonpositive -> pos or zero -> None is returned
pos * neg * nonnegative -> neg or zero -> False is returned
"""
return self._eval_pos_neg(1)
def _eval_pos_neg(self, sign):
saw_NON = saw_NOT = False
for t in self.args:
if t.is_extended_positive:
continue
elif t.is_extended_negative:
sign = -sign
elif t.is_zero:
if all(a.is_finite for a in self.args):
return False
return
elif t.is_extended_nonpositive:
sign = -sign
saw_NON = True
elif t.is_extended_nonnegative:
saw_NON = True
# FIXME: is_positive/is_negative is False doesn't take account of
# Symbol('x', infinite=True, extended_real=True) which has
# e.g. is_positive is False but has uncertain sign.
elif t.is_positive is False:
sign = -sign
if saw_NOT:
return
saw_NOT = True
elif t.is_negative is False:
if saw_NOT:
return
saw_NOT = True
else:
return
if sign == 1 and saw_NON is False and saw_NOT is False:
return True
if sign < 0:
return False
def _eval_is_extended_negative(self):
return self._eval_pos_neg(-1)
def _eval_is_odd(self):
is_integer = self.is_integer
if is_integer:
r, acc = True, 1
for t in self.args:
if not t.is_integer:
return None
elif t.is_even:
r = False
elif t.is_integer:
if r is False:
pass
elif acc != 1 and (acc + t).is_odd:
r = False
elif t.is_odd is None:
r = None
acc = t
return r
# !integer -> !odd
elif is_integer is False:
return False
def _eval_is_even(self):
is_integer = self.is_integer
if is_integer:
return fuzzy_not(self.is_odd)
elif is_integer is False:
return False
def _eval_is_composite(self):
"""
Here we count the number of arguments that have a minimum value
greater than two.
If there are more than one of such a symbol then the result is composite.
Else, the result cannot be determined.
"""
number_of_args = 0 # count of symbols with minimum value greater than one
for arg in self.args:
if not (arg.is_integer and arg.is_positive):
return None
if (arg-1).is_positive:
number_of_args += 1
if number_of_args > 1:
return True
def _eval_subs(self, old, new):
from sympy.functions.elementary.complexes import sign
from sympy.ntheory.factor_ import multiplicity
from sympy.simplify.powsimp import powdenest
from sympy.simplify.radsimp import fraction
if not old.is_Mul:
return None
# try keep replacement literal so -2*x doesn't replace 4*x
if old.args[0].is_Number and old.args[0] < 0:
if self.args[0].is_Number:
if self.args[0] < 0:
return self._subs(-old, -new)
return None
def base_exp(a):
# if I and -1 are in a Mul, they get both end up with
# a -1 base (see issue 6421); all we want here are the
# true Pow or exp separated into base and exponent
from sympy import exp
if a.is_Pow or isinstance(a, exp):
return a.as_base_exp()
return a, S.One
def breakup(eq):
"""break up powers of eq when treated as a Mul:
b**(Rational*e) -> b**e, Rational
commutatives come back as a dictionary {b**e: Rational}
noncommutatives come back as a list [(b**e, Rational)]
"""
(c, nc) = (defaultdict(int), list())
for a in Mul.make_args(eq):
a = powdenest(a)
(b, e) = base_exp(a)
if e is not S.One:
(co, _) = e.as_coeff_mul()
b = Pow(b, e/co)
e = co
if a.is_commutative:
c[b] += e
else:
nc.append([b, e])
return (c, nc)
def rejoin(b, co):
"""
Put rational back with exponent; in general this is not ok, but
since we took it from the exponent for analysis, it's ok to put
it back.
"""
(b, e) = base_exp(b)
return Pow(b, e*co)
def ndiv(a, b):
"""if b divides a in an extractive way (like 1/4 divides 1/2
but not vice versa, and 2/5 does not divide 1/3) then return
the integer number of times it divides, else return 0.
"""
if not b.q % a.q or not a.q % b.q:
return int(a/b)
return 0
# give Muls in the denominator a chance to be changed (see issue 5651)
# rv will be the default return value
rv = None
n, d = fraction(self)
self2 = self
if d is not S.One:
self2 = n._subs(old, new)/d._subs(old, new)
if not self2.is_Mul:
return self2._subs(old, new)
if self2 != self:
rv = self2
# Now continue with regular substitution.
# handle the leading coefficient and use it to decide if anything
# should even be started; we always know where to find the Rational
# so it's a quick test
co_self = self2.args[0]
co_old = old.args[0]
co_xmul = None
if co_old.is_Rational and co_self.is_Rational:
# if coeffs are the same there will be no updating to do
# below after breakup() step; so skip (and keep co_xmul=None)
if co_old != co_self:
co_xmul = co_self.extract_multiplicatively(co_old)
elif co_old.is_Rational:
return rv
# break self and old into factors
(c, nc) = breakup(self2)
(old_c, old_nc) = breakup(old)
# update the coefficients if we had an extraction
# e.g. if co_self were 2*(3/35*x)**2 and co_old = 3/5
# then co_self in c is replaced by (3/5)**2 and co_residual
# is 2*(1/7)**2
if co_xmul and co_xmul.is_Rational and abs(co_old) != 1:
mult = S(multiplicity(abs(co_old), co_self))
c.pop(co_self)
if co_old in c:
c[co_old] += mult
else:
c[co_old] = mult
co_residual = co_self/co_old**mult
else:
co_residual = 1
# do quick tests to see if we can't succeed
ok = True
if len(old_nc) > len(nc):
# more non-commutative terms
ok = False
elif len(old_c) > len(c):
# more commutative terms
ok = False
elif {i[0] for i in old_nc}.difference({i[0] for i in nc}):
# unmatched non-commutative bases
ok = False
elif set(old_c).difference(set(c)):
# unmatched commutative terms
ok = False
elif any(sign(c[b]) != sign(old_c[b]) for b in old_c):
# differences in sign
ok = False
if not ok:
return rv
if not old_c:
cdid = None
else:
rat = []
for (b, old_e) in old_c.items():
c_e = c[b]
rat.append(ndiv(c_e, old_e))
if not rat[-1]:
return rv
cdid = min(rat)
if not old_nc:
ncdid = None
for i in range(len(nc)):
nc[i] = rejoin(*nc[i])
else:
ncdid = 0 # number of nc replacements we did
take = len(old_nc) # how much to look at each time
limit = cdid or S.Infinity # max number that we can take
failed = [] # failed terms will need subs if other terms pass
i = 0
while limit and i + take <= len(nc):
hit = False
# the bases must be equivalent in succession, and
# the powers must be extractively compatible on the
# first and last factor but equal in between.
rat = []
for j in range(take):
if nc[i + j][0] != old_nc[j][0]:
break
elif j == 0:
rat.append(ndiv(nc[i + j][1], old_nc[j][1]))
elif j == take - 1:
rat.append(ndiv(nc[i + j][1], old_nc[j][1]))
elif nc[i + j][1] != old_nc[j][1]:
break
else:
rat.append(1)
j += 1
else:
ndo = min(rat)
if ndo:
if take == 1:
if cdid:
ndo = min(cdid, ndo)
nc[i] = Pow(new, ndo)*rejoin(nc[i][0],
nc[i][1] - ndo*old_nc[0][1])
else:
ndo = 1
# the left residual
l = rejoin(nc[i][0], nc[i][1] - ndo*
old_nc[0][1])
# eliminate all middle terms
mid = new
# the right residual (which may be the same as the middle if take == 2)
ir = i + take - 1
r = (nc[ir][0], nc[ir][1] - ndo*
old_nc[-1][1])
if r[1]:
if i + take < len(nc):
nc[i:i + take] = [l*mid, r]
else:
r = rejoin(*r)
nc[i:i + take] = [l*mid*r]
else:
# there was nothing left on the right
nc[i:i + take] = [l*mid]
limit -= ndo
ncdid += ndo
hit = True
if not hit:
# do the subs on this failing factor
failed.append(i)
i += 1
else:
if not ncdid:
return rv
# although we didn't fail, certain nc terms may have
# failed so we rebuild them after attempting a partial
# subs on them
failed.extend(range(i, len(nc)))
for i in failed:
nc[i] = rejoin(*nc[i]).subs(old, new)
# rebuild the expression
if cdid is None:
do = ncdid
elif ncdid is None:
do = cdid
else:
do = min(ncdid, cdid)
margs = []
for b in c:
if b in old_c:
# calculate the new exponent
e = c[b] - old_c[b]*do
margs.append(rejoin(b, e))
else:
margs.append(rejoin(b.subs(old, new), c[b]))
if cdid and not ncdid:
# in case we are replacing commutative with non-commutative,
# we want the new term to come at the front just like the
# rest of this routine
margs = [Pow(new, cdid)] + margs
return co_residual*self2.func(*margs)*self2.func(*nc)
def _eval_nseries(self, x, n, logx, cdir=0):
from sympy import degree, Mul, Order, ceiling, powsimp, PolynomialError
from itertools import product
def coeff_exp(term, x):
coeff, exp = S.One, S.Zero
for factor in Mul.make_args(term):
if factor.has(x):
base, exp = factor.as_base_exp()
if base != x:
try:
return term.leadterm(x)
except ValueError:
return term, S.Zero
else:
coeff *= factor
return coeff, exp
ords = []
try:
for t in self.args:
coeff, exp = t.leadterm(x)
if not coeff.has(x):
ords.append((t, exp))
else:
raise ValueError
n0 = sum(t[1] for t in ords)
facs = []
for t, m in ords:
n1 = ceiling(n - n0 + m)
s = t.nseries(x, n=n1, logx=logx, cdir=cdir)
ns = s.getn()
if ns is not None:
if ns < n1: # less than expected
n -= n1 - ns # reduce n
facs.append(s.removeO())
except (ValueError, NotImplementedError, TypeError, AttributeError):
facs = [t.nseries(x, n=n, logx=logx, cdir=cdir) for t in self.args]
res = powsimp(self.func(*facs).expand(), combine='exp', deep=True)
if res.has(Order):
res += Order(x**n, x)
return res
res = S.Zero
ords2 = [Add.make_args(factor) for factor in facs]
for fac in product(*ords2):
ords3 = [coeff_exp(term, x) for term in fac]
coeffs, powers = zip(*ords3)
power = sum(powers)
if power < n:
res += Mul(*coeffs)*(x**power)
if self.is_polynomial(x):
try:
if degree(self, x) != degree(res, x):
res += Order(x**n, x)
except PolynomialError:
pass
else:
return res
for i in (1, 2, 3):
if (res - self).subs(x, i) is not S.Zero:
res += Order(x**n, x)
break
return res
def _eval_as_leading_term(self, x, cdir=0):
return self.func(*[t.as_leading_term(x, cdir=cdir) for t in self.args])
def _eval_conjugate(self):
return self.func(*[t.conjugate() for t in self.args])
def _eval_transpose(self):
return self.func(*[t.transpose() for t in self.args[::-1]])
def _eval_adjoint(self):
return self.func(*[t.adjoint() for t in self.args[::-1]])
def _sage_(self):
s = 1
for x in self.args:
s *= x._sage_()
return s
def as_content_primitive(self, radical=False, clear=True):
"""Return the tuple (R, self/R) where R is the positive Rational
extracted from self.
Examples
========
>>> from sympy import sqrt
>>> (-3*sqrt(2)*(2 - 2*sqrt(2))).as_content_primitive()
(6, -sqrt(2)*(1 - sqrt(2)))
See docstring of Expr.as_content_primitive for more examples.
"""
coef = S.One
args = []
for i, a in enumerate(self.args):
c, p = a.as_content_primitive(radical=radical, clear=clear)
coef *= c
if p is not S.One:
args.append(p)
# don't use self._from_args here to reconstruct args
# since there may be identical args now that should be combined
# e.g. (2+2*x)*(3+3*x) should be (6, (1 + x)**2) not (6, (1+x)*(1+x))
return coef, self.func(*args)
def as_ordered_factors(self, order=None):
"""Transform an expression into an ordered list of factors.
Examples
========
>>> from sympy import sin, cos
>>> from sympy.abc import x, y
>>> (2*x*y*sin(x)*cos(x)).as_ordered_factors()
[2, x, y, sin(x), cos(x)]
"""
cpart, ncpart = self.args_cnc()
cpart.sort(key=lambda expr: expr.sort_key(order=order))
return cpart + ncpart
@property
def _sorted_args(self):
return tuple(self.as_ordered_factors())
mul = AssocOpDispatcher('mul')
def prod(a, start=1):
"""Return product of elements of a. Start with int 1 so if only
ints are included then an int result is returned.
Examples
========
>>> from sympy import prod, S
>>> prod(range(3))
0
>>> type(_) is int
True
>>> prod([S(2), 3])
6
>>> _.is_Integer
True
You can start the product at something other than 1:
>>> prod([1, 2], 3)
6
"""
return reduce(operator.mul, a, start)
def _keep_coeff(coeff, factors, clear=True, sign=False):
"""Return ``coeff*factors`` unevaluated if necessary.
If ``clear`` is False, do not keep the coefficient as a factor
if it can be distributed on a single factor such that one or
more terms will still have integer coefficients.
If ``sign`` is True, allow a coefficient of -1 to remain factored out.
Examples
========
>>> from sympy.core.mul import _keep_coeff
>>> from sympy.abc import x, y
>>> from sympy import S
>>> _keep_coeff(S.Half, x + 2)
(x + 2)/2
>>> _keep_coeff(S.Half, x + 2, clear=False)
x/2 + 1
>>> _keep_coeff(S.Half, (x + 2)*y, clear=False)
y*(x + 2)/2
>>> _keep_coeff(S(-1), x + y)
-x - y
>>> _keep_coeff(S(-1), x + y, sign=True)
-(x + y)
"""
if not coeff.is_Number:
if factors.is_Number:
factors, coeff = coeff, factors
else:
return coeff*factors
if coeff is S.One:
return factors
elif coeff is S.NegativeOne and not sign:
return -factors
elif factors.is_Add:
if not clear and coeff.is_Rational and coeff.q != 1:
q = S(coeff.q)
for i in factors.args:
c, t = i.as_coeff_Mul()
r = c/q
if r == int(r):
return coeff*factors
return Mul(coeff, factors, evaluate=False)
elif factors.is_Mul:
margs = list(factors.args)
if margs[0].is_Number:
margs[0] *= coeff
if margs[0] == 1:
margs.pop(0)
else:
margs.insert(0, coeff)
return Mul._from_args(margs)
else:
return coeff*factors
def expand_2arg(e):
from sympy.simplify.simplify import bottom_up
def do(e):
if e.is_Mul:
c, r = e.as_coeff_Mul()
if c.is_Number and r.is_Add:
return _unevaluated_Add(*[c*ri for ri in r.args])
return e
return bottom_up(e, do)
from .numbers import Rational
from .power import Pow
from .add import Add, _addsort, _unevaluated_Add
|
9511ea934497f5f0df4cc2783fea551a30000567841024462472289c9e7fe54a | """Tools for setting up interactive sessions. """
from distutils.version import LooseVersion as V
from sympy.interactive.printing import init_printing
preexec_source = """\
from __future__ import division
from sympy import *
x, y, z, t = symbols('x y z t')
k, m, n = symbols('k m n', integer=True)
f, g, h = symbols('f g h', cls=Function)
init_printing()
"""
verbose_message = """\
These commands were executed:
%(source)s
Documentation can be found at https://docs.sympy.org/%(version)s
"""
no_ipython = """\
Couldn't locate IPython. Having IPython installed is greatly recommended.
See http://ipython.scipy.org for more details. If you use Debian/Ubuntu,
just install the 'ipython' package and start isympy again.
"""
def _make_message(ipython=True, quiet=False, source=None):
"""Create a banner for an interactive session. """
from sympy import __version__ as sympy_version
from sympy.external.gmpy import GROUND_TYPES
from sympy.utilities.misc import ARCH
from sympy import SYMPY_DEBUG
import sys
import os
if quiet:
return ""
python_version = "%d.%d.%d" % sys.version_info[:3]
if ipython:
shell_name = "IPython"
else:
shell_name = "Python"
info = ['ground types: %s' % GROUND_TYPES]
cache = os.getenv('SYMPY_USE_CACHE')
if cache is not None and cache.lower() == 'no':
info.append('cache: off')
if SYMPY_DEBUG:
info.append('debugging: on')
args = shell_name, sympy_version, python_version, ARCH, ', '.join(info)
message = "%s console for SymPy %s (Python %s-%s) (%s)\n" % args
if source is None:
source = preexec_source
_source = ""
for line in source.split('\n')[:-1]:
if not line:
_source += '\n'
else:
_source += '>>> ' + line + '\n'
doc_version = sympy_version
if 'dev' in doc_version:
doc_version = "dev"
else:
doc_version = "%s/" % doc_version
message += '\n' + verbose_message % {'source': _source,
'version': doc_version}
return message
def int_to_Integer(s):
"""
Wrap integer literals with Integer.
This is based on the decistmt example from
http://docs.python.org/library/tokenize.html.
Only integer literals are converted. Float literals are left alone.
Examples
========
>>> from __future__ import division
>>> from sympy import Integer # noqa: F401
>>> from sympy.interactive.session import int_to_Integer
>>> s = '1.2 + 1/2 - 0x12 + a1'
>>> int_to_Integer(s)
'1.2 +Integer (1 )/Integer (2 )-Integer (0x12 )+a1 '
>>> s = 'print (1/2)'
>>> int_to_Integer(s)
'print (Integer (1 )/Integer (2 ))'
>>> exec(s)
0.5
>>> exec(int_to_Integer(s))
1/2
"""
from tokenize import generate_tokens, untokenize, NUMBER, NAME, OP
from io import StringIO
def _is_int(num):
"""
Returns true if string value num (with token NUMBER) represents an integer.
"""
# XXX: Is there something in the standard library that will do this?
if '.' in num or 'j' in num.lower() or 'e' in num.lower():
return False
return True
result = []
g = generate_tokens(StringIO(s).readline) # tokenize the string
for toknum, tokval, _, _, _ in g:
if toknum == NUMBER and _is_int(tokval): # replace NUMBER tokens
result.extend([
(NAME, 'Integer'),
(OP, '('),
(NUMBER, tokval),
(OP, ')')
])
else:
result.append((toknum, tokval))
return untokenize(result)
def enable_automatic_int_sympification(shell):
"""
Allow IPython to automatically convert integer literals to Integer.
"""
import ast
old_run_cell = shell.run_cell
def my_run_cell(cell, *args, **kwargs):
try:
# Check the cell for syntax errors. This way, the syntax error
# will show the original input, not the transformed input. The
# downside here is that IPython magic like %timeit will not work
# with transformed input (but on the other hand, IPython magic
# that doesn't expect transformed input will continue to work).
ast.parse(cell)
except SyntaxError:
pass
else:
cell = int_to_Integer(cell)
old_run_cell(cell, *args, **kwargs)
shell.run_cell = my_run_cell
def enable_automatic_symbols(shell):
"""Allow IPython to automatically create symbols (``isympy -a``). """
# XXX: This should perhaps use tokenize, like int_to_Integer() above.
# This would avoid re-executing the code, which can lead to subtle
# issues. For example:
#
# In [1]: a = 1
#
# In [2]: for i in range(10):
# ...: a += 1
# ...:
#
# In [3]: a
# Out[3]: 11
#
# In [4]: a = 1
#
# In [5]: for i in range(10):
# ...: a += 1
# ...: print b
# ...:
# b
# b
# b
# b
# b
# b
# b
# b
# b
# b
#
# In [6]: a
# Out[6]: 12
#
# Note how the for loop is executed again because `b` was not defined, but `a`
# was already incremented once, so the result is that it is incremented
# multiple times.
import re
re_nameerror = re.compile(
"name '(?P<symbol>[A-Za-z_][A-Za-z0-9_]*)' is not defined")
def _handler(self, etype, value, tb, tb_offset=None):
"""Handle :exc:`NameError` exception and allow injection of missing symbols. """
if etype is NameError and tb.tb_next and not tb.tb_next.tb_next:
match = re_nameerror.match(str(value))
if match is not None:
# XXX: Make sure Symbol is in scope. Otherwise you'll get infinite recursion.
self.run_cell("%(symbol)s = Symbol('%(symbol)s')" %
{'symbol': match.group("symbol")}, store_history=False)
try:
code = self.user_ns['In'][-1]
except (KeyError, IndexError):
pass
else:
self.run_cell(code, store_history=False)
return None
finally:
self.run_cell("del %s" % match.group("symbol"),
store_history=False)
stb = self.InteractiveTB.structured_traceback(
etype, value, tb, tb_offset=tb_offset)
self._showtraceback(etype, value, stb)
shell.set_custom_exc((NameError,), _handler)
def init_ipython_session(shell=None, argv=[], auto_symbols=False, auto_int_to_Integer=False):
"""Construct new IPython session. """
import IPython
if V(IPython.__version__) >= '0.11':
if not shell:
# use an app to parse the command line, and init config
# IPython 1.0 deprecates the frontend module, so we import directly
# from the terminal module to prevent a deprecation message from being
# shown.
if V(IPython.__version__) >= '1.0':
from IPython.terminal import ipapp
else:
from IPython.frontend.terminal import ipapp
app = ipapp.TerminalIPythonApp()
# don't draw IPython banner during initialization:
app.display_banner = False
app.initialize(argv)
shell = app.shell
if auto_symbols:
enable_automatic_symbols(shell)
if auto_int_to_Integer:
enable_automatic_int_sympification(shell)
return shell
else:
from IPython.Shell import make_IPython
return make_IPython(argv)
def init_python_session():
"""Construct new Python session. """
from code import InteractiveConsole
class SymPyConsole(InteractiveConsole):
"""An interactive console with readline support. """
def __init__(self):
InteractiveConsole.__init__(self)
try:
import readline
except ImportError:
pass
else:
import os
import atexit
readline.parse_and_bind('tab: complete')
if hasattr(readline, 'read_history_file'):
history = os.path.expanduser('~/.sympy-history')
try:
readline.read_history_file(history)
except OSError:
pass
atexit.register(readline.write_history_file, history)
return SymPyConsole()
def init_session(ipython=None, pretty_print=True, order=None,
use_unicode=None, use_latex=None, quiet=False, auto_symbols=False,
auto_int_to_Integer=False, str_printer=None, pretty_printer=None,
latex_printer=None, argv=[]):
"""
Initialize an embedded IPython or Python session. The IPython session is
initiated with the --pylab option, without the numpy imports, so that
matplotlib plotting can be interactive.
Parameters
==========
pretty_print: boolean
If True, use pretty_print to stringify;
if False, use sstrrepr to stringify.
order: string or None
There are a few different settings for this parameter:
lex (default), which is lexographic order;
grlex, which is graded lexographic order;
grevlex, which is reversed graded lexographic order;
old, which is used for compatibility reasons and for long expressions;
None, which sets it to lex.
use_unicode: boolean or None
If True, use unicode characters;
if False, do not use unicode characters.
use_latex: boolean or None
If True, use latex rendering if IPython GUI's;
if False, do not use latex rendering.
quiet: boolean
If True, init_session will not print messages regarding its status;
if False, init_session will print messages regarding its status.
auto_symbols: boolean
If True, IPython will automatically create symbols for you.
If False, it will not.
The default is False.
auto_int_to_Integer: boolean
If True, IPython will automatically wrap int literals with Integer, so
that things like 1/2 give Rational(1, 2).
If False, it will not.
The default is False.
ipython: boolean or None
If True, printing will initialize for an IPython console;
if False, printing will initialize for a normal console;
The default is None, which automatically determines whether we are in
an ipython instance or not.
str_printer: function, optional, default=None
A custom string printer function. This should mimic
sympy.printing.sstrrepr().
pretty_printer: function, optional, default=None
A custom pretty printer. This should mimic sympy.printing.pretty().
latex_printer: function, optional, default=None
A custom LaTeX printer. This should mimic sympy.printing.latex()
This should mimic sympy.printing.latex().
argv: list of arguments for IPython
See sympy.bin.isympy for options that can be used to initialize IPython.
See Also
========
sympy.interactive.printing.init_printing: for examples and the rest of the parameters.
Examples
========
>>> from sympy import init_session, Symbol, sin, sqrt
>>> sin(x) #doctest: +SKIP
NameError: name 'x' is not defined
>>> init_session() #doctest: +SKIP
>>> sin(x) #doctest: +SKIP
sin(x)
>>> sqrt(5) #doctest: +SKIP
___
\\/ 5
>>> init_session(pretty_print=False) #doctest: +SKIP
>>> sqrt(5) #doctest: +SKIP
sqrt(5)
>>> y + x + y**2 + x**2 #doctest: +SKIP
x**2 + x + y**2 + y
>>> init_session(order='grlex') #doctest: +SKIP
>>> y + x + y**2 + x**2 #doctest: +SKIP
x**2 + y**2 + x + y
>>> init_session(order='grevlex') #doctest: +SKIP
>>> y * x**2 + x * y**2 #doctest: +SKIP
x**2*y + x*y**2
>>> init_session(order='old') #doctest: +SKIP
>>> x**2 + y**2 + x + y #doctest: +SKIP
x + y + x**2 + y**2
>>> theta = Symbol('theta') #doctest: +SKIP
>>> theta #doctest: +SKIP
theta
>>> init_session(use_unicode=True) #doctest: +SKIP
>>> theta # doctest: +SKIP
\u03b8
"""
import sys
in_ipython = False
if ipython is not False:
try:
import IPython
except ImportError:
if ipython is True:
raise RuntimeError("IPython is not available on this system")
ip = None
else:
try:
from IPython import get_ipython
ip = get_ipython()
except ImportError:
ip = None
in_ipython = bool(ip)
if ipython is None:
ipython = in_ipython
if ipython is False:
ip = init_python_session()
mainloop = ip.interact
else:
ip = init_ipython_session(ip, argv=argv, auto_symbols=auto_symbols,
auto_int_to_Integer=auto_int_to_Integer)
if V(IPython.__version__) >= '0.11':
# runsource is gone, use run_cell instead, which doesn't
# take a symbol arg. The second arg is `store_history`,
# and False means don't add the line to IPython's history.
ip.runsource = lambda src, symbol='exec': ip.run_cell(src, False)
#Enable interactive plotting using pylab.
try:
ip.enable_pylab(import_all=False)
except Exception:
# Causes an import error if matplotlib is not installed.
# Causes other errors (depending on the backend) if there
# is no display, or if there is some problem in the
# backend, so we have a bare "except Exception" here
pass
if not in_ipython:
mainloop = ip.mainloop
if auto_symbols and (not ipython or V(IPython.__version__) < '0.11'):
raise RuntimeError("automatic construction of symbols is possible only in IPython 0.11 or above")
if auto_int_to_Integer and (not ipython or V(IPython.__version__) < '0.11'):
raise RuntimeError("automatic int to Integer transformation is possible only in IPython 0.11 or above")
_preexec_source = preexec_source
ip.runsource(_preexec_source, symbol='exec')
init_printing(pretty_print=pretty_print, order=order,
use_unicode=use_unicode, use_latex=use_latex, ip=ip,
str_printer=str_printer, pretty_printer=pretty_printer,
latex_printer=latex_printer)
message = _make_message(ipython, quiet, _preexec_source)
if not in_ipython:
print(message)
mainloop()
sys.exit('Exiting ...')
else:
print(message)
import atexit
atexit.register(lambda: print("Exiting ...\n"))
|
c34f03d7a04e55712d0ed66d2de13ce96998df9876c5b4d699c54afc10786cfd | """OO layer for several polynomial representations. """
from sympy import oo
from sympy.core.sympify import CantSympify
from sympy.polys.polyerrors import CoercionFailed, NotReversible, NotInvertible
from sympy.polys.polyutils import PicklableWithSlots
class GenericPoly(PicklableWithSlots):
"""Base class for low-level polynomial representations. """
def ground_to_ring(f):
"""Make the ground domain a ring. """
return f.set_domain(f.dom.get_ring())
def ground_to_field(f):
"""Make the ground domain a field. """
return f.set_domain(f.dom.get_field())
def ground_to_exact(f):
"""Make the ground domain exact. """
return f.set_domain(f.dom.get_exact())
@classmethod
def _perify_factors(per, result, include):
if include:
coeff, factors = result
else:
coeff = result
factors = [ (per(g), k) for g, k in factors ]
if include:
return coeff, factors
else:
return factors
from sympy.polys.densebasic import (
dmp_validate,
dup_normal, dmp_normal,
dup_convert, dmp_convert,
dmp_from_sympy,
dup_strip,
dup_degree, dmp_degree_in,
dmp_degree_list,
dmp_negative_p,
dup_LC, dmp_ground_LC,
dup_TC, dmp_ground_TC,
dmp_ground_nth,
dmp_one, dmp_ground,
dmp_zero_p, dmp_one_p, dmp_ground_p,
dup_from_dict, dmp_from_dict,
dmp_to_dict,
dmp_deflate,
dmp_inject, dmp_eject,
dmp_terms_gcd,
dmp_list_terms, dmp_exclude,
dmp_slice_in, dmp_permute,
dmp_to_tuple,)
from sympy.polys.densearith import (
dmp_add_ground,
dmp_sub_ground,
dmp_mul_ground,
dmp_quo_ground,
dmp_exquo_ground,
dmp_abs,
dup_neg, dmp_neg,
dup_add, dmp_add,
dup_sub, dmp_sub,
dup_mul, dmp_mul,
dmp_sqr,
dup_pow, dmp_pow,
dmp_pdiv,
dmp_prem,
dmp_pquo,
dmp_pexquo,
dmp_div,
dup_rem, dmp_rem,
dmp_quo,
dmp_exquo,
dmp_add_mul, dmp_sub_mul,
dmp_max_norm,
dmp_l1_norm)
from sympy.polys.densetools import (
dmp_clear_denoms,
dmp_integrate_in,
dmp_diff_in,
dmp_eval_in,
dup_revert,
dmp_ground_trunc,
dmp_ground_content,
dmp_ground_primitive,
dmp_ground_monic,
dmp_compose,
dup_decompose,
dup_shift,
dup_transform,
dmp_lift)
from sympy.polys.euclidtools import (
dup_half_gcdex, dup_gcdex, dup_invert,
dmp_subresultants,
dmp_resultant,
dmp_discriminant,
dmp_inner_gcd,
dmp_gcd,
dmp_lcm,
dmp_cancel)
from sympy.polys.sqfreetools import (
dup_gff_list,
dmp_norm,
dmp_sqf_p,
dmp_sqf_norm,
dmp_sqf_part,
dmp_sqf_list, dmp_sqf_list_include)
from sympy.polys.factortools import (
dup_cyclotomic_p, dmp_irreducible_p,
dmp_factor_list, dmp_factor_list_include)
from sympy.polys.rootisolation import (
dup_isolate_real_roots_sqf,
dup_isolate_real_roots,
dup_isolate_all_roots_sqf,
dup_isolate_all_roots,
dup_refine_real_root,
dup_count_real_roots,
dup_count_complex_roots,
dup_sturm)
from sympy.polys.polyerrors import (
UnificationFailed,
PolynomialError)
def init_normal_DMP(rep, lev, dom):
return DMP(dmp_normal(rep, lev, dom), dom, lev)
class DMP(PicklableWithSlots, CantSympify):
"""Dense Multivariate Polynomials over `K`. """
__slots__ = ('rep', 'lev', 'dom', 'ring')
def __init__(self, rep, dom, lev=None, ring=None):
if lev is not None:
if type(rep) is dict:
rep = dmp_from_dict(rep, lev, dom)
elif type(rep) is not list:
rep = dmp_ground(dom.convert(rep), lev)
else:
rep, lev = dmp_validate(rep)
self.rep = rep
self.lev = lev
self.dom = dom
self.ring = ring
def __repr__(f):
return "%s(%s, %s, %s)" % (f.__class__.__name__, f.rep, f.dom, f.ring)
def __hash__(f):
return hash((f.__class__.__name__, f.to_tuple(), f.lev, f.dom, f.ring))
def unify(f, g):
"""Unify representations of two multivariate polynomials. """
if not isinstance(g, DMP) or f.lev != g.lev:
raise UnificationFailed("can't unify %s with %s" % (f, g))
if f.dom == g.dom and f.ring == g.ring:
return f.lev, f.dom, f.per, f.rep, g.rep
else:
lev, dom = f.lev, f.dom.unify(g.dom)
ring = f.ring
if g.ring is not None:
if ring is not None:
ring = ring.unify(g.ring)
else:
ring = g.ring
F = dmp_convert(f.rep, lev, f.dom, dom)
G = dmp_convert(g.rep, lev, g.dom, dom)
def per(rep, dom=dom, lev=lev, kill=False):
if kill:
if not lev:
return rep
else:
lev -= 1
return DMP(rep, dom, lev, ring)
return lev, dom, per, F, G
def per(f, rep, dom=None, kill=False, ring=None):
"""Create a DMP out of the given representation. """
lev = f.lev
if kill:
if not lev:
return rep
else:
lev -= 1
if dom is None:
dom = f.dom
if ring is None:
ring = f.ring
return DMP(rep, dom, lev, ring)
@classmethod
def zero(cls, lev, dom, ring=None):
return DMP(0, dom, lev, ring)
@classmethod
def one(cls, lev, dom, ring=None):
return DMP(1, dom, lev, ring)
@classmethod
def from_list(cls, rep, lev, dom):
"""Create an instance of ``cls`` given a list of native coefficients. """
return cls(dmp_convert(rep, lev, None, dom), dom, lev)
@classmethod
def from_sympy_list(cls, rep, lev, dom):
"""Create an instance of ``cls`` given a list of SymPy coefficients. """
return cls(dmp_from_sympy(rep, lev, dom), dom, lev)
def to_dict(f, zero=False):
"""Convert ``f`` to a dict representation with native coefficients. """
return dmp_to_dict(f.rep, f.lev, f.dom, zero=zero)
def to_sympy_dict(f, zero=False):
"""Convert ``f`` to a dict representation with SymPy coefficients. """
rep = dmp_to_dict(f.rep, f.lev, f.dom, zero=zero)
for k, v in rep.items():
rep[k] = f.dom.to_sympy(v)
return rep
def to_list(f):
"""Convert ``f`` to a list representation with native coefficients. """
return f.rep
def to_sympy_list(f):
"""Convert ``f`` to a list representation with SymPy coefficients. """
def sympify_nested_list(rep):
out = []
for val in rep:
if isinstance(val, list):
out.append(sympify_nested_list(val))
else:
out.append(f.dom.to_sympy(val))
return out
return sympify_nested_list(f.rep)
def to_tuple(f):
"""
Convert ``f`` to a tuple representation with native coefficients.
This is needed for hashing.
"""
return dmp_to_tuple(f.rep, f.lev)
@classmethod
def from_dict(cls, rep, lev, dom):
"""Construct and instance of ``cls`` from a ``dict`` representation. """
return cls(dmp_from_dict(rep, lev, dom), dom, lev)
@classmethod
def from_monoms_coeffs(cls, monoms, coeffs, lev, dom, ring=None):
return DMP(dict(list(zip(monoms, coeffs))), dom, lev, ring)
def to_ring(f):
"""Make the ground domain a ring. """
return f.convert(f.dom.get_ring())
def to_field(f):
"""Make the ground domain a field. """
return f.convert(f.dom.get_field())
def to_exact(f):
"""Make the ground domain exact. """
return f.convert(f.dom.get_exact())
def convert(f, dom):
"""Convert the ground domain of ``f``. """
if f.dom == dom:
return f
else:
return DMP(dmp_convert(f.rep, f.lev, f.dom, dom), dom, f.lev)
def slice(f, m, n, j=0):
"""Take a continuous subsequence of terms of ``f``. """
return f.per(dmp_slice_in(f.rep, m, n, j, f.lev, f.dom))
def coeffs(f, order=None):
"""Returns all non-zero coefficients from ``f`` in lex order. """
return [ c for _, c in dmp_list_terms(f.rep, f.lev, f.dom, order=order) ]
def monoms(f, order=None):
"""Returns all non-zero monomials from ``f`` in lex order. """
return [ m for m, _ in dmp_list_terms(f.rep, f.lev, f.dom, order=order) ]
def terms(f, order=None):
"""Returns all non-zero terms from ``f`` in lex order. """
return dmp_list_terms(f.rep, f.lev, f.dom, order=order)
def all_coeffs(f):
"""Returns all coefficients from ``f``. """
if not f.lev:
if not f:
return [f.dom.zero]
else:
return [ c for c in f.rep ]
else:
raise PolynomialError('multivariate polynomials not supported')
def all_monoms(f):
"""Returns all monomials from ``f``. """
if not f.lev:
n = dup_degree(f.rep)
if n < 0:
return [(0,)]
else:
return [ (n - i,) for i, c in enumerate(f.rep) ]
else:
raise PolynomialError('multivariate polynomials not supported')
def all_terms(f):
"""Returns all terms from a ``f``. """
if not f.lev:
n = dup_degree(f.rep)
if n < 0:
return [((0,), f.dom.zero)]
else:
return [ ((n - i,), c) for i, c in enumerate(f.rep) ]
else:
raise PolynomialError('multivariate polynomials not supported')
def lift(f):
"""Convert algebraic coefficients to rationals. """
return f.per(dmp_lift(f.rep, f.lev, f.dom), dom=f.dom.dom)
def deflate(f):
"""Reduce degree of `f` by mapping `x_i^m` to `y_i`. """
J, F = dmp_deflate(f.rep, f.lev, f.dom)
return J, f.per(F)
def inject(f, front=False):
"""Inject ground domain generators into ``f``. """
F, lev = dmp_inject(f.rep, f.lev, f.dom, front=front)
return f.__class__(F, f.dom.dom, lev)
def eject(f, dom, front=False):
"""Eject selected generators into the ground domain. """
F = dmp_eject(f.rep, f.lev, dom, front=front)
return f.__class__(F, dom, f.lev - len(dom.symbols))
def exclude(f):
r"""
Remove useless generators from ``f``.
Returns the removed generators and the new excluded ``f``.
Examples
========
>>> from sympy.polys.polyclasses import DMP
>>> from sympy.polys.domains import ZZ
>>> DMP([[[ZZ(1)]], [[ZZ(1)], [ZZ(2)]]], ZZ).exclude()
([2], DMP([[1], [1, 2]], ZZ, None))
"""
J, F, u = dmp_exclude(f.rep, f.lev, f.dom)
return J, f.__class__(F, f.dom, u)
def permute(f, P):
r"""
Returns a polynomial in `K[x_{P(1)}, ..., x_{P(n)}]`.
Examples
========
>>> from sympy.polys.polyclasses import DMP
>>> from sympy.polys.domains import ZZ
>>> DMP([[[ZZ(2)], [ZZ(1), ZZ(0)]], [[]]], ZZ).permute([1, 0, 2])
DMP([[[2], []], [[1, 0], []]], ZZ, None)
>>> DMP([[[ZZ(2)], [ZZ(1), ZZ(0)]], [[]]], ZZ).permute([1, 2, 0])
DMP([[[1], []], [[2, 0], []]], ZZ, None)
"""
return f.per(dmp_permute(f.rep, P, f.lev, f.dom))
def terms_gcd(f):
"""Remove GCD of terms from the polynomial ``f``. """
J, F = dmp_terms_gcd(f.rep, f.lev, f.dom)
return J, f.per(F)
def add_ground(f, c):
"""Add an element of the ground domain to ``f``. """
return f.per(dmp_add_ground(f.rep, f.dom.convert(c), f.lev, f.dom))
def sub_ground(f, c):
"""Subtract an element of the ground domain from ``f``. """
return f.per(dmp_sub_ground(f.rep, f.dom.convert(c), f.lev, f.dom))
def mul_ground(f, c):
"""Multiply ``f`` by a an element of the ground domain. """
return f.per(dmp_mul_ground(f.rep, f.dom.convert(c), f.lev, f.dom))
def quo_ground(f, c):
"""Quotient of ``f`` by a an element of the ground domain. """
return f.per(dmp_quo_ground(f.rep, f.dom.convert(c), f.lev, f.dom))
def exquo_ground(f, c):
"""Exact quotient of ``f`` by a an element of the ground domain. """
return f.per(dmp_exquo_ground(f.rep, f.dom.convert(c), f.lev, f.dom))
def abs(f):
"""Make all coefficients in ``f`` positive. """
return f.per(dmp_abs(f.rep, f.lev, f.dom))
def neg(f):
"""Negate all coefficients in ``f``. """
return f.per(dmp_neg(f.rep, f.lev, f.dom))
def add(f, g):
"""Add two multivariate polynomials ``f`` and ``g``. """
lev, dom, per, F, G = f.unify(g)
return per(dmp_add(F, G, lev, dom))
def sub(f, g):
"""Subtract two multivariate polynomials ``f`` and ``g``. """
lev, dom, per, F, G = f.unify(g)
return per(dmp_sub(F, G, lev, dom))
def mul(f, g):
"""Multiply two multivariate polynomials ``f`` and ``g``. """
lev, dom, per, F, G = f.unify(g)
return per(dmp_mul(F, G, lev, dom))
def sqr(f):
"""Square a multivariate polynomial ``f``. """
return f.per(dmp_sqr(f.rep, f.lev, f.dom))
def pow(f, n):
"""Raise ``f`` to a non-negative power ``n``. """
if isinstance(n, int):
return f.per(dmp_pow(f.rep, n, f.lev, f.dom))
else:
raise TypeError("``int`` expected, got %s" % type(n))
def pdiv(f, g):
"""Polynomial pseudo-division of ``f`` and ``g``. """
lev, dom, per, F, G = f.unify(g)
q, r = dmp_pdiv(F, G, lev, dom)
return per(q), per(r)
def prem(f, g):
"""Polynomial pseudo-remainder of ``f`` and ``g``. """
lev, dom, per, F, G = f.unify(g)
return per(dmp_prem(F, G, lev, dom))
def pquo(f, g):
"""Polynomial pseudo-quotient of ``f`` and ``g``. """
lev, dom, per, F, G = f.unify(g)
return per(dmp_pquo(F, G, lev, dom))
def pexquo(f, g):
"""Polynomial exact pseudo-quotient of ``f`` and ``g``. """
lev, dom, per, F, G = f.unify(g)
return per(dmp_pexquo(F, G, lev, dom))
def div(f, g):
"""Polynomial division with remainder of ``f`` and ``g``. """
lev, dom, per, F, G = f.unify(g)
q, r = dmp_div(F, G, lev, dom)
return per(q), per(r)
def rem(f, g):
"""Computes polynomial remainder of ``f`` and ``g``. """
lev, dom, per, F, G = f.unify(g)
return per(dmp_rem(F, G, lev, dom))
def quo(f, g):
"""Computes polynomial quotient of ``f`` and ``g``. """
lev, dom, per, F, G = f.unify(g)
return per(dmp_quo(F, G, lev, dom))
def exquo(f, g):
"""Computes polynomial exact quotient of ``f`` and ``g``. """
lev, dom, per, F, G = f.unify(g)
res = per(dmp_exquo(F, G, lev, dom))
if f.ring and res not in f.ring:
from sympy.polys.polyerrors import ExactQuotientFailed
raise ExactQuotientFailed(f, g, f.ring)
return res
def degree(f, j=0):
"""Returns the leading degree of ``f`` in ``x_j``. """
if isinstance(j, int):
return dmp_degree_in(f.rep, j, f.lev)
else:
raise TypeError("``int`` expected, got %s" % type(j))
def degree_list(f):
"""Returns a list of degrees of ``f``. """
return dmp_degree_list(f.rep, f.lev)
def total_degree(f):
"""Returns the total degree of ``f``. """
return max(sum(m) for m in f.monoms())
def homogenize(f, s):
"""Return homogeneous polynomial of ``f``"""
td = f.total_degree()
result = {}
new_symbol = (s == len(f.terms()[0][0]))
for term in f.terms():
d = sum(term[0])
if d < td:
i = td - d
else:
i = 0
if new_symbol:
result[term[0] + (i,)] = term[1]
else:
l = list(term[0])
l[s] += i
result[tuple(l)] = term[1]
return DMP(result, f.dom, f.lev + int(new_symbol), f.ring)
def homogeneous_order(f):
"""Returns the homogeneous order of ``f``. """
if f.is_zero:
return -oo
monoms = f.monoms()
tdeg = sum(monoms[0])
for monom in monoms:
_tdeg = sum(monom)
if _tdeg != tdeg:
return None
return tdeg
def LC(f):
"""Returns the leading coefficient of ``f``. """
return dmp_ground_LC(f.rep, f.lev, f.dom)
def TC(f):
"""Returns the trailing coefficient of ``f``. """
return dmp_ground_TC(f.rep, f.lev, f.dom)
def nth(f, *N):
"""Returns the ``n``-th coefficient of ``f``. """
if all(isinstance(n, int) for n in N):
return dmp_ground_nth(f.rep, N, f.lev, f.dom)
else:
raise TypeError("a sequence of integers expected")
def max_norm(f):
"""Returns maximum norm of ``f``. """
return dmp_max_norm(f.rep, f.lev, f.dom)
def l1_norm(f):
"""Returns l1 norm of ``f``. """
return dmp_l1_norm(f.rep, f.lev, f.dom)
def clear_denoms(f):
"""Clear denominators, but keep the ground domain. """
coeff, F = dmp_clear_denoms(f.rep, f.lev, f.dom)
return coeff, f.per(F)
def integrate(f, m=1, j=0):
"""Computes the ``m``-th order indefinite integral of ``f`` in ``x_j``. """
if not isinstance(m, int):
raise TypeError("``int`` expected, got %s" % type(m))
if not isinstance(j, int):
raise TypeError("``int`` expected, got %s" % type(j))
return f.per(dmp_integrate_in(f.rep, m, j, f.lev, f.dom))
def diff(f, m=1, j=0):
"""Computes the ``m``-th order derivative of ``f`` in ``x_j``. """
if not isinstance(m, int):
raise TypeError("``int`` expected, got %s" % type(m))
if not isinstance(j, int):
raise TypeError("``int`` expected, got %s" % type(j))
return f.per(dmp_diff_in(f.rep, m, j, f.lev, f.dom))
def eval(f, a, j=0):
"""Evaluates ``f`` at the given point ``a`` in ``x_j``. """
if not isinstance(j, int):
raise TypeError("``int`` expected, got %s" % type(j))
return f.per(dmp_eval_in(f.rep,
f.dom.convert(a), j, f.lev, f.dom), kill=True)
def half_gcdex(f, g):
"""Half extended Euclidean algorithm, if univariate. """
lev, dom, per, F, G = f.unify(g)
if not lev:
s, h = dup_half_gcdex(F, G, dom)
return per(s), per(h)
else:
raise ValueError('univariate polynomial expected')
def gcdex(f, g):
"""Extended Euclidean algorithm, if univariate. """
lev, dom, per, F, G = f.unify(g)
if not lev:
s, t, h = dup_gcdex(F, G, dom)
return per(s), per(t), per(h)
else:
raise ValueError('univariate polynomial expected')
def invert(f, g):
"""Invert ``f`` modulo ``g``, if possible. """
lev, dom, per, F, G = f.unify(g)
if not lev:
return per(dup_invert(F, G, dom))
else:
raise ValueError('univariate polynomial expected')
def revert(f, n):
"""Compute ``f**(-1)`` mod ``x**n``. """
if not f.lev:
return f.per(dup_revert(f.rep, n, f.dom))
else:
raise ValueError('univariate polynomial expected')
def subresultants(f, g):
"""Computes subresultant PRS sequence of ``f`` and ``g``. """
lev, dom, per, F, G = f.unify(g)
R = dmp_subresultants(F, G, lev, dom)
return list(map(per, R))
def resultant(f, g, includePRS=False):
"""Computes resultant of ``f`` and ``g`` via PRS. """
lev, dom, per, F, G = f.unify(g)
if includePRS:
res, R = dmp_resultant(F, G, lev, dom, includePRS=includePRS)
return per(res, kill=True), list(map(per, R))
return per(dmp_resultant(F, G, lev, dom), kill=True)
def discriminant(f):
"""Computes discriminant of ``f``. """
return f.per(dmp_discriminant(f.rep, f.lev, f.dom), kill=True)
def cofactors(f, g):
"""Returns GCD of ``f`` and ``g`` and their cofactors. """
lev, dom, per, F, G = f.unify(g)
h, cff, cfg = dmp_inner_gcd(F, G, lev, dom)
return per(h), per(cff), per(cfg)
def gcd(f, g):
"""Returns polynomial GCD of ``f`` and ``g``. """
lev, dom, per, F, G = f.unify(g)
return per(dmp_gcd(F, G, lev, dom))
def lcm(f, g):
"""Returns polynomial LCM of ``f`` and ``g``. """
lev, dom, per, F, G = f.unify(g)
return per(dmp_lcm(F, G, lev, dom))
def cancel(f, g, include=True):
"""Cancel common factors in a rational function ``f/g``. """
lev, dom, per, F, G = f.unify(g)
if include:
F, G = dmp_cancel(F, G, lev, dom, include=True)
else:
cF, cG, F, G = dmp_cancel(F, G, lev, dom, include=False)
F, G = per(F), per(G)
if include:
return F, G
else:
return cF, cG, F, G
def trunc(f, p):
"""Reduce ``f`` modulo a constant ``p``. """
return f.per(dmp_ground_trunc(f.rep, f.dom.convert(p), f.lev, f.dom))
def monic(f):
"""Divides all coefficients by ``LC(f)``. """
return f.per(dmp_ground_monic(f.rep, f.lev, f.dom))
def content(f):
"""Returns GCD of polynomial coefficients. """
return dmp_ground_content(f.rep, f.lev, f.dom)
def primitive(f):
"""Returns content and a primitive form of ``f``. """
cont, F = dmp_ground_primitive(f.rep, f.lev, f.dom)
return cont, f.per(F)
def compose(f, g):
"""Computes functional composition of ``f`` and ``g``. """
lev, dom, per, F, G = f.unify(g)
return per(dmp_compose(F, G, lev, dom))
def decompose(f):
"""Computes functional decomposition of ``f``. """
if not f.lev:
return list(map(f.per, dup_decompose(f.rep, f.dom)))
else:
raise ValueError('univariate polynomial expected')
def shift(f, a):
"""Efficiently compute Taylor shift ``f(x + a)``. """
if not f.lev:
return f.per(dup_shift(f.rep, f.dom.convert(a), f.dom))
else:
raise ValueError('univariate polynomial expected')
def transform(f, p, q):
"""Evaluate functional transformation ``q**n * f(p/q)``."""
if f.lev:
raise ValueError('univariate polynomial expected')
lev, dom, per, P, Q = p.unify(q)
lev, dom, per, F, P = f.unify(per(P, dom, lev))
lev, dom, per, F, Q = per(F, dom, lev).unify(per(Q, dom, lev))
if not lev:
return per(dup_transform(F, P, Q, dom))
else:
raise ValueError('univariate polynomial expected')
def sturm(f):
"""Computes the Sturm sequence of ``f``. """
if not f.lev:
return list(map(f.per, dup_sturm(f.rep, f.dom)))
else:
raise ValueError('univariate polynomial expected')
def gff_list(f):
"""Computes greatest factorial factorization of ``f``. """
if not f.lev:
return [ (f.per(g), k) for g, k in dup_gff_list(f.rep, f.dom) ]
else:
raise ValueError('univariate polynomial expected')
def norm(f):
"""Computes ``Norm(f)``."""
r = dmp_norm(f.rep, f.lev, f.dom)
return f.per(r, dom=f.dom.dom)
def sqf_norm(f):
"""Computes square-free norm of ``f``. """
s, g, r = dmp_sqf_norm(f.rep, f.lev, f.dom)
return s, f.per(g), f.per(r, dom=f.dom.dom)
def sqf_part(f):
"""Computes square-free part of ``f``. """
return f.per(dmp_sqf_part(f.rep, f.lev, f.dom))
def sqf_list(f, all=False):
"""Returns a list of square-free factors of ``f``. """
coeff, factors = dmp_sqf_list(f.rep, f.lev, f.dom, all)
return coeff, [ (f.per(g), k) for g, k in factors ]
def sqf_list_include(f, all=False):
"""Returns a list of square-free factors of ``f``. """
factors = dmp_sqf_list_include(f.rep, f.lev, f.dom, all)
return [ (f.per(g), k) for g, k in factors ]
def factor_list(f):
"""Returns a list of irreducible factors of ``f``. """
coeff, factors = dmp_factor_list(f.rep, f.lev, f.dom)
return coeff, [ (f.per(g), k) for g, k in factors ]
def factor_list_include(f):
"""Returns a list of irreducible factors of ``f``. """
factors = dmp_factor_list_include(f.rep, f.lev, f.dom)
return [ (f.per(g), k) for g, k in factors ]
def intervals(f, all=False, eps=None, inf=None, sup=None, fast=False, sqf=False):
"""Compute isolating intervals for roots of ``f``. """
if not f.lev:
if not all:
if not sqf:
return dup_isolate_real_roots(f.rep, f.dom, eps=eps, inf=inf, sup=sup, fast=fast)
else:
return dup_isolate_real_roots_sqf(f.rep, f.dom, eps=eps, inf=inf, sup=sup, fast=fast)
else:
if not sqf:
return dup_isolate_all_roots(f.rep, f.dom, eps=eps, inf=inf, sup=sup, fast=fast)
else:
return dup_isolate_all_roots_sqf(f.rep, f.dom, eps=eps, inf=inf, sup=sup, fast=fast)
else:
raise PolynomialError(
"can't isolate roots of a multivariate polynomial")
def refine_root(f, s, t, eps=None, steps=None, fast=False):
"""
Refine an isolating interval to the given precision.
``eps`` should be a rational number.
"""
if not f.lev:
return dup_refine_real_root(f.rep, s, t, f.dom, eps=eps, steps=steps, fast=fast)
else:
raise PolynomialError(
"can't refine a root of a multivariate polynomial")
def count_real_roots(f, inf=None, sup=None):
"""Return the number of real roots of ``f`` in ``[inf, sup]``. """
return dup_count_real_roots(f.rep, f.dom, inf=inf, sup=sup)
def count_complex_roots(f, inf=None, sup=None):
"""Return the number of complex roots of ``f`` in ``[inf, sup]``. """
return dup_count_complex_roots(f.rep, f.dom, inf=inf, sup=sup)
@property
def is_zero(f):
"""Returns ``True`` if ``f`` is a zero polynomial. """
return dmp_zero_p(f.rep, f.lev)
@property
def is_one(f):
"""Returns ``True`` if ``f`` is a unit polynomial. """
return dmp_one_p(f.rep, f.lev, f.dom)
@property
def is_ground(f):
"""Returns ``True`` if ``f`` is an element of the ground domain. """
return dmp_ground_p(f.rep, None, f.lev)
@property
def is_sqf(f):
"""Returns ``True`` if ``f`` is a square-free polynomial. """
return dmp_sqf_p(f.rep, f.lev, f.dom)
@property
def is_monic(f):
"""Returns ``True`` if the leading coefficient of ``f`` is one. """
return f.dom.is_one(dmp_ground_LC(f.rep, f.lev, f.dom))
@property
def is_primitive(f):
"""Returns ``True`` if the GCD of the coefficients of ``f`` is one. """
return f.dom.is_one(dmp_ground_content(f.rep, f.lev, f.dom))
@property
def is_linear(f):
"""Returns ``True`` if ``f`` is linear in all its variables. """
return all(sum(monom) <= 1 for monom in dmp_to_dict(f.rep, f.lev, f.dom).keys())
@property
def is_quadratic(f):
"""Returns ``True`` if ``f`` is quadratic in all its variables. """
return all(sum(monom) <= 2 for monom in dmp_to_dict(f.rep, f.lev, f.dom).keys())
@property
def is_monomial(f):
"""Returns ``True`` if ``f`` is zero or has only one term. """
return len(f.to_dict()) <= 1
@property
def is_homogeneous(f):
"""Returns ``True`` if ``f`` is a homogeneous polynomial. """
return f.homogeneous_order() is not None
@property
def is_irreducible(f):
"""Returns ``True`` if ``f`` has no factors over its domain. """
return dmp_irreducible_p(f.rep, f.lev, f.dom)
@property
def is_cyclotomic(f):
"""Returns ``True`` if ``f`` is a cyclotomic polynomial. """
if not f.lev:
return dup_cyclotomic_p(f.rep, f.dom)
else:
return False
def __abs__(f):
return f.abs()
def __neg__(f):
return f.neg()
def __add__(f, g):
if not isinstance(g, DMP):
try:
g = f.per(dmp_ground(f.dom.convert(g), f.lev))
except TypeError:
return NotImplemented
except (CoercionFailed, NotImplementedError):
if f.ring is not None:
try:
g = f.ring.convert(g)
except (CoercionFailed, NotImplementedError):
return NotImplemented
return f.add(g)
def __radd__(f, g):
return f.__add__(g)
def __sub__(f, g):
if not isinstance(g, DMP):
try:
g = f.per(dmp_ground(f.dom.convert(g), f.lev))
except TypeError:
return NotImplemented
except (CoercionFailed, NotImplementedError):
if f.ring is not None:
try:
g = f.ring.convert(g)
except (CoercionFailed, NotImplementedError):
return NotImplemented
return f.sub(g)
def __rsub__(f, g):
return (-f).__add__(g)
def __mul__(f, g):
if isinstance(g, DMP):
return f.mul(g)
else:
try:
return f.mul_ground(g)
except TypeError:
return NotImplemented
except (CoercionFailed, NotImplementedError):
if f.ring is not None:
try:
return f.mul(f.ring.convert(g))
except (CoercionFailed, NotImplementedError):
pass
return NotImplemented
def __truediv__(f, g):
if isinstance(g, DMP):
return f.exquo(g)
else:
try:
return f.mul_ground(g)
except TypeError:
return NotImplemented
except (CoercionFailed, NotImplementedError):
if f.ring is not None:
try:
return f.exquo(f.ring.convert(g))
except (CoercionFailed, NotImplementedError):
pass
return NotImplemented
def __rtruediv__(f, g):
if isinstance(g, DMP):
return g.exquo(f)
elif f.ring is not None:
try:
return f.ring.convert(g).exquo(f)
except (CoercionFailed, NotImplementedError):
pass
return NotImplemented
def __rmul__(f, g):
return f.__mul__(g)
def __pow__(f, n):
return f.pow(n)
def __divmod__(f, g):
return f.div(g)
def __mod__(f, g):
return f.rem(g)
def __floordiv__(f, g):
if isinstance(g, DMP):
return f.quo(g)
else:
try:
return f.quo_ground(g)
except TypeError:
return NotImplemented
def __eq__(f, g):
try:
_, _, _, F, G = f.unify(g)
if f.lev == g.lev:
return F == G
except UnificationFailed:
pass
return False
def __ne__(f, g):
return not f == g
def eq(f, g, strict=False):
if not strict:
return f == g
else:
return f._strict_eq(g)
def ne(f, g, strict=False):
return not f.eq(g, strict=strict)
def _strict_eq(f, g):
return isinstance(g, f.__class__) and f.lev == g.lev \
and f.dom == g.dom \
and f.rep == g.rep
def __lt__(f, g):
_, _, _, F, G = f.unify(g)
return F < G
def __le__(f, g):
_, _, _, F, G = f.unify(g)
return F <= G
def __gt__(f, g):
_, _, _, F, G = f.unify(g)
return F > G
def __ge__(f, g):
_, _, _, F, G = f.unify(g)
return F >= G
def __bool__(f):
return not dmp_zero_p(f.rep, f.lev)
def init_normal_DMF(num, den, lev, dom):
return DMF(dmp_normal(num, lev, dom),
dmp_normal(den, lev, dom), dom, lev)
class DMF(PicklableWithSlots, CantSympify):
"""Dense Multivariate Fractions over `K`. """
__slots__ = ('num', 'den', 'lev', 'dom', 'ring')
def __init__(self, rep, dom, lev=None, ring=None):
num, den, lev = self._parse(rep, dom, lev)
num, den = dmp_cancel(num, den, lev, dom)
self.num = num
self.den = den
self.lev = lev
self.dom = dom
self.ring = ring
@classmethod
def new(cls, rep, dom, lev=None, ring=None):
num, den, lev = cls._parse(rep, dom, lev)
obj = object.__new__(cls)
obj.num = num
obj.den = den
obj.lev = lev
obj.dom = dom
obj.ring = ring
return obj
@classmethod
def _parse(cls, rep, dom, lev=None):
if type(rep) is tuple:
num, den = rep
if lev is not None:
if type(num) is dict:
num = dmp_from_dict(num, lev, dom)
if type(den) is dict:
den = dmp_from_dict(den, lev, dom)
else:
num, num_lev = dmp_validate(num)
den, den_lev = dmp_validate(den)
if num_lev == den_lev:
lev = num_lev
else:
raise ValueError('inconsistent number of levels')
if dmp_zero_p(den, lev):
raise ZeroDivisionError('fraction denominator')
if dmp_zero_p(num, lev):
den = dmp_one(lev, dom)
else:
if dmp_negative_p(den, lev, dom):
num = dmp_neg(num, lev, dom)
den = dmp_neg(den, lev, dom)
else:
num = rep
if lev is not None:
if type(num) is dict:
num = dmp_from_dict(num, lev, dom)
elif type(num) is not list:
num = dmp_ground(dom.convert(num), lev)
else:
num, lev = dmp_validate(num)
den = dmp_one(lev, dom)
return num, den, lev
def __repr__(f):
return "%s((%s, %s), %s, %s)" % (f.__class__.__name__, f.num, f.den,
f.dom, f.ring)
def __hash__(f):
return hash((f.__class__.__name__, dmp_to_tuple(f.num, f.lev),
dmp_to_tuple(f.den, f.lev), f.lev, f.dom, f.ring))
def poly_unify(f, g):
"""Unify a multivariate fraction and a polynomial. """
if not isinstance(g, DMP) or f.lev != g.lev:
raise UnificationFailed("can't unify %s with %s" % (f, g))
if f.dom == g.dom and f.ring == g.ring:
return (f.lev, f.dom, f.per, (f.num, f.den), g.rep)
else:
lev, dom = f.lev, f.dom.unify(g.dom)
ring = f.ring
if g.ring is not None:
if ring is not None:
ring = ring.unify(g.ring)
else:
ring = g.ring
F = (dmp_convert(f.num, lev, f.dom, dom),
dmp_convert(f.den, lev, f.dom, dom))
G = dmp_convert(g.rep, lev, g.dom, dom)
def per(num, den, cancel=True, kill=False, lev=lev):
if kill:
if not lev:
return num/den
else:
lev = lev - 1
if cancel:
num, den = dmp_cancel(num, den, lev, dom)
return f.__class__.new((num, den), dom, lev, ring=ring)
return lev, dom, per, F, G
def frac_unify(f, g):
"""Unify representations of two multivariate fractions. """
if not isinstance(g, DMF) or f.lev != g.lev:
raise UnificationFailed("can't unify %s with %s" % (f, g))
if f.dom == g.dom and f.ring == g.ring:
return (f.lev, f.dom, f.per, (f.num, f.den),
(g.num, g.den))
else:
lev, dom = f.lev, f.dom.unify(g.dom)
ring = f.ring
if g.ring is not None:
if ring is not None:
ring = ring.unify(g.ring)
else:
ring = g.ring
F = (dmp_convert(f.num, lev, f.dom, dom),
dmp_convert(f.den, lev, f.dom, dom))
G = (dmp_convert(g.num, lev, g.dom, dom),
dmp_convert(g.den, lev, g.dom, dom))
def per(num, den, cancel=True, kill=False, lev=lev):
if kill:
if not lev:
return num/den
else:
lev = lev - 1
if cancel:
num, den = dmp_cancel(num, den, lev, dom)
return f.__class__.new((num, den), dom, lev, ring=ring)
return lev, dom, per, F, G
def per(f, num, den, cancel=True, kill=False, ring=None):
"""Create a DMF out of the given representation. """
lev, dom = f.lev, f.dom
if kill:
if not lev:
return num/den
else:
lev -= 1
if cancel:
num, den = dmp_cancel(num, den, lev, dom)
if ring is None:
ring = f.ring
return f.__class__.new((num, den), dom, lev, ring=ring)
def half_per(f, rep, kill=False):
"""Create a DMP out of the given representation. """
lev = f.lev
if kill:
if not lev:
return rep
else:
lev -= 1
return DMP(rep, f.dom, lev)
@classmethod
def zero(cls, lev, dom, ring=None):
return cls.new(0, dom, lev, ring=ring)
@classmethod
def one(cls, lev, dom, ring=None):
return cls.new(1, dom, lev, ring=ring)
def numer(f):
"""Returns the numerator of ``f``. """
return f.half_per(f.num)
def denom(f):
"""Returns the denominator of ``f``. """
return f.half_per(f.den)
def cancel(f):
"""Remove common factors from ``f.num`` and ``f.den``. """
return f.per(f.num, f.den)
def neg(f):
"""Negate all coefficients in ``f``. """
return f.per(dmp_neg(f.num, f.lev, f.dom), f.den, cancel=False)
def add(f, g):
"""Add two multivariate fractions ``f`` and ``g``. """
if isinstance(g, DMP):
lev, dom, per, (F_num, F_den), G = f.poly_unify(g)
num, den = dmp_add_mul(F_num, F_den, G, lev, dom), F_den
else:
lev, dom, per, F, G = f.frac_unify(g)
(F_num, F_den), (G_num, G_den) = F, G
num = dmp_add(dmp_mul(F_num, G_den, lev, dom),
dmp_mul(F_den, G_num, lev, dom), lev, dom)
den = dmp_mul(F_den, G_den, lev, dom)
return per(num, den)
def sub(f, g):
"""Subtract two multivariate fractions ``f`` and ``g``. """
if isinstance(g, DMP):
lev, dom, per, (F_num, F_den), G = f.poly_unify(g)
num, den = dmp_sub_mul(F_num, F_den, G, lev, dom), F_den
else:
lev, dom, per, F, G = f.frac_unify(g)
(F_num, F_den), (G_num, G_den) = F, G
num = dmp_sub(dmp_mul(F_num, G_den, lev, dom),
dmp_mul(F_den, G_num, lev, dom), lev, dom)
den = dmp_mul(F_den, G_den, lev, dom)
return per(num, den)
def mul(f, g):
"""Multiply two multivariate fractions ``f`` and ``g``. """
if isinstance(g, DMP):
lev, dom, per, (F_num, F_den), G = f.poly_unify(g)
num, den = dmp_mul(F_num, G, lev, dom), F_den
else:
lev, dom, per, F, G = f.frac_unify(g)
(F_num, F_den), (G_num, G_den) = F, G
num = dmp_mul(F_num, G_num, lev, dom)
den = dmp_mul(F_den, G_den, lev, dom)
return per(num, den)
def pow(f, n):
"""Raise ``f`` to a non-negative power ``n``. """
if isinstance(n, int):
return f.per(dmp_pow(f.num, n, f.lev, f.dom),
dmp_pow(f.den, n, f.lev, f.dom), cancel=False)
else:
raise TypeError("``int`` expected, got %s" % type(n))
def quo(f, g):
"""Computes quotient of fractions ``f`` and ``g``. """
if isinstance(g, DMP):
lev, dom, per, (F_num, F_den), G = f.poly_unify(g)
num, den = F_num, dmp_mul(F_den, G, lev, dom)
else:
lev, dom, per, F, G = f.frac_unify(g)
(F_num, F_den), (G_num, G_den) = F, G
num = dmp_mul(F_num, G_den, lev, dom)
den = dmp_mul(F_den, G_num, lev, dom)
res = per(num, den)
if f.ring is not None and res not in f.ring:
from sympy.polys.polyerrors import ExactQuotientFailed
raise ExactQuotientFailed(f, g, f.ring)
return res
exquo = quo
def invert(f, check=True):
"""Computes inverse of a fraction ``f``. """
if check and f.ring is not None and not f.ring.is_unit(f):
raise NotReversible(f, f.ring)
res = f.per(f.den, f.num, cancel=False)
return res
@property
def is_zero(f):
"""Returns ``True`` if ``f`` is a zero fraction. """
return dmp_zero_p(f.num, f.lev)
@property
def is_one(f):
"""Returns ``True`` if ``f`` is a unit fraction. """
return dmp_one_p(f.num, f.lev, f.dom) and \
dmp_one_p(f.den, f.lev, f.dom)
def __neg__(f):
return f.neg()
def __add__(f, g):
if isinstance(g, (DMP, DMF)):
return f.add(g)
try:
return f.add(f.half_per(g))
except TypeError:
return NotImplemented
except (CoercionFailed, NotImplementedError):
if f.ring is not None:
try:
return f.add(f.ring.convert(g))
except (CoercionFailed, NotImplementedError):
pass
return NotImplemented
def __radd__(f, g):
return f.__add__(g)
def __sub__(f, g):
if isinstance(g, (DMP, DMF)):
return f.sub(g)
try:
return f.sub(f.half_per(g))
except TypeError:
return NotImplemented
except (CoercionFailed, NotImplementedError):
if f.ring is not None:
try:
return f.sub(f.ring.convert(g))
except (CoercionFailed, NotImplementedError):
pass
return NotImplemented
def __rsub__(f, g):
return (-f).__add__(g)
def __mul__(f, g):
if isinstance(g, (DMP, DMF)):
return f.mul(g)
try:
return f.mul(f.half_per(g))
except TypeError:
return NotImplemented
except (CoercionFailed, NotImplementedError):
if f.ring is not None:
try:
return f.mul(f.ring.convert(g))
except (CoercionFailed, NotImplementedError):
pass
return NotImplemented
def __rmul__(f, g):
return f.__mul__(g)
def __pow__(f, n):
return f.pow(n)
def __truediv__(f, g):
if isinstance(g, (DMP, DMF)):
return f.quo(g)
try:
return f.quo(f.half_per(g))
except TypeError:
return NotImplemented
except (CoercionFailed, NotImplementedError):
if f.ring is not None:
try:
return f.quo(f.ring.convert(g))
except (CoercionFailed, NotImplementedError):
pass
return NotImplemented
def __rtruediv__(self, g):
r = self.invert(check=False)*g
if self.ring and r not in self.ring:
from sympy.polys.polyerrors import ExactQuotientFailed
raise ExactQuotientFailed(g, self, self.ring)
return r
def __eq__(f, g):
try:
if isinstance(g, DMP):
_, _, _, (F_num, F_den), G = f.poly_unify(g)
if f.lev == g.lev:
return dmp_one_p(F_den, f.lev, f.dom) and F_num == G
else:
_, _, _, F, G = f.frac_unify(g)
if f.lev == g.lev:
return F == G
except UnificationFailed:
pass
return False
def __ne__(f, g):
try:
if isinstance(g, DMP):
_, _, _, (F_num, F_den), G = f.poly_unify(g)
if f.lev == g.lev:
return not (dmp_one_p(F_den, f.lev, f.dom) and F_num == G)
else:
_, _, _, F, G = f.frac_unify(g)
if f.lev == g.lev:
return F != G
except UnificationFailed:
pass
return True
def __lt__(f, g):
_, _, _, F, G = f.frac_unify(g)
return F < G
def __le__(f, g):
_, _, _, F, G = f.frac_unify(g)
return F <= G
def __gt__(f, g):
_, _, _, F, G = f.frac_unify(g)
return F > G
def __ge__(f, g):
_, _, _, F, G = f.frac_unify(g)
return F >= G
def __bool__(f):
return not dmp_zero_p(f.num, f.lev)
def init_normal_ANP(rep, mod, dom):
return ANP(dup_normal(rep, dom),
dup_normal(mod, dom), dom)
class ANP(PicklableWithSlots, CantSympify):
"""Dense Algebraic Number Polynomials over a field. """
__slots__ = ('rep', 'mod', 'dom')
def __init__(self, rep, mod, dom):
if type(rep) is dict:
self.rep = dup_from_dict(rep, dom)
else:
if type(rep) is not list:
rep = [dom.convert(rep)]
self.rep = dup_strip(rep)
if isinstance(mod, DMP):
self.mod = mod.rep
else:
if type(mod) is dict:
self.mod = dup_from_dict(mod, dom)
else:
self.mod = dup_strip(mod)
self.dom = dom
def __repr__(f):
return "%s(%s, %s, %s)" % (f.__class__.__name__, f.rep, f.mod, f.dom)
def __hash__(f):
return hash((f.__class__.__name__, f.to_tuple(), dmp_to_tuple(f.mod, 0), f.dom))
def unify(f, g):
"""Unify representations of two algebraic numbers. """
if not isinstance(g, ANP) or f.mod != g.mod:
raise UnificationFailed("can't unify %s with %s" % (f, g))
if f.dom == g.dom:
return f.dom, f.per, f.rep, g.rep, f.mod
else:
dom = f.dom.unify(g.dom)
F = dup_convert(f.rep, f.dom, dom)
G = dup_convert(g.rep, g.dom, dom)
if dom != f.dom and dom != g.dom:
mod = dup_convert(f.mod, f.dom, dom)
else:
if dom == f.dom:
mod = f.mod
else:
mod = g.mod
per = lambda rep: ANP(rep, mod, dom)
return dom, per, F, G, mod
def per(f, rep, mod=None, dom=None):
return ANP(rep, mod or f.mod, dom or f.dom)
@classmethod
def zero(cls, mod, dom):
return ANP(0, mod, dom)
@classmethod
def one(cls, mod, dom):
return ANP(1, mod, dom)
def to_dict(f):
"""Convert ``f`` to a dict representation with native coefficients. """
return dmp_to_dict(f.rep, 0, f.dom)
def to_sympy_dict(f):
"""Convert ``f`` to a dict representation with SymPy coefficients. """
rep = dmp_to_dict(f.rep, 0, f.dom)
for k, v in rep.items():
rep[k] = f.dom.to_sympy(v)
return rep
def to_list(f):
"""Convert ``f`` to a list representation with native coefficients. """
return f.rep
def to_sympy_list(f):
"""Convert ``f`` to a list representation with SymPy coefficients. """
return [ f.dom.to_sympy(c) for c in f.rep ]
def to_tuple(f):
"""
Convert ``f`` to a tuple representation with native coefficients.
This is needed for hashing.
"""
return dmp_to_tuple(f.rep, 0)
@classmethod
def from_list(cls, rep, mod, dom):
return ANP(dup_strip(list(map(dom.convert, rep))), mod, dom)
def neg(f):
return f.per(dup_neg(f.rep, f.dom))
def add(f, g):
dom, per, F, G, mod = f.unify(g)
return per(dup_add(F, G, dom))
def sub(f, g):
dom, per, F, G, mod = f.unify(g)
return per(dup_sub(F, G, dom))
def mul(f, g):
dom, per, F, G, mod = f.unify(g)
return per(dup_rem(dup_mul(F, G, dom), mod, dom))
def pow(f, n):
"""Raise ``f`` to a non-negative power ``n``. """
if isinstance(n, int):
if n < 0:
F, n = dup_invert(f.rep, f.mod, f.dom), -n
else:
F = f.rep
return f.per(dup_rem(dup_pow(F, n, f.dom), f.mod, f.dom))
else:
raise TypeError("``int`` expected, got %s" % type(n))
def div(f, g):
dom, per, F, G, mod = f.unify(g)
return (per(dup_rem(dup_mul(F, dup_invert(G, mod, dom), dom), mod, dom)), f.zero(mod, dom))
def rem(f, g):
dom, _, _, G, mod = f.unify(g)
s, h = dup_half_gcdex(G, mod, dom)
if h == [dom.one]:
return f.zero(mod, dom)
else:
raise NotInvertible("zero divisor")
def quo(f, g):
dom, per, F, G, mod = f.unify(g)
return per(dup_rem(dup_mul(F, dup_invert(G, mod, dom), dom), mod, dom))
exquo = quo
def LC(f):
"""Returns the leading coefficient of ``f``. """
return dup_LC(f.rep, f.dom)
def TC(f):
"""Returns the trailing coefficient of ``f``. """
return dup_TC(f.rep, f.dom)
@property
def is_zero(f):
"""Returns ``True`` if ``f`` is a zero algebraic number. """
return not f
@property
def is_one(f):
"""Returns ``True`` if ``f`` is a unit algebraic number. """
return f.rep == [f.dom.one]
@property
def is_ground(f):
"""Returns ``True`` if ``f`` is an element of the ground domain. """
return not f.rep or len(f.rep) == 1
def __pos__(f):
return f
def __neg__(f):
return f.neg()
def __add__(f, g):
if isinstance(g, ANP):
return f.add(g)
else:
try:
return f.add(f.per(g))
except (CoercionFailed, TypeError):
return NotImplemented
def __radd__(f, g):
return f.__add__(g)
def __sub__(f, g):
if isinstance(g, ANP):
return f.sub(g)
else:
try:
return f.sub(f.per(g))
except (CoercionFailed, TypeError):
return NotImplemented
def __rsub__(f, g):
return (-f).__add__(g)
def __mul__(f, g):
if isinstance(g, ANP):
return f.mul(g)
else:
try:
return f.mul(f.per(g))
except (CoercionFailed, TypeError):
return NotImplemented
def __rmul__(f, g):
return f.__mul__(g)
def __pow__(f, n):
return f.pow(n)
def __divmod__(f, g):
return f.div(g)
def __mod__(f, g):
return f.rem(g)
def __truediv__(f, g):
if isinstance(g, ANP):
return f.quo(g)
else:
try:
return f.quo(f.per(g))
except (CoercionFailed, TypeError):
return NotImplemented
def __eq__(f, g):
try:
_, _, F, G, _ = f.unify(g)
return F == G
except UnificationFailed:
return False
def __ne__(f, g):
try:
_, _, F, G, _ = f.unify(g)
return F != G
except UnificationFailed:
return True
def __lt__(f, g):
_, _, F, G, _ = f.unify(g)
return F < G
def __le__(f, g):
_, _, F, G, _ = f.unify(g)
return F <= G
def __gt__(f, g):
_, _, F, G, _ = f.unify(g)
return F > G
def __ge__(f, g):
_, _, F, G, _ = f.unify(g)
return F >= G
def __bool__(f):
return bool(f.rep)
|
e29fe7957497d163cae7b41d1fa1ff48f10a66f4f95b62e894b94909d519f96d | """User-friendly public interface to polynomial functions. """
from functools import wraps, reduce
from operator import mul
from sympy.core import (
S, Basic, Expr, I, Integer, Add, Mul, Dummy, Tuple
)
from sympy.core.basic import preorder_traversal
from sympy.core.compatibility import iterable, ordered
from sympy.core.decorators import _sympifyit
from sympy.core.evalf import pure_complex
from sympy.core.function import Derivative
from sympy.core.mul import _keep_coeff
from sympy.core.relational import Relational
from sympy.core.symbol import Symbol
from sympy.core.sympify import sympify, _sympify
from sympy.logic.boolalg import BooleanAtom
from sympy.polys import polyoptions as options
from sympy.polys.constructor import construct_domain
from sympy.polys.domains import FF, QQ, ZZ
from sympy.polys.domains.domainelement import DomainElement
from sympy.polys.fglmtools import matrix_fglm
from sympy.polys.groebnertools import groebner as _groebner
from sympy.polys.monomials import Monomial
from sympy.polys.orderings import monomial_key
from sympy.polys.polyclasses import DMP, DMF, ANP
from sympy.polys.polyerrors import (
OperationNotSupported, DomainError,
CoercionFailed, UnificationFailed,
GeneratorsNeeded, PolynomialError,
MultivariatePolynomialError,
ExactQuotientFailed,
PolificationFailed,
ComputationFailed,
GeneratorsError,
)
from sympy.polys.polyutils import (
basic_from_dict,
_sort_gens,
_unify_gens,
_dict_reorder,
_dict_from_expr,
_parallel_dict_from_expr,
)
from sympy.polys.rationaltools import together
from sympy.polys.rootisolation import dup_isolate_real_roots_list
from sympy.utilities import group, sift, public, filldedent
from sympy.utilities.exceptions import SymPyDeprecationWarning
# Required to avoid errors
import sympy.polys
import mpmath
from mpmath.libmp.libhyper import NoConvergence
def _polifyit(func):
@wraps(func)
def wrapper(f, g):
g = _sympify(g)
if isinstance(g, Poly):
return func(f, g)
elif isinstance(g, Expr):
try:
g = f.from_expr(g, *f.gens)
except PolynomialError:
if g.is_Matrix:
return NotImplemented
expr_method = getattr(f.as_expr(), func.__name__)
result = expr_method(g)
if result is not NotImplemented:
SymPyDeprecationWarning(
feature="Mixing Poly with non-polynomial expressions in binary operations",
issue=18613,
deprecated_since_version="1.6",
useinstead="the as_expr or as_poly method to convert types").warn()
return result
else:
return func(f, g)
else:
return NotImplemented
return wrapper
@public
class Poly(Basic):
"""
Generic class for representing and operating on polynomial expressions.
See :ref:`polys-docs` for general documentation.
Poly is a subclass of Basic rather than Expr but instances can be
converted to Expr with the :py:meth:`~.Poly.as_expr` method.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
Create a univariate polynomial:
>>> Poly(x*(x**2 + x - 1)**2)
Poly(x**5 + 2*x**4 - x**3 - 2*x**2 + x, x, domain='ZZ')
Create a univariate polynomial with specific domain:
>>> from sympy import sqrt
>>> Poly(x**2 + 2*x + sqrt(3), domain='R')
Poly(1.0*x**2 + 2.0*x + 1.73205080756888, x, domain='RR')
Create a multivariate polynomial:
>>> Poly(y*x**2 + x*y + 1)
Poly(x**2*y + x*y + 1, x, y, domain='ZZ')
Create a univariate polynomial, where y is a constant:
>>> Poly(y*x**2 + x*y + 1,x)
Poly(y*x**2 + y*x + 1, x, domain='ZZ[y]')
You can evaluate the above polynomial as a function of y:
>>> Poly(y*x**2 + x*y + 1,x).eval(2)
6*y + 1
See Also
========
sympy.core.expr.Expr
"""
__slots__ = ('rep', 'gens')
is_commutative = True
is_Poly = True
_op_priority = 10.001
def __new__(cls, rep, *gens, **args):
"""Create a new polynomial instance out of something useful. """
opt = options.build_options(gens, args)
if 'order' in opt:
raise NotImplementedError("'order' keyword is not implemented yet")
if isinstance(rep, (DMP, DMF, ANP, DomainElement)):
return cls._from_domain_element(rep, opt)
elif iterable(rep, exclude=str):
if isinstance(rep, dict):
return cls._from_dict(rep, opt)
else:
return cls._from_list(list(rep), opt)
else:
rep = sympify(rep)
if rep.is_Poly:
return cls._from_poly(rep, opt)
else:
return cls._from_expr(rep, opt)
# Poly does not pass its args to Basic.__new__ to be stored in _args so we
# have to emulate them here with an args property that derives from rep
# and gens which are instance attributes. This also means we need to
# define _hashable_content. The _hashable_content is rep and gens but args
# uses expr instead of rep (expr is the Basic version of rep). Passing
# expr in args means that Basic methods like subs should work. Using rep
# otherwise means that Poly can remain more efficient than Basic by
# avoiding creating a Basic instance just to be hashable.
@classmethod
def new(cls, rep, *gens):
"""Construct :class:`Poly` instance from raw representation. """
if not isinstance(rep, DMP):
raise PolynomialError(
"invalid polynomial representation: %s" % rep)
elif rep.lev != len(gens) - 1:
raise PolynomialError("invalid arguments: %s, %s" % (rep, gens))
obj = Basic.__new__(cls)
obj.rep = rep
obj.gens = gens
return obj
@property
def expr(self):
return basic_from_dict(self.rep.to_sympy_dict(), *self.gens)
@property
def args(self):
return (self.expr,) + self.gens
def _hashable_content(self):
return (self.rep,) + self.gens
@classmethod
def from_dict(cls, rep, *gens, **args):
"""Construct a polynomial from a ``dict``. """
opt = options.build_options(gens, args)
return cls._from_dict(rep, opt)
@classmethod
def from_list(cls, rep, *gens, **args):
"""Construct a polynomial from a ``list``. """
opt = options.build_options(gens, args)
return cls._from_list(rep, opt)
@classmethod
def from_poly(cls, rep, *gens, **args):
"""Construct a polynomial from a polynomial. """
opt = options.build_options(gens, args)
return cls._from_poly(rep, opt)
@classmethod
def from_expr(cls, rep, *gens, **args):
"""Construct a polynomial from an expression. """
opt = options.build_options(gens, args)
return cls._from_expr(rep, opt)
@classmethod
def _from_dict(cls, rep, opt):
"""Construct a polynomial from a ``dict``. """
gens = opt.gens
if not gens:
raise GeneratorsNeeded(
"can't initialize from 'dict' without generators")
level = len(gens) - 1
domain = opt.domain
if domain is None:
domain, rep = construct_domain(rep, opt=opt)
else:
for monom, coeff in rep.items():
rep[monom] = domain.convert(coeff)
return cls.new(DMP.from_dict(rep, level, domain), *gens)
@classmethod
def _from_list(cls, rep, opt):
"""Construct a polynomial from a ``list``. """
gens = opt.gens
if not gens:
raise GeneratorsNeeded(
"can't initialize from 'list' without generators")
elif len(gens) != 1:
raise MultivariatePolynomialError(
"'list' representation not supported")
level = len(gens) - 1
domain = opt.domain
if domain is None:
domain, rep = construct_domain(rep, opt=opt)
else:
rep = list(map(domain.convert, rep))
return cls.new(DMP.from_list(rep, level, domain), *gens)
@classmethod
def _from_poly(cls, rep, opt):
"""Construct a polynomial from a polynomial. """
if cls != rep.__class__:
rep = cls.new(rep.rep, *rep.gens)
gens = opt.gens
field = opt.field
domain = opt.domain
if gens and rep.gens != gens:
if set(rep.gens) != set(gens):
return cls._from_expr(rep.as_expr(), opt)
else:
rep = rep.reorder(*gens)
if 'domain' in opt and domain:
rep = rep.set_domain(domain)
elif field is True:
rep = rep.to_field()
return rep
@classmethod
def _from_expr(cls, rep, opt):
"""Construct a polynomial from an expression. """
rep, opt = _dict_from_expr(rep, opt)
return cls._from_dict(rep, opt)
@classmethod
def _from_domain_element(cls, rep, opt):
gens = opt.gens
domain = opt.domain
level = len(gens) - 1
rep = [domain.convert(rep)]
return cls.new(DMP.from_list(rep, level, domain), *gens)
def __hash__(self):
return super().__hash__()
@property
def free_symbols(self):
"""
Free symbols of a polynomial expression.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y, z
>>> Poly(x**2 + 1).free_symbols
{x}
>>> Poly(x**2 + y).free_symbols
{x, y}
>>> Poly(x**2 + y, x).free_symbols
{x, y}
>>> Poly(x**2 + y, x, z).free_symbols
{x, y}
"""
symbols = set()
gens = self.gens
for i in range(len(gens)):
for monom in self.monoms():
if monom[i]:
symbols |= gens[i].free_symbols
break
return symbols | self.free_symbols_in_domain
@property
def free_symbols_in_domain(self):
"""
Free symbols of the domain of ``self``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + 1).free_symbols_in_domain
set()
>>> Poly(x**2 + y).free_symbols_in_domain
set()
>>> Poly(x**2 + y, x).free_symbols_in_domain
{y}
"""
domain, symbols = self.rep.dom, set()
if domain.is_Composite:
for gen in domain.symbols:
symbols |= gen.free_symbols
elif domain.is_EX:
for coeff in self.coeffs():
symbols |= coeff.free_symbols
return symbols
@property
def gen(self):
"""
Return the principal generator.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).gen
x
"""
return self.gens[0]
@property
def domain(self):
"""Get the ground domain of a :py:class:`~.Poly`
Returns
=======
:py:class:`~.Domain`:
Ground domain of the :py:class:`~.Poly`.
Examples
========
>>> from sympy import Poly, Symbol
>>> x = Symbol('x')
>>> p = Poly(x**2 + x)
>>> p
Poly(x**2 + x, x, domain='ZZ')
>>> p.domain
ZZ
"""
return self.get_domain()
@property
def zero(self):
"""Return zero polynomial with ``self``'s properties. """
return self.new(self.rep.zero(self.rep.lev, self.rep.dom), *self.gens)
@property
def one(self):
"""Return one polynomial with ``self``'s properties. """
return self.new(self.rep.one(self.rep.lev, self.rep.dom), *self.gens)
@property
def unit(self):
"""Return unit polynomial with ``self``'s properties. """
return self.new(self.rep.unit(self.rep.lev, self.rep.dom), *self.gens)
def unify(f, g):
"""
Make ``f`` and ``g`` belong to the same domain.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> f, g = Poly(x/2 + 1), Poly(2*x + 1)
>>> f
Poly(1/2*x + 1, x, domain='QQ')
>>> g
Poly(2*x + 1, x, domain='ZZ')
>>> F, G = f.unify(g)
>>> F
Poly(1/2*x + 1, x, domain='QQ')
>>> G
Poly(2*x + 1, x, domain='QQ')
"""
_, per, F, G = f._unify(g)
return per(F), per(G)
def _unify(f, g):
g = sympify(g)
if not g.is_Poly:
try:
return f.rep.dom, f.per, f.rep, f.rep.per(f.rep.dom.from_sympy(g))
except CoercionFailed:
raise UnificationFailed("can't unify %s with %s" % (f, g))
if isinstance(f.rep, DMP) and isinstance(g.rep, DMP):
gens = _unify_gens(f.gens, g.gens)
dom, lev = f.rep.dom.unify(g.rep.dom, gens), len(gens) - 1
if f.gens != gens:
f_monoms, f_coeffs = _dict_reorder(
f.rep.to_dict(), f.gens, gens)
if f.rep.dom != dom:
f_coeffs = [dom.convert(c, f.rep.dom) for c in f_coeffs]
F = DMP(dict(list(zip(f_monoms, f_coeffs))), dom, lev)
else:
F = f.rep.convert(dom)
if g.gens != gens:
g_monoms, g_coeffs = _dict_reorder(
g.rep.to_dict(), g.gens, gens)
if g.rep.dom != dom:
g_coeffs = [dom.convert(c, g.rep.dom) for c in g_coeffs]
G = DMP(dict(list(zip(g_monoms, g_coeffs))), dom, lev)
else:
G = g.rep.convert(dom)
else:
raise UnificationFailed("can't unify %s with %s" % (f, g))
cls = f.__class__
def per(rep, dom=dom, gens=gens, remove=None):
if remove is not None:
gens = gens[:remove] + gens[remove + 1:]
if not gens:
return dom.to_sympy(rep)
return cls.new(rep, *gens)
return dom, per, F, G
def per(f, rep, gens=None, remove=None):
"""
Create a Poly out of the given representation.
Examples
========
>>> from sympy import Poly, ZZ
>>> from sympy.abc import x, y
>>> from sympy.polys.polyclasses import DMP
>>> a = Poly(x**2 + 1)
>>> a.per(DMP([ZZ(1), ZZ(1)], ZZ), gens=[y])
Poly(y + 1, y, domain='ZZ')
"""
if gens is None:
gens = f.gens
if remove is not None:
gens = gens[:remove] + gens[remove + 1:]
if not gens:
return f.rep.dom.to_sympy(rep)
return f.__class__.new(rep, *gens)
def set_domain(f, domain):
"""Set the ground domain of ``f``. """
opt = options.build_options(f.gens, {'domain': domain})
return f.per(f.rep.convert(opt.domain))
def get_domain(f):
"""Get the ground domain of ``f``. """
return f.rep.dom
def set_modulus(f, modulus):
"""
Set the modulus of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(5*x**2 + 2*x - 1, x).set_modulus(2)
Poly(x**2 + 1, x, modulus=2)
"""
modulus = options.Modulus.preprocess(modulus)
return f.set_domain(FF(modulus))
def get_modulus(f):
"""
Get the modulus of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, modulus=2).get_modulus()
2
"""
domain = f.get_domain()
if domain.is_FiniteField:
return Integer(domain.characteristic())
else:
raise PolynomialError("not a polynomial over a Galois field")
def _eval_subs(f, old, new):
"""Internal implementation of :func:`subs`. """
if old in f.gens:
if new.is_number:
return f.eval(old, new)
else:
try:
return f.replace(old, new)
except PolynomialError:
pass
return f.as_expr().subs(old, new)
def exclude(f):
"""
Remove unnecessary generators from ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import a, b, c, d, x
>>> Poly(a + x, a, b, c, d, x).exclude()
Poly(a + x, a, x, domain='ZZ')
"""
J, new = f.rep.exclude()
gens = []
for j in range(len(f.gens)):
if j not in J:
gens.append(f.gens[j])
return f.per(new, gens=gens)
def replace(f, x, y=None, **_ignore):
# XXX this does not match Basic's signature
"""
Replace ``x`` with ``y`` in generators list.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + 1, x).replace(x, y)
Poly(y**2 + 1, y, domain='ZZ')
"""
if y is None:
if f.is_univariate:
x, y = f.gen, x
else:
raise PolynomialError(
"syntax supported only in univariate case")
if x == y or x not in f.gens:
return f
if x in f.gens and y not in f.gens:
dom = f.get_domain()
if not dom.is_Composite or y not in dom.symbols:
gens = list(f.gens)
gens[gens.index(x)] = y
return f.per(f.rep, gens=gens)
raise PolynomialError("can't replace %s with %s in %s" % (x, y, f))
def match(f, *args, **kwargs):
"""Match expression from Poly. See Basic.match()"""
return f.as_expr().match(*args, **kwargs)
def reorder(f, *gens, **args):
"""
Efficiently apply new order of generators.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + x*y**2, x, y).reorder(y, x)
Poly(y**2*x + x**2, y, x, domain='ZZ')
"""
opt = options.Options((), args)
if not gens:
gens = _sort_gens(f.gens, opt=opt)
elif set(f.gens) != set(gens):
raise PolynomialError(
"generators list can differ only up to order of elements")
rep = dict(list(zip(*_dict_reorder(f.rep.to_dict(), f.gens, gens))))
return f.per(DMP(rep, f.rep.dom, len(gens) - 1), gens=gens)
def ltrim(f, gen):
"""
Remove dummy generators from ``f`` that are to the left of
specified ``gen`` in the generators as ordered. When ``gen``
is an integer, it refers to the generator located at that
position within the tuple of generators of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y, z
>>> Poly(y**2 + y*z**2, x, y, z).ltrim(y)
Poly(y**2 + y*z**2, y, z, domain='ZZ')
>>> Poly(z, x, y, z).ltrim(-1)
Poly(z, z, domain='ZZ')
"""
rep = f.as_dict(native=True)
j = f._gen_to_level(gen)
terms = {}
for monom, coeff in rep.items():
if any(monom[:j]):
# some generator is used in the portion to be trimmed
raise PolynomialError("can't left trim %s" % f)
terms[monom[j:]] = coeff
gens = f.gens[j:]
return f.new(DMP.from_dict(terms, len(gens) - 1, f.rep.dom), *gens)
def has_only_gens(f, *gens):
"""
Return ``True`` if ``Poly(f, *gens)`` retains ground domain.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y, z
>>> Poly(x*y + 1, x, y, z).has_only_gens(x, y)
True
>>> Poly(x*y + z, x, y, z).has_only_gens(x, y)
False
"""
indices = set()
for gen in gens:
try:
index = f.gens.index(gen)
except ValueError:
raise GeneratorsError(
"%s doesn't have %s as generator" % (f, gen))
else:
indices.add(index)
for monom in f.monoms():
for i, elt in enumerate(monom):
if i not in indices and elt:
return False
return True
def to_ring(f):
"""
Make the ground domain a ring.
Examples
========
>>> from sympy import Poly, QQ
>>> from sympy.abc import x
>>> Poly(x**2 + 1, domain=QQ).to_ring()
Poly(x**2 + 1, x, domain='ZZ')
"""
if hasattr(f.rep, 'to_ring'):
result = f.rep.to_ring()
else: # pragma: no cover
raise OperationNotSupported(f, 'to_ring')
return f.per(result)
def to_field(f):
"""
Make the ground domain a field.
Examples
========
>>> from sympy import Poly, ZZ
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x, domain=ZZ).to_field()
Poly(x**2 + 1, x, domain='QQ')
"""
if hasattr(f.rep, 'to_field'):
result = f.rep.to_field()
else: # pragma: no cover
raise OperationNotSupported(f, 'to_field')
return f.per(result)
def to_exact(f):
"""
Make the ground domain exact.
Examples
========
>>> from sympy import Poly, RR
>>> from sympy.abc import x
>>> Poly(x**2 + 1.0, x, domain=RR).to_exact()
Poly(x**2 + 1, x, domain='QQ')
"""
if hasattr(f.rep, 'to_exact'):
result = f.rep.to_exact()
else: # pragma: no cover
raise OperationNotSupported(f, 'to_exact')
return f.per(result)
def retract(f, field=None):
"""
Recalculate the ground domain of a polynomial.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> f = Poly(x**2 + 1, x, domain='QQ[y]')
>>> f
Poly(x**2 + 1, x, domain='QQ[y]')
>>> f.retract()
Poly(x**2 + 1, x, domain='ZZ')
>>> f.retract(field=True)
Poly(x**2 + 1, x, domain='QQ')
"""
dom, rep = construct_domain(f.as_dict(zero=True),
field=field, composite=f.domain.is_Composite or None)
return f.from_dict(rep, f.gens, domain=dom)
def slice(f, x, m, n=None):
"""Take a continuous subsequence of terms of ``f``. """
if n is None:
j, m, n = 0, x, m
else:
j = f._gen_to_level(x)
m, n = int(m), int(n)
if hasattr(f.rep, 'slice'):
result = f.rep.slice(m, n, j)
else: # pragma: no cover
raise OperationNotSupported(f, 'slice')
return f.per(result)
def coeffs(f, order=None):
"""
Returns all non-zero coefficients from ``f`` in lex order.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**3 + 2*x + 3, x).coeffs()
[1, 2, 3]
See Also
========
all_coeffs
coeff_monomial
nth
"""
return [f.rep.dom.to_sympy(c) for c in f.rep.coeffs(order=order)]
def monoms(f, order=None):
"""
Returns all non-zero monomials from ``f`` in lex order.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + 2*x*y**2 + x*y + 3*y, x, y).monoms()
[(2, 0), (1, 2), (1, 1), (0, 1)]
See Also
========
all_monoms
"""
return f.rep.monoms(order=order)
def terms(f, order=None):
"""
Returns all non-zero terms from ``f`` in lex order.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + 2*x*y**2 + x*y + 3*y, x, y).terms()
[((2, 0), 1), ((1, 2), 2), ((1, 1), 1), ((0, 1), 3)]
See Also
========
all_terms
"""
return [(m, f.rep.dom.to_sympy(c)) for m, c in f.rep.terms(order=order)]
def all_coeffs(f):
"""
Returns all coefficients from a univariate polynomial ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**3 + 2*x - 1, x).all_coeffs()
[1, 0, 2, -1]
"""
return [f.rep.dom.to_sympy(c) for c in f.rep.all_coeffs()]
def all_monoms(f):
"""
Returns all monomials from a univariate polynomial ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**3 + 2*x - 1, x).all_monoms()
[(3,), (2,), (1,), (0,)]
See Also
========
all_terms
"""
return f.rep.all_monoms()
def all_terms(f):
"""
Returns all terms from a univariate polynomial ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**3 + 2*x - 1, x).all_terms()
[((3,), 1), ((2,), 0), ((1,), 2), ((0,), -1)]
"""
return [(m, f.rep.dom.to_sympy(c)) for m, c in f.rep.all_terms()]
def termwise(f, func, *gens, **args):
"""
Apply a function to all terms of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> def func(k, coeff):
... k = k[0]
... return coeff//10**(2-k)
>>> Poly(x**2 + 20*x + 400).termwise(func)
Poly(x**2 + 2*x + 4, x, domain='ZZ')
"""
terms = {}
for monom, coeff in f.terms():
result = func(monom, coeff)
if isinstance(result, tuple):
monom, coeff = result
else:
coeff = result
if coeff:
if monom not in terms:
terms[monom] = coeff
else:
raise PolynomialError(
"%s monomial was generated twice" % monom)
return f.from_dict(terms, *(gens or f.gens), **args)
def length(f):
"""
Returns the number of non-zero terms in ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 2*x - 1).length()
3
"""
return len(f.as_dict())
def as_dict(f, native=False, zero=False):
"""
Switch to a ``dict`` representation.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + 2*x*y**2 - y, x, y).as_dict()
{(0, 1): -1, (1, 2): 2, (2, 0): 1}
"""
if native:
return f.rep.to_dict(zero=zero)
else:
return f.rep.to_sympy_dict(zero=zero)
def as_list(f, native=False):
"""Switch to a ``list`` representation. """
if native:
return f.rep.to_list()
else:
return f.rep.to_sympy_list()
def as_expr(f, *gens):
"""
Convert a Poly instance to an Expr instance.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> f = Poly(x**2 + 2*x*y**2 - y, x, y)
>>> f.as_expr()
x**2 + 2*x*y**2 - y
>>> f.as_expr({x: 5})
10*y**2 - y + 25
>>> f.as_expr(5, 6)
379
"""
if not gens:
return f.expr
if len(gens) == 1 and isinstance(gens[0], dict):
mapping = gens[0]
gens = list(f.gens)
for gen, value in mapping.items():
try:
index = gens.index(gen)
except ValueError:
raise GeneratorsError(
"%s doesn't have %s as generator" % (f, gen))
else:
gens[index] = value
return basic_from_dict(f.rep.to_sympy_dict(), *gens)
def as_poly(self, *gens, **args):
"""Converts ``self`` to a polynomial or returns ``None``.
>>> from sympy import sin
>>> from sympy.abc import x, y
>>> print((x**2 + x*y).as_poly())
Poly(x**2 + x*y, x, y, domain='ZZ')
>>> print((x**2 + x*y).as_poly(x, y))
Poly(x**2 + x*y, x, y, domain='ZZ')
>>> print((x**2 + sin(y)).as_poly(x, y))
None
"""
try:
poly = Poly(self, *gens, **args)
if not poly.is_Poly:
return None
else:
return poly
except PolynomialError:
return None
def lift(f):
"""
Convert algebraic coefficients to rationals.
Examples
========
>>> from sympy import Poly, I
>>> from sympy.abc import x
>>> Poly(x**2 + I*x + 1, x, extension=I).lift()
Poly(x**4 + 3*x**2 + 1, x, domain='QQ')
"""
if hasattr(f.rep, 'lift'):
result = f.rep.lift()
else: # pragma: no cover
raise OperationNotSupported(f, 'lift')
return f.per(result)
def deflate(f):
"""
Reduce degree of ``f`` by mapping ``x_i**m`` to ``y_i``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**6*y**2 + x**3 + 1, x, y).deflate()
((3, 2), Poly(x**2*y + x + 1, x, y, domain='ZZ'))
"""
if hasattr(f.rep, 'deflate'):
J, result = f.rep.deflate()
else: # pragma: no cover
raise OperationNotSupported(f, 'deflate')
return J, f.per(result)
def inject(f, front=False):
"""
Inject ground domain generators into ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> f = Poly(x**2*y + x*y**3 + x*y + 1, x)
>>> f.inject()
Poly(x**2*y + x*y**3 + x*y + 1, x, y, domain='ZZ')
>>> f.inject(front=True)
Poly(y**3*x + y*x**2 + y*x + 1, y, x, domain='ZZ')
"""
dom = f.rep.dom
if dom.is_Numerical:
return f
elif not dom.is_Poly:
raise DomainError("can't inject generators over %s" % dom)
if hasattr(f.rep, 'inject'):
result = f.rep.inject(front=front)
else: # pragma: no cover
raise OperationNotSupported(f, 'inject')
if front:
gens = dom.symbols + f.gens
else:
gens = f.gens + dom.symbols
return f.new(result, *gens)
def eject(f, *gens):
"""
Eject selected generators into the ground domain.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> f = Poly(x**2*y + x*y**3 + x*y + 1, x, y)
>>> f.eject(x)
Poly(x*y**3 + (x**2 + x)*y + 1, y, domain='ZZ[x]')
>>> f.eject(y)
Poly(y*x**2 + (y**3 + y)*x + 1, x, domain='ZZ[y]')
"""
dom = f.rep.dom
if not dom.is_Numerical:
raise DomainError("can't eject generators over %s" % dom)
k = len(gens)
if f.gens[:k] == gens:
_gens, front = f.gens[k:], True
elif f.gens[-k:] == gens:
_gens, front = f.gens[:-k], False
else:
raise NotImplementedError(
"can only eject front or back generators")
dom = dom.inject(*gens)
if hasattr(f.rep, 'eject'):
result = f.rep.eject(dom, front=front)
else: # pragma: no cover
raise OperationNotSupported(f, 'eject')
return f.new(result, *_gens)
def terms_gcd(f):
"""
Remove GCD of terms from the polynomial ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**6*y**2 + x**3*y, x, y).terms_gcd()
((3, 1), Poly(x**3*y + 1, x, y, domain='ZZ'))
"""
if hasattr(f.rep, 'terms_gcd'):
J, result = f.rep.terms_gcd()
else: # pragma: no cover
raise OperationNotSupported(f, 'terms_gcd')
return J, f.per(result)
def add_ground(f, coeff):
"""
Add an element of the ground domain to ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x + 1).add_ground(2)
Poly(x + 3, x, domain='ZZ')
"""
if hasattr(f.rep, 'add_ground'):
result = f.rep.add_ground(coeff)
else: # pragma: no cover
raise OperationNotSupported(f, 'add_ground')
return f.per(result)
def sub_ground(f, coeff):
"""
Subtract an element of the ground domain from ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x + 1).sub_ground(2)
Poly(x - 1, x, domain='ZZ')
"""
if hasattr(f.rep, 'sub_ground'):
result = f.rep.sub_ground(coeff)
else: # pragma: no cover
raise OperationNotSupported(f, 'sub_ground')
return f.per(result)
def mul_ground(f, coeff):
"""
Multiply ``f`` by a an element of the ground domain.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x + 1).mul_ground(2)
Poly(2*x + 2, x, domain='ZZ')
"""
if hasattr(f.rep, 'mul_ground'):
result = f.rep.mul_ground(coeff)
else: # pragma: no cover
raise OperationNotSupported(f, 'mul_ground')
return f.per(result)
def quo_ground(f, coeff):
"""
Quotient of ``f`` by a an element of the ground domain.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(2*x + 4).quo_ground(2)
Poly(x + 2, x, domain='ZZ')
>>> Poly(2*x + 3).quo_ground(2)
Poly(x + 1, x, domain='ZZ')
"""
if hasattr(f.rep, 'quo_ground'):
result = f.rep.quo_ground(coeff)
else: # pragma: no cover
raise OperationNotSupported(f, 'quo_ground')
return f.per(result)
def exquo_ground(f, coeff):
"""
Exact quotient of ``f`` by a an element of the ground domain.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(2*x + 4).exquo_ground(2)
Poly(x + 2, x, domain='ZZ')
>>> Poly(2*x + 3).exquo_ground(2)
Traceback (most recent call last):
...
ExactQuotientFailed: 2 does not divide 3 in ZZ
"""
if hasattr(f.rep, 'exquo_ground'):
result = f.rep.exquo_ground(coeff)
else: # pragma: no cover
raise OperationNotSupported(f, 'exquo_ground')
return f.per(result)
def abs(f):
"""
Make all coefficients in ``f`` positive.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 1, x).abs()
Poly(x**2 + 1, x, domain='ZZ')
"""
if hasattr(f.rep, 'abs'):
result = f.rep.abs()
else: # pragma: no cover
raise OperationNotSupported(f, 'abs')
return f.per(result)
def neg(f):
"""
Negate all coefficients in ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 1, x).neg()
Poly(-x**2 + 1, x, domain='ZZ')
>>> -Poly(x**2 - 1, x)
Poly(-x**2 + 1, x, domain='ZZ')
"""
if hasattr(f.rep, 'neg'):
result = f.rep.neg()
else: # pragma: no cover
raise OperationNotSupported(f, 'neg')
return f.per(result)
def add(f, g):
"""
Add two polynomials ``f`` and ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).add(Poly(x - 2, x))
Poly(x**2 + x - 1, x, domain='ZZ')
>>> Poly(x**2 + 1, x) + Poly(x - 2, x)
Poly(x**2 + x - 1, x, domain='ZZ')
"""
g = sympify(g)
if not g.is_Poly:
return f.add_ground(g)
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'add'):
result = F.add(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'add')
return per(result)
def sub(f, g):
"""
Subtract two polynomials ``f`` and ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).sub(Poly(x - 2, x))
Poly(x**2 - x + 3, x, domain='ZZ')
>>> Poly(x**2 + 1, x) - Poly(x - 2, x)
Poly(x**2 - x + 3, x, domain='ZZ')
"""
g = sympify(g)
if not g.is_Poly:
return f.sub_ground(g)
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'sub'):
result = F.sub(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'sub')
return per(result)
def mul(f, g):
"""
Multiply two polynomials ``f`` and ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).mul(Poly(x - 2, x))
Poly(x**3 - 2*x**2 + x - 2, x, domain='ZZ')
>>> Poly(x**2 + 1, x)*Poly(x - 2, x)
Poly(x**3 - 2*x**2 + x - 2, x, domain='ZZ')
"""
g = sympify(g)
if not g.is_Poly:
return f.mul_ground(g)
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'mul'):
result = F.mul(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'mul')
return per(result)
def sqr(f):
"""
Square a polynomial ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x - 2, x).sqr()
Poly(x**2 - 4*x + 4, x, domain='ZZ')
>>> Poly(x - 2, x)**2
Poly(x**2 - 4*x + 4, x, domain='ZZ')
"""
if hasattr(f.rep, 'sqr'):
result = f.rep.sqr()
else: # pragma: no cover
raise OperationNotSupported(f, 'sqr')
return f.per(result)
def pow(f, n):
"""
Raise ``f`` to a non-negative power ``n``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x - 2, x).pow(3)
Poly(x**3 - 6*x**2 + 12*x - 8, x, domain='ZZ')
>>> Poly(x - 2, x)**3
Poly(x**3 - 6*x**2 + 12*x - 8, x, domain='ZZ')
"""
n = int(n)
if hasattr(f.rep, 'pow'):
result = f.rep.pow(n)
else: # pragma: no cover
raise OperationNotSupported(f, 'pow')
return f.per(result)
def pdiv(f, g):
"""
Polynomial pseudo-division of ``f`` by ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).pdiv(Poly(2*x - 4, x))
(Poly(2*x + 4, x, domain='ZZ'), Poly(20, x, domain='ZZ'))
"""
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'pdiv'):
q, r = F.pdiv(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'pdiv')
return per(q), per(r)
def prem(f, g):
"""
Polynomial pseudo-remainder of ``f`` by ``g``.
Caveat: The function prem(f, g, x) can be safely used to compute
in Z[x] _only_ subresultant polynomial remainder sequences (prs's).
To safely compute Euclidean and Sturmian prs's in Z[x]
employ anyone of the corresponding functions found in
the module sympy.polys.subresultants_qq_zz. The functions
in the module with suffix _pg compute prs's in Z[x] employing
rem(f, g, x), whereas the functions with suffix _amv
compute prs's in Z[x] employing rem_z(f, g, x).
The function rem_z(f, g, x) differs from prem(f, g, x) in that
to compute the remainder polynomials in Z[x] it premultiplies
the divident times the absolute value of the leading coefficient
of the divisor raised to the power degree(f, x) - degree(g, x) + 1.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).prem(Poly(2*x - 4, x))
Poly(20, x, domain='ZZ')
"""
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'prem'):
result = F.prem(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'prem')
return per(result)
def pquo(f, g):
"""
Polynomial pseudo-quotient of ``f`` by ``g``.
See the Caveat note in the function prem(f, g).
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).pquo(Poly(2*x - 4, x))
Poly(2*x + 4, x, domain='ZZ')
>>> Poly(x**2 - 1, x).pquo(Poly(2*x - 2, x))
Poly(2*x + 2, x, domain='ZZ')
"""
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'pquo'):
result = F.pquo(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'pquo')
return per(result)
def pexquo(f, g):
"""
Polynomial exact pseudo-quotient of ``f`` by ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 1, x).pexquo(Poly(2*x - 2, x))
Poly(2*x + 2, x, domain='ZZ')
>>> Poly(x**2 + 1, x).pexquo(Poly(2*x - 4, x))
Traceback (most recent call last):
...
ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1
"""
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'pexquo'):
try:
result = F.pexquo(G)
except ExactQuotientFailed as exc:
raise exc.new(f.as_expr(), g.as_expr())
else: # pragma: no cover
raise OperationNotSupported(f, 'pexquo')
return per(result)
def div(f, g, auto=True):
"""
Polynomial division with remainder of ``f`` by ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).div(Poly(2*x - 4, x))
(Poly(1/2*x + 1, x, domain='QQ'), Poly(5, x, domain='QQ'))
>>> Poly(x**2 + 1, x).div(Poly(2*x - 4, x), auto=False)
(Poly(0, x, domain='ZZ'), Poly(x**2 + 1, x, domain='ZZ'))
"""
dom, per, F, G = f._unify(g)
retract = False
if auto and dom.is_Ring and not dom.is_Field:
F, G = F.to_field(), G.to_field()
retract = True
if hasattr(f.rep, 'div'):
q, r = F.div(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'div')
if retract:
try:
Q, R = q.to_ring(), r.to_ring()
except CoercionFailed:
pass
else:
q, r = Q, R
return per(q), per(r)
def rem(f, g, auto=True):
"""
Computes the polynomial remainder of ``f`` by ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).rem(Poly(2*x - 4, x))
Poly(5, x, domain='ZZ')
>>> Poly(x**2 + 1, x).rem(Poly(2*x - 4, x), auto=False)
Poly(x**2 + 1, x, domain='ZZ')
"""
dom, per, F, G = f._unify(g)
retract = False
if auto and dom.is_Ring and not dom.is_Field:
F, G = F.to_field(), G.to_field()
retract = True
if hasattr(f.rep, 'rem'):
r = F.rem(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'rem')
if retract:
try:
r = r.to_ring()
except CoercionFailed:
pass
return per(r)
def quo(f, g, auto=True):
"""
Computes polynomial quotient of ``f`` by ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).quo(Poly(2*x - 4, x))
Poly(1/2*x + 1, x, domain='QQ')
>>> Poly(x**2 - 1, x).quo(Poly(x - 1, x))
Poly(x + 1, x, domain='ZZ')
"""
dom, per, F, G = f._unify(g)
retract = False
if auto and dom.is_Ring and not dom.is_Field:
F, G = F.to_field(), G.to_field()
retract = True
if hasattr(f.rep, 'quo'):
q = F.quo(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'quo')
if retract:
try:
q = q.to_ring()
except CoercionFailed:
pass
return per(q)
def exquo(f, g, auto=True):
"""
Computes polynomial exact quotient of ``f`` by ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 1, x).exquo(Poly(x - 1, x))
Poly(x + 1, x, domain='ZZ')
>>> Poly(x**2 + 1, x).exquo(Poly(2*x - 4, x))
Traceback (most recent call last):
...
ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1
"""
dom, per, F, G = f._unify(g)
retract = False
if auto and dom.is_Ring and not dom.is_Field:
F, G = F.to_field(), G.to_field()
retract = True
if hasattr(f.rep, 'exquo'):
try:
q = F.exquo(G)
except ExactQuotientFailed as exc:
raise exc.new(f.as_expr(), g.as_expr())
else: # pragma: no cover
raise OperationNotSupported(f, 'exquo')
if retract:
try:
q = q.to_ring()
except CoercionFailed:
pass
return per(q)
def _gen_to_level(f, gen):
"""Returns level associated with the given generator. """
if isinstance(gen, int):
length = len(f.gens)
if -length <= gen < length:
if gen < 0:
return length + gen
else:
return gen
else:
raise PolynomialError("-%s <= gen < %s expected, got %s" %
(length, length, gen))
else:
try:
return f.gens.index(sympify(gen))
except ValueError:
raise PolynomialError(
"a valid generator expected, got %s" % gen)
def degree(f, gen=0):
"""
Returns degree of ``f`` in ``x_j``.
The degree of 0 is negative infinity.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + y*x + 1, x, y).degree()
2
>>> Poly(x**2 + y*x + y, x, y).degree(y)
1
>>> Poly(0, x).degree()
-oo
"""
j = f._gen_to_level(gen)
if hasattr(f.rep, 'degree'):
return f.rep.degree(j)
else: # pragma: no cover
raise OperationNotSupported(f, 'degree')
def degree_list(f):
"""
Returns a list of degrees of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + y*x + 1, x, y).degree_list()
(2, 1)
"""
if hasattr(f.rep, 'degree_list'):
return f.rep.degree_list()
else: # pragma: no cover
raise OperationNotSupported(f, 'degree_list')
def total_degree(f):
"""
Returns the total degree of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + y*x + 1, x, y).total_degree()
2
>>> Poly(x + y**5, x, y).total_degree()
5
"""
if hasattr(f.rep, 'total_degree'):
return f.rep.total_degree()
else: # pragma: no cover
raise OperationNotSupported(f, 'total_degree')
def homogenize(f, s):
"""
Returns the homogeneous polynomial of ``f``.
A homogeneous polynomial is a polynomial whose all monomials with
non-zero coefficients have the same total degree. If you only
want to check if a polynomial is homogeneous, then use
:func:`Poly.is_homogeneous`. If you want not only to check if a
polynomial is homogeneous but also compute its homogeneous order,
then use :func:`Poly.homogeneous_order`.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y, z
>>> f = Poly(x**5 + 2*x**2*y**2 + 9*x*y**3)
>>> f.homogenize(z)
Poly(x**5 + 2*x**2*y**2*z + 9*x*y**3*z, x, y, z, domain='ZZ')
"""
if not isinstance(s, Symbol):
raise TypeError("``Symbol`` expected, got %s" % type(s))
if s in f.gens:
i = f.gens.index(s)
gens = f.gens
else:
i = len(f.gens)
gens = f.gens + (s,)
if hasattr(f.rep, 'homogenize'):
return f.per(f.rep.homogenize(i), gens=gens)
raise OperationNotSupported(f, 'homogeneous_order')
def homogeneous_order(f):
"""
Returns the homogeneous order of ``f``.
A homogeneous polynomial is a polynomial whose all monomials with
non-zero coefficients have the same total degree. This degree is
the homogeneous order of ``f``. If you only want to check if a
polynomial is homogeneous, then use :func:`Poly.is_homogeneous`.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> f = Poly(x**5 + 2*x**3*y**2 + 9*x*y**4)
>>> f.homogeneous_order()
5
"""
if hasattr(f.rep, 'homogeneous_order'):
return f.rep.homogeneous_order()
else: # pragma: no cover
raise OperationNotSupported(f, 'homogeneous_order')
def LC(f, order=None):
"""
Returns the leading coefficient of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(4*x**3 + 2*x**2 + 3*x, x).LC()
4
"""
if order is not None:
return f.coeffs(order)[0]
if hasattr(f.rep, 'LC'):
result = f.rep.LC()
else: # pragma: no cover
raise OperationNotSupported(f, 'LC')
return f.rep.dom.to_sympy(result)
def TC(f):
"""
Returns the trailing coefficient of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**3 + 2*x**2 + 3*x, x).TC()
0
"""
if hasattr(f.rep, 'TC'):
result = f.rep.TC()
else: # pragma: no cover
raise OperationNotSupported(f, 'TC')
return f.rep.dom.to_sympy(result)
def EC(f, order=None):
"""
Returns the last non-zero coefficient of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**3 + 2*x**2 + 3*x, x).EC()
3
"""
if hasattr(f.rep, 'coeffs'):
return f.coeffs(order)[-1]
else: # pragma: no cover
raise OperationNotSupported(f, 'EC')
def coeff_monomial(f, monom):
"""
Returns the coefficient of ``monom`` in ``f`` if there, else None.
Examples
========
>>> from sympy import Poly, exp
>>> from sympy.abc import x, y
>>> p = Poly(24*x*y*exp(8) + 23*x, x, y)
>>> p.coeff_monomial(x)
23
>>> p.coeff_monomial(y)
0
>>> p.coeff_monomial(x*y)
24*exp(8)
Note that ``Expr.coeff()`` behaves differently, collecting terms
if possible; the Poly must be converted to an Expr to use that
method, however:
>>> p.as_expr().coeff(x)
24*y*exp(8) + 23
>>> p.as_expr().coeff(y)
24*x*exp(8)
>>> p.as_expr().coeff(x*y)
24*exp(8)
See Also
========
nth: more efficient query using exponents of the monomial's generators
"""
return f.nth(*Monomial(monom, f.gens).exponents)
def nth(f, *N):
"""
Returns the ``n``-th coefficient of ``f`` where ``N`` are the
exponents of the generators in the term of interest.
Examples
========
>>> from sympy import Poly, sqrt
>>> from sympy.abc import x, y
>>> Poly(x**3 + 2*x**2 + 3*x, x).nth(2)
2
>>> Poly(x**3 + 2*x*y**2 + y**2, x, y).nth(1, 2)
2
>>> Poly(4*sqrt(x)*y)
Poly(4*y*(sqrt(x)), y, sqrt(x), domain='ZZ')
>>> _.nth(1, 1)
4
See Also
========
coeff_monomial
"""
if hasattr(f.rep, 'nth'):
if len(N) != len(f.gens):
raise ValueError('exponent of each generator must be specified')
result = f.rep.nth(*list(map(int, N)))
else: # pragma: no cover
raise OperationNotSupported(f, 'nth')
return f.rep.dom.to_sympy(result)
def coeff(f, x, n=1, right=False):
# the semantics of coeff_monomial and Expr.coeff are different;
# if someone is working with a Poly, they should be aware of the
# differences and chose the method best suited for the query.
# Alternatively, a pure-polys method could be written here but
# at this time the ``right`` keyword would be ignored because Poly
# doesn't work with non-commutatives.
raise NotImplementedError(
'Either convert to Expr with `as_expr` method '
'to use Expr\'s coeff method or else use the '
'`coeff_monomial` method of Polys.')
def LM(f, order=None):
"""
Returns the leading monomial of ``f``.
The Leading monomial signifies the monomial having
the highest power of the principal generator in the
expression f.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).LM()
x**2*y**0
"""
return Monomial(f.monoms(order)[0], f.gens)
def EM(f, order=None):
"""
Returns the last non-zero monomial of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).EM()
x**0*y**1
"""
return Monomial(f.monoms(order)[-1], f.gens)
def LT(f, order=None):
"""
Returns the leading term of ``f``.
The Leading term signifies the term having
the highest power of the principal generator in the
expression f along with its coefficient.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).LT()
(x**2*y**0, 4)
"""
monom, coeff = f.terms(order)[0]
return Monomial(monom, f.gens), coeff
def ET(f, order=None):
"""
Returns the last non-zero term of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).ET()
(x**0*y**1, 3)
"""
monom, coeff = f.terms(order)[-1]
return Monomial(monom, f.gens), coeff
def max_norm(f):
"""
Returns maximum norm of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(-x**2 + 2*x - 3, x).max_norm()
3
"""
if hasattr(f.rep, 'max_norm'):
result = f.rep.max_norm()
else: # pragma: no cover
raise OperationNotSupported(f, 'max_norm')
return f.rep.dom.to_sympy(result)
def l1_norm(f):
"""
Returns l1 norm of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(-x**2 + 2*x - 3, x).l1_norm()
6
"""
if hasattr(f.rep, 'l1_norm'):
result = f.rep.l1_norm()
else: # pragma: no cover
raise OperationNotSupported(f, 'l1_norm')
return f.rep.dom.to_sympy(result)
def clear_denoms(self, convert=False):
"""
Clear denominators, but keep the ground domain.
Examples
========
>>> from sympy import Poly, S, QQ
>>> from sympy.abc import x
>>> f = Poly(x/2 + S(1)/3, x, domain=QQ)
>>> f.clear_denoms()
(6, Poly(3*x + 2, x, domain='QQ'))
>>> f.clear_denoms(convert=True)
(6, Poly(3*x + 2, x, domain='ZZ'))
"""
f = self
if not f.rep.dom.is_Field:
return S.One, f
dom = f.get_domain()
if dom.has_assoc_Ring:
dom = f.rep.dom.get_ring()
if hasattr(f.rep, 'clear_denoms'):
coeff, result = f.rep.clear_denoms()
else: # pragma: no cover
raise OperationNotSupported(f, 'clear_denoms')
coeff, f = dom.to_sympy(coeff), f.per(result)
if not convert or not dom.has_assoc_Ring:
return coeff, f
else:
return coeff, f.to_ring()
def rat_clear_denoms(self, g):
"""
Clear denominators in a rational function ``f/g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> f = Poly(x**2/y + 1, x)
>>> g = Poly(x**3 + y, x)
>>> p, q = f.rat_clear_denoms(g)
>>> p
Poly(x**2 + y, x, domain='ZZ[y]')
>>> q
Poly(y*x**3 + y**2, x, domain='ZZ[y]')
"""
f = self
dom, per, f, g = f._unify(g)
f = per(f)
g = per(g)
if not (dom.is_Field and dom.has_assoc_Ring):
return f, g
a, f = f.clear_denoms(convert=True)
b, g = g.clear_denoms(convert=True)
f = f.mul_ground(b)
g = g.mul_ground(a)
return f, g
def integrate(self, *specs, **args):
"""
Computes indefinite integral of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + 2*x + 1, x).integrate()
Poly(1/3*x**3 + x**2 + x, x, domain='QQ')
>>> Poly(x*y**2 + x, x, y).integrate((0, 1), (1, 0))
Poly(1/2*x**2*y**2 + 1/2*x**2, x, y, domain='QQ')
"""
f = self
if args.get('auto', True) and f.rep.dom.is_Ring:
f = f.to_field()
if hasattr(f.rep, 'integrate'):
if not specs:
return f.per(f.rep.integrate(m=1))
rep = f.rep
for spec in specs:
if type(spec) is tuple:
gen, m = spec
else:
gen, m = spec, 1
rep = rep.integrate(int(m), f._gen_to_level(gen))
return f.per(rep)
else: # pragma: no cover
raise OperationNotSupported(f, 'integrate')
def diff(f, *specs, **kwargs):
"""
Computes partial derivative of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + 2*x + 1, x).diff()
Poly(2*x + 2, x, domain='ZZ')
>>> Poly(x*y**2 + x, x, y).diff((0, 0), (1, 1))
Poly(2*x*y, x, y, domain='ZZ')
"""
if not kwargs.get('evaluate', True):
return Derivative(f, *specs, **kwargs)
if hasattr(f.rep, 'diff'):
if not specs:
return f.per(f.rep.diff(m=1))
rep = f.rep
for spec in specs:
if type(spec) is tuple:
gen, m = spec
else:
gen, m = spec, 1
rep = rep.diff(int(m), f._gen_to_level(gen))
return f.per(rep)
else: # pragma: no cover
raise OperationNotSupported(f, 'diff')
_eval_derivative = diff
def eval(self, x, a=None, auto=True):
"""
Evaluate ``f`` at ``a`` in the given variable.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y, z
>>> Poly(x**2 + 2*x + 3, x).eval(2)
11
>>> Poly(2*x*y + 3*x + y + 2, x, y).eval(x, 2)
Poly(5*y + 8, y, domain='ZZ')
>>> f = Poly(2*x*y + 3*x + y + 2*z, x, y, z)
>>> f.eval({x: 2})
Poly(5*y + 2*z + 6, y, z, domain='ZZ')
>>> f.eval({x: 2, y: 5})
Poly(2*z + 31, z, domain='ZZ')
>>> f.eval({x: 2, y: 5, z: 7})
45
>>> f.eval((2, 5))
Poly(2*z + 31, z, domain='ZZ')
>>> f(2, 5)
Poly(2*z + 31, z, domain='ZZ')
"""
f = self
if a is None:
if isinstance(x, dict):
mapping = x
for gen, value in mapping.items():
f = f.eval(gen, value)
return f
elif isinstance(x, (tuple, list)):
values = x
if len(values) > len(f.gens):
raise ValueError("too many values provided")
for gen, value in zip(f.gens, values):
f = f.eval(gen, value)
return f
else:
j, a = 0, x
else:
j = f._gen_to_level(x)
if not hasattr(f.rep, 'eval'): # pragma: no cover
raise OperationNotSupported(f, 'eval')
try:
result = f.rep.eval(a, j)
except CoercionFailed:
if not auto:
raise DomainError("can't evaluate at %s in %s" % (a, f.rep.dom))
else:
a_domain, [a] = construct_domain([a])
new_domain = f.get_domain().unify_with_symbols(a_domain, f.gens)
f = f.set_domain(new_domain)
a = new_domain.convert(a, a_domain)
result = f.rep.eval(a, j)
return f.per(result, remove=j)
def __call__(f, *values):
"""
Evaluate ``f`` at the give values.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y, z
>>> f = Poly(2*x*y + 3*x + y + 2*z, x, y, z)
>>> f(2)
Poly(5*y + 2*z + 6, y, z, domain='ZZ')
>>> f(2, 5)
Poly(2*z + 31, z, domain='ZZ')
>>> f(2, 5, 7)
45
"""
return f.eval(values)
def half_gcdex(f, g, auto=True):
"""
Half extended Euclidean algorithm of ``f`` and ``g``.
Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
>>> g = x**3 + x**2 - 4*x - 4
>>> Poly(f).half_gcdex(Poly(g))
(Poly(-1/5*x + 3/5, x, domain='QQ'), Poly(x + 1, x, domain='QQ'))
"""
dom, per, F, G = f._unify(g)
if auto and dom.is_Ring:
F, G = F.to_field(), G.to_field()
if hasattr(f.rep, 'half_gcdex'):
s, h = F.half_gcdex(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'half_gcdex')
return per(s), per(h)
def gcdex(f, g, auto=True):
"""
Extended Euclidean algorithm of ``f`` and ``g``.
Returns ``(s, t, h)`` such that ``h = gcd(f, g)`` and ``s*f + t*g = h``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
>>> g = x**3 + x**2 - 4*x - 4
>>> Poly(f).gcdex(Poly(g))
(Poly(-1/5*x + 3/5, x, domain='QQ'),
Poly(1/5*x**2 - 6/5*x + 2, x, domain='QQ'),
Poly(x + 1, x, domain='QQ'))
"""
dom, per, F, G = f._unify(g)
if auto and dom.is_Ring:
F, G = F.to_field(), G.to_field()
if hasattr(f.rep, 'gcdex'):
s, t, h = F.gcdex(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'gcdex')
return per(s), per(t), per(h)
def invert(f, g, auto=True):
"""
Invert ``f`` modulo ``g`` when possible.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 1, x).invert(Poly(2*x - 1, x))
Poly(-4/3, x, domain='QQ')
>>> Poly(x**2 - 1, x).invert(Poly(x - 1, x))
Traceback (most recent call last):
...
NotInvertible: zero divisor
"""
dom, per, F, G = f._unify(g)
if auto and dom.is_Ring:
F, G = F.to_field(), G.to_field()
if hasattr(f.rep, 'invert'):
result = F.invert(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'invert')
return per(result)
def revert(f, n):
"""
Compute ``f**(-1)`` mod ``x**n``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(1, x).revert(2)
Poly(1, x, domain='ZZ')
>>> Poly(1 + x, x).revert(1)
Poly(1, x, domain='ZZ')
>>> Poly(x**2 - 2, x).revert(2)
Traceback (most recent call last):
...
NotReversible: only units are reversible in a ring
>>> Poly(1/x, x).revert(1)
Traceback (most recent call last):
...
PolynomialError: 1/x contains an element of the generators set
"""
if hasattr(f.rep, 'revert'):
result = f.rep.revert(int(n))
else: # pragma: no cover
raise OperationNotSupported(f, 'revert')
return f.per(result)
def subresultants(f, g):
"""
Computes the subresultant PRS of ``f`` and ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).subresultants(Poly(x**2 - 1, x))
[Poly(x**2 + 1, x, domain='ZZ'),
Poly(x**2 - 1, x, domain='ZZ'),
Poly(-2, x, domain='ZZ')]
"""
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'subresultants'):
result = F.subresultants(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'subresultants')
return list(map(per, result))
def resultant(f, g, includePRS=False):
"""
Computes the resultant of ``f`` and ``g`` via PRS.
If includePRS=True, it includes the subresultant PRS in the result.
Because the PRS is used to calculate the resultant, this is more
efficient than calling :func:`subresultants` separately.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> f = Poly(x**2 + 1, x)
>>> f.resultant(Poly(x**2 - 1, x))
4
>>> f.resultant(Poly(x**2 - 1, x), includePRS=True)
(4, [Poly(x**2 + 1, x, domain='ZZ'), Poly(x**2 - 1, x, domain='ZZ'),
Poly(-2, x, domain='ZZ')])
"""
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'resultant'):
if includePRS:
result, R = F.resultant(G, includePRS=includePRS)
else:
result = F.resultant(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'resultant')
if includePRS:
return (per(result, remove=0), list(map(per, R)))
return per(result, remove=0)
def discriminant(f):
"""
Computes the discriminant of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 2*x + 3, x).discriminant()
-8
"""
if hasattr(f.rep, 'discriminant'):
result = f.rep.discriminant()
else: # pragma: no cover
raise OperationNotSupported(f, 'discriminant')
return f.per(result, remove=0)
def dispersionset(f, g=None):
r"""Compute the *dispersion set* of two polynomials.
For two polynomials `f(x)` and `g(x)` with `\deg f > 0`
and `\deg g > 0` the dispersion set `\operatorname{J}(f, g)` is defined as:
.. math::
\operatorname{J}(f, g)
& := \{a \in \mathbb{N}_0 | \gcd(f(x), g(x+a)) \neq 1\} \\
& = \{a \in \mathbb{N}_0 | \deg \gcd(f(x), g(x+a)) \geq 1\}
For a single polynomial one defines `\operatorname{J}(f) := \operatorname{J}(f, f)`.
Examples
========
>>> from sympy import poly
>>> from sympy.polys.dispersion import dispersion, dispersionset
>>> from sympy.abc import x
Dispersion set and dispersion of a simple polynomial:
>>> fp = poly((x - 3)*(x + 3), x)
>>> sorted(dispersionset(fp))
[0, 6]
>>> dispersion(fp)
6
Note that the definition of the dispersion is not symmetric:
>>> fp = poly(x**4 - 3*x**2 + 1, x)
>>> gp = fp.shift(-3)
>>> sorted(dispersionset(fp, gp))
[2, 3, 4]
>>> dispersion(fp, gp)
4
>>> sorted(dispersionset(gp, fp))
[]
>>> dispersion(gp, fp)
-oo
Computing the dispersion also works over field extensions:
>>> from sympy import sqrt
>>> fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ<sqrt(5)>')
>>> gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ<sqrt(5)>')
>>> sorted(dispersionset(fp, gp))
[2]
>>> sorted(dispersionset(gp, fp))
[1, 4]
We can even perform the computations for polynomials
having symbolic coefficients:
>>> from sympy.abc import a
>>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x)
>>> sorted(dispersionset(fp))
[0, 1]
See Also
========
dispersion
References
==========
1. [ManWright94]_
2. [Koepf98]_
3. [Abramov71]_
4. [Man93]_
"""
from sympy.polys.dispersion import dispersionset
return dispersionset(f, g)
def dispersion(f, g=None):
r"""Compute the *dispersion* of polynomials.
For two polynomials `f(x)` and `g(x)` with `\deg f > 0`
and `\deg g > 0` the dispersion `\operatorname{dis}(f, g)` is defined as:
.. math::
\operatorname{dis}(f, g)
& := \max\{ J(f,g) \cup \{0\} \} \\
& = \max\{ \{a \in \mathbb{N} | \gcd(f(x), g(x+a)) \neq 1\} \cup \{0\} \}
and for a single polynomial `\operatorname{dis}(f) := \operatorname{dis}(f, f)`.
Examples
========
>>> from sympy import poly
>>> from sympy.polys.dispersion import dispersion, dispersionset
>>> from sympy.abc import x
Dispersion set and dispersion of a simple polynomial:
>>> fp = poly((x - 3)*(x + 3), x)
>>> sorted(dispersionset(fp))
[0, 6]
>>> dispersion(fp)
6
Note that the definition of the dispersion is not symmetric:
>>> fp = poly(x**4 - 3*x**2 + 1, x)
>>> gp = fp.shift(-3)
>>> sorted(dispersionset(fp, gp))
[2, 3, 4]
>>> dispersion(fp, gp)
4
>>> sorted(dispersionset(gp, fp))
[]
>>> dispersion(gp, fp)
-oo
Computing the dispersion also works over field extensions:
>>> from sympy import sqrt
>>> fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ<sqrt(5)>')
>>> gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ<sqrt(5)>')
>>> sorted(dispersionset(fp, gp))
[2]
>>> sorted(dispersionset(gp, fp))
[1, 4]
We can even perform the computations for polynomials
having symbolic coefficients:
>>> from sympy.abc import a
>>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x)
>>> sorted(dispersionset(fp))
[0, 1]
See Also
========
dispersionset
References
==========
1. [ManWright94]_
2. [Koepf98]_
3. [Abramov71]_
4. [Man93]_
"""
from sympy.polys.dispersion import dispersion
return dispersion(f, g)
def cofactors(f, g):
"""
Returns the GCD of ``f`` and ``g`` and their cofactors.
Returns polynomials ``(h, cff, cfg)`` such that ``h = gcd(f, g)``, and
``cff = quo(f, h)`` and ``cfg = quo(g, h)`` are, so called, cofactors
of ``f`` and ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 1, x).cofactors(Poly(x**2 - 3*x + 2, x))
(Poly(x - 1, x, domain='ZZ'),
Poly(x + 1, x, domain='ZZ'),
Poly(x - 2, x, domain='ZZ'))
"""
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'cofactors'):
h, cff, cfg = F.cofactors(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'cofactors')
return per(h), per(cff), per(cfg)
def gcd(f, g):
"""
Returns the polynomial GCD of ``f`` and ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 1, x).gcd(Poly(x**2 - 3*x + 2, x))
Poly(x - 1, x, domain='ZZ')
"""
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'gcd'):
result = F.gcd(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'gcd')
return per(result)
def lcm(f, g):
"""
Returns polynomial LCM of ``f`` and ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 1, x).lcm(Poly(x**2 - 3*x + 2, x))
Poly(x**3 - 2*x**2 - x + 2, x, domain='ZZ')
"""
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'lcm'):
result = F.lcm(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'lcm')
return per(result)
def trunc(f, p):
"""
Reduce ``f`` modulo a constant ``p``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(2*x**3 + 3*x**2 + 5*x + 7, x).trunc(3)
Poly(-x**3 - x + 1, x, domain='ZZ')
"""
p = f.rep.dom.convert(p)
if hasattr(f.rep, 'trunc'):
result = f.rep.trunc(p)
else: # pragma: no cover
raise OperationNotSupported(f, 'trunc')
return f.per(result)
def monic(self, auto=True):
"""
Divides all coefficients by ``LC(f)``.
Examples
========
>>> from sympy import Poly, ZZ
>>> from sympy.abc import x
>>> Poly(3*x**2 + 6*x + 9, x, domain=ZZ).monic()
Poly(x**2 + 2*x + 3, x, domain='QQ')
>>> Poly(3*x**2 + 4*x + 2, x, domain=ZZ).monic()
Poly(x**2 + 4/3*x + 2/3, x, domain='QQ')
"""
f = self
if auto and f.rep.dom.is_Ring:
f = f.to_field()
if hasattr(f.rep, 'monic'):
result = f.rep.monic()
else: # pragma: no cover
raise OperationNotSupported(f, 'monic')
return f.per(result)
def content(f):
"""
Returns the GCD of polynomial coefficients.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(6*x**2 + 8*x + 12, x).content()
2
"""
if hasattr(f.rep, 'content'):
result = f.rep.content()
else: # pragma: no cover
raise OperationNotSupported(f, 'content')
return f.rep.dom.to_sympy(result)
def primitive(f):
"""
Returns the content and a primitive form of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(2*x**2 + 8*x + 12, x).primitive()
(2, Poly(x**2 + 4*x + 6, x, domain='ZZ'))
"""
if hasattr(f.rep, 'primitive'):
cont, result = f.rep.primitive()
else: # pragma: no cover
raise OperationNotSupported(f, 'primitive')
return f.rep.dom.to_sympy(cont), f.per(result)
def compose(f, g):
"""
Computes the functional composition of ``f`` and ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + x, x).compose(Poly(x - 1, x))
Poly(x**2 - x, x, domain='ZZ')
"""
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'compose'):
result = F.compose(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'compose')
return per(result)
def decompose(f):
"""
Computes a functional decomposition of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**4 + 2*x**3 - x - 1, x, domain='ZZ').decompose()
[Poly(x**2 - x - 1, x, domain='ZZ'), Poly(x**2 + x, x, domain='ZZ')]
"""
if hasattr(f.rep, 'decompose'):
result = f.rep.decompose()
else: # pragma: no cover
raise OperationNotSupported(f, 'decompose')
return list(map(f.per, result))
def shift(f, a):
"""
Efficiently compute Taylor shift ``f(x + a)``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 2*x + 1, x).shift(2)
Poly(x**2 + 2*x + 1, x, domain='ZZ')
"""
if hasattr(f.rep, 'shift'):
result = f.rep.shift(a)
else: # pragma: no cover
raise OperationNotSupported(f, 'shift')
return f.per(result)
def transform(f, p, q):
"""
Efficiently evaluate the functional transformation ``q**n * f(p/q)``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 2*x + 1, x).transform(Poly(x + 1, x), Poly(x - 1, x))
Poly(4, x, domain='ZZ')
"""
P, Q = p.unify(q)
F, P = f.unify(P)
F, Q = F.unify(Q)
if hasattr(F.rep, 'transform'):
result = F.rep.transform(P.rep, Q.rep)
else: # pragma: no cover
raise OperationNotSupported(F, 'transform')
return F.per(result)
def sturm(self, auto=True):
"""
Computes the Sturm sequence of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**3 - 2*x**2 + x - 3, x).sturm()
[Poly(x**3 - 2*x**2 + x - 3, x, domain='QQ'),
Poly(3*x**2 - 4*x + 1, x, domain='QQ'),
Poly(2/9*x + 25/9, x, domain='QQ'),
Poly(-2079/4, x, domain='QQ')]
"""
f = self
if auto and f.rep.dom.is_Ring:
f = f.to_field()
if hasattr(f.rep, 'sturm'):
result = f.rep.sturm()
else: # pragma: no cover
raise OperationNotSupported(f, 'sturm')
return list(map(f.per, result))
def gff_list(f):
"""
Computes greatest factorial factorization of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> f = x**5 + 2*x**4 - x**3 - 2*x**2
>>> Poly(f).gff_list()
[(Poly(x, x, domain='ZZ'), 1), (Poly(x + 2, x, domain='ZZ'), 4)]
"""
if hasattr(f.rep, 'gff_list'):
result = f.rep.gff_list()
else: # pragma: no cover
raise OperationNotSupported(f, 'gff_list')
return [(f.per(g), k) for g, k in result]
def norm(f):
"""
Computes the product, ``Norm(f)``, of the conjugates of
a polynomial ``f`` defined over a number field ``K``.
Examples
========
>>> from sympy import Poly, sqrt
>>> from sympy.abc import x
>>> a, b = sqrt(2), sqrt(3)
A polynomial over a quadratic extension.
Two conjugates x - a and x + a.
>>> f = Poly(x - a, x, extension=a)
>>> f.norm()
Poly(x**2 - 2, x, domain='QQ')
A polynomial over a quartic extension.
Four conjugates x - a, x - a, x + a and x + a.
>>> f = Poly(x - a, x, extension=(a, b))
>>> f.norm()
Poly(x**4 - 4*x**2 + 4, x, domain='QQ')
"""
if hasattr(f.rep, 'norm'):
r = f.rep.norm()
else: # pragma: no cover
raise OperationNotSupported(f, 'norm')
return f.per(r)
def sqf_norm(f):
"""
Computes square-free norm of ``f``.
Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and
``r(x) = Norm(g(x))`` is a square-free polynomial over ``K``,
where ``a`` is the algebraic extension of the ground domain.
Examples
========
>>> from sympy import Poly, sqrt
>>> from sympy.abc import x
>>> s, f, r = Poly(x**2 + 1, x, extension=[sqrt(3)]).sqf_norm()
>>> s
1
>>> f
Poly(x**2 - 2*sqrt(3)*x + 4, x, domain='QQ<sqrt(3)>')
>>> r
Poly(x**4 - 4*x**2 + 16, x, domain='QQ')
"""
if hasattr(f.rep, 'sqf_norm'):
s, g, r = f.rep.sqf_norm()
else: # pragma: no cover
raise OperationNotSupported(f, 'sqf_norm')
return s, f.per(g), f.per(r)
def sqf_part(f):
"""
Computes square-free part of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**3 - 3*x - 2, x).sqf_part()
Poly(x**2 - x - 2, x, domain='ZZ')
"""
if hasattr(f.rep, 'sqf_part'):
result = f.rep.sqf_part()
else: # pragma: no cover
raise OperationNotSupported(f, 'sqf_part')
return f.per(result)
def sqf_list(f, all=False):
"""
Returns a list of square-free factors of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> f = 2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16
>>> Poly(f).sqf_list()
(2, [(Poly(x + 1, x, domain='ZZ'), 2),
(Poly(x + 2, x, domain='ZZ'), 3)])
>>> Poly(f).sqf_list(all=True)
(2, [(Poly(1, x, domain='ZZ'), 1),
(Poly(x + 1, x, domain='ZZ'), 2),
(Poly(x + 2, x, domain='ZZ'), 3)])
"""
if hasattr(f.rep, 'sqf_list'):
coeff, factors = f.rep.sqf_list(all)
else: # pragma: no cover
raise OperationNotSupported(f, 'sqf_list')
return f.rep.dom.to_sympy(coeff), [(f.per(g), k) for g, k in factors]
def sqf_list_include(f, all=False):
"""
Returns a list of square-free factors of ``f``.
Examples
========
>>> from sympy import Poly, expand
>>> from sympy.abc import x
>>> f = expand(2*(x + 1)**3*x**4)
>>> f
2*x**7 + 6*x**6 + 6*x**5 + 2*x**4
>>> Poly(f).sqf_list_include()
[(Poly(2, x, domain='ZZ'), 1),
(Poly(x + 1, x, domain='ZZ'), 3),
(Poly(x, x, domain='ZZ'), 4)]
>>> Poly(f).sqf_list_include(all=True)
[(Poly(2, x, domain='ZZ'), 1),
(Poly(1, x, domain='ZZ'), 2),
(Poly(x + 1, x, domain='ZZ'), 3),
(Poly(x, x, domain='ZZ'), 4)]
"""
if hasattr(f.rep, 'sqf_list_include'):
factors = f.rep.sqf_list_include(all)
else: # pragma: no cover
raise OperationNotSupported(f, 'sqf_list_include')
return [(f.per(g), k) for g, k in factors]
def factor_list(f):
"""
Returns a list of irreducible factors of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> f = 2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y
>>> Poly(f).factor_list()
(2, [(Poly(x + y, x, y, domain='ZZ'), 1),
(Poly(x**2 + 1, x, y, domain='ZZ'), 2)])
"""
if hasattr(f.rep, 'factor_list'):
try:
coeff, factors = f.rep.factor_list()
except DomainError:
return S.One, [(f, 1)]
else: # pragma: no cover
raise OperationNotSupported(f, 'factor_list')
return f.rep.dom.to_sympy(coeff), [(f.per(g), k) for g, k in factors]
def factor_list_include(f):
"""
Returns a list of irreducible factors of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> f = 2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y
>>> Poly(f).factor_list_include()
[(Poly(2*x + 2*y, x, y, domain='ZZ'), 1),
(Poly(x**2 + 1, x, y, domain='ZZ'), 2)]
"""
if hasattr(f.rep, 'factor_list_include'):
try:
factors = f.rep.factor_list_include()
except DomainError:
return [(f, 1)]
else: # pragma: no cover
raise OperationNotSupported(f, 'factor_list_include')
return [(f.per(g), k) for g, k in factors]
def intervals(f, all=False, eps=None, inf=None, sup=None, fast=False, sqf=False):
"""
Compute isolating intervals for roots of ``f``.
For real roots the Vincent-Akritas-Strzebonski (VAS) continued fractions method is used.
References
==========
.. [#] Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative Study of Two Real Root
Isolation Methods . Nonlinear Analysis: Modelling and Control, Vol. 10, No. 4, 297-304, 2005.
.. [#] Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S. Vigklas: Improving the
Performance of the Continued Fractions Method Using new Bounds of Positive Roots. Nonlinear
Analysis: Modelling and Control, Vol. 13, No. 3, 265-279, 2008.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 3, x).intervals()
[((-2, -1), 1), ((1, 2), 1)]
>>> Poly(x**2 - 3, x).intervals(eps=1e-2)
[((-26/15, -19/11), 1), ((19/11, 26/15), 1)]
"""
if eps is not None:
eps = QQ.convert(eps)
if eps <= 0:
raise ValueError("'eps' must be a positive rational")
if inf is not None:
inf = QQ.convert(inf)
if sup is not None:
sup = QQ.convert(sup)
if hasattr(f.rep, 'intervals'):
result = f.rep.intervals(
all=all, eps=eps, inf=inf, sup=sup, fast=fast, sqf=sqf)
else: # pragma: no cover
raise OperationNotSupported(f, 'intervals')
if sqf:
def _real(interval):
s, t = interval
return (QQ.to_sympy(s), QQ.to_sympy(t))
if not all:
return list(map(_real, result))
def _complex(rectangle):
(u, v), (s, t) = rectangle
return (QQ.to_sympy(u) + I*QQ.to_sympy(v),
QQ.to_sympy(s) + I*QQ.to_sympy(t))
real_part, complex_part = result
return list(map(_real, real_part)), list(map(_complex, complex_part))
else:
def _real(interval):
(s, t), k = interval
return ((QQ.to_sympy(s), QQ.to_sympy(t)), k)
if not all:
return list(map(_real, result))
def _complex(rectangle):
((u, v), (s, t)), k = rectangle
return ((QQ.to_sympy(u) + I*QQ.to_sympy(v),
QQ.to_sympy(s) + I*QQ.to_sympy(t)), k)
real_part, complex_part = result
return list(map(_real, real_part)), list(map(_complex, complex_part))
def refine_root(f, s, t, eps=None, steps=None, fast=False, check_sqf=False):
"""
Refine an isolating interval of a root to the given precision.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 3, x).refine_root(1, 2, eps=1e-2)
(19/11, 26/15)
"""
if check_sqf and not f.is_sqf:
raise PolynomialError("only square-free polynomials supported")
s, t = QQ.convert(s), QQ.convert(t)
if eps is not None:
eps = QQ.convert(eps)
if eps <= 0:
raise ValueError("'eps' must be a positive rational")
if steps is not None:
steps = int(steps)
elif eps is None:
steps = 1
if hasattr(f.rep, 'refine_root'):
S, T = f.rep.refine_root(s, t, eps=eps, steps=steps, fast=fast)
else: # pragma: no cover
raise OperationNotSupported(f, 'refine_root')
return QQ.to_sympy(S), QQ.to_sympy(T)
def count_roots(f, inf=None, sup=None):
"""
Return the number of roots of ``f`` in ``[inf, sup]`` interval.
Examples
========
>>> from sympy import Poly, I
>>> from sympy.abc import x
>>> Poly(x**4 - 4, x).count_roots(-3, 3)
2
>>> Poly(x**4 - 4, x).count_roots(0, 1 + 3*I)
1
"""
inf_real, sup_real = True, True
if inf is not None:
inf = sympify(inf)
if inf is S.NegativeInfinity:
inf = None
else:
re, im = inf.as_real_imag()
if not im:
inf = QQ.convert(inf)
else:
inf, inf_real = list(map(QQ.convert, (re, im))), False
if sup is not None:
sup = sympify(sup)
if sup is S.Infinity:
sup = None
else:
re, im = sup.as_real_imag()
if not im:
sup = QQ.convert(sup)
else:
sup, sup_real = list(map(QQ.convert, (re, im))), False
if inf_real and sup_real:
if hasattr(f.rep, 'count_real_roots'):
count = f.rep.count_real_roots(inf=inf, sup=sup)
else: # pragma: no cover
raise OperationNotSupported(f, 'count_real_roots')
else:
if inf_real and inf is not None:
inf = (inf, QQ.zero)
if sup_real and sup is not None:
sup = (sup, QQ.zero)
if hasattr(f.rep, 'count_complex_roots'):
count = f.rep.count_complex_roots(inf=inf, sup=sup)
else: # pragma: no cover
raise OperationNotSupported(f, 'count_complex_roots')
return Integer(count)
def root(f, index, radicals=True):
"""
Get an indexed root of a polynomial.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> f = Poly(2*x**3 - 7*x**2 + 4*x + 4)
>>> f.root(0)
-1/2
>>> f.root(1)
2
>>> f.root(2)
2
>>> f.root(3)
Traceback (most recent call last):
...
IndexError: root index out of [-3, 2] range, got 3
>>> Poly(x**5 + x + 1).root(0)
CRootOf(x**3 - x**2 + 1, 0)
"""
return sympy.polys.rootoftools.rootof(f, index, radicals=radicals)
def real_roots(f, multiple=True, radicals=True):
"""
Return a list of real roots with multiplicities.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(2*x**3 - 7*x**2 + 4*x + 4).real_roots()
[-1/2, 2, 2]
>>> Poly(x**3 + x + 1).real_roots()
[CRootOf(x**3 + x + 1, 0)]
"""
reals = sympy.polys.rootoftools.CRootOf.real_roots(f, radicals=radicals)
if multiple:
return reals
else:
return group(reals, multiple=False)
def all_roots(f, multiple=True, radicals=True):
"""
Return a list of real and complex roots with multiplicities.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(2*x**3 - 7*x**2 + 4*x + 4).all_roots()
[-1/2, 2, 2]
>>> Poly(x**3 + x + 1).all_roots()
[CRootOf(x**3 + x + 1, 0),
CRootOf(x**3 + x + 1, 1),
CRootOf(x**3 + x + 1, 2)]
"""
roots = sympy.polys.rootoftools.CRootOf.all_roots(f, radicals=radicals)
if multiple:
return roots
else:
return group(roots, multiple=False)
def nroots(f, n=15, maxsteps=50, cleanup=True):
"""
Compute numerical approximations of roots of ``f``.
Parameters
==========
n ... the number of digits to calculate
maxsteps ... the maximum number of iterations to do
If the accuracy `n` cannot be reached in `maxsteps`, it will raise an
exception. You need to rerun with higher maxsteps.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 3).nroots(n=15)
[-1.73205080756888, 1.73205080756888]
>>> Poly(x**2 - 3).nroots(n=30)
[-1.73205080756887729352744634151, 1.73205080756887729352744634151]
"""
from sympy.functions.elementary.complexes import sign
if f.is_multivariate:
raise MultivariatePolynomialError(
"can't compute numerical roots of %s" % f)
if f.degree() <= 0:
return []
# For integer and rational coefficients, convert them to integers only
# (for accuracy). Otherwise just try to convert the coefficients to
# mpmath.mpc and raise an exception if the conversion fails.
if f.rep.dom is ZZ:
coeffs = [int(coeff) for coeff in f.all_coeffs()]
elif f.rep.dom is QQ:
denoms = [coeff.q for coeff in f.all_coeffs()]
from sympy.core.numbers import ilcm
fac = ilcm(*denoms)
coeffs = [int(coeff*fac) for coeff in f.all_coeffs()]
else:
coeffs = [coeff.evalf(n=n).as_real_imag()
for coeff in f.all_coeffs()]
try:
coeffs = [mpmath.mpc(*coeff) for coeff in coeffs]
except TypeError:
raise DomainError("Numerical domain expected, got %s" % \
f.rep.dom)
dps = mpmath.mp.dps
mpmath.mp.dps = n
try:
# We need to add extra precision to guard against losing accuracy.
# 10 times the degree of the polynomial seems to work well.
roots = mpmath.polyroots(coeffs, maxsteps=maxsteps,
cleanup=cleanup, error=False, extraprec=f.degree()*10)
# Mpmath puts real roots first, then complex ones (as does all_roots)
# so we make sure this convention holds here, too.
roots = list(map(sympify,
sorted(roots, key=lambda r: (1 if r.imag else 0, r.real, abs(r.imag), sign(r.imag)))))
except NoConvergence:
raise NoConvergence(
'convergence to root failed; try n < %s or maxsteps > %s' % (
n, maxsteps))
finally:
mpmath.mp.dps = dps
return roots
def ground_roots(f):
"""
Compute roots of ``f`` by factorization in the ground domain.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**6 - 4*x**4 + 4*x**3 - x**2).ground_roots()
{0: 2, 1: 2}
"""
if f.is_multivariate:
raise MultivariatePolynomialError(
"can't compute ground roots of %s" % f)
roots = {}
for factor, k in f.factor_list()[1]:
if factor.is_linear:
a, b = factor.all_coeffs()
roots[-b/a] = k
return roots
def nth_power_roots_poly(f, n):
"""
Construct a polynomial with n-th powers of roots of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> f = Poly(x**4 - x**2 + 1)
>>> f.nth_power_roots_poly(2)
Poly(x**4 - 2*x**3 + 3*x**2 - 2*x + 1, x, domain='ZZ')
>>> f.nth_power_roots_poly(3)
Poly(x**4 + 2*x**2 + 1, x, domain='ZZ')
>>> f.nth_power_roots_poly(4)
Poly(x**4 + 2*x**3 + 3*x**2 + 2*x + 1, x, domain='ZZ')
>>> f.nth_power_roots_poly(12)
Poly(x**4 - 4*x**3 + 6*x**2 - 4*x + 1, x, domain='ZZ')
"""
if f.is_multivariate:
raise MultivariatePolynomialError(
"must be a univariate polynomial")
N = sympify(n)
if N.is_Integer and N >= 1:
n = int(N)
else:
raise ValueError("'n' must an integer and n >= 1, got %s" % n)
x = f.gen
t = Dummy('t')
r = f.resultant(f.__class__.from_expr(x**n - t, x, t))
return r.replace(t, x)
def cancel(f, g, include=False):
"""
Cancel common factors in a rational function ``f/g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(2*x**2 - 2, x).cancel(Poly(x**2 - 2*x + 1, x))
(1, Poly(2*x + 2, x, domain='ZZ'), Poly(x - 1, x, domain='ZZ'))
>>> Poly(2*x**2 - 2, x).cancel(Poly(x**2 - 2*x + 1, x), include=True)
(Poly(2*x + 2, x, domain='ZZ'), Poly(x - 1, x, domain='ZZ'))
"""
dom, per, F, G = f._unify(g)
if hasattr(F, 'cancel'):
result = F.cancel(G, include=include)
else: # pragma: no cover
raise OperationNotSupported(f, 'cancel')
if not include:
if dom.has_assoc_Ring:
dom = dom.get_ring()
cp, cq, p, q = result
cp = dom.to_sympy(cp)
cq = dom.to_sympy(cq)
return cp/cq, per(p), per(q)
else:
return tuple(map(per, result))
@property
def is_zero(f):
"""
Returns ``True`` if ``f`` is a zero polynomial.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(0, x).is_zero
True
>>> Poly(1, x).is_zero
False
"""
return f.rep.is_zero
@property
def is_one(f):
"""
Returns ``True`` if ``f`` is a unit polynomial.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(0, x).is_one
False
>>> Poly(1, x).is_one
True
"""
return f.rep.is_one
@property
def is_sqf(f):
"""
Returns ``True`` if ``f`` is a square-free polynomial.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 2*x + 1, x).is_sqf
False
>>> Poly(x**2 - 1, x).is_sqf
True
"""
return f.rep.is_sqf
@property
def is_monic(f):
"""
Returns ``True`` if the leading coefficient of ``f`` is one.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x + 2, x).is_monic
True
>>> Poly(2*x + 2, x).is_monic
False
"""
return f.rep.is_monic
@property
def is_primitive(f):
"""
Returns ``True`` if GCD of the coefficients of ``f`` is one.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(2*x**2 + 6*x + 12, x).is_primitive
False
>>> Poly(x**2 + 3*x + 6, x).is_primitive
True
"""
return f.rep.is_primitive
@property
def is_ground(f):
"""
Returns ``True`` if ``f`` is an element of the ground domain.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x, x).is_ground
False
>>> Poly(2, x).is_ground
True
>>> Poly(y, x).is_ground
True
"""
return f.rep.is_ground
@property
def is_linear(f):
"""
Returns ``True`` if ``f`` is linear in all its variables.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x + y + 2, x, y).is_linear
True
>>> Poly(x*y + 2, x, y).is_linear
False
"""
return f.rep.is_linear
@property
def is_quadratic(f):
"""
Returns ``True`` if ``f`` is quadratic in all its variables.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x*y + 2, x, y).is_quadratic
True
>>> Poly(x*y**2 + 2, x, y).is_quadratic
False
"""
return f.rep.is_quadratic
@property
def is_monomial(f):
"""
Returns ``True`` if ``f`` is zero or has only one term.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(3*x**2, x).is_monomial
True
>>> Poly(3*x**2 + 1, x).is_monomial
False
"""
return f.rep.is_monomial
@property
def is_homogeneous(f):
"""
Returns ``True`` if ``f`` is a homogeneous polynomial.
A homogeneous polynomial is a polynomial whose all monomials with
non-zero coefficients have the same total degree. If you want not
only to check if a polynomial is homogeneous but also compute its
homogeneous order, then use :func:`Poly.homogeneous_order`.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + x*y, x, y).is_homogeneous
True
>>> Poly(x**3 + x*y, x, y).is_homogeneous
False
"""
return f.rep.is_homogeneous
@property
def is_irreducible(f):
"""
Returns ``True`` if ``f`` has no factors over its domain.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + x + 1, x, modulus=2).is_irreducible
True
>>> Poly(x**2 + 1, x, modulus=2).is_irreducible
False
"""
return f.rep.is_irreducible
@property
def is_univariate(f):
"""
Returns ``True`` if ``f`` is a univariate polynomial.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + x + 1, x).is_univariate
True
>>> Poly(x*y**2 + x*y + 1, x, y).is_univariate
False
>>> Poly(x*y**2 + x*y + 1, x).is_univariate
True
>>> Poly(x**2 + x + 1, x, y).is_univariate
False
"""
return len(f.gens) == 1
@property
def is_multivariate(f):
"""
Returns ``True`` if ``f`` is a multivariate polynomial.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + x + 1, x).is_multivariate
False
>>> Poly(x*y**2 + x*y + 1, x, y).is_multivariate
True
>>> Poly(x*y**2 + x*y + 1, x).is_multivariate
False
>>> Poly(x**2 + x + 1, x, y).is_multivariate
True
"""
return len(f.gens) != 1
@property
def is_cyclotomic(f):
"""
Returns ``True`` if ``f`` is a cyclotomic polnomial.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> f = x**16 + x**14 - x**10 + x**8 - x**6 + x**2 + 1
>>> Poly(f).is_cyclotomic
False
>>> g = x**16 + x**14 - x**10 - x**8 - x**6 + x**2 + 1
>>> Poly(g).is_cyclotomic
True
"""
return f.rep.is_cyclotomic
def __abs__(f):
return f.abs()
def __neg__(f):
return f.neg()
@_polifyit
def __add__(f, g):
return f.add(g)
@_polifyit
def __radd__(f, g):
return g.add(f)
@_polifyit
def __sub__(f, g):
return f.sub(g)
@_polifyit
def __rsub__(f, g):
return g.sub(f)
@_polifyit
def __mul__(f, g):
return f.mul(g)
@_polifyit
def __rmul__(f, g):
return g.mul(f)
@_sympifyit('n', NotImplemented)
def __pow__(f, n):
if n.is_Integer and n >= 0:
return f.pow(n)
else:
return NotImplemented
@_polifyit
def __divmod__(f, g):
return f.div(g)
@_polifyit
def __rdivmod__(f, g):
return g.div(f)
@_polifyit
def __mod__(f, g):
return f.rem(g)
@_polifyit
def __rmod__(f, g):
return g.rem(f)
@_polifyit
def __floordiv__(f, g):
return f.quo(g)
@_polifyit
def __rfloordiv__(f, g):
return g.quo(f)
@_sympifyit('g', NotImplemented)
def __truediv__(f, g):
return f.as_expr()/g.as_expr()
@_sympifyit('g', NotImplemented)
def __rtruediv__(f, g):
return g.as_expr()/f.as_expr()
@_sympifyit('other', NotImplemented)
def __eq__(self, other):
f, g = self, other
if not g.is_Poly:
try:
g = f.__class__(g, f.gens, domain=f.get_domain())
except (PolynomialError, DomainError, CoercionFailed):
return False
if f.gens != g.gens:
return False
if f.rep.dom != g.rep.dom:
return False
return f.rep == g.rep
@_sympifyit('g', NotImplemented)
def __ne__(f, g):
return not f == g
def __bool__(f):
return not f.is_zero
def eq(f, g, strict=False):
if not strict:
return f == g
else:
return f._strict_eq(sympify(g))
def ne(f, g, strict=False):
return not f.eq(g, strict=strict)
def _strict_eq(f, g):
return isinstance(g, f.__class__) and f.gens == g.gens and f.rep.eq(g.rep, strict=True)
@public
class PurePoly(Poly):
"""Class for representing pure polynomials. """
def _hashable_content(self):
"""Allow SymPy to hash Poly instances. """
return (self.rep,)
def __hash__(self):
return super().__hash__()
@property
def free_symbols(self):
"""
Free symbols of a polynomial.
Examples
========
>>> from sympy import PurePoly
>>> from sympy.abc import x, y
>>> PurePoly(x**2 + 1).free_symbols
set()
>>> PurePoly(x**2 + y).free_symbols
set()
>>> PurePoly(x**2 + y, x).free_symbols
{y}
"""
return self.free_symbols_in_domain
@_sympifyit('other', NotImplemented)
def __eq__(self, other):
f, g = self, other
if not g.is_Poly:
try:
g = f.__class__(g, f.gens, domain=f.get_domain())
except (PolynomialError, DomainError, CoercionFailed):
return False
if len(f.gens) != len(g.gens):
return False
if f.rep.dom != g.rep.dom:
try:
dom = f.rep.dom.unify(g.rep.dom, f.gens)
except UnificationFailed:
return False
f = f.set_domain(dom)
g = g.set_domain(dom)
return f.rep == g.rep
def _strict_eq(f, g):
return isinstance(g, f.__class__) and f.rep.eq(g.rep, strict=True)
def _unify(f, g):
g = sympify(g)
if not g.is_Poly:
try:
return f.rep.dom, f.per, f.rep, f.rep.per(f.rep.dom.from_sympy(g))
except CoercionFailed:
raise UnificationFailed("can't unify %s with %s" % (f, g))
if len(f.gens) != len(g.gens):
raise UnificationFailed("can't unify %s with %s" % (f, g))
if not (isinstance(f.rep, DMP) and isinstance(g.rep, DMP)):
raise UnificationFailed("can't unify %s with %s" % (f, g))
cls = f.__class__
gens = f.gens
dom = f.rep.dom.unify(g.rep.dom, gens)
F = f.rep.convert(dom)
G = g.rep.convert(dom)
def per(rep, dom=dom, gens=gens, remove=None):
if remove is not None:
gens = gens[:remove] + gens[remove + 1:]
if not gens:
return dom.to_sympy(rep)
return cls.new(rep, *gens)
return dom, per, F, G
@public
def poly_from_expr(expr, *gens, **args):
"""Construct a polynomial from an expression. """
opt = options.build_options(gens, args)
return _poly_from_expr(expr, opt)
def _poly_from_expr(expr, opt):
"""Construct a polynomial from an expression. """
orig, expr = expr, sympify(expr)
if not isinstance(expr, Basic):
raise PolificationFailed(opt, orig, expr)
elif expr.is_Poly:
poly = expr.__class__._from_poly(expr, opt)
opt.gens = poly.gens
opt.domain = poly.domain
if opt.polys is None:
opt.polys = True
return poly, opt
elif opt.expand:
expr = expr.expand()
rep, opt = _dict_from_expr(expr, opt)
if not opt.gens:
raise PolificationFailed(opt, orig, expr)
monoms, coeffs = list(zip(*list(rep.items())))
domain = opt.domain
if domain is None:
opt.domain, coeffs = construct_domain(coeffs, opt=opt)
else:
coeffs = list(map(domain.from_sympy, coeffs))
rep = dict(list(zip(monoms, coeffs)))
poly = Poly._from_dict(rep, opt)
if opt.polys is None:
opt.polys = False
return poly, opt
@public
def parallel_poly_from_expr(exprs, *gens, **args):
"""Construct polynomials from expressions. """
opt = options.build_options(gens, args)
return _parallel_poly_from_expr(exprs, opt)
def _parallel_poly_from_expr(exprs, opt):
"""Construct polynomials from expressions. """
from sympy.functions.elementary.piecewise import Piecewise
if len(exprs) == 2:
f, g = exprs
if isinstance(f, Poly) and isinstance(g, Poly):
f = f.__class__._from_poly(f, opt)
g = g.__class__._from_poly(g, opt)
f, g = f.unify(g)
opt.gens = f.gens
opt.domain = f.domain
if opt.polys is None:
opt.polys = True
return [f, g], opt
origs, exprs = list(exprs), []
_exprs, _polys = [], []
failed = False
for i, expr in enumerate(origs):
expr = sympify(expr)
if isinstance(expr, Basic):
if expr.is_Poly:
_polys.append(i)
else:
_exprs.append(i)
if opt.expand:
expr = expr.expand()
else:
failed = True
exprs.append(expr)
if failed:
raise PolificationFailed(opt, origs, exprs, True)
if _polys:
# XXX: this is a temporary solution
for i in _polys:
exprs[i] = exprs[i].as_expr()
reps, opt = _parallel_dict_from_expr(exprs, opt)
if not opt.gens:
raise PolificationFailed(opt, origs, exprs, True)
for k in opt.gens:
if isinstance(k, Piecewise):
raise PolynomialError("Piecewise generators do not make sense")
coeffs_list, lengths = [], []
all_monoms = []
all_coeffs = []
for rep in reps:
monoms, coeffs = list(zip(*list(rep.items())))
coeffs_list.extend(coeffs)
all_monoms.append(monoms)
lengths.append(len(coeffs))
domain = opt.domain
if domain is None:
opt.domain, coeffs_list = construct_domain(coeffs_list, opt=opt)
else:
coeffs_list = list(map(domain.from_sympy, coeffs_list))
for k in lengths:
all_coeffs.append(coeffs_list[:k])
coeffs_list = coeffs_list[k:]
polys = []
for monoms, coeffs in zip(all_monoms, all_coeffs):
rep = dict(list(zip(monoms, coeffs)))
poly = Poly._from_dict(rep, opt)
polys.append(poly)
if opt.polys is None:
opt.polys = bool(_polys)
return polys, opt
def _update_args(args, key, value):
"""Add a new ``(key, value)`` pair to arguments ``dict``. """
args = dict(args)
if key not in args:
args[key] = value
return args
@public
def degree(f, gen=0):
"""
Return the degree of ``f`` in the given variable.
The degree of 0 is negative infinity.
Examples
========
>>> from sympy import degree
>>> from sympy.abc import x, y
>>> degree(x**2 + y*x + 1, gen=x)
2
>>> degree(x**2 + y*x + 1, gen=y)
1
>>> degree(0, x)
-oo
See also
========
sympy.polys.polytools.Poly.total_degree
degree_list
"""
f = sympify(f, strict=True)
gen_is_Num = sympify(gen, strict=True).is_Number
if f.is_Poly:
p = f
isNum = p.as_expr().is_Number
else:
isNum = f.is_Number
if not isNum:
if gen_is_Num:
p, _ = poly_from_expr(f)
else:
p, _ = poly_from_expr(f, gen)
if isNum:
return S.Zero if f else S.NegativeInfinity
if not gen_is_Num:
if f.is_Poly and gen not in p.gens:
# try recast without explicit gens
p, _ = poly_from_expr(f.as_expr())
if gen not in p.gens:
return S.Zero
elif not f.is_Poly and len(f.free_symbols) > 1:
raise TypeError(filldedent('''
A symbolic generator of interest is required for a multivariate
expression like func = %s, e.g. degree(func, gen = %s) instead of
degree(func, gen = %s).
''' % (f, next(ordered(f.free_symbols)), gen)))
result = p.degree(gen)
return Integer(result) if isinstance(result, int) else S.NegativeInfinity
@public
def total_degree(f, *gens):
"""
Return the total_degree of ``f`` in the given variables.
Examples
========
>>> from sympy import total_degree, Poly
>>> from sympy.abc import x, y
>>> total_degree(1)
0
>>> total_degree(x + x*y)
2
>>> total_degree(x + x*y, x)
1
If the expression is a Poly and no variables are given
then the generators of the Poly will be used:
>>> p = Poly(x + x*y, y)
>>> total_degree(p)
1
To deal with the underlying expression of the Poly, convert
it to an Expr:
>>> total_degree(p.as_expr())
2
This is done automatically if any variables are given:
>>> total_degree(p, x)
1
See also
========
degree
"""
p = sympify(f)
if p.is_Poly:
p = p.as_expr()
if p.is_Number:
rv = 0
else:
if f.is_Poly:
gens = gens or f.gens
rv = Poly(p, gens).total_degree()
return Integer(rv)
@public
def degree_list(f, *gens, **args):
"""
Return a list of degrees of ``f`` in all variables.
Examples
========
>>> from sympy import degree_list
>>> from sympy.abc import x, y
>>> degree_list(x**2 + y*x + 1)
(2, 1)
"""
options.allowed_flags(args, ['polys'])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('degree_list', 1, exc)
degrees = F.degree_list()
return tuple(map(Integer, degrees))
@public
def LC(f, *gens, **args):
"""
Return the leading coefficient of ``f``.
Examples
========
>>> from sympy import LC
>>> from sympy.abc import x, y
>>> LC(4*x**2 + 2*x*y**2 + x*y + 3*y)
4
"""
options.allowed_flags(args, ['polys'])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('LC', 1, exc)
return F.LC(order=opt.order)
@public
def LM(f, *gens, **args):
"""
Return the leading monomial of ``f``.
Examples
========
>>> from sympy import LM
>>> from sympy.abc import x, y
>>> LM(4*x**2 + 2*x*y**2 + x*y + 3*y)
x**2
"""
options.allowed_flags(args, ['polys'])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('LM', 1, exc)
monom = F.LM(order=opt.order)
return monom.as_expr()
@public
def LT(f, *gens, **args):
"""
Return the leading term of ``f``.
Examples
========
>>> from sympy import LT
>>> from sympy.abc import x, y
>>> LT(4*x**2 + 2*x*y**2 + x*y + 3*y)
4*x**2
"""
options.allowed_flags(args, ['polys'])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('LT', 1, exc)
monom, coeff = F.LT(order=opt.order)
return coeff*monom.as_expr()
@public
def pdiv(f, g, *gens, **args):
"""
Compute polynomial pseudo-division of ``f`` and ``g``.
Examples
========
>>> from sympy import pdiv
>>> from sympy.abc import x
>>> pdiv(x**2 + 1, 2*x - 4)
(2*x + 4, 20)
"""
options.allowed_flags(args, ['polys'])
try:
(F, G), opt = parallel_poly_from_expr((f, g), *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('pdiv', 2, exc)
q, r = F.pdiv(G)
if not opt.polys:
return q.as_expr(), r.as_expr()
else:
return q, r
@public
def prem(f, g, *gens, **args):
"""
Compute polynomial pseudo-remainder of ``f`` and ``g``.
Examples
========
>>> from sympy import prem
>>> from sympy.abc import x
>>> prem(x**2 + 1, 2*x - 4)
20
"""
options.allowed_flags(args, ['polys'])
try:
(F, G), opt = parallel_poly_from_expr((f, g), *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('prem', 2, exc)
r = F.prem(G)
if not opt.polys:
return r.as_expr()
else:
return r
@public
def pquo(f, g, *gens, **args):
"""
Compute polynomial pseudo-quotient of ``f`` and ``g``.
Examples
========
>>> from sympy import pquo
>>> from sympy.abc import x
>>> pquo(x**2 + 1, 2*x - 4)
2*x + 4
>>> pquo(x**2 - 1, 2*x - 1)
2*x + 1
"""
options.allowed_flags(args, ['polys'])
try:
(F, G), opt = parallel_poly_from_expr((f, g), *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('pquo', 2, exc)
try:
q = F.pquo(G)
except ExactQuotientFailed:
raise ExactQuotientFailed(f, g)
if not opt.polys:
return q.as_expr()
else:
return q
@public
def pexquo(f, g, *gens, **args):
"""
Compute polynomial exact pseudo-quotient of ``f`` and ``g``.
Examples
========
>>> from sympy import pexquo
>>> from sympy.abc import x
>>> pexquo(x**2 - 1, 2*x - 2)
2*x + 2
>>> pexquo(x**2 + 1, 2*x - 4)
Traceback (most recent call last):
...
ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1
"""
options.allowed_flags(args, ['polys'])
try:
(F, G), opt = parallel_poly_from_expr((f, g), *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('pexquo', 2, exc)
q = F.pexquo(G)
if not opt.polys:
return q.as_expr()
else:
return q
@public
def div(f, g, *gens, **args):
"""
Compute polynomial division of ``f`` and ``g``.
Examples
========
>>> from sympy import div, ZZ, QQ
>>> from sympy.abc import x
>>> div(x**2 + 1, 2*x - 4, domain=ZZ)
(0, x**2 + 1)
>>> div(x**2 + 1, 2*x - 4, domain=QQ)
(x/2 + 1, 5)
"""
options.allowed_flags(args, ['auto', 'polys'])
try:
(F, G), opt = parallel_poly_from_expr((f, g), *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('div', 2, exc)
q, r = F.div(G, auto=opt.auto)
if not opt.polys:
return q.as_expr(), r.as_expr()
else:
return q, r
@public
def rem(f, g, *gens, **args):
"""
Compute polynomial remainder of ``f`` and ``g``.
Examples
========
>>> from sympy import rem, ZZ, QQ
>>> from sympy.abc import x
>>> rem(x**2 + 1, 2*x - 4, domain=ZZ)
x**2 + 1
>>> rem(x**2 + 1, 2*x - 4, domain=QQ)
5
"""
options.allowed_flags(args, ['auto', 'polys'])
try:
(F, G), opt = parallel_poly_from_expr((f, g), *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('rem', 2, exc)
r = F.rem(G, auto=opt.auto)
if not opt.polys:
return r.as_expr()
else:
return r
@public
def quo(f, g, *gens, **args):
"""
Compute polynomial quotient of ``f`` and ``g``.
Examples
========
>>> from sympy import quo
>>> from sympy.abc import x
>>> quo(x**2 + 1, 2*x - 4)
x/2 + 1
>>> quo(x**2 - 1, x - 1)
x + 1
"""
options.allowed_flags(args, ['auto', 'polys'])
try:
(F, G), opt = parallel_poly_from_expr((f, g), *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('quo', 2, exc)
q = F.quo(G, auto=opt.auto)
if not opt.polys:
return q.as_expr()
else:
return q
@public
def exquo(f, g, *gens, **args):
"""
Compute polynomial exact quotient of ``f`` and ``g``.
Examples
========
>>> from sympy import exquo
>>> from sympy.abc import x
>>> exquo(x**2 - 1, x - 1)
x + 1
>>> exquo(x**2 + 1, 2*x - 4)
Traceback (most recent call last):
...
ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1
"""
options.allowed_flags(args, ['auto', 'polys'])
try:
(F, G), opt = parallel_poly_from_expr((f, g), *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('exquo', 2, exc)
q = F.exquo(G, auto=opt.auto)
if not opt.polys:
return q.as_expr()
else:
return q
@public
def half_gcdex(f, g, *gens, **args):
"""
Half extended Euclidean algorithm of ``f`` and ``g``.
Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``.
Examples
========
>>> from sympy import half_gcdex
>>> from sympy.abc import x
>>> half_gcdex(x**4 - 2*x**3 - 6*x**2 + 12*x + 15, x**3 + x**2 - 4*x - 4)
(3/5 - x/5, x + 1)
"""
options.allowed_flags(args, ['auto', 'polys'])
try:
(F, G), opt = parallel_poly_from_expr((f, g), *gens, **args)
except PolificationFailed as exc:
domain, (a, b) = construct_domain(exc.exprs)
try:
s, h = domain.half_gcdex(a, b)
except NotImplementedError:
raise ComputationFailed('half_gcdex', 2, exc)
else:
return domain.to_sympy(s), domain.to_sympy(h)
s, h = F.half_gcdex(G, auto=opt.auto)
if not opt.polys:
return s.as_expr(), h.as_expr()
else:
return s, h
@public
def gcdex(f, g, *gens, **args):
"""
Extended Euclidean algorithm of ``f`` and ``g``.
Returns ``(s, t, h)`` such that ``h = gcd(f, g)`` and ``s*f + t*g = h``.
Examples
========
>>> from sympy import gcdex
>>> from sympy.abc import x
>>> gcdex(x**4 - 2*x**3 - 6*x**2 + 12*x + 15, x**3 + x**2 - 4*x - 4)
(3/5 - x/5, x**2/5 - 6*x/5 + 2, x + 1)
"""
options.allowed_flags(args, ['auto', 'polys'])
try:
(F, G), opt = parallel_poly_from_expr((f, g), *gens, **args)
except PolificationFailed as exc:
domain, (a, b) = construct_domain(exc.exprs)
try:
s, t, h = domain.gcdex(a, b)
except NotImplementedError:
raise ComputationFailed('gcdex', 2, exc)
else:
return domain.to_sympy(s), domain.to_sympy(t), domain.to_sympy(h)
s, t, h = F.gcdex(G, auto=opt.auto)
if not opt.polys:
return s.as_expr(), t.as_expr(), h.as_expr()
else:
return s, t, h
@public
def invert(f, g, *gens, **args):
"""
Invert ``f`` modulo ``g`` when possible.
Examples
========
>>> from sympy import invert, S
>>> from sympy.core.numbers import mod_inverse
>>> from sympy.abc import x
>>> invert(x**2 - 1, 2*x - 1)
-4/3
>>> invert(x**2 - 1, x - 1)
Traceback (most recent call last):
...
NotInvertible: zero divisor
For more efficient inversion of Rationals,
use the :obj:`~.mod_inverse` function:
>>> mod_inverse(3, 5)
2
>>> (S(2)/5).invert(S(7)/3)
5/2
See Also
========
sympy.core.numbers.mod_inverse
"""
options.allowed_flags(args, ['auto', 'polys'])
try:
(F, G), opt = parallel_poly_from_expr((f, g), *gens, **args)
except PolificationFailed as exc:
domain, (a, b) = construct_domain(exc.exprs)
try:
return domain.to_sympy(domain.invert(a, b))
except NotImplementedError:
raise ComputationFailed('invert', 2, exc)
h = F.invert(G, auto=opt.auto)
if not opt.polys:
return h.as_expr()
else:
return h
@public
def subresultants(f, g, *gens, **args):
"""
Compute subresultant PRS of ``f`` and ``g``.
Examples
========
>>> from sympy import subresultants
>>> from sympy.abc import x
>>> subresultants(x**2 + 1, x**2 - 1)
[x**2 + 1, x**2 - 1, -2]
"""
options.allowed_flags(args, ['polys'])
try:
(F, G), opt = parallel_poly_from_expr((f, g), *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('subresultants', 2, exc)
result = F.subresultants(G)
if not opt.polys:
return [r.as_expr() for r in result]
else:
return result
@public
def resultant(f, g, *gens, includePRS=False, **args):
"""
Compute resultant of ``f`` and ``g``.
Examples
========
>>> from sympy import resultant
>>> from sympy.abc import x
>>> resultant(x**2 + 1, x**2 - 1)
4
"""
options.allowed_flags(args, ['polys'])
try:
(F, G), opt = parallel_poly_from_expr((f, g), *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('resultant', 2, exc)
if includePRS:
result, R = F.resultant(G, includePRS=includePRS)
else:
result = F.resultant(G)
if not opt.polys:
if includePRS:
return result.as_expr(), [r.as_expr() for r in R]
return result.as_expr()
else:
if includePRS:
return result, R
return result
@public
def discriminant(f, *gens, **args):
"""
Compute discriminant of ``f``.
Examples
========
>>> from sympy import discriminant
>>> from sympy.abc import x
>>> discriminant(x**2 + 2*x + 3)
-8
"""
options.allowed_flags(args, ['polys'])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('discriminant', 1, exc)
result = F.discriminant()
if not opt.polys:
return result.as_expr()
else:
return result
@public
def cofactors(f, g, *gens, **args):
"""
Compute GCD and cofactors of ``f`` and ``g``.
Returns polynomials ``(h, cff, cfg)`` such that ``h = gcd(f, g)``, and
``cff = quo(f, h)`` and ``cfg = quo(g, h)`` are, so called, cofactors
of ``f`` and ``g``.
Examples
========
>>> from sympy import cofactors
>>> from sympy.abc import x
>>> cofactors(x**2 - 1, x**2 - 3*x + 2)
(x - 1, x + 1, x - 2)
"""
options.allowed_flags(args, ['polys'])
try:
(F, G), opt = parallel_poly_from_expr((f, g), *gens, **args)
except PolificationFailed as exc:
domain, (a, b) = construct_domain(exc.exprs)
try:
h, cff, cfg = domain.cofactors(a, b)
except NotImplementedError:
raise ComputationFailed('cofactors', 2, exc)
else:
return domain.to_sympy(h), domain.to_sympy(cff), domain.to_sympy(cfg)
h, cff, cfg = F.cofactors(G)
if not opt.polys:
return h.as_expr(), cff.as_expr(), cfg.as_expr()
else:
return h, cff, cfg
@public
def gcd_list(seq, *gens, **args):
"""
Compute GCD of a list of polynomials.
Examples
========
>>> from sympy import gcd_list
>>> from sympy.abc import x
>>> gcd_list([x**3 - 1, x**2 - 1, x**2 - 3*x + 2])
x - 1
"""
seq = sympify(seq)
def try_non_polynomial_gcd(seq):
if not gens and not args:
domain, numbers = construct_domain(seq)
if not numbers:
return domain.zero
elif domain.is_Numerical:
result, numbers = numbers[0], numbers[1:]
for number in numbers:
result = domain.gcd(result, number)
if domain.is_one(result):
break
return domain.to_sympy(result)
return None
result = try_non_polynomial_gcd(seq)
if result is not None:
return result
options.allowed_flags(args, ['polys'])
try:
polys, opt = parallel_poly_from_expr(seq, *gens, **args)
# gcd for domain Q[irrational] (purely algebraic irrational)
if len(seq) > 1 and all(elt.is_algebraic and elt.is_irrational for elt in seq):
a = seq[-1]
lst = [ (a/elt).ratsimp() for elt in seq[:-1] ]
if all(frc.is_rational for frc in lst):
lc = 1
for frc in lst:
lc = lcm(lc, frc.as_numer_denom()[0])
# abs ensures that the gcd is always non-negative
return abs(a/lc)
except PolificationFailed as exc:
result = try_non_polynomial_gcd(exc.exprs)
if result is not None:
return result
else:
raise ComputationFailed('gcd_list', len(seq), exc)
if not polys:
if not opt.polys:
return S.Zero
else:
return Poly(0, opt=opt)
result, polys = polys[0], polys[1:]
for poly in polys:
result = result.gcd(poly)
if result.is_one:
break
if not opt.polys:
return result.as_expr()
else:
return result
@public
def gcd(f, g=None, *gens, **args):
"""
Compute GCD of ``f`` and ``g``.
Examples
========
>>> from sympy import gcd
>>> from sympy.abc import x
>>> gcd(x**2 - 1, x**2 - 3*x + 2)
x - 1
"""
if hasattr(f, '__iter__'):
if g is not None:
gens = (g,) + gens
return gcd_list(f, *gens, **args)
elif g is None:
raise TypeError("gcd() takes 2 arguments or a sequence of arguments")
options.allowed_flags(args, ['polys'])
try:
(F, G), opt = parallel_poly_from_expr((f, g), *gens, **args)
# gcd for domain Q[irrational] (purely algebraic irrational)
a, b = map(sympify, (f, g))
if a.is_algebraic and a.is_irrational and b.is_algebraic and b.is_irrational:
frc = (a/b).ratsimp()
if frc.is_rational:
# abs ensures that the returned gcd is always non-negative
return abs(a/frc.as_numer_denom()[0])
except PolificationFailed as exc:
domain, (a, b) = construct_domain(exc.exprs)
try:
return domain.to_sympy(domain.gcd(a, b))
except NotImplementedError:
raise ComputationFailed('gcd', 2, exc)
result = F.gcd(G)
if not opt.polys:
return result.as_expr()
else:
return result
@public
def lcm_list(seq, *gens, **args):
"""
Compute LCM of a list of polynomials.
Examples
========
>>> from sympy import lcm_list
>>> from sympy.abc import x
>>> lcm_list([x**3 - 1, x**2 - 1, x**2 - 3*x + 2])
x**5 - x**4 - 2*x**3 - x**2 + x + 2
"""
seq = sympify(seq)
def try_non_polynomial_lcm(seq):
if not gens and not args:
domain, numbers = construct_domain(seq)
if not numbers:
return domain.one
elif domain.is_Numerical:
result, numbers = numbers[0], numbers[1:]
for number in numbers:
result = domain.lcm(result, number)
return domain.to_sympy(result)
return None
result = try_non_polynomial_lcm(seq)
if result is not None:
return result
options.allowed_flags(args, ['polys'])
try:
polys, opt = parallel_poly_from_expr(seq, *gens, **args)
# lcm for domain Q[irrational] (purely algebraic irrational)
if len(seq) > 1 and all(elt.is_algebraic and elt.is_irrational for elt in seq):
a = seq[-1]
lst = [ (a/elt).ratsimp() for elt in seq[:-1] ]
if all(frc.is_rational for frc in lst):
lc = 1
for frc in lst:
lc = lcm(lc, frc.as_numer_denom()[1])
return a*lc
except PolificationFailed as exc:
result = try_non_polynomial_lcm(exc.exprs)
if result is not None:
return result
else:
raise ComputationFailed('lcm_list', len(seq), exc)
if not polys:
if not opt.polys:
return S.One
else:
return Poly(1, opt=opt)
result, polys = polys[0], polys[1:]
for poly in polys:
result = result.lcm(poly)
if not opt.polys:
return result.as_expr()
else:
return result
@public
def lcm(f, g=None, *gens, **args):
"""
Compute LCM of ``f`` and ``g``.
Examples
========
>>> from sympy import lcm
>>> from sympy.abc import x
>>> lcm(x**2 - 1, x**2 - 3*x + 2)
x**3 - 2*x**2 - x + 2
"""
if hasattr(f, '__iter__'):
if g is not None:
gens = (g,) + gens
return lcm_list(f, *gens, **args)
elif g is None:
raise TypeError("lcm() takes 2 arguments or a sequence of arguments")
options.allowed_flags(args, ['polys'])
try:
(F, G), opt = parallel_poly_from_expr((f, g), *gens, **args)
# lcm for domain Q[irrational] (purely algebraic irrational)
a, b = map(sympify, (f, g))
if a.is_algebraic and a.is_irrational and b.is_algebraic and b.is_irrational:
frc = (a/b).ratsimp()
if frc.is_rational:
return a*frc.as_numer_denom()[1]
except PolificationFailed as exc:
domain, (a, b) = construct_domain(exc.exprs)
try:
return domain.to_sympy(domain.lcm(a, b))
except NotImplementedError:
raise ComputationFailed('lcm', 2, exc)
result = F.lcm(G)
if not opt.polys:
return result.as_expr()
else:
return result
@public
def terms_gcd(f, *gens, **args):
"""
Remove GCD of terms from ``f``.
If the ``deep`` flag is True, then the arguments of ``f`` will have
terms_gcd applied to them.
If a fraction is factored out of ``f`` and ``f`` is an Add, then
an unevaluated Mul will be returned so that automatic simplification
does not redistribute it. The hint ``clear``, when set to False, can be
used to prevent such factoring when all coefficients are not fractions.
Examples
========
>>> from sympy import terms_gcd, cos
>>> from sympy.abc import x, y
>>> terms_gcd(x**6*y**2 + x**3*y, x, y)
x**3*y*(x**3*y + 1)
The default action of polys routines is to expand the expression
given to them. terms_gcd follows this behavior:
>>> terms_gcd((3+3*x)*(x+x*y))
3*x*(x*y + x + y + 1)
If this is not desired then the hint ``expand`` can be set to False.
In this case the expression will be treated as though it were comprised
of one or more terms:
>>> terms_gcd((3+3*x)*(x+x*y), expand=False)
(3*x + 3)*(x*y + x)
In order to traverse factors of a Mul or the arguments of other
functions, the ``deep`` hint can be used:
>>> terms_gcd((3 + 3*x)*(x + x*y), expand=False, deep=True)
3*x*(x + 1)*(y + 1)
>>> terms_gcd(cos(x + x*y), deep=True)
cos(x*(y + 1))
Rationals are factored out by default:
>>> terms_gcd(x + y/2)
(2*x + y)/2
Only the y-term had a coefficient that was a fraction; if one
does not want to factor out the 1/2 in cases like this, the
flag ``clear`` can be set to False:
>>> terms_gcd(x + y/2, clear=False)
x + y/2
>>> terms_gcd(x*y/2 + y**2, clear=False)
y*(x/2 + y)
The ``clear`` flag is ignored if all coefficients are fractions:
>>> terms_gcd(x/3 + y/2, clear=False)
(2*x + 3*y)/6
See Also
========
sympy.core.exprtools.gcd_terms, sympy.core.exprtools.factor_terms
"""
from sympy.core.relational import Equality
orig = sympify(f)
if isinstance(f, Equality):
return Equality(*(terms_gcd(s, *gens, **args) for s in [f.lhs, f.rhs]))
elif isinstance(f, Relational):
raise TypeError("Inequalities can not be used with terms_gcd. Found: %s" %(f,))
if not isinstance(f, Expr) or f.is_Atom:
return orig
if args.get('deep', False):
new = f.func(*[terms_gcd(a, *gens, **args) for a in f.args])
args.pop('deep')
args['expand'] = False
return terms_gcd(new, *gens, **args)
clear = args.pop('clear', True)
options.allowed_flags(args, ['polys'])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed as exc:
return exc.expr
J, f = F.terms_gcd()
if opt.domain.is_Ring:
if opt.domain.is_Field:
denom, f = f.clear_denoms(convert=True)
coeff, f = f.primitive()
if opt.domain.is_Field:
coeff /= denom
else:
coeff = S.One
term = Mul(*[x**j for x, j in zip(f.gens, J)])
if coeff == 1:
coeff = S.One
if term == 1:
return orig
if clear:
return _keep_coeff(coeff, term*f.as_expr())
# base the clearing on the form of the original expression, not
# the (perhaps) Mul that we have now
coeff, f = _keep_coeff(coeff, f.as_expr(), clear=False).as_coeff_Mul()
return _keep_coeff(coeff, term*f, clear=False)
@public
def trunc(f, p, *gens, **args):
"""
Reduce ``f`` modulo a constant ``p``.
Examples
========
>>> from sympy import trunc
>>> from sympy.abc import x
>>> trunc(2*x**3 + 3*x**2 + 5*x + 7, 3)
-x**3 - x + 1
"""
options.allowed_flags(args, ['auto', 'polys'])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('trunc', 1, exc)
result = F.trunc(sympify(p))
if not opt.polys:
return result.as_expr()
else:
return result
@public
def monic(f, *gens, **args):
"""
Divide all coefficients of ``f`` by ``LC(f)``.
Examples
========
>>> from sympy import monic
>>> from sympy.abc import x
>>> monic(3*x**2 + 4*x + 2)
x**2 + 4*x/3 + 2/3
"""
options.allowed_flags(args, ['auto', 'polys'])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('monic', 1, exc)
result = F.monic(auto=opt.auto)
if not opt.polys:
return result.as_expr()
else:
return result
@public
def content(f, *gens, **args):
"""
Compute GCD of coefficients of ``f``.
Examples
========
>>> from sympy import content
>>> from sympy.abc import x
>>> content(6*x**2 + 8*x + 12)
2
"""
options.allowed_flags(args, ['polys'])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('content', 1, exc)
return F.content()
@public
def primitive(f, *gens, **args):
"""
Compute content and the primitive form of ``f``.
Examples
========
>>> from sympy.polys.polytools import primitive
>>> from sympy.abc import x
>>> primitive(6*x**2 + 8*x + 12)
(2, 3*x**2 + 4*x + 6)
>>> eq = (2 + 2*x)*x + 2
Expansion is performed by default:
>>> primitive(eq)
(2, x**2 + x + 1)
Set ``expand`` to False to shut this off. Note that the
extraction will not be recursive; use the as_content_primitive method
for recursive, non-destructive Rational extraction.
>>> primitive(eq, expand=False)
(1, x*(2*x + 2) + 2)
>>> eq.as_content_primitive()
(2, x*(x + 1) + 1)
"""
options.allowed_flags(args, ['polys'])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('primitive', 1, exc)
cont, result = F.primitive()
if not opt.polys:
return cont, result.as_expr()
else:
return cont, result
@public
def compose(f, g, *gens, **args):
"""
Compute functional composition ``f(g)``.
Examples
========
>>> from sympy import compose
>>> from sympy.abc import x
>>> compose(x**2 + x, x - 1)
x**2 - x
"""
options.allowed_flags(args, ['polys'])
try:
(F, G), opt = parallel_poly_from_expr((f, g), *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('compose', 2, exc)
result = F.compose(G)
if not opt.polys:
return result.as_expr()
else:
return result
@public
def decompose(f, *gens, **args):
"""
Compute functional decomposition of ``f``.
Examples
========
>>> from sympy import decompose
>>> from sympy.abc import x
>>> decompose(x**4 + 2*x**3 - x - 1)
[x**2 - x - 1, x**2 + x]
"""
options.allowed_flags(args, ['polys'])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('decompose', 1, exc)
result = F.decompose()
if not opt.polys:
return [r.as_expr() for r in result]
else:
return result
@public
def sturm(f, *gens, **args):
"""
Compute Sturm sequence of ``f``.
Examples
========
>>> from sympy import sturm
>>> from sympy.abc import x
>>> sturm(x**3 - 2*x**2 + x - 3)
[x**3 - 2*x**2 + x - 3, 3*x**2 - 4*x + 1, 2*x/9 + 25/9, -2079/4]
"""
options.allowed_flags(args, ['auto', 'polys'])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('sturm', 1, exc)
result = F.sturm(auto=opt.auto)
if not opt.polys:
return [r.as_expr() for r in result]
else:
return result
@public
def gff_list(f, *gens, **args):
"""
Compute a list of greatest factorial factors of ``f``.
Note that the input to ff() and rf() should be Poly instances to use the
definitions here.
Examples
========
>>> from sympy import gff_list, ff, Poly
>>> from sympy.abc import x
>>> f = Poly(x**5 + 2*x**4 - x**3 - 2*x**2, x)
>>> gff_list(f)
[(Poly(x, x, domain='ZZ'), 1), (Poly(x + 2, x, domain='ZZ'), 4)]
>>> (ff(Poly(x), 1)*ff(Poly(x + 2), 4)) == f
True
>>> f = Poly(x**12 + 6*x**11 - 11*x**10 - 56*x**9 + 220*x**8 + 208*x**7 - \
1401*x**6 + 1090*x**5 + 2715*x**4 - 6720*x**3 - 1092*x**2 + 5040*x, x)
>>> gff_list(f)
[(Poly(x**3 + 7, x, domain='ZZ'), 2), (Poly(x**2 + 5*x, x, domain='ZZ'), 3)]
>>> ff(Poly(x**3 + 7, x), 2)*ff(Poly(x**2 + 5*x, x), 3) == f
True
"""
options.allowed_flags(args, ['polys'])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('gff_list', 1, exc)
factors = F.gff_list()
if not opt.polys:
return [(g.as_expr(), k) for g, k in factors]
else:
return factors
@public
def gff(f, *gens, **args):
"""Compute greatest factorial factorization of ``f``. """
raise NotImplementedError('symbolic falling factorial')
@public
def sqf_norm(f, *gens, **args):
"""
Compute square-free norm of ``f``.
Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and
``r(x) = Norm(g(x))`` is a square-free polynomial over ``K``,
where ``a`` is the algebraic extension of the ground domain.
Examples
========
>>> from sympy import sqf_norm, sqrt
>>> from sympy.abc import x
>>> sqf_norm(x**2 + 1, extension=[sqrt(3)])
(1, x**2 - 2*sqrt(3)*x + 4, x**4 - 4*x**2 + 16)
"""
options.allowed_flags(args, ['polys'])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('sqf_norm', 1, exc)
s, g, r = F.sqf_norm()
if not opt.polys:
return Integer(s), g.as_expr(), r.as_expr()
else:
return Integer(s), g, r
@public
def sqf_part(f, *gens, **args):
"""
Compute square-free part of ``f``.
Examples
========
>>> from sympy import sqf_part
>>> from sympy.abc import x
>>> sqf_part(x**3 - 3*x - 2)
x**2 - x - 2
"""
options.allowed_flags(args, ['polys'])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('sqf_part', 1, exc)
result = F.sqf_part()
if not opt.polys:
return result.as_expr()
else:
return result
def _sorted_factors(factors, method):
"""Sort a list of ``(expr, exp)`` pairs. """
if method == 'sqf':
def key(obj):
poly, exp = obj
rep = poly.rep.rep
return (exp, len(rep), len(poly.gens), rep)
else:
def key(obj):
poly, exp = obj
rep = poly.rep.rep
return (len(rep), len(poly.gens), exp, rep)
return sorted(factors, key=key)
def _factors_product(factors):
"""Multiply a list of ``(expr, exp)`` pairs. """
return Mul(*[f.as_expr()**k for f, k in factors])
def _symbolic_factor_list(expr, opt, method):
"""Helper function for :func:`_symbolic_factor`. """
coeff, factors = S.One, []
args = [i._eval_factor() if hasattr(i, '_eval_factor') else i
for i in Mul.make_args(expr)]
for arg in args:
if arg.is_Number or (isinstance(arg, Expr) and pure_complex(arg)):
coeff *= arg
continue
elif arg.is_Pow and arg.base != S.Exp1:
base, exp = arg.args
if base.is_Number and exp.is_Number:
coeff *= arg
continue
if base.is_Number:
factors.append((base, exp))
continue
else:
base, exp = arg, S.One
try:
poly, _ = _poly_from_expr(base, opt)
except PolificationFailed as exc:
factors.append((exc.expr, exp))
else:
func = getattr(poly, method + '_list')
_coeff, _factors = func()
if _coeff is not S.One:
if exp.is_Integer:
coeff *= _coeff**exp
elif _coeff.is_positive:
factors.append((_coeff, exp))
else:
_factors.append((_coeff, S.One))
if exp is S.One:
factors.extend(_factors)
elif exp.is_integer:
factors.extend([(f, k*exp) for f, k in _factors])
else:
other = []
for f, k in _factors:
if f.as_expr().is_positive:
factors.append((f, k*exp))
else:
other.append((f, k))
factors.append((_factors_product(other), exp))
if method == 'sqf':
factors = [(reduce(mul, (f for f, _ in factors if _ == k)), k)
for k in {i for _, i in factors}]
return coeff, factors
def _symbolic_factor(expr, opt, method):
"""Helper function for :func:`_factor`. """
if isinstance(expr, Expr):
if hasattr(expr,'_eval_factor'):
return expr._eval_factor()
coeff, factors = _symbolic_factor_list(together(expr, fraction=opt['fraction']), opt, method)
return _keep_coeff(coeff, _factors_product(factors))
elif hasattr(expr, 'args'):
return expr.func(*[_symbolic_factor(arg, opt, method) for arg in expr.args])
elif hasattr(expr, '__iter__'):
return expr.__class__([_symbolic_factor(arg, opt, method) for arg in expr])
else:
return expr
def _generic_factor_list(expr, gens, args, method):
"""Helper function for :func:`sqf_list` and :func:`factor_list`. """
options.allowed_flags(args, ['frac', 'polys'])
opt = options.build_options(gens, args)
expr = sympify(expr)
if isinstance(expr, (Expr, Poly)):
if isinstance(expr, Poly):
numer, denom = expr, 1
else:
numer, denom = together(expr).as_numer_denom()
cp, fp = _symbolic_factor_list(numer, opt, method)
cq, fq = _symbolic_factor_list(denom, opt, method)
if fq and not opt.frac:
raise PolynomialError("a polynomial expected, got %s" % expr)
_opt = opt.clone(dict(expand=True))
for factors in (fp, fq):
for i, (f, k) in enumerate(factors):
if not f.is_Poly:
f, _ = _poly_from_expr(f, _opt)
factors[i] = (f, k)
fp = _sorted_factors(fp, method)
fq = _sorted_factors(fq, method)
if not opt.polys:
fp = [(f.as_expr(), k) for f, k in fp]
fq = [(f.as_expr(), k) for f, k in fq]
coeff = cp/cq
if not opt.frac:
return coeff, fp
else:
return coeff, fp, fq
else:
raise PolynomialError("a polynomial expected, got %s" % expr)
def _generic_factor(expr, gens, args, method):
"""Helper function for :func:`sqf` and :func:`factor`. """
fraction = args.pop('fraction', True)
options.allowed_flags(args, [])
opt = options.build_options(gens, args)
opt['fraction'] = fraction
return _symbolic_factor(sympify(expr), opt, method)
def to_rational_coeffs(f):
"""
try to transform a polynomial to have rational coefficients
try to find a transformation ``x = alpha*y``
``f(x) = lc*alpha**n * g(y)`` where ``g`` is a polynomial with
rational coefficients, ``lc`` the leading coefficient.
If this fails, try ``x = y + beta``
``f(x) = g(y)``
Returns ``None`` if ``g`` not found;
``(lc, alpha, None, g)`` in case of rescaling
``(None, None, beta, g)`` in case of translation
Notes
=====
Currently it transforms only polynomials without roots larger than 2.
Examples
========
>>> from sympy import sqrt, Poly, simplify
>>> from sympy.polys.polytools import to_rational_coeffs
>>> from sympy.abc import x
>>> p = Poly(((x**2-1)*(x-2)).subs({x:x*(1 + sqrt(2))}), x, domain='EX')
>>> lc, r, _, g = to_rational_coeffs(p)
>>> lc, r
(7 + 5*sqrt(2), 2 - 2*sqrt(2))
>>> g
Poly(x**3 + x**2 - 1/4*x - 1/4, x, domain='QQ')
>>> r1 = simplify(1/r)
>>> Poly(lc*r**3*(g.as_expr()).subs({x:x*r1}), x, domain='EX') == p
True
"""
from sympy.simplify.simplify import simplify
def _try_rescale(f, f1=None):
"""
try rescaling ``x -> alpha*x`` to convert f to a polynomial
with rational coefficients.
Returns ``alpha, f``; if the rescaling is successful,
``alpha`` is the rescaling factor, and ``f`` is the rescaled
polynomial; else ``alpha`` is ``None``.
"""
from sympy.core.add import Add
if not len(f.gens) == 1 or not (f.gens[0]).is_Atom:
return None, f
n = f.degree()
lc = f.LC()
f1 = f1 or f1.monic()
coeffs = f1.all_coeffs()[1:]
coeffs = [simplify(coeffx) for coeffx in coeffs]
if len(coeffs) > 1 and coeffs[-2]:
rescale1_x = simplify(coeffs[-2]/coeffs[-1])
coeffs1 = []
for i in range(len(coeffs)):
coeffx = simplify(coeffs[i]*rescale1_x**(i + 1))
if not coeffx.is_rational:
break
coeffs1.append(coeffx)
else:
rescale_x = simplify(1/rescale1_x)
x = f.gens[0]
v = [x**n]
for i in range(1, n + 1):
v.append(coeffs1[i - 1]*x**(n - i))
f = Add(*v)
f = Poly(f)
return lc, rescale_x, f
return None
def _try_translate(f, f1=None):
"""
try translating ``x -> x + alpha`` to convert f to a polynomial
with rational coefficients.
Returns ``alpha, f``; if the translating is successful,
``alpha`` is the translating factor, and ``f`` is the shifted
polynomial; else ``alpha`` is ``None``.
"""
if not len(f.gens) == 1 or not (f.gens[0]).is_Atom:
return None, f
n = f.degree()
f1 = f1 or f1.monic()
coeffs = f1.all_coeffs()[1:]
c = simplify(coeffs[0])
if c.is_Add and not c.is_rational:
rat, nonrat = sift(c.args,
lambda z: z.is_rational is True, binary=True)
alpha = -c.func(*nonrat)/n
f2 = f1.shift(alpha)
return alpha, f2
return None
def _has_square_roots(p):
"""
Return True if ``f`` is a sum with square roots but no other root
"""
from sympy.core.exprtools import Factors
coeffs = p.coeffs()
has_sq = False
for y in coeffs:
for x in Add.make_args(y):
f = Factors(x).factors
r = [wx.q for b, wx in f.items() if
b.is_number and wx.is_Rational and wx.q >= 2]
if not r:
continue
if min(r) == 2:
has_sq = True
if max(r) > 2:
return False
return has_sq
if f.get_domain().is_EX and _has_square_roots(f):
f1 = f.monic()
r = _try_rescale(f, f1)
if r:
return r[0], r[1], None, r[2]
else:
r = _try_translate(f, f1)
if r:
return None, None, r[0], r[1]
return None
def _torational_factor_list(p, x):
"""
helper function to factor polynomial using to_rational_coeffs
Examples
========
>>> from sympy.polys.polytools import _torational_factor_list
>>> from sympy.abc import x
>>> from sympy import sqrt, expand, Mul
>>> p = expand(((x**2-1)*(x-2)).subs({x:x*(1 + sqrt(2))}))
>>> factors = _torational_factor_list(p, x); factors
(-2, [(-x*(1 + sqrt(2))/2 + 1, 1), (-x*(1 + sqrt(2)) - 1, 1), (-x*(1 + sqrt(2)) + 1, 1)])
>>> expand(factors[0]*Mul(*[z[0] for z in factors[1]])) == p
True
>>> p = expand(((x**2-1)*(x-2)).subs({x:x + sqrt(2)}))
>>> factors = _torational_factor_list(p, x); factors
(1, [(x - 2 + sqrt(2), 1), (x - 1 + sqrt(2), 1), (x + 1 + sqrt(2), 1)])
>>> expand(factors[0]*Mul(*[z[0] for z in factors[1]])) == p
True
"""
from sympy.simplify.simplify import simplify
p1 = Poly(p, x, domain='EX')
n = p1.degree()
res = to_rational_coeffs(p1)
if not res:
return None
lc, r, t, g = res
factors = factor_list(g.as_expr())
if lc:
c = simplify(factors[0]*lc*r**n)
r1 = simplify(1/r)
a = []
for z in factors[1:][0]:
a.append((simplify(z[0].subs({x: x*r1})), z[1]))
else:
c = factors[0]
a = []
for z in factors[1:][0]:
a.append((z[0].subs({x: x - t}), z[1]))
return (c, a)
@public
def sqf_list(f, *gens, **args):
"""
Compute a list of square-free factors of ``f``.
Examples
========
>>> from sympy import sqf_list
>>> from sympy.abc import x
>>> sqf_list(2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16)
(2, [(x + 1, 2), (x + 2, 3)])
"""
return _generic_factor_list(f, gens, args, method='sqf')
@public
def sqf(f, *gens, **args):
"""
Compute square-free factorization of ``f``.
Examples
========
>>> from sympy import sqf
>>> from sympy.abc import x
>>> sqf(2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16)
2*(x + 1)**2*(x + 2)**3
"""
return _generic_factor(f, gens, args, method='sqf')
@public
def factor_list(f, *gens, **args):
"""
Compute a list of irreducible factors of ``f``.
Examples
========
>>> from sympy import factor_list
>>> from sympy.abc import x, y
>>> factor_list(2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y)
(2, [(x + y, 1), (x**2 + 1, 2)])
"""
return _generic_factor_list(f, gens, args, method='factor')
@public
def factor(f, *gens, deep=False, **args):
"""
Compute the factorization of expression, ``f``, into irreducibles. (To
factor an integer into primes, use ``factorint``.)
There two modes implemented: symbolic and formal. If ``f`` is not an
instance of :class:`Poly` and generators are not specified, then the
former mode is used. Otherwise, the formal mode is used.
In symbolic mode, :func:`factor` will traverse the expression tree and
factor its components without any prior expansion, unless an instance
of :class:`~.Add` is encountered (in this case formal factorization is
used). This way :func:`factor` can handle large or symbolic exponents.
By default, the factorization is computed over the rationals. To factor
over other domain, e.g. an algebraic or finite field, use appropriate
options: ``extension``, ``modulus`` or ``domain``.
Examples
========
>>> from sympy import factor, sqrt, exp
>>> from sympy.abc import x, y
>>> factor(2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y)
2*(x + y)*(x**2 + 1)**2
>>> factor(x**2 + 1)
x**2 + 1
>>> factor(x**2 + 1, modulus=2)
(x + 1)**2
>>> factor(x**2 + 1, gaussian=True)
(x - I)*(x + I)
>>> factor(x**2 - 2, extension=sqrt(2))
(x - sqrt(2))*(x + sqrt(2))
>>> factor((x**2 - 1)/(x**2 + 4*x + 4))
(x - 1)*(x + 1)/(x + 2)**2
>>> factor((x**2 + 4*x + 4)**10000000*(x**2 + 1))
(x + 2)**20000000*(x**2 + 1)
By default, factor deals with an expression as a whole:
>>> eq = 2**(x**2 + 2*x + 1)
>>> factor(eq)
2**(x**2 + 2*x + 1)
If the ``deep`` flag is True then subexpressions will
be factored:
>>> factor(eq, deep=True)
2**((x + 1)**2)
If the ``fraction`` flag is False then rational expressions
won't be combined. By default it is True.
>>> factor(5*x + 3*exp(2 - 7*x), deep=True)
(5*x*exp(7*x) + 3*exp(2))*exp(-7*x)
>>> factor(5*x + 3*exp(2 - 7*x), deep=True, fraction=False)
5*x + 3*exp(2)*exp(-7*x)
See Also
========
sympy.ntheory.factor_.factorint
"""
f = sympify(f)
if deep:
from sympy.simplify.simplify import bottom_up
def _try_factor(expr):
"""
Factor, but avoid changing the expression when unable to.
"""
fac = factor(expr, *gens, **args)
if fac.is_Mul or fac.is_Pow:
return fac
return expr
f = bottom_up(f, _try_factor)
# clean up any subexpressions that may have been expanded
# while factoring out a larger expression
partials = {}
muladd = f.atoms(Mul, Add)
for p in muladd:
fac = factor(p, *gens, **args)
if (fac.is_Mul or fac.is_Pow) and fac != p:
partials[p] = fac
return f.xreplace(partials)
try:
return _generic_factor(f, gens, args, method='factor')
except PolynomialError as msg:
if not f.is_commutative:
from sympy.core.exprtools import factor_nc
return factor_nc(f)
else:
raise PolynomialError(msg)
@public
def intervals(F, all=False, eps=None, inf=None, sup=None, strict=False, fast=False, sqf=False):
"""
Compute isolating intervals for roots of ``f``.
Examples
========
>>> from sympy import intervals
>>> from sympy.abc import x
>>> intervals(x**2 - 3)
[((-2, -1), 1), ((1, 2), 1)]
>>> intervals(x**2 - 3, eps=1e-2)
[((-26/15, -19/11), 1), ((19/11, 26/15), 1)]
"""
if not hasattr(F, '__iter__'):
try:
F = Poly(F)
except GeneratorsNeeded:
return []
return F.intervals(all=all, eps=eps, inf=inf, sup=sup, fast=fast, sqf=sqf)
else:
polys, opt = parallel_poly_from_expr(F, domain='QQ')
if len(opt.gens) > 1:
raise MultivariatePolynomialError
for i, poly in enumerate(polys):
polys[i] = poly.rep.rep
if eps is not None:
eps = opt.domain.convert(eps)
if eps <= 0:
raise ValueError("'eps' must be a positive rational")
if inf is not None:
inf = opt.domain.convert(inf)
if sup is not None:
sup = opt.domain.convert(sup)
intervals = dup_isolate_real_roots_list(polys, opt.domain,
eps=eps, inf=inf, sup=sup, strict=strict, fast=fast)
result = []
for (s, t), indices in intervals:
s, t = opt.domain.to_sympy(s), opt.domain.to_sympy(t)
result.append(((s, t), indices))
return result
@public
def refine_root(f, s, t, eps=None, steps=None, fast=False, check_sqf=False):
"""
Refine an isolating interval of a root to the given precision.
Examples
========
>>> from sympy import refine_root
>>> from sympy.abc import x
>>> refine_root(x**2 - 3, 1, 2, eps=1e-2)
(19/11, 26/15)
"""
try:
F = Poly(f)
if not isinstance(f, Poly) and not F.gen.is_Symbol:
# root of sin(x) + 1 is -1 but when someone
# passes an Expr instead of Poly they may not expect
# that the generator will be sin(x), not x
raise PolynomialError("generator must be a Symbol")
except GeneratorsNeeded:
raise PolynomialError(
"can't refine a root of %s, not a polynomial" % f)
return F.refine_root(s, t, eps=eps, steps=steps, fast=fast, check_sqf=check_sqf)
@public
def count_roots(f, inf=None, sup=None):
"""
Return the number of roots of ``f`` in ``[inf, sup]`` interval.
If one of ``inf`` or ``sup`` is complex, it will return the number of roots
in the complex rectangle with corners at ``inf`` and ``sup``.
Examples
========
>>> from sympy import count_roots, I
>>> from sympy.abc import x
>>> count_roots(x**4 - 4, -3, 3)
2
>>> count_roots(x**4 - 4, 0, 1 + 3*I)
1
"""
try:
F = Poly(f, greedy=False)
if not isinstance(f, Poly) and not F.gen.is_Symbol:
# root of sin(x) + 1 is -1 but when someone
# passes an Expr instead of Poly they may not expect
# that the generator will be sin(x), not x
raise PolynomialError("generator must be a Symbol")
except GeneratorsNeeded:
raise PolynomialError("can't count roots of %s, not a polynomial" % f)
return F.count_roots(inf=inf, sup=sup)
@public
def real_roots(f, multiple=True):
"""
Return a list of real roots with multiplicities of ``f``.
Examples
========
>>> from sympy import real_roots
>>> from sympy.abc import x
>>> real_roots(2*x**3 - 7*x**2 + 4*x + 4)
[-1/2, 2, 2]
"""
try:
F = Poly(f, greedy=False)
if not isinstance(f, Poly) and not F.gen.is_Symbol:
# root of sin(x) + 1 is -1 but when someone
# passes an Expr instead of Poly they may not expect
# that the generator will be sin(x), not x
raise PolynomialError("generator must be a Symbol")
except GeneratorsNeeded:
raise PolynomialError(
"can't compute real roots of %s, not a polynomial" % f)
return F.real_roots(multiple=multiple)
@public
def nroots(f, n=15, maxsteps=50, cleanup=True):
"""
Compute numerical approximations of roots of ``f``.
Examples
========
>>> from sympy import nroots
>>> from sympy.abc import x
>>> nroots(x**2 - 3, n=15)
[-1.73205080756888, 1.73205080756888]
>>> nroots(x**2 - 3, n=30)
[-1.73205080756887729352744634151, 1.73205080756887729352744634151]
"""
try:
F = Poly(f, greedy=False)
if not isinstance(f, Poly) and not F.gen.is_Symbol:
# root of sin(x) + 1 is -1 but when someone
# passes an Expr instead of Poly they may not expect
# that the generator will be sin(x), not x
raise PolynomialError("generator must be a Symbol")
except GeneratorsNeeded:
raise PolynomialError(
"can't compute numerical roots of %s, not a polynomial" % f)
return F.nroots(n=n, maxsteps=maxsteps, cleanup=cleanup)
@public
def ground_roots(f, *gens, **args):
"""
Compute roots of ``f`` by factorization in the ground domain.
Examples
========
>>> from sympy import ground_roots
>>> from sympy.abc import x
>>> ground_roots(x**6 - 4*x**4 + 4*x**3 - x**2)
{0: 2, 1: 2}
"""
options.allowed_flags(args, [])
try:
F, opt = poly_from_expr(f, *gens, **args)
if not isinstance(f, Poly) and not F.gen.is_Symbol:
# root of sin(x) + 1 is -1 but when someone
# passes an Expr instead of Poly they may not expect
# that the generator will be sin(x), not x
raise PolynomialError("generator must be a Symbol")
except PolificationFailed as exc:
raise ComputationFailed('ground_roots', 1, exc)
return F.ground_roots()
@public
def nth_power_roots_poly(f, n, *gens, **args):
"""
Construct a polynomial with n-th powers of roots of ``f``.
Examples
========
>>> from sympy import nth_power_roots_poly, factor, roots
>>> from sympy.abc import x
>>> f = x**4 - x**2 + 1
>>> g = factor(nth_power_roots_poly(f, 2))
>>> g
(x**2 - x + 1)**2
>>> R_f = [ (r**2).expand() for r in roots(f) ]
>>> R_g = roots(g).keys()
>>> set(R_f) == set(R_g)
True
"""
options.allowed_flags(args, [])
try:
F, opt = poly_from_expr(f, *gens, **args)
if not isinstance(f, Poly) and not F.gen.is_Symbol:
# root of sin(x) + 1 is -1 but when someone
# passes an Expr instead of Poly they may not expect
# that the generator will be sin(x), not x
raise PolynomialError("generator must be a Symbol")
except PolificationFailed as exc:
raise ComputationFailed('nth_power_roots_poly', 1, exc)
result = F.nth_power_roots_poly(n)
if not opt.polys:
return result.as_expr()
else:
return result
@public
def cancel(f, *gens, **args):
"""
Cancel common factors in a rational function ``f``.
Examples
========
>>> from sympy import cancel, sqrt, Symbol, together
>>> from sympy.abc import x
>>> A = Symbol('A', commutative=False)
>>> cancel((2*x**2 - 2)/(x**2 - 2*x + 1))
(2*x + 2)/(x - 1)
>>> cancel((sqrt(3) + sqrt(15)*A)/(sqrt(2) + sqrt(10)*A))
sqrt(6)/2
Note: due to automatic distribution of Rationals, a sum divided by an integer
will appear as a sum. To recover a rational form use `together` on the result:
>>> cancel(x/2 + 1)
x/2 + 1
>>> together(_)
(x + 2)/2
"""
from sympy.core.exprtools import factor_terms
from sympy.functions.elementary.piecewise import Piecewise
from sympy.polys.rings import sring
options.allowed_flags(args, ['polys'])
f = sympify(f)
opt = {}
if 'polys' in args:
opt['polys'] = args['polys']
if not isinstance(f, (tuple, Tuple)):
if f.is_Number or isinstance(f, Relational) or not isinstance(f, Expr):
return f
f = factor_terms(f, radical=True)
p, q = f.as_numer_denom()
elif len(f) == 2:
p, q = f
if isinstance(p, Poly) and isinstance(q, Poly):
opt['gens'] = p.gens
opt['domain'] = p.domain
opt['polys'] = opt.get('polys', True)
p, q = p.as_expr(), q.as_expr()
elif isinstance(f, Tuple):
return factor_terms(f)
else:
raise ValueError('unexpected argument: %s' % f)
try:
if f.has(Piecewise):
raise PolynomialError()
R, (F, G) = sring((p, q), *gens, **args)
if not R.ngens:
if not isinstance(f, (tuple, Tuple)):
return f.expand()
else:
return S.One, p, q
except PolynomialError as msg:
if f.is_commutative and not f.has(Piecewise):
raise PolynomialError(msg)
# Handling of noncommutative and/or piecewise expressions
if f.is_Add or f.is_Mul:
c, nc = sift(f.args, lambda x:
x.is_commutative is True and not x.has(Piecewise),
binary=True)
nc = [cancel(i) for i in nc]
return f.func(cancel(f.func(*c)), *nc)
else:
reps = []
pot = preorder_traversal(f)
next(pot)
for e in pot:
# XXX: This should really skip anything that's not Expr.
if isinstance(e, (tuple, Tuple, BooleanAtom)):
continue
try:
reps.append((e, cancel(e)))
pot.skip() # this was handled successfully
except NotImplementedError:
pass
return f.xreplace(dict(reps))
c, (P, Q) = 1, F.cancel(G)
if opt.get('polys', False) and not 'gens' in opt:
opt['gens'] = R.symbols
if not isinstance(f, (tuple, Tuple)):
return c*(P.as_expr()/Q.as_expr())
else:
P, Q = P.as_expr(), Q.as_expr()
if not opt.get('polys', False):
return c, P, Q
else:
return c, Poly(P, *gens, **opt), Poly(Q, *gens, **opt)
@public
def reduced(f, G, *gens, **args):
"""
Reduces a polynomial ``f`` modulo a set of polynomials ``G``.
Given a polynomial ``f`` and a set of polynomials ``G = (g_1, ..., g_n)``,
computes a set of quotients ``q = (q_1, ..., q_n)`` and the remainder ``r``
such that ``f = q_1*g_1 + ... + q_n*g_n + r``, where ``r`` vanishes or ``r``
is a completely reduced polynomial with respect to ``G``.
Examples
========
>>> from sympy import reduced
>>> from sympy.abc import x, y
>>> reduced(2*x**4 + y**2 - x**2 + y**3, [x**3 - x, y**3 - y])
([2*x, 1], x**2 + y**2 + y)
"""
options.allowed_flags(args, ['polys', 'auto'])
try:
polys, opt = parallel_poly_from_expr([f] + list(G), *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('reduced', 0, exc)
domain = opt.domain
retract = False
if opt.auto and domain.is_Ring and not domain.is_Field:
opt = opt.clone(dict(domain=domain.get_field()))
retract = True
from sympy.polys.rings import xring
_ring, _ = xring(opt.gens, opt.domain, opt.order)
for i, poly in enumerate(polys):
poly = poly.set_domain(opt.domain).rep.to_dict()
polys[i] = _ring.from_dict(poly)
Q, r = polys[0].div(polys[1:])
Q = [Poly._from_dict(dict(q), opt) for q in Q]
r = Poly._from_dict(dict(r), opt)
if retract:
try:
_Q, _r = [q.to_ring() for q in Q], r.to_ring()
except CoercionFailed:
pass
else:
Q, r = _Q, _r
if not opt.polys:
return [q.as_expr() for q in Q], r.as_expr()
else:
return Q, r
@public
def groebner(F, *gens, **args):
"""
Computes the reduced Groebner basis for a set of polynomials.
Use the ``order`` argument to set the monomial ordering that will be
used to compute the basis. Allowed orders are ``lex``, ``grlex`` and
``grevlex``. If no order is specified, it defaults to ``lex``.
For more information on Groebner bases, see the references and the docstring
of :func:`~.solve_poly_system`.
Examples
========
Example taken from [1].
>>> from sympy import groebner
>>> from sympy.abc import x, y
>>> F = [x*y - 2*y, 2*y**2 - x**2]
>>> groebner(F, x, y, order='lex')
GroebnerBasis([x**2 - 2*y**2, x*y - 2*y, y**3 - 2*y], x, y,
domain='ZZ', order='lex')
>>> groebner(F, x, y, order='grlex')
GroebnerBasis([y**3 - 2*y, x**2 - 2*y**2, x*y - 2*y], x, y,
domain='ZZ', order='grlex')
>>> groebner(F, x, y, order='grevlex')
GroebnerBasis([y**3 - 2*y, x**2 - 2*y**2, x*y - 2*y], x, y,
domain='ZZ', order='grevlex')
By default, an improved implementation of the Buchberger algorithm is
used. Optionally, an implementation of the F5B algorithm can be used. The
algorithm can be set using the ``method`` flag or with the
:func:`sympy.polys.polyconfig.setup` function.
>>> F = [x**2 - x - 1, (2*x - 1) * y - (x**10 - (1 - x)**10)]
>>> groebner(F, x, y, method='buchberger')
GroebnerBasis([x**2 - x - 1, y - 55], x, y, domain='ZZ', order='lex')
>>> groebner(F, x, y, method='f5b')
GroebnerBasis([x**2 - x - 1, y - 55], x, y, domain='ZZ', order='lex')
References
==========
1. [Buchberger01]_
2. [Cox97]_
"""
return GroebnerBasis(F, *gens, **args)
@public
def is_zero_dimensional(F, *gens, **args):
"""
Checks if the ideal generated by a Groebner basis is zero-dimensional.
The algorithm checks if the set of monomials not divisible by the
leading monomial of any element of ``F`` is bounded.
References
==========
David A. Cox, John B. Little, Donal O'Shea. Ideals, Varieties and
Algorithms, 3rd edition, p. 230
"""
return GroebnerBasis(F, *gens, **args).is_zero_dimensional
@public
class GroebnerBasis(Basic):
"""Represents a reduced Groebner basis. """
def __new__(cls, F, *gens, **args):
"""Compute a reduced Groebner basis for a system of polynomials. """
options.allowed_flags(args, ['polys', 'method'])
try:
polys, opt = parallel_poly_from_expr(F, *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('groebner', len(F), exc)
from sympy.polys.rings import PolyRing
ring = PolyRing(opt.gens, opt.domain, opt.order)
polys = [ring.from_dict(poly.rep.to_dict()) for poly in polys if poly]
G = _groebner(polys, ring, method=opt.method)
G = [Poly._from_dict(g, opt) for g in G]
return cls._new(G, opt)
@classmethod
def _new(cls, basis, options):
obj = Basic.__new__(cls)
obj._basis = tuple(basis)
obj._options = options
return obj
@property
def args(self):
basis = (p.as_expr() for p in self._basis)
return (Tuple(*basis), Tuple(*self._options.gens))
@property
def exprs(self):
return [poly.as_expr() for poly in self._basis]
@property
def polys(self):
return list(self._basis)
@property
def gens(self):
return self._options.gens
@property
def domain(self):
return self._options.domain
@property
def order(self):
return self._options.order
def __len__(self):
return len(self._basis)
def __iter__(self):
if self._options.polys:
return iter(self.polys)
else:
return iter(self.exprs)
def __getitem__(self, item):
if self._options.polys:
basis = self.polys
else:
basis = self.exprs
return basis[item]
def __hash__(self):
return hash((self._basis, tuple(self._options.items())))
def __eq__(self, other):
if isinstance(other, self.__class__):
return self._basis == other._basis and self._options == other._options
elif iterable(other):
return self.polys == list(other) or self.exprs == list(other)
else:
return False
def __ne__(self, other):
return not self == other
@property
def is_zero_dimensional(self):
"""
Checks if the ideal generated by a Groebner basis is zero-dimensional.
The algorithm checks if the set of monomials not divisible by the
leading monomial of any element of ``F`` is bounded.
References
==========
David A. Cox, John B. Little, Donal O'Shea. Ideals, Varieties and
Algorithms, 3rd edition, p. 230
"""
def single_var(monomial):
return sum(map(bool, monomial)) == 1
exponents = Monomial([0]*len(self.gens))
order = self._options.order
for poly in self.polys:
monomial = poly.LM(order=order)
if single_var(monomial):
exponents *= monomial
# If any element of the exponents vector is zero, then there's
# a variable for which there's no degree bound and the ideal
# generated by this Groebner basis isn't zero-dimensional.
return all(exponents)
def fglm(self, order):
"""
Convert a Groebner basis from one ordering to another.
The FGLM algorithm converts reduced Groebner bases of zero-dimensional
ideals from one ordering to another. This method is often used when it
is infeasible to compute a Groebner basis with respect to a particular
ordering directly.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy import groebner
>>> F = [x**2 - 3*y - x + 1, y**2 - 2*x + y - 1]
>>> G = groebner(F, x, y, order='grlex')
>>> list(G.fglm('lex'))
[2*x - y**2 - y + 1, y**4 + 2*y**3 - 3*y**2 - 16*y + 7]
>>> list(groebner(F, x, y, order='lex'))
[2*x - y**2 - y + 1, y**4 + 2*y**3 - 3*y**2 - 16*y + 7]
References
==========
.. [1] J.C. Faugere, P. Gianni, D. Lazard, T. Mora (1994). Efficient
Computation of Zero-dimensional Groebner Bases by Change of
Ordering
"""
opt = self._options
src_order = opt.order
dst_order = monomial_key(order)
if src_order == dst_order:
return self
if not self.is_zero_dimensional:
raise NotImplementedError("can't convert Groebner bases of ideals with positive dimension")
polys = list(self._basis)
domain = opt.domain
opt = opt.clone(dict(
domain=domain.get_field(),
order=dst_order,
))
from sympy.polys.rings import xring
_ring, _ = xring(opt.gens, opt.domain, src_order)
for i, poly in enumerate(polys):
poly = poly.set_domain(opt.domain).rep.to_dict()
polys[i] = _ring.from_dict(poly)
G = matrix_fglm(polys, _ring, dst_order)
G = [Poly._from_dict(dict(g), opt) for g in G]
if not domain.is_Field:
G = [g.clear_denoms(convert=True)[1] for g in G]
opt.domain = domain
return self._new(G, opt)
def reduce(self, expr, auto=True):
"""
Reduces a polynomial modulo a Groebner basis.
Given a polynomial ``f`` and a set of polynomials ``G = (g_1, ..., g_n)``,
computes a set of quotients ``q = (q_1, ..., q_n)`` and the remainder ``r``
such that ``f = q_1*f_1 + ... + q_n*f_n + r``, where ``r`` vanishes or ``r``
is a completely reduced polynomial with respect to ``G``.
Examples
========
>>> from sympy import groebner, expand
>>> from sympy.abc import x, y
>>> f = 2*x**4 - x**2 + y**3 + y**2
>>> G = groebner([x**3 - x, y**3 - y])
>>> G.reduce(f)
([2*x, 1], x**2 + y**2 + y)
>>> Q, r = _
>>> expand(sum(q*g for q, g in zip(Q, G)) + r)
2*x**4 - x**2 + y**3 + y**2
>>> _ == f
True
"""
poly = Poly._from_expr(expr, self._options)
polys = [poly] + list(self._basis)
opt = self._options
domain = opt.domain
retract = False
if auto and domain.is_Ring and not domain.is_Field:
opt = opt.clone(dict(domain=domain.get_field()))
retract = True
from sympy.polys.rings import xring
_ring, _ = xring(opt.gens, opt.domain, opt.order)
for i, poly in enumerate(polys):
poly = poly.set_domain(opt.domain).rep.to_dict()
polys[i] = _ring.from_dict(poly)
Q, r = polys[0].div(polys[1:])
Q = [Poly._from_dict(dict(q), opt) for q in Q]
r = Poly._from_dict(dict(r), opt)
if retract:
try:
_Q, _r = [q.to_ring() for q in Q], r.to_ring()
except CoercionFailed:
pass
else:
Q, r = _Q, _r
if not opt.polys:
return [q.as_expr() for q in Q], r.as_expr()
else:
return Q, r
def contains(self, poly):
"""
Check if ``poly`` belongs the ideal generated by ``self``.
Examples
========
>>> from sympy import groebner
>>> from sympy.abc import x, y
>>> f = 2*x**3 + y**3 + 3*y
>>> G = groebner([x**2 + y**2 - 1, x*y - 2])
>>> G.contains(f)
True
>>> G.contains(f + 1)
False
"""
return self.reduce(poly)[1] == 0
@public
def poly(expr, *gens, **args):
"""
Efficiently transform an expression into a polynomial.
Examples
========
>>> from sympy import poly
>>> from sympy.abc import x
>>> poly(x*(x**2 + x - 1)**2)
Poly(x**5 + 2*x**4 - x**3 - 2*x**2 + x, x, domain='ZZ')
"""
options.allowed_flags(args, [])
def _poly(expr, opt):
terms, poly_terms = [], []
for term in Add.make_args(expr):
factors, poly_factors = [], []
for factor in Mul.make_args(term):
if factor.is_Add:
poly_factors.append(_poly(factor, opt))
elif factor.is_Pow and factor.base.is_Add and \
factor.exp.is_Integer and factor.exp >= 0:
poly_factors.append(
_poly(factor.base, opt).pow(factor.exp))
else:
factors.append(factor)
if not poly_factors:
terms.append(term)
else:
product = poly_factors[0]
for factor in poly_factors[1:]:
product = product.mul(factor)
if factors:
factor = Mul(*factors)
if factor.is_Number:
product = product.mul(factor)
else:
product = product.mul(Poly._from_expr(factor, opt))
poly_terms.append(product)
if not poly_terms:
result = Poly._from_expr(expr, opt)
else:
result = poly_terms[0]
for term in poly_terms[1:]:
result = result.add(term)
if terms:
term = Add(*terms)
if term.is_Number:
result = result.add(term)
else:
result = result.add(Poly._from_expr(term, opt))
return result.reorder(*opt.get('gens', ()), **args)
expr = sympify(expr)
if expr.is_Poly:
return Poly(expr, *gens, **args)
if 'expand' not in args:
args['expand'] = False
opt = options.build_options(gens, args)
return _poly(expr, opt)
|
19198ffc6f0b6fa2f28a1a92a8aac2d208f6b0a963ca699aa70e302e854b74d0 | """Algorithms for computing symbolic roots of polynomials. """
import math
from functools import reduce
from sympy.core import S, I, pi
from sympy.core.compatibility import ordered
from sympy.core.exprtools import factor_terms
from sympy.core.function import _mexpand
from sympy.core.logic import fuzzy_not
from sympy.core.mul import expand_2arg, Mul
from sympy.core.numbers import Rational, igcd, comp
from sympy.core.power import Pow
from sympy.core.relational import Eq
from sympy.core.symbol import Dummy, Symbol, symbols
from sympy.core.sympify import sympify
from sympy.functions import exp, sqrt, im, cos, acos, Piecewise
from sympy.functions.elementary.miscellaneous import root
from sympy.ntheory import divisors, isprime, nextprime
from sympy.polys.domains import EX
from sympy.polys.polyerrors import (PolynomialError, GeneratorsNeeded,
DomainError)
from sympy.polys.polyquinticconst import PolyQuintic
from sympy.polys.polytools import Poly, cancel, factor, gcd_list, discriminant
from sympy.polys.rationaltools import together
from sympy.polys.specialpolys import cyclotomic_poly
from sympy.simplify import simplify, powsimp
from sympy.utilities import public
def roots_linear(f):
"""Returns a list of roots of a linear polynomial."""
r = -f.nth(0)/f.nth(1)
dom = f.get_domain()
if not dom.is_Numerical:
if dom.is_Composite:
r = factor(r)
else:
r = simplify(r)
return [r]
def roots_quadratic(f):
"""Returns a list of roots of a quadratic polynomial. If the domain is ZZ
then the roots will be sorted with negatives coming before positives.
The ordering will be the same for any numerical coefficients as long as
the assumptions tested are correct, otherwise the ordering will not be
sorted (but will be canonical).
"""
a, b, c = f.all_coeffs()
dom = f.get_domain()
def _sqrt(d):
# remove squares from square root since both will be represented
# in the results; a similar thing is happening in roots() but
# must be duplicated here because not all quadratics are binomials
co = []
other = []
for di in Mul.make_args(d):
if di.is_Pow and di.exp.is_Integer and di.exp % 2 == 0:
co.append(Pow(di.base, di.exp//2))
else:
other.append(di)
if co:
d = Mul(*other)
co = Mul(*co)
return co*sqrt(d)
return sqrt(d)
def _simplify(expr):
if dom.is_Composite:
return factor(expr)
else:
return simplify(expr)
if c is S.Zero:
r0, r1 = S.Zero, -b/a
if not dom.is_Numerical:
r1 = _simplify(r1)
elif r1.is_negative:
r0, r1 = r1, r0
elif b is S.Zero:
r = -c/a
if not dom.is_Numerical:
r = _simplify(r)
R = _sqrt(r)
r0 = -R
r1 = R
else:
d = b**2 - 4*a*c
A = 2*a
B = -b/A
if not dom.is_Numerical:
d = _simplify(d)
B = _simplify(B)
D = factor_terms(_sqrt(d)/A)
r0 = B - D
r1 = B + D
if a.is_negative:
r0, r1 = r1, r0
elif not dom.is_Numerical:
r0, r1 = [expand_2arg(i) for i in (r0, r1)]
return [r0, r1]
def roots_cubic(f, trig=False):
"""Returns a list of roots of a cubic polynomial.
References
==========
[1] https://en.wikipedia.org/wiki/Cubic_function, General formula for roots,
(accessed November 17, 2014).
"""
if trig:
a, b, c, d = f.all_coeffs()
p = (3*a*c - b**2)/3/a**2
q = (2*b**3 - 9*a*b*c + 27*a**2*d)/(27*a**3)
D = 18*a*b*c*d - 4*b**3*d + b**2*c**2 - 4*a*c**3 - 27*a**2*d**2
if (D > 0) == True:
rv = []
for k in range(3):
rv.append(2*sqrt(-p/3)*cos(acos(q/p*sqrt(-3/p)*Rational(3, 2))/3 - k*pi*Rational(2, 3)))
return [i - b/3/a for i in rv]
# a*x**3 + b*x**2 + c*x + d -> x**3 + a*x**2 + b*x + c
_, a, b, c = f.monic().all_coeffs()
if c is S.Zero:
x1, x2 = roots([1, a, b], multiple=True)
return [x1, S.Zero, x2]
# x**3 + a*x**2 + b*x + c -> u**3 + p*u + q
p = b - a**2/3
q = c - a*b/3 + 2*a**3/27
pon3 = p/3
aon3 = a/3
u1 = None
if p is S.Zero:
if q is S.Zero:
return [-aon3]*3
u1 = -root(q, 3) if q.is_positive else root(-q, 3)
elif q is S.Zero:
y1, y2 = roots([1, 0, p], multiple=True)
return [tmp - aon3 for tmp in [y1, S.Zero, y2]]
elif q.is_real and q.is_negative:
u1 = -root(-q/2 + sqrt(q**2/4 + pon3**3), 3)
coeff = I*sqrt(3)/2
if u1 is None:
u1 = S.One
u2 = Rational(-1, 2) + coeff
u3 = Rational(-1, 2) - coeff
a, b, c, d = S(1), a, b, c
D0 = b**2 - 3*a*c
D1 = 2*b**3 - 9*a*b*c + 27*a**2*d
C = root((D1 + sqrt(D1**2 - 4*D0**3))/2, 3)
return [-(b + uk*C + D0/C/uk)/3/a for uk in [u1, u2, u3]]
u2 = u1*(Rational(-1, 2) + coeff)
u3 = u1*(Rational(-1, 2) - coeff)
if p is S.Zero:
return [u1 - aon3, u2 - aon3, u3 - aon3]
soln = [
-u1 + pon3/u1 - aon3,
-u2 + pon3/u2 - aon3,
-u3 + pon3/u3 - aon3
]
return soln
def _roots_quartic_euler(p, q, r, a):
"""
Descartes-Euler solution of the quartic equation
Parameters
==========
p, q, r: coefficients of ``x**4 + p*x**2 + q*x + r``
a: shift of the roots
Notes
=====
This is a helper function for ``roots_quartic``.
Look for solutions of the form ::
``x1 = sqrt(R) - sqrt(A + B*sqrt(R))``
``x2 = -sqrt(R) - sqrt(A - B*sqrt(R))``
``x3 = -sqrt(R) + sqrt(A - B*sqrt(R))``
``x4 = sqrt(R) + sqrt(A + B*sqrt(R))``
To satisfy the quartic equation one must have
``p = -2*(R + A); q = -4*B*R; r = (R - A)**2 - B**2*R``
so that ``R`` must satisfy the Descartes-Euler resolvent equation
``64*R**3 + 32*p*R**2 + (4*p**2 - 16*r)*R - q**2 = 0``
If the resolvent does not have a rational solution, return None;
in that case it is likely that the Ferrari method gives a simpler
solution.
Examples
========
>>> from sympy import S
>>> from sympy.polys.polyroots import _roots_quartic_euler
>>> p, q, r = -S(64)/5, -S(512)/125, -S(1024)/3125
>>> _roots_quartic_euler(p, q, r, S(0))[0]
-sqrt(32*sqrt(5)/125 + 16/5) + 4*sqrt(5)/5
"""
# solve the resolvent equation
x = Dummy('x')
eq = 64*x**3 + 32*p*x**2 + (4*p**2 - 16*r)*x - q**2
xsols = list(roots(Poly(eq, x), cubics=False).keys())
xsols = [sol for sol in xsols if sol.is_rational and sol.is_nonzero]
if not xsols:
return None
R = max(xsols)
c1 = sqrt(R)
B = -q*c1/(4*R)
A = -R - p/2
c2 = sqrt(A + B)
c3 = sqrt(A - B)
return [c1 - c2 - a, -c1 - c3 - a, -c1 + c3 - a, c1 + c2 - a]
def roots_quartic(f):
r"""
Returns a list of roots of a quartic polynomial.
There are many references for solving quartic expressions available [1-5].
This reviewer has found that many of them require one to select from among
2 or more possible sets of solutions and that some solutions work when one
is searching for real roots but don't work when searching for complex roots
(though this is not always stated clearly). The following routine has been
tested and found to be correct for 0, 2 or 4 complex roots.
The quasisymmetric case solution [6] looks for quartics that have the form
`x**4 + A*x**3 + B*x**2 + C*x + D = 0` where `(C/A)**2 = D`.
Although no general solution that is always applicable for all
coefficients is known to this reviewer, certain conditions are tested
to determine the simplest 4 expressions that can be returned:
1) `f = c + a*(a**2/8 - b/2) == 0`
2) `g = d - a*(a*(3*a**2/256 - b/16) + c/4) = 0`
3) if `f != 0` and `g != 0` and `p = -d + a*c/4 - b**2/12` then
a) `p == 0`
b) `p != 0`
Examples
========
>>> from sympy import Poly
>>> from sympy.polys.polyroots import roots_quartic
>>> r = roots_quartic(Poly('x**4-6*x**3+17*x**2-26*x+20'))
>>> # 4 complex roots: 1+-I*sqrt(3), 2+-I
>>> sorted(str(tmp.evalf(n=2)) for tmp in r)
['1.0 + 1.7*I', '1.0 - 1.7*I', '2.0 + 1.0*I', '2.0 - 1.0*I']
References
==========
1. http://mathforum.org/dr.math/faq/faq.cubic.equations.html
2. https://en.wikipedia.org/wiki/Quartic_function#Summary_of_Ferrari.27s_method
3. http://planetmath.org/encyclopedia/GaloisTheoreticDerivationOfTheQuarticFormula.html
4. http://staff.bath.ac.uk/masjhd/JHD-CA.pdf
5. http://www.albmath.org/files/Math_5713.pdf
6. http://www.statemaster.com/encyclopedia/Quartic-equation
7. eqworld.ipmnet.ru/en/solutions/ae/ae0108.pdf
"""
_, a, b, c, d = f.monic().all_coeffs()
if not d:
return [S.Zero] + roots([1, a, b, c], multiple=True)
elif (c/a)**2 == d:
x, m = f.gen, c/a
g = Poly(x**2 + a*x + b - 2*m, x)
z1, z2 = roots_quadratic(g)
h1 = Poly(x**2 - z1*x + m, x)
h2 = Poly(x**2 - z2*x + m, x)
r1 = roots_quadratic(h1)
r2 = roots_quadratic(h2)
return r1 + r2
else:
a2 = a**2
e = b - 3*a2/8
f = _mexpand(c + a*(a2/8 - b/2))
g = _mexpand(d - a*(a*(3*a2/256 - b/16) + c/4))
aon4 = a/4
if f is S.Zero:
y1, y2 = [sqrt(tmp) for tmp in
roots([1, e, g], multiple=True)]
return [tmp - aon4 for tmp in [-y1, -y2, y1, y2]]
if g is S.Zero:
y = [S.Zero] + roots([1, 0, e, f], multiple=True)
return [tmp - aon4 for tmp in y]
else:
# Descartes-Euler method, see [7]
sols = _roots_quartic_euler(e, f, g, aon4)
if sols:
return sols
# Ferrari method, see [1, 2]
a2 = a**2
e = b - 3*a2/8
f = c + a*(a2/8 - b/2)
g = d - a*(a*(3*a2/256 - b/16) + c/4)
p = -e**2/12 - g
q = -e**3/108 + e*g/3 - f**2/8
TH = Rational(1, 3)
def _ans(y):
w = sqrt(e + 2*y)
arg1 = 3*e + 2*y
arg2 = 2*f/w
ans = []
for s in [-1, 1]:
root = sqrt(-(arg1 + s*arg2))
for t in [-1, 1]:
ans.append((s*w - t*root)/2 - aon4)
return ans
# whether a Piecewise is returned or not
# depends on knowing p, so try to put
# in a simple form
p = _mexpand(p)
# p == 0 case
y1 = e*Rational(-5, 6) - q**TH
if p.is_zero:
return _ans(y1)
# if p != 0 then u below is not 0
root = sqrt(q**2/4 + p**3/27)
r = -q/2 + root # or -q/2 - root
u = r**TH # primary root of solve(x**3 - r, x)
y2 = e*Rational(-5, 6) + u - p/u/3
if fuzzy_not(p.is_zero):
return _ans(y2)
# sort it out once they know the values of the coefficients
return [Piecewise((a1, Eq(p, 0)), (a2, True))
for a1, a2 in zip(_ans(y1), _ans(y2))]
def roots_binomial(f):
"""Returns a list of roots of a binomial polynomial. If the domain is ZZ
then the roots will be sorted with negatives coming before positives.
The ordering will be the same for any numerical coefficients as long as
the assumptions tested are correct, otherwise the ordering will not be
sorted (but will be canonical).
"""
n = f.degree()
a, b = f.nth(n), f.nth(0)
base = -cancel(b/a)
alpha = root(base, n)
if alpha.is_number:
alpha = alpha.expand(complex=True)
# define some parameters that will allow us to order the roots.
# If the domain is ZZ this is guaranteed to return roots sorted
# with reals before non-real roots and non-real sorted according
# to real part and imaginary part, e.g. -1, 1, -1 + I, 2 - I
neg = base.is_negative
even = n % 2 == 0
if neg:
if even == True and (base + 1).is_positive:
big = True
else:
big = False
# get the indices in the right order so the computed
# roots will be sorted when the domain is ZZ
ks = []
imax = n//2
if even:
ks.append(imax)
imax -= 1
if not neg:
ks.append(0)
for i in range(imax, 0, -1):
if neg:
ks.extend([i, -i])
else:
ks.extend([-i, i])
if neg:
ks.append(0)
if big:
for i in range(0, len(ks), 2):
pair = ks[i: i + 2]
pair = list(reversed(pair))
# compute the roots
roots, d = [], 2*I*pi/n
for k in ks:
zeta = exp(k*d).expand(complex=True)
roots.append((alpha*zeta).expand(power_base=False))
return roots
def _inv_totient_estimate(m):
"""
Find ``(L, U)`` such that ``L <= phi^-1(m) <= U``.
Examples
========
>>> from sympy.polys.polyroots import _inv_totient_estimate
>>> _inv_totient_estimate(192)
(192, 840)
>>> _inv_totient_estimate(400)
(400, 1750)
"""
primes = [ d + 1 for d in divisors(m) if isprime(d + 1) ]
a, b = 1, 1
for p in primes:
a *= p
b *= p - 1
L = m
U = int(math.ceil(m*(float(a)/b)))
P = p = 2
primes = []
while P <= U:
p = nextprime(p)
primes.append(p)
P *= p
P //= p
b = 1
for p in primes[:-1]:
b *= p - 1
U = int(math.ceil(m*(float(P)/b)))
return L, U
def roots_cyclotomic(f, factor=False):
"""Compute roots of cyclotomic polynomials. """
L, U = _inv_totient_estimate(f.degree())
for n in range(L, U + 1):
g = cyclotomic_poly(n, f.gen, polys=True)
if f.expr == g.expr:
break
else: # pragma: no cover
raise RuntimeError("failed to find index of a cyclotomic polynomial")
roots = []
if not factor:
# get the indices in the right order so the computed
# roots will be sorted
h = n//2
ks = [i for i in range(1, n + 1) if igcd(i, n) == 1]
ks.sort(key=lambda x: (x, -1) if x <= h else (abs(x - n), 1))
d = 2*I*pi/n
for k in reversed(ks):
roots.append(exp(k*d).expand(complex=True))
else:
g = Poly(f, extension=root(-1, n))
for h, _ in ordered(g.factor_list()[1]):
roots.append(-h.TC())
return roots
def roots_quintic(f):
"""
Calculate exact roots of a solvable quintic
"""
result = []
coeff_5, coeff_4, p, q, r, s = f.all_coeffs()
# Eqn must be of the form x^5 + px^3 + qx^2 + rx + s
if coeff_4:
return result
if coeff_5 != 1:
l = [p/coeff_5, q/coeff_5, r/coeff_5, s/coeff_5]
if not all(coeff.is_Rational for coeff in l):
return result
f = Poly(f/coeff_5)
elif not all(coeff.is_Rational for coeff in (p, q, r, s)):
return result
quintic = PolyQuintic(f)
# Eqn standardized. Algo for solving starts here
if not f.is_irreducible:
return result
f20 = quintic.f20
# Check if f20 has linear factors over domain Z
if f20.is_irreducible:
return result
# Now, we know that f is solvable
for _factor in f20.factor_list()[1]:
if _factor[0].is_linear:
theta = _factor[0].root(0)
break
d = discriminant(f)
delta = sqrt(d)
# zeta = a fifth root of unity
zeta1, zeta2, zeta3, zeta4 = quintic.zeta
T = quintic.T(theta, d)
tol = S(1e-10)
alpha = T[1] + T[2]*delta
alpha_bar = T[1] - T[2]*delta
beta = T[3] + T[4]*delta
beta_bar = T[3] - T[4]*delta
disc = alpha**2 - 4*beta
disc_bar = alpha_bar**2 - 4*beta_bar
l0 = quintic.l0(theta)
l1 = _quintic_simplify((-alpha + sqrt(disc)) / S(2))
l4 = _quintic_simplify((-alpha - sqrt(disc)) / S(2))
l2 = _quintic_simplify((-alpha_bar + sqrt(disc_bar)) / S(2))
l3 = _quintic_simplify((-alpha_bar - sqrt(disc_bar)) / S(2))
order = quintic.order(theta, d)
test = (order*delta.n()) - ( (l1.n() - l4.n())*(l2.n() - l3.n()) )
# Comparing floats
if not comp(test, 0, tol):
l2, l3 = l3, l2
# Now we have correct order of l's
R1 = l0 + l1*zeta1 + l2*zeta2 + l3*zeta3 + l4*zeta4
R2 = l0 + l3*zeta1 + l1*zeta2 + l4*zeta3 + l2*zeta4
R3 = l0 + l2*zeta1 + l4*zeta2 + l1*zeta3 + l3*zeta4
R4 = l0 + l4*zeta1 + l3*zeta2 + l2*zeta3 + l1*zeta4
Res = [None, [None]*5, [None]*5, [None]*5, [None]*5]
Res_n = [None, [None]*5, [None]*5, [None]*5, [None]*5]
sol = Symbol('sol')
# Simplifying improves performance a lot for exact expressions
R1 = _quintic_simplify(R1)
R2 = _quintic_simplify(R2)
R3 = _quintic_simplify(R3)
R4 = _quintic_simplify(R4)
# Solve imported here. Causing problems if imported as 'solve'
# and hence the changed name
from sympy.solvers.solvers import solve as _solve
a, b = symbols('a b', cls=Dummy)
_sol = _solve( sol**5 - a - I*b, sol)
for i in range(5):
_sol[i] = factor(_sol[i])
R1 = R1.as_real_imag()
R2 = R2.as_real_imag()
R3 = R3.as_real_imag()
R4 = R4.as_real_imag()
for i, currentroot in enumerate(_sol):
Res[1][i] = _quintic_simplify(currentroot.subs({ a: R1[0], b: R1[1] }))
Res[2][i] = _quintic_simplify(currentroot.subs({ a: R2[0], b: R2[1] }))
Res[3][i] = _quintic_simplify(currentroot.subs({ a: R3[0], b: R3[1] }))
Res[4][i] = _quintic_simplify(currentroot.subs({ a: R4[0], b: R4[1] }))
for i in range(1, 5):
for j in range(5):
Res_n[i][j] = Res[i][j].n()
Res[i][j] = _quintic_simplify(Res[i][j])
r1 = Res[1][0]
r1_n = Res_n[1][0]
for i in range(5):
if comp(im(r1_n*Res_n[4][i]), 0, tol):
r4 = Res[4][i]
break
# Now we have various Res values. Each will be a list of five
# values. We have to pick one r value from those five for each Res
u, v = quintic.uv(theta, d)
testplus = (u + v*delta*sqrt(5)).n()
testminus = (u - v*delta*sqrt(5)).n()
# Evaluated numbers suffixed with _n
# We will use evaluated numbers for calculation. Much faster.
r4_n = r4.n()
r2 = r3 = None
for i in range(5):
r2temp_n = Res_n[2][i]
for j in range(5):
# Again storing away the exact number and using
# evaluated numbers in computations
r3temp_n = Res_n[3][j]
if (comp((r1_n*r2temp_n**2 + r4_n*r3temp_n**2 - testplus).n(), 0, tol) and
comp((r3temp_n*r1_n**2 + r2temp_n*r4_n**2 - testminus).n(), 0, tol)):
r2 = Res[2][i]
r3 = Res[3][j]
break
if r2:
break
else:
return [] # fall back to normal solve
# Now, we have r's so we can get roots
x1 = (r1 + r2 + r3 + r4)/5
x2 = (r1*zeta4 + r2*zeta3 + r3*zeta2 + r4*zeta1)/5
x3 = (r1*zeta3 + r2*zeta1 + r3*zeta4 + r4*zeta2)/5
x4 = (r1*zeta2 + r2*zeta4 + r3*zeta1 + r4*zeta3)/5
x5 = (r1*zeta1 + r2*zeta2 + r3*zeta3 + r4*zeta4)/5
result = [x1, x2, x3, x4, x5]
# Now check if solutions are distinct
saw = set()
for r in result:
r = r.n(2)
if r in saw:
# Roots were identical. Abort, return []
# and fall back to usual solve
return []
saw.add(r)
return result
def _quintic_simplify(expr):
expr = powsimp(expr)
expr = cancel(expr)
return together(expr)
def _integer_basis(poly):
"""Compute coefficient basis for a polynomial over integers.
Returns the integer ``div`` such that substituting ``x = div*y``
``p(x) = m*q(y)`` where the coefficients of ``q`` are smaller
than those of ``p``.
For example ``x**5 + 512*x + 1024 = 0``
with ``div = 4`` becomes ``y**5 + 2*y + 1 = 0``
Returns the integer ``div`` or ``None`` if there is no possible scaling.
Examples
========
>>> from sympy.polys import Poly
>>> from sympy.abc import x
>>> from sympy.polys.polyroots import _integer_basis
>>> p = Poly(x**5 + 512*x + 1024, x, domain='ZZ')
>>> _integer_basis(p)
4
"""
monoms, coeffs = list(zip(*poly.terms()))
monoms, = list(zip(*monoms))
coeffs = list(map(abs, coeffs))
if coeffs[0] < coeffs[-1]:
coeffs = list(reversed(coeffs))
n = monoms[0]
monoms = [n - i for i in reversed(monoms)]
else:
return None
monoms = monoms[:-1]
coeffs = coeffs[:-1]
divs = reversed(divisors(gcd_list(coeffs))[1:])
try:
div = next(divs)
except StopIteration:
return None
while True:
for monom, coeff in zip(monoms, coeffs):
if coeff % div**monom != 0:
try:
div = next(divs)
except StopIteration:
return None
else:
break
else:
return div
def preprocess_roots(poly):
"""Try to get rid of symbolic coefficients from ``poly``. """
coeff = S.One
poly_func = poly.func
try:
_, poly = poly.clear_denoms(convert=True)
except DomainError:
return coeff, poly
poly = poly.primitive()[1]
poly = poly.retract()
# TODO: This is fragile. Figure out how to make this independent of construct_domain().
if poly.get_domain().is_Poly and all(c.is_term for c in poly.rep.coeffs()):
poly = poly.inject()
strips = list(zip(*poly.monoms()))
gens = list(poly.gens[1:])
base, strips = strips[0], strips[1:]
for gen, strip in zip(list(gens), strips):
reverse = False
if strip[0] < strip[-1]:
strip = reversed(strip)
reverse = True
ratio = None
for a, b in zip(base, strip):
if not a and not b:
continue
elif not a or not b:
break
elif b % a != 0:
break
else:
_ratio = b // a
if ratio is None:
ratio = _ratio
elif ratio != _ratio:
break
else:
if reverse:
ratio = -ratio
poly = poly.eval(gen, 1)
coeff *= gen**(-ratio)
gens.remove(gen)
if gens:
poly = poly.eject(*gens)
if poly.is_univariate and poly.get_domain().is_ZZ:
basis = _integer_basis(poly)
if basis is not None:
n = poly.degree()
def func(k, coeff):
return coeff//basis**(n - k[0])
poly = poly.termwise(func)
coeff *= basis
if not isinstance(poly, poly_func):
poly = poly_func(poly)
return coeff, poly
@public
def roots(f, *gens,
auto=True,
cubics=True,
trig=False,
quartics=True,
quintics=False,
multiple=False,
filter=None,
predicate=None,
**flags):
"""
Computes symbolic roots of a univariate polynomial.
Given a univariate polynomial f with symbolic coefficients (or
a list of the polynomial's coefficients), returns a dictionary
with its roots and their multiplicities.
Only roots expressible via radicals will be returned. To get
a complete set of roots use RootOf class or numerical methods
instead. By default cubic and quartic formulas are used in
the algorithm. To disable them because of unreadable output
set ``cubics=False`` or ``quartics=False`` respectively. If cubic
roots are real but are expressed in terms of complex numbers
(casus irreducibilis [1]) the ``trig`` flag can be set to True to
have the solutions returned in terms of cosine and inverse cosine
functions.
To get roots from a specific domain set the ``filter`` flag with
one of the following specifiers: Z, Q, R, I, C. By default all
roots are returned (this is equivalent to setting ``filter='C'``).
By default a dictionary is returned giving a compact result in
case of multiple roots. However to get a list containing all
those roots set the ``multiple`` flag to True; the list will
have identical roots appearing next to each other in the result.
(For a given Poly, the all_roots method will give the roots in
sorted numerical order.)
Examples
========
>>> from sympy import Poly, roots
>>> from sympy.abc import x, y
>>> roots(x**2 - 1, x)
{-1: 1, 1: 1}
>>> p = Poly(x**2-1, x)
>>> roots(p)
{-1: 1, 1: 1}
>>> p = Poly(x**2-y, x, y)
>>> roots(Poly(p, x))
{-sqrt(y): 1, sqrt(y): 1}
>>> roots(x**2 - y, x)
{-sqrt(y): 1, sqrt(y): 1}
>>> roots([1, 0, -1])
{-1: 1, 1: 1}
References
==========
.. [1] https://en.wikipedia.org/wiki/Cubic_function#Trigonometric_.28and_hyperbolic.29_method
"""
from sympy.polys.polytools import to_rational_coeffs
flags = dict(flags)
if isinstance(f, list):
if gens:
raise ValueError('redundant generators given')
x = Dummy('x')
poly, i = {}, len(f) - 1
for coeff in f:
poly[i], i = sympify(coeff), i - 1
f = Poly(poly, x, field=True)
else:
try:
F = Poly(f, *gens, **flags)
if not isinstance(f, Poly) and not F.gen.is_Symbol:
raise PolynomialError("generator must be a Symbol")
else:
f = F
if f.length == 2 and f.degree() != 1:
# check for foo**n factors in the constant
n = f.degree()
npow_bases = []
others = []
expr = f.as_expr()
con = expr.as_independent(*gens)[0]
for p in Mul.make_args(con):
if p.is_Pow and not p.exp % n:
npow_bases.append(p.base**(p.exp/n))
else:
others.append(p)
if npow_bases:
b = Mul(*npow_bases)
B = Dummy()
d = roots(Poly(expr - con + B**n*Mul(*others), *gens,
**flags), *gens, **flags)
rv = {}
for k, v in d.items():
rv[k.subs(B, b)] = v
return rv
except GeneratorsNeeded:
if multiple:
return []
else:
return {}
if f.is_multivariate:
raise PolynomialError('multivariate polynomials are not supported')
def _update_dict(result, currentroot, k):
if currentroot in result:
result[currentroot] += k
else:
result[currentroot] = k
def _try_decompose(f):
"""Find roots using functional decomposition. """
factors, roots = f.decompose(), []
for currentroot in _try_heuristics(factors[0]):
roots.append(currentroot)
for currentfactor in factors[1:]:
previous, roots = list(roots), []
for currentroot in previous:
g = currentfactor - Poly(currentroot, f.gen)
for currentroot in _try_heuristics(g):
roots.append(currentroot)
return roots
def _try_heuristics(f):
"""Find roots using formulas and some tricks. """
if f.is_ground:
return []
if f.is_monomial:
return [S.Zero]*f.degree()
if f.length() == 2:
if f.degree() == 1:
return list(map(cancel, roots_linear(f)))
else:
return roots_binomial(f)
result = []
for i in [-1, 1]:
if not f.eval(i):
f = f.quo(Poly(f.gen - i, f.gen))
result.append(i)
break
n = f.degree()
if n == 1:
result += list(map(cancel, roots_linear(f)))
elif n == 2:
result += list(map(cancel, roots_quadratic(f)))
elif f.is_cyclotomic:
result += roots_cyclotomic(f)
elif n == 3 and cubics:
result += roots_cubic(f, trig=trig)
elif n == 4 and quartics:
result += roots_quartic(f)
elif n == 5 and quintics:
result += roots_quintic(f)
return result
# Convert the generators to symbols
dumgens = symbols('x:%d' % len(f.gens), cls=Dummy)
f = f.per(f.rep, dumgens)
(k,), f = f.terms_gcd()
if not k:
zeros = {}
else:
zeros = {S.Zero: k}
coeff, f = preprocess_roots(f)
if auto and f.get_domain().is_Ring:
f = f.to_field()
# Use EX instead of ZZ_I or QQ_I
if f.get_domain().is_QQ_I:
f = f.per(f.rep.convert(EX))
rescale_x = None
translate_x = None
result = {}
if not f.is_ground:
dom = f.get_domain()
if not dom.is_Exact and dom.is_Numerical:
for r in f.nroots():
_update_dict(result, r, 1)
elif f.degree() == 1:
result[roots_linear(f)[0]] = 1
elif f.length() == 2:
roots_fun = roots_quadratic if f.degree() == 2 else roots_binomial
for r in roots_fun(f):
_update_dict(result, r, 1)
else:
_, factors = Poly(f.as_expr()).factor_list()
if len(factors) == 1 and f.degree() == 2:
for r in roots_quadratic(f):
_update_dict(result, r, 1)
else:
if len(factors) == 1 and factors[0][1] == 1:
if f.get_domain().is_EX:
res = to_rational_coeffs(f)
if res:
if res[0] is None:
translate_x, f = res[2:]
else:
rescale_x, f = res[1], res[-1]
result = roots(f)
if not result:
for currentroot in _try_decompose(f):
_update_dict(result, currentroot, 1)
else:
for r in _try_heuristics(f):
_update_dict(result, r, 1)
else:
for currentroot in _try_decompose(f):
_update_dict(result, currentroot, 1)
else:
for currentfactor, k in factors:
for r in _try_heuristics(Poly(currentfactor, f.gen, field=True)):
_update_dict(result, r, k)
if coeff is not S.One:
_result, result, = result, {}
for currentroot, k in _result.items():
result[coeff*currentroot] = k
if filter not in [None, 'C']:
handlers = {
'Z': lambda r: r.is_Integer,
'Q': lambda r: r.is_Rational,
'R': lambda r: all(a.is_real for a in r.as_numer_denom()),
'I': lambda r: r.is_imaginary,
}
try:
query = handlers[filter]
except KeyError:
raise ValueError("Invalid filter: %s" % filter)
for zero in dict(result).keys():
if not query(zero):
del result[zero]
if predicate is not None:
for zero in dict(result).keys():
if not predicate(zero):
del result[zero]
if rescale_x:
result1 = {}
for k, v in result.items():
result1[k*rescale_x] = v
result = result1
if translate_x:
result1 = {}
for k, v in result.items():
result1[k + translate_x] = v
result = result1
# adding zero roots after non-trivial roots have been translated
result.update(zeros)
if not multiple:
return result
else:
zeros = []
for zero in ordered(result):
zeros.extend([zero]*result[zero])
return zeros
def root_factors(f, *gens, filter=None, **args):
"""
Returns all factors of a univariate polynomial.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.polys.polyroots import root_factors
>>> root_factors(x**2 - y, x)
[x - sqrt(y), x + sqrt(y)]
"""
args = dict(args)
F = Poly(f, *gens, **args)
if not F.is_Poly:
return [f]
if F.is_multivariate:
raise ValueError('multivariate polynomials are not supported')
x = F.gens[0]
zeros = roots(F, filter=filter)
if not zeros:
factors = [F]
else:
factors, N = [], 0
for r, n in ordered(zeros.items()):
factors, N = factors + [Poly(x - r, x)]*n, N + n
if N < F.degree():
G = reduce(lambda p, q: p*q, factors)
factors.append(F.quo(G))
if not isinstance(f, Poly):
factors = [ f.as_expr() for f in factors ]
return factors
|
cb34651abb2e900afc2da3796db08d26a2c5b4b07cb283811c33365452a5958e | """Sparse polynomial rings. """
from typing import Any, Dict
from operator import add, mul, lt, le, gt, ge
from functools import reduce
from types import GeneratorType
from sympy.core.compatibility import is_sequence
from sympy.core.expr import Expr
from sympy.core.numbers import igcd, oo
from sympy.core.symbol import Symbol, symbols as _symbols
from sympy.core.sympify import CantSympify, sympify
from sympy.ntheory.multinomial import multinomial_coefficients
from sympy.polys.compatibility import IPolys
from sympy.polys.constructor import construct_domain
from sympy.polys.densebasic import dmp_to_dict, dmp_from_dict
from sympy.polys.domains.domainelement import DomainElement
from sympy.polys.domains.polynomialring import PolynomialRing
from sympy.polys.heuristicgcd import heugcd
from sympy.polys.monomials import MonomialOps
from sympy.polys.orderings import lex
from sympy.polys.polyerrors import (
CoercionFailed, GeneratorsError,
ExactQuotientFailed, MultivariatePolynomialError)
from sympy.polys.polyoptions import (Domain as DomainOpt,
Order as OrderOpt, build_options)
from sympy.polys.polyutils import (expr_from_dict, _dict_reorder,
_parallel_dict_from_expr)
from sympy.printing.defaults import DefaultPrinting
from sympy.utilities import public
from sympy.utilities.magic import pollute
@public
def ring(symbols, domain, order=lex):
"""Construct a polynomial ring returning ``(ring, x_1, ..., x_n)``.
Parameters
==========
symbols : str
Symbol/Expr or sequence of str, Symbol/Expr (non-empty)
domain : :class:`~.Domain` or coercible
order : :class:`~.MonomialOrder` or coercible, optional, defaults to ``lex``
Examples
========
>>> from sympy.polys.rings import ring
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.orderings import lex
>>> R, x, y, z = ring("x,y,z", ZZ, lex)
>>> R
Polynomial ring in x, y, z over ZZ with lex order
>>> x + y + z
x + y + z
>>> type(_)
<class 'sympy.polys.rings.PolyElement'>
"""
_ring = PolyRing(symbols, domain, order)
return (_ring,) + _ring.gens
@public
def xring(symbols, domain, order=lex):
"""Construct a polynomial ring returning ``(ring, (x_1, ..., x_n))``.
Parameters
==========
symbols : str
Symbol/Expr or sequence of str, Symbol/Expr (non-empty)
domain : :class:`~.Domain` or coercible
order : :class:`~.MonomialOrder` or coercible, optional, defaults to ``lex``
Examples
========
>>> from sympy.polys.rings import xring
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.orderings import lex
>>> R, (x, y, z) = xring("x,y,z", ZZ, lex)
>>> R
Polynomial ring in x, y, z over ZZ with lex order
>>> x + y + z
x + y + z
>>> type(_)
<class 'sympy.polys.rings.PolyElement'>
"""
_ring = PolyRing(symbols, domain, order)
return (_ring, _ring.gens)
@public
def vring(symbols, domain, order=lex):
"""Construct a polynomial ring and inject ``x_1, ..., x_n`` into the global namespace.
Parameters
==========
symbols : str
Symbol/Expr or sequence of str, Symbol/Expr (non-empty)
domain : :class:`~.Domain` or coercible
order : :class:`~.MonomialOrder` or coercible, optional, defaults to ``lex``
Examples
========
>>> from sympy.polys.rings import vring
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.orderings import lex
>>> vring("x,y,z", ZZ, lex)
Polynomial ring in x, y, z over ZZ with lex order
>>> x + y + z # noqa:
x + y + z
>>> type(_)
<class 'sympy.polys.rings.PolyElement'>
"""
_ring = PolyRing(symbols, domain, order)
pollute([ sym.name for sym in _ring.symbols ], _ring.gens)
return _ring
@public
def sring(exprs, *symbols, **options):
"""Construct a ring deriving generators and domain from options and input expressions.
Parameters
==========
exprs : :class:`~.Expr` or sequence of :class:`~.Expr` (sympifiable)
symbols : sequence of :class:`~.Symbol`/:class:`~.Expr`
options : keyword arguments understood by :class:`~.Options`
Examples
========
>>> from sympy.core import symbols
>>> from sympy.polys.rings import sring
>>> x, y, z = symbols("x,y,z")
>>> R, f = sring(x + 2*y + 3*z)
>>> R
Polynomial ring in x, y, z over ZZ with lex order
>>> f
x + 2*y + 3*z
>>> type(_)
<class 'sympy.polys.rings.PolyElement'>
"""
single = False
if not is_sequence(exprs):
exprs, single = [exprs], True
exprs = list(map(sympify, exprs))
opt = build_options(symbols, options)
# TODO: rewrite this so that it doesn't use expand() (see poly()).
reps, opt = _parallel_dict_from_expr(exprs, opt)
if opt.domain is None:
coeffs = sum([ list(rep.values()) for rep in reps ], [])
opt.domain, coeffs_dom = construct_domain(coeffs, opt=opt)
coeff_map = dict(zip(coeffs, coeffs_dom))
reps = [{m: coeff_map[c] for m, c in rep.items()} for rep in reps]
_ring = PolyRing(opt.gens, opt.domain, opt.order)
polys = list(map(_ring.from_dict, reps))
if single:
return (_ring, polys[0])
else:
return (_ring, polys)
def _parse_symbols(symbols):
if isinstance(symbols, str):
return _symbols(symbols, seq=True) if symbols else ()
elif isinstance(symbols, Expr):
return (symbols,)
elif is_sequence(symbols):
if all(isinstance(s, str) for s in symbols):
return _symbols(symbols)
elif all(isinstance(s, Expr) for s in symbols):
return symbols
raise GeneratorsError("expected a string, Symbol or expression or a non-empty sequence of strings, Symbols or expressions")
_ring_cache = {} # type: Dict[Any, Any]
class PolyRing(DefaultPrinting, IPolys):
"""Multivariate distributed polynomial ring. """
def __new__(cls, symbols, domain, order=lex):
symbols = tuple(_parse_symbols(symbols))
ngens = len(symbols)
domain = DomainOpt.preprocess(domain)
order = OrderOpt.preprocess(order)
_hash_tuple = (cls.__name__, symbols, ngens, domain, order)
obj = _ring_cache.get(_hash_tuple)
if obj is None:
if domain.is_Composite and set(symbols) & set(domain.symbols):
raise GeneratorsError("polynomial ring and it's ground domain share generators")
obj = object.__new__(cls)
obj._hash_tuple = _hash_tuple
obj._hash = hash(_hash_tuple)
obj.dtype = type("PolyElement", (PolyElement,), {"ring": obj})
obj.symbols = symbols
obj.ngens = ngens
obj.domain = domain
obj.order = order
obj.zero_monom = (0,)*ngens
obj.gens = obj._gens()
obj._gens_set = set(obj.gens)
obj._one = [(obj.zero_monom, domain.one)]
if ngens:
# These expect monomials in at least one variable
codegen = MonomialOps(ngens)
obj.monomial_mul = codegen.mul()
obj.monomial_pow = codegen.pow()
obj.monomial_mulpow = codegen.mulpow()
obj.monomial_ldiv = codegen.ldiv()
obj.monomial_div = codegen.div()
obj.monomial_lcm = codegen.lcm()
obj.monomial_gcd = codegen.gcd()
else:
monunit = lambda a, b: ()
obj.monomial_mul = monunit
obj.monomial_pow = monunit
obj.monomial_mulpow = lambda a, b, c: ()
obj.monomial_ldiv = monunit
obj.monomial_div = monunit
obj.monomial_lcm = monunit
obj.monomial_gcd = monunit
if order is lex:
obj.leading_expv = lambda f: max(f)
else:
obj.leading_expv = lambda f: max(f, key=order)
for symbol, generator in zip(obj.symbols, obj.gens):
if isinstance(symbol, Symbol):
name = symbol.name
if not hasattr(obj, name):
setattr(obj, name, generator)
_ring_cache[_hash_tuple] = obj
return obj
def _gens(self):
"""Return a list of polynomial generators. """
one = self.domain.one
_gens = []
for i in range(self.ngens):
expv = self.monomial_basis(i)
poly = self.zero
poly[expv] = one
_gens.append(poly)
return tuple(_gens)
def __getnewargs__(self):
return (self.symbols, self.domain, self.order)
def __getstate__(self):
state = self.__dict__.copy()
del state["leading_expv"]
for key, value in state.items():
if key.startswith("monomial_"):
del state[key]
return state
def __hash__(self):
return self._hash
def __eq__(self, other):
return isinstance(other, PolyRing) and \
(self.symbols, self.domain, self.ngens, self.order) == \
(other.symbols, other.domain, other.ngens, other.order)
def __ne__(self, other):
return not self == other
def clone(self, symbols=None, domain=None, order=None):
return self.__class__(symbols or self.symbols, domain or self.domain, order or self.order)
def monomial_basis(self, i):
"""Return the ith-basis element. """
basis = [0]*self.ngens
basis[i] = 1
return tuple(basis)
@property
def zero(self):
return self.dtype()
@property
def one(self):
return self.dtype(self._one)
def domain_new(self, element, orig_domain=None):
return self.domain.convert(element, orig_domain)
def ground_new(self, coeff):
return self.term_new(self.zero_monom, coeff)
def term_new(self, monom, coeff):
coeff = self.domain_new(coeff)
poly = self.zero
if coeff:
poly[monom] = coeff
return poly
def ring_new(self, element):
if isinstance(element, PolyElement):
if self == element.ring:
return element
elif isinstance(self.domain, PolynomialRing) and self.domain.ring == element.ring:
return self.ground_new(element)
else:
raise NotImplementedError("conversion")
elif isinstance(element, str):
raise NotImplementedError("parsing")
elif isinstance(element, dict):
return self.from_dict(element)
elif isinstance(element, list):
try:
return self.from_terms(element)
except ValueError:
return self.from_list(element)
elif isinstance(element, Expr):
return self.from_expr(element)
else:
return self.ground_new(element)
__call__ = ring_new
def from_dict(self, element, orig_domain=None):
domain_new = self.domain_new
poly = self.zero
for monom, coeff in element.items():
coeff = domain_new(coeff, orig_domain)
if coeff:
poly[monom] = coeff
return poly
def from_terms(self, element, orig_domain=None):
return self.from_dict(dict(element), orig_domain)
def from_list(self, element):
return self.from_dict(dmp_to_dict(element, self.ngens-1, self.domain))
def _rebuild_expr(self, expr, mapping):
domain = self.domain
def _rebuild(expr):
generator = mapping.get(expr)
if generator is not None:
return generator
elif expr.is_Add:
return reduce(add, list(map(_rebuild, expr.args)))
elif expr.is_Mul:
return reduce(mul, list(map(_rebuild, expr.args)))
elif expr.is_Pow and expr.exp.is_Integer and expr.exp >= 0:
return _rebuild(expr.base)**int(expr.exp)
else:
return self.ground_new(domain.convert(expr))
return _rebuild(sympify(expr))
def from_expr(self, expr):
mapping = dict(list(zip(self.symbols, self.gens)))
try:
poly = self._rebuild_expr(expr, mapping)
except CoercionFailed:
raise ValueError("expected an expression convertible to a polynomial in %s, got %s" % (self, expr))
else:
return self.ring_new(poly)
def index(self, gen):
"""Compute index of ``gen`` in ``self.gens``. """
if gen is None:
if self.ngens:
i = 0
else:
i = -1 # indicate impossible choice
elif isinstance(gen, int):
i = gen
if 0 <= i and i < self.ngens:
pass
elif -self.ngens <= i and i <= -1:
i = -i - 1
else:
raise ValueError("invalid generator index: %s" % gen)
elif isinstance(gen, self.dtype):
try:
i = self.gens.index(gen)
except ValueError:
raise ValueError("invalid generator: %s" % gen)
elif isinstance(gen, str):
try:
i = self.symbols.index(gen)
except ValueError:
raise ValueError("invalid generator: %s" % gen)
else:
raise ValueError("expected a polynomial generator, an integer, a string or None, got %s" % gen)
return i
def drop(self, *gens):
"""Remove specified generators from this ring. """
indices = set(map(self.index, gens))
symbols = [ s for i, s in enumerate(self.symbols) if i not in indices ]
if not symbols:
return self.domain
else:
return self.clone(symbols=symbols)
def __getitem__(self, key):
symbols = self.symbols[key]
if not symbols:
return self.domain
else:
return self.clone(symbols=symbols)
def to_ground(self):
# TODO: should AlgebraicField be a Composite domain?
if self.domain.is_Composite or hasattr(self.domain, 'domain'):
return self.clone(domain=self.domain.domain)
else:
raise ValueError("%s is not a composite domain" % self.domain)
def to_domain(self):
return PolynomialRing(self)
def to_field(self):
from sympy.polys.fields import FracField
return FracField(self.symbols, self.domain, self.order)
@property
def is_univariate(self):
return len(self.gens) == 1
@property
def is_multivariate(self):
return len(self.gens) > 1
def add(self, *objs):
"""
Add a sequence of polynomials or containers of polynomials.
Examples
========
>>> from sympy.polys.rings import ring
>>> from sympy.polys.domains import ZZ
>>> R, x = ring("x", ZZ)
>>> R.add([ x**2 + 2*i + 3 for i in range(4) ])
4*x**2 + 24
>>> _.factor_list()
(4, [(x**2 + 6, 1)])
"""
p = self.zero
for obj in objs:
if is_sequence(obj, include=GeneratorType):
p += self.add(*obj)
else:
p += obj
return p
def mul(self, *objs):
"""
Multiply a sequence of polynomials or containers of polynomials.
Examples
========
>>> from sympy.polys.rings import ring
>>> from sympy.polys.domains import ZZ
>>> R, x = ring("x", ZZ)
>>> R.mul([ x**2 + 2*i + 3 for i in range(4) ])
x**8 + 24*x**6 + 206*x**4 + 744*x**2 + 945
>>> _.factor_list()
(1, [(x**2 + 3, 1), (x**2 + 5, 1), (x**2 + 7, 1), (x**2 + 9, 1)])
"""
p = self.one
for obj in objs:
if is_sequence(obj, include=GeneratorType):
p *= self.mul(*obj)
else:
p *= obj
return p
def drop_to_ground(self, *gens):
r"""
Remove specified generators from the ring and inject them into
its domain.
"""
indices = set(map(self.index, gens))
symbols = [s for i, s in enumerate(self.symbols) if i not in indices]
gens = [gen for i, gen in enumerate(self.gens) if i not in indices]
if not symbols:
return self
else:
return self.clone(symbols=symbols, domain=self.drop(*gens))
def compose(self, other):
"""Add the generators of ``other`` to ``self``"""
if self != other:
syms = set(self.symbols).union(set(other.symbols))
return self.clone(symbols=list(syms))
else:
return self
def add_gens(self, symbols):
"""Add the elements of ``symbols`` as generators to ``self``"""
syms = set(self.symbols).union(set(symbols))
return self.clone(symbols=list(syms))
class PolyElement(DomainElement, DefaultPrinting, CantSympify, dict):
"""Element of multivariate distributed polynomial ring. """
def new(self, init):
return self.__class__(init)
def parent(self):
return self.ring.to_domain()
def __getnewargs__(self):
return (self.ring, list(self.iterterms()))
_hash = None
def __hash__(self):
# XXX: This computes a hash of a dictionary, but currently we don't
# protect dictionary from being changed so any use site modifications
# will make hashing go wrong. Use this feature with caution until we
# figure out how to make a safe API without compromising speed of this
# low-level class.
_hash = self._hash
if _hash is None:
self._hash = _hash = hash((self.ring, frozenset(self.items())))
return _hash
def copy(self):
"""Return a copy of polynomial self.
Polynomials are mutable; if one is interested in preserving
a polynomial, and one plans to use inplace operations, one
can copy the polynomial. This method makes a shallow copy.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.rings import ring
>>> R, x, y = ring('x, y', ZZ)
>>> p = (x + y)**2
>>> p1 = p.copy()
>>> p2 = p
>>> p[R.zero_monom] = 3
>>> p
x**2 + 2*x*y + y**2 + 3
>>> p1
x**2 + 2*x*y + y**2
>>> p2
x**2 + 2*x*y + y**2 + 3
"""
return self.new(self)
def set_ring(self, new_ring):
if self.ring == new_ring:
return self
elif self.ring.symbols != new_ring.symbols:
terms = list(zip(*_dict_reorder(self, self.ring.symbols, new_ring.symbols)))
return new_ring.from_terms(terms, self.ring.domain)
else:
return new_ring.from_dict(self, self.ring.domain)
def as_expr(self, *symbols):
if symbols and len(symbols) != self.ring.ngens:
raise ValueError("not enough symbols, expected %s got %s" % (self.ring.ngens, len(symbols)))
else:
symbols = self.ring.symbols
return expr_from_dict(self.as_expr_dict(), *symbols)
def as_expr_dict(self):
to_sympy = self.ring.domain.to_sympy
return {monom: to_sympy(coeff) for monom, coeff in self.iterterms()}
def clear_denoms(self):
domain = self.ring.domain
if not domain.is_Field or not domain.has_assoc_Ring:
return domain.one, self
ground_ring = domain.get_ring()
common = ground_ring.one
lcm = ground_ring.lcm
denom = domain.denom
for coeff in self.values():
common = lcm(common, denom(coeff))
poly = self.new([ (k, v*common) for k, v in self.items() ])
return common, poly
def strip_zero(self):
"""Eliminate monomials with zero coefficient. """
for k, v in list(self.items()):
if not v:
del self[k]
def __eq__(p1, p2):
"""Equality test for polynomials.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.rings import ring
>>> _, x, y = ring('x, y', ZZ)
>>> p1 = (x + y)**2 + (x - y)**2
>>> p1 == 4*x*y
False
>>> p1 == 2*(x**2 + y**2)
True
"""
if not p2:
return not p1
elif isinstance(p2, PolyElement) and p2.ring == p1.ring:
return dict.__eq__(p1, p2)
elif len(p1) > 1:
return False
else:
return p1.get(p1.ring.zero_monom) == p2
def __ne__(p1, p2):
return not p1 == p2
def almosteq(p1, p2, tolerance=None):
"""Approximate equality test for polynomials. """
ring = p1.ring
if isinstance(p2, ring.dtype):
if set(p1.keys()) != set(p2.keys()):
return False
almosteq = ring.domain.almosteq
for k in p1.keys():
if not almosteq(p1[k], p2[k], tolerance):
return False
return True
elif len(p1) > 1:
return False
else:
try:
p2 = ring.domain.convert(p2)
except CoercionFailed:
return False
else:
return ring.domain.almosteq(p1.const(), p2, tolerance)
def sort_key(self):
return (len(self), self.terms())
def _cmp(p1, p2, op):
if isinstance(p2, p1.ring.dtype):
return op(p1.sort_key(), p2.sort_key())
else:
return NotImplemented
def __lt__(p1, p2):
return p1._cmp(p2, lt)
def __le__(p1, p2):
return p1._cmp(p2, le)
def __gt__(p1, p2):
return p1._cmp(p2, gt)
def __ge__(p1, p2):
return p1._cmp(p2, ge)
def _drop(self, gen):
ring = self.ring
i = ring.index(gen)
if ring.ngens == 1:
return i, ring.domain
else:
symbols = list(ring.symbols)
del symbols[i]
return i, ring.clone(symbols=symbols)
def drop(self, gen):
i, ring = self._drop(gen)
if self.ring.ngens == 1:
if self.is_ground:
return self.coeff(1)
else:
raise ValueError("can't drop %s" % gen)
else:
poly = ring.zero
for k, v in self.items():
if k[i] == 0:
K = list(k)
del K[i]
poly[tuple(K)] = v
else:
raise ValueError("can't drop %s" % gen)
return poly
def _drop_to_ground(self, gen):
ring = self.ring
i = ring.index(gen)
symbols = list(ring.symbols)
del symbols[i]
return i, ring.clone(symbols=symbols, domain=ring[i])
def drop_to_ground(self, gen):
if self.ring.ngens == 1:
raise ValueError("can't drop only generator to ground")
i, ring = self._drop_to_ground(gen)
poly = ring.zero
gen = ring.domain.gens[0]
for monom, coeff in self.iterterms():
mon = monom[:i] + monom[i+1:]
if not mon in poly:
poly[mon] = (gen**monom[i]).mul_ground(coeff)
else:
poly[mon] += (gen**monom[i]).mul_ground(coeff)
return poly
def to_dense(self):
return dmp_from_dict(self, self.ring.ngens-1, self.ring.domain)
def to_dict(self):
return dict(self)
def str(self, printer, precedence, exp_pattern, mul_symbol):
if not self:
return printer._print(self.ring.domain.zero)
prec_mul = precedence["Mul"]
prec_atom = precedence["Atom"]
ring = self.ring
symbols = ring.symbols
ngens = ring.ngens
zm = ring.zero_monom
sexpvs = []
for expv, coeff in self.terms():
negative = ring.domain.is_negative(coeff)
sign = " - " if negative else " + "
sexpvs.append(sign)
if expv == zm:
scoeff = printer._print(coeff)
if negative and scoeff.startswith("-"):
scoeff = scoeff[1:]
else:
if negative:
coeff = -coeff
if coeff != self.ring.one:
scoeff = printer.parenthesize(coeff, prec_mul, strict=True)
else:
scoeff = ''
sexpv = []
for i in range(ngens):
exp = expv[i]
if not exp:
continue
symbol = printer.parenthesize(symbols[i], prec_atom, strict=True)
if exp != 1:
if exp != int(exp) or exp < 0:
sexp = printer.parenthesize(exp, prec_atom, strict=False)
else:
sexp = exp
sexpv.append(exp_pattern % (symbol, sexp))
else:
sexpv.append('%s' % symbol)
if scoeff:
sexpv = [scoeff] + sexpv
sexpvs.append(mul_symbol.join(sexpv))
if sexpvs[0] in [" + ", " - "]:
head = sexpvs.pop(0)
if head == " - ":
sexpvs.insert(0, "-")
return "".join(sexpvs)
@property
def is_generator(self):
return self in self.ring._gens_set
@property
def is_ground(self):
return not self or (len(self) == 1 and self.ring.zero_monom in self)
@property
def is_monomial(self):
return not self or (len(self) == 1 and self.LC == 1)
@property
def is_term(self):
return len(self) <= 1
@property
def is_negative(self):
return self.ring.domain.is_negative(self.LC)
@property
def is_positive(self):
return self.ring.domain.is_positive(self.LC)
@property
def is_nonnegative(self):
return self.ring.domain.is_nonnegative(self.LC)
@property
def is_nonpositive(self):
return self.ring.domain.is_nonpositive(self.LC)
@property
def is_zero(f):
return not f
@property
def is_one(f):
return f == f.ring.one
@property
def is_monic(f):
return f.ring.domain.is_one(f.LC)
@property
def is_primitive(f):
return f.ring.domain.is_one(f.content())
@property
def is_linear(f):
return all(sum(monom) <= 1 for monom in f.itermonoms())
@property
def is_quadratic(f):
return all(sum(monom) <= 2 for monom in f.itermonoms())
@property
def is_squarefree(f):
if not f.ring.ngens:
return True
return f.ring.dmp_sqf_p(f)
@property
def is_irreducible(f):
if not f.ring.ngens:
return True
return f.ring.dmp_irreducible_p(f)
@property
def is_cyclotomic(f):
if f.ring.is_univariate:
return f.ring.dup_cyclotomic_p(f)
else:
raise MultivariatePolynomialError("cyclotomic polynomial")
def __neg__(self):
return self.new([ (monom, -coeff) for monom, coeff in self.iterterms() ])
def __pos__(self):
return self
def __add__(p1, p2):
"""Add two polynomials.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.rings import ring
>>> _, x, y = ring('x, y', ZZ)
>>> (x + y)**2 + (x - y)**2
2*x**2 + 2*y**2
"""
if not p2:
return p1.copy()
ring = p1.ring
if isinstance(p2, ring.dtype):
p = p1.copy()
get = p.get
zero = ring.domain.zero
for k, v in p2.items():
v = get(k, zero) + v
if v:
p[k] = v
else:
del p[k]
return p
elif isinstance(p2, PolyElement):
if isinstance(ring.domain, PolynomialRing) and ring.domain.ring == p2.ring:
pass
elif isinstance(p2.ring.domain, PolynomialRing) and p2.ring.domain.ring == ring:
return p2.__radd__(p1)
else:
return NotImplemented
try:
cp2 = ring.domain_new(p2)
except CoercionFailed:
return NotImplemented
else:
p = p1.copy()
if not cp2:
return p
zm = ring.zero_monom
if zm not in p1.keys():
p[zm] = cp2
else:
if p2 == -p[zm]:
del p[zm]
else:
p[zm] += cp2
return p
def __radd__(p1, n):
p = p1.copy()
if not n:
return p
ring = p1.ring
try:
n = ring.domain_new(n)
except CoercionFailed:
return NotImplemented
else:
zm = ring.zero_monom
if zm not in p1.keys():
p[zm] = n
else:
if n == -p[zm]:
del p[zm]
else:
p[zm] += n
return p
def __sub__(p1, p2):
"""Subtract polynomial p2 from p1.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.rings import ring
>>> _, x, y = ring('x, y', ZZ)
>>> p1 = x + y**2
>>> p2 = x*y + y**2
>>> p1 - p2
-x*y + x
"""
if not p2:
return p1.copy()
ring = p1.ring
if isinstance(p2, ring.dtype):
p = p1.copy()
get = p.get
zero = ring.domain.zero
for k, v in p2.items():
v = get(k, zero) - v
if v:
p[k] = v
else:
del p[k]
return p
elif isinstance(p2, PolyElement):
if isinstance(ring.domain, PolynomialRing) and ring.domain.ring == p2.ring:
pass
elif isinstance(p2.ring.domain, PolynomialRing) and p2.ring.domain.ring == ring:
return p2.__rsub__(p1)
else:
return NotImplemented
try:
p2 = ring.domain_new(p2)
except CoercionFailed:
return NotImplemented
else:
p = p1.copy()
zm = ring.zero_monom
if zm not in p1.keys():
p[zm] = -p2
else:
if p2 == p[zm]:
del p[zm]
else:
p[zm] -= p2
return p
def __rsub__(p1, n):
"""n - p1 with n convertible to the coefficient domain.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.rings import ring
>>> _, x, y = ring('x, y', ZZ)
>>> p = x + y
>>> 4 - p
-x - y + 4
"""
ring = p1.ring
try:
n = ring.domain_new(n)
except CoercionFailed:
return NotImplemented
else:
p = ring.zero
for expv in p1:
p[expv] = -p1[expv]
p += n
return p
def __mul__(p1, p2):
"""Multiply two polynomials.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.rings import ring
>>> _, x, y = ring('x, y', QQ)
>>> p1 = x + y
>>> p2 = x - y
>>> p1*p2
x**2 - y**2
"""
ring = p1.ring
p = ring.zero
if not p1 or not p2:
return p
elif isinstance(p2, ring.dtype):
get = p.get
zero = ring.domain.zero
monomial_mul = ring.monomial_mul
p2it = list(p2.items())
for exp1, v1 in p1.items():
for exp2, v2 in p2it:
exp = monomial_mul(exp1, exp2)
p[exp] = get(exp, zero) + v1*v2
p.strip_zero()
return p
elif isinstance(p2, PolyElement):
if isinstance(ring.domain, PolynomialRing) and ring.domain.ring == p2.ring:
pass
elif isinstance(p2.ring.domain, PolynomialRing) and p2.ring.domain.ring == ring:
return p2.__rmul__(p1)
else:
return NotImplemented
try:
p2 = ring.domain_new(p2)
except CoercionFailed:
return NotImplemented
else:
for exp1, v1 in p1.items():
v = v1*p2
if v:
p[exp1] = v
return p
def __rmul__(p1, p2):
"""p2 * p1 with p2 in the coefficient domain of p1.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.rings import ring
>>> _, x, y = ring('x, y', ZZ)
>>> p = x + y
>>> 4 * p
4*x + 4*y
"""
p = p1.ring.zero
if not p2:
return p
try:
p2 = p.ring.domain_new(p2)
except CoercionFailed:
return NotImplemented
else:
for exp1, v1 in p1.items():
v = p2*v1
if v:
p[exp1] = v
return p
def __pow__(self, n):
"""raise polynomial to power `n`
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.rings import ring
>>> _, x, y = ring('x, y', ZZ)
>>> p = x + y**2
>>> p**3
x**3 + 3*x**2*y**2 + 3*x*y**4 + y**6
"""
ring = self.ring
if not n:
if self:
return ring.one
else:
raise ValueError("0**0")
elif len(self) == 1:
monom, coeff = list(self.items())[0]
p = ring.zero
if coeff == 1:
p[ring.monomial_pow(monom, n)] = coeff
else:
p[ring.monomial_pow(monom, n)] = coeff**n
return p
# For ring series, we need negative and rational exponent support only
# with monomials.
n = int(n)
if n < 0:
raise ValueError("Negative exponent")
elif n == 1:
return self.copy()
elif n == 2:
return self.square()
elif n == 3:
return self*self.square()
elif len(self) <= 5: # TODO: use an actual density measure
return self._pow_multinomial(n)
else:
return self._pow_generic(n)
def _pow_generic(self, n):
p = self.ring.one
c = self
while True:
if n & 1:
p = p*c
n -= 1
if not n:
break
c = c.square()
n = n // 2
return p
def _pow_multinomial(self, n):
multinomials = list(multinomial_coefficients(len(self), n).items())
monomial_mulpow = self.ring.monomial_mulpow
zero_monom = self.ring.zero_monom
terms = list(self.iterterms())
zero = self.ring.domain.zero
poly = self.ring.zero
for multinomial, multinomial_coeff in multinomials:
product_monom = zero_monom
product_coeff = multinomial_coeff
for exp, (monom, coeff) in zip(multinomial, terms):
if exp:
product_monom = monomial_mulpow(product_monom, monom, exp)
product_coeff *= coeff**exp
monom = tuple(product_monom)
coeff = product_coeff
coeff = poly.get(monom, zero) + coeff
if coeff:
poly[monom] = coeff
else:
del poly[monom]
return poly
def square(self):
"""square of a polynomial
Examples
========
>>> from sympy.polys.rings import ring
>>> from sympy.polys.domains import ZZ
>>> _, x, y = ring('x, y', ZZ)
>>> p = x + y**2
>>> p.square()
x**2 + 2*x*y**2 + y**4
"""
ring = self.ring
p = ring.zero
get = p.get
keys = list(self.keys())
zero = ring.domain.zero
monomial_mul = ring.monomial_mul
for i in range(len(keys)):
k1 = keys[i]
pk = self[k1]
for j in range(i):
k2 = keys[j]
exp = monomial_mul(k1, k2)
p[exp] = get(exp, zero) + pk*self[k2]
p = p.imul_num(2)
get = p.get
for k, v in self.items():
k2 = monomial_mul(k, k)
p[k2] = get(k2, zero) + v**2
p.strip_zero()
return p
def __divmod__(p1, p2):
ring = p1.ring
if not p2:
raise ZeroDivisionError("polynomial division")
elif isinstance(p2, ring.dtype):
return p1.div(p2)
elif isinstance(p2, PolyElement):
if isinstance(ring.domain, PolynomialRing) and ring.domain.ring == p2.ring:
pass
elif isinstance(p2.ring.domain, PolynomialRing) and p2.ring.domain.ring == ring:
return p2.__rdivmod__(p1)
else:
return NotImplemented
try:
p2 = ring.domain_new(p2)
except CoercionFailed:
return NotImplemented
else:
return (p1.quo_ground(p2), p1.rem_ground(p2))
def __rdivmod__(p1, p2):
return NotImplemented
def __mod__(p1, p2):
ring = p1.ring
if not p2:
raise ZeroDivisionError("polynomial division")
elif isinstance(p2, ring.dtype):
return p1.rem(p2)
elif isinstance(p2, PolyElement):
if isinstance(ring.domain, PolynomialRing) and ring.domain.ring == p2.ring:
pass
elif isinstance(p2.ring.domain, PolynomialRing) and p2.ring.domain.ring == ring:
return p2.__rmod__(p1)
else:
return NotImplemented
try:
p2 = ring.domain_new(p2)
except CoercionFailed:
return NotImplemented
else:
return p1.rem_ground(p2)
def __rmod__(p1, p2):
return NotImplemented
def __truediv__(p1, p2):
ring = p1.ring
if not p2:
raise ZeroDivisionError("polynomial division")
elif isinstance(p2, ring.dtype):
if p2.is_monomial:
return p1*(p2**(-1))
else:
return p1.quo(p2)
elif isinstance(p2, PolyElement):
if isinstance(ring.domain, PolynomialRing) and ring.domain.ring == p2.ring:
pass
elif isinstance(p2.ring.domain, PolynomialRing) and p2.ring.domain.ring == ring:
return p2.__rtruediv__(p1)
else:
return NotImplemented
try:
p2 = ring.domain_new(p2)
except CoercionFailed:
return NotImplemented
else:
return p1.quo_ground(p2)
def __rtruediv__(p1, p2):
return NotImplemented
__floordiv__ = __truediv__
__rfloordiv__ = __rtruediv__
# TODO: use // (__floordiv__) for exquo()?
def _term_div(self):
zm = self.ring.zero_monom
domain = self.ring.domain
domain_quo = domain.quo
monomial_div = self.ring.monomial_div
if domain.is_Field:
def term_div(a_lm_a_lc, b_lm_b_lc):
a_lm, a_lc = a_lm_a_lc
b_lm, b_lc = b_lm_b_lc
if b_lm == zm: # apparently this is a very common case
monom = a_lm
else:
monom = monomial_div(a_lm, b_lm)
if monom is not None:
return monom, domain_quo(a_lc, b_lc)
else:
return None
else:
def term_div(a_lm_a_lc, b_lm_b_lc):
a_lm, a_lc = a_lm_a_lc
b_lm, b_lc = b_lm_b_lc
if b_lm == zm: # apparently this is a very common case
monom = a_lm
else:
monom = monomial_div(a_lm, b_lm)
if not (monom is None or a_lc % b_lc):
return monom, domain_quo(a_lc, b_lc)
else:
return None
return term_div
def div(self, fv):
"""Division algorithm, see [CLO] p64.
fv array of polynomials
return qv, r such that
self = sum(fv[i]*qv[i]) + r
All polynomials are required not to be Laurent polynomials.
Examples
========
>>> from sympy.polys.rings import ring
>>> from sympy.polys.domains import ZZ
>>> _, x, y = ring('x, y', ZZ)
>>> f = x**3
>>> f0 = x - y**2
>>> f1 = x - y
>>> qv, r = f.div((f0, f1))
>>> qv[0]
x**2 + x*y**2 + y**4
>>> qv[1]
0
>>> r
y**6
"""
ring = self.ring
ret_single = False
if isinstance(fv, PolyElement):
ret_single = True
fv = [fv]
if any(not f for f in fv):
raise ZeroDivisionError("polynomial division")
if not self:
if ret_single:
return ring.zero, ring.zero
else:
return [], ring.zero
for f in fv:
if f.ring != ring:
raise ValueError('self and f must have the same ring')
s = len(fv)
qv = [ring.zero for i in range(s)]
p = self.copy()
r = ring.zero
term_div = self._term_div()
expvs = [fx.leading_expv() for fx in fv]
while p:
i = 0
divoccurred = 0
while i < s and divoccurred == 0:
expv = p.leading_expv()
term = term_div((expv, p[expv]), (expvs[i], fv[i][expvs[i]]))
if term is not None:
expv1, c = term
qv[i] = qv[i]._iadd_monom((expv1, c))
p = p._iadd_poly_monom(fv[i], (expv1, -c))
divoccurred = 1
else:
i += 1
if not divoccurred:
expv = p.leading_expv()
r = r._iadd_monom((expv, p[expv]))
del p[expv]
if expv == ring.zero_monom:
r += p
if ret_single:
if not qv:
return ring.zero, r
else:
return qv[0], r
else:
return qv, r
def rem(self, G):
f = self
if isinstance(G, PolyElement):
G = [G]
if any(not g for g in G):
raise ZeroDivisionError("polynomial division")
ring = f.ring
domain = ring.domain
zero = domain.zero
monomial_mul = ring.monomial_mul
r = ring.zero
term_div = f._term_div()
ltf = f.LT
f = f.copy()
get = f.get
while f:
for g in G:
tq = term_div(ltf, g.LT)
if tq is not None:
m, c = tq
for mg, cg in g.iterterms():
m1 = monomial_mul(mg, m)
c1 = get(m1, zero) - c*cg
if not c1:
del f[m1]
else:
f[m1] = c1
ltm = f.leading_expv()
if ltm is not None:
ltf = ltm, f[ltm]
break
else:
ltm, ltc = ltf
if ltm in r:
r[ltm] += ltc
else:
r[ltm] = ltc
del f[ltm]
ltm = f.leading_expv()
if ltm is not None:
ltf = ltm, f[ltm]
return r
def quo(f, G):
return f.div(G)[0]
def exquo(f, G):
q, r = f.div(G)
if not r:
return q
else:
raise ExactQuotientFailed(f, G)
def _iadd_monom(self, mc):
"""add to self the monomial coeff*x0**i0*x1**i1*...
unless self is a generator -- then just return the sum of the two.
mc is a tuple, (monom, coeff), where monomial is (i0, i1, ...)
Examples
========
>>> from sympy.polys.rings import ring
>>> from sympy.polys.domains import ZZ
>>> _, x, y = ring('x, y', ZZ)
>>> p = x**4 + 2*y
>>> m = (1, 2)
>>> p1 = p._iadd_monom((m, 5))
>>> p1
x**4 + 5*x*y**2 + 2*y
>>> p1 is p
True
>>> p = x
>>> p1 = p._iadd_monom((m, 5))
>>> p1
5*x*y**2 + x
>>> p1 is p
False
"""
if self in self.ring._gens_set:
cpself = self.copy()
else:
cpself = self
expv, coeff = mc
c = cpself.get(expv)
if c is None:
cpself[expv] = coeff
else:
c += coeff
if c:
cpself[expv] = c
else:
del cpself[expv]
return cpself
def _iadd_poly_monom(self, p2, mc):
"""add to self the product of (p)*(coeff*x0**i0*x1**i1*...)
unless self is a generator -- then just return the sum of the two.
mc is a tuple, (monom, coeff), where monomial is (i0, i1, ...)
Examples
========
>>> from sympy.polys.rings import ring
>>> from sympy.polys.domains import ZZ
>>> _, x, y, z = ring('x, y, z', ZZ)
>>> p1 = x**4 + 2*y
>>> p2 = y + z
>>> m = (1, 2, 3)
>>> p1 = p1._iadd_poly_monom(p2, (m, 3))
>>> p1
x**4 + 3*x*y**3*z**3 + 3*x*y**2*z**4 + 2*y
"""
p1 = self
if p1 in p1.ring._gens_set:
p1 = p1.copy()
(m, c) = mc
get = p1.get
zero = p1.ring.domain.zero
monomial_mul = p1.ring.monomial_mul
for k, v in p2.items():
ka = monomial_mul(k, m)
coeff = get(ka, zero) + v*c
if coeff:
p1[ka] = coeff
else:
del p1[ka]
return p1
def degree(f, x=None):
"""
The leading degree in ``x`` or the main variable.
Note that the degree of 0 is negative infinity (the SymPy object -oo).
"""
i = f.ring.index(x)
if not f:
return -oo
elif i < 0:
return 0
else:
return max([ monom[i] for monom in f.itermonoms() ])
def degrees(f):
"""
A tuple containing leading degrees in all variables.
Note that the degree of 0 is negative infinity (the SymPy object -oo)
"""
if not f:
return (-oo,)*f.ring.ngens
else:
return tuple(map(max, list(zip(*f.itermonoms()))))
def tail_degree(f, x=None):
"""
The tail degree in ``x`` or the main variable.
Note that the degree of 0 is negative infinity (the SymPy object -oo)
"""
i = f.ring.index(x)
if not f:
return -oo
elif i < 0:
return 0
else:
return min([ monom[i] for monom in f.itermonoms() ])
def tail_degrees(f):
"""
A tuple containing tail degrees in all variables.
Note that the degree of 0 is negative infinity (the SymPy object -oo)
"""
if not f:
return (-oo,)*f.ring.ngens
else:
return tuple(map(min, list(zip(*f.itermonoms()))))
def leading_expv(self):
"""Leading monomial tuple according to the monomial ordering.
Examples
========
>>> from sympy.polys.rings import ring
>>> from sympy.polys.domains import ZZ
>>> _, x, y, z = ring('x, y, z', ZZ)
>>> p = x**4 + x**3*y + x**2*z**2 + z**7
>>> p.leading_expv()
(4, 0, 0)
"""
if self:
return self.ring.leading_expv(self)
else:
return None
def _get_coeff(self, expv):
return self.get(expv, self.ring.domain.zero)
def coeff(self, element):
"""
Returns the coefficient that stands next to the given monomial.
Parameters
==========
element : PolyElement (with ``is_monomial = True``) or 1
Examples
========
>>> from sympy.polys.rings import ring
>>> from sympy.polys.domains import ZZ
>>> _, x, y, z = ring("x,y,z", ZZ)
>>> f = 3*x**2*y - x*y*z + 7*z**3 + 23
>>> f.coeff(x**2*y)
3
>>> f.coeff(x*y)
0
>>> f.coeff(1)
23
"""
if element == 1:
return self._get_coeff(self.ring.zero_monom)
elif isinstance(element, self.ring.dtype):
terms = list(element.iterterms())
if len(terms) == 1:
monom, coeff = terms[0]
if coeff == self.ring.domain.one:
return self._get_coeff(monom)
raise ValueError("expected a monomial, got %s" % element)
def const(self):
"""Returns the constant coeffcient. """
return self._get_coeff(self.ring.zero_monom)
@property
def LC(self):
return self._get_coeff(self.leading_expv())
@property
def LM(self):
expv = self.leading_expv()
if expv is None:
return self.ring.zero_monom
else:
return expv
def leading_monom(self):
"""
Leading monomial as a polynomial element.
Examples
========
>>> from sympy.polys.rings import ring
>>> from sympy.polys.domains import ZZ
>>> _, x, y = ring('x, y', ZZ)
>>> (3*x*y + y**2).leading_monom()
x*y
"""
p = self.ring.zero
expv = self.leading_expv()
if expv:
p[expv] = self.ring.domain.one
return p
@property
def LT(self):
expv = self.leading_expv()
if expv is None:
return (self.ring.zero_monom, self.ring.domain.zero)
else:
return (expv, self._get_coeff(expv))
def leading_term(self):
"""Leading term as a polynomial element.
Examples
========
>>> from sympy.polys.rings import ring
>>> from sympy.polys.domains import ZZ
>>> _, x, y = ring('x, y', ZZ)
>>> (3*x*y + y**2).leading_term()
3*x*y
"""
p = self.ring.zero
expv = self.leading_expv()
if expv is not None:
p[expv] = self[expv]
return p
def _sorted(self, seq, order):
if order is None:
order = self.ring.order
else:
order = OrderOpt.preprocess(order)
if order is lex:
return sorted(seq, key=lambda monom: monom[0], reverse=True)
else:
return sorted(seq, key=lambda monom: order(monom[0]), reverse=True)
def coeffs(self, order=None):
"""Ordered list of polynomial coefficients.
Parameters
==========
order : :class:`~.MonomialOrder` or coercible, optional
Examples
========
>>> from sympy.polys.rings import ring
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.orderings import lex, grlex
>>> _, x, y = ring("x, y", ZZ, lex)
>>> f = x*y**7 + 2*x**2*y**3
>>> f.coeffs()
[2, 1]
>>> f.coeffs(grlex)
[1, 2]
"""
return [ coeff for _, coeff in self.terms(order) ]
def monoms(self, order=None):
"""Ordered list of polynomial monomials.
Parameters
==========
order : :class:`~.MonomialOrder` or coercible, optional
Examples
========
>>> from sympy.polys.rings import ring
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.orderings import lex, grlex
>>> _, x, y = ring("x, y", ZZ, lex)
>>> f = x*y**7 + 2*x**2*y**3
>>> f.monoms()
[(2, 3), (1, 7)]
>>> f.monoms(grlex)
[(1, 7), (2, 3)]
"""
return [ monom for monom, _ in self.terms(order) ]
def terms(self, order=None):
"""Ordered list of polynomial terms.
Parameters
==========
order : :class:`~.MonomialOrder` or coercible, optional
Examples
========
>>> from sympy.polys.rings import ring
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.orderings import lex, grlex
>>> _, x, y = ring("x, y", ZZ, lex)
>>> f = x*y**7 + 2*x**2*y**3
>>> f.terms()
[((2, 3), 2), ((1, 7), 1)]
>>> f.terms(grlex)
[((1, 7), 1), ((2, 3), 2)]
"""
return self._sorted(list(self.items()), order)
def itercoeffs(self):
"""Iterator over coefficients of a polynomial. """
return iter(self.values())
def itermonoms(self):
"""Iterator over monomials of a polynomial. """
return iter(self.keys())
def iterterms(self):
"""Iterator over terms of a polynomial. """
return iter(self.items())
def listcoeffs(self):
"""Unordered list of polynomial coefficients. """
return list(self.values())
def listmonoms(self):
"""Unordered list of polynomial monomials. """
return list(self.keys())
def listterms(self):
"""Unordered list of polynomial terms. """
return list(self.items())
def imul_num(p, c):
"""multiply inplace the polynomial p by an element in the
coefficient ring, provided p is not one of the generators;
else multiply not inplace
Examples
========
>>> from sympy.polys.rings import ring
>>> from sympy.polys.domains import ZZ
>>> _, x, y = ring('x, y', ZZ)
>>> p = x + y**2
>>> p1 = p.imul_num(3)
>>> p1
3*x + 3*y**2
>>> p1 is p
True
>>> p = x
>>> p1 = p.imul_num(3)
>>> p1
3*x
>>> p1 is p
False
"""
if p in p.ring._gens_set:
return p*c
if not c:
p.clear()
return
for exp in p:
p[exp] *= c
return p
def content(f):
"""Returns GCD of polynomial's coefficients. """
domain = f.ring.domain
cont = domain.zero
gcd = domain.gcd
for coeff in f.itercoeffs():
cont = gcd(cont, coeff)
return cont
def primitive(f):
"""Returns content and a primitive polynomial. """
cont = f.content()
return cont, f.quo_ground(cont)
def monic(f):
"""Divides all coefficients by the leading coefficient. """
if not f:
return f
else:
return f.quo_ground(f.LC)
def mul_ground(f, x):
if not x:
return f.ring.zero
terms = [ (monom, coeff*x) for monom, coeff in f.iterterms() ]
return f.new(terms)
def mul_monom(f, monom):
monomial_mul = f.ring.monomial_mul
terms = [ (monomial_mul(f_monom, monom), f_coeff) for f_monom, f_coeff in f.items() ]
return f.new(terms)
def mul_term(f, term):
monom, coeff = term
if not f or not coeff:
return f.ring.zero
elif monom == f.ring.zero_monom:
return f.mul_ground(coeff)
monomial_mul = f.ring.monomial_mul
terms = [ (monomial_mul(f_monom, monom), f_coeff*coeff) for f_monom, f_coeff in f.items() ]
return f.new(terms)
def quo_ground(f, x):
domain = f.ring.domain
if not x:
raise ZeroDivisionError('polynomial division')
if not f or x == domain.one:
return f
if domain.is_Field:
quo = domain.quo
terms = [ (monom, quo(coeff, x)) for monom, coeff in f.iterterms() ]
else:
terms = [ (monom, coeff // x) for monom, coeff in f.iterterms() if not (coeff % x) ]
return f.new(terms)
def quo_term(f, term):
monom, coeff = term
if not coeff:
raise ZeroDivisionError("polynomial division")
elif not f:
return f.ring.zero
elif monom == f.ring.zero_monom:
return f.quo_ground(coeff)
term_div = f._term_div()
terms = [ term_div(t, term) for t in f.iterterms() ]
return f.new([ t for t in terms if t is not None ])
def trunc_ground(f, p):
if f.ring.domain.is_ZZ:
terms = []
for monom, coeff in f.iterterms():
coeff = coeff % p
if coeff > p // 2:
coeff = coeff - p
terms.append((monom, coeff))
else:
terms = [ (monom, coeff % p) for monom, coeff in f.iterterms() ]
poly = f.new(terms)
poly.strip_zero()
return poly
rem_ground = trunc_ground
def extract_ground(self, g):
f = self
fc = f.content()
gc = g.content()
gcd = f.ring.domain.gcd(fc, gc)
f = f.quo_ground(gcd)
g = g.quo_ground(gcd)
return gcd, f, g
def _norm(f, norm_func):
if not f:
return f.ring.domain.zero
else:
ground_abs = f.ring.domain.abs
return norm_func([ ground_abs(coeff) for coeff in f.itercoeffs() ])
def max_norm(f):
return f._norm(max)
def l1_norm(f):
return f._norm(sum)
def deflate(f, *G):
ring = f.ring
polys = [f] + list(G)
J = [0]*ring.ngens
for p in polys:
for monom in p.itermonoms():
for i, m in enumerate(monom):
J[i] = igcd(J[i], m)
for i, b in enumerate(J):
if not b:
J[i] = 1
J = tuple(J)
if all(b == 1 for b in J):
return J, polys
H = []
for p in polys:
h = ring.zero
for I, coeff in p.iterterms():
N = [ i // j for i, j in zip(I, J) ]
h[tuple(N)] = coeff
H.append(h)
return J, H
def inflate(f, J):
poly = f.ring.zero
for I, coeff in f.iterterms():
N = [ i*j for i, j in zip(I, J) ]
poly[tuple(N)] = coeff
return poly
def lcm(self, g):
f = self
domain = f.ring.domain
if not domain.is_Field:
fc, f = f.primitive()
gc, g = g.primitive()
c = domain.lcm(fc, gc)
h = (f*g).quo(f.gcd(g))
if not domain.is_Field:
return h.mul_ground(c)
else:
return h.monic()
def gcd(f, g):
return f.cofactors(g)[0]
def cofactors(f, g):
if not f and not g:
zero = f.ring.zero
return zero, zero, zero
elif not f:
h, cff, cfg = f._gcd_zero(g)
return h, cff, cfg
elif not g:
h, cfg, cff = g._gcd_zero(f)
return h, cff, cfg
elif len(f) == 1:
h, cff, cfg = f._gcd_monom(g)
return h, cff, cfg
elif len(g) == 1:
h, cfg, cff = g._gcd_monom(f)
return h, cff, cfg
J, (f, g) = f.deflate(g)
h, cff, cfg = f._gcd(g)
return (h.inflate(J), cff.inflate(J), cfg.inflate(J))
def _gcd_zero(f, g):
one, zero = f.ring.one, f.ring.zero
if g.is_nonnegative:
return g, zero, one
else:
return -g, zero, -one
def _gcd_monom(f, g):
ring = f.ring
ground_gcd = ring.domain.gcd
ground_quo = ring.domain.quo
monomial_gcd = ring.monomial_gcd
monomial_ldiv = ring.monomial_ldiv
mf, cf = list(f.iterterms())[0]
_mgcd, _cgcd = mf, cf
for mg, cg in g.iterterms():
_mgcd = monomial_gcd(_mgcd, mg)
_cgcd = ground_gcd(_cgcd, cg)
h = f.new([(_mgcd, _cgcd)])
cff = f.new([(monomial_ldiv(mf, _mgcd), ground_quo(cf, _cgcd))])
cfg = f.new([(monomial_ldiv(mg, _mgcd), ground_quo(cg, _cgcd)) for mg, cg in g.iterterms()])
return h, cff, cfg
def _gcd(f, g):
ring = f.ring
if ring.domain.is_QQ:
return f._gcd_QQ(g)
elif ring.domain.is_ZZ:
return f._gcd_ZZ(g)
else: # TODO: don't use dense representation (port PRS algorithms)
return ring.dmp_inner_gcd(f, g)
def _gcd_ZZ(f, g):
return heugcd(f, g)
def _gcd_QQ(self, g):
f = self
ring = f.ring
new_ring = ring.clone(domain=ring.domain.get_ring())
cf, f = f.clear_denoms()
cg, g = g.clear_denoms()
f = f.set_ring(new_ring)
g = g.set_ring(new_ring)
h, cff, cfg = f._gcd_ZZ(g)
h = h.set_ring(ring)
c, h = h.LC, h.monic()
cff = cff.set_ring(ring).mul_ground(ring.domain.quo(c, cf))
cfg = cfg.set_ring(ring).mul_ground(ring.domain.quo(c, cg))
return h, cff, cfg
def cancel(self, g):
"""
Cancel common factors in a rational function ``f/g``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> (2*x**2 - 2).cancel(x**2 - 2*x + 1)
(2*x + 2, x - 1)
"""
f = self
ring = f.ring
if not f:
return f, ring.one
domain = ring.domain
if not (domain.is_Field and domain.has_assoc_Ring):
_, p, q = f.cofactors(g)
else:
new_ring = ring.clone(domain=domain.get_ring())
cq, f = f.clear_denoms()
cp, g = g.clear_denoms()
f = f.set_ring(new_ring)
g = g.set_ring(new_ring)
_, p, q = f.cofactors(g)
_, cp, cq = new_ring.domain.cofactors(cp, cq)
p = p.set_ring(ring)
q = q.set_ring(ring)
p = p.mul_ground(cp)
q = q.mul_ground(cq)
# Make canonical with respect to sign or quadrant in the case of ZZ_I
# or QQ_I. This ensures that the LC of the denominator is canonical by
# multiplying top and bottom by a unit of the ring.
u = q.canonical_unit()
if u == domain.one:
p, q = p, q
elif u == -domain.one:
p, q = -p, -q
else:
p = p.mul_ground(u)
q = q.mul_ground(u)
return p, q
def canonical_unit(f):
domain = f.ring.domain
return domain.canonical_unit(f.LC)
def diff(f, x):
"""Computes partial derivative in ``x``.
Examples
========
>>> from sympy.polys.rings import ring
>>> from sympy.polys.domains import ZZ
>>> _, x, y = ring("x,y", ZZ)
>>> p = x + x**2*y**3
>>> p.diff(x)
2*x*y**3 + 1
"""
ring = f.ring
i = ring.index(x)
m = ring.monomial_basis(i)
g = ring.zero
for expv, coeff in f.iterterms():
if expv[i]:
e = ring.monomial_ldiv(expv, m)
g[e] = ring.domain_new(coeff*expv[i])
return g
def __call__(f, *values):
if 0 < len(values) <= f.ring.ngens:
return f.evaluate(list(zip(f.ring.gens, values)))
else:
raise ValueError("expected at least 1 and at most %s values, got %s" % (f.ring.ngens, len(values)))
def evaluate(self, x, a=None):
f = self
if isinstance(x, list) and a is None:
(X, a), x = x[0], x[1:]
f = f.evaluate(X, a)
if not x:
return f
else:
x = [ (Y.drop(X), a) for (Y, a) in x ]
return f.evaluate(x)
ring = f.ring
i = ring.index(x)
a = ring.domain.convert(a)
if ring.ngens == 1:
result = ring.domain.zero
for (n,), coeff in f.iterterms():
result += coeff*a**n
return result
else:
poly = ring.drop(x).zero
for monom, coeff in f.iterterms():
n, monom = monom[i], monom[:i] + monom[i+1:]
coeff = coeff*a**n
if monom in poly:
coeff = coeff + poly[monom]
if coeff:
poly[monom] = coeff
else:
del poly[monom]
else:
if coeff:
poly[monom] = coeff
return poly
def subs(self, x, a=None):
f = self
if isinstance(x, list) and a is None:
for X, a in x:
f = f.subs(X, a)
return f
ring = f.ring
i = ring.index(x)
a = ring.domain.convert(a)
if ring.ngens == 1:
result = ring.domain.zero
for (n,), coeff in f.iterterms():
result += coeff*a**n
return ring.ground_new(result)
else:
poly = ring.zero
for monom, coeff in f.iterterms():
n, monom = monom[i], monom[:i] + (0,) + monom[i+1:]
coeff = coeff*a**n
if monom in poly:
coeff = coeff + poly[monom]
if coeff:
poly[monom] = coeff
else:
del poly[monom]
else:
if coeff:
poly[monom] = coeff
return poly
def compose(f, x, a=None):
ring = f.ring
poly = ring.zero
gens_map = dict(list(zip(ring.gens, list(range(ring.ngens)))))
if a is not None:
replacements = [(x, a)]
else:
if isinstance(x, list):
replacements = list(x)
elif isinstance(x, dict):
replacements = sorted(list(x.items()), key=lambda k: gens_map[k[0]])
else:
raise ValueError("expected a generator, value pair a sequence of such pairs")
for k, (x, g) in enumerate(replacements):
replacements[k] = (gens_map[x], ring.ring_new(g))
for monom, coeff in f.iterterms():
monom = list(monom)
subpoly = ring.one
for i, g in replacements:
n, monom[i] = monom[i], 0
if n:
subpoly *= g**n
subpoly = subpoly.mul_term((tuple(monom), coeff))
poly += subpoly
return poly
# TODO: following methods should point to polynomial
# representation independent algorithm implementations.
def pdiv(f, g):
return f.ring.dmp_pdiv(f, g)
def prem(f, g):
return f.ring.dmp_prem(f, g)
def pquo(f, g):
return f.ring.dmp_quo(f, g)
def pexquo(f, g):
return f.ring.dmp_exquo(f, g)
def half_gcdex(f, g):
return f.ring.dmp_half_gcdex(f, g)
def gcdex(f, g):
return f.ring.dmp_gcdex(f, g)
def subresultants(f, g):
return f.ring.dmp_subresultants(f, g)
def resultant(f, g):
return f.ring.dmp_resultant(f, g)
def discriminant(f):
return f.ring.dmp_discriminant(f)
def decompose(f):
if f.ring.is_univariate:
return f.ring.dup_decompose(f)
else:
raise MultivariatePolynomialError("polynomial decomposition")
def shift(f, a):
if f.ring.is_univariate:
return f.ring.dup_shift(f, a)
else:
raise MultivariatePolynomialError("polynomial shift")
def sturm(f):
if f.ring.is_univariate:
return f.ring.dup_sturm(f)
else:
raise MultivariatePolynomialError("sturm sequence")
def gff_list(f):
return f.ring.dmp_gff_list(f)
def sqf_norm(f):
return f.ring.dmp_sqf_norm(f)
def sqf_part(f):
return f.ring.dmp_sqf_part(f)
def sqf_list(f, all=False):
return f.ring.dmp_sqf_list(f, all=all)
def factor_list(f):
return f.ring.dmp_factor_list(f)
|
a913e1f14ed39320d24446a01a19da552348947ad008a6aacf3fbcb5d8bce453 | """Tools and arithmetics for monomials of distributed polynomials. """
from itertools import combinations_with_replacement, product
from textwrap import dedent
from sympy.core import Mul, S, Tuple, sympify
from sympy.core.compatibility import iterable
from sympy.polys.polyerrors import ExactQuotientFailed
from sympy.polys.polyutils import PicklableWithSlots, dict_from_expr
from sympy.utilities import public
from sympy.core.compatibility import is_sequence
@public
def itermonomials(variables, max_degrees, min_degrees=None):
r"""
``max_degrees`` and ``min_degrees`` are either both integers or both lists.
Unless otherwise specified, ``min_degrees`` is either ``0`` or
``[0, ..., 0]``.
A generator of all monomials ``monom`` is returned, such that
either
``min_degree <= total_degree(monom) <= max_degree``,
or
``min_degrees[i] <= degree_list(monom)[i] <= max_degrees[i]``,
for all ``i``.
Case I. ``max_degrees`` and ``min_degrees`` are both integers
=============================================================
Given a set of variables $V$ and a min_degree $N$ and a max_degree $M$
generate a set of monomials of degree less than or equal to $N$ and greater
than or equal to $M$. The total number of monomials in commutative
variables is huge and is given by the following formula if $M = 0$:
.. math::
\frac{(\#V + N)!}{\#V! N!}
For example if we would like to generate a dense polynomial of
a total degree $N = 50$ and $M = 0$, which is the worst case, in 5
variables, assuming that exponents and all of coefficients are 32-bit long
and stored in an array we would need almost 80 GiB of memory! Fortunately
most polynomials, that we will encounter, are sparse.
Consider monomials in commutative variables $x$ and $y$
and non-commutative variables $a$ and $b$::
>>> from sympy import symbols
>>> from sympy.polys.monomials import itermonomials
>>> from sympy.polys.orderings import monomial_key
>>> from sympy.abc import x, y
>>> sorted(itermonomials([x, y], 2), key=monomial_key('grlex', [y, x]))
[1, x, y, x**2, x*y, y**2]
>>> sorted(itermonomials([x, y], 3), key=monomial_key('grlex', [y, x]))
[1, x, y, x**2, x*y, y**2, x**3, x**2*y, x*y**2, y**3]
>>> a, b = symbols('a, b', commutative=False)
>>> set(itermonomials([a, b, x], 2))
{1, a, a**2, b, b**2, x, x**2, a*b, b*a, x*a, x*b}
>>> sorted(itermonomials([x, y], 2, 1), key=monomial_key('grlex', [y, x]))
[x, y, x**2, x*y, y**2]
Case II. ``max_degrees`` and ``min_degrees`` are both lists
===========================================================
If ``max_degrees = [d_1, ..., d_n]`` and
``min_degrees = [e_1, ..., e_n]``, the number of monomials generated
is:
.. math::
(d_1 - e_1 + 1) (d_2 - e_2 + 1) \cdots (d_n - e_n + 1)
Let us generate all monomials ``monom`` in variables $x$ and $y$
such that ``[1, 2][i] <= degree_list(monom)[i] <= [2, 4][i]``,
``i = 0, 1`` ::
>>> from sympy import symbols
>>> from sympy.polys.monomials import itermonomials
>>> from sympy.polys.orderings import monomial_key
>>> from sympy.abc import x, y
>>> sorted(itermonomials([x, y], [2, 4], [1, 2]), reverse=True, key=monomial_key('lex', [x, y]))
[x**2*y**4, x**2*y**3, x**2*y**2, x*y**4, x*y**3, x*y**2]
"""
n = len(variables)
if is_sequence(max_degrees):
if len(max_degrees) != n:
raise ValueError('Argument sizes do not match')
if min_degrees is None:
min_degrees = [0]*n
elif not is_sequence(min_degrees):
raise ValueError('min_degrees is not a list')
else:
if len(min_degrees) != n:
raise ValueError('Argument sizes do not match')
if any(i < 0 for i in min_degrees):
raise ValueError("min_degrees can't contain negative numbers")
total_degree = False
else:
max_degree = max_degrees
if max_degree < 0:
raise ValueError("max_degrees can't be negative")
if min_degrees is None:
min_degree = 0
else:
if min_degrees < 0:
raise ValueError("min_degrees can't be negative")
min_degree = min_degrees
total_degree = True
if total_degree:
if min_degree > max_degree:
return
if not variables or max_degree == 0:
yield S.One
return
# Force to list in case of passed tuple or other incompatible collection
variables = list(variables) + [S.One]
if all(variable.is_commutative for variable in variables):
monomials_list_comm = []
for item in combinations_with_replacement(variables, max_degree):
powers = dict()
for variable in variables:
powers[variable] = 0
for variable in item:
if variable != 1:
powers[variable] += 1
if max(powers.values()) >= min_degree:
monomials_list_comm.append(Mul(*item))
yield from set(monomials_list_comm)
else:
monomials_list_non_comm = []
for item in product(variables, repeat=max_degree):
powers = dict()
for variable in variables:
powers[variable] = 0
for variable in item:
if variable != 1:
powers[variable] += 1
if max(powers.values()) >= min_degree:
monomials_list_non_comm.append(Mul(*item))
yield from set(monomials_list_non_comm)
else:
if any(min_degrees[i] > max_degrees[i] for i in range(n)):
raise ValueError('min_degrees[i] must be <= max_degrees[i] for all i')
power_lists = []
for var, min_d, max_d in zip(variables, min_degrees, max_degrees):
power_lists.append([var**i for i in range(min_d, max_d + 1)])
for powers in product(*power_lists):
yield Mul(*powers)
def monomial_count(V, N):
r"""
Computes the number of monomials.
The number of monomials is given by the following formula:
.. math::
\frac{(\#V + N)!}{\#V! N!}
where `N` is a total degree and `V` is a set of variables.
Examples
========
>>> from sympy.polys.monomials import itermonomials, monomial_count
>>> from sympy.polys.orderings import monomial_key
>>> from sympy.abc import x, y
>>> monomial_count(2, 2)
6
>>> M = list(itermonomials([x, y], 2))
>>> sorted(M, key=monomial_key('grlex', [y, x]))
[1, x, y, x**2, x*y, y**2]
>>> len(M)
6
"""
from sympy import factorial
return factorial(V + N) / factorial(V) / factorial(N)
def monomial_mul(A, B):
"""
Multiplication of tuples representing monomials.
Examples
========
Lets multiply `x**3*y**4*z` with `x*y**2`::
>>> from sympy.polys.monomials import monomial_mul
>>> monomial_mul((3, 4, 1), (1, 2, 0))
(4, 6, 1)
which gives `x**4*y**5*z`.
"""
return tuple([ a + b for a, b in zip(A, B) ])
def monomial_div(A, B):
"""
Division of tuples representing monomials.
Examples
========
Lets divide `x**3*y**4*z` by `x*y**2`::
>>> from sympy.polys.monomials import monomial_div
>>> monomial_div((3, 4, 1), (1, 2, 0))
(2, 2, 1)
which gives `x**2*y**2*z`. However::
>>> monomial_div((3, 4, 1), (1, 2, 2)) is None
True
`x*y**2*z**2` does not divide `x**3*y**4*z`.
"""
C = monomial_ldiv(A, B)
if all(c >= 0 for c in C):
return tuple(C)
else:
return None
def monomial_ldiv(A, B):
"""
Division of tuples representing monomials.
Examples
========
Lets divide `x**3*y**4*z` by `x*y**2`::
>>> from sympy.polys.monomials import monomial_ldiv
>>> monomial_ldiv((3, 4, 1), (1, 2, 0))
(2, 2, 1)
which gives `x**2*y**2*z`.
>>> monomial_ldiv((3, 4, 1), (1, 2, 2))
(2, 2, -1)
which gives `x**2*y**2*z**-1`.
"""
return tuple([ a - b for a, b in zip(A, B) ])
def monomial_pow(A, n):
"""Return the n-th pow of the monomial. """
return tuple([ a*n for a in A ])
def monomial_gcd(A, B):
"""
Greatest common divisor of tuples representing monomials.
Examples
========
Lets compute GCD of `x*y**4*z` and `x**3*y**2`::
>>> from sympy.polys.monomials import monomial_gcd
>>> monomial_gcd((1, 4, 1), (3, 2, 0))
(1, 2, 0)
which gives `x*y**2`.
"""
return tuple([ min(a, b) for a, b in zip(A, B) ])
def monomial_lcm(A, B):
"""
Least common multiple of tuples representing monomials.
Examples
========
Lets compute LCM of `x*y**4*z` and `x**3*y**2`::
>>> from sympy.polys.monomials import monomial_lcm
>>> monomial_lcm((1, 4, 1), (3, 2, 0))
(3, 4, 1)
which gives `x**3*y**4*z`.
"""
return tuple([ max(a, b) for a, b in zip(A, B) ])
def monomial_divides(A, B):
"""
Does there exist a monomial X such that XA == B?
Examples
========
>>> from sympy.polys.monomials import monomial_divides
>>> monomial_divides((1, 2), (3, 4))
True
>>> monomial_divides((1, 2), (0, 2))
False
"""
return all(a <= b for a, b in zip(A, B))
def monomial_max(*monoms):
"""
Returns maximal degree for each variable in a set of monomials.
Examples
========
Consider monomials `x**3*y**4*z**5`, `y**5*z` and `x**6*y**3*z**9`.
We wish to find out what is the maximal degree for each of `x`, `y`
and `z` variables::
>>> from sympy.polys.monomials import monomial_max
>>> monomial_max((3,4,5), (0,5,1), (6,3,9))
(6, 5, 9)
"""
M = list(monoms[0])
for N in monoms[1:]:
for i, n in enumerate(N):
M[i] = max(M[i], n)
return tuple(M)
def monomial_min(*monoms):
"""
Returns minimal degree for each variable in a set of monomials.
Examples
========
Consider monomials `x**3*y**4*z**5`, `y**5*z` and `x**6*y**3*z**9`.
We wish to find out what is the minimal degree for each of `x`, `y`
and `z` variables::
>>> from sympy.polys.monomials import monomial_min
>>> monomial_min((3,4,5), (0,5,1), (6,3,9))
(0, 3, 1)
"""
M = list(monoms[0])
for N in monoms[1:]:
for i, n in enumerate(N):
M[i] = min(M[i], n)
return tuple(M)
def monomial_deg(M):
"""
Returns the total degree of a monomial.
Examples
========
The total degree of `xy^2` is 3:
>>> from sympy.polys.monomials import monomial_deg
>>> monomial_deg((1, 2))
3
"""
return sum(M)
def term_div(a, b, domain):
"""Division of two terms in over a ring/field. """
a_lm, a_lc = a
b_lm, b_lc = b
monom = monomial_div(a_lm, b_lm)
if domain.is_Field:
if monom is not None:
return monom, domain.quo(a_lc, b_lc)
else:
return None
else:
if not (monom is None or a_lc % b_lc):
return monom, domain.quo(a_lc, b_lc)
else:
return None
class MonomialOps:
"""Code generator of fast monomial arithmetic functions. """
def __init__(self, ngens):
self.ngens = ngens
def _build(self, code, name):
ns = {}
exec(code, ns)
return ns[name]
def _vars(self, name):
return [ "%s%s" % (name, i) for i in range(self.ngens) ]
def mul(self):
name = "monomial_mul"
template = dedent("""\
def %(name)s(A, B):
(%(A)s,) = A
(%(B)s,) = B
return (%(AB)s,)
""")
A = self._vars("a")
B = self._vars("b")
AB = [ "%s + %s" % (a, b) for a, b in zip(A, B) ]
code = template % dict(name=name, A=", ".join(A), B=", ".join(B), AB=", ".join(AB))
return self._build(code, name)
def pow(self):
name = "monomial_pow"
template = dedent("""\
def %(name)s(A, k):
(%(A)s,) = A
return (%(Ak)s,)
""")
A = self._vars("a")
Ak = [ "%s*k" % a for a in A ]
code = template % dict(name=name, A=", ".join(A), Ak=", ".join(Ak))
return self._build(code, name)
def mulpow(self):
name = "monomial_mulpow"
template = dedent("""\
def %(name)s(A, B, k):
(%(A)s,) = A
(%(B)s,) = B
return (%(ABk)s,)
""")
A = self._vars("a")
B = self._vars("b")
ABk = [ "%s + %s*k" % (a, b) for a, b in zip(A, B) ]
code = template % dict(name=name, A=", ".join(A), B=", ".join(B), ABk=", ".join(ABk))
return self._build(code, name)
def ldiv(self):
name = "monomial_ldiv"
template = dedent("""\
def %(name)s(A, B):
(%(A)s,) = A
(%(B)s,) = B
return (%(AB)s,)
""")
A = self._vars("a")
B = self._vars("b")
AB = [ "%s - %s" % (a, b) for a, b in zip(A, B) ]
code = template % dict(name=name, A=", ".join(A), B=", ".join(B), AB=", ".join(AB))
return self._build(code, name)
def div(self):
name = "monomial_div"
template = dedent("""\
def %(name)s(A, B):
(%(A)s,) = A
(%(B)s,) = B
%(RAB)s
return (%(R)s,)
""")
A = self._vars("a")
B = self._vars("b")
RAB = [ "r%(i)s = a%(i)s - b%(i)s\n if r%(i)s < 0: return None" % dict(i=i) for i in range(self.ngens) ]
R = self._vars("r")
code = template % dict(name=name, A=", ".join(A), B=", ".join(B), RAB="\n ".join(RAB), R=", ".join(R))
return self._build(code, name)
def lcm(self):
name = "monomial_lcm"
template = dedent("""\
def %(name)s(A, B):
(%(A)s,) = A
(%(B)s,) = B
return (%(AB)s,)
""")
A = self._vars("a")
B = self._vars("b")
AB = [ "%s if %s >= %s else %s" % (a, a, b, b) for a, b in zip(A, B) ]
code = template % dict(name=name, A=", ".join(A), B=", ".join(B), AB=", ".join(AB))
return self._build(code, name)
def gcd(self):
name = "monomial_gcd"
template = dedent("""\
def %(name)s(A, B):
(%(A)s,) = A
(%(B)s,) = B
return (%(AB)s,)
""")
A = self._vars("a")
B = self._vars("b")
AB = [ "%s if %s <= %s else %s" % (a, a, b, b) for a, b in zip(A, B) ]
code = template % dict(name=name, A=", ".join(A), B=", ".join(B), AB=", ".join(AB))
return self._build(code, name)
@public
class Monomial(PicklableWithSlots):
"""Class representing a monomial, i.e. a product of powers. """
__slots__ = ('exponents', 'gens')
def __init__(self, monom, gens=None):
if not iterable(monom):
rep, gens = dict_from_expr(sympify(monom), gens=gens)
if len(rep) == 1 and list(rep.values())[0] == 1:
monom = list(rep.keys())[0]
else:
raise ValueError("Expected a monomial got {}".format(monom))
self.exponents = tuple(map(int, monom))
self.gens = gens
def rebuild(self, exponents, gens=None):
return self.__class__(exponents, gens or self.gens)
def __len__(self):
return len(self.exponents)
def __iter__(self):
return iter(self.exponents)
def __getitem__(self, item):
return self.exponents[item]
def __hash__(self):
return hash((self.__class__.__name__, self.exponents, self.gens))
def __str__(self):
if self.gens:
return "*".join([ "%s**%s" % (gen, exp) for gen, exp in zip(self.gens, self.exponents) ])
else:
return "%s(%s)" % (self.__class__.__name__, self.exponents)
def as_expr(self, *gens):
"""Convert a monomial instance to a SymPy expression. """
gens = gens or self.gens
if not gens:
raise ValueError(
"can't convert %s to an expression without generators" % self)
return Mul(*[ gen**exp for gen, exp in zip(gens, self.exponents) ])
def __eq__(self, other):
if isinstance(other, Monomial):
exponents = other.exponents
elif isinstance(other, (tuple, Tuple)):
exponents = other
else:
return False
return self.exponents == exponents
def __ne__(self, other):
return not self == other
def __mul__(self, other):
if isinstance(other, Monomial):
exponents = other.exponents
elif isinstance(other, (tuple, Tuple)):
exponents = other
else:
raise NotImplementedError
return self.rebuild(monomial_mul(self.exponents, exponents))
def __truediv__(self, other):
if isinstance(other, Monomial):
exponents = other.exponents
elif isinstance(other, (tuple, Tuple)):
exponents = other
else:
raise NotImplementedError
result = monomial_div(self.exponents, exponents)
if result is not None:
return self.rebuild(result)
else:
raise ExactQuotientFailed(self, Monomial(other))
__floordiv__ = __truediv__
def __pow__(self, other):
n = int(other)
if not n:
return self.rebuild([0]*len(self))
elif n > 0:
exponents = self.exponents
for i in range(1, n):
exponents = monomial_mul(exponents, self.exponents)
return self.rebuild(exponents)
else:
raise ValueError("a non-negative integer expected, got %s" % other)
def gcd(self, other):
"""Greatest common divisor of monomials. """
if isinstance(other, Monomial):
exponents = other.exponents
elif isinstance(other, (tuple, Tuple)):
exponents = other
else:
raise TypeError(
"an instance of Monomial class expected, got %s" % other)
return self.rebuild(monomial_gcd(self.exponents, exponents))
def lcm(self, other):
"""Least common multiple of monomials. """
if isinstance(other, Monomial):
exponents = other.exponents
elif isinstance(other, (tuple, Tuple)):
exponents = other
else:
raise TypeError(
"an instance of Monomial class expected, got %s" % other)
return self.rebuild(monomial_lcm(self.exponents, exponents))
|
c27d176ac4239e55983e059ebfa0aa184b5b65ce7ad768640a3ec2c89ea0c4f1 | """Sparse rational function fields. """
from typing import Any, Dict
from functools import reduce
from operator import add, mul, lt, le, gt, ge
from sympy.core.compatibility import is_sequence
from sympy.core.expr import Expr
from sympy.core.mod import Mod
from sympy.core.numbers import Exp1
from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.core.sympify import CantSympify, sympify
from sympy.functions.elementary.exponential import ExpBase
from sympy.polys.domains.domainelement import DomainElement
from sympy.polys.domains.fractionfield import FractionField
from sympy.polys.domains.polynomialring import PolynomialRing
from sympy.polys.constructor import construct_domain
from sympy.polys.orderings import lex
from sympy.polys.polyerrors import CoercionFailed
from sympy.polys.polyoptions import build_options
from sympy.polys.polyutils import _parallel_dict_from_expr
from sympy.polys.rings import PolyElement
from sympy.printing.defaults import DefaultPrinting
from sympy.utilities import public
from sympy.utilities.magic import pollute
@public
def field(symbols, domain, order=lex):
"""Construct new rational function field returning (field, x1, ..., xn). """
_field = FracField(symbols, domain, order)
return (_field,) + _field.gens
@public
def xfield(symbols, domain, order=lex):
"""Construct new rational function field returning (field, (x1, ..., xn)). """
_field = FracField(symbols, domain, order)
return (_field, _field.gens)
@public
def vfield(symbols, domain, order=lex):
"""Construct new rational function field and inject generators into global namespace. """
_field = FracField(symbols, domain, order)
pollute([ sym.name for sym in _field.symbols ], _field.gens)
return _field
@public
def sfield(exprs, *symbols, **options):
"""Construct a field deriving generators and domain
from options and input expressions.
Parameters
==========
exprs : py:class:`~.Expr` or sequence of :py:class:`~.Expr` (sympifiable)
symbols : sequence of :py:class:`~.Symbol`/:py:class:`~.Expr`
options : keyword arguments understood by :py:class:`~.Options`
Examples
========
>>> from sympy.core import symbols
>>> from sympy.functions import exp, log
>>> from sympy.polys.fields import sfield
>>> x = symbols("x")
>>> K, f = sfield((x*log(x) + 4*x**2)*exp(1/x + log(x)/3)/x**2)
>>> K
Rational function field in x, exp(1/x), log(x), x**(1/3) over ZZ with lex order
>>> f
(4*x**2*(exp(1/x)) + x*(exp(1/x))*(log(x)))/((x**(1/3))**5)
"""
single = False
if not is_sequence(exprs):
exprs, single = [exprs], True
exprs = list(map(sympify, exprs))
opt = build_options(symbols, options)
numdens = []
for expr in exprs:
numdens.extend(expr.as_numer_denom())
reps, opt = _parallel_dict_from_expr(numdens, opt)
if opt.domain is None:
# NOTE: this is inefficient because construct_domain() automatically
# performs conversion to the target domain. It shouldn't do this.
coeffs = sum([list(rep.values()) for rep in reps], [])
opt.domain, _ = construct_domain(coeffs, opt=opt)
_field = FracField(opt.gens, opt.domain, opt.order)
fracs = []
for i in range(0, len(reps), 2):
fracs.append(_field(tuple(reps[i:i+2])))
if single:
return (_field, fracs[0])
else:
return (_field, fracs)
_field_cache = {} # type: Dict[Any, Any]
class FracField(DefaultPrinting):
"""Multivariate distributed rational function field. """
def __new__(cls, symbols, domain, order=lex):
from sympy.polys.rings import PolyRing
ring = PolyRing(symbols, domain, order)
symbols = ring.symbols
ngens = ring.ngens
domain = ring.domain
order = ring.order
_hash_tuple = (cls.__name__, symbols, ngens, domain, order)
obj = _field_cache.get(_hash_tuple)
if obj is None:
obj = object.__new__(cls)
obj._hash_tuple = _hash_tuple
obj._hash = hash(_hash_tuple)
obj.ring = ring
obj.dtype = type("FracElement", (FracElement,), {"field": obj})
obj.symbols = symbols
obj.ngens = ngens
obj.domain = domain
obj.order = order
obj.zero = obj.dtype(ring.zero)
obj.one = obj.dtype(ring.one)
obj.gens = obj._gens()
for symbol, generator in zip(obj.symbols, obj.gens):
if isinstance(symbol, Symbol):
name = symbol.name
if not hasattr(obj, name):
setattr(obj, name, generator)
_field_cache[_hash_tuple] = obj
return obj
def _gens(self):
"""Return a list of polynomial generators. """
return tuple([ self.dtype(gen) for gen in self.ring.gens ])
def __getnewargs__(self):
return (self.symbols, self.domain, self.order)
def __hash__(self):
return self._hash
def index(self, gen):
if isinstance(gen, self.dtype):
return self.ring.index(gen.to_poly())
else:
raise ValueError("expected a %s, got %s instead" % (self.dtype,gen))
def __eq__(self, other):
return isinstance(other, FracField) and \
(self.symbols, self.ngens, self.domain, self.order) == \
(other.symbols, other.ngens, other.domain, other.order)
def __ne__(self, other):
return not self == other
def raw_new(self, numer, denom=None):
return self.dtype(numer, denom)
def new(self, numer, denom=None):
if denom is None: denom = self.ring.one
numer, denom = numer.cancel(denom)
return self.raw_new(numer, denom)
def domain_new(self, element):
return self.domain.convert(element)
def ground_new(self, element):
try:
return self.new(self.ring.ground_new(element))
except CoercionFailed:
domain = self.domain
if not domain.is_Field and domain.has_assoc_Field:
ring = self.ring
ground_field = domain.get_field()
element = ground_field.convert(element)
numer = ring.ground_new(ground_field.numer(element))
denom = ring.ground_new(ground_field.denom(element))
return self.raw_new(numer, denom)
else:
raise
def field_new(self, element):
if isinstance(element, FracElement):
if self == element.field:
return element
if isinstance(self.domain, FractionField) and \
self.domain.field == element.field:
return self.ground_new(element)
elif isinstance(self.domain, PolynomialRing) and \
self.domain.ring.to_field() == element.field:
return self.ground_new(element)
else:
raise NotImplementedError("conversion")
elif isinstance(element, PolyElement):
denom, numer = element.clear_denoms()
if isinstance(self.domain, PolynomialRing) and \
numer.ring == self.domain.ring:
numer = self.ring.ground_new(numer)
elif isinstance(self.domain, FractionField) and \
numer.ring == self.domain.field.to_ring():
numer = self.ring.ground_new(numer)
else:
numer = numer.set_ring(self.ring)
denom = self.ring.ground_new(denom)
return self.raw_new(numer, denom)
elif isinstance(element, tuple) and len(element) == 2:
numer, denom = list(map(self.ring.ring_new, element))
return self.new(numer, denom)
elif isinstance(element, str):
raise NotImplementedError("parsing")
elif isinstance(element, Expr):
return self.from_expr(element)
else:
return self.ground_new(element)
__call__ = field_new
def _rebuild_expr(self, expr, mapping):
domain = self.domain
powers = tuple((gen, gen.as_base_exp()) for gen in mapping.keys()
if gen.is_Pow or isinstance(gen, ExpBase))
def _rebuild(expr):
generator = mapping.get(expr)
if generator is not None:
return generator
elif expr.is_Add:
return reduce(add, list(map(_rebuild, expr.args)))
elif expr.is_Mul:
return reduce(mul, list(map(_rebuild, expr.args)))
elif expr.is_Pow or isinstance(expr, (ExpBase, Exp1)):
b, e = expr.as_base_exp()
# look for bg**eg whose integer power may be b**e
for gen, (bg, eg) in powers:
if bg == b and Mod(e, eg) == 0:
return mapping.get(gen)**int(e/eg)
if e.is_Integer and e is not S.One:
return _rebuild(b)**int(e)
try:
return domain.convert(expr)
except CoercionFailed:
if not domain.is_Field and domain.has_assoc_Field:
return domain.get_field().convert(expr)
else:
raise
return _rebuild(sympify(expr))
def from_expr(self, expr):
mapping = dict(list(zip(self.symbols, self.gens)))
try:
frac = self._rebuild_expr(expr, mapping)
except CoercionFailed:
raise ValueError("expected an expression convertible to a rational function in %s, got %s" % (self, expr))
else:
return self.field_new(frac)
def to_domain(self):
return FractionField(self)
def to_ring(self):
from sympy.polys.rings import PolyRing
return PolyRing(self.symbols, self.domain, self.order)
class FracElement(DomainElement, DefaultPrinting, CantSympify):
"""Element of multivariate distributed rational function field. """
def __init__(self, numer, denom=None):
if denom is None:
denom = self.field.ring.one
elif not denom:
raise ZeroDivisionError("zero denominator")
self.numer = numer
self.denom = denom
def raw_new(f, numer, denom):
return f.__class__(numer, denom)
def new(f, numer, denom):
return f.raw_new(*numer.cancel(denom))
def to_poly(f):
if f.denom != 1:
raise ValueError("f.denom should be 1")
return f.numer
def parent(self):
return self.field.to_domain()
def __getnewargs__(self):
return (self.field, self.numer, self.denom)
_hash = None
def __hash__(self):
_hash = self._hash
if _hash is None:
self._hash = _hash = hash((self.field, self.numer, self.denom))
return _hash
def copy(self):
return self.raw_new(self.numer.copy(), self.denom.copy())
def set_field(self, new_field):
if self.field == new_field:
return self
else:
new_ring = new_field.ring
numer = self.numer.set_ring(new_ring)
denom = self.denom.set_ring(new_ring)
return new_field.new(numer, denom)
def as_expr(self, *symbols):
return self.numer.as_expr(*symbols)/self.denom.as_expr(*symbols)
def __eq__(f, g):
if isinstance(g, FracElement) and f.field == g.field:
return f.numer == g.numer and f.denom == g.denom
else:
return f.numer == g and f.denom == f.field.ring.one
def __ne__(f, g):
return not f == g
def __bool__(f):
return bool(f.numer)
def sort_key(self):
return (self.denom.sort_key(), self.numer.sort_key())
def _cmp(f1, f2, op):
if isinstance(f2, f1.field.dtype):
return op(f1.sort_key(), f2.sort_key())
else:
return NotImplemented
def __lt__(f1, f2):
return f1._cmp(f2, lt)
def __le__(f1, f2):
return f1._cmp(f2, le)
def __gt__(f1, f2):
return f1._cmp(f2, gt)
def __ge__(f1, f2):
return f1._cmp(f2, ge)
def __pos__(f):
"""Negate all coefficients in ``f``. """
return f.raw_new(f.numer, f.denom)
def __neg__(f):
"""Negate all coefficients in ``f``. """
return f.raw_new(-f.numer, f.denom)
def _extract_ground(self, element):
domain = self.field.domain
try:
element = domain.convert(element)
except CoercionFailed:
if not domain.is_Field and domain.has_assoc_Field:
ground_field = domain.get_field()
try:
element = ground_field.convert(element)
except CoercionFailed:
pass
else:
return -1, ground_field.numer(element), ground_field.denom(element)
return 0, None, None
else:
return 1, element, None
def __add__(f, g):
"""Add rational functions ``f`` and ``g``. """
field = f.field
if not g:
return f
elif not f:
return g
elif isinstance(g, field.dtype):
if f.denom == g.denom:
return f.new(f.numer + g.numer, f.denom)
else:
return f.new(f.numer*g.denom + f.denom*g.numer, f.denom*g.denom)
elif isinstance(g, field.ring.dtype):
return f.new(f.numer + f.denom*g, f.denom)
else:
if isinstance(g, FracElement):
if isinstance(field.domain, FractionField) and field.domain.field == g.field:
pass
elif isinstance(g.field.domain, FractionField) and g.field.domain.field == field:
return g.__radd__(f)
else:
return NotImplemented
elif isinstance(g, PolyElement):
if isinstance(field.domain, PolynomialRing) and field.domain.ring == g.ring:
pass
else:
return g.__radd__(f)
return f.__radd__(g)
def __radd__(f, c):
if isinstance(c, f.field.ring.dtype):
return f.new(f.numer + f.denom*c, f.denom)
op, g_numer, g_denom = f._extract_ground(c)
if op == 1:
return f.new(f.numer + f.denom*g_numer, f.denom)
elif not op:
return NotImplemented
else:
return f.new(f.numer*g_denom + f.denom*g_numer, f.denom*g_denom)
def __sub__(f, g):
"""Subtract rational functions ``f`` and ``g``. """
field = f.field
if not g:
return f
elif not f:
return -g
elif isinstance(g, field.dtype):
if f.denom == g.denom:
return f.new(f.numer - g.numer, f.denom)
else:
return f.new(f.numer*g.denom - f.denom*g.numer, f.denom*g.denom)
elif isinstance(g, field.ring.dtype):
return f.new(f.numer - f.denom*g, f.denom)
else:
if isinstance(g, FracElement):
if isinstance(field.domain, FractionField) and field.domain.field == g.field:
pass
elif isinstance(g.field.domain, FractionField) and g.field.domain.field == field:
return g.__rsub__(f)
else:
return NotImplemented
elif isinstance(g, PolyElement):
if isinstance(field.domain, PolynomialRing) and field.domain.ring == g.ring:
pass
else:
return g.__rsub__(f)
op, g_numer, g_denom = f._extract_ground(g)
if op == 1:
return f.new(f.numer - f.denom*g_numer, f.denom)
elif not op:
return NotImplemented
else:
return f.new(f.numer*g_denom - f.denom*g_numer, f.denom*g_denom)
def __rsub__(f, c):
if isinstance(c, f.field.ring.dtype):
return f.new(-f.numer + f.denom*c, f.denom)
op, g_numer, g_denom = f._extract_ground(c)
if op == 1:
return f.new(-f.numer + f.denom*g_numer, f.denom)
elif not op:
return NotImplemented
else:
return f.new(-f.numer*g_denom + f.denom*g_numer, f.denom*g_denom)
def __mul__(f, g):
"""Multiply rational functions ``f`` and ``g``. """
field = f.field
if not f or not g:
return field.zero
elif isinstance(g, field.dtype):
return f.new(f.numer*g.numer, f.denom*g.denom)
elif isinstance(g, field.ring.dtype):
return f.new(f.numer*g, f.denom)
else:
if isinstance(g, FracElement):
if isinstance(field.domain, FractionField) and field.domain.field == g.field:
pass
elif isinstance(g.field.domain, FractionField) and g.field.domain.field == field:
return g.__rmul__(f)
else:
return NotImplemented
elif isinstance(g, PolyElement):
if isinstance(field.domain, PolynomialRing) and field.domain.ring == g.ring:
pass
else:
return g.__rmul__(f)
return f.__rmul__(g)
def __rmul__(f, c):
if isinstance(c, f.field.ring.dtype):
return f.new(f.numer*c, f.denom)
op, g_numer, g_denom = f._extract_ground(c)
if op == 1:
return f.new(f.numer*g_numer, f.denom)
elif not op:
return NotImplemented
else:
return f.new(f.numer*g_numer, f.denom*g_denom)
def __truediv__(f, g):
"""Computes quotient of fractions ``f`` and ``g``. """
field = f.field
if not g:
raise ZeroDivisionError
elif isinstance(g, field.dtype):
return f.new(f.numer*g.denom, f.denom*g.numer)
elif isinstance(g, field.ring.dtype):
return f.new(f.numer, f.denom*g)
else:
if isinstance(g, FracElement):
if isinstance(field.domain, FractionField) and field.domain.field == g.field:
pass
elif isinstance(g.field.domain, FractionField) and g.field.domain.field == field:
return g.__rtruediv__(f)
else:
return NotImplemented
elif isinstance(g, PolyElement):
if isinstance(field.domain, PolynomialRing) and field.domain.ring == g.ring:
pass
else:
return g.__rtruediv__(f)
op, g_numer, g_denom = f._extract_ground(g)
if op == 1:
return f.new(f.numer, f.denom*g_numer)
elif not op:
return NotImplemented
else:
return f.new(f.numer*g_denom, f.denom*g_numer)
def __rtruediv__(f, c):
if not f:
raise ZeroDivisionError
elif isinstance(c, f.field.ring.dtype):
return f.new(f.denom*c, f.numer)
op, g_numer, g_denom = f._extract_ground(c)
if op == 1:
return f.new(f.denom*g_numer, f.numer)
elif not op:
return NotImplemented
else:
return f.new(f.denom*g_numer, f.numer*g_denom)
def __pow__(f, n):
"""Raise ``f`` to a non-negative power ``n``. """
if n >= 0:
return f.raw_new(f.numer**n, f.denom**n)
elif not f:
raise ZeroDivisionError
else:
return f.raw_new(f.denom**-n, f.numer**-n)
def diff(f, x):
"""Computes partial derivative in ``x``.
Examples
========
>>> from sympy.polys.fields import field
>>> from sympy.polys.domains import ZZ
>>> _, x, y, z = field("x,y,z", ZZ)
>>> ((x**2 + y)/(z + 1)).diff(x)
2*x/(z + 1)
"""
x = x.to_poly()
return f.new(f.numer.diff(x)*f.denom - f.numer*f.denom.diff(x), f.denom**2)
def __call__(f, *values):
if 0 < len(values) <= f.field.ngens:
return f.evaluate(list(zip(f.field.gens, values)))
else:
raise ValueError("expected at least 1 and at most %s values, got %s" % (f.field.ngens, len(values)))
def evaluate(f, x, a=None):
if isinstance(x, list) and a is None:
x = [ (X.to_poly(), a) for X, a in x ]
numer, denom = f.numer.evaluate(x), f.denom.evaluate(x)
else:
x = x.to_poly()
numer, denom = f.numer.evaluate(x, a), f.denom.evaluate(x, a)
field = numer.ring.to_field()
return field.new(numer, denom)
def subs(f, x, a=None):
if isinstance(x, list) and a is None:
x = [ (X.to_poly(), a) for X, a in x ]
numer, denom = f.numer.subs(x), f.denom.subs(x)
else:
x = x.to_poly()
numer, denom = f.numer.subs(x, a), f.denom.subs(x, a)
return f.new(numer, denom)
def compose(f, x, a=None):
raise NotImplementedError
|
34711d751dccbcae7d9085b3968acc2603faf70ccbfd3b781c74f31be5ab309a | """Low-level linear systems solver. """
from sympy.utilities.iterables import connected_components
from sympy.matrices import MutableDenseMatrix
from sympy.polys.domains import EX
from sympy.polys.rings import sring
from sympy.polys.polyerrors import NotInvertible
from sympy.polys.domainmatrix import DomainMatrix
class PolyNonlinearError(Exception):
"""Raised by solve_lin_sys for nonlinear equations"""
pass
class RawMatrix(MutableDenseMatrix):
_sympify = staticmethod(lambda x: x)
def eqs_to_matrix(eqs_coeffs, eqs_rhs, gens, domain):
"""Get matrix from linear equations in dict format.
Explanation
===========
Get the matrix representation of a system of linear equations represented
as dicts with low-level DomainElement coefficients. This is an
*internal* function that is used by solve_lin_sys.
Parameters
==========
eqs_coeffs: list[dict[Symbol, DomainElement]]
The left hand sides of the equations as dicts mapping from symbols to
coefficients where the coefficients are instances of
DomainElement.
eqs_rhs: list[DomainElements]
The right hand sides of the equations as instances of
DomainElement.
gens: list[Symbol]
The unknowns in the system of equations.
domain: Domain
The domain for coefficients of both lhs and rhs.
Returns
=======
The augmented matrix representation of the system as a DomainMatrix.
Examples
========
>>> from sympy import symbols, ZZ
>>> from sympy.polys.solvers import eqs_to_matrix
>>> x, y = symbols('x, y')
>>> eqs_coeff = [{x:ZZ(1), y:ZZ(1)}, {x:ZZ(1), y:ZZ(-1)}]
>>> eqs_rhs = [ZZ(0), ZZ(-1)]
>>> eqs_to_matrix(eqs_coeff, eqs_rhs, [x, y], ZZ)
DomainMatrix([[1, 1, 0], [1, -1, 1]], (2, 3), ZZ)
See also
========
solve_lin_sys: Uses :func:`~eqs_to_matrix` internally
"""
sym2index = {x: n for n, x in enumerate(gens)}
nrows = len(eqs_coeffs)
ncols = len(gens) + 1
rows = [[domain.zero] * ncols for _ in range(nrows)]
for row, eq_coeff, eq_rhs in zip(rows, eqs_coeffs, eqs_rhs):
for sym, coeff in eq_coeff.items():
row[sym2index[sym]] = domain.convert(coeff)
row[-1] = -domain.convert(eq_rhs)
return DomainMatrix(rows, (nrows, ncols), domain)
def sympy_eqs_to_ring(eqs, symbols):
"""Convert a system of equations from Expr to a PolyRing
Explanation
===========
High-level functions like ``solve`` expect Expr as inputs but can use
``solve_lin_sys`` internally. This function converts equations from
``Expr`` to the low-level poly types used by the ``solve_lin_sys``
function.
Parameters
==========
eqs: List of Expr
A list of equations as Expr instances
symbols: List of Symbol
A list of the symbols that are the unknowns in the system of
equations.
Returns
=======
Tuple[List[PolyElement], Ring]: The equations as PolyElement instances
and the ring of polynomials within which each equation is represented.
Examples
========
>>> from sympy import symbols
>>> from sympy.polys.solvers import sympy_eqs_to_ring
>>> a, x, y = symbols('a, x, y')
>>> eqs = [x-y, x+a*y]
>>> eqs_ring, ring = sympy_eqs_to_ring(eqs, [x, y])
>>> eqs_ring
[x - y, x + a*y]
>>> type(eqs_ring[0])
<class 'sympy.polys.rings.PolyElement'>
>>> ring
ZZ(a)[x,y]
With the equations in this form they can be passed to ``solve_lin_sys``:
>>> from sympy.polys.solvers import solve_lin_sys
>>> solve_lin_sys(eqs_ring, ring)
{y: 0, x: 0}
"""
try:
K, eqs_K = sring(eqs, symbols, field=True, extension=True)
except NotInvertible:
# https://github.com/sympy/sympy/issues/18874
K, eqs_K = sring(eqs, symbols, domain=EX)
return eqs_K, K.to_domain()
def solve_lin_sys(eqs, ring, _raw=True):
"""Solve a system of linear equations from a PolynomialRing
Explanation
===========
Solves a system of linear equations given as PolyElement instances of a
PolynomialRing. The basic arithmetic is carried out using instance of
DomainElement which is more efficient than :class:`~sympy.core.expr.Expr`
for the most common inputs.
While this is a public function it is intended primarily for internal use
so its interface is not necessarily convenient. Users are suggested to use
the :func:`sympy.solvers.solveset.linsolve` function (which uses this
function internally) instead.
Parameters
==========
eqs: list[PolyElement]
The linear equations to be solved as elements of a
PolynomialRing (assumed equal to zero).
ring: PolynomialRing
The polynomial ring from which eqs are drawn. The generators of this
ring are the unkowns to be solved for and the domain of the ring is
the domain of the coefficients of the system of equations.
_raw: bool
If *_raw* is False, the keys and values in the returned dictionary
will be of type Expr (and the unit of the field will be removed from
the keys) otherwise the low-level polys types will be returned, e.g.
PolyElement: PythonRational.
Returns
=======
``None`` if the system has no solution.
dict[Symbol, Expr] if _raw=False
dict[Symbol, DomainElement] if _raw=True.
Examples
========
>>> from sympy import symbols
>>> from sympy.polys.solvers import solve_lin_sys, sympy_eqs_to_ring
>>> x, y = symbols('x, y')
>>> eqs = [x - y, x + y - 2]
>>> eqs_ring, ring = sympy_eqs_to_ring(eqs, [x, y])
>>> solve_lin_sys(eqs_ring, ring)
{y: 1, x: 1}
Passing ``_raw=False`` returns the same result except that the keys are
``Expr`` rather than low-level poly types.
>>> solve_lin_sys(eqs_ring, ring, _raw=False)
{x: 1, y: 1}
See also
========
sympy_eqs_to_ring: prepares the inputs to ``solve_lin_sys``.
linsolve: ``linsolve`` uses ``solve_lin_sys`` internally.
sympy.solvers.solvers.solve: ``solve`` uses ``solve_lin_sys`` internally.
"""
as_expr = not _raw
assert ring.domain.is_Field
eqs_dict = [dict(eq) for eq in eqs]
one_monom = ring.one.monoms()[0]
zero = ring.domain.zero
eqs_rhs = []
eqs_coeffs = []
for eq_dict in eqs_dict:
eq_rhs = eq_dict.pop(one_monom, zero)
eq_coeffs = {}
for monom, coeff in eq_dict.items():
if sum(monom) != 1:
msg = "Nonlinear term encountered in solve_lin_sys"
raise PolyNonlinearError(msg)
eq_coeffs[ring.gens[monom.index(1)]] = coeff
if not eq_coeffs:
if not eq_rhs:
continue
else:
return None
eqs_rhs.append(eq_rhs)
eqs_coeffs.append(eq_coeffs)
result = _solve_lin_sys(eqs_coeffs, eqs_rhs, ring)
if result is not None and as_expr:
def to_sympy(x):
as_expr = getattr(x, 'as_expr', None)
if as_expr:
return as_expr()
else:
return ring.domain.to_sympy(x)
tresult = {to_sympy(sym): to_sympy(val) for sym, val in result.items()}
# Remove 1.0x
result = {}
for k, v in tresult.items():
if k.is_Mul:
c, s = k.as_coeff_Mul()
result[s] = v/c
else:
result[k] = v
return result
def _solve_lin_sys(eqs_coeffs, eqs_rhs, ring):
"""Solve a linear system from dict of PolynomialRing coefficients
Explanation
===========
This is an **internal** function used by :func:`solve_lin_sys` after the
equations have been preprocessed. The role of this function is to split
the system into connected components and pass those to
:func:`_solve_lin_sys_component`.
Examples
========
Setup a system for $x-y=0$ and $x+y=2$ and solve:
>>> from sympy import symbols, sring
>>> from sympy.polys.solvers import _solve_lin_sys
>>> x, y = symbols('x, y')
>>> R, (xr, yr) = sring([x, y], [x, y])
>>> eqs = [{xr:R.one, yr:-R.one}, {xr:R.one, yr:R.one}]
>>> eqs_rhs = [R.zero, -2*R.one]
>>> _solve_lin_sys(eqs, eqs_rhs, R)
{y: 1, x: 1}
See also
========
solve_lin_sys: This function is used internally by :func:`solve_lin_sys`.
"""
V = ring.gens
E = []
for eq_coeffs in eqs_coeffs:
syms = list(eq_coeffs)
E.extend(zip(syms[:-1], syms[1:]))
G = V, E
components = connected_components(G)
sym2comp = {}
for n, component in enumerate(components):
for sym in component:
sym2comp[sym] = n
subsystems = [([], []) for _ in range(len(components))]
for eq_coeff, eq_rhs in zip(eqs_coeffs, eqs_rhs):
sym = next(iter(eq_coeff), None)
sub_coeff, sub_rhs = subsystems[sym2comp[sym]]
sub_coeff.append(eq_coeff)
sub_rhs.append(eq_rhs)
sol = {}
for subsystem in subsystems:
subsol = _solve_lin_sys_component(subsystem[0], subsystem[1], ring)
if subsol is None:
return None
sol.update(subsol)
return sol
def _solve_lin_sys_component(eqs_coeffs, eqs_rhs, ring):
"""Solve a linear system from dict of PolynomialRing coefficients
Explanation
===========
This is an **internal** function used by :func:`solve_lin_sys` after the
equations have been preprocessed. After :func:`_solve_lin_sys` splits the
system into connected components this function is called for each
component. The system of equations is solved using Gauss-Jordan
elimination with division followed by back-substitution.
Examples
========
Setup a system for $x-y=0$ and $x+y=2$ and solve:
>>> from sympy import symbols, sring
>>> from sympy.polys.solvers import _solve_lin_sys_component
>>> x, y = symbols('x, y')
>>> R, (xr, yr) = sring([x, y], [x, y])
>>> eqs = [{xr:R.one, yr:-R.one}, {xr:R.one, yr:R.one}]
>>> eqs_rhs = [R.zero, -2*R.one]
>>> _solve_lin_sys_component(eqs, eqs_rhs, R)
{y: 1, x: 1}
See also
========
solve_lin_sys: This function is used internally by :func:`solve_lin_sys`.
"""
# transform from equations to matrix form
matrix = eqs_to_matrix(eqs_coeffs, eqs_rhs, ring.gens, ring.domain)
# convert to a field for rref
if not matrix.domain.is_Field:
matrix = matrix.to_field()
# solve by row-reduction
echelon, pivots = matrix.rref()
# construct the returnable form of the solutions
keys = ring.gens
if pivots and pivots[-1] == len(keys):
return None
if len(pivots) == len(keys):
sol = []
for s in [row[-1] for row in echelon.rep.to_ddm()]:
a = s
sol.append(a)
sols = dict(zip(keys, sol))
else:
sols = {}
g = ring.gens
# Extract ground domain coefficients and convert to the ring:
if hasattr(ring, 'ring'):
convert = ring.ring.ground_new
else:
convert = ring.ground_new
echelon = echelon.rep.to_ddm()
vals_set = {v for row in echelon for v in row}
vals_map = {v: convert(v) for v in vals_set}
echelon = [[vals_map[eij] for eij in ei] for ei in echelon]
for i, p in enumerate(pivots):
v = echelon[i][-1] - sum(echelon[i][j]*g[j] for j in range(p+1, len(g)) if echelon[i][j])
sols[keys[p]] = v
return sols
|
d4c2498430fbc456b0e7b167bbb0bebc46857ac4ac27f9ec29857e80efa2e2f2 | """Real and complex root isolation and refinement algorithms. """
from sympy.polys.densearith import (
dup_neg, dup_rshift, dup_rem)
from sympy.polys.densebasic import (
dup_LC, dup_TC, dup_degree,
dup_strip, dup_reverse,
dup_convert,
dup_terms_gcd)
from sympy.polys.densetools import (
dup_clear_denoms,
dup_mirror, dup_scale, dup_shift,
dup_transform,
dup_diff,
dup_eval, dmp_eval_in,
dup_sign_variations,
dup_real_imag)
from sympy.polys.factortools import (
dup_factor_list)
from sympy.polys.polyerrors import (
RefinementFailed,
DomainError)
from sympy.polys.sqfreetools import (
dup_sqf_part, dup_sqf_list)
def dup_sturm(f, K):
"""
Computes the Sturm sequence of ``f`` in ``F[x]``.
Given a univariate, square-free polynomial ``f(x)`` returns the
associated Sturm sequence ``f_0(x), ..., f_n(x)`` defined by::
f_0(x), f_1(x) = f(x), f'(x)
f_n = -rem(f_{n-2}(x), f_{n-1}(x))
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> R.dup_sturm(x**3 - 2*x**2 + x - 3)
[x**3 - 2*x**2 + x - 3, 3*x**2 - 4*x + 1, 2/9*x + 25/9, -2079/4]
References
==========
.. [1] [Davenport88]_
"""
if not K.is_Field:
raise DomainError("can't compute Sturm sequence over %s" % K)
f = dup_sqf_part(f, K)
sturm = [f, dup_diff(f, 1, K)]
while sturm[-1]:
s = dup_rem(sturm[-2], sturm[-1], K)
sturm.append(dup_neg(s, K))
return sturm[:-1]
def dup_root_upper_bound(f, K):
"""Compute the LMQ upper bound for the positive roots of `f`;
LMQ (Local Max Quadratic) was developed by Akritas-Strzebonski-Vigklas.
References
==========
.. [1] Alkiviadis G. Akritas: "Linear and Quadratic Complexity Bounds on the
Values of the Positive Roots of Polynomials"
Journal of Universal Computer Science, Vol. 15, No. 3, 523-537, 2009.
"""
n, P = len(f), []
t = n * [K.one]
if dup_LC(f, K) < 0:
f = dup_neg(f, K)
f = list(reversed(f))
for i in range(0, n):
if f[i] >= 0:
continue
a, QL = K.log(-f[i], 2), []
for j in range(i + 1, n):
if f[j] <= 0:
continue
q = t[j] + a - K.log(f[j], 2)
QL.append([q // (j - i) , j])
if not QL:
continue
q = min(QL)
t[q[1]] = t[q[1]] + 1
P.append(q[0])
if not P:
return None
else:
return K.get_field()(2)**(max(P) + 1)
def dup_root_lower_bound(f, K):
"""Compute the LMQ lower bound for the positive roots of `f`;
LMQ (Local Max Quadratic) was developed by Akritas-Strzebonski-Vigklas.
References
==========
.. [1] Alkiviadis G. Akritas: "Linear and Quadratic Complexity Bounds on the
Values of the Positive Roots of Polynomials"
Journal of Universal Computer Science, Vol. 15, No. 3, 523-537, 2009.
"""
bound = dup_root_upper_bound(dup_reverse(f), K)
if bound is not None:
return 1/bound
else:
return None
def _mobius_from_interval(I, field):
"""Convert an open interval to a Mobius transform. """
s, t = I
a, c = field.numer(s), field.denom(s)
b, d = field.numer(t), field.denom(t)
return a, b, c, d
def _mobius_to_interval(M, field):
"""Convert a Mobius transform to an open interval. """
a, b, c, d = M
s, t = field(a, c), field(b, d)
if s <= t:
return (s, t)
else:
return (t, s)
def dup_step_refine_real_root(f, M, K, fast=False):
"""One step of positive real root refinement algorithm. """
a, b, c, d = M
if a == b and c == d:
return f, (a, b, c, d)
A = dup_root_lower_bound(f, K)
if A is not None:
A = K(int(A))
else:
A = K.zero
if fast and A > 16:
f = dup_scale(f, A, K)
a, c, A = A*a, A*c, K.one
if A >= K.one:
f = dup_shift(f, A, K)
b, d = A*a + b, A*c + d
if not dup_eval(f, K.zero, K):
return f, (b, b, d, d)
f, g = dup_shift(f, K.one, K), f
a1, b1, c1, d1 = a, a + b, c, c + d
if not dup_eval(f, K.zero, K):
return f, (b1, b1, d1, d1)
k = dup_sign_variations(f, K)
if k == 1:
a, b, c, d = a1, b1, c1, d1
else:
f = dup_shift(dup_reverse(g), K.one, K)
if not dup_eval(f, K.zero, K):
f = dup_rshift(f, 1, K)
a, b, c, d = b, a + b, d, c + d
return f, (a, b, c, d)
def dup_inner_refine_real_root(f, M, K, eps=None, steps=None, disjoint=None, fast=False, mobius=False):
"""Refine a positive root of `f` given a Mobius transform or an interval. """
F = K.get_field()
if len(M) == 2:
a, b, c, d = _mobius_from_interval(M, F)
else:
a, b, c, d = M
while not c:
f, (a, b, c, d) = dup_step_refine_real_root(f, (a, b, c,
d), K, fast=fast)
if eps is not None and steps is not None:
for i in range(0, steps):
if abs(F(a, c) - F(b, d)) >= eps:
f, (a, b, c, d) = dup_step_refine_real_root(f, (a, b, c, d), K, fast=fast)
else:
break
else:
if eps is not None:
while abs(F(a, c) - F(b, d)) >= eps:
f, (a, b, c, d) = dup_step_refine_real_root(f, (a, b, c, d), K, fast=fast)
if steps is not None:
for i in range(0, steps):
f, (a, b, c, d) = dup_step_refine_real_root(f, (a, b, c, d), K, fast=fast)
if disjoint is not None:
while True:
u, v = _mobius_to_interval((a, b, c, d), F)
if v <= disjoint or disjoint <= u:
break
else:
f, (a, b, c, d) = dup_step_refine_real_root(f, (a, b, c, d), K, fast=fast)
if not mobius:
return _mobius_to_interval((a, b, c, d), F)
else:
return f, (a, b, c, d)
def dup_outer_refine_real_root(f, s, t, K, eps=None, steps=None, disjoint=None, fast=False):
"""Refine a positive root of `f` given an interval `(s, t)`. """
a, b, c, d = _mobius_from_interval((s, t), K.get_field())
f = dup_transform(f, dup_strip([a, b]),
dup_strip([c, d]), K)
if dup_sign_variations(f, K) != 1:
raise RefinementFailed("there should be exactly one root in (%s, %s) interval" % (s, t))
return dup_inner_refine_real_root(f, (a, b, c, d), K, eps=eps, steps=steps, disjoint=disjoint, fast=fast)
def dup_refine_real_root(f, s, t, K, eps=None, steps=None, disjoint=None, fast=False):
"""Refine real root's approximating interval to the given precision. """
if K.is_QQ:
(_, f), K = dup_clear_denoms(f, K, convert=True), K.get_ring()
elif not K.is_ZZ:
raise DomainError("real root refinement not supported over %s" % K)
if s == t:
return (s, t)
if s > t:
s, t = t, s
negative = False
if s < 0:
if t <= 0:
f, s, t, negative = dup_mirror(f, K), -t, -s, True
else:
raise ValueError("can't refine a real root in (%s, %s)" % (s, t))
if negative and disjoint is not None:
if disjoint < 0:
disjoint = -disjoint
else:
disjoint = None
s, t = dup_outer_refine_real_root(
f, s, t, K, eps=eps, steps=steps, disjoint=disjoint, fast=fast)
if negative:
return (-t, -s)
else:
return ( s, t)
def dup_inner_isolate_real_roots(f, K, eps=None, fast=False):
"""Internal function for isolation positive roots up to given precision.
References
==========
1. Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative Study of Two Real Root
Isolation Methods . Nonlinear Analysis: Modelling and Control, Vol. 10, No. 4, 297-304, 2005.
2. Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S. Vigklas: Improving the
Performance of the Continued Fractions Method Using new Bounds of Positive Roots. Nonlinear
Analysis: Modelling and Control, Vol. 13, No. 3, 265-279, 2008.
"""
a, b, c, d = K.one, K.zero, K.zero, K.one
k = dup_sign_variations(f, K)
if k == 0:
return []
if k == 1:
roots = [dup_inner_refine_real_root(
f, (a, b, c, d), K, eps=eps, fast=fast, mobius=True)]
else:
roots, stack = [], [(a, b, c, d, f, k)]
while stack:
a, b, c, d, f, k = stack.pop()
A = dup_root_lower_bound(f, K)
if A is not None:
A = K(int(A))
else:
A = K.zero
if fast and A > 16:
f = dup_scale(f, A, K)
a, c, A = A*a, A*c, K.one
if A >= K.one:
f = dup_shift(f, A, K)
b, d = A*a + b, A*c + d
if not dup_TC(f, K):
roots.append((f, (b, b, d, d)))
f = dup_rshift(f, 1, K)
k = dup_sign_variations(f, K)
if k == 0:
continue
if k == 1:
roots.append(dup_inner_refine_real_root(
f, (a, b, c, d), K, eps=eps, fast=fast, mobius=True))
continue
f1 = dup_shift(f, K.one, K)
a1, b1, c1, d1, r = a, a + b, c, c + d, 0
if not dup_TC(f1, K):
roots.append((f1, (b1, b1, d1, d1)))
f1, r = dup_rshift(f1, 1, K), 1
k1 = dup_sign_variations(f1, K)
k2 = k - k1 - r
a2, b2, c2, d2 = b, a + b, d, c + d
if k2 > 1:
f2 = dup_shift(dup_reverse(f), K.one, K)
if not dup_TC(f2, K):
f2 = dup_rshift(f2, 1, K)
k2 = dup_sign_variations(f2, K)
else:
f2 = None
if k1 < k2:
a1, a2, b1, b2 = a2, a1, b2, b1
c1, c2, d1, d2 = c2, c1, d2, d1
f1, f2, k1, k2 = f2, f1, k2, k1
if not k1:
continue
if f1 is None:
f1 = dup_shift(dup_reverse(f), K.one, K)
if not dup_TC(f1, K):
f1 = dup_rshift(f1, 1, K)
if k1 == 1:
roots.append(dup_inner_refine_real_root(
f1, (a1, b1, c1, d1), K, eps=eps, fast=fast, mobius=True))
else:
stack.append((a1, b1, c1, d1, f1, k1))
if not k2:
continue
if f2 is None:
f2 = dup_shift(dup_reverse(f), K.one, K)
if not dup_TC(f2, K):
f2 = dup_rshift(f2, 1, K)
if k2 == 1:
roots.append(dup_inner_refine_real_root(
f2, (a2, b2, c2, d2), K, eps=eps, fast=fast, mobius=True))
else:
stack.append((a2, b2, c2, d2, f2, k2))
return roots
def _discard_if_outside_interval(f, M, inf, sup, K, negative, fast, mobius):
"""Discard an isolating interval if outside ``(inf, sup)``. """
F = K.get_field()
while True:
u, v = _mobius_to_interval(M, F)
if negative:
u, v = -v, -u
if (inf is None or u >= inf) and (sup is None or v <= sup):
if not mobius:
return u, v
else:
return f, M
elif (sup is not None and u > sup) or (inf is not None and v < inf):
return None
else:
f, M = dup_step_refine_real_root(f, M, K, fast=fast)
def dup_inner_isolate_positive_roots(f, K, eps=None, inf=None, sup=None, fast=False, mobius=False):
"""Iteratively compute disjoint positive root isolation intervals. """
if sup is not None and sup < 0:
return []
roots = dup_inner_isolate_real_roots(f, K, eps=eps, fast=fast)
F, results = K.get_field(), []
if inf is not None or sup is not None:
for f, M in roots:
result = _discard_if_outside_interval(f, M, inf, sup, K, False, fast, mobius)
if result is not None:
results.append(result)
elif not mobius:
for f, M in roots:
u, v = _mobius_to_interval(M, F)
results.append((u, v))
else:
results = roots
return results
def dup_inner_isolate_negative_roots(f, K, inf=None, sup=None, eps=None, fast=False, mobius=False):
"""Iteratively compute disjoint negative root isolation intervals. """
if inf is not None and inf >= 0:
return []
roots = dup_inner_isolate_real_roots(dup_mirror(f, K), K, eps=eps, fast=fast)
F, results = K.get_field(), []
if inf is not None or sup is not None:
for f, M in roots:
result = _discard_if_outside_interval(f, M, inf, sup, K, True, fast, mobius)
if result is not None:
results.append(result)
elif not mobius:
for f, M in roots:
u, v = _mobius_to_interval(M, F)
results.append((-v, -u))
else:
results = roots
return results
def _isolate_zero(f, K, inf, sup, basis=False, sqf=False):
"""Handle special case of CF algorithm when ``f`` is homogeneous. """
j, f = dup_terms_gcd(f, K)
if j > 0:
F = K.get_field()
if (inf is None or inf <= 0) and (sup is None or 0 <= sup):
if not sqf:
if not basis:
return [((F.zero, F.zero), j)], f
else:
return [((F.zero, F.zero), j, [K.one, K.zero])], f
else:
return [(F.zero, F.zero)], f
return [], f
def dup_isolate_real_roots_sqf(f, K, eps=None, inf=None, sup=None, fast=False, blackbox=False):
"""Isolate real roots of a square-free polynomial using the Vincent-Akritas-Strzebonski (VAS) CF approach.
References
==========
.. [1] Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative
Study of Two Real Root Isolation Methods. Nonlinear Analysis:
Modelling and Control, Vol. 10, No. 4, 297-304, 2005.
.. [2] Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S.
Vigklas: Improving the Performance of the Continued Fractions
Method Using New Bounds of Positive Roots. Nonlinear Analysis:
Modelling and Control, Vol. 13, No. 3, 265-279, 2008.
"""
if K.is_QQ:
(_, f), K = dup_clear_denoms(f, K, convert=True), K.get_ring()
elif not K.is_ZZ:
raise DomainError("isolation of real roots not supported over %s" % K)
if dup_degree(f) <= 0:
return []
I_zero, f = _isolate_zero(f, K, inf, sup, basis=False, sqf=True)
I_neg = dup_inner_isolate_negative_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast)
I_pos = dup_inner_isolate_positive_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast)
roots = sorted(I_neg + I_zero + I_pos)
if not blackbox:
return roots
else:
return [ RealInterval((a, b), f, K) for (a, b) in roots ]
def dup_isolate_real_roots(f, K, eps=None, inf=None, sup=None, basis=False, fast=False):
"""Isolate real roots using Vincent-Akritas-Strzebonski (VAS) continued fractions approach.
References
==========
.. [1] Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative
Study of Two Real Root Isolation Methods. Nonlinear Analysis:
Modelling and Control, Vol. 10, No. 4, 297-304, 2005.
.. [2] Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S.
Vigklas: Improving the Performance of the Continued Fractions
Method Using New Bounds of Positive Roots.
Nonlinear Analysis: Modelling and Control, Vol. 13, No. 3, 265-279, 2008.
"""
if K.is_QQ:
(_, f), K = dup_clear_denoms(f, K, convert=True), K.get_ring()
elif not K.is_ZZ:
raise DomainError("isolation of real roots not supported over %s" % K)
if dup_degree(f) <= 0:
return []
I_zero, f = _isolate_zero(f, K, inf, sup, basis=basis, sqf=False)
_, factors = dup_sqf_list(f, K)
if len(factors) == 1:
((f, k),) = factors
I_neg = dup_inner_isolate_negative_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast)
I_pos = dup_inner_isolate_positive_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast)
I_neg = [ ((u, v), k) for u, v in I_neg ]
I_pos = [ ((u, v), k) for u, v in I_pos ]
else:
I_neg, I_pos = _real_isolate_and_disjoin(factors, K,
eps=eps, inf=inf, sup=sup, basis=basis, fast=fast)
return sorted(I_neg + I_zero + I_pos)
def dup_isolate_real_roots_list(polys, K, eps=None, inf=None, sup=None, strict=False, basis=False, fast=False):
"""Isolate real roots of a list of square-free polynomial using Vincent-Akritas-Strzebonski (VAS) CF approach.
References
==========
.. [1] Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative
Study of Two Real Root Isolation Methods. Nonlinear Analysis:
Modelling and Control, Vol. 10, No. 4, 297-304, 2005.
.. [2] Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S.
Vigklas: Improving the Performance of the Continued Fractions
Method Using New Bounds of Positive Roots.
Nonlinear Analysis: Modelling and Control, Vol. 13, No. 3, 265-279, 2008.
"""
if K.is_QQ:
K, F, polys = K.get_ring(), K, polys[:]
for i, p in enumerate(polys):
polys[i] = dup_clear_denoms(p, F, K, convert=True)[1]
elif not K.is_ZZ:
raise DomainError("isolation of real roots not supported over %s" % K)
zeros, factors_dict = False, {}
if (inf is None or inf <= 0) and (sup is None or 0 <= sup):
zeros, zero_indices = True, {}
for i, p in enumerate(polys):
j, p = dup_terms_gcd(p, K)
if zeros and j > 0:
zero_indices[i] = j
for f, k in dup_factor_list(p, K)[1]:
f = tuple(f)
if f not in factors_dict:
factors_dict[f] = {i: k}
else:
factors_dict[f][i] = k
factors_list = []
for f, indices in factors_dict.items():
factors_list.append((list(f), indices))
I_neg, I_pos = _real_isolate_and_disjoin(factors_list, K, eps=eps,
inf=inf, sup=sup, strict=strict, basis=basis, fast=fast)
F = K.get_field()
if not zeros or not zero_indices:
I_zero = []
else:
if not basis:
I_zero = [((F.zero, F.zero), zero_indices)]
else:
I_zero = [((F.zero, F.zero), zero_indices, [K.one, K.zero])]
return sorted(I_neg + I_zero + I_pos)
def _disjoint_p(M, N, strict=False):
"""Check if Mobius transforms define disjoint intervals. """
a1, b1, c1, d1 = M
a2, b2, c2, d2 = N
a1d1, b1c1 = a1*d1, b1*c1
a2d2, b2c2 = a2*d2, b2*c2
if a1d1 == b1c1 and a2d2 == b2c2:
return True
if a1d1 > b1c1:
a1, c1, b1, d1 = b1, d1, a1, c1
if a2d2 > b2c2:
a2, c2, b2, d2 = b2, d2, a2, c2
if not strict:
return a2*d1 >= c2*b1 or b2*c1 <= d2*a1
else:
return a2*d1 > c2*b1 or b2*c1 < d2*a1
def _real_isolate_and_disjoin(factors, K, eps=None, inf=None, sup=None, strict=False, basis=False, fast=False):
"""Isolate real roots of a list of polynomials and disjoin intervals. """
I_pos, I_neg = [], []
for i, (f, k) in enumerate(factors):
for F, M in dup_inner_isolate_positive_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast, mobius=True):
I_pos.append((F, M, k, f))
for G, N in dup_inner_isolate_negative_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast, mobius=True):
I_neg.append((G, N, k, f))
for i, (f, M, k, F) in enumerate(I_pos):
for j, (g, N, m, G) in enumerate(I_pos[i + 1:]):
while not _disjoint_p(M, N, strict=strict):
f, M = dup_inner_refine_real_root(f, M, K, steps=1, fast=fast, mobius=True)
g, N = dup_inner_refine_real_root(g, N, K, steps=1, fast=fast, mobius=True)
I_pos[i + j + 1] = (g, N, m, G)
I_pos[i] = (f, M, k, F)
for i, (f, M, k, F) in enumerate(I_neg):
for j, (g, N, m, G) in enumerate(I_neg[i + 1:]):
while not _disjoint_p(M, N, strict=strict):
f, M = dup_inner_refine_real_root(f, M, K, steps=1, fast=fast, mobius=True)
g, N = dup_inner_refine_real_root(g, N, K, steps=1, fast=fast, mobius=True)
I_neg[i + j + 1] = (g, N, m, G)
I_neg[i] = (f, M, k, F)
if strict:
for i, (f, M, k, F) in enumerate(I_neg):
if not M[0]:
while not M[0]:
f, M = dup_inner_refine_real_root(f, M, K, steps=1, fast=fast, mobius=True)
I_neg[i] = (f, M, k, F)
break
for j, (g, N, m, G) in enumerate(I_pos):
if not N[0]:
while not N[0]:
g, N = dup_inner_refine_real_root(g, N, K, steps=1, fast=fast, mobius=True)
I_pos[j] = (g, N, m, G)
break
field = K.get_field()
I_neg = [ (_mobius_to_interval(M, field), k, f) for (_, M, k, f) in I_neg ]
I_pos = [ (_mobius_to_interval(M, field), k, f) for (_, M, k, f) in I_pos ]
if not basis:
I_neg = [ ((-v, -u), k) for ((u, v), k, _) in I_neg ]
I_pos = [ (( u, v), k) for ((u, v), k, _) in I_pos ]
else:
I_neg = [ ((-v, -u), k, f) for ((u, v), k, f) in I_neg ]
I_pos = [ (( u, v), k, f) for ((u, v), k, f) in I_pos ]
return I_neg, I_pos
def dup_count_real_roots(f, K, inf=None, sup=None):
"""Returns the number of distinct real roots of ``f`` in ``[inf, sup]``. """
if dup_degree(f) <= 0:
return 0
if not K.is_Field:
R, K = K, K.get_field()
f = dup_convert(f, R, K)
sturm = dup_sturm(f, K)
if inf is None:
signs_inf = dup_sign_variations([ dup_LC(s, K)*(-1)**dup_degree(s) for s in sturm ], K)
else:
signs_inf = dup_sign_variations([ dup_eval(s, inf, K) for s in sturm ], K)
if sup is None:
signs_sup = dup_sign_variations([ dup_LC(s, K) for s in sturm ], K)
else:
signs_sup = dup_sign_variations([ dup_eval(s, sup, K) for s in sturm ], K)
count = abs(signs_inf - signs_sup)
if inf is not None and not dup_eval(f, inf, K):
count += 1
return count
OO = 'OO' # Origin of (re, im) coordinate system
Q1 = 'Q1' # Quadrant #1 (++): re > 0 and im > 0
Q2 = 'Q2' # Quadrant #2 (-+): re < 0 and im > 0
Q3 = 'Q3' # Quadrant #3 (--): re < 0 and im < 0
Q4 = 'Q4' # Quadrant #4 (+-): re > 0 and im < 0
A1 = 'A1' # Axis #1 (+0): re > 0 and im = 0
A2 = 'A2' # Axis #2 (0+): re = 0 and im > 0
A3 = 'A3' # Axis #3 (-0): re < 0 and im = 0
A4 = 'A4' # Axis #4 (0-): re = 0 and im < 0
_rules_simple = {
# Q --> Q (same) => no change
(Q1, Q1): 0,
(Q2, Q2): 0,
(Q3, Q3): 0,
(Q4, Q4): 0,
# A -- CCW --> Q => +1/4 (CCW)
(A1, Q1): 1,
(A2, Q2): 1,
(A3, Q3): 1,
(A4, Q4): 1,
# A -- CW --> Q => -1/4 (CCW)
(A1, Q4): 2,
(A2, Q1): 2,
(A3, Q2): 2,
(A4, Q3): 2,
# Q -- CCW --> A => +1/4 (CCW)
(Q1, A2): 3,
(Q2, A3): 3,
(Q3, A4): 3,
(Q4, A1): 3,
# Q -- CW --> A => -1/4 (CCW)
(Q1, A1): 4,
(Q2, A2): 4,
(Q3, A3): 4,
(Q4, A4): 4,
# Q -- CCW --> Q => +1/2 (CCW)
(Q1, Q2): +5,
(Q2, Q3): +5,
(Q3, Q4): +5,
(Q4, Q1): +5,
# Q -- CW --> Q => -1/2 (CW)
(Q1, Q4): -5,
(Q2, Q1): -5,
(Q3, Q2): -5,
(Q4, Q3): -5,
}
_rules_ambiguous = {
# A -- CCW --> Q => { +1/4 (CCW), -9/4 (CW) }
(A1, OO, Q1): -1,
(A2, OO, Q2): -1,
(A3, OO, Q3): -1,
(A4, OO, Q4): -1,
# A -- CW --> Q => { -1/4 (CCW), +7/4 (CW) }
(A1, OO, Q4): -2,
(A2, OO, Q1): -2,
(A3, OO, Q2): -2,
(A4, OO, Q3): -2,
# Q -- CCW --> A => { +1/4 (CCW), -9/4 (CW) }
(Q1, OO, A2): -3,
(Q2, OO, A3): -3,
(Q3, OO, A4): -3,
(Q4, OO, A1): -3,
# Q -- CW --> A => { -1/4 (CCW), +7/4 (CW) }
(Q1, OO, A1): -4,
(Q2, OO, A2): -4,
(Q3, OO, A3): -4,
(Q4, OO, A4): -4,
# A -- OO --> A => { +1 (CCW), -1 (CW) }
(A1, A3): 7,
(A2, A4): 7,
(A3, A1): 7,
(A4, A2): 7,
(A1, OO, A3): 7,
(A2, OO, A4): 7,
(A3, OO, A1): 7,
(A4, OO, A2): 7,
# Q -- DIA --> Q => { +1 (CCW), -1 (CW) }
(Q1, Q3): 8,
(Q2, Q4): 8,
(Q3, Q1): 8,
(Q4, Q2): 8,
(Q1, OO, Q3): 8,
(Q2, OO, Q4): 8,
(Q3, OO, Q1): 8,
(Q4, OO, Q2): 8,
# A --- R ---> A => { +1/2 (CCW), -3/2 (CW) }
(A1, A2): 9,
(A2, A3): 9,
(A3, A4): 9,
(A4, A1): 9,
(A1, OO, A2): 9,
(A2, OO, A3): 9,
(A3, OO, A4): 9,
(A4, OO, A1): 9,
# A --- L ---> A => { +3/2 (CCW), -1/2 (CW) }
(A1, A4): 10,
(A2, A1): 10,
(A3, A2): 10,
(A4, A3): 10,
(A1, OO, A4): 10,
(A2, OO, A1): 10,
(A3, OO, A2): 10,
(A4, OO, A3): 10,
# Q --- 1 ---> A => { +3/4 (CCW), -5/4 (CW) }
(Q1, A3): 11,
(Q2, A4): 11,
(Q3, A1): 11,
(Q4, A2): 11,
(Q1, OO, A3): 11,
(Q2, OO, A4): 11,
(Q3, OO, A1): 11,
(Q4, OO, A2): 11,
# Q --- 2 ---> A => { +5/4 (CCW), -3/4 (CW) }
(Q1, A4): 12,
(Q2, A1): 12,
(Q3, A2): 12,
(Q4, A3): 12,
(Q1, OO, A4): 12,
(Q2, OO, A1): 12,
(Q3, OO, A2): 12,
(Q4, OO, A3): 12,
# A --- 1 ---> Q => { +5/4 (CCW), -3/4 (CW) }
(A1, Q3): 13,
(A2, Q4): 13,
(A3, Q1): 13,
(A4, Q2): 13,
(A1, OO, Q3): 13,
(A2, OO, Q4): 13,
(A3, OO, Q1): 13,
(A4, OO, Q2): 13,
# A --- 2 ---> Q => { +3/4 (CCW), -5/4 (CW) }
(A1, Q2): 14,
(A2, Q3): 14,
(A3, Q4): 14,
(A4, Q1): 14,
(A1, OO, Q2): 14,
(A2, OO, Q3): 14,
(A3, OO, Q4): 14,
(A4, OO, Q1): 14,
# Q --> OO --> Q => { +1/2 (CCW), -3/2 (CW) }
(Q1, OO, Q2): 15,
(Q2, OO, Q3): 15,
(Q3, OO, Q4): 15,
(Q4, OO, Q1): 15,
# Q --> OO --> Q => { +3/2 (CCW), -1/2 (CW) }
(Q1, OO, Q4): 16,
(Q2, OO, Q1): 16,
(Q3, OO, Q2): 16,
(Q4, OO, Q3): 16,
# A --> OO --> A => { +2 (CCW), 0 (CW) }
(A1, OO, A1): 17,
(A2, OO, A2): 17,
(A3, OO, A3): 17,
(A4, OO, A4): 17,
# Q --> OO --> Q => { +2 (CCW), 0 (CW) }
(Q1, OO, Q1): 18,
(Q2, OO, Q2): 18,
(Q3, OO, Q3): 18,
(Q4, OO, Q4): 18,
}
_values = {
0: [( 0, 1)],
1: [(+1, 4)],
2: [(-1, 4)],
3: [(+1, 4)],
4: [(-1, 4)],
-1: [(+9, 4), (+1, 4)],
-2: [(+7, 4), (-1, 4)],
-3: [(+9, 4), (+1, 4)],
-4: [(+7, 4), (-1, 4)],
+5: [(+1, 2)],
-5: [(-1, 2)],
7: [(+1, 1), (-1, 1)],
8: [(+1, 1), (-1, 1)],
9: [(+1, 2), (-3, 2)],
10: [(+3, 2), (-1, 2)],
11: [(+3, 4), (-5, 4)],
12: [(+5, 4), (-3, 4)],
13: [(+5, 4), (-3, 4)],
14: [(+3, 4), (-5, 4)],
15: [(+1, 2), (-3, 2)],
16: [(+3, 2), (-1, 2)],
17: [(+2, 1), ( 0, 1)],
18: [(+2, 1), ( 0, 1)],
}
def _classify_point(re, im):
"""Return the half-axis (or origin) on which (re, im) point is located. """
if not re and not im:
return OO
if not re:
if im > 0:
return A2
else:
return A4
elif not im:
if re > 0:
return A1
else:
return A3
def _intervals_to_quadrants(intervals, f1, f2, s, t, F):
"""Generate a sequence of extended quadrants from a list of critical points. """
if not intervals:
return []
Q = []
if not f1:
(a, b), _, _ = intervals[0]
if a == b == s:
if len(intervals) == 1:
if dup_eval(f2, t, F) > 0:
return [OO, A2]
else:
return [OO, A4]
else:
(a, _), _, _ = intervals[1]
if dup_eval(f2, (s + a)/2, F) > 0:
Q.extend([OO, A2])
f2_sgn = +1
else:
Q.extend([OO, A4])
f2_sgn = -1
intervals = intervals[1:]
else:
if dup_eval(f2, s, F) > 0:
Q.append(A2)
f2_sgn = +1
else:
Q.append(A4)
f2_sgn = -1
for (a, _), indices, _ in intervals:
Q.append(OO)
if indices[1] % 2 == 1:
f2_sgn = -f2_sgn
if a != t:
if f2_sgn > 0:
Q.append(A2)
else:
Q.append(A4)
return Q
if not f2:
(a, b), _, _ = intervals[0]
if a == b == s:
if len(intervals) == 1:
if dup_eval(f1, t, F) > 0:
return [OO, A1]
else:
return [OO, A3]
else:
(a, _), _, _ = intervals[1]
if dup_eval(f1, (s + a)/2, F) > 0:
Q.extend([OO, A1])
f1_sgn = +1
else:
Q.extend([OO, A3])
f1_sgn = -1
intervals = intervals[1:]
else:
if dup_eval(f1, s, F) > 0:
Q.append(A1)
f1_sgn = +1
else:
Q.append(A3)
f1_sgn = -1
for (a, _), indices, _ in intervals:
Q.append(OO)
if indices[0] % 2 == 1:
f1_sgn = -f1_sgn
if a != t:
if f1_sgn > 0:
Q.append(A1)
else:
Q.append(A3)
return Q
re = dup_eval(f1, s, F)
im = dup_eval(f2, s, F)
if not re or not im:
Q.append(_classify_point(re, im))
if len(intervals) == 1:
re = dup_eval(f1, t, F)
im = dup_eval(f2, t, F)
else:
(a, _), _, _ = intervals[1]
re = dup_eval(f1, (s + a)/2, F)
im = dup_eval(f2, (s + a)/2, F)
intervals = intervals[1:]
if re > 0:
f1_sgn = +1
else:
f1_sgn = -1
if im > 0:
f2_sgn = +1
else:
f2_sgn = -1
sgn = {
(+1, +1): Q1,
(-1, +1): Q2,
(-1, -1): Q3,
(+1, -1): Q4,
}
Q.append(sgn[(f1_sgn, f2_sgn)])
for (a, b), indices, _ in intervals:
if a == b:
re = dup_eval(f1, a, F)
im = dup_eval(f2, a, F)
cls = _classify_point(re, im)
if cls is not None:
Q.append(cls)
if 0 in indices:
if indices[0] % 2 == 1:
f1_sgn = -f1_sgn
if 1 in indices:
if indices[1] % 2 == 1:
f2_sgn = -f2_sgn
if not (a == b and b == t):
Q.append(sgn[(f1_sgn, f2_sgn)])
return Q
def _traverse_quadrants(Q_L1, Q_L2, Q_L3, Q_L4, exclude=None):
"""Transform sequences of quadrants to a sequence of rules. """
if exclude is True:
edges = [1, 1, 0, 0]
corners = {
(0, 1): 1,
(1, 2): 1,
(2, 3): 0,
(3, 0): 1,
}
else:
edges = [0, 0, 0, 0]
corners = {
(0, 1): 0,
(1, 2): 0,
(2, 3): 0,
(3, 0): 0,
}
if exclude is not None and exclude is not True:
exclude = set(exclude)
for i, edge in enumerate(['S', 'E', 'N', 'W']):
if edge in exclude:
edges[i] = 1
for i, corner in enumerate(['SW', 'SE', 'NE', 'NW']):
if corner in exclude:
corners[((i - 1) % 4, i)] = 1
QQ, rules = [Q_L1, Q_L2, Q_L3, Q_L4], []
for i, Q in enumerate(QQ):
if not Q:
continue
if Q[-1] == OO:
Q = Q[:-1]
if Q[0] == OO:
j, Q = (i - 1) % 4, Q[1:]
qq = (QQ[j][-2], OO, Q[0])
if qq in _rules_ambiguous:
rules.append((_rules_ambiguous[qq], corners[(j, i)]))
else:
raise NotImplementedError("3 element rule (corner): " + str(qq))
q1, k = Q[0], 1
while k < len(Q):
q2, k = Q[k], k + 1
if q2 != OO:
qq = (q1, q2)
if qq in _rules_simple:
rules.append((_rules_simple[qq], 0))
elif qq in _rules_ambiguous:
rules.append((_rules_ambiguous[qq], edges[i]))
else:
raise NotImplementedError("2 element rule (inside): " + str(qq))
else:
qq, k = (q1, q2, Q[k]), k + 1
if qq in _rules_ambiguous:
rules.append((_rules_ambiguous[qq], edges[i]))
else:
raise NotImplementedError("3 element rule (edge): " + str(qq))
q1 = qq[-1]
return rules
def _reverse_intervals(intervals):
"""Reverse intervals for traversal from right to left and from top to bottom. """
return [ ((b, a), indices, f) for (a, b), indices, f in reversed(intervals) ]
def _winding_number(T, field):
"""Compute the winding number of the input polynomial, i.e. the number of roots. """
return int(sum([ field(*_values[t][i]) for t, i in T ]) / field(2))
def dup_count_complex_roots(f, K, inf=None, sup=None, exclude=None):
"""Count all roots in [u + v*I, s + t*I] rectangle using Collins-Krandick algorithm. """
if not K.is_ZZ and not K.is_QQ:
raise DomainError("complex root counting is not supported over %s" % K)
if K.is_ZZ:
R, F = K, K.get_field()
else:
R, F = K.get_ring(), K
f = dup_convert(f, K, F)
if inf is None or sup is None:
_, lc = dup_degree(f), abs(dup_LC(f, F))
B = 2*max([ F.quo(abs(c), lc) for c in f ])
if inf is None:
(u, v) = (-B, -B)
else:
(u, v) = inf
if sup is None:
(s, t) = (+B, +B)
else:
(s, t) = sup
f1, f2 = dup_real_imag(f, F)
f1L1F = dmp_eval_in(f1, v, 1, 1, F)
f2L1F = dmp_eval_in(f2, v, 1, 1, F)
_, f1L1R = dup_clear_denoms(f1L1F, F, R, convert=True)
_, f2L1R = dup_clear_denoms(f2L1F, F, R, convert=True)
f1L2F = dmp_eval_in(f1, s, 0, 1, F)
f2L2F = dmp_eval_in(f2, s, 0, 1, F)
_, f1L2R = dup_clear_denoms(f1L2F, F, R, convert=True)
_, f2L2R = dup_clear_denoms(f2L2F, F, R, convert=True)
f1L3F = dmp_eval_in(f1, t, 1, 1, F)
f2L3F = dmp_eval_in(f2, t, 1, 1, F)
_, f1L3R = dup_clear_denoms(f1L3F, F, R, convert=True)
_, f2L3R = dup_clear_denoms(f2L3F, F, R, convert=True)
f1L4F = dmp_eval_in(f1, u, 0, 1, F)
f2L4F = dmp_eval_in(f2, u, 0, 1, F)
_, f1L4R = dup_clear_denoms(f1L4F, F, R, convert=True)
_, f2L4R = dup_clear_denoms(f2L4F, F, R, convert=True)
S_L1 = [f1L1R, f2L1R]
S_L2 = [f1L2R, f2L2R]
S_L3 = [f1L3R, f2L3R]
S_L4 = [f1L4R, f2L4R]
I_L1 = dup_isolate_real_roots_list(S_L1, R, inf=u, sup=s, fast=True, basis=True, strict=True)
I_L2 = dup_isolate_real_roots_list(S_L2, R, inf=v, sup=t, fast=True, basis=True, strict=True)
I_L3 = dup_isolate_real_roots_list(S_L3, R, inf=u, sup=s, fast=True, basis=True, strict=True)
I_L4 = dup_isolate_real_roots_list(S_L4, R, inf=v, sup=t, fast=True, basis=True, strict=True)
I_L3 = _reverse_intervals(I_L3)
I_L4 = _reverse_intervals(I_L4)
Q_L1 = _intervals_to_quadrants(I_L1, f1L1F, f2L1F, u, s, F)
Q_L2 = _intervals_to_quadrants(I_L2, f1L2F, f2L2F, v, t, F)
Q_L3 = _intervals_to_quadrants(I_L3, f1L3F, f2L3F, s, u, F)
Q_L4 = _intervals_to_quadrants(I_L4, f1L4F, f2L4F, t, v, F)
T = _traverse_quadrants(Q_L1, Q_L2, Q_L3, Q_L4, exclude=exclude)
return _winding_number(T, F)
def _vertical_bisection(N, a, b, I, Q, F1, F2, f1, f2, F):
"""Vertical bisection step in Collins-Krandick root isolation algorithm. """
(u, v), (s, t) = a, b
I_L1, I_L2, I_L3, I_L4 = I
Q_L1, Q_L2, Q_L3, Q_L4 = Q
f1L1F, f1L2F, f1L3F, f1L4F = F1
f2L1F, f2L2F, f2L3F, f2L4F = F2
x = (u + s) / 2
f1V = dmp_eval_in(f1, x, 0, 1, F)
f2V = dmp_eval_in(f2, x, 0, 1, F)
I_V = dup_isolate_real_roots_list([f1V, f2V], F, inf=v, sup=t, fast=True, strict=True, basis=True)
I_L1_L, I_L1_R = [], []
I_L2_L, I_L2_R = I_V, I_L2
I_L3_L, I_L3_R = [], []
I_L4_L, I_L4_R = I_L4, _reverse_intervals(I_V)
for I in I_L1:
(a, b), indices, h = I
if a == b:
if a == x:
I_L1_L.append(I)
I_L1_R.append(I)
elif a < x:
I_L1_L.append(I)
else:
I_L1_R.append(I)
else:
if b <= x:
I_L1_L.append(I)
elif a >= x:
I_L1_R.append(I)
else:
a, b = dup_refine_real_root(h, a, b, F.get_ring(), disjoint=x, fast=True)
if b <= x:
I_L1_L.append(((a, b), indices, h))
if a >= x:
I_L1_R.append(((a, b), indices, h))
for I in I_L3:
(b, a), indices, h = I
if a == b:
if a == x:
I_L3_L.append(I)
I_L3_R.append(I)
elif a < x:
I_L3_L.append(I)
else:
I_L3_R.append(I)
else:
if b <= x:
I_L3_L.append(I)
elif a >= x:
I_L3_R.append(I)
else:
a, b = dup_refine_real_root(h, a, b, F.get_ring(), disjoint=x, fast=True)
if b <= x:
I_L3_L.append(((b, a), indices, h))
if a >= x:
I_L3_R.append(((b, a), indices, h))
Q_L1_L = _intervals_to_quadrants(I_L1_L, f1L1F, f2L1F, u, x, F)
Q_L2_L = _intervals_to_quadrants(I_L2_L, f1V, f2V, v, t, F)
Q_L3_L = _intervals_to_quadrants(I_L3_L, f1L3F, f2L3F, x, u, F)
Q_L4_L = Q_L4
Q_L1_R = _intervals_to_quadrants(I_L1_R, f1L1F, f2L1F, x, s, F)
Q_L2_R = Q_L2
Q_L3_R = _intervals_to_quadrants(I_L3_R, f1L3F, f2L3F, s, x, F)
Q_L4_R = _intervals_to_quadrants(I_L4_R, f1V, f2V, t, v, F)
T_L = _traverse_quadrants(Q_L1_L, Q_L2_L, Q_L3_L, Q_L4_L, exclude=True)
T_R = _traverse_quadrants(Q_L1_R, Q_L2_R, Q_L3_R, Q_L4_R, exclude=True)
N_L = _winding_number(T_L, F)
N_R = _winding_number(T_R, F)
I_L = (I_L1_L, I_L2_L, I_L3_L, I_L4_L)
Q_L = (Q_L1_L, Q_L2_L, Q_L3_L, Q_L4_L)
I_R = (I_L1_R, I_L2_R, I_L3_R, I_L4_R)
Q_R = (Q_L1_R, Q_L2_R, Q_L3_R, Q_L4_R)
F1_L = (f1L1F, f1V, f1L3F, f1L4F)
F2_L = (f2L1F, f2V, f2L3F, f2L4F)
F1_R = (f1L1F, f1L2F, f1L3F, f1V)
F2_R = (f2L1F, f2L2F, f2L3F, f2V)
a, b = (u, v), (x, t)
c, d = (x, v), (s, t)
D_L = (N_L, a, b, I_L, Q_L, F1_L, F2_L)
D_R = (N_R, c, d, I_R, Q_R, F1_R, F2_R)
return D_L, D_R
def _horizontal_bisection(N, a, b, I, Q, F1, F2, f1, f2, F):
"""Horizontal bisection step in Collins-Krandick root isolation algorithm. """
(u, v), (s, t) = a, b
I_L1, I_L2, I_L3, I_L4 = I
Q_L1, Q_L2, Q_L3, Q_L4 = Q
f1L1F, f1L2F, f1L3F, f1L4F = F1
f2L1F, f2L2F, f2L3F, f2L4F = F2
y = (v + t) / 2
f1H = dmp_eval_in(f1, y, 1, 1, F)
f2H = dmp_eval_in(f2, y, 1, 1, F)
I_H = dup_isolate_real_roots_list([f1H, f2H], F, inf=u, sup=s, fast=True, strict=True, basis=True)
I_L1_B, I_L1_U = I_L1, I_H
I_L2_B, I_L2_U = [], []
I_L3_B, I_L3_U = _reverse_intervals(I_H), I_L3
I_L4_B, I_L4_U = [], []
for I in I_L2:
(a, b), indices, h = I
if a == b:
if a == y:
I_L2_B.append(I)
I_L2_U.append(I)
elif a < y:
I_L2_B.append(I)
else:
I_L2_U.append(I)
else:
if b <= y:
I_L2_B.append(I)
elif a >= y:
I_L2_U.append(I)
else:
a, b = dup_refine_real_root(h, a, b, F.get_ring(), disjoint=y, fast=True)
if b <= y:
I_L2_B.append(((a, b), indices, h))
if a >= y:
I_L2_U.append(((a, b), indices, h))
for I in I_L4:
(b, a), indices, h = I
if a == b:
if a == y:
I_L4_B.append(I)
I_L4_U.append(I)
elif a < y:
I_L4_B.append(I)
else:
I_L4_U.append(I)
else:
if b <= y:
I_L4_B.append(I)
elif a >= y:
I_L4_U.append(I)
else:
a, b = dup_refine_real_root(h, a, b, F.get_ring(), disjoint=y, fast=True)
if b <= y:
I_L4_B.append(((b, a), indices, h))
if a >= y:
I_L4_U.append(((b, a), indices, h))
Q_L1_B = Q_L1
Q_L2_B = _intervals_to_quadrants(I_L2_B, f1L2F, f2L2F, v, y, F)
Q_L3_B = _intervals_to_quadrants(I_L3_B, f1H, f2H, s, u, F)
Q_L4_B = _intervals_to_quadrants(I_L4_B, f1L4F, f2L4F, y, v, F)
Q_L1_U = _intervals_to_quadrants(I_L1_U, f1H, f2H, u, s, F)
Q_L2_U = _intervals_to_quadrants(I_L2_U, f1L2F, f2L2F, y, t, F)
Q_L3_U = Q_L3
Q_L4_U = _intervals_to_quadrants(I_L4_U, f1L4F, f2L4F, t, y, F)
T_B = _traverse_quadrants(Q_L1_B, Q_L2_B, Q_L3_B, Q_L4_B, exclude=True)
T_U = _traverse_quadrants(Q_L1_U, Q_L2_U, Q_L3_U, Q_L4_U, exclude=True)
N_B = _winding_number(T_B, F)
N_U = _winding_number(T_U, F)
I_B = (I_L1_B, I_L2_B, I_L3_B, I_L4_B)
Q_B = (Q_L1_B, Q_L2_B, Q_L3_B, Q_L4_B)
I_U = (I_L1_U, I_L2_U, I_L3_U, I_L4_U)
Q_U = (Q_L1_U, Q_L2_U, Q_L3_U, Q_L4_U)
F1_B = (f1L1F, f1L2F, f1H, f1L4F)
F2_B = (f2L1F, f2L2F, f2H, f2L4F)
F1_U = (f1H, f1L2F, f1L3F, f1L4F)
F2_U = (f2H, f2L2F, f2L3F, f2L4F)
a, b = (u, v), (s, y)
c, d = (u, y), (s, t)
D_B = (N_B, a, b, I_B, Q_B, F1_B, F2_B)
D_U = (N_U, c, d, I_U, Q_U, F1_U, F2_U)
return D_B, D_U
def _depth_first_select(rectangles):
"""Find a rectangle of minimum area for bisection. """
min_area, j = None, None
for i, (_, (u, v), (s, t), _, _, _, _) in enumerate(rectangles):
area = (s - u)*(t - v)
if min_area is None or area < min_area:
min_area, j = area, i
return rectangles.pop(j)
def _rectangle_small_p(a, b, eps):
"""Return ``True`` if the given rectangle is small enough. """
(u, v), (s, t) = a, b
if eps is not None:
return s - u < eps and t - v < eps
else:
return True
def dup_isolate_complex_roots_sqf(f, K, eps=None, inf=None, sup=None, blackbox=False):
"""Isolate complex roots of a square-free polynomial using Collins-Krandick algorithm. """
if not K.is_ZZ and not K.is_QQ:
raise DomainError("isolation of complex roots is not supported over %s" % K)
if dup_degree(f) <= 0:
return []
if K.is_ZZ:
F = K.get_field()
else:
F = K
f = dup_convert(f, K, F)
lc = abs(dup_LC(f, F))
B = 2*max([ F.quo(abs(c), lc) for c in f ])
(u, v), (s, t) = (-B, F.zero), (B, B)
if inf is not None:
u = inf
if sup is not None:
s = sup
if v < 0 or t <= v or s <= u:
raise ValueError("not a valid complex isolation rectangle")
f1, f2 = dup_real_imag(f, F)
f1L1 = dmp_eval_in(f1, v, 1, 1, F)
f2L1 = dmp_eval_in(f2, v, 1, 1, F)
f1L2 = dmp_eval_in(f1, s, 0, 1, F)
f2L2 = dmp_eval_in(f2, s, 0, 1, F)
f1L3 = dmp_eval_in(f1, t, 1, 1, F)
f2L3 = dmp_eval_in(f2, t, 1, 1, F)
f1L4 = dmp_eval_in(f1, u, 0, 1, F)
f2L4 = dmp_eval_in(f2, u, 0, 1, F)
S_L1 = [f1L1, f2L1]
S_L2 = [f1L2, f2L2]
S_L3 = [f1L3, f2L3]
S_L4 = [f1L4, f2L4]
I_L1 = dup_isolate_real_roots_list(S_L1, F, inf=u, sup=s, fast=True, strict=True, basis=True)
I_L2 = dup_isolate_real_roots_list(S_L2, F, inf=v, sup=t, fast=True, strict=True, basis=True)
I_L3 = dup_isolate_real_roots_list(S_L3, F, inf=u, sup=s, fast=True, strict=True, basis=True)
I_L4 = dup_isolate_real_roots_list(S_L4, F, inf=v, sup=t, fast=True, strict=True, basis=True)
I_L3 = _reverse_intervals(I_L3)
I_L4 = _reverse_intervals(I_L4)
Q_L1 = _intervals_to_quadrants(I_L1, f1L1, f2L1, u, s, F)
Q_L2 = _intervals_to_quadrants(I_L2, f1L2, f2L2, v, t, F)
Q_L3 = _intervals_to_quadrants(I_L3, f1L3, f2L3, s, u, F)
Q_L4 = _intervals_to_quadrants(I_L4, f1L4, f2L4, t, v, F)
T = _traverse_quadrants(Q_L1, Q_L2, Q_L3, Q_L4)
N = _winding_number(T, F)
if not N:
return []
I = (I_L1, I_L2, I_L3, I_L4)
Q = (Q_L1, Q_L2, Q_L3, Q_L4)
F1 = (f1L1, f1L2, f1L3, f1L4)
F2 = (f2L1, f2L2, f2L3, f2L4)
rectangles, roots = [(N, (u, v), (s, t), I, Q, F1, F2)], []
while rectangles:
N, (u, v), (s, t), I, Q, F1, F2 = _depth_first_select(rectangles)
if s - u > t - v:
D_L, D_R = _vertical_bisection(N, (u, v), (s, t), I, Q, F1, F2, f1, f2, F)
N_L, a, b, I_L, Q_L, F1_L, F2_L = D_L
N_R, c, d, I_R, Q_R, F1_R, F2_R = D_R
if N_L >= 1:
if N_L == 1 and _rectangle_small_p(a, b, eps):
roots.append(ComplexInterval(a, b, I_L, Q_L, F1_L, F2_L, f1, f2, F))
else:
rectangles.append(D_L)
if N_R >= 1:
if N_R == 1 and _rectangle_small_p(c, d, eps):
roots.append(ComplexInterval(c, d, I_R, Q_R, F1_R, F2_R, f1, f2, F))
else:
rectangles.append(D_R)
else:
D_B, D_U = _horizontal_bisection(N, (u, v), (s, t), I, Q, F1, F2, f1, f2, F)
N_B, a, b, I_B, Q_B, F1_B, F2_B = D_B
N_U, c, d, I_U, Q_U, F1_U, F2_U = D_U
if N_B >= 1:
if N_B == 1 and _rectangle_small_p(a, b, eps):
roots.append(ComplexInterval(
a, b, I_B, Q_B, F1_B, F2_B, f1, f2, F))
else:
rectangles.append(D_B)
if N_U >= 1:
if N_U == 1 and _rectangle_small_p(c, d, eps):
roots.append(ComplexInterval(
c, d, I_U, Q_U, F1_U, F2_U, f1, f2, F))
else:
rectangles.append(D_U)
_roots, roots = sorted(roots, key=lambda r: (r.ax, r.ay)), []
for root in _roots:
roots.extend([root.conjugate(), root])
if blackbox:
return roots
else:
return [ r.as_tuple() for r in roots ]
def dup_isolate_all_roots_sqf(f, K, eps=None, inf=None, sup=None, fast=False, blackbox=False):
"""Isolate real and complex roots of a square-free polynomial ``f``. """
return (
dup_isolate_real_roots_sqf( f, K, eps=eps, inf=inf, sup=sup, fast=fast, blackbox=blackbox),
dup_isolate_complex_roots_sqf(f, K, eps=eps, inf=inf, sup=sup, blackbox=blackbox))
def dup_isolate_all_roots(f, K, eps=None, inf=None, sup=None, fast=False):
"""Isolate real and complex roots of a non-square-free polynomial ``f``. """
if not K.is_ZZ and not K.is_QQ:
raise DomainError("isolation of real and complex roots is not supported over %s" % K)
_, factors = dup_sqf_list(f, K)
if len(factors) == 1:
((f, k),) = factors
real_part, complex_part = dup_isolate_all_roots_sqf(
f, K, eps=eps, inf=inf, sup=sup, fast=fast)
real_part = [ ((a, b), k) for (a, b) in real_part ]
complex_part = [ ((a, b), k) for (a, b) in complex_part ]
return real_part, complex_part
else:
raise NotImplementedError( "only trivial square-free polynomials are supported")
class RealInterval:
"""A fully qualified representation of a real isolation interval. """
def __init__(self, data, f, dom):
"""Initialize new real interval with complete information. """
if len(data) == 2:
s, t = data
self.neg = False
if s < 0:
if t <= 0:
f, s, t, self.neg = dup_mirror(f, dom), -t, -s, True
else:
raise ValueError("can't refine a real root in (%s, %s)" % (s, t))
a, b, c, d = _mobius_from_interval((s, t), dom.get_field())
f = dup_transform(f, dup_strip([a, b]),
dup_strip([c, d]), dom)
self.mobius = a, b, c, d
else:
self.mobius = data[:-1]
self.neg = data[-1]
self.f, self.dom = f, dom
@property
def func(self):
return RealInterval
@property
def args(self):
i = self
return (i.mobius + (i.neg,), i.f, i.dom)
def __eq__(self, other):
if type(other) != type(self):
return False
return self.args == other.args
@property
def a(self):
"""Return the position of the left end. """
field = self.dom.get_field()
a, b, c, d = self.mobius
if not self.neg:
if a*d < b*c:
return field(a, c)
return field(b, d)
else:
if a*d > b*c:
return -field(a, c)
return -field(b, d)
@property
def b(self):
"""Return the position of the right end. """
was = self.neg
self.neg = not was
rv = -self.a
self.neg = was
return rv
@property
def dx(self):
"""Return width of the real isolating interval. """
return self.b - self.a
@property
def center(self):
"""Return the center of the real isolating interval. """
return (self.a + self.b)/2
def as_tuple(self):
"""Return tuple representation of real isolating interval. """
return (self.a, self.b)
def __repr__(self):
return "(%s, %s)" % (self.a, self.b)
def is_disjoint(self, other):
"""Return ``True`` if two isolation intervals are disjoint. """
if isinstance(other, RealInterval):
return (self.b < other.a or other.b < self.a)
assert isinstance(other, ComplexInterval)
return (self.b < other.ax or other.bx < self.a
or other.ay*other.by > 0)
def _inner_refine(self):
"""Internal one step real root refinement procedure. """
if self.mobius is None:
return self
f, mobius = dup_inner_refine_real_root(
self.f, self.mobius, self.dom, steps=1, mobius=True)
return RealInterval(mobius + (self.neg,), f, self.dom)
def refine_disjoint(self, other):
"""Refine an isolating interval until it is disjoint with another one. """
expr = self
while not expr.is_disjoint(other):
expr, other = expr._inner_refine(), other._inner_refine()
return expr, other
def refine_size(self, dx):
"""Refine an isolating interval until it is of sufficiently small size. """
expr = self
while not (expr.dx < dx):
expr = expr._inner_refine()
return expr
def refine_step(self, steps=1):
"""Perform several steps of real root refinement algorithm. """
expr = self
for _ in range(steps):
expr = expr._inner_refine()
return expr
def refine(self):
"""Perform one step of real root refinement algorithm. """
return self._inner_refine()
class ComplexInterval:
"""A fully qualified representation of a complex isolation interval.
The printed form is shown as (ax, bx) x (ay, by) where (ax, ay)
and (bx, by) are the coordinates of the southwest and northeast
corners of the interval's rectangle, respectively.
Examples
========
>>> from sympy import CRootOf, S
>>> from sympy.abc import x
>>> CRootOf.clear_cache() # for doctest reproducibility
>>> root = CRootOf(x**10 - 2*x + 3, 9)
>>> i = root._get_interval(); i
(3/64, 3/32) x (9/8, 75/64)
The real part of the root lies within the range [0, 3/4] while
the imaginary part lies within the range [9/8, 3/2]:
>>> root.n(3)
0.0766 + 1.14*I
The width of the ranges in the x and y directions on the complex
plane are:
>>> i.dx, i.dy
(3/64, 3/64)
The center of the range is
>>> i.center
(9/128, 147/128)
The northeast coordinate of the rectangle bounding the root in the
complex plane is given by attribute b and the x and y components
are accessed by bx and by:
>>> i.b, i.bx, i.by
((3/32, 75/64), 3/32, 75/64)
The southwest coordinate is similarly given by i.a
>>> i.a, i.ax, i.ay
((3/64, 9/8), 3/64, 9/8)
Although the interval prints to show only the real and imaginary
range of the root, all the information of the underlying root
is contained as properties of the interval.
For example, an interval with a nonpositive imaginary range is
considered to be the conjugate. Since the y values of y are in the
range [0, 1/4] it is not the conjugate:
>>> i.conj
False
The conjugate's interval is
>>> ic = i.conjugate(); ic
(3/64, 3/32) x (-75/64, -9/8)
NOTE: the values printed still represent the x and y range
in which the root -- conjugate, in this case -- is located,
but the underlying a and b values of a root and its conjugate
are the same:
>>> assert i.a == ic.a and i.b == ic.b
What changes are the reported coordinates of the bounding rectangle:
>>> (i.ax, i.ay), (i.bx, i.by)
((3/64, 9/8), (3/32, 75/64))
>>> (ic.ax, ic.ay), (ic.bx, ic.by)
((3/64, -75/64), (3/32, -9/8))
The interval can be refined once:
>>> i # for reference, this is the current interval
(3/64, 3/32) x (9/8, 75/64)
>>> i.refine()
(3/64, 3/32) x (9/8, 147/128)
Several refinement steps can be taken:
>>> i.refine_step(2) # 2 steps
(9/128, 3/32) x (9/8, 147/128)
It is also possible to refine to a given tolerance:
>>> tol = min(i.dx, i.dy)/2
>>> i.refine_size(tol)
(9/128, 21/256) x (9/8, 291/256)
A disjoint interval is one whose bounding rectangle does not
overlap with another. An interval, necessarily, is not disjoint with
itself, but any interval is disjoint with a conjugate since the
conjugate rectangle will always be in the lower half of the complex
plane and the non-conjugate in the upper half:
>>> i.is_disjoint(i), i.is_disjoint(i.conjugate())
(False, True)
The following interval j is not disjoint from i:
>>> close = CRootOf(x**10 - 2*x + 300/S(101), 9)
>>> j = close._get_interval(); j
(75/1616, 75/808) x (225/202, 1875/1616)
>>> i.is_disjoint(j)
False
The two can be made disjoint, however:
>>> newi, newj = i.refine_disjoint(j)
>>> newi
(39/512, 159/2048) x (2325/2048, 4653/4096)
>>> newj
(3975/51712, 2025/25856) x (29325/25856, 117375/103424)
Even though the real ranges overlap, the imaginary do not, so
the roots have been resolved as distinct. Intervals are disjoint
when either the real or imaginary component of the intervals is
distinct. In the case above, the real components have not been
resolved (so we don't know, yet, which root has the smaller real
part) but the imaginary part of ``close`` is larger than ``root``:
>>> close.n(3)
0.0771 + 1.13*I
>>> root.n(3)
0.0766 + 1.14*I
"""
def __init__(self, a, b, I, Q, F1, F2, f1, f2, dom, conj=False):
"""Initialize new complex interval with complete information. """
# a and b are the SW and NE corner of the bounding interval,
# (ax, ay) and (bx, by), respectively, for the NON-CONJUGATE
# root (the one with the positive imaginary part); when working
# with the conjugate, the a and b value are still non-negative
# but the ay, by are reversed and have oppositite sign
self.a, self.b = a, b
self.I, self.Q = I, Q
self.f1, self.F1 = f1, F1
self.f2, self.F2 = f2, F2
self.dom = dom
self.conj = conj
@property
def func(self):
return ComplexInterval
@property
def args(self):
i = self
return (i.a, i.b, i.I, i.Q, i.F1, i.F2, i.f1, i.f2, i.dom, i.conj)
def __eq__(self, other):
if type(other) != type(self):
return False
return self.args == other.args
@property
def ax(self):
"""Return ``x`` coordinate of south-western corner. """
return self.a[0]
@property
def ay(self):
"""Return ``y`` coordinate of south-western corner. """
if not self.conj:
return self.a[1]
else:
return -self.b[1]
@property
def bx(self):
"""Return ``x`` coordinate of north-eastern corner. """
return self.b[0]
@property
def by(self):
"""Return ``y`` coordinate of north-eastern corner. """
if not self.conj:
return self.b[1]
else:
return -self.a[1]
@property
def dx(self):
"""Return width of the complex isolating interval. """
return self.b[0] - self.a[0]
@property
def dy(self):
"""Return height of the complex isolating interval. """
return self.b[1] - self.a[1]
@property
def center(self):
"""Return the center of the complex isolating interval. """
return ((self.ax + self.bx)/2, (self.ay + self.by)/2)
def as_tuple(self):
"""Return tuple representation of the complex isolating
interval's SW and NE corners, respectively. """
return ((self.ax, self.ay), (self.bx, self.by))
def __repr__(self):
return "(%s, %s) x (%s, %s)" % (self.ax, self.bx, self.ay, self.by)
def conjugate(self):
"""This complex interval really is located in lower half-plane. """
return ComplexInterval(self.a, self.b, self.I, self.Q,
self.F1, self.F2, self.f1, self.f2, self.dom, conj=True)
def is_disjoint(self, other):
"""Return ``True`` if two isolation intervals are disjoint. """
if isinstance(other, RealInterval):
return other.is_disjoint(self)
if self.conj != other.conj: # above and below real axis
return True
re_distinct = (self.bx < other.ax or other.bx < self.ax)
if re_distinct:
return True
im_distinct = (self.by < other.ay or other.by < self.ay)
return im_distinct
def _inner_refine(self):
"""Internal one step complex root refinement procedure. """
(u, v), (s, t) = self.a, self.b
I, Q = self.I, self.Q
f1, F1 = self.f1, self.F1
f2, F2 = self.f2, self.F2
dom = self.dom
if s - u > t - v:
D_L, D_R = _vertical_bisection(1, (u, v), (s, t), I, Q, F1, F2, f1, f2, dom)
if D_L[0] == 1:
_, a, b, I, Q, F1, F2 = D_L
else:
_, a, b, I, Q, F1, F2 = D_R
else:
D_B, D_U = _horizontal_bisection(1, (u, v), (s, t), I, Q, F1, F2, f1, f2, dom)
if D_B[0] == 1:
_, a, b, I, Q, F1, F2 = D_B
else:
_, a, b, I, Q, F1, F2 = D_U
return ComplexInterval(a, b, I, Q, F1, F2, f1, f2, dom, self.conj)
def refine_disjoint(self, other):
"""Refine an isolating interval until it is disjoint with another one. """
expr = self
while not expr.is_disjoint(other):
expr, other = expr._inner_refine(), other._inner_refine()
return expr, other
def refine_size(self, dx, dy=None):
"""Refine an isolating interval until it is of sufficiently small size. """
if dy is None:
dy = dx
expr = self
while not (expr.dx < dx and expr.dy < dy):
expr = expr._inner_refine()
return expr
def refine_step(self, steps=1):
"""Perform several steps of complex root refinement algorithm. """
expr = self
for _ in range(steps):
expr = expr._inner_refine()
return expr
def refine(self):
"""Perform one step of complex root refinement algorithm. """
return self._inner_refine()
|
389a05bb797d3799fe7d86f3348815a19ddfda5694f462d279890cf8c54adaba | """Useful utilities for higher level polynomial classes. """
from sympy.core import (S, Add, Mul, Pow, Eq, Expr,
expand_mul, expand_multinomial)
from sympy.core.exprtools import decompose_power, decompose_power_rat
from sympy.polys.polyerrors import PolynomialError, GeneratorsError
from sympy.polys.polyoptions import build_options
import re
_gens_order = {
'a': 301, 'b': 302, 'c': 303, 'd': 304,
'e': 305, 'f': 306, 'g': 307, 'h': 308,
'i': 309, 'j': 310, 'k': 311, 'l': 312,
'm': 313, 'n': 314, 'o': 315, 'p': 216,
'q': 217, 'r': 218, 's': 219, 't': 220,
'u': 221, 'v': 222, 'w': 223, 'x': 124,
'y': 125, 'z': 126,
}
_max_order = 1000
_re_gen = re.compile(r"^(.+?)(\d*)$")
def _nsort(roots, separated=False):
"""Sort the numerical roots putting the real roots first, then sorting
according to real and imaginary parts. If ``separated`` is True, then
the real and imaginary roots will be returned in two lists, respectively.
This routine tries to avoid issue 6137 by separating the roots into real
and imaginary parts before evaluation. In addition, the sorting will raise
an error if any computation cannot be done with precision.
"""
if not all(r.is_number for r in roots):
raise NotImplementedError
# see issue 6137:
# get the real part of the evaluated real and imaginary parts of each root
key = [[i.n(2).as_real_imag()[0] for i in r.as_real_imag()] for r in roots]
# make sure the parts were computed with precision
if len(roots) > 1 and any(i._prec == 1 for k in key for i in k):
raise NotImplementedError("could not compute root with precision")
# insert a key to indicate if the root has an imaginary part
key = [(1 if i else 0, r, i) for r, i in key]
key = sorted(zip(key, roots))
# return the real and imaginary roots separately if desired
if separated:
r = []
i = []
for (im, _, _), v in key:
if im:
i.append(v)
else:
r.append(v)
return r, i
_, roots = zip(*key)
return list(roots)
def _sort_gens(gens, **args):
"""Sort generators in a reasonably intelligent way. """
opt = build_options(args)
gens_order, wrt = {}, None
if opt is not None:
gens_order, wrt = {}, opt.wrt
for i, gen in enumerate(opt.sort):
gens_order[gen] = i + 1
def order_key(gen):
gen = str(gen)
if wrt is not None:
try:
return (-len(wrt) + wrt.index(gen), gen, 0)
except ValueError:
pass
name, index = _re_gen.match(gen).groups()
if index:
index = int(index)
else:
index = 0
try:
return ( gens_order[name], name, index)
except KeyError:
pass
try:
return (_gens_order[name], name, index)
except KeyError:
pass
return (_max_order, name, index)
try:
gens = sorted(gens, key=order_key)
except TypeError: # pragma: no cover
pass
return tuple(gens)
def _unify_gens(f_gens, g_gens):
"""Unify generators in a reasonably intelligent way. """
f_gens = list(f_gens)
g_gens = list(g_gens)
if f_gens == g_gens:
return tuple(f_gens)
gens, common, k = [], [], 0
for gen in f_gens:
if gen in g_gens:
common.append(gen)
for i, gen in enumerate(g_gens):
if gen in common:
g_gens[i], k = common[k], k + 1
for gen in common:
i = f_gens.index(gen)
gens.extend(f_gens[:i])
f_gens = f_gens[i + 1:]
i = g_gens.index(gen)
gens.extend(g_gens[:i])
g_gens = g_gens[i + 1:]
gens.append(gen)
gens.extend(f_gens)
gens.extend(g_gens)
return tuple(gens)
def _analyze_gens(gens):
"""Support for passing generators as `*gens` and `[gens]`. """
if len(gens) == 1 and hasattr(gens[0], '__iter__'):
return tuple(gens[0])
else:
return tuple(gens)
def _sort_factors(factors, **args):
"""Sort low-level factors in increasing 'complexity' order. """
def order_if_multiple_key(factor):
(f, n) = factor
return (len(f), n, f)
def order_no_multiple_key(f):
return (len(f), f)
if args.get('multiple', True):
return sorted(factors, key=order_if_multiple_key)
else:
return sorted(factors, key=order_no_multiple_key)
illegal = [S.NaN, S.Infinity, S.NegativeInfinity, S.ComplexInfinity]
illegal_types = [type(obj) for obj in illegal]
finf = [float(i) for i in illegal[1:3]]
def _not_a_coeff(expr):
"""Do not treat NaN and infinities as valid polynomial coefficients. """
if type(expr) in illegal_types or expr in finf:
return True
if type(expr) is float and float(expr) != expr:
return True # nan
return # could be
def _parallel_dict_from_expr_if_gens(exprs, opt):
"""Transform expressions into a multinomial form given generators. """
k, indices = len(opt.gens), {}
for i, g in enumerate(opt.gens):
indices[g] = i
polys = []
for expr in exprs:
poly = {}
if expr.is_Equality:
expr = expr.lhs - expr.rhs
for term in Add.make_args(expr):
coeff, monom = [], [0]*k
for factor in Mul.make_args(term):
if not _not_a_coeff(factor) and factor.is_Number:
coeff.append(factor)
else:
try:
if opt.series is False:
base, exp = decompose_power(factor)
if exp < 0:
exp, base = -exp, Pow(base, -S.One)
else:
base, exp = decompose_power_rat(factor)
monom[indices[base]] = exp
except KeyError:
if not factor.free_symbols.intersection(opt.gens):
coeff.append(factor)
else:
raise PolynomialError("%s contains an element of "
"the set of generators." % factor)
monom = tuple(monom)
if monom in poly:
poly[monom] += Mul(*coeff)
else:
poly[monom] = Mul(*coeff)
polys.append(poly)
return polys, opt.gens
def _parallel_dict_from_expr_no_gens(exprs, opt):
"""Transform expressions into a multinomial form and figure out generators. """
if opt.domain is not None:
def _is_coeff(factor):
return factor in opt.domain
elif opt.extension is True:
def _is_coeff(factor):
return factor.is_algebraic
elif opt.greedy is not False:
def _is_coeff(factor):
return factor is S.ImaginaryUnit
else:
def _is_coeff(factor):
return factor.is_number
gens, reprs = set(), []
for expr in exprs:
terms = []
if expr.is_Equality:
expr = expr.lhs - expr.rhs
for term in Add.make_args(expr):
coeff, elements = [], {}
for factor in Mul.make_args(term):
if not _not_a_coeff(factor) and (factor.is_Number or _is_coeff(factor)):
coeff.append(factor)
else:
if opt.series is False:
base, exp = decompose_power(factor)
if exp < 0:
exp, base = -exp, Pow(base, -S.One)
else:
base, exp = decompose_power_rat(factor)
elements[base] = elements.setdefault(base, 0) + exp
gens.add(base)
terms.append((coeff, elements))
reprs.append(terms)
gens = _sort_gens(gens, opt=opt)
k, indices = len(gens), {}
for i, g in enumerate(gens):
indices[g] = i
polys = []
for terms in reprs:
poly = {}
for coeff, term in terms:
monom = [0]*k
for base, exp in term.items():
monom[indices[base]] = exp
monom = tuple(monom)
if monom in poly:
poly[monom] += Mul(*coeff)
else:
poly[monom] = Mul(*coeff)
polys.append(poly)
return polys, tuple(gens)
def _dict_from_expr_if_gens(expr, opt):
"""Transform an expression into a multinomial form given generators. """
(poly,), gens = _parallel_dict_from_expr_if_gens((expr,), opt)
return poly, gens
def _dict_from_expr_no_gens(expr, opt):
"""Transform an expression into a multinomial form and figure out generators. """
(poly,), gens = _parallel_dict_from_expr_no_gens((expr,), opt)
return poly, gens
def parallel_dict_from_expr(exprs, **args):
"""Transform expressions into a multinomial form. """
reps, opt = _parallel_dict_from_expr(exprs, build_options(args))
return reps, opt.gens
def _parallel_dict_from_expr(exprs, opt):
"""Transform expressions into a multinomial form. """
if opt.expand is not False:
exprs = [ expr.expand() for expr in exprs ]
if any(expr.is_commutative is False for expr in exprs):
raise PolynomialError('non-commutative expressions are not supported')
if opt.gens:
reps, gens = _parallel_dict_from_expr_if_gens(exprs, opt)
else:
reps, gens = _parallel_dict_from_expr_no_gens(exprs, opt)
return reps, opt.clone({'gens': gens})
def dict_from_expr(expr, **args):
"""Transform an expression into a multinomial form. """
rep, opt = _dict_from_expr(expr, build_options(args))
return rep, opt.gens
def _dict_from_expr(expr, opt):
"""Transform an expression into a multinomial form. """
if expr.is_commutative is False:
raise PolynomialError('non-commutative expressions are not supported')
def _is_expandable_pow(expr):
return (expr.is_Pow and expr.exp.is_positive and expr.exp.is_Integer
and expr.base.is_Add)
if opt.expand is not False:
if not isinstance(expr, (Expr, Eq)):
raise PolynomialError('expression must be of type Expr')
expr = expr.expand()
# TODO: Integrate this into expand() itself
while any(_is_expandable_pow(i) or i.is_Mul and
any(_is_expandable_pow(j) for j in i.args) for i in
Add.make_args(expr)):
expr = expand_multinomial(expr)
while any(i.is_Mul and any(j.is_Add for j in i.args) for i in Add.make_args(expr)):
expr = expand_mul(expr)
if opt.gens:
rep, gens = _dict_from_expr_if_gens(expr, opt)
else:
rep, gens = _dict_from_expr_no_gens(expr, opt)
return rep, opt.clone({'gens': gens})
def expr_from_dict(rep, *gens):
"""Convert a multinomial form into an expression. """
result = []
for monom, coeff in rep.items():
term = [coeff]
for g, m in zip(gens, monom):
if m:
term.append(Pow(g, m))
result.append(Mul(*term))
return Add(*result)
parallel_dict_from_basic = parallel_dict_from_expr
dict_from_basic = dict_from_expr
basic_from_dict = expr_from_dict
def _dict_reorder(rep, gens, new_gens):
"""Reorder levels using dict representation. """
gens = list(gens)
monoms = rep.keys()
coeffs = rep.values()
new_monoms = [ [] for _ in range(len(rep)) ]
used_indices = set()
for gen in new_gens:
try:
j = gens.index(gen)
used_indices.add(j)
for M, new_M in zip(monoms, new_monoms):
new_M.append(M[j])
except ValueError:
for new_M in new_monoms:
new_M.append(0)
for i, _ in enumerate(gens):
if i not in used_indices:
for monom in monoms:
if monom[i]:
raise GeneratorsError("unable to drop generators")
return map(tuple, new_monoms), coeffs
class PicklableWithSlots:
"""
Mixin class that allows to pickle objects with ``__slots__``.
Examples
========
First define a class that mixes :class:`PicklableWithSlots` in::
>>> from sympy.polys.polyutils import PicklableWithSlots
>>> class Some(PicklableWithSlots):
... __slots__ = ('foo', 'bar')
...
... def __init__(self, foo, bar):
... self.foo = foo
... self.bar = bar
To make :mod:`pickle` happy in doctest we have to use these hacks::
>>> import builtins
>>> builtins.Some = Some
>>> from sympy.polys import polyutils
>>> polyutils.Some = Some
Next lets see if we can create an instance, pickle it and unpickle::
>>> some = Some('abc', 10)
>>> some.foo, some.bar
('abc', 10)
>>> from pickle import dumps, loads
>>> some2 = loads(dumps(some))
>>> some2.foo, some2.bar
('abc', 10)
"""
__slots__ = ()
def __getstate__(self, cls=None):
if cls is None:
# This is the case for the instance that gets pickled
cls = self.__class__
d = {}
# Get all data that should be stored from super classes
for c in cls.__bases__:
if hasattr(c, "__getstate__"):
d.update(c.__getstate__(self, c))
# Get all information that should be stored from cls and return the dict
for name in cls.__slots__:
if hasattr(self, name):
d[name] = getattr(self, name)
return d
def __setstate__(self, d):
# All values that were pickled are now assigned to a fresh instance
for name, value in d.items():
try:
setattr(self, name, value)
except AttributeError: # This is needed in cases like Rational :> Half
pass
|
e8de2206168eea48aca7da0133694840e1dfbf735f0c873eb671db0be79d5b61 | """Euclidean algorithms, GCDs, LCMs and polynomial remainder sequences. """
from sympy.ntheory import nextprime
from sympy.polys.densearith import (
dup_sub_mul,
dup_neg, dmp_neg,
dmp_add,
dmp_sub,
dup_mul, dmp_mul,
dmp_pow,
dup_div, dmp_div,
dup_rem,
dup_quo, dmp_quo,
dup_prem, dmp_prem,
dup_mul_ground, dmp_mul_ground,
dmp_mul_term,
dup_quo_ground, dmp_quo_ground,
dup_max_norm, dmp_max_norm)
from sympy.polys.densebasic import (
dup_strip, dmp_raise,
dmp_zero, dmp_one, dmp_ground,
dmp_one_p, dmp_zero_p,
dmp_zeros,
dup_degree, dmp_degree, dmp_degree_in,
dup_LC, dmp_LC, dmp_ground_LC,
dmp_multi_deflate, dmp_inflate,
dup_convert, dmp_convert,
dmp_apply_pairs)
from sympy.polys.densetools import (
dup_clear_denoms, dmp_clear_denoms,
dup_diff, dmp_diff,
dup_eval, dmp_eval, dmp_eval_in,
dup_trunc, dmp_ground_trunc,
dup_monic, dmp_ground_monic,
dup_primitive, dmp_ground_primitive,
dup_extract, dmp_ground_extract)
from sympy.polys.galoistools import (
gf_int, gf_crt)
from sympy.polys.polyconfig import query
from sympy.polys.polyerrors import (
MultivariatePolynomialError,
HeuristicGCDFailed,
HomomorphismFailed,
NotInvertible,
DomainError)
def dup_half_gcdex(f, g, K):
"""
Half extended Euclidean algorithm in `F[x]`.
Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
>>> g = x**3 + x**2 - 4*x - 4
>>> R.dup_half_gcdex(f, g)
(-1/5*x + 3/5, x + 1)
"""
if not K.is_Field:
raise DomainError("can't compute half extended GCD over %s" % K)
a, b = [K.one], []
while g:
q, r = dup_div(f, g, K)
f, g = g, r
a, b = b, dup_sub_mul(a, q, b, K)
a = dup_quo_ground(a, dup_LC(f, K), K)
f = dup_monic(f, K)
return a, f
def dmp_half_gcdex(f, g, u, K):
"""
Half extended Euclidean algorithm in `F[X]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
"""
if not u:
return dup_half_gcdex(f, g, K)
else:
raise MultivariatePolynomialError(f, g)
def dup_gcdex(f, g, K):
"""
Extended Euclidean algorithm in `F[x]`.
Returns ``(s, t, h)`` such that ``h = gcd(f, g)`` and ``s*f + t*g = h``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
>>> g = x**3 + x**2 - 4*x - 4
>>> R.dup_gcdex(f, g)
(-1/5*x + 3/5, 1/5*x**2 - 6/5*x + 2, x + 1)
"""
s, h = dup_half_gcdex(f, g, K)
F = dup_sub_mul(h, s, f, K)
t = dup_quo(F, g, K)
return s, t, h
def dmp_gcdex(f, g, u, K):
"""
Extended Euclidean algorithm in `F[X]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
"""
if not u:
return dup_gcdex(f, g, K)
else:
raise MultivariatePolynomialError(f, g)
def dup_invert(f, g, K):
"""
Compute multiplicative inverse of `f` modulo `g` in `F[x]`.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> f = x**2 - 1
>>> g = 2*x - 1
>>> h = x - 1
>>> R.dup_invert(f, g)
-4/3
>>> R.dup_invert(f, h)
Traceback (most recent call last):
...
NotInvertible: zero divisor
"""
s, h = dup_half_gcdex(f, g, K)
if h == [K.one]:
return dup_rem(s, g, K)
else:
raise NotInvertible("zero divisor")
def dmp_invert(f, g, u, K):
"""
Compute multiplicative inverse of `f` modulo `g` in `F[X]`.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
"""
if not u:
return dup_invert(f, g, K)
else:
raise MultivariatePolynomialError(f, g)
def dup_euclidean_prs(f, g, K):
"""
Euclidean polynomial remainder sequence (PRS) in `K[x]`.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
>>> g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
>>> prs = R.dup_euclidean_prs(f, g)
>>> prs[0]
x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
>>> prs[1]
3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
>>> prs[2]
-5/9*x**4 + 1/9*x**2 - 1/3
>>> prs[3]
-117/25*x**2 - 9*x + 441/25
>>> prs[4]
233150/19773*x - 102500/6591
>>> prs[5]
-1288744821/543589225
"""
prs = [f, g]
h = dup_rem(f, g, K)
while h:
prs.append(h)
f, g = g, h
h = dup_rem(f, g, K)
return prs
def dmp_euclidean_prs(f, g, u, K):
"""
Euclidean polynomial remainder sequence (PRS) in `K[X]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
"""
if not u:
return dup_euclidean_prs(f, g, K)
else:
raise MultivariatePolynomialError(f, g)
def dup_primitive_prs(f, g, K):
"""
Primitive polynomial remainder sequence (PRS) in `K[x]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
>>> g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
>>> prs = R.dup_primitive_prs(f, g)
>>> prs[0]
x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
>>> prs[1]
3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
>>> prs[2]
-5*x**4 + x**2 - 3
>>> prs[3]
13*x**2 + 25*x - 49
>>> prs[4]
4663*x - 6150
>>> prs[5]
1
"""
prs = [f, g]
_, h = dup_primitive(dup_prem(f, g, K), K)
while h:
prs.append(h)
f, g = g, h
_, h = dup_primitive(dup_prem(f, g, K), K)
return prs
def dmp_primitive_prs(f, g, u, K):
"""
Primitive polynomial remainder sequence (PRS) in `K[X]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
"""
if not u:
return dup_primitive_prs(f, g, K)
else:
raise MultivariatePolynomialError(f, g)
def dup_inner_subresultants(f, g, K):
"""
Subresultant PRS algorithm in `K[x]`.
Computes the subresultant polynomial remainder sequence (PRS)
and the non-zero scalar subresultants of `f` and `g`.
By [1] Thm. 3, these are the constants '-c' (- to optimize
computation of sign).
The first subdeterminant is set to 1 by convention to match
the polynomial and the scalar subdeterminants.
If 'deg(f) < deg(g)', the subresultants of '(g,f)' are computed.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_inner_subresultants(x**2 + 1, x**2 - 1)
([x**2 + 1, x**2 - 1, -2], [1, 1, 4])
References
==========
.. [1] W.S. Brown, The Subresultant PRS Algorithm.
ACM Transaction of Mathematical Software 4 (1978) 237-249
"""
n = dup_degree(f)
m = dup_degree(g)
if n < m:
f, g = g, f
n, m = m, n
if not f:
return [], []
if not g:
return [f], [K.one]
R = [f, g]
d = n - m
b = (-K.one)**(d + 1)
h = dup_prem(f, g, K)
h = dup_mul_ground(h, b, K)
lc = dup_LC(g, K)
c = lc**d
# Conventional first scalar subdeterminant is 1
S = [K.one, c]
c = -c
while h:
k = dup_degree(h)
R.append(h)
f, g, m, d = g, h, k, m - k
b = -lc * c**d
h = dup_prem(f, g, K)
h = dup_quo_ground(h, b, K)
lc = dup_LC(g, K)
if d > 1: # abnormal case
q = c**(d - 1)
c = K.quo((-lc)**d, q)
else:
c = -lc
S.append(-c)
return R, S
def dup_subresultants(f, g, K):
"""
Computes subresultant PRS of two polynomials in `K[x]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_subresultants(x**2 + 1, x**2 - 1)
[x**2 + 1, x**2 - 1, -2]
"""
return dup_inner_subresultants(f, g, K)[0]
def dup_prs_resultant(f, g, K):
"""
Resultant algorithm in `K[x]` using subresultant PRS.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_prs_resultant(x**2 + 1, x**2 - 1)
(4, [x**2 + 1, x**2 - 1, -2])
"""
if not f or not g:
return (K.zero, [])
R, S = dup_inner_subresultants(f, g, K)
if dup_degree(R[-1]) > 0:
return (K.zero, R)
return S[-1], R
def dup_resultant(f, g, K, includePRS=False):
"""
Computes resultant of two polynomials in `K[x]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_resultant(x**2 + 1, x**2 - 1)
4
"""
if includePRS:
return dup_prs_resultant(f, g, K)
return dup_prs_resultant(f, g, K)[0]
def dmp_inner_subresultants(f, g, u, K):
"""
Subresultant PRS algorithm in `K[X]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> f = 3*x**2*y - y**3 - 4
>>> g = x**2 + x*y**3 - 9
>>> a = 3*x*y**4 + y**3 - 27*y + 4
>>> b = -3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16
>>> prs = [f, g, a, b]
>>> sres = [[1], [1], [3, 0, 0, 0, 0], [-3, 0, 0, -12, 1, 0, -54, 8, 729, -216, 16]]
>>> R.dmp_inner_subresultants(f, g) == (prs, sres)
True
"""
if not u:
return dup_inner_subresultants(f, g, K)
n = dmp_degree(f, u)
m = dmp_degree(g, u)
if n < m:
f, g = g, f
n, m = m, n
if dmp_zero_p(f, u):
return [], []
v = u - 1
if dmp_zero_p(g, u):
return [f], [dmp_ground(K.one, v)]
R = [f, g]
d = n - m
b = dmp_pow(dmp_ground(-K.one, v), d + 1, v, K)
h = dmp_prem(f, g, u, K)
h = dmp_mul_term(h, b, 0, u, K)
lc = dmp_LC(g, K)
c = dmp_pow(lc, d, v, K)
S = [dmp_ground(K.one, v), c]
c = dmp_neg(c, v, K)
while not dmp_zero_p(h, u):
k = dmp_degree(h, u)
R.append(h)
f, g, m, d = g, h, k, m - k
b = dmp_mul(dmp_neg(lc, v, K),
dmp_pow(c, d, v, K), v, K)
h = dmp_prem(f, g, u, K)
h = [ dmp_quo(ch, b, v, K) for ch in h ]
lc = dmp_LC(g, K)
if d > 1:
p = dmp_pow(dmp_neg(lc, v, K), d, v, K)
q = dmp_pow(c, d - 1, v, K)
c = dmp_quo(p, q, v, K)
else:
c = dmp_neg(lc, v, K)
S.append(dmp_neg(c, v, K))
return R, S
def dmp_subresultants(f, g, u, K):
"""
Computes subresultant PRS of two polynomials in `K[X]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> f = 3*x**2*y - y**3 - 4
>>> g = x**2 + x*y**3 - 9
>>> a = 3*x*y**4 + y**3 - 27*y + 4
>>> b = -3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16
>>> R.dmp_subresultants(f, g) == [f, g, a, b]
True
"""
return dmp_inner_subresultants(f, g, u, K)[0]
def dmp_prs_resultant(f, g, u, K):
"""
Resultant algorithm in `K[X]` using subresultant PRS.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> f = 3*x**2*y - y**3 - 4
>>> g = x**2 + x*y**3 - 9
>>> a = 3*x*y**4 + y**3 - 27*y + 4
>>> b = -3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16
>>> res, prs = R.dmp_prs_resultant(f, g)
>>> res == b # resultant has n-1 variables
False
>>> res == b.drop(x)
True
>>> prs == [f, g, a, b]
True
"""
if not u:
return dup_prs_resultant(f, g, K)
if dmp_zero_p(f, u) or dmp_zero_p(g, u):
return (dmp_zero(u - 1), [])
R, S = dmp_inner_subresultants(f, g, u, K)
if dmp_degree(R[-1], u) > 0:
return (dmp_zero(u - 1), R)
return S[-1], R
def dmp_zz_modular_resultant(f, g, p, u, K):
"""
Compute resultant of `f` and `g` modulo a prime `p`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> f = x + y + 2
>>> g = 2*x*y + x + 3
>>> R.dmp_zz_modular_resultant(f, g, 5)
-2*y**2 + 1
"""
if not u:
return gf_int(dup_prs_resultant(f, g, K)[0] % p, p)
v = u - 1
n = dmp_degree(f, u)
m = dmp_degree(g, u)
N = dmp_degree_in(f, 1, u)
M = dmp_degree_in(g, 1, u)
B = n*M + m*N
D, a = [K.one], -K.one
r = dmp_zero(v)
while dup_degree(D) <= B:
while True:
a += K.one
if a == p:
raise HomomorphismFailed('no luck')
F = dmp_eval_in(f, gf_int(a, p), 1, u, K)
if dmp_degree(F, v) == n:
G = dmp_eval_in(g, gf_int(a, p), 1, u, K)
if dmp_degree(G, v) == m:
break
R = dmp_zz_modular_resultant(F, G, p, v, K)
e = dmp_eval(r, a, v, K)
if not v:
R = dup_strip([R])
e = dup_strip([e])
else:
R = [R]
e = [e]
d = K.invert(dup_eval(D, a, K), p)
d = dup_mul_ground(D, d, K)
d = dmp_raise(d, v, 0, K)
c = dmp_mul(d, dmp_sub(R, e, v, K), v, K)
r = dmp_add(r, c, v, K)
r = dmp_ground_trunc(r, p, v, K)
D = dup_mul(D, [K.one, -a], K)
D = dup_trunc(D, p, K)
return r
def _collins_crt(r, R, P, p, K):
"""Wrapper of CRT for Collins's resultant algorithm. """
return gf_int(gf_crt([r, R], [P, p], K), P*p)
def dmp_zz_collins_resultant(f, g, u, K):
"""
Collins's modular resultant algorithm in `Z[X]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> f = x + y + 2
>>> g = 2*x*y + x + 3
>>> R.dmp_zz_collins_resultant(f, g)
-2*y**2 - 5*y + 1
"""
n = dmp_degree(f, u)
m = dmp_degree(g, u)
if n < 0 or m < 0:
return dmp_zero(u - 1)
A = dmp_max_norm(f, u, K)
B = dmp_max_norm(g, u, K)
a = dmp_ground_LC(f, u, K)
b = dmp_ground_LC(g, u, K)
v = u - 1
B = K(2)*K.factorial(K(n + m))*A**m*B**n
r, p, P = dmp_zero(v), K.one, K.one
while P <= B:
p = K(nextprime(p))
while not (a % p) or not (b % p):
p = K(nextprime(p))
F = dmp_ground_trunc(f, p, u, K)
G = dmp_ground_trunc(g, p, u, K)
try:
R = dmp_zz_modular_resultant(F, G, p, u, K)
except HomomorphismFailed:
continue
if K.is_one(P):
r = R
else:
r = dmp_apply_pairs(r, R, _collins_crt, (P, p, K), v, K)
P *= p
return r
def dmp_qq_collins_resultant(f, g, u, K0):
"""
Collins's modular resultant algorithm in `Q[X]`.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x,y = ring("x,y", QQ)
>>> f = QQ(1,2)*x + y + QQ(2,3)
>>> g = 2*x*y + x + 3
>>> R.dmp_qq_collins_resultant(f, g)
-2*y**2 - 7/3*y + 5/6
"""
n = dmp_degree(f, u)
m = dmp_degree(g, u)
if n < 0 or m < 0:
return dmp_zero(u - 1)
K1 = K0.get_ring()
cf, f = dmp_clear_denoms(f, u, K0, K1)
cg, g = dmp_clear_denoms(g, u, K0, K1)
f = dmp_convert(f, u, K0, K1)
g = dmp_convert(g, u, K0, K1)
r = dmp_zz_collins_resultant(f, g, u, K1)
r = dmp_convert(r, u - 1, K1, K0)
c = K0.convert(cf**m * cg**n, K1)
return dmp_quo_ground(r, c, u - 1, K0)
def dmp_resultant(f, g, u, K, includePRS=False):
"""
Computes resultant of two polynomials in `K[X]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> f = 3*x**2*y - y**3 - 4
>>> g = x**2 + x*y**3 - 9
>>> R.dmp_resultant(f, g)
-3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16
"""
if not u:
return dup_resultant(f, g, K, includePRS=includePRS)
if includePRS:
return dmp_prs_resultant(f, g, u, K)
if K.is_Field:
if K.is_QQ and query('USE_COLLINS_RESULTANT'):
return dmp_qq_collins_resultant(f, g, u, K)
else:
if K.is_ZZ and query('USE_COLLINS_RESULTANT'):
return dmp_zz_collins_resultant(f, g, u, K)
return dmp_prs_resultant(f, g, u, K)[0]
def dup_discriminant(f, K):
"""
Computes discriminant of a polynomial in `K[x]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_discriminant(x**2 + 2*x + 3)
-8
"""
d = dup_degree(f)
if d <= 0:
return K.zero
else:
s = (-1)**((d*(d - 1)) // 2)
c = dup_LC(f, K)
r = dup_resultant(f, dup_diff(f, 1, K), K)
return K.quo(r, c*K(s))
def dmp_discriminant(f, u, K):
"""
Computes discriminant of a polynomial in `K[X]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y,z,t = ring("x,y,z,t", ZZ)
>>> R.dmp_discriminant(x**2*y + x*z + t)
-4*y*t + z**2
"""
if not u:
return dup_discriminant(f, K)
d, v = dmp_degree(f, u), u - 1
if d <= 0:
return dmp_zero(v)
else:
s = (-1)**((d*(d - 1)) // 2)
c = dmp_LC(f, K)
r = dmp_resultant(f, dmp_diff(f, 1, u, K), u, K)
c = dmp_mul_ground(c, K(s), v, K)
return dmp_quo(r, c, v, K)
def _dup_rr_trivial_gcd(f, g, K):
"""Handle trivial cases in GCD algorithm over a ring. """
if not (f or g):
return [], [], []
elif not f:
if K.is_nonnegative(dup_LC(g, K)):
return g, [], [K.one]
else:
return dup_neg(g, K), [], [-K.one]
elif not g:
if K.is_nonnegative(dup_LC(f, K)):
return f, [K.one], []
else:
return dup_neg(f, K), [-K.one], []
return None
def _dup_ff_trivial_gcd(f, g, K):
"""Handle trivial cases in GCD algorithm over a field. """
if not (f or g):
return [], [], []
elif not f:
return dup_monic(g, K), [], [dup_LC(g, K)]
elif not g:
return dup_monic(f, K), [dup_LC(f, K)], []
else:
return None
def _dmp_rr_trivial_gcd(f, g, u, K):
"""Handle trivial cases in GCD algorithm over a ring. """
zero_f = dmp_zero_p(f, u)
zero_g = dmp_zero_p(g, u)
if_contain_one = dmp_one_p(f, u, K) or dmp_one_p(g, u, K)
if zero_f and zero_g:
return tuple(dmp_zeros(3, u, K))
elif zero_f:
if K.is_nonnegative(dmp_ground_LC(g, u, K)):
return g, dmp_zero(u), dmp_one(u, K)
else:
return dmp_neg(g, u, K), dmp_zero(u), dmp_ground(-K.one, u)
elif zero_g:
if K.is_nonnegative(dmp_ground_LC(f, u, K)):
return f, dmp_one(u, K), dmp_zero(u)
else:
return dmp_neg(f, u, K), dmp_ground(-K.one, u), dmp_zero(u)
elif if_contain_one:
return dmp_one(u, K), f, g
elif query('USE_SIMPLIFY_GCD'):
return _dmp_simplify_gcd(f, g, u, K)
else:
return None
def _dmp_ff_trivial_gcd(f, g, u, K):
"""Handle trivial cases in GCD algorithm over a field. """
zero_f = dmp_zero_p(f, u)
zero_g = dmp_zero_p(g, u)
if zero_f and zero_g:
return tuple(dmp_zeros(3, u, K))
elif zero_f:
return (dmp_ground_monic(g, u, K),
dmp_zero(u),
dmp_ground(dmp_ground_LC(g, u, K), u))
elif zero_g:
return (dmp_ground_monic(f, u, K),
dmp_ground(dmp_ground_LC(f, u, K), u),
dmp_zero(u))
elif query('USE_SIMPLIFY_GCD'):
return _dmp_simplify_gcd(f, g, u, K)
else:
return None
def _dmp_simplify_gcd(f, g, u, K):
"""Try to eliminate `x_0` from GCD computation in `K[X]`. """
df = dmp_degree(f, u)
dg = dmp_degree(g, u)
if df > 0 and dg > 0:
return None
if not (df or dg):
F = dmp_LC(f, K)
G = dmp_LC(g, K)
else:
if not df:
F = dmp_LC(f, K)
G = dmp_content(g, u, K)
else:
F = dmp_content(f, u, K)
G = dmp_LC(g, K)
v = u - 1
h = dmp_gcd(F, G, v, K)
cff = [ dmp_quo(cf, h, v, K) for cf in f ]
cfg = [ dmp_quo(cg, h, v, K) for cg in g ]
return [h], cff, cfg
def dup_rr_prs_gcd(f, g, K):
"""
Computes polynomial GCD using subresultants over a ring.
Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``,
and ``cfg = quo(g, h)``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_rr_prs_gcd(x**2 - 1, x**2 - 3*x + 2)
(x - 1, x + 1, x - 2)
"""
result = _dup_rr_trivial_gcd(f, g, K)
if result is not None:
return result
fc, F = dup_primitive(f, K)
gc, G = dup_primitive(g, K)
c = K.gcd(fc, gc)
h = dup_subresultants(F, G, K)[-1]
_, h = dup_primitive(h, K)
c *= K.canonical_unit(dup_LC(h, K))
h = dup_mul_ground(h, c, K)
cff = dup_quo(f, h, K)
cfg = dup_quo(g, h, K)
return h, cff, cfg
def dup_ff_prs_gcd(f, g, K):
"""
Computes polynomial GCD using subresultants over a field.
Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``,
and ``cfg = quo(g, h)``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> R.dup_ff_prs_gcd(x**2 - 1, x**2 - 3*x + 2)
(x - 1, x + 1, x - 2)
"""
result = _dup_ff_trivial_gcd(f, g, K)
if result is not None:
return result
h = dup_subresultants(f, g, K)[-1]
h = dup_monic(h, K)
cff = dup_quo(f, h, K)
cfg = dup_quo(g, h, K)
return h, cff, cfg
def dmp_rr_prs_gcd(f, g, u, K):
"""
Computes polynomial GCD using subresultants over a ring.
Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``,
and ``cfg = quo(g, h)``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y, = ring("x,y", ZZ)
>>> f = x**2 + 2*x*y + y**2
>>> g = x**2 + x*y
>>> R.dmp_rr_prs_gcd(f, g)
(x + y, x + y, x)
"""
if not u:
return dup_rr_prs_gcd(f, g, K)
result = _dmp_rr_trivial_gcd(f, g, u, K)
if result is not None:
return result
fc, F = dmp_primitive(f, u, K)
gc, G = dmp_primitive(g, u, K)
h = dmp_subresultants(F, G, u, K)[-1]
c, _, _ = dmp_rr_prs_gcd(fc, gc, u - 1, K)
if K.is_negative(dmp_ground_LC(h, u, K)):
h = dmp_neg(h, u, K)
_, h = dmp_primitive(h, u, K)
h = dmp_mul_term(h, c, 0, u, K)
cff = dmp_quo(f, h, u, K)
cfg = dmp_quo(g, h, u, K)
return h, cff, cfg
def dmp_ff_prs_gcd(f, g, u, K):
"""
Computes polynomial GCD using subresultants over a field.
Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``,
and ``cfg = quo(g, h)``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x,y, = ring("x,y", QQ)
>>> f = QQ(1,2)*x**2 + x*y + QQ(1,2)*y**2
>>> g = x**2 + x*y
>>> R.dmp_ff_prs_gcd(f, g)
(x + y, 1/2*x + 1/2*y, x)
"""
if not u:
return dup_ff_prs_gcd(f, g, K)
result = _dmp_ff_trivial_gcd(f, g, u, K)
if result is not None:
return result
fc, F = dmp_primitive(f, u, K)
gc, G = dmp_primitive(g, u, K)
h = dmp_subresultants(F, G, u, K)[-1]
c, _, _ = dmp_ff_prs_gcd(fc, gc, u - 1, K)
_, h = dmp_primitive(h, u, K)
h = dmp_mul_term(h, c, 0, u, K)
h = dmp_ground_monic(h, u, K)
cff = dmp_quo(f, h, u, K)
cfg = dmp_quo(g, h, u, K)
return h, cff, cfg
HEU_GCD_MAX = 6
def _dup_zz_gcd_interpolate(h, x, K):
"""Interpolate polynomial GCD from integer GCD. """
f = []
while h:
g = h % x
if g > x // 2:
g -= x
f.insert(0, g)
h = (h - g) // x
return f
def dup_zz_heu_gcd(f, g, K):
"""
Heuristic polynomial GCD in `Z[x]`.
Given univariate polynomials `f` and `g` in `Z[x]`, returns
their GCD and cofactors, i.e. polynomials ``h``, ``cff`` and ``cfg``
such that::
h = gcd(f, g), cff = quo(f, h) and cfg = quo(g, h)
The algorithm is purely heuristic which means it may fail to compute
the GCD. This will be signaled by raising an exception. In this case
you will need to switch to another GCD method.
The algorithm computes the polynomial GCD by evaluating polynomials
f and g at certain points and computing (fast) integer GCD of those
evaluations. The polynomial GCD is recovered from the integer image
by interpolation. The final step is to verify if the result is the
correct GCD. This gives cofactors as a side effect.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_zz_heu_gcd(x**2 - 1, x**2 - 3*x + 2)
(x - 1, x + 1, x - 2)
References
==========
.. [1] [Liao95]_
"""
result = _dup_rr_trivial_gcd(f, g, K)
if result is not None:
return result
df = dup_degree(f)
dg = dup_degree(g)
gcd, f, g = dup_extract(f, g, K)
if df == 0 or dg == 0:
return [gcd], f, g
f_norm = dup_max_norm(f, K)
g_norm = dup_max_norm(g, K)
B = K(2*min(f_norm, g_norm) + 29)
x = max(min(B, 99*K.sqrt(B)),
2*min(f_norm // abs(dup_LC(f, K)),
g_norm // abs(dup_LC(g, K))) + 2)
for i in range(0, HEU_GCD_MAX):
ff = dup_eval(f, x, K)
gg = dup_eval(g, x, K)
if ff and gg:
h = K.gcd(ff, gg)
cff = ff // h
cfg = gg // h
h = _dup_zz_gcd_interpolate(h, x, K)
h = dup_primitive(h, K)[1]
cff_, r = dup_div(f, h, K)
if not r:
cfg_, r = dup_div(g, h, K)
if not r:
h = dup_mul_ground(h, gcd, K)
return h, cff_, cfg_
cff = _dup_zz_gcd_interpolate(cff, x, K)
h, r = dup_div(f, cff, K)
if not r:
cfg_, r = dup_div(g, h, K)
if not r:
h = dup_mul_ground(h, gcd, K)
return h, cff, cfg_
cfg = _dup_zz_gcd_interpolate(cfg, x, K)
h, r = dup_div(g, cfg, K)
if not r:
cff_, r = dup_div(f, h, K)
if not r:
h = dup_mul_ground(h, gcd, K)
return h, cff_, cfg
x = 73794*x * K.sqrt(K.sqrt(x)) // 27011
raise HeuristicGCDFailed('no luck')
def _dmp_zz_gcd_interpolate(h, x, v, K):
"""Interpolate polynomial GCD from integer GCD. """
f = []
while not dmp_zero_p(h, v):
g = dmp_ground_trunc(h, x, v, K)
f.insert(0, g)
h = dmp_sub(h, g, v, K)
h = dmp_quo_ground(h, x, v, K)
if K.is_negative(dmp_ground_LC(f, v + 1, K)):
return dmp_neg(f, v + 1, K)
else:
return f
def dmp_zz_heu_gcd(f, g, u, K):
"""
Heuristic polynomial GCD in `Z[X]`.
Given univariate polynomials `f` and `g` in `Z[X]`, returns
their GCD and cofactors, i.e. polynomials ``h``, ``cff`` and ``cfg``
such that::
h = gcd(f, g), cff = quo(f, h) and cfg = quo(g, h)
The algorithm is purely heuristic which means it may fail to compute
the GCD. This will be signaled by raising an exception. In this case
you will need to switch to another GCD method.
The algorithm computes the polynomial GCD by evaluating polynomials
f and g at certain points and computing (fast) integer GCD of those
evaluations. The polynomial GCD is recovered from the integer image
by interpolation. The evaluation process reduces f and g variable by
variable into a large integer. The final step is to verify if the
interpolated polynomial is the correct GCD. This gives cofactors of
the input polynomials as a side effect.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y, = ring("x,y", ZZ)
>>> f = x**2 + 2*x*y + y**2
>>> g = x**2 + x*y
>>> R.dmp_zz_heu_gcd(f, g)
(x + y, x + y, x)
References
==========
.. [1] [Liao95]_
"""
if not u:
return dup_zz_heu_gcd(f, g, K)
result = _dmp_rr_trivial_gcd(f, g, u, K)
if result is not None:
return result
gcd, f, g = dmp_ground_extract(f, g, u, K)
f_norm = dmp_max_norm(f, u, K)
g_norm = dmp_max_norm(g, u, K)
B = K(2*min(f_norm, g_norm) + 29)
x = max(min(B, 99*K.sqrt(B)),
2*min(f_norm // abs(dmp_ground_LC(f, u, K)),
g_norm // abs(dmp_ground_LC(g, u, K))) + 2)
for i in range(0, HEU_GCD_MAX):
ff = dmp_eval(f, x, u, K)
gg = dmp_eval(g, x, u, K)
v = u - 1
if not (dmp_zero_p(ff, v) or dmp_zero_p(gg, v)):
h, cff, cfg = dmp_zz_heu_gcd(ff, gg, v, K)
h = _dmp_zz_gcd_interpolate(h, x, v, K)
h = dmp_ground_primitive(h, u, K)[1]
cff_, r = dmp_div(f, h, u, K)
if dmp_zero_p(r, u):
cfg_, r = dmp_div(g, h, u, K)
if dmp_zero_p(r, u):
h = dmp_mul_ground(h, gcd, u, K)
return h, cff_, cfg_
cff = _dmp_zz_gcd_interpolate(cff, x, v, K)
h, r = dmp_div(f, cff, u, K)
if dmp_zero_p(r, u):
cfg_, r = dmp_div(g, h, u, K)
if dmp_zero_p(r, u):
h = dmp_mul_ground(h, gcd, u, K)
return h, cff, cfg_
cfg = _dmp_zz_gcd_interpolate(cfg, x, v, K)
h, r = dmp_div(g, cfg, u, K)
if dmp_zero_p(r, u):
cff_, r = dmp_div(f, h, u, K)
if dmp_zero_p(r, u):
h = dmp_mul_ground(h, gcd, u, K)
return h, cff_, cfg
x = 73794*x * K.sqrt(K.sqrt(x)) // 27011
raise HeuristicGCDFailed('no luck')
def dup_qq_heu_gcd(f, g, K0):
"""
Heuristic polynomial GCD in `Q[x]`.
Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``,
``cff = quo(f, h)``, and ``cfg = quo(g, h)``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> f = QQ(1,2)*x**2 + QQ(7,4)*x + QQ(3,2)
>>> g = QQ(1,2)*x**2 + x
>>> R.dup_qq_heu_gcd(f, g)
(x + 2, 1/2*x + 3/4, 1/2*x)
"""
result = _dup_ff_trivial_gcd(f, g, K0)
if result is not None:
return result
K1 = K0.get_ring()
cf, f = dup_clear_denoms(f, K0, K1)
cg, g = dup_clear_denoms(g, K0, K1)
f = dup_convert(f, K0, K1)
g = dup_convert(g, K0, K1)
h, cff, cfg = dup_zz_heu_gcd(f, g, K1)
h = dup_convert(h, K1, K0)
c = dup_LC(h, K0)
h = dup_monic(h, K0)
cff = dup_convert(cff, K1, K0)
cfg = dup_convert(cfg, K1, K0)
cff = dup_mul_ground(cff, K0.quo(c, cf), K0)
cfg = dup_mul_ground(cfg, K0.quo(c, cg), K0)
return h, cff, cfg
def dmp_qq_heu_gcd(f, g, u, K0):
"""
Heuristic polynomial GCD in `Q[X]`.
Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``,
``cff = quo(f, h)``, and ``cfg = quo(g, h)``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x,y, = ring("x,y", QQ)
>>> f = QQ(1,4)*x**2 + x*y + y**2
>>> g = QQ(1,2)*x**2 + x*y
>>> R.dmp_qq_heu_gcd(f, g)
(x + 2*y, 1/4*x + 1/2*y, 1/2*x)
"""
result = _dmp_ff_trivial_gcd(f, g, u, K0)
if result is not None:
return result
K1 = K0.get_ring()
cf, f = dmp_clear_denoms(f, u, K0, K1)
cg, g = dmp_clear_denoms(g, u, K0, K1)
f = dmp_convert(f, u, K0, K1)
g = dmp_convert(g, u, K0, K1)
h, cff, cfg = dmp_zz_heu_gcd(f, g, u, K1)
h = dmp_convert(h, u, K1, K0)
c = dmp_ground_LC(h, u, K0)
h = dmp_ground_monic(h, u, K0)
cff = dmp_convert(cff, u, K1, K0)
cfg = dmp_convert(cfg, u, K1, K0)
cff = dmp_mul_ground(cff, K0.quo(c, cf), u, K0)
cfg = dmp_mul_ground(cfg, K0.quo(c, cg), u, K0)
return h, cff, cfg
def dup_inner_gcd(f, g, K):
"""
Computes polynomial GCD and cofactors of `f` and `g` in `K[x]`.
Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``,
``cff = quo(f, h)``, and ``cfg = quo(g, h)``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_inner_gcd(x**2 - 1, x**2 - 3*x + 2)
(x - 1, x + 1, x - 2)
"""
if not K.is_Exact:
try:
exact = K.get_exact()
except DomainError:
return [K.one], f, g
f = dup_convert(f, K, exact)
g = dup_convert(g, K, exact)
h, cff, cfg = dup_inner_gcd(f, g, exact)
h = dup_convert(h, exact, K)
cff = dup_convert(cff, exact, K)
cfg = dup_convert(cfg, exact, K)
return h, cff, cfg
elif K.is_Field:
if K.is_QQ and query('USE_HEU_GCD'):
try:
return dup_qq_heu_gcd(f, g, K)
except HeuristicGCDFailed:
pass
return dup_ff_prs_gcd(f, g, K)
else:
if K.is_ZZ and query('USE_HEU_GCD'):
try:
return dup_zz_heu_gcd(f, g, K)
except HeuristicGCDFailed:
pass
return dup_rr_prs_gcd(f, g, K)
def _dmp_inner_gcd(f, g, u, K):
"""Helper function for `dmp_inner_gcd()`. """
if not K.is_Exact:
try:
exact = K.get_exact()
except DomainError:
return dmp_one(u, K), f, g
f = dmp_convert(f, u, K, exact)
g = dmp_convert(g, u, K, exact)
h, cff, cfg = _dmp_inner_gcd(f, g, u, exact)
h = dmp_convert(h, u, exact, K)
cff = dmp_convert(cff, u, exact, K)
cfg = dmp_convert(cfg, u, exact, K)
return h, cff, cfg
elif K.is_Field:
if K.is_QQ and query('USE_HEU_GCD'):
try:
return dmp_qq_heu_gcd(f, g, u, K)
except HeuristicGCDFailed:
pass
return dmp_ff_prs_gcd(f, g, u, K)
else:
if K.is_ZZ and query('USE_HEU_GCD'):
try:
return dmp_zz_heu_gcd(f, g, u, K)
except HeuristicGCDFailed:
pass
return dmp_rr_prs_gcd(f, g, u, K)
def dmp_inner_gcd(f, g, u, K):
"""
Computes polynomial GCD and cofactors of `f` and `g` in `K[X]`.
Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``,
``cff = quo(f, h)``, and ``cfg = quo(g, h)``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y, = ring("x,y", ZZ)
>>> f = x**2 + 2*x*y + y**2
>>> g = x**2 + x*y
>>> R.dmp_inner_gcd(f, g)
(x + y, x + y, x)
"""
if not u:
return dup_inner_gcd(f, g, K)
J, (f, g) = dmp_multi_deflate((f, g), u, K)
h, cff, cfg = _dmp_inner_gcd(f, g, u, K)
return (dmp_inflate(h, J, u, K),
dmp_inflate(cff, J, u, K),
dmp_inflate(cfg, J, u, K))
def dup_gcd(f, g, K):
"""
Computes polynomial GCD of `f` and `g` in `K[x]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_gcd(x**2 - 1, x**2 - 3*x + 2)
x - 1
"""
return dup_inner_gcd(f, g, K)[0]
def dmp_gcd(f, g, u, K):
"""
Computes polynomial GCD of `f` and `g` in `K[X]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y, = ring("x,y", ZZ)
>>> f = x**2 + 2*x*y + y**2
>>> g = x**2 + x*y
>>> R.dmp_gcd(f, g)
x + y
"""
return dmp_inner_gcd(f, g, u, K)[0]
def dup_rr_lcm(f, g, K):
"""
Computes polynomial LCM over a ring in `K[x]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_rr_lcm(x**2 - 1, x**2 - 3*x + 2)
x**3 - 2*x**2 - x + 2
"""
fc, f = dup_primitive(f, K)
gc, g = dup_primitive(g, K)
c = K.lcm(fc, gc)
h = dup_quo(dup_mul(f, g, K),
dup_gcd(f, g, K), K)
return dup_mul_ground(h, c, K)
def dup_ff_lcm(f, g, K):
"""
Computes polynomial LCM over a field in `K[x]`.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> f = QQ(1,2)*x**2 + QQ(7,4)*x + QQ(3,2)
>>> g = QQ(1,2)*x**2 + x
>>> R.dup_ff_lcm(f, g)
x**3 + 7/2*x**2 + 3*x
"""
h = dup_quo(dup_mul(f, g, K),
dup_gcd(f, g, K), K)
return dup_monic(h, K)
def dup_lcm(f, g, K):
"""
Computes polynomial LCM of `f` and `g` in `K[x]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_lcm(x**2 - 1, x**2 - 3*x + 2)
x**3 - 2*x**2 - x + 2
"""
if K.is_Field:
return dup_ff_lcm(f, g, K)
else:
return dup_rr_lcm(f, g, K)
def dmp_rr_lcm(f, g, u, K):
"""
Computes polynomial LCM over a ring in `K[X]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y, = ring("x,y", ZZ)
>>> f = x**2 + 2*x*y + y**2
>>> g = x**2 + x*y
>>> R.dmp_rr_lcm(f, g)
x**3 + 2*x**2*y + x*y**2
"""
fc, f = dmp_ground_primitive(f, u, K)
gc, g = dmp_ground_primitive(g, u, K)
c = K.lcm(fc, gc)
h = dmp_quo(dmp_mul(f, g, u, K),
dmp_gcd(f, g, u, K), u, K)
return dmp_mul_ground(h, c, u, K)
def dmp_ff_lcm(f, g, u, K):
"""
Computes polynomial LCM over a field in `K[X]`.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x,y, = ring("x,y", QQ)
>>> f = QQ(1,4)*x**2 + x*y + y**2
>>> g = QQ(1,2)*x**2 + x*y
>>> R.dmp_ff_lcm(f, g)
x**3 + 4*x**2*y + 4*x*y**2
"""
h = dmp_quo(dmp_mul(f, g, u, K),
dmp_gcd(f, g, u, K), u, K)
return dmp_ground_monic(h, u, K)
def dmp_lcm(f, g, u, K):
"""
Computes polynomial LCM of `f` and `g` in `K[X]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y, = ring("x,y", ZZ)
>>> f = x**2 + 2*x*y + y**2
>>> g = x**2 + x*y
>>> R.dmp_lcm(f, g)
x**3 + 2*x**2*y + x*y**2
"""
if not u:
return dup_lcm(f, g, K)
if K.is_Field:
return dmp_ff_lcm(f, g, u, K)
else:
return dmp_rr_lcm(f, g, u, K)
def dmp_content(f, u, K):
"""
Returns GCD of multivariate coefficients.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y, = ring("x,y", ZZ)
>>> R.dmp_content(2*x*y + 6*x + 4*y + 12)
2*y + 6
"""
cont, v = dmp_LC(f, K), u - 1
if dmp_zero_p(f, u):
return cont
for c in f[1:]:
cont = dmp_gcd(cont, c, v, K)
if dmp_one_p(cont, v, K):
break
if K.is_negative(dmp_ground_LC(cont, v, K)):
return dmp_neg(cont, v, K)
else:
return cont
def dmp_primitive(f, u, K):
"""
Returns multivariate content and a primitive polynomial.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y, = ring("x,y", ZZ)
>>> R.dmp_primitive(2*x*y + 6*x + 4*y + 12)
(2*y + 6, x + 2)
"""
cont, v = dmp_content(f, u, K), u - 1
if dmp_zero_p(f, u) or dmp_one_p(cont, v, K):
return cont, f
else:
return cont, [ dmp_quo(c, cont, v, K) for c in f ]
def dup_cancel(f, g, K, include=True):
"""
Cancel common factors in a rational function `f/g`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_cancel(2*x**2 - 2, x**2 - 2*x + 1)
(2*x + 2, x - 1)
"""
return dmp_cancel(f, g, 0, K, include=include)
def dmp_cancel(f, g, u, K, include=True):
"""
Cancel common factors in a rational function `f/g`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> R.dmp_cancel(2*x**2 - 2, x**2 - 2*x + 1)
(2*x + 2, x - 1)
"""
K0 = None
if K.is_Field and K.has_assoc_Ring:
K0, K = K, K.get_ring()
cq, f = dmp_clear_denoms(f, u, K0, K, convert=True)
cp, g = dmp_clear_denoms(g, u, K0, K, convert=True)
else:
cp, cq = K.one, K.one
_, p, q = dmp_inner_gcd(f, g, u, K)
if K0 is not None:
_, cp, cq = K.cofactors(cp, cq)
p = dmp_convert(p, u, K, K0)
q = dmp_convert(q, u, K, K0)
K = K0
p_neg = K.is_negative(dmp_ground_LC(p, u, K))
q_neg = K.is_negative(dmp_ground_LC(q, u, K))
if p_neg and q_neg:
p, q = dmp_neg(p, u, K), dmp_neg(q, u, K)
elif p_neg:
cp, p = -cp, dmp_neg(p, u, K)
elif q_neg:
cp, q = -cp, dmp_neg(q, u, K)
if not include:
return cp, cq, p, q
p = dmp_mul_ground(p, cp, u, K)
q = dmp_mul_ground(q, cq, u, K)
return p, q
|
3090d232b637cebb4ace093e201022e5fead6d099aa620826c51e0b6a328446d | """Tools for constructing domains for expressions. """
from sympy.core import sympify
from sympy.core.compatibility import ordered
from sympy.core.evalf import pure_complex
from sympy.polys.domains import ZZ, QQ, ZZ_I, QQ_I, EX
from sympy.polys.domains.complexfield import ComplexField
from sympy.polys.domains.realfield import RealField
from sympy.polys.polyoptions import build_options
from sympy.polys.polyutils import parallel_dict_from_basic
from sympy.utilities import public
def _construct_simple(coeffs, opt):
"""Handle simple domains, e.g.: ZZ, QQ, RR and algebraic domains. """
rationals = floats = complexes = algebraics = False
float_numbers = []
if opt.extension is True:
is_algebraic = lambda coeff: coeff.is_number and coeff.is_algebraic
else:
is_algebraic = lambda coeff: False
for coeff in coeffs:
if coeff.is_Rational:
if not coeff.is_Integer:
rationals = True
elif coeff.is_Float:
if algebraics:
# there are both reals and algebraics -> EX
return False
else:
floats = True
float_numbers.append(coeff)
else:
is_complex = pure_complex(coeff)
if is_complex:
complexes = True
x, y = is_complex
if x.is_Rational and y.is_Rational:
if not (x.is_Integer and y.is_Integer):
rationals = True
continue
else:
floats = True
if x.is_Float:
float_numbers.append(x)
if y.is_Float:
float_numbers.append(y)
if is_algebraic(coeff):
if floats:
# there are both algebraics and reals -> EX
return False
algebraics = True
else:
# this is a composite domain, e.g. ZZ[X], EX
return None
# Use the maximum precision of all coefficients for the RR or CC
# precision
max_prec = max(c._prec for c in float_numbers) if float_numbers else 53
if algebraics:
domain, result = _construct_algebraic(coeffs, opt)
else:
if floats and complexes:
domain = ComplexField(prec=max_prec)
elif floats:
domain = RealField(prec=max_prec)
elif rationals or opt.field:
domain = QQ_I if complexes else QQ
else:
domain = ZZ_I if complexes else ZZ
result = [domain.from_sympy(coeff) for coeff in coeffs]
return domain, result
def _construct_algebraic(coeffs, opt):
"""We know that coefficients are algebraic so construct the extension. """
from sympy.polys.numberfields import primitive_element
exts = set()
def build_trees(args):
trees = []
for a in args:
if a.is_Rational:
tree = ('Q', QQ.from_sympy(a))
elif a.is_Add:
tree = ('+', build_trees(a.args))
elif a.is_Mul:
tree = ('*', build_trees(a.args))
else:
tree = ('e', a)
exts.add(a)
trees.append(tree)
return trees
trees = build_trees(coeffs)
exts = list(ordered(exts))
g, span, H = primitive_element(exts, ex=True, polys=True)
root = sum([ s*ext for s, ext in zip(span, exts) ])
domain, g = QQ.algebraic_field((g, root)), g.rep.rep
exts_dom = [domain.dtype.from_list(h, g, QQ) for h in H]
exts_map = dict(zip(exts, exts_dom))
def convert_tree(tree):
op, args = tree
if op == 'Q':
return domain.dtype.from_list([args], g, QQ)
elif op == '+':
return sum((convert_tree(a) for a in args), domain.zero)
elif op == '*':
# return prod(convert(a) for a in args)
t = convert_tree(args[0])
for a in args[1:]:
t *= convert_tree(a)
return t
elif op == 'e':
return exts_map[args]
else:
raise RuntimeError
result = [convert_tree(tree) for tree in trees]
return domain, result
def _construct_composite(coeffs, opt):
"""Handle composite domains, e.g.: ZZ[X], QQ[X], ZZ(X), QQ(X). """
numers, denoms = [], []
for coeff in coeffs:
numer, denom = coeff.as_numer_denom()
numers.append(numer)
denoms.append(denom)
polys, gens = parallel_dict_from_basic(numers + denoms) # XXX: sorting
if not gens:
return None
if opt.composite is None:
if any(gen.is_number and gen.is_algebraic for gen in gens):
return None # generators are number-like so lets better use EX
all_symbols = set()
for gen in gens:
symbols = gen.free_symbols
if all_symbols & symbols:
return None # there could be algebraic relations between generators
else:
all_symbols |= symbols
n = len(gens)
k = len(polys)//2
numers = polys[:k]
denoms = polys[k:]
if opt.field:
fractions = True
else:
fractions, zeros = False, (0,)*n
for denom in denoms:
if len(denom) > 1 or zeros not in denom:
fractions = True
break
coeffs = set()
if not fractions:
for numer, denom in zip(numers, denoms):
denom = denom[zeros]
for monom, coeff in numer.items():
coeff /= denom
coeffs.add(coeff)
numer[monom] = coeff
else:
for numer, denom in zip(numers, denoms):
coeffs.update(list(numer.values()))
coeffs.update(list(denom.values()))
rationals = floats = complexes = False
float_numbers = []
for coeff in coeffs:
if coeff.is_Rational:
if not coeff.is_Integer:
rationals = True
elif coeff.is_Float:
floats = True
float_numbers.append(coeff)
else:
is_complex = pure_complex(coeff)
if is_complex is not None:
complexes = True
x, y = is_complex
if x.is_Rational and y.is_Rational:
if not (x.is_Integer and y.is_Integer):
rationals = True
else:
floats = True
if x.is_Float:
float_numbers.append(x)
if y.is_Float:
float_numbers.append(y)
max_prec = max(c._prec for c in float_numbers) if float_numbers else 53
if floats and complexes:
ground = ComplexField(prec=max_prec)
elif floats:
ground = RealField(prec=max_prec)
elif complexes:
if rationals:
ground = QQ_I
else:
ground = ZZ_I
elif rationals:
ground = QQ
else:
ground = ZZ
result = []
if not fractions:
domain = ground.poly_ring(*gens)
for numer in numers:
for monom, coeff in numer.items():
numer[monom] = ground.from_sympy(coeff)
result.append(domain(numer))
else:
domain = ground.frac_field(*gens)
for numer, denom in zip(numers, denoms):
for monom, coeff in numer.items():
numer[monom] = ground.from_sympy(coeff)
for monom, coeff in denom.items():
denom[monom] = ground.from_sympy(coeff)
result.append(domain((numer, denom)))
return domain, result
def _construct_expression(coeffs, opt):
"""The last resort case, i.e. use the expression domain. """
domain, result = EX, []
for coeff in coeffs:
result.append(domain.from_sympy(coeff))
return domain, result
@public
def construct_domain(obj, **args):
"""Construct a minimal domain for a list of expressions.
Explanation
===========
Given a list of normal SymPy expressions (of type :py:class:`~.Expr`)
``construct_domain`` will find a minimal :py:class:`~.Domain` that can
represent those expressions. The expressions will be converted to elements
of the domain and both the domain and the domain elements are returned.
Parameters
==========
obj: list or dict
The expressions to build a domain for.
**args: keyword arguments
Options that affect the choice of domain.
Returns
=======
(K, elements): Domain and list of domain elements
The domain K that can represent the expressions and the list or dict
of domain elements representing the same expressions as elements of K.
Examples
========
Given a list of :py:class:`~.Integer` ``construct_domain`` will return the
domain :ref:`ZZ` and a list of integers as elements of :ref:`ZZ`.
>>> from sympy import construct_domain, S
>>> expressions = [S(2), S(3), S(4)]
>>> K, elements = construct_domain(expressions)
>>> K
ZZ
>>> elements
[2, 3, 4]
>>> type(elements[0]) # doctest: +SKIP
<class 'int'>
>>> type(expressions[0])
<class 'sympy.core.numbers.Integer'>
If there are any :py:class:`~.Rational` then :ref:`QQ` is returned
instead.
>>> construct_domain([S(1)/2, S(3)/4])
(QQ, [1/2, 3/4])
If there are symbols then a polynomial ring :ref:`K[x]` is returned.
>>> from sympy import symbols
>>> x, y = symbols('x, y')
>>> construct_domain([2*x + 1, S(3)/4])
(QQ[x], [2*x + 1, 3/4])
>>> construct_domain([2*x + 1, y])
(ZZ[x,y], [2*x + 1, y])
If any symbols appear with negative powers then a rational function field
:ref:`K(x)` will be returned.
>>> construct_domain([y/x, x/(1 - y)])
(ZZ(x,y), [y/x, -x/(y - 1)])
Irrational algebraic numbers will result in the :ref:`EX` domain by
default. The keyword argument ``extension=True`` leads to the construction
of an algebraic number field :ref:`QQ(a)`.
>>> from sympy import sqrt
>>> construct_domain([sqrt(2)])
(EX, [EX(sqrt(2))])
>>> construct_domain([sqrt(2)], extension=True) # doctest: +SKIP
(QQ<sqrt(2)>, [ANP([1, 0], [1, 0, -2], QQ)])
See also
========
Domain
Expr
"""
opt = build_options(args)
if hasattr(obj, '__iter__'):
if isinstance(obj, dict):
if not obj:
monoms, coeffs = [], []
else:
monoms, coeffs = list(zip(*list(obj.items())))
else:
coeffs = obj
else:
coeffs = [obj]
coeffs = list(map(sympify, coeffs))
result = _construct_simple(coeffs, opt)
if result is not None:
if result is not False:
domain, coeffs = result
else:
domain, coeffs = _construct_expression(coeffs, opt)
else:
if opt.composite is False:
result = None
else:
result = _construct_composite(coeffs, opt)
if result is not None:
domain, coeffs = result
else:
domain, coeffs = _construct_expression(coeffs, opt)
if hasattr(obj, '__iter__'):
if isinstance(obj, dict):
return domain, dict(list(zip(monoms, coeffs)))
else:
return domain, coeffs
else:
return domain, coeffs[0]
|
a87d4ed83e98efe3a2742c8949c6e6d9f0d709a064e4294b4ff88d0b95fafcee | import re
import fnmatch
# XXX Python 2 unicode import test.
# May remove after deprecating python 2.7.
message_unicode_A = \
"File contains a unicode character : %s, line %s. " \
"But with no encoding header. " \
"See https://www.python.org/dev/peps/pep-0263/ " \
"and add '# coding=utf-8'"
message_unicode_B = \
"File contains a unicode character : %s, line %s. " \
"But not in the whitelist. " \
"Add the file to the whitelist in " + __file__
message_unicode_C = \
"File contains a unicode character : %s, line %s. " \
"And is in the whitelist, but without the encoding header. " \
"See https://www.python.org/dev/peps/pep-0263/ " \
"and add '# coding=utf-8'."
message_unicode_D = \
"File does not contain a unicode character : %s." \
"but is in the whitelist. " \
"Remove the file from the whitelist in " + __file__
message_unicode_E = \
"File does not contain a unicode character : %s." \
"but contains the header '# coding=utf-8' or equivalent." \
"Remove the header."
encoding_header_re = re.compile(
r'^[ \t\f]*#.*?coding[:=][ \t]*([-_.a-zA-Z0-9]+)')
# Whitelist pattern for files which can have unicode.
unicode_whitelist = [
# Author names can include non-ASCII characters
r'*/bin/authors_update.py',
# These files have functions and test functions for unicode input and
# output.
r'*/sympy/testing/tests/test_code_quality.py',
r'*/sympy/physics/vector/tests/test_printing.py',
r'*/physics/quantum/tests/test_printing.py',
r'*/sympy/vector/tests/test_printing.py',
r'*/sympy/parsing/tests/test_sympy_parser.py',
r'*/sympy/printing/pretty/tests/test_pretty.py',
r'*/sympy/printing/tests/test_conventions.py',
r'*/sympy/printing/tests/test_preview.py',
r'*/liealgebras/type_g.py',
r'*/liealgebras/weyl_group.py',
r'*/liealgebras/tests/test_type_G.py',
# wigner.py and polarization.py have unicode doctests. These probably
# don't need to be there but some of the examples that are there are
# pretty ugly without use_unicode (matrices need to be wrapped across
# multiple lines etc)
r'*/sympy/physics/wigner.py',
r'*/sympy/physics/optics/polarization.py',
]
unicode_strict_whitelist = [
r'*/sympy/parsing/latex/_antlr/__init__.py',
]
def _test_this_file_encoding(
fname, test_file,
unicode_whitelist=unicode_whitelist,
unicode_strict_whitelist=unicode_strict_whitelist):
"""Test helper function for python 2 importability test
This test checks whether the file has
# coding=utf-8
or
# -*- coding: utf-8 -*-
line if there is a unicode character in the code
The test may have to operate on filewise manner, so it had moved
to a separate process.
May remove after deprecating python 2.7.
"""
has_coding_utf8 = False
has_unicode = False
is_in_whitelist = False
is_in_strict_whitelist = False
for patt in unicode_whitelist:
if fnmatch.fnmatch(fname, patt):
is_in_whitelist = True
break
for patt in unicode_strict_whitelist:
if fnmatch.fnmatch(fname, patt):
is_in_strict_whitelist = True
is_in_whitelist = True
break
if is_in_whitelist:
for idx, line in enumerate(test_file):
if idx in (0, 1):
match = encoding_header_re.match(line)
if match and match.group(1).lower() == 'utf-8':
has_coding_utf8 = True
try:
line.encode(encoding='ascii')
except (UnicodeEncodeError, UnicodeDecodeError):
has_unicode = True
if has_coding_utf8 is False:
assert False, \
message_unicode_C % (fname, idx + 1)
if not has_unicode and not is_in_strict_whitelist:
assert False, message_unicode_D % fname
else:
for idx, line in enumerate(test_file):
if idx in (0, 1):
match = encoding_header_re.match(line)
if match and match.group(1).lower() == 'utf-8':
has_coding_utf8 = True
try:
line.encode(encoding='ascii')
except (UnicodeEncodeError, UnicodeDecodeError):
has_unicode = True
if has_coding_utf8:
assert False, \
message_unicode_B % (fname, idx + 1)
else:
assert False, \
message_unicode_A % (fname, idx + 1)
if not has_unicode and has_coding_utf8:
assert False, \
message_unicode_E % fname
|
89261d2c6dbd2a0aa0cd90bc2f72a86346d0c0e20b6f8b7118f6ba57c3a1303e | """
This is our testing framework.
Goals:
* it should be compatible with py.test and operate very similarly
(or identically)
* doesn't require any external dependencies
* preferably all the functionality should be in this file only
* no magic, just import the test file and execute the test functions, that's it
* portable
"""
from __future__ import print_function, division
import os
import sys
import platform
import inspect
import traceback
import pdb
import re
import linecache
import time
from fnmatch import fnmatch
from timeit import default_timer as clock
import doctest as pdoctest # avoid clashing with our doctest() function
from doctest import DocTestFinder, DocTestRunner
import random
import subprocess
import shutil
import signal
import stat
import tempfile
import warnings
from contextlib import contextmanager
from sympy.core.cache import clear_cache
from sympy.core.compatibility import (PY3, unwrap)
from sympy.external import import_module
IS_WINDOWS = (os.name == 'nt')
ON_TRAVIS = os.getenv('TRAVIS_BUILD_NUMBER', None)
# emperically generated list of the proportion of time spent running
# an even split of tests. This should periodically be regenerated.
# A list of [.6, .1, .3] would mean that if the tests are evenly split
# into '1/3', '2/3', '3/3', the first split would take 60% of the time,
# the second 10% and the third 30%. These lists are normalized to sum
# to 1, so [60, 10, 30] has the same behavior as [6, 1, 3] or [.6, .1, .3].
#
# This list can be generated with the code:
# from time import time
# import sympy
# import os
# os.environ["TRAVIS_BUILD_NUMBER"] = '2' # Mock travis to get more correct densities
# delays, num_splits = [], 30
# for i in range(1, num_splits + 1):
# tic = time()
# sympy.test(split='{}/{}'.format(i, num_splits), time_balance=False) # Add slow=True for slow tests
# delays.append(time() - tic)
# tot = sum(delays)
# print([round(x / tot, 4) for x in delays])
SPLIT_DENSITY = [
0.0059, 0.0027, 0.0068, 0.0011, 0.0006,
0.0058, 0.0047, 0.0046, 0.004, 0.0257,
0.0017, 0.0026, 0.004, 0.0032, 0.0016,
0.0015, 0.0004, 0.0011, 0.0016, 0.0014,
0.0077, 0.0137, 0.0217, 0.0074, 0.0043,
0.0067, 0.0236, 0.0004, 0.1189, 0.0142,
0.0234, 0.0003, 0.0003, 0.0047, 0.0006,
0.0013, 0.0004, 0.0008, 0.0007, 0.0006,
0.0139, 0.0013, 0.0007, 0.0051, 0.002,
0.0004, 0.0005, 0.0213, 0.0048, 0.0016,
0.0012, 0.0014, 0.0024, 0.0015, 0.0004,
0.0005, 0.0007, 0.011, 0.0062, 0.0015,
0.0021, 0.0049, 0.0006, 0.0006, 0.0011,
0.0006, 0.0019, 0.003, 0.0044, 0.0054,
0.0057, 0.0049, 0.0016, 0.0006, 0.0009,
0.0006, 0.0012, 0.0006, 0.0149, 0.0532,
0.0076, 0.0041, 0.0024, 0.0135, 0.0081,
0.2209, 0.0459, 0.0438, 0.0488, 0.0137,
0.002, 0.0003, 0.0008, 0.0039, 0.0024,
0.0005, 0.0004, 0.003, 0.056, 0.0026]
SPLIT_DENSITY_SLOW = [0.0086, 0.0004, 0.0568, 0.0003, 0.0032, 0.0005, 0.0004, 0.0013, 0.0016, 0.0648, 0.0198, 0.1285, 0.098, 0.0005, 0.0064, 0.0003, 0.0004, 0.0026, 0.0007, 0.0051, 0.0089, 0.0024, 0.0033, 0.0057, 0.0005, 0.0003, 0.001, 0.0045, 0.0091, 0.0006, 0.0005, 0.0321, 0.0059, 0.1105, 0.216, 0.1489, 0.0004, 0.0003, 0.0006, 0.0483]
class Skipped(Exception):
pass
class TimeOutError(Exception):
pass
class DependencyError(Exception):
pass
# add more flags ??
future_flags = division.compiler_flag
def _indent(s, indent=4):
"""
Add the given number of space characters to the beginning of
every non-blank line in ``s``, and return the result.
If the string ``s`` is Unicode, it is encoded using the stdout
encoding and the ``backslashreplace`` error handler.
"""
# This regexp matches the start of non-blank lines:
return re.sub('(?m)^(?!$)', indent*' ', s)
pdoctest._indent = _indent # type: ignore
# override reporter to maintain windows and python3
def _report_failure(self, out, test, example, got):
"""
Report that the given example failed.
"""
s = self._checker.output_difference(example, got, self.optionflags)
s = s.encode('raw_unicode_escape').decode('utf8', 'ignore')
out(self._failure_header(test, example) + s)
if PY3 and IS_WINDOWS:
DocTestRunner.report_failure = _report_failure # type: ignore
def convert_to_native_paths(lst):
"""
Converts a list of '/' separated paths into a list of
native (os.sep separated) paths and converts to lowercase
if the system is case insensitive.
"""
newlst = []
for i, rv in enumerate(lst):
rv = os.path.join(*rv.split("/"))
# on windows the slash after the colon is dropped
if sys.platform == "win32":
pos = rv.find(':')
if pos != -1:
if rv[pos + 1] != '\\':
rv = rv[:pos + 1] + '\\' + rv[pos + 1:]
newlst.append(os.path.normcase(rv))
return newlst
def get_sympy_dir():
"""
Returns the root sympy directory and set the global value
indicating whether the system is case sensitive or not.
"""
this_file = os.path.abspath(__file__)
sympy_dir = os.path.join(os.path.dirname(this_file), "..", "..")
sympy_dir = os.path.normpath(sympy_dir)
return os.path.normcase(sympy_dir)
def setup_pprint():
from sympy import pprint_use_unicode, init_printing
import sympy.interactive.printing as interactive_printing
# force pprint to be in ascii mode in doctests
use_unicode_prev = pprint_use_unicode(False)
# hook our nice, hash-stable strprinter
init_printing(pretty_print=False)
# Prevent init_printing() in doctests from affecting other doctests
interactive_printing.NO_GLOBAL = True
return use_unicode_prev
@contextmanager
def raise_on_deprecated():
"""Context manager to make DeprecationWarning raise an error
This is to catch SymPyDeprecationWarning from library code while running
tests and doctests. It is important to use this context manager around
each individual test/doctest in case some tests modify the warning
filters.
"""
with warnings.catch_warnings():
warnings.filterwarnings('error', '.*', DeprecationWarning, module='sympy.*')
yield
def run_in_subprocess_with_hash_randomization(
function, function_args=(),
function_kwargs=None, command=sys.executable,
module='sympy.testing.runtests', force=False):
"""
Run a function in a Python subprocess with hash randomization enabled.
If hash randomization is not supported by the version of Python given, it
returns False. Otherwise, it returns the exit value of the command. The
function is passed to sys.exit(), so the return value of the function will
be the return value.
The environment variable PYTHONHASHSEED is used to seed Python's hash
randomization. If it is set, this function will return False, because
starting a new subprocess is unnecessary in that case. If it is not set,
one is set at random, and the tests are run. Note that if this
environment variable is set when Python starts, hash randomization is
automatically enabled. To force a subprocess to be created even if
PYTHONHASHSEED is set, pass ``force=True``. This flag will not force a
subprocess in Python versions that do not support hash randomization (see
below), because those versions of Python do not support the ``-R`` flag.
``function`` should be a string name of a function that is importable from
the module ``module``, like "_test". The default for ``module`` is
"sympy.testing.runtests". ``function_args`` and ``function_kwargs``
should be a repr-able tuple and dict, respectively. The default Python
command is sys.executable, which is the currently running Python command.
This function is necessary because the seed for hash randomization must be
set by the environment variable before Python starts. Hence, in order to
use a predetermined seed for tests, we must start Python in a separate
subprocess.
Hash randomization was added in the minor Python versions 2.6.8, 2.7.3,
3.1.5, and 3.2.3, and is enabled by default in all Python versions after
and including 3.3.0.
Examples
========
>>> from sympy.testing.runtests import (
... run_in_subprocess_with_hash_randomization)
>>> # run the core tests in verbose mode
>>> run_in_subprocess_with_hash_randomization("_test",
... function_args=("core",),
... function_kwargs={'verbose': True}) # doctest: +SKIP
# Will return 0 if sys.executable supports hash randomization and tests
# pass, 1 if they fail, and False if it does not support hash
# randomization.
"""
cwd = get_sympy_dir()
# Note, we must return False everywhere, not None, as subprocess.call will
# sometimes return None.
# First check if the Python version supports hash randomization
# If it doesn't have this support, it won't recognize the -R flag
p = subprocess.Popen([command, "-RV"], stdout=subprocess.PIPE,
stderr=subprocess.STDOUT, cwd=cwd)
p.communicate()
if p.returncode != 0:
return False
hash_seed = os.getenv("PYTHONHASHSEED")
if not hash_seed:
os.environ["PYTHONHASHSEED"] = str(random.randrange(2**32))
else:
if not force:
return False
function_kwargs = function_kwargs or {}
# Now run the command
commandstring = ("import sys; from %s import %s;sys.exit(%s(*%s, **%s))" %
(module, function, function, repr(function_args),
repr(function_kwargs)))
try:
p = subprocess.Popen([command, "-R", "-c", commandstring], cwd=cwd)
p.communicate()
except KeyboardInterrupt:
p.wait()
finally:
# Put the environment variable back, so that it reads correctly for
# the current Python process.
if hash_seed is None:
del os.environ["PYTHONHASHSEED"]
else:
os.environ["PYTHONHASHSEED"] = hash_seed
return p.returncode
def run_all_tests(test_args=(), test_kwargs=None,
doctest_args=(), doctest_kwargs=None,
examples_args=(), examples_kwargs=None):
"""
Run all tests.
Right now, this runs the regular tests (bin/test), the doctests
(bin/doctest), the examples (examples/all.py), and the sage tests (see
sympy/external/tests/test_sage.py).
This is what ``setup.py test`` uses.
You can pass arguments and keyword arguments to the test functions that
support them (for now, test, doctest, and the examples). See the
docstrings of those functions for a description of the available options.
For example, to run the solvers tests with colors turned off:
>>> from sympy.testing.runtests import run_all_tests
>>> run_all_tests(test_args=("solvers",),
... test_kwargs={"colors:False"}) # doctest: +SKIP
"""
cwd = get_sympy_dir()
tests_successful = True
test_kwargs = test_kwargs or {}
doctest_kwargs = doctest_kwargs or {}
examples_kwargs = examples_kwargs or {'quiet': True}
try:
# Regular tests
if not test(*test_args, **test_kwargs):
# some regular test fails, so set the tests_successful
# flag to false and continue running the doctests
tests_successful = False
# Doctests
print()
if not doctest(*doctest_args, **doctest_kwargs):
tests_successful = False
# Examples
print()
sys.path.append("examples") # examples/all.py
from all import run_examples # type: ignore
if not run_examples(*examples_args, **examples_kwargs):
tests_successful = False
# Sage tests
if sys.platform != "win32" and not PY3 and os.path.exists("bin/test"):
# run Sage tests; Sage currently doesn't support Windows or Python 3
# Only run Sage tests if 'bin/test' is present (it is missing from
# our release because everything in the 'bin' directory gets
# installed).
dev_null = open(os.devnull, 'w')
if subprocess.call("sage -v", shell=True, stdout=dev_null,
stderr=dev_null) == 0:
if subprocess.call("sage -python bin/test "
"sympy/external/tests/test_sage.py",
shell=True, cwd=cwd) != 0:
tests_successful = False
if tests_successful:
return
else:
# Return nonzero exit code
sys.exit(1)
except KeyboardInterrupt:
print()
print("DO *NOT* COMMIT!")
sys.exit(1)
def test(*paths, subprocess=True, rerun=0, **kwargs):
"""
Run tests in the specified test_*.py files.
Tests in a particular test_*.py file are run if any of the given strings
in ``paths`` matches a part of the test file's path. If ``paths=[]``,
tests in all test_*.py files are run.
Notes:
- If sort=False, tests are run in random order (not default).
- Paths can be entered in native system format or in unix,
forward-slash format.
- Files that are on the blacklist can be tested by providing
their path; they are only excluded if no paths are given.
**Explanation of test results**
====== ===============================================================
Output Meaning
====== ===============================================================
. passed
F failed
X XPassed (expected to fail but passed)
f XFAILed (expected to fail and indeed failed)
s skipped
w slow
T timeout (e.g., when ``--timeout`` is used)
K KeyboardInterrupt (when running the slow tests with ``--slow``,
you can interrupt one of them without killing the test runner)
====== ===============================================================
Colors have no additional meaning and are used just to facilitate
interpreting the output.
Examples
========
>>> import sympy
Run all tests:
>>> sympy.test() # doctest: +SKIP
Run one file:
>>> sympy.test("sympy/core/tests/test_basic.py") # doctest: +SKIP
>>> sympy.test("_basic") # doctest: +SKIP
Run all tests in sympy/functions/ and some particular file:
>>> sympy.test("sympy/core/tests/test_basic.py",
... "sympy/functions") # doctest: +SKIP
Run all tests in sympy/core and sympy/utilities:
>>> sympy.test("/core", "/util") # doctest: +SKIP
Run specific test from a file:
>>> sympy.test("sympy/core/tests/test_basic.py",
... kw="test_equality") # doctest: +SKIP
Run specific test from any file:
>>> sympy.test(kw="subs") # doctest: +SKIP
Run the tests with verbose mode on:
>>> sympy.test(verbose=True) # doctest: +SKIP
Don't sort the test output:
>>> sympy.test(sort=False) # doctest: +SKIP
Turn on post-mortem pdb:
>>> sympy.test(pdb=True) # doctest: +SKIP
Turn off colors:
>>> sympy.test(colors=False) # doctest: +SKIP
Force colors, even when the output is not to a terminal (this is useful,
e.g., if you are piping to ``less -r`` and you still want colors)
>>> sympy.test(force_colors=False) # doctest: +SKIP
The traceback verboseness can be set to "short" or "no" (default is
"short")
>>> sympy.test(tb='no') # doctest: +SKIP
The ``split`` option can be passed to split the test run into parts. The
split currently only splits the test files, though this may change in the
future. ``split`` should be a string of the form 'a/b', which will run
part ``a`` of ``b``. For instance, to run the first half of the test suite:
>>> sympy.test(split='1/2') # doctest: +SKIP
The ``time_balance`` option can be passed in conjunction with ``split``.
If ``time_balance=True`` (the default for ``sympy.test``), sympy will attempt
to split the tests such that each split takes equal time. This heuristic
for balancing is based on pre-recorded test data.
>>> sympy.test(split='1/2', time_balance=True) # doctest: +SKIP
You can disable running the tests in a separate subprocess using
``subprocess=False``. This is done to support seeding hash randomization,
which is enabled by default in the Python versions where it is supported.
If subprocess=False, hash randomization is enabled/disabled according to
whether it has been enabled or not in the calling Python process.
However, even if it is enabled, the seed cannot be printed unless it is
called from a new Python process.
Hash randomization was added in the minor Python versions 2.6.8, 2.7.3,
3.1.5, and 3.2.3, and is enabled by default in all Python versions after
and including 3.3.0.
If hash randomization is not supported ``subprocess=False`` is used
automatically.
>>> sympy.test(subprocess=False) # doctest: +SKIP
To set the hash randomization seed, set the environment variable
``PYTHONHASHSEED`` before running the tests. This can be done from within
Python using
>>> import os
>>> os.environ['PYTHONHASHSEED'] = '42' # doctest: +SKIP
Or from the command line using
$ PYTHONHASHSEED=42 ./bin/test
If the seed is not set, a random seed will be chosen.
Note that to reproduce the same hash values, you must use both the same seed
as well as the same architecture (32-bit vs. 64-bit).
"""
# count up from 0, do not print 0
print_counter = lambda i : (print("rerun %d" % (rerun-i))
if rerun-i else None)
if subprocess:
# loop backwards so last i is 0
for i in range(rerun, -1, -1):
print_counter(i)
ret = run_in_subprocess_with_hash_randomization("_test",
function_args=paths, function_kwargs=kwargs)
if ret is False:
break
val = not bool(ret)
# exit on the first failure or if done
if not val or i == 0:
return val
# rerun even if hash randomization is not supported
for i in range(rerun, -1, -1):
print_counter(i)
val = not bool(_test(*paths, **kwargs))
if not val or i == 0:
return val
def _test(*paths,
verbose=False, tb="short", kw=None, pdb=False, colors=True,
force_colors=False, sort=True, seed=None, timeout=False,
fail_on_timeout=False, slow=False, enhance_asserts=False, split=None,
time_balance=True, blacklist=('sympy/integrals/rubi/rubi_tests/tests',),
fast_threshold=None, slow_threshold=None):
"""
Internal function that actually runs the tests.
All keyword arguments from ``test()`` are passed to this function except for
``subprocess``.
Returns 0 if tests passed and 1 if they failed. See the docstring of
``test()`` for more information.
"""
kw = kw or ()
# ensure that kw is a tuple
if isinstance(kw, str):
kw = (kw,)
post_mortem = pdb
if seed is None:
seed = random.randrange(100000000)
if ON_TRAVIS and timeout is False:
# Travis times out if no activity is seen for 10 minutes.
timeout = 595
fail_on_timeout = True
if ON_TRAVIS:
# pyglet does not work on Travis
blacklist = list(blacklist) + ['sympy/plotting/pygletplot/tests']
blacklist = convert_to_native_paths(blacklist)
r = PyTestReporter(verbose=verbose, tb=tb, colors=colors,
force_colors=force_colors, split=split)
t = SymPyTests(r, kw, post_mortem, seed,
fast_threshold=fast_threshold,
slow_threshold=slow_threshold)
test_files = t.get_test_files('sympy')
not_blacklisted = [f for f in test_files
if not any(b in f for b in blacklist)]
if len(paths) == 0:
matched = not_blacklisted
else:
paths = convert_to_native_paths(paths)
matched = []
for f in not_blacklisted:
basename = os.path.basename(f)
for p in paths:
if p in f or fnmatch(basename, p):
matched.append(f)
break
density = None
if time_balance:
if slow:
density = SPLIT_DENSITY_SLOW
else:
density = SPLIT_DENSITY
if split:
matched = split_list(matched, split, density=density)
t._testfiles.extend(matched)
return int(not t.test(sort=sort, timeout=timeout, slow=slow,
enhance_asserts=enhance_asserts, fail_on_timeout=fail_on_timeout))
def doctest(*paths, subprocess=True, rerun=0, **kwargs):
r"""
Runs doctests in all \*.py files in the sympy directory which match
any of the given strings in ``paths`` or all tests if paths=[].
Notes:
- Paths can be entered in native system format or in unix,
forward-slash format.
- Files that are on the blacklist can be tested by providing
their path; they are only excluded if no paths are given.
Examples
========
>>> import sympy
Run all tests:
>>> sympy.doctest() # doctest: +SKIP
Run one file:
>>> sympy.doctest("sympy/core/basic.py") # doctest: +SKIP
>>> sympy.doctest("polynomial.rst") # doctest: +SKIP
Run all tests in sympy/functions/ and some particular file:
>>> sympy.doctest("/functions", "basic.py") # doctest: +SKIP
Run any file having polynomial in its name, doc/src/modules/polynomial.rst,
sympy/functions/special/polynomials.py, and sympy/polys/polynomial.py:
>>> sympy.doctest("polynomial") # doctest: +SKIP
The ``split`` option can be passed to split the test run into parts. The
split currently only splits the test files, though this may change in the
future. ``split`` should be a string of the form 'a/b', which will run
part ``a`` of ``b``. Note that the regular doctests and the Sphinx
doctests are split independently. For instance, to run the first half of
the test suite:
>>> sympy.doctest(split='1/2') # doctest: +SKIP
The ``subprocess`` and ``verbose`` options are the same as with the function
``test()``. See the docstring of that function for more information.
"""
# count up from 0, do not print 0
print_counter = lambda i : (print("rerun %d" % (rerun-i))
if rerun-i else None)
if subprocess:
# loop backwards so last i is 0
for i in range(rerun, -1, -1):
print_counter(i)
ret = run_in_subprocess_with_hash_randomization("_doctest",
function_args=paths, function_kwargs=kwargs)
if ret is False:
break
val = not bool(ret)
# exit on the first failure or if done
if not val or i == 0:
return val
# rerun even if hash randomization is not supported
for i in range(rerun, -1, -1):
print_counter(i)
val = not bool(_doctest(*paths, **kwargs))
if not val or i == 0:
return val
def _get_doctest_blacklist():
'''Get the default blacklist for the doctests'''
blacklist = []
blacklist.extend([
"doc/src/modules/plotting.rst", # generates live plots
"doc/src/modules/physics/mechanics/autolev_parser.rst",
"sympy/galgebra.py", # no longer part of SymPy
"sympy/this.py", # prints text
"sympy/physics/gaussopt.py", # raises deprecation warning
"sympy/matrices/densearith.py", # raises deprecation warning
"sympy/matrices/densesolve.py", # raises deprecation warning
"sympy/matrices/densetools.py", # raises deprecation warning
"sympy/printing/ccode.py", # backwards compatibility shim, importing it breaks the codegen doctests
"sympy/printing/fcode.py", # backwards compatibility shim, importing it breaks the codegen doctests
"sympy/printing/cxxcode.py", # backwards compatibility shim, importing it breaks the codegen doctests
"sympy/parsing/autolev/_antlr/autolevlexer.py", # generated code
"sympy/parsing/autolev/_antlr/autolevparser.py", # generated code
"sympy/parsing/autolev/_antlr/autolevlistener.py", # generated code
"sympy/parsing/latex/_antlr/latexlexer.py", # generated code
"sympy/parsing/latex/_antlr/latexparser.py", # generated code
"sympy/integrals/rubi/rubi.py",
"sympy/plotting/pygletplot/__init__.py", # crashes on some systems
"sympy/plotting/pygletplot/plot.py", # crashes on some systems
])
# autolev parser tests
num = 12
for i in range (1, num+1):
blacklist.append("sympy/parsing/autolev/test-examples/ruletest" + str(i) + ".py")
blacklist.extend(["sympy/parsing/autolev/test-examples/pydy-example-repo/mass_spring_damper.py",
"sympy/parsing/autolev/test-examples/pydy-example-repo/chaos_pendulum.py",
"sympy/parsing/autolev/test-examples/pydy-example-repo/double_pendulum.py",
"sympy/parsing/autolev/test-examples/pydy-example-repo/non_min_pendulum.py"])
if import_module('numpy') is None:
blacklist.extend([
"sympy/plotting/experimental_lambdify.py",
"sympy/plotting/plot_implicit.py",
"examples/advanced/autowrap_integrators.py",
"examples/advanced/autowrap_ufuncify.py",
"examples/intermediate/sample.py",
"examples/intermediate/mplot2d.py",
"examples/intermediate/mplot3d.py",
"doc/src/modules/numeric-computation.rst"
])
else:
if import_module('matplotlib') is None:
blacklist.extend([
"examples/intermediate/mplot2d.py",
"examples/intermediate/mplot3d.py"
])
else:
# Use a non-windowed backend, so that the tests work on Travis
import matplotlib
matplotlib.use('Agg')
if ON_TRAVIS or import_module('pyglet') is None:
blacklist.extend(["sympy/plotting/pygletplot"])
if import_module('aesara') is None:
blacklist.extend([
"sympy/printing/aesaracode.py",
"doc/src/modules/numeric-computation.rst",
])
if import_module('cupy') is None:
blacklist.extend([
"doc/src/modules/numeric-computation.rst",
])
if import_module('antlr4') is None:
blacklist.extend([
"sympy/parsing/autolev/__init__.py",
"sympy/parsing/latex/_parse_latex_antlr.py",
])
if import_module('lfortran') is None:
#throws ImportError when lfortran not installed
blacklist.extend([
"sympy/parsing/sym_expr.py",
])
# disabled because of doctest failures in asmeurer's bot
blacklist.extend([
"sympy/utilities/autowrap.py",
"examples/advanced/autowrap_integrators.py",
"examples/advanced/autowrap_ufuncify.py"
])
# blacklist these modules until issue 4840 is resolved
blacklist.extend([
"sympy/conftest.py", # Python 2.7 issues
"sympy/testing/benchmarking.py",
])
# These are deprecated stubs to be removed:
blacklist.extend([
"sympy/utilities/benchmarking.py",
"sympy/utilities/tmpfiles.py",
"sympy/utilities/pytest.py",
"sympy/utilities/runtests.py",
"sympy/utilities/quality_unicode.py",
"sympy/utilities/randtest.py",
])
blacklist = convert_to_native_paths(blacklist)
return blacklist
def _doctest(*paths, **kwargs):
"""
Internal function that actually runs the doctests.
All keyword arguments from ``doctest()`` are passed to this function
except for ``subprocess``.
Returns 0 if tests passed and 1 if they failed. See the docstrings of
``doctest()`` and ``test()`` for more information.
"""
from sympy import pprint_use_unicode
normal = kwargs.get("normal", False)
verbose = kwargs.get("verbose", False)
colors = kwargs.get("colors", True)
force_colors = kwargs.get("force_colors", False)
blacklist = kwargs.get("blacklist", [])
split = kwargs.get('split', None)
blacklist.extend(_get_doctest_blacklist())
# Use a non-windowed backend, so that the tests work on Travis
if import_module('matplotlib') is not None:
import matplotlib
matplotlib.use('Agg')
# Disable warnings for external modules
import sympy.external
sympy.external.importtools.WARN_OLD_VERSION = False
sympy.external.importtools.WARN_NOT_INSTALLED = False
# Disable showing up of plots
from sympy.plotting.plot import unset_show
unset_show()
r = PyTestReporter(verbose, split=split, colors=colors,\
force_colors=force_colors)
t = SymPyDocTests(r, normal)
test_files = t.get_test_files('sympy')
test_files.extend(t.get_test_files('examples', init_only=False))
not_blacklisted = [f for f in test_files
if not any(b in f for b in blacklist)]
if len(paths) == 0:
matched = not_blacklisted
else:
# take only what was requested...but not blacklisted items
# and allow for partial match anywhere or fnmatch of name
paths = convert_to_native_paths(paths)
matched = []
for f in not_blacklisted:
basename = os.path.basename(f)
for p in paths:
if p in f or fnmatch(basename, p):
matched.append(f)
break
if split:
matched = split_list(matched, split)
t._testfiles.extend(matched)
# run the tests and record the result for this *py portion of the tests
if t._testfiles:
failed = not t.test()
else:
failed = False
# N.B.
# --------------------------------------------------------------------
# Here we test *.rst files at or below doc/src. Code from these must
# be self supporting in terms of imports since there is no importing
# of necessary modules by doctest.testfile. If you try to pass *.py
# files through this they might fail because they will lack the needed
# imports and smarter parsing that can be done with source code.
#
test_files = t.get_test_files('doc/src', '*.rst', init_only=False)
test_files.sort()
not_blacklisted = [f for f in test_files
if not any(b in f for b in blacklist)]
if len(paths) == 0:
matched = not_blacklisted
else:
# Take only what was requested as long as it's not on the blacklist.
# Paths were already made native in *py tests so don't repeat here.
# There's no chance of having a *py file slip through since we
# only have *rst files in test_files.
matched = []
for f in not_blacklisted:
basename = os.path.basename(f)
for p in paths:
if p in f or fnmatch(basename, p):
matched.append(f)
break
if split:
matched = split_list(matched, split)
first_report = True
for rst_file in matched:
if not os.path.isfile(rst_file):
continue
old_displayhook = sys.displayhook
try:
use_unicode_prev = setup_pprint()
out = sympytestfile(
rst_file, module_relative=False, encoding='utf-8',
optionflags=pdoctest.ELLIPSIS | pdoctest.NORMALIZE_WHITESPACE |
pdoctest.IGNORE_EXCEPTION_DETAIL)
finally:
# make sure we return to the original displayhook in case some
# doctest has changed that
sys.displayhook = old_displayhook
# The NO_GLOBAL flag overrides the no_global flag to init_printing
# if True
import sympy.interactive.printing as interactive_printing
interactive_printing.NO_GLOBAL = False
pprint_use_unicode(use_unicode_prev)
rstfailed, tested = out
if tested:
failed = rstfailed or failed
if first_report:
first_report = False
msg = 'rst doctests start'
if not t._testfiles:
r.start(msg=msg)
else:
r.write_center(msg)
print()
# use as the id, everything past the first 'sympy'
file_id = rst_file[rst_file.find('sympy') + len('sympy') + 1:]
print(file_id, end=" ")
# get at least the name out so it is know who is being tested
wid = r.terminal_width - len(file_id) - 1 # update width
test_file = '[%s]' % (tested)
report = '[%s]' % (rstfailed or 'OK')
print(''.join(
[test_file, ' '*(wid - len(test_file) - len(report)), report])
)
# the doctests for *py will have printed this message already if there was
# a failure, so now only print it if there was intervening reporting by
# testing the *rst as evidenced by first_report no longer being True.
if not first_report and failed:
print()
print("DO *NOT* COMMIT!")
return int(failed)
sp = re.compile(r'([0-9]+)/([1-9][0-9]*)')
def split_list(l, split, density=None):
"""
Splits a list into part a of b
split should be a string of the form 'a/b'. For instance, '1/3' would give
the split one of three.
If the length of the list is not divisible by the number of splits, the
last split will have more items.
`density` may be specified as a list. If specified,
tests will be balanced so that each split has as equal-as-possible
amount of mass according to `density`.
>>> from sympy.testing.runtests import split_list
>>> a = list(range(10))
>>> split_list(a, '1/3')
[0, 1, 2]
>>> split_list(a, '2/3')
[3, 4, 5]
>>> split_list(a, '3/3')
[6, 7, 8, 9]
"""
m = sp.match(split)
if not m:
raise ValueError("split must be a string of the form a/b where a and b are ints")
i, t = map(int, m.groups())
if not density:
return l[(i - 1)*len(l)//t : i*len(l)//t]
# normalize density
tot = sum(density)
density = [x / tot for x in density]
def density_inv(x):
"""Interpolate the inverse to the cumulative
distribution function given by density"""
if x <= 0:
return 0
if x >= sum(density):
return 1
# find the first time the cumulative sum surpasses x
# and linearly interpolate
cumm = 0
for i, d in enumerate(density):
cumm += d
if cumm >= x:
break
frac = (d - (cumm - x)) / d
return (i + frac) / len(density)
lower_frac = density_inv((i - 1) / t)
higher_frac = density_inv(i / t)
return l[int(lower_frac*len(l)) : int(higher_frac*len(l))]
from collections import namedtuple
SymPyTestResults = namedtuple('SymPyTestResults', 'failed attempted')
def sympytestfile(filename, module_relative=True, name=None, package=None,
globs=None, verbose=None, report=True, optionflags=0,
extraglobs=None, raise_on_error=False,
parser=pdoctest.DocTestParser(), encoding=None):
"""
Test examples in the given file. Return (#failures, #tests).
Optional keyword arg ``module_relative`` specifies how filenames
should be interpreted:
- If ``module_relative`` is True (the default), then ``filename``
specifies a module-relative path. By default, this path is
relative to the calling module's directory; but if the
``package`` argument is specified, then it is relative to that
package. To ensure os-independence, ``filename`` should use
"/" characters to separate path segments, and should not
be an absolute path (i.e., it may not begin with "/").
- If ``module_relative`` is False, then ``filename`` specifies an
os-specific path. The path may be absolute or relative (to
the current working directory).
Optional keyword arg ``name`` gives the name of the test; by default
use the file's basename.
Optional keyword argument ``package`` is a Python package or the
name of a Python package whose directory should be used as the
base directory for a module relative filename. If no package is
specified, then the calling module's directory is used as the base
directory for module relative filenames. It is an error to
specify ``package`` if ``module_relative`` is False.
Optional keyword arg ``globs`` gives a dict to be used as the globals
when executing examples; by default, use {}. A copy of this dict
is actually used for each docstring, so that each docstring's
examples start with a clean slate.
Optional keyword arg ``extraglobs`` gives a dictionary that should be
merged into the globals that are used to execute examples. By
default, no extra globals are used.
Optional keyword arg ``verbose`` prints lots of stuff if true, prints
only failures if false; by default, it's true iff "-v" is in sys.argv.
Optional keyword arg ``report`` prints a summary at the end when true,
else prints nothing at the end. In verbose mode, the summary is
detailed, else very brief (in fact, empty if all tests passed).
Optional keyword arg ``optionflags`` or's together module constants,
and defaults to 0. Possible values (see the docs for details):
- DONT_ACCEPT_TRUE_FOR_1
- DONT_ACCEPT_BLANKLINE
- NORMALIZE_WHITESPACE
- ELLIPSIS
- SKIP
- IGNORE_EXCEPTION_DETAIL
- REPORT_UDIFF
- REPORT_CDIFF
- REPORT_NDIFF
- REPORT_ONLY_FIRST_FAILURE
Optional keyword arg ``raise_on_error`` raises an exception on the
first unexpected exception or failure. This allows failures to be
post-mortem debugged.
Optional keyword arg ``parser`` specifies a DocTestParser (or
subclass) that should be used to extract tests from the files.
Optional keyword arg ``encoding`` specifies an encoding that should
be used to convert the file to unicode.
Advanced tomfoolery: testmod runs methods of a local instance of
class doctest.Tester, then merges the results into (or creates)
global Tester instance doctest.master. Methods of doctest.master
can be called directly too, if you want to do something unusual.
Passing report=0 to testmod is especially useful then, to delay
displaying a summary. Invoke doctest.master.summarize(verbose)
when you're done fiddling.
"""
if package and not module_relative:
raise ValueError("Package may only be specified for module-"
"relative paths.")
# Relativize the path
if not PY3:
text, filename = pdoctest._load_testfile(
filename, package, module_relative)
if encoding is not None:
text = text.decode(encoding)
else:
text, filename = pdoctest._load_testfile(
filename, package, module_relative, encoding)
# If no name was given, then use the file's name.
if name is None:
name = os.path.basename(filename)
# Assemble the globals.
if globs is None:
globs = {}
else:
globs = globs.copy()
if extraglobs is not None:
globs.update(extraglobs)
if '__name__' not in globs:
globs['__name__'] = '__main__'
if raise_on_error:
runner = pdoctest.DebugRunner(verbose=verbose, optionflags=optionflags)
else:
runner = SymPyDocTestRunner(verbose=verbose, optionflags=optionflags)
runner._checker = SymPyOutputChecker()
# Read the file, convert it to a test, and run it.
test = parser.get_doctest(text, globs, name, filename, 0)
runner.run(test, compileflags=future_flags)
if report:
runner.summarize()
if pdoctest.master is None:
pdoctest.master = runner
else:
pdoctest.master.merge(runner)
return SymPyTestResults(runner.failures, runner.tries)
class SymPyTests:
def __init__(self, reporter, kw="", post_mortem=False,
seed=None, fast_threshold=None, slow_threshold=None):
self._post_mortem = post_mortem
self._kw = kw
self._count = 0
self._root_dir = get_sympy_dir()
self._reporter = reporter
self._reporter.root_dir(self._root_dir)
self._testfiles = []
self._seed = seed if seed is not None else random.random()
# Defaults in seconds, from human / UX design limits
# http://www.nngroup.com/articles/response-times-3-important-limits/
#
# These defaults are *NOT* set in stone as we are measuring different
# things, so others feel free to come up with a better yardstick :)
if fast_threshold:
self._fast_threshold = float(fast_threshold)
else:
self._fast_threshold = 8
if slow_threshold:
self._slow_threshold = float(slow_threshold)
else:
self._slow_threshold = 10
def test(self, sort=False, timeout=False, slow=False,
enhance_asserts=False, fail_on_timeout=False):
"""
Runs the tests returning True if all tests pass, otherwise False.
If sort=False run tests in random order.
"""
if sort:
self._testfiles.sort()
elif slow:
pass
else:
random.seed(self._seed)
random.shuffle(self._testfiles)
self._reporter.start(self._seed)
for f in self._testfiles:
try:
self.test_file(f, sort, timeout, slow,
enhance_asserts, fail_on_timeout)
except KeyboardInterrupt:
print(" interrupted by user")
self._reporter.finish()
raise
return self._reporter.finish()
def _enhance_asserts(self, source):
from ast import (NodeTransformer, Compare, Name, Store, Load, Tuple,
Assign, BinOp, Str, Mod, Assert, parse, fix_missing_locations)
ops = {"Eq": '==', "NotEq": '!=', "Lt": '<', "LtE": '<=',
"Gt": '>', "GtE": '>=', "Is": 'is', "IsNot": 'is not',
"In": 'in', "NotIn": 'not in'}
class Transform(NodeTransformer):
def visit_Assert(self, stmt):
if isinstance(stmt.test, Compare):
compare = stmt.test
values = [compare.left] + compare.comparators
names = [ "_%s" % i for i, _ in enumerate(values) ]
names_store = [ Name(n, Store()) for n in names ]
names_load = [ Name(n, Load()) for n in names ]
target = Tuple(names_store, Store())
value = Tuple(values, Load())
assign = Assign([target], value)
new_compare = Compare(names_load[0], compare.ops, names_load[1:])
msg_format = "\n%s " + "\n%s ".join([ ops[op.__class__.__name__] for op in compare.ops ]) + "\n%s"
msg = BinOp(Str(msg_format), Mod(), Tuple(names_load, Load()))
test = Assert(new_compare, msg, lineno=stmt.lineno, col_offset=stmt.col_offset)
return [assign, test]
else:
return stmt
tree = parse(source)
new_tree = Transform().visit(tree)
return fix_missing_locations(new_tree)
def test_file(self, filename, sort=True, timeout=False, slow=False,
enhance_asserts=False, fail_on_timeout=False):
reporter = self._reporter
funcs = []
try:
gl = {'__file__': filename}
try:
if PY3:
open_file = lambda: open(filename, encoding="utf8")
else:
open_file = lambda: open(filename)
with open_file() as f:
source = f.read()
if self._kw:
for l in source.splitlines():
if l.lstrip().startswith('def '):
if any(l.find(k) != -1 for k in self._kw):
break
else:
return
if enhance_asserts:
try:
source = self._enhance_asserts(source)
except ImportError:
pass
code = compile(source, filename, "exec", flags=0, dont_inherit=True)
exec(code, gl)
except (SystemExit, KeyboardInterrupt):
raise
except ImportError:
reporter.import_error(filename, sys.exc_info())
return
except Exception:
reporter.test_exception(sys.exc_info())
clear_cache()
self._count += 1
random.seed(self._seed)
disabled = gl.get("disabled", False)
if not disabled:
# we need to filter only those functions that begin with 'test_'
# We have to be careful about decorated functions. As long as
# the decorator uses functools.wraps, we can detect it.
funcs = []
for f in gl:
if (f.startswith("test_") and (inspect.isfunction(gl[f])
or inspect.ismethod(gl[f]))):
func = gl[f]
# Handle multiple decorators
while hasattr(func, '__wrapped__'):
func = func.__wrapped__
if inspect.getsourcefile(func) == filename:
funcs.append(gl[f])
if slow:
funcs = [f for f in funcs if getattr(f, '_slow', False)]
# Sorting of XFAILed functions isn't fixed yet :-(
funcs.sort(key=lambda x: inspect.getsourcelines(x)[1])
i = 0
while i < len(funcs):
if inspect.isgeneratorfunction(funcs[i]):
# some tests can be generators, that return the actual
# test functions. We unpack it below:
f = funcs.pop(i)
for fg in f():
func = fg[0]
args = fg[1:]
fgw = lambda: func(*args)
funcs.insert(i, fgw)
i += 1
else:
i += 1
# drop functions that are not selected with the keyword expression:
funcs = [x for x in funcs if self.matches(x)]
if not funcs:
return
except Exception:
reporter.entering_filename(filename, len(funcs))
raise
reporter.entering_filename(filename, len(funcs))
if not sort:
random.shuffle(funcs)
for f in funcs:
start = time.time()
reporter.entering_test(f)
try:
if getattr(f, '_slow', False) and not slow:
raise Skipped("Slow")
with raise_on_deprecated():
if timeout:
self._timeout(f, timeout, fail_on_timeout)
else:
random.seed(self._seed)
f()
except KeyboardInterrupt:
if getattr(f, '_slow', False):
reporter.test_skip("KeyboardInterrupt")
else:
raise
except Exception:
if timeout:
signal.alarm(0) # Disable the alarm. It could not be handled before.
t, v, tr = sys.exc_info()
if t is AssertionError:
reporter.test_fail((t, v, tr))
if self._post_mortem:
pdb.post_mortem(tr)
elif t.__name__ == "Skipped":
reporter.test_skip(v)
elif t.__name__ == "XFail":
reporter.test_xfail()
elif t.__name__ == "XPass":
reporter.test_xpass(v)
else:
reporter.test_exception((t, v, tr))
if self._post_mortem:
pdb.post_mortem(tr)
else:
reporter.test_pass()
taken = time.time() - start
if taken > self._slow_threshold:
filename = os.path.relpath(filename, reporter._root_dir)
reporter.slow_test_functions.append(
(filename + "::" + f.__name__, taken))
if getattr(f, '_slow', False) and slow:
if taken < self._fast_threshold:
filename = os.path.relpath(filename, reporter._root_dir)
reporter.fast_test_functions.append(
(filename + "::" + f.__name__, taken))
reporter.leaving_filename()
def _timeout(self, function, timeout, fail_on_timeout):
def callback(x, y):
signal.alarm(0)
if fail_on_timeout:
raise TimeOutError("Timed out after %d seconds" % timeout)
else:
raise Skipped("Timeout")
signal.signal(signal.SIGALRM, callback)
signal.alarm(timeout) # Set an alarm with a given timeout
function()
signal.alarm(0) # Disable the alarm
def matches(self, x):
"""
Does the keyword expression self._kw match "x"? Returns True/False.
Always returns True if self._kw is "".
"""
if not self._kw:
return True
for kw in self._kw:
if x.__name__.find(kw) != -1:
return True
return False
def get_test_files(self, dir, pat='test_*.py'):
"""
Returns the list of test_*.py (default) files at or below directory
``dir`` relative to the sympy home directory.
"""
dir = os.path.join(self._root_dir, convert_to_native_paths([dir])[0])
g = []
for path, folders, files in os.walk(dir):
g.extend([os.path.join(path, f) for f in files if fnmatch(f, pat)])
return sorted([os.path.normcase(gi) for gi in g])
class SymPyDocTests:
def __init__(self, reporter, normal):
self._count = 0
self._root_dir = get_sympy_dir()
self._reporter = reporter
self._reporter.root_dir(self._root_dir)
self._normal = normal
self._testfiles = []
def test(self):
"""
Runs the tests and returns True if all tests pass, otherwise False.
"""
self._reporter.start()
for f in self._testfiles:
try:
self.test_file(f)
except KeyboardInterrupt:
print(" interrupted by user")
self._reporter.finish()
raise
return self._reporter.finish()
def test_file(self, filename):
clear_cache()
from io import StringIO
import sympy.interactive.printing as interactive_printing
from sympy import pprint_use_unicode
rel_name = filename[len(self._root_dir) + 1:]
dirname, file = os.path.split(filename)
module = rel_name.replace(os.sep, '.')[:-3]
if rel_name.startswith("examples"):
# Examples files do not have __init__.py files,
# So we have to temporarily extend sys.path to import them
sys.path.insert(0, dirname)
module = file[:-3] # remove ".py"
try:
module = pdoctest._normalize_module(module)
tests = SymPyDocTestFinder().find(module)
except (SystemExit, KeyboardInterrupt):
raise
except ImportError:
self._reporter.import_error(filename, sys.exc_info())
return
finally:
if rel_name.startswith("examples"):
del sys.path[0]
tests = [test for test in tests if len(test.examples) > 0]
# By default tests are sorted by alphabetical order by function name.
# We sort by line number so one can edit the file sequentially from
# bottom to top. However, if there are decorated functions, their line
# numbers will be too large and for now one must just search for these
# by text and function name.
tests.sort(key=lambda x: -x.lineno)
if not tests:
return
self._reporter.entering_filename(filename, len(tests))
for test in tests:
assert len(test.examples) != 0
if self._reporter._verbose:
self._reporter.write("\n{} ".format(test.name))
# check if there are external dependencies which need to be met
if '_doctest_depends_on' in test.globs:
try:
self._check_dependencies(**test.globs['_doctest_depends_on'])
except DependencyError as e:
self._reporter.test_skip(v=str(e))
continue
runner = SymPyDocTestRunner(optionflags=pdoctest.ELLIPSIS |
pdoctest.NORMALIZE_WHITESPACE |
pdoctest.IGNORE_EXCEPTION_DETAIL)
runner._checker = SymPyOutputChecker()
old = sys.stdout
new = StringIO()
sys.stdout = new
# If the testing is normal, the doctests get importing magic to
# provide the global namespace. If not normal (the default) then
# then must run on their own; all imports must be explicit within
# a function's docstring. Once imported that import will be
# available to the rest of the tests in a given function's
# docstring (unless clear_globs=True below).
if not self._normal:
test.globs = {}
# if this is uncommented then all the test would get is what
# comes by default with a "from sympy import *"
#exec('from sympy import *') in test.globs
test.globs['print_function'] = print_function
old_displayhook = sys.displayhook
use_unicode_prev = setup_pprint()
try:
f, t = runner.run(test, compileflags=future_flags,
out=new.write, clear_globs=False)
except KeyboardInterrupt:
raise
finally:
sys.stdout = old
if f > 0:
self._reporter.doctest_fail(test.name, new.getvalue())
else:
self._reporter.test_pass()
sys.displayhook = old_displayhook
interactive_printing.NO_GLOBAL = False
pprint_use_unicode(use_unicode_prev)
self._reporter.leaving_filename()
def get_test_files(self, dir, pat='*.py', init_only=True):
r"""
Returns the list of \*.py files (default) from which docstrings
will be tested which are at or below directory ``dir``. By default,
only those that have an __init__.py in their parent directory
and do not start with ``test_`` will be included.
"""
def importable(x):
"""
Checks if given pathname x is an importable module by checking for
__init__.py file.
Returns True/False.
Currently we only test if the __init__.py file exists in the
directory with the file "x" (in theory we should also test all the
parent dirs).
"""
init_py = os.path.join(os.path.dirname(x), "__init__.py")
return os.path.exists(init_py)
dir = os.path.join(self._root_dir, convert_to_native_paths([dir])[0])
g = []
for path, folders, files in os.walk(dir):
g.extend([os.path.join(path, f) for f in files
if not f.startswith('test_') and fnmatch(f, pat)])
if init_only:
# skip files that are not importable (i.e. missing __init__.py)
g = [x for x in g if importable(x)]
return [os.path.normcase(gi) for gi in g]
def _check_dependencies(self,
executables=(),
modules=(),
disable_viewers=(),
python_version=(3, 5)):
"""
Checks if the dependencies for the test are installed.
Raises ``DependencyError`` it at least one dependency is not installed.
"""
for executable in executables:
if not shutil.which(executable):
raise DependencyError("Could not find %s" % executable)
for module in modules:
if module == 'matplotlib':
matplotlib = import_module(
'matplotlib',
import_kwargs={'fromlist':
['pyplot', 'cm', 'collections']},
min_module_version='1.0.0', catch=(RuntimeError,))
if matplotlib is None:
raise DependencyError("Could not import matplotlib")
else:
if not import_module(module):
raise DependencyError("Could not import %s" % module)
if disable_viewers:
tempdir = tempfile.mkdtemp()
os.environ['PATH'] = '%s:%s' % (tempdir, os.environ['PATH'])
vw = ('#!/usr/bin/env {}\n'
'import sys\n'
'if len(sys.argv) <= 1:\n'
' exit("wrong number of args")\n').format(
'python3' if PY3 else 'python')
for viewer in disable_viewers:
with open(os.path.join(tempdir, viewer), 'w') as fh:
fh.write(vw)
# make the file executable
os.chmod(os.path.join(tempdir, viewer),
stat.S_IREAD | stat.S_IWRITE | stat.S_IXUSR)
if python_version:
if sys.version_info < python_version:
raise DependencyError("Requires Python >= " + '.'.join(map(str, python_version)))
if 'pyglet' in modules:
# monkey-patch pyglet s.t. it does not open a window during
# doctesting
import pyglet
class DummyWindow:
def __init__(self, *args, **kwargs):
self.has_exit = True
self.width = 600
self.height = 400
def set_vsync(self, x):
pass
def switch_to(self):
pass
def push_handlers(self, x):
pass
def close(self):
pass
pyglet.window.Window = DummyWindow
class SymPyDocTestFinder(DocTestFinder):
"""
A class used to extract the DocTests that are relevant to a given
object, from its docstring and the docstrings of its contained
objects. Doctests can currently be extracted from the following
object types: modules, functions, classes, methods, staticmethods,
classmethods, and properties.
Modified from doctest's version to look harder for code that
appears comes from a different module. For example, the @vectorize
decorator makes it look like functions come from multidimensional.py
even though their code exists elsewhere.
"""
def _find(self, tests, obj, name, module, source_lines, globs, seen):
"""
Find tests for the given object and any contained objects, and
add them to ``tests``.
"""
if self._verbose:
print('Finding tests in %s' % name)
# If we've already processed this object, then ignore it.
if id(obj) in seen:
return
seen[id(obj)] = 1
# Make sure we don't run doctests for classes outside of sympy, such
# as in numpy or scipy.
if inspect.isclass(obj):
if obj.__module__.split('.')[0] != 'sympy':
return
# Find a test for this object, and add it to the list of tests.
test = self._get_test(obj, name, module, globs, source_lines)
if test is not None:
tests.append(test)
if not self._recurse:
return
# Look for tests in a module's contained objects.
if inspect.ismodule(obj):
for rawname, val in obj.__dict__.items():
# Recurse to functions & classes.
if inspect.isfunction(val) or inspect.isclass(val):
# Make sure we don't run doctests functions or classes
# from different modules
if val.__module__ != module.__name__:
continue
assert self._from_module(module, val), \
"%s is not in module %s (rawname %s)" % (val, module, rawname)
try:
valname = '%s.%s' % (name, rawname)
self._find(tests, val, valname, module,
source_lines, globs, seen)
except KeyboardInterrupt:
raise
# Look for tests in a module's __test__ dictionary.
for valname, val in getattr(obj, '__test__', {}).items():
if not isinstance(valname, str):
raise ValueError("SymPyDocTestFinder.find: __test__ keys "
"must be strings: %r" %
(type(valname),))
if not (inspect.isfunction(val) or inspect.isclass(val) or
inspect.ismethod(val) or inspect.ismodule(val) or
isinstance(val, str)):
raise ValueError("SymPyDocTestFinder.find: __test__ values "
"must be strings, functions, methods, "
"classes, or modules: %r" %
(type(val),))
valname = '%s.__test__.%s' % (name, valname)
self._find(tests, val, valname, module, source_lines,
globs, seen)
# Look for tests in a class's contained objects.
if inspect.isclass(obj):
for valname, val in obj.__dict__.items():
# Special handling for staticmethod/classmethod.
if isinstance(val, staticmethod):
val = getattr(obj, valname)
if isinstance(val, classmethod):
val = getattr(obj, valname).__func__
# Recurse to methods, properties, and nested classes.
if ((inspect.isfunction(unwrap(val)) or
inspect.isclass(val) or
isinstance(val, property)) and
self._from_module(module, val)):
# Make sure we don't run doctests functions or classes
# from different modules
if isinstance(val, property):
if hasattr(val.fget, '__module__'):
if val.fget.__module__ != module.__name__:
continue
else:
if val.__module__ != module.__name__:
continue
assert self._from_module(module, val), \
"%s is not in module %s (valname %s)" % (
val, module, valname)
valname = '%s.%s' % (name, valname)
self._find(tests, val, valname, module, source_lines,
globs, seen)
def _get_test(self, obj, name, module, globs, source_lines):
"""
Return a DocTest for the given object, if it defines a docstring;
otherwise, return None.
"""
lineno = None
# Extract the object's docstring. If it doesn't have one,
# then return None (no test for this object).
if isinstance(obj, str):
# obj is a string in the case for objects in the polys package.
# Note that source_lines is a binary string (compiled polys
# modules), which can't be handled by _find_lineno so determine
# the line number here.
docstring = obj
matches = re.findall(r"line \d+", name)
assert len(matches) == 1, \
"string '%s' does not contain lineno " % name
# NOTE: this is not the exact linenumber but its better than no
# lineno ;)
lineno = int(matches[0][5:])
else:
try:
if obj.__doc__ is None:
docstring = ''
else:
docstring = obj.__doc__
if not isinstance(docstring, str):
docstring = str(docstring)
except (TypeError, AttributeError):
docstring = ''
# Don't bother if the docstring is empty.
if self._exclude_empty and not docstring:
return None
# check that properties have a docstring because _find_lineno
# assumes it
if isinstance(obj, property):
if obj.fget.__doc__ is None:
return None
# Find the docstring's location in the file.
if lineno is None:
obj = unwrap(obj)
# handling of properties is not implemented in _find_lineno so do
# it here
if hasattr(obj, 'func_closure') and obj.func_closure is not None:
tobj = obj.func_closure[0].cell_contents
elif isinstance(obj, property):
tobj = obj.fget
else:
tobj = obj
lineno = self._find_lineno(tobj, source_lines)
if lineno is None:
return None
# Return a DocTest for this object.
if module is None:
filename = None
else:
filename = getattr(module, '__file__', module.__name__)
if filename[-4:] in (".pyc", ".pyo"):
filename = filename[:-1]
globs['_doctest_depends_on'] = getattr(obj, '_doctest_depends_on', {})
return self._parser.get_doctest(docstring, globs, name,
filename, lineno)
class SymPyDocTestRunner(DocTestRunner):
"""
A class used to run DocTest test cases, and accumulate statistics.
The ``run`` method is used to process a single DocTest case. It
returns a tuple ``(f, t)``, where ``t`` is the number of test cases
tried, and ``f`` is the number of test cases that failed.
Modified from the doctest version to not reset the sys.displayhook (see
issue 5140).
See the docstring of the original DocTestRunner for more information.
"""
def run(self, test, compileflags=None, out=None, clear_globs=True):
"""
Run the examples in ``test``, and display the results using the
writer function ``out``.
The examples are run in the namespace ``test.globs``. If
``clear_globs`` is true (the default), then this namespace will
be cleared after the test runs, to help with garbage
collection. If you would like to examine the namespace after
the test completes, then use ``clear_globs=False``.
``compileflags`` gives the set of flags that should be used by
the Python compiler when running the examples. If not
specified, then it will default to the set of future-import
flags that apply to ``globs``.
The output of each example is checked using
``SymPyDocTestRunner.check_output``, and the results are
formatted by the ``SymPyDocTestRunner.report_*`` methods.
"""
self.test = test
if compileflags is None:
compileflags = pdoctest._extract_future_flags(test.globs)
save_stdout = sys.stdout
if out is None:
out = save_stdout.write
sys.stdout = self._fakeout
# Patch pdb.set_trace to restore sys.stdout during interactive
# debugging (so it's not still redirected to self._fakeout).
# Note that the interactive output will go to *our*
# save_stdout, even if that's not the real sys.stdout; this
# allows us to write test cases for the set_trace behavior.
save_set_trace = pdb.set_trace
self.debugger = pdoctest._OutputRedirectingPdb(save_stdout)
self.debugger.reset()
pdb.set_trace = self.debugger.set_trace
# Patch linecache.getlines, so we can see the example's source
# when we're inside the debugger.
self.save_linecache_getlines = pdoctest.linecache.getlines
linecache.getlines = self.__patched_linecache_getlines
# Fail for deprecation warnings
with raise_on_deprecated():
try:
test.globs['print_function'] = print_function
return self.__run(test, compileflags, out)
finally:
sys.stdout = save_stdout
pdb.set_trace = save_set_trace
linecache.getlines = self.save_linecache_getlines
if clear_globs:
test.globs.clear()
# We have to override the name mangled methods.
monkeypatched_methods = [
'patched_linecache_getlines',
'run',
'record_outcome'
]
for method in monkeypatched_methods:
oldname = '_DocTestRunner__' + method
newname = '_SymPyDocTestRunner__' + method
setattr(SymPyDocTestRunner, newname, getattr(DocTestRunner, oldname))
class SymPyOutputChecker(pdoctest.OutputChecker):
"""
Compared to the OutputChecker from the stdlib our OutputChecker class
supports numerical comparison of floats occurring in the output of the
doctest examples
"""
def __init__(self):
# NOTE OutputChecker is an old-style class with no __init__ method,
# so we can't call the base class version of __init__ here
got_floats = r'(\d+\.\d*|\.\d+)'
# floats in the 'want' string may contain ellipses
want_floats = got_floats + r'(\.{3})?'
front_sep = r'\s|\+|\-|\*|,'
back_sep = front_sep + r'|j|e'
fbeg = r'^%s(?=%s|$)' % (got_floats, back_sep)
fmidend = r'(?<=%s)%s(?=%s|$)' % (front_sep, got_floats, back_sep)
self.num_got_rgx = re.compile(r'(%s|%s)' %(fbeg, fmidend))
fbeg = r'^%s(?=%s|$)' % (want_floats, back_sep)
fmidend = r'(?<=%s)%s(?=%s|$)' % (front_sep, want_floats, back_sep)
self.num_want_rgx = re.compile(r'(%s|%s)' %(fbeg, fmidend))
def check_output(self, want, got, optionflags):
"""
Return True iff the actual output from an example (`got`)
matches the expected output (`want`). These strings are
always considered to match if they are identical; but
depending on what option flags the test runner is using,
several non-exact match types are also possible. See the
documentation for `TestRunner` for more information about
option flags.
"""
# Handle the common case first, for efficiency:
# if they're string-identical, always return true.
if got == want:
return True
# TODO parse integers as well ?
# Parse floats and compare them. If some of the parsed floats contain
# ellipses, skip the comparison.
matches = self.num_got_rgx.finditer(got)
numbers_got = [match.group(1) for match in matches] # list of strs
matches = self.num_want_rgx.finditer(want)
numbers_want = [match.group(1) for match in matches] # list of strs
if len(numbers_got) != len(numbers_want):
return False
if len(numbers_got) > 0:
nw_ = []
for ng, nw in zip(numbers_got, numbers_want):
if '...' in nw:
nw_.append(ng)
continue
else:
nw_.append(nw)
if abs(float(ng)-float(nw)) > 1e-5:
return False
got = self.num_got_rgx.sub(r'%s', got)
got = got % tuple(nw_)
# <BLANKLINE> can be used as a special sequence to signify a
# blank line, unless the DONT_ACCEPT_BLANKLINE flag is used.
if not (optionflags & pdoctest.DONT_ACCEPT_BLANKLINE):
# Replace <BLANKLINE> in want with a blank line.
want = re.sub(r'(?m)^%s\s*?$' % re.escape(pdoctest.BLANKLINE_MARKER),
'', want)
# If a line in got contains only spaces, then remove the
# spaces.
got = re.sub(r'(?m)^\s*?$', '', got)
if got == want:
return True
# This flag causes doctest to ignore any differences in the
# contents of whitespace strings. Note that this can be used
# in conjunction with the ELLIPSIS flag.
if optionflags & pdoctest.NORMALIZE_WHITESPACE:
got = ' '.join(got.split())
want = ' '.join(want.split())
if got == want:
return True
# The ELLIPSIS flag says to let the sequence "..." in `want`
# match any substring in `got`.
if optionflags & pdoctest.ELLIPSIS:
if pdoctest._ellipsis_match(want, got):
return True
# We didn't find any match; return false.
return False
class Reporter:
"""
Parent class for all reporters.
"""
pass
class PyTestReporter(Reporter):
"""
Py.test like reporter. Should produce output identical to py.test.
"""
def __init__(self, verbose=False, tb="short", colors=True,
force_colors=False, split=None):
self._verbose = verbose
self._tb_style = tb
self._colors = colors
self._force_colors = force_colors
self._xfailed = 0
self._xpassed = []
self._failed = []
self._failed_doctest = []
self._passed = 0
self._skipped = 0
self._exceptions = []
self._terminal_width = None
self._default_width = 80
self._split = split
self._active_file = ''
self._active_f = None
# TODO: Should these be protected?
self.slow_test_functions = []
self.fast_test_functions = []
# this tracks the x-position of the cursor (useful for positioning
# things on the screen), without the need for any readline library:
self._write_pos = 0
self._line_wrap = False
def root_dir(self, dir):
self._root_dir = dir
@property
def terminal_width(self):
if self._terminal_width is not None:
return self._terminal_width
def findout_terminal_width():
if sys.platform == "win32":
# Windows support is based on:
#
# http://code.activestate.com/recipes/
# 440694-determine-size-of-console-window-on-windows/
from ctypes import windll, create_string_buffer
h = windll.kernel32.GetStdHandle(-12)
csbi = create_string_buffer(22)
res = windll.kernel32.GetConsoleScreenBufferInfo(h, csbi)
if res:
import struct
(_, _, _, _, _, left, _, right, _, _, _) = \
struct.unpack("hhhhHhhhhhh", csbi.raw)
return right - left
else:
return self._default_width
if hasattr(sys.stdout, 'isatty') and not sys.stdout.isatty():
return self._default_width # leave PIPEs alone
try:
process = subprocess.Popen(['stty', '-a'],
stdout=subprocess.PIPE,
stderr=subprocess.PIPE)
stdout = process.stdout.read()
if PY3:
stdout = stdout.decode("utf-8")
except OSError:
pass
else:
# We support the following output formats from stty:
#
# 1) Linux -> columns 80
# 2) OS X -> 80 columns
# 3) Solaris -> columns = 80
re_linux = r"columns\s+(?P<columns>\d+);"
re_osx = r"(?P<columns>\d+)\s*columns;"
re_solaris = r"columns\s+=\s+(?P<columns>\d+);"
for regex in (re_linux, re_osx, re_solaris):
match = re.search(regex, stdout)
if match is not None:
columns = match.group('columns')
try:
width = int(columns)
except ValueError:
pass
if width != 0:
return width
return self._default_width
width = findout_terminal_width()
self._terminal_width = width
return width
def write(self, text, color="", align="left", width=None,
force_colors=False):
"""
Prints a text on the screen.
It uses sys.stdout.write(), so no readline library is necessary.
Parameters
==========
color : choose from the colors below, "" means default color
align : "left"/"right", "left" is a normal print, "right" is aligned on
the right-hand side of the screen, filled with spaces if
necessary
width : the screen width
"""
color_templates = (
("Black", "0;30"),
("Red", "0;31"),
("Green", "0;32"),
("Brown", "0;33"),
("Blue", "0;34"),
("Purple", "0;35"),
("Cyan", "0;36"),
("LightGray", "0;37"),
("DarkGray", "1;30"),
("LightRed", "1;31"),
("LightGreen", "1;32"),
("Yellow", "1;33"),
("LightBlue", "1;34"),
("LightPurple", "1;35"),
("LightCyan", "1;36"),
("White", "1;37"),
)
colors = {}
for name, value in color_templates:
colors[name] = value
c_normal = '\033[0m'
c_color = '\033[%sm'
if width is None:
width = self.terminal_width
if align == "right":
if self._write_pos + len(text) > width:
# we don't fit on the current line, create a new line
self.write("\n")
self.write(" "*(width - self._write_pos - len(text)))
if not self._force_colors and hasattr(sys.stdout, 'isatty') and not \
sys.stdout.isatty():
# the stdout is not a terminal, this for example happens if the
# output is piped to less, e.g. "bin/test | less". In this case,
# the terminal control sequences would be printed verbatim, so
# don't use any colors.
color = ""
elif sys.platform == "win32":
# Windows consoles don't support ANSI escape sequences
color = ""
elif not self._colors:
color = ""
if self._line_wrap:
if text[0] != "\n":
sys.stdout.write("\n")
# Avoid UnicodeEncodeError when printing out test failures
if PY3 and IS_WINDOWS:
text = text.encode('raw_unicode_escape').decode('utf8', 'ignore')
elif PY3 and not sys.stdout.encoding.lower().startswith('utf'):
text = text.encode(sys.stdout.encoding, 'backslashreplace'
).decode(sys.stdout.encoding)
if color == "":
sys.stdout.write(text)
else:
sys.stdout.write("%s%s%s" %
(c_color % colors[color], text, c_normal))
sys.stdout.flush()
l = text.rfind("\n")
if l == -1:
self._write_pos += len(text)
else:
self._write_pos = len(text) - l - 1
self._line_wrap = self._write_pos >= width
self._write_pos %= width
def write_center(self, text, delim="="):
width = self.terminal_width
if text != "":
text = " %s " % text
idx = (width - len(text)) // 2
t = delim*idx + text + delim*(width - idx - len(text))
self.write(t + "\n")
def write_exception(self, e, val, tb):
# remove the first item, as that is always runtests.py
tb = tb.tb_next
t = traceback.format_exception(e, val, tb)
self.write("".join(t))
def start(self, seed=None, msg="test process starts"):
self.write_center(msg)
executable = sys.executable
v = tuple(sys.version_info)
python_version = "%s.%s.%s-%s-%s" % v
implementation = platform.python_implementation()
if implementation == 'PyPy':
implementation += " %s.%s.%s-%s-%s" % sys.pypy_version_info
self.write("executable: %s (%s) [%s]\n" %
(executable, python_version, implementation))
from sympy.utilities.misc import ARCH
self.write("architecture: %s\n" % ARCH)
from sympy.core.cache import USE_CACHE
self.write("cache: %s\n" % USE_CACHE)
from sympy.external.gmpy import GROUND_TYPES, HAS_GMPY
version = ''
if GROUND_TYPES =='gmpy':
if HAS_GMPY == 1:
import gmpy
elif HAS_GMPY == 2:
import gmpy2 as gmpy
version = gmpy.version()
self.write("ground types: %s %s\n" % (GROUND_TYPES, version))
numpy = import_module('numpy')
self.write("numpy: %s\n" % (None if not numpy else numpy.__version__))
if seed is not None:
self.write("random seed: %d\n" % seed)
from sympy.utilities.misc import HASH_RANDOMIZATION
self.write("hash randomization: ")
hash_seed = os.getenv("PYTHONHASHSEED") or '0'
if HASH_RANDOMIZATION and (hash_seed == "random" or int(hash_seed)):
self.write("on (PYTHONHASHSEED=%s)\n" % hash_seed)
else:
self.write("off\n")
if self._split:
self.write("split: %s\n" % self._split)
self.write('\n')
self._t_start = clock()
def finish(self):
self._t_end = clock()
self.write("\n")
global text, linelen
text = "tests finished: %d passed, " % self._passed
linelen = len(text)
def add_text(mytext):
global text, linelen
"""Break new text if too long."""
if linelen + len(mytext) > self.terminal_width:
text += '\n'
linelen = 0
text += mytext
linelen += len(mytext)
if len(self._failed) > 0:
add_text("%d failed, " % len(self._failed))
if len(self._failed_doctest) > 0:
add_text("%d failed, " % len(self._failed_doctest))
if self._skipped > 0:
add_text("%d skipped, " % self._skipped)
if self._xfailed > 0:
add_text("%d expected to fail, " % self._xfailed)
if len(self._xpassed) > 0:
add_text("%d expected to fail but passed, " % len(self._xpassed))
if len(self._exceptions) > 0:
add_text("%d exceptions, " % len(self._exceptions))
add_text("in %.2f seconds" % (self._t_end - self._t_start))
if self.slow_test_functions:
self.write_center('slowest tests', '_')
sorted_slow = sorted(self.slow_test_functions, key=lambda r: r[1])
for slow_func_name, taken in sorted_slow:
print('%s - Took %.3f seconds' % (slow_func_name, taken))
if self.fast_test_functions:
self.write_center('unexpectedly fast tests', '_')
sorted_fast = sorted(self.fast_test_functions,
key=lambda r: r[1])
for fast_func_name, taken in sorted_fast:
print('%s - Took %.3f seconds' % (fast_func_name, taken))
if len(self._xpassed) > 0:
self.write_center("xpassed tests", "_")
for e in self._xpassed:
self.write("%s: %s\n" % (e[0], e[1]))
self.write("\n")
if self._tb_style != "no" and len(self._exceptions) > 0:
for e in self._exceptions:
filename, f, (t, val, tb) = e
self.write_center("", "_")
if f is None:
s = "%s" % filename
else:
s = "%s:%s" % (filename, f.__name__)
self.write_center(s, "_")
self.write_exception(t, val, tb)
self.write("\n")
if self._tb_style != "no" and len(self._failed) > 0:
for e in self._failed:
filename, f, (t, val, tb) = e
self.write_center("", "_")
self.write_center("%s:%s" % (filename, f.__name__), "_")
self.write_exception(t, val, tb)
self.write("\n")
if self._tb_style != "no" and len(self._failed_doctest) > 0:
for e in self._failed_doctest:
filename, msg = e
self.write_center("", "_")
self.write_center("%s" % filename, "_")
self.write(msg)
self.write("\n")
self.write_center(text)
ok = len(self._failed) == 0 and len(self._exceptions) == 0 and \
len(self._failed_doctest) == 0
if not ok:
self.write("DO *NOT* COMMIT!\n")
return ok
def entering_filename(self, filename, n):
rel_name = filename[len(self._root_dir) + 1:]
self._active_file = rel_name
self._active_file_error = False
self.write(rel_name)
self.write("[%d] " % n)
def leaving_filename(self):
self.write(" ")
if self._active_file_error:
self.write("[FAIL]", "Red", align="right")
else:
self.write("[OK]", "Green", align="right")
self.write("\n")
if self._verbose:
self.write("\n")
def entering_test(self, f):
self._active_f = f
if self._verbose:
self.write("\n" + f.__name__ + " ")
def test_xfail(self):
self._xfailed += 1
self.write("f", "Green")
def test_xpass(self, v):
message = str(v)
self._xpassed.append((self._active_file, message))
self.write("X", "Green")
def test_fail(self, exc_info):
self._failed.append((self._active_file, self._active_f, exc_info))
self.write("F", "Red")
self._active_file_error = True
def doctest_fail(self, name, error_msg):
# the first line contains "******", remove it:
error_msg = "\n".join(error_msg.split("\n")[1:])
self._failed_doctest.append((name, error_msg))
self.write("F", "Red")
self._active_file_error = True
def test_pass(self, char="."):
self._passed += 1
if self._verbose:
self.write("ok", "Green")
else:
self.write(char, "Green")
def test_skip(self, v=None):
char = "s"
self._skipped += 1
if v is not None:
message = str(v)
if message == "KeyboardInterrupt":
char = "K"
elif message == "Timeout":
char = "T"
elif message == "Slow":
char = "w"
if self._verbose:
if v is not None:
self.write(message + ' ', "Blue")
else:
self.write(" - ", "Blue")
self.write(char, "Blue")
def test_exception(self, exc_info):
self._exceptions.append((self._active_file, self._active_f, exc_info))
if exc_info[0] is TimeOutError:
self.write("T", "Red")
else:
self.write("E", "Red")
self._active_file_error = True
def import_error(self, filename, exc_info):
self._exceptions.append((filename, None, exc_info))
rel_name = filename[len(self._root_dir) + 1:]
self.write(rel_name)
self.write("[?] Failed to import", "Red")
self.write(" ")
self.write("[FAIL]", "Red", align="right")
self.write("\n")
|
da8182e570b39c318aab8c8dad80be4d54c295acfec19a3290ebc50f33157ce6 | from sympy import S, simplify
from sympy.core import Basic, diff
from sympy.matrices import Matrix
from sympy.vector import (CoordSys3D, Vector, ParametricRegion,
parametric_region_list, ImplicitRegion)
from sympy.vector.operators import _get_coord_sys_from_expr
from sympy.integrals import Integral, integrate
from sympy.utilities.iterables import topological_sort, default_sort_key
from sympy.geometry.entity import GeometryEntity
class ParametricIntegral(Basic):
"""
Represents integral of a scalar or vector field
over a Parametric Region
Examples
========
>>> from sympy import cos, sin, pi
>>> from sympy.vector import CoordSys3D, ParametricRegion, ParametricIntegral
>>> from sympy.abc import r, t, theta, phi
>>> C = CoordSys3D('C')
>>> curve = ParametricRegion((3*t - 2, t + 1), (t, 1, 2))
>>> ParametricIntegral(C.x, curve)
5*sqrt(10)/2
>>> length = ParametricIntegral(1, curve)
>>> length
sqrt(10)
>>> semisphere = ParametricRegion((2*sin(phi)*cos(theta), 2*sin(phi)*sin(theta), 2*cos(phi)),\
(theta, 0, 2*pi), (phi, 0, pi/2))
>>> ParametricIntegral(C.z, semisphere)
8*pi
>>> ParametricIntegral(C.j + C.k, ParametricRegion((r*cos(theta), r*sin(theta)), r, theta))
0
"""
def __new__(cls, field, parametricregion):
coord_set = _get_coord_sys_from_expr(field)
if len(coord_set) == 0:
coord_sys = CoordSys3D('C')
elif len(coord_set) > 1:
raise ValueError
else:
coord_sys = next(iter(coord_set))
if parametricregion.dimensions == 0:
return S.Zero
base_vectors = coord_sys.base_vectors()
base_scalars = coord_sys.base_scalars()
parametricfield = field
r = Vector.zero
for i in range(len(parametricregion.definition)):
r += base_vectors[i]*parametricregion.definition[i]
if len(coord_set) != 0:
for i in range(len(parametricregion.definition)):
parametricfield = parametricfield.subs(base_scalars[i], parametricregion.definition[i])
if parametricregion.dimensions == 1:
parameter = parametricregion.parameters[0]
r_diff = diff(r, parameter)
lower, upper = parametricregion.limits[parameter][0], parametricregion.limits[parameter][1]
if isinstance(parametricfield, Vector):
integrand = simplify(r_diff.dot(parametricfield))
else:
integrand = simplify(r_diff.magnitude()*parametricfield)
result = integrate(integrand, (parameter, lower, upper))
elif parametricregion.dimensions == 2:
u, v = cls._bounds_case(parametricregion.parameters, parametricregion.limits)
r_u = diff(r, u)
r_v = diff(r, v)
normal_vector = simplify(r_u.cross(r_v))
if isinstance(parametricfield, Vector):
integrand = parametricfield.dot(normal_vector)
else:
integrand = parametricfield*normal_vector.magnitude()
integrand = simplify(integrand)
lower_u, upper_u = parametricregion.limits[u][0], parametricregion.limits[u][1]
lower_v, upper_v = parametricregion.limits[v][0], parametricregion.limits[v][1]
result = integrate(integrand, (u, lower_u, upper_u), (v, lower_v, upper_v))
else:
variables = cls._bounds_case(parametricregion.parameters, parametricregion.limits)
coeff = Matrix(parametricregion.definition).jacobian(variables).det()
integrand = simplify(parametricfield*coeff)
l = [(var, parametricregion.limits[var][0], parametricregion.limits[var][1]) for var in variables]
result = integrate(integrand, *l)
if not isinstance(result, Integral):
return result
else:
return super().__new__(cls, field, parametricregion)
@classmethod
def _bounds_case(cls, parameters, limits):
V = list(limits.keys())
E = list()
for p in V:
lower_p = limits[p][0]
upper_p = limits[p][1]
lower_p = lower_p.atoms()
upper_p = upper_p.atoms()
for q in V:
if p == q:
continue
if lower_p.issuperset({q}) or upper_p.issuperset({q}):
E.append((p, q))
if not E:
return parameters
else:
return topological_sort((V, E), key=default_sort_key)
@property
def field(self):
return self.args[0]
@property
def parametricregion(self):
return self.args[1]
def vector_integrate(field, *region):
"""
Compute the integral of a vector/scalar field
over a a region or a set of parameters.
Examples
========
>>> from sympy.vector import CoordSys3D, ParametricRegion, vector_integrate
>>> from sympy.abc import x, y, t
>>> C = CoordSys3D('C')
>>> region = ParametricRegion((t, t**2), (t, 1, 5))
>>> vector_integrate(C.x*C.i, region)
12
Integrals over some objects of geometry module can also be calculated.
>>> from sympy.geometry import Point, Circle, Triangle
>>> c = Circle(Point(0, 2), 5)
>>> vector_integrate(C.x**2 + C.y**2, c)
290*pi
>>> triangle = Triangle(Point(-2, 3), Point(2, 3), Point(0, 5))
>>> vector_integrate(3*C.x**2*C.y*C.i + C.j, triangle)
-8
Integrals over some simple implicit regions can be computed. But in most cases,
it takes too long to compute over them. This is due to the expressions of parametric
representation becoming large.
>>> from sympy.vector import ImplicitRegion
>>> c2 = ImplicitRegion((x, y), (x - 2)**2 + (y - 1)**2 - 9)
>>> vector_integrate(1, c2)
6*pi
Integral of fields with respect to base scalars:
>>> vector_integrate(12*C.y**3, (C.y, 1, 3))
240
>>> vector_integrate(C.x**2*C.z, C.x)
C.x**3*C.z/3
>>> vector_integrate(C.x*C.i - C.y*C.k, C.x)
(Integral(C.x, C.x))*C.i + (Integral(-C.y, C.x))*C.k
>>> _.doit()
C.x**2/2*C.i + (-C.x*C.y)*C.k
"""
if len(region) == 1:
if isinstance(region[0], ParametricRegion):
return ParametricIntegral(field, region[0])
if isinstance(region[0], ImplicitRegion):
region = parametric_region_list(region[0])[0]
return vector_integrate(field, region)
if isinstance(region[0], GeometryEntity):
regions_list = parametric_region_list(region[0])
result = 0
for reg in regions_list:
result += vector_integrate(field, reg)
return result
return integrate(field, *region)
|
60e769daf63f5d942a2bd503ed07df3c556dea3bbede71f36949de07effa6026 | from functools import singledispatch
from sympy import pi, tan
from sympy.simplify import trigsimp
from sympy.core import Basic, Tuple
from sympy.core.symbol import _symbol
from sympy.solvers import solve
from sympy.geometry import Point, Segment, Curve, Ellipse, Polygon
from sympy.vector import ImplicitRegion
class ParametricRegion(Basic):
"""
Represents a parametric region in space.
Examples
========
>>> from sympy import cos, sin, pi
>>> from sympy.abc import r, theta, t, a, b, x, y
>>> from sympy.vector import ParametricRegion
>>> ParametricRegion((t, t**2), (t, -1, 2))
ParametricRegion((t, t**2), (t, -1, 2))
>>> ParametricRegion((x, y), (x, 3, 4), (y, 5, 6))
ParametricRegion((x, y), (x, 3, 4), (y, 5, 6))
>>> ParametricRegion((r*cos(theta), r*sin(theta)), (r, -2, 2), (theta, 0, pi))
ParametricRegion((r*cos(theta), r*sin(theta)), (r, -2, 2), (theta, 0, pi))
>>> ParametricRegion((a*cos(t), b*sin(t)), t)
ParametricRegion((a*cos(t), b*sin(t)), t)
>>> circle = ParametricRegion((r*cos(theta), r*sin(theta)), r, (theta, 0, pi))
>>> circle.parameters
(r, theta)
>>> circle.definition
(r*cos(theta), r*sin(theta))
>>> circle.limits
{theta: (0, pi)}
Dimension of a parametric region determines whether a region is a curve, surface
or volume region. It does not represent its dimensions in space.
>>> circle.dimensions
1
Parameters
==========
definition : tuple to define base scalars in terms of parameters.
bounds : Parameter or a tuple of length 3 to define parameter and corresponding lower and upper bound.
"""
def __new__(cls, definition, *bounds):
parameters = ()
limits = {}
if not isinstance(bounds, Tuple):
bounds = Tuple(*bounds)
for bound in bounds:
if isinstance(bound, tuple) or isinstance(bound, Tuple):
if len(bound) != 3:
raise ValueError("Tuple should be in the form (parameter, lowerbound, upperbound)")
parameters += (bound[0],)
limits[bound[0]] = (bound[1], bound[2])
else:
parameters += (bound,)
if not (isinstance(definition, tuple) or isinstance(definition, Tuple)):
definition = (definition,)
obj = super().__new__(cls, Tuple(*definition), *bounds)
obj._parameters = parameters
obj._limits = limits
return obj
@property
def definition(self):
return self.args[0]
@property
def limits(self):
return self._limits
@property
def parameters(self):
return self._parameters
@property
def dimensions(self):
return len(self.limits)
@singledispatch
def parametric_region_list(reg):
"""
Returns a list of ParametricRegion objects representing the geometric region.
Examples
========
>>> from sympy.abc import t
>>> from sympy.vector import parametric_region_list
>>> from sympy.geometry import Point, Curve, Ellipse, Segment, Polygon
>>> p = Point(2, 5)
>>> parametric_region_list(p)
[ParametricRegion((2, 5))]
>>> c = Curve((t**3, 4*t), (t, -3, 4))
>>> parametric_region_list(c)
[ParametricRegion((t**3, 4*t), (t, -3, 4))]
>>> e = Ellipse(Point(1, 3), 2, 3)
>>> parametric_region_list(e)
[ParametricRegion((2*cos(t) + 1, 3*sin(t) + 3), (t, 0, 2*pi))]
>>> s = Segment(Point(1, 3), Point(2, 6))
>>> parametric_region_list(s)
[ParametricRegion((t + 1, 3*t + 3), (t, 0, 1))]
>>> p1, p2, p3, p4 = [(0, 1), (2, -3), (5, 3), (-2, 3)]
>>> poly = Polygon(p1, p2, p3, p4)
>>> parametric_region_list(poly)
[ParametricRegion((2*t, 1 - 4*t), (t, 0, 1)), ParametricRegion((3*t + 2, 6*t - 3), (t, 0, 1)),\
ParametricRegion((5 - 7*t, 3), (t, 0, 1)), ParametricRegion((2*t - 2, 3 - 2*t), (t, 0, 1))]
"""
raise ValueError("SymPy cannot determine parametric representation of the region.")
@parametric_region_list.register(Point)
def _(obj):
return [ParametricRegion(obj.args)]
@parametric_region_list.register(Curve) # type: ignore
def _(obj):
definition = obj.arbitrary_point(obj.parameter).args
bounds = obj.limits
return [ParametricRegion(definition, bounds)]
@parametric_region_list.register(Ellipse) # type: ignore
def _(obj, parameter='t'):
definition = obj.arbitrary_point(parameter).args
t = _symbol(parameter, real=True)
bounds = (t, 0, 2*pi)
return [ParametricRegion(definition, bounds)]
@parametric_region_list.register(Segment) # type: ignore
def _(obj, parameter='t'):
t = _symbol(parameter, real=True)
definition = obj.arbitrary_point(t).args
for i in range(0, 3):
lower_bound = solve(definition[i] - obj.points[0].args[i], t)
upper_bound = solve(definition[i] - obj.points[1].args[i], t)
if len(lower_bound) == 1 and len(upper_bound) == 1:
bounds = t, lower_bound[0], upper_bound[0]
break
definition_tuple = obj.arbitrary_point(parameter).args
return [ParametricRegion(definition_tuple, bounds)]
@parametric_region_list.register(Polygon) # type: ignore
def _(obj, parameter='t'):
l = [parametric_region_list(side, parameter)[0] for side in obj.sides]
return l
@parametric_region_list.register(ImplicitRegion) # type: ignore
def _(obj, parameters=('t', 's')):
definition = obj.rational_parametrization(parameters)
bounds = []
for i in range(len(obj.variables) - 1):
# Each parameter is replaced by its tangent to simplify intergation
parameter = _symbol(parameters[i], real=True)
definition = [trigsimp(elem.subs(parameter, tan(parameter/2))) for elem in definition]
bounds.append((parameter, 0, 2*pi),)
definition = Tuple(*definition)
return [ParametricRegion(definition, *bounds)]
|
17a4799090ef5c1f351486b2687b020d697b021fd97c5001ea0778048b026ecb | """Elliptical geometrical entities.
Contains
* Ellipse
* Circle
"""
from sympy import Expr, Eq
from sympy.core import S, pi, sympify
from sympy.core.parameters import global_parameters
from sympy.core.logic import fuzzy_bool
from sympy.core.numbers import Rational, oo
from sympy.core.compatibility import ordered
from sympy.core.symbol import Dummy, uniquely_named_symbol, _symbol
from sympy.simplify import simplify, trigsimp
from sympy.functions.elementary.miscellaneous import sqrt, Max
from sympy.functions.elementary.trigonometric import cos, sin
from sympy.functions.special.elliptic_integrals import elliptic_e
from sympy.geometry.exceptions import GeometryError
from sympy.geometry.line import Ray2D, Segment2D, Line2D, LinearEntity3D
from sympy.polys import DomainError, Poly, PolynomialError
from sympy.polys.polyutils import _not_a_coeff, _nsort
from sympy.solvers import solve
from sympy.solvers.solveset import linear_coeffs
from sympy.utilities.misc import filldedent, func_name
from .entity import GeometryEntity, GeometrySet
from .point import Point, Point2D, Point3D
from .line import Line, Segment
from .util import idiff
import random
class Ellipse(GeometrySet):
"""An elliptical GeometryEntity.
Parameters
==========
center : Point, optional
Default value is Point(0, 0)
hradius : number or SymPy expression, optional
vradius : number or SymPy expression, optional
eccentricity : number or SymPy expression, optional
Two of `hradius`, `vradius` and `eccentricity` must be supplied to
create an Ellipse. The third is derived from the two supplied.
Attributes
==========
center
hradius
vradius
area
circumference
eccentricity
periapsis
apoapsis
focus_distance
foci
Raises
======
GeometryError
When `hradius`, `vradius` and `eccentricity` are incorrectly supplied
as parameters.
TypeError
When `center` is not a Point.
See Also
========
Circle
Notes
-----
Constructed from a center and two radii, the first being the horizontal
radius (along the x-axis) and the second being the vertical radius (along
the y-axis).
When symbolic value for hradius and vradius are used, any calculation that
refers to the foci or the major or minor axis will assume that the ellipse
has its major radius on the x-axis. If this is not true then a manual
rotation is necessary.
Examples
========
>>> from sympy import Ellipse, Point, Rational
>>> e1 = Ellipse(Point(0, 0), 5, 1)
>>> e1.hradius, e1.vradius
(5, 1)
>>> e2 = Ellipse(Point(3, 1), hradius=3, eccentricity=Rational(4, 5))
>>> e2
Ellipse(Point2D(3, 1), 3, 9/5)
"""
def __contains__(self, o):
if isinstance(o, Point):
x = Dummy('x', real=True)
y = Dummy('y', real=True)
res = self.equation(x, y).subs({x: o.x, y: o.y})
return trigsimp(simplify(res)) is S.Zero
elif isinstance(o, Ellipse):
return self == o
return False
def __eq__(self, o):
"""Is the other GeometryEntity the same as this ellipse?"""
return isinstance(o, Ellipse) and (self.center == o.center and
self.hradius == o.hradius and
self.vradius == o.vradius)
def __hash__(self):
return super().__hash__()
def __new__(
cls, center=None, hradius=None, vradius=None, eccentricity=None, **kwargs):
hradius = sympify(hradius)
vradius = sympify(vradius)
eccentricity = sympify(eccentricity)
if center is None:
center = Point(0, 0)
else:
center = Point(center, dim=2)
if len(center) != 2:
raise ValueError('The center of "{}" must be a two dimensional point'.format(cls))
if len(list(filter(lambda x: x is not None, (hradius, vradius, eccentricity)))) != 2:
raise ValueError(filldedent('''
Exactly two arguments of "hradius", "vradius", and
"eccentricity" must not be None.'''))
if eccentricity is not None:
if eccentricity.is_negative:
raise GeometryError("Eccentricity of ellipse/circle should lie between [0, 1)")
elif hradius is None:
hradius = vradius / sqrt(1 - eccentricity**2)
elif vradius is None:
vradius = hradius * sqrt(1 - eccentricity**2)
if hradius == vradius:
return Circle(center, hradius, **kwargs)
if hradius == 0 or vradius == 0:
return Segment(Point(center[0] - hradius, center[1] - vradius), Point(center[0] + hradius, center[1] + vradius))
if hradius.is_real is False or vradius.is_real is False:
raise GeometryError("Invalid value encountered when computing hradius / vradius.")
return GeometryEntity.__new__(cls, center, hradius, vradius, **kwargs)
def _svg(self, scale_factor=1., fill_color="#66cc99"):
"""Returns SVG ellipse element for the Ellipse.
Parameters
==========
scale_factor : float
Multiplication factor for the SVG stroke-width. Default is 1.
fill_color : str, optional
Hex string for fill color. Default is "#66cc99".
"""
from sympy.core.evalf import N
c = N(self.center)
h, v = N(self.hradius), N(self.vradius)
return (
'<ellipse fill="{1}" stroke="#555555" '
'stroke-width="{0}" opacity="0.6" cx="{2}" cy="{3}" rx="{4}" ry="{5}"/>'
).format(2. * scale_factor, fill_color, c.x, c.y, h, v)
@property
def ambient_dimension(self):
return 2
@property
def apoapsis(self):
"""The apoapsis of the ellipse.
The greatest distance between the focus and the contour.
Returns
=======
apoapsis : number
See Also
========
periapsis : Returns shortest distance between foci and contour
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.apoapsis
2*sqrt(2) + 3
"""
return self.major * (1 + self.eccentricity)
def arbitrary_point(self, parameter='t'):
"""A parameterized point on the ellipse.
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
arbitrary_point : Point
Raises
======
ValueError
When `parameter` already appears in the functions.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.arbitrary_point()
Point2D(3*cos(t), 2*sin(t))
"""
t = _symbol(parameter, real=True)
if t.name in (f.name for f in self.free_symbols):
raise ValueError(filldedent('Symbol %s already appears in object '
'and cannot be used as a parameter.' % t.name))
return Point(self.center.x + self.hradius*cos(t),
self.center.y + self.vradius*sin(t))
@property
def area(self):
"""The area of the ellipse.
Returns
=======
area : number
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.area
3*pi
"""
return simplify(S.Pi * self.hradius * self.vradius)
@property
def bounds(self):
"""Return a tuple (xmin, ymin, xmax, ymax) representing the bounding
rectangle for the geometric figure.
"""
h, v = self.hradius, self.vradius
return (self.center.x - h, self.center.y - v, self.center.x + h, self.center.y + v)
@property
def center(self):
"""The center of the ellipse.
Returns
=======
center : number
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.center
Point2D(0, 0)
"""
return self.args[0]
@property
def circumference(self):
"""The circumference of the ellipse.
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.circumference
12*elliptic_e(8/9)
"""
if self.eccentricity == 1:
# degenerate
return 4*self.major
elif self.eccentricity == 0:
# circle
return 2*pi*self.hradius
else:
return 4*self.major*elliptic_e(self.eccentricity**2)
@property
def eccentricity(self):
"""The eccentricity of the ellipse.
Returns
=======
eccentricity : number
Examples
========
>>> from sympy import Point, Ellipse, sqrt
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, sqrt(2))
>>> e1.eccentricity
sqrt(7)/3
"""
return self.focus_distance / self.major
def encloses_point(self, p):
"""
Return True if p is enclosed by (is inside of) self.
Notes
-----
Being on the border of self is considered False.
Parameters
==========
p : Point
Returns
=======
encloses_point : True, False or None
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Ellipse, S
>>> from sympy.abc import t
>>> e = Ellipse((0, 0), 3, 2)
>>> e.encloses_point((0, 0))
True
>>> e.encloses_point(e.arbitrary_point(t).subs(t, S.Half))
False
>>> e.encloses_point((4, 0))
False
"""
p = Point(p, dim=2)
if p in self:
return False
if len(self.foci) == 2:
# if the combined distance from the foci to p (h1 + h2) is less
# than the combined distance from the foci to the minor axis
# (which is the same as the major axis length) then p is inside
# the ellipse
h1, h2 = [f.distance(p) for f in self.foci]
test = 2*self.major - (h1 + h2)
else:
test = self.radius - self.center.distance(p)
return fuzzy_bool(test.is_positive)
def equation(self, x='x', y='y', _slope=None):
"""
Returns the equation of an ellipse aligned with the x and y axes;
when slope is given, the equation returned corresponds to an ellipse
with a major axis having that slope.
Parameters
==========
x : str, optional
Label for the x-axis. Default value is 'x'.
y : str, optional
Label for the y-axis. Default value is 'y'.
_slope : Expr, optional
The slope of the major axis. Ignored when 'None'.
Returns
=======
equation : sympy expression
See Also
========
arbitrary_point : Returns parameterized point on ellipse
Examples
========
>>> from sympy import Point, Ellipse, pi
>>> from sympy.abc import x, y
>>> e1 = Ellipse(Point(1, 0), 3, 2)
>>> eq1 = e1.equation(x, y); eq1
y**2/4 + (x/3 - 1/3)**2 - 1
>>> eq2 = e1.equation(x, y, _slope=1); eq2
(-x + y + 1)**2/8 + (x + y - 1)**2/18 - 1
A point on e1 satisfies eq1. Let's use one on the x-axis:
>>> p1 = e1.center + Point(e1.major, 0)
>>> assert eq1.subs(x, p1.x).subs(y, p1.y) == 0
When rotated the same as the rotated ellipse, about the center
point of the ellipse, it will satisfy the rotated ellipse's
equation, too:
>>> r1 = p1.rotate(pi/4, e1.center)
>>> assert eq2.subs(x, r1.x).subs(y, r1.y) == 0
References
==========
.. [1] https://math.stackexchange.com/questions/108270/what-is-the-equation-of-an-ellipse-that-is-not-aligned-with-the-axis
.. [2] https://en.wikipedia.org/wiki/Ellipse#Equation_of_a_shifted_ellipse
"""
x = _symbol(x, real=True)
y = _symbol(y, real=True)
dx = x - self.center.x
dy = y - self.center.y
if _slope is not None:
L = (dy - _slope*dx)**2
l = (_slope*dy + dx)**2
h = 1 + _slope**2
b = h*self.major**2
a = h*self.minor**2
return l/b + L/a - 1
else:
t1 = (dx/self.hradius)**2
t2 = (dy/self.vradius)**2
return t1 + t2 - 1
def evolute(self, x='x', y='y'):
"""The equation of evolute of the ellipse.
Parameters
==========
x : str, optional
Label for the x-axis. Default value is 'x'.
y : str, optional
Label for the y-axis. Default value is 'y'.
Returns
=======
equation : sympy expression
Examples
========
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(1, 0), 3, 2)
>>> e1.evolute()
2**(2/3)*y**(2/3) + (3*x - 3)**(2/3) - 5**(2/3)
"""
if len(self.args) != 3:
raise NotImplementedError('Evolute of arbitrary Ellipse is not supported.')
x = _symbol(x, real=True)
y = _symbol(y, real=True)
t1 = (self.hradius*(x - self.center.x))**Rational(2, 3)
t2 = (self.vradius*(y - self.center.y))**Rational(2, 3)
return t1 + t2 - (self.hradius**2 - self.vradius**2)**Rational(2, 3)
@property
def foci(self):
"""The foci of the ellipse.
Notes
-----
The foci can only be calculated if the major/minor axes are known.
Raises
======
ValueError
When the major and minor axis cannot be determined.
See Also
========
sympy.geometry.point.Point
focus_distance : Returns the distance between focus and center
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.foci
(Point2D(-2*sqrt(2), 0), Point2D(2*sqrt(2), 0))
"""
c = self.center
hr, vr = self.hradius, self.vradius
if hr == vr:
return (c, c)
# calculate focus distance manually, since focus_distance calls this
# routine
fd = sqrt(self.major**2 - self.minor**2)
if hr == self.minor:
# foci on the y-axis
return (c + Point(0, -fd), c + Point(0, fd))
elif hr == self.major:
# foci on the x-axis
return (c + Point(-fd, 0), c + Point(fd, 0))
@property
def focus_distance(self):
"""The focal distance of the ellipse.
The distance between the center and one focus.
Returns
=======
focus_distance : number
See Also
========
foci
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.focus_distance
2*sqrt(2)
"""
return Point.distance(self.center, self.foci[0])
@property
def hradius(self):
"""The horizontal radius of the ellipse.
Returns
=======
hradius : number
See Also
========
vradius, major, minor
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.hradius
3
"""
return self.args[1]
def intersection(self, o):
"""The intersection of this ellipse and another geometrical entity
`o`.
Parameters
==========
o : GeometryEntity
Returns
=======
intersection : list of GeometryEntity objects
Notes
-----
Currently supports intersections with Point, Line, Segment, Ray,
Circle and Ellipse types.
See Also
========
sympy.geometry.entity.GeometryEntity
Examples
========
>>> from sympy import Ellipse, Point, Line
>>> e = Ellipse(Point(0, 0), 5, 7)
>>> e.intersection(Point(0, 0))
[]
>>> e.intersection(Point(5, 0))
[Point2D(5, 0)]
>>> e.intersection(Line(Point(0,0), Point(0, 1)))
[Point2D(0, -7), Point2D(0, 7)]
>>> e.intersection(Line(Point(5,0), Point(5, 1)))
[Point2D(5, 0)]
>>> e.intersection(Line(Point(6,0), Point(6, 1)))
[]
>>> e = Ellipse(Point(-1, 0), 4, 3)
>>> e.intersection(Ellipse(Point(1, 0), 4, 3))
[Point2D(0, -3*sqrt(15)/4), Point2D(0, 3*sqrt(15)/4)]
>>> e.intersection(Ellipse(Point(5, 0), 4, 3))
[Point2D(2, -3*sqrt(7)/4), Point2D(2, 3*sqrt(7)/4)]
>>> e.intersection(Ellipse(Point(100500, 0), 4, 3))
[]
>>> e.intersection(Ellipse(Point(0, 0), 3, 4))
[Point2D(3, 0), Point2D(-363/175, -48*sqrt(111)/175), Point2D(-363/175, 48*sqrt(111)/175)]
>>> e.intersection(Ellipse(Point(-1, 0), 3, 4))
[Point2D(-17/5, -12/5), Point2D(-17/5, 12/5), Point2D(7/5, -12/5), Point2D(7/5, 12/5)]
"""
# TODO: Replace solve with nonlinsolve, when nonlinsolve will be able to solve in real domain
x = Dummy('x', real=True)
y = Dummy('y', real=True)
if isinstance(o, Point):
if o in self:
return [o]
else:
return []
elif isinstance(o, (Segment2D, Ray2D)):
ellipse_equation = self.equation(x, y)
result = solve([ellipse_equation, Line(o.points[0], o.points[1]).equation(x, y)], [x, y])
return list(ordered([Point(i) for i in result if i in o]))
elif isinstance(o, Polygon):
return o.intersection(self)
elif isinstance(o, (Ellipse, Line2D)):
if o == self:
return self
else:
ellipse_equation = self.equation(x, y)
return list(ordered([Point(i) for i in solve([ellipse_equation, o.equation(x, y)], [x, y])]))
elif isinstance(o, LinearEntity3D):
raise TypeError('Entity must be two dimensional, not three dimensional')
else:
raise TypeError('Intersection not handled for %s' % func_name(o))
def is_tangent(self, o):
"""Is `o` tangent to the ellipse?
Parameters
==========
o : GeometryEntity
An Ellipse, LinearEntity or Polygon
Raises
======
NotImplementedError
When the wrong type of argument is supplied.
Returns
=======
is_tangent: boolean
True if o is tangent to the ellipse, False otherwise.
See Also
========
tangent_lines
Examples
========
>>> from sympy import Point, Ellipse, Line
>>> p0, p1, p2 = Point(0, 0), Point(3, 0), Point(3, 3)
>>> e1 = Ellipse(p0, 3, 2)
>>> l1 = Line(p1, p2)
>>> e1.is_tangent(l1)
True
"""
if isinstance(o, Point2D):
return False
elif isinstance(o, Ellipse):
intersect = self.intersection(o)
if isinstance(intersect, Ellipse):
return True
elif intersect:
return all((self.tangent_lines(i)[0]).equals(o.tangent_lines(i)[0]) for i in intersect)
else:
return False
elif isinstance(o, Line2D):
hit = self.intersection(o)
if not hit:
return False
if len(hit) == 1:
return True
# might return None if it can't decide
return hit[0].equals(hit[1])
elif isinstance(o, Ray2D):
intersect = self.intersection(o)
if len(intersect) == 1:
return intersect[0] != o.source and not self.encloses_point(o.source)
else:
return False
elif isinstance(o, (Segment2D, Polygon)):
all_tangents = False
segments = o.sides if isinstance(o, Polygon) else [o]
for segment in segments:
intersect = self.intersection(segment)
if len(intersect) == 1:
if not any(intersect[0] in i for i in segment.points) \
and all(not self.encloses_point(i) for i in segment.points):
all_tangents = True
continue
else:
return False
else:
return all_tangents
return all_tangents
elif isinstance(o, (LinearEntity3D, Point3D)):
raise TypeError('Entity must be two dimensional, not three dimensional')
else:
raise TypeError('Is_tangent not handled for %s' % func_name(o))
@property
def major(self):
"""Longer axis of the ellipse (if it can be determined) else hradius.
Returns
=======
major : number or expression
See Also
========
hradius, vradius, minor
Examples
========
>>> from sympy import Point, Ellipse, Symbol
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.major
3
>>> a = Symbol('a')
>>> b = Symbol('b')
>>> Ellipse(p1, a, b).major
a
>>> Ellipse(p1, b, a).major
b
>>> m = Symbol('m')
>>> M = m + 1
>>> Ellipse(p1, m, M).major
m + 1
"""
ab = self.args[1:3]
if len(ab) == 1:
return ab[0]
a, b = ab
o = b - a < 0
if o == True:
return a
elif o == False:
return b
return self.hradius
@property
def minor(self):
"""Shorter axis of the ellipse (if it can be determined) else vradius.
Returns
=======
minor : number or expression
See Also
========
hradius, vradius, major
Examples
========
>>> from sympy import Point, Ellipse, Symbol
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.minor
1
>>> a = Symbol('a')
>>> b = Symbol('b')
>>> Ellipse(p1, a, b).minor
b
>>> Ellipse(p1, b, a).minor
a
>>> m = Symbol('m')
>>> M = m + 1
>>> Ellipse(p1, m, M).minor
m
"""
ab = self.args[1:3]
if len(ab) == 1:
return ab[0]
a, b = ab
o = a - b < 0
if o == True:
return a
elif o == False:
return b
return self.vradius
def normal_lines(self, p, prec=None):
"""Normal lines between `p` and the ellipse.
Parameters
==========
p : Point
Returns
=======
normal_lines : list with 1, 2 or 4 Lines
Examples
========
>>> from sympy import Point, Ellipse
>>> e = Ellipse((0, 0), 2, 3)
>>> c = e.center
>>> e.normal_lines(c + Point(1, 0))
[Line2D(Point2D(0, 0), Point2D(1, 0))]
>>> e.normal_lines(c)
[Line2D(Point2D(0, 0), Point2D(0, 1)), Line2D(Point2D(0, 0), Point2D(1, 0))]
Off-axis points require the solution of a quartic equation. This
often leads to very large expressions that may be of little practical
use. An approximate solution of `prec` digits can be obtained by
passing in the desired value:
>>> e.normal_lines((3, 3), prec=2)
[Line2D(Point2D(-0.81, -2.7), Point2D(0.19, -1.2)),
Line2D(Point2D(1.5, -2.0), Point2D(2.5, -2.7))]
Whereas the above solution has an operation count of 12, the exact
solution has an operation count of 2020.
"""
p = Point(p, dim=2)
# XXX change True to something like self.angle == 0 if the arbitrarily
# rotated ellipse is introduced.
# https://github.com/sympy/sympy/issues/2815)
if True:
rv = []
if p.x == self.center.x:
rv.append(Line(self.center, slope=oo))
if p.y == self.center.y:
rv.append(Line(self.center, slope=0))
if rv:
# at these special orientations of p either 1 or 2 normals
# exist and we are done
return rv
# find the 4 normal points and construct lines through them with
# the corresponding slope
x, y = Dummy('x', real=True), Dummy('y', real=True)
eq = self.equation(x, y)
dydx = idiff(eq, y, x)
norm = -1/dydx
slope = Line(p, (x, y)).slope
seq = slope - norm
# TODO: Replace solve with solveset, when this line is tested
yis = solve(seq, y)[0]
xeq = eq.subs(y, yis).as_numer_denom()[0].expand()
if len(xeq.free_symbols) == 1:
try:
# this is so much faster, it's worth a try
xsol = Poly(xeq, x).real_roots()
except (DomainError, PolynomialError, NotImplementedError):
# TODO: Replace solve with solveset, when these lines are tested
xsol = _nsort(solve(xeq, x), separated=True)[0]
points = [Point(i, solve(eq.subs(x, i), y)[0]) for i in xsol]
else:
raise NotImplementedError(
'intersections for the general ellipse are not supported')
slopes = [norm.subs(zip((x, y), pt.args)) for pt in points]
if prec is not None:
points = [pt.n(prec) for pt in points]
slopes = [i if _not_a_coeff(i) else i.n(prec) for i in slopes]
return [Line(pt, slope=s) for pt, s in zip(points, slopes)]
@property
def periapsis(self):
"""The periapsis of the ellipse.
The shortest distance between the focus and the contour.
Returns
=======
periapsis : number
See Also
========
apoapsis : Returns greatest distance between focus and contour
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.periapsis
3 - 2*sqrt(2)
"""
return self.major * (1 - self.eccentricity)
@property
def semilatus_rectum(self):
"""
Calculates the semi-latus rectum of the Ellipse.
Semi-latus rectum is defined as one half of the the chord through a
focus parallel to the conic section directrix of a conic section.
Returns
=======
semilatus_rectum : number
See Also
========
apoapsis : Returns greatest distance between focus and contour
periapsis : The shortest distance between the focus and the contour
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.semilatus_rectum
1/3
References
==========
[1] http://mathworld.wolfram.com/SemilatusRectum.html
[2] https://en.wikipedia.org/wiki/Ellipse#Semi-latus_rectum
"""
return self.major * (1 - self.eccentricity ** 2)
def auxiliary_circle(self):
"""Returns a Circle whose diameter is the major axis of the ellipse.
Examples
========
>>> from sympy import Ellipse, Point, symbols
>>> c = Point(1, 2)
>>> Ellipse(c, 8, 7).auxiliary_circle()
Circle(Point2D(1, 2), 8)
>>> a, b = symbols('a b')
>>> Ellipse(c, a, b).auxiliary_circle()
Circle(Point2D(1, 2), Max(a, b))
"""
return Circle(self.center, Max(self.hradius, self.vradius))
def director_circle(self):
"""
Returns a Circle consisting of all points where two perpendicular
tangent lines to the ellipse cross each other.
Returns
=======
Circle
A director circle returned as a geometric object.
Examples
========
>>> from sympy import Ellipse, Point, symbols
>>> c = Point(3,8)
>>> Ellipse(c, 7, 9).director_circle()
Circle(Point2D(3, 8), sqrt(130))
>>> a, b = symbols('a b')
>>> Ellipse(c, a, b).director_circle()
Circle(Point2D(3, 8), sqrt(a**2 + b**2))
References
==========
.. [1] https://en.wikipedia.org/wiki/Director_circle
"""
return Circle(self.center, sqrt(self.hradius**2 + self.vradius**2))
def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of the Ellipse.
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
plot_interval : list
[parameter, lower_bound, upper_bound]
Examples
========
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.plot_interval()
[t, -pi, pi]
"""
t = _symbol(parameter, real=True)
return [t, -S.Pi, S.Pi]
def random_point(self, seed=None):
"""A random point on the ellipse.
Returns
=======
point : Point
Examples
========
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.random_point() # gives some random point
Point2D(...)
>>> p1 = e1.random_point(seed=0); p1.n(2)
Point2D(2.1, 1.4)
Notes
=====
When creating a random point, one may simply replace the
parameter with a random number. When doing so, however, the
random number should be made a Rational or else the point
may not test as being in the ellipse:
>>> from sympy.abc import t
>>> from sympy import Rational
>>> arb = e1.arbitrary_point(t); arb
Point2D(3*cos(t), 2*sin(t))
>>> arb.subs(t, .1) in e1
False
>>> arb.subs(t, Rational(.1)) in e1
True
>>> arb.subs(t, Rational('.1')) in e1
True
See Also
========
sympy.geometry.point.Point
arbitrary_point : Returns parameterized point on ellipse
"""
from sympy import sin, cos, Rational
t = _symbol('t', real=True)
x, y = self.arbitrary_point(t).args
# get a random value in [-1, 1) corresponding to cos(t)
# and confirm that it will test as being in the ellipse
if seed is not None:
rng = random.Random(seed)
else:
rng = random
# simplify this now or else the Float will turn s into a Float
r = Rational(rng.random())
c = 2*r - 1
s = sqrt(1 - c**2)
return Point(x.subs(cos(t), c), y.subs(sin(t), s))
def reflect(self, line):
"""Override GeometryEntity.reflect since the radius
is not a GeometryEntity.
Examples
========
>>> from sympy import Circle, Line
>>> Circle((0, 1), 1).reflect(Line((0, 0), (1, 1)))
Circle(Point2D(1, 0), -1)
>>> from sympy import Ellipse, Line, Point
>>> Ellipse(Point(3, 4), 1, 3).reflect(Line(Point(0, -4), Point(5, 0)))
Traceback (most recent call last):
...
NotImplementedError:
General Ellipse is not supported but the equation of the reflected
Ellipse is given by the zeros of: f(x, y) = (9*x/41 + 40*y/41 +
37/41)**2 + (40*x/123 - 3*y/41 - 364/123)**2 - 1
Notes
=====
Until the general ellipse (with no axis parallel to the x-axis) is
supported a NotImplemented error is raised and the equation whose
zeros define the rotated ellipse is given.
"""
if line.slope in (0, oo):
c = self.center
c = c.reflect(line)
return self.func(c, -self.hradius, self.vradius)
else:
x, y = [uniquely_named_symbol(
name, (self, line), modify=lambda s: '_' + s, real=True)
for name in 'xy']
expr = self.equation(x, y)
p = Point(x, y).reflect(line)
result = expr.subs(zip((x, y), p.args
), simultaneous=True)
raise NotImplementedError(filldedent(
'General Ellipse is not supported but the equation '
'of the reflected Ellipse is given by the zeros of: ' +
"f(%s, %s) = %s" % (str(x), str(y), str(result))))
def rotate(self, angle=0, pt=None):
"""Rotate ``angle`` radians counterclockwise about Point ``pt``.
Note: since the general ellipse is not supported, only rotations that
are integer multiples of pi/2 are allowed.
Examples
========
>>> from sympy import Ellipse, pi
>>> Ellipse((1, 0), 2, 1).rotate(pi/2)
Ellipse(Point2D(0, 1), 1, 2)
>>> Ellipse((1, 0), 2, 1).rotate(pi)
Ellipse(Point2D(-1, 0), 2, 1)
"""
if self.hradius == self.vradius:
return self.func(self.center.rotate(angle, pt), self.hradius)
if (angle/S.Pi).is_integer:
return super().rotate(angle, pt)
if (2*angle/S.Pi).is_integer:
return self.func(self.center.rotate(angle, pt), self.vradius, self.hradius)
# XXX see https://github.com/sympy/sympy/issues/2815 for general ellipes
raise NotImplementedError('Only rotations of pi/2 are currently supported for Ellipse.')
def scale(self, x=1, y=1, pt=None):
"""Override GeometryEntity.scale since it is the major and minor
axes which must be scaled and they are not GeometryEntities.
Examples
========
>>> from sympy import Ellipse
>>> Ellipse((0, 0), 2, 1).scale(2, 4)
Circle(Point2D(0, 0), 4)
>>> Ellipse((0, 0), 2, 1).scale(2)
Ellipse(Point2D(0, 0), 4, 1)
"""
c = self.center
if pt:
pt = Point(pt, dim=2)
return self.translate(*(-pt).args).scale(x, y).translate(*pt.args)
h = self.hradius
v = self.vradius
return self.func(c.scale(x, y), hradius=h*x, vradius=v*y)
def tangent_lines(self, p):
"""Tangent lines between `p` and the ellipse.
If `p` is on the ellipse, returns the tangent line through point `p`.
Otherwise, returns the tangent line(s) from `p` to the ellipse, or
None if no tangent line is possible (e.g., `p` inside ellipse).
Parameters
==========
p : Point
Returns
=======
tangent_lines : list with 1 or 2 Lines
Raises
======
NotImplementedError
Can only find tangent lines for a point, `p`, on the ellipse.
See Also
========
sympy.geometry.point.Point, sympy.geometry.line.Line
Examples
========
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.tangent_lines(Point(3, 0))
[Line2D(Point2D(3, 0), Point2D(3, -12))]
"""
p = Point(p, dim=2)
if self.encloses_point(p):
return []
if p in self:
delta = self.center - p
rise = (self.vradius**2)*delta.x
run = -(self.hradius**2)*delta.y
p2 = Point(simplify(p.x + run),
simplify(p.y + rise))
return [Line(p, p2)]
else:
if len(self.foci) == 2:
f1, f2 = self.foci
maj = self.hradius
test = (2*maj -
Point.distance(f1, p) -
Point.distance(f2, p))
else:
test = self.radius - Point.distance(self.center, p)
if test.is_number and test.is_positive:
return []
# else p is outside the ellipse or we can't tell. In case of the
# latter, the solutions returned will only be valid if
# the point is not inside the ellipse; if it is, nan will result.
x, y = Dummy('x'), Dummy('y')
eq = self.equation(x, y)
dydx = idiff(eq, y, x)
slope = Line(p, Point(x, y)).slope
# TODO: Replace solve with solveset, when this line is tested
tangent_points = solve([slope - dydx, eq], [x, y])
# handle horizontal and vertical tangent lines
if len(tangent_points) == 1:
if tangent_points[0][
0] == p.x or tangent_points[0][1] == p.y:
return [Line(p, p + Point(1, 0)), Line(p, p + Point(0, 1))]
else:
return [Line(p, p + Point(0, 1)), Line(p, tangent_points[0])]
# others
return [Line(p, tangent_points[0]), Line(p, tangent_points[1])]
@property
def vradius(self):
"""The vertical radius of the ellipse.
Returns
=======
vradius : number
See Also
========
hradius, major, minor
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.vradius
1
"""
return self.args[2]
def second_moment_of_area(self, point=None):
"""Returns the second moment and product moment area of an ellipse.
Parameters
==========
point : Point, two-tuple of sympifiable objects, or None(default=None)
point is the point about which second moment of area is to be found.
If "point=None" it will be calculated about the axis passing through the
centroid of the ellipse.
Returns
=======
I_xx, I_yy, I_xy : number or sympy expression
I_xx, I_yy are second moment of area of an ellise.
I_xy is product moment of area of an ellipse.
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.second_moment_of_area()
(3*pi/4, 27*pi/4, 0)
References
==========
https://en.wikipedia.org/wiki/List_of_second_moments_of_area
"""
I_xx = (S.Pi*(self.hradius)*(self.vradius**3))/4
I_yy = (S.Pi*(self.hradius**3)*(self.vradius))/4
I_xy = 0
if point is None:
return I_xx, I_yy, I_xy
# parallel axis theorem
I_xx = I_xx + self.area*((point[1] - self.center.y)**2)
I_yy = I_yy + self.area*((point[0] - self.center.x)**2)
I_xy = I_xy + self.area*(point[0] - self.center.x)*(point[1] - self.center.y)
return I_xx, I_yy, I_xy
def polar_second_moment_of_area(self):
"""Returns the polar second moment of area of an Ellipse
It is a constituent of the second moment of area, linked through
the perpendicular axis theorem. While the planar second moment of
area describes an object's resistance to deflection (bending) when
subjected to a force applied to a plane parallel to the central
axis, the polar second moment of area describes an object's
resistance to deflection when subjected to a moment applied in a
plane perpendicular to the object's central axis (i.e. parallel to
the cross-section)
References
==========
https://en.wikipedia.org/wiki/Polar_moment_of_inertia
Examples
========
>>> from sympy import symbols, Circle, Ellipse
>>> c = Circle((5, 5), 4)
>>> c.polar_second_moment_of_area()
128*pi
>>> a, b = symbols('a, b')
>>> e = Ellipse((0, 0), a, b)
>>> e.polar_second_moment_of_area()
pi*a**3*b/4 + pi*a*b**3/4
"""
second_moment = self.second_moment_of_area()
return second_moment[0] + second_moment[1]
def section_modulus(self, point=None):
"""Returns a tuple with the section modulus of an ellipse
Section modulus is a geometric property of an ellipse defined as the
ratio of second moment of area to the distance of the extreme end of
the ellipse from the centroidal axis.
References
==========
https://en.wikipedia.org/wiki/Section_modulus
Parameters
==========
point : Point, two-tuple of sympifyable objects, or None(default=None)
point is the point at which section modulus is to be found.
If "point=None" section modulus will be calculated for the
point farthest from the centroidal axis of the ellipse.
Returns
=======
S_x, S_y: numbers or SymPy expressions
S_x is the section modulus with respect to the x-axis
S_y is the section modulus with respect to the y-axis
A negative sign indicates that the section modulus is
determined for a point below the centroidal axis.
Examples
========
>>> from sympy import Symbol, Ellipse, Circle, Point2D
>>> d = Symbol('d', positive=True)
>>> c = Circle((0, 0), d/2)
>>> c.section_modulus()
(pi*d**3/32, pi*d**3/32)
>>> e = Ellipse(Point2D(0, 0), 2, 4)
>>> e.section_modulus()
(8*pi, 4*pi)
>>> e.section_modulus((2, 2))
(16*pi, 4*pi)
"""
x_c, y_c = self.center
if point is None:
# taking x and y as maximum distances from centroid
x_min, y_min, x_max, y_max = self.bounds
y = max(y_c - y_min, y_max - y_c)
x = max(x_c - x_min, x_max - x_c)
else:
# taking x and y as distances of the given point from the center
point = Point2D(point)
y = point.y - y_c
x = point.x - x_c
second_moment = self.second_moment_of_area()
S_x = second_moment[0]/y
S_y = second_moment[1]/x
return S_x, S_y
class Circle(Ellipse):
"""A circle in space.
Constructed simply from a center and a radius, from three
non-collinear points, or the equation of a circle.
Parameters
==========
center : Point
radius : number or sympy expression
points : sequence of three Points
equation : equation of a circle
Attributes
==========
radius (synonymous with hradius, vradius, major and minor)
circumference
equation
Raises
======
GeometryError
When the given equation is not that of a circle.
When trying to construct circle from incorrect parameters.
See Also
========
Ellipse, sympy.geometry.point.Point
Examples
========
>>> from sympy import Eq
>>> from sympy.geometry import Point, Circle
>>> from sympy.abc import x, y, a, b
A circle constructed from a center and radius:
>>> c1 = Circle(Point(0, 0), 5)
>>> c1.hradius, c1.vradius, c1.radius
(5, 5, 5)
A circle constructed from three points:
>>> c2 = Circle(Point(0, 0), Point(1, 1), Point(1, 0))
>>> c2.hradius, c2.vradius, c2.radius, c2.center
(sqrt(2)/2, sqrt(2)/2, sqrt(2)/2, Point2D(1/2, 1/2))
A circle can be constructed from an equation in the form
`a*x**2 + by**2 + gx + hy + c = 0`, too:
>>> Circle(x**2 + y**2 - 25)
Circle(Point2D(0, 0), 5)
If the variables corresponding to x and y are named something
else, their name or symbol can be supplied:
>>> Circle(Eq(a**2 + b**2, 25), x='a', y=b)
Circle(Point2D(0, 0), 5)
"""
def __new__(cls, *args, **kwargs):
from sympy.geometry.util import find
from .polygon import Triangle
evaluate = kwargs.get('evaluate', global_parameters.evaluate)
if len(args) == 1 and isinstance(args[0], (Expr, Eq)):
x = kwargs.get('x', 'x')
y = kwargs.get('y', 'y')
equation = args[0]
if isinstance(equation, Eq):
equation = equation.lhs - equation.rhs
x = find(x, equation)
y = find(y, equation)
try:
a, b, c, d, e = linear_coeffs(equation, x**2, y**2, x, y)
except ValueError:
raise GeometryError("The given equation is not that of a circle.")
if a == 0 or b == 0 or a != b:
raise GeometryError("The given equation is not that of a circle.")
center_x = -c/a/2
center_y = -d/b/2
r2 = (center_x**2) + (center_y**2) - e
return Circle((center_x, center_y), sqrt(r2), evaluate=evaluate)
else:
c, r = None, None
if len(args) == 3:
args = [Point(a, dim=2, evaluate=evaluate) for a in args]
t = Triangle(*args)
if not isinstance(t, Triangle):
return t
c = t.circumcenter
r = t.circumradius
elif len(args) == 2:
# Assume (center, radius) pair
c = Point(args[0], dim=2, evaluate=evaluate)
r = args[1]
# this will prohibit imaginary radius
try:
r = Point(r, 0, evaluate=evaluate).x
except ValueError:
raise GeometryError("Circle with imaginary radius is not permitted")
if not (c is None or r is None):
if r == 0:
return c
return GeometryEntity.__new__(cls, c, r, **kwargs)
raise GeometryError("Circle.__new__ received unknown arguments")
@property
def circumference(self):
"""The circumference of the circle.
Returns
=======
circumference : number or SymPy expression
Examples
========
>>> from sympy import Point, Circle
>>> c1 = Circle(Point(3, 4), 6)
>>> c1.circumference
12*pi
"""
return 2 * S.Pi * self.radius
def equation(self, x='x', y='y'):
"""The equation of the circle.
Parameters
==========
x : str or Symbol, optional
Default value is 'x'.
y : str or Symbol, optional
Default value is 'y'.
Returns
=======
equation : SymPy expression
Examples
========
>>> from sympy import Point, Circle
>>> c1 = Circle(Point(0, 0), 5)
>>> c1.equation()
x**2 + y**2 - 25
"""
x = _symbol(x, real=True)
y = _symbol(y, real=True)
t1 = (x - self.center.x)**2
t2 = (y - self.center.y)**2
return t1 + t2 - self.major**2
def intersection(self, o):
"""The intersection of this circle with another geometrical entity.
Parameters
==========
o : GeometryEntity
Returns
=======
intersection : list of GeometryEntities
Examples
========
>>> from sympy import Point, Circle, Line, Ray
>>> p1, p2, p3 = Point(0, 0), Point(5, 5), Point(6, 0)
>>> p4 = Point(5, 0)
>>> c1 = Circle(p1, 5)
>>> c1.intersection(p2)
[]
>>> c1.intersection(p4)
[Point2D(5, 0)]
>>> c1.intersection(Ray(p1, p2))
[Point2D(5*sqrt(2)/2, 5*sqrt(2)/2)]
>>> c1.intersection(Line(p2, p3))
[]
"""
return Ellipse.intersection(self, o)
@property
def radius(self):
"""The radius of the circle.
Returns
=======
radius : number or sympy expression
See Also
========
Ellipse.major, Ellipse.minor, Ellipse.hradius, Ellipse.vradius
Examples
========
>>> from sympy import Point, Circle
>>> c1 = Circle(Point(3, 4), 6)
>>> c1.radius
6
"""
return self.args[1]
def reflect(self, line):
"""Override GeometryEntity.reflect since the radius
is not a GeometryEntity.
Examples
========
>>> from sympy import Circle, Line
>>> Circle((0, 1), 1).reflect(Line((0, 0), (1, 1)))
Circle(Point2D(1, 0), -1)
"""
c = self.center
c = c.reflect(line)
return self.func(c, -self.radius)
def scale(self, x=1, y=1, pt=None):
"""Override GeometryEntity.scale since the radius
is not a GeometryEntity.
Examples
========
>>> from sympy import Circle
>>> Circle((0, 0), 1).scale(2, 2)
Circle(Point2D(0, 0), 2)
>>> Circle((0, 0), 1).scale(2, 4)
Ellipse(Point2D(0, 0), 2, 4)
"""
c = self.center
if pt:
pt = Point(pt, dim=2)
return self.translate(*(-pt).args).scale(x, y).translate(*pt.args)
c = c.scale(x, y)
x, y = [abs(i) for i in (x, y)]
if x == y:
return self.func(c, x*self.radius)
h = v = self.radius
return Ellipse(c, hradius=h*x, vradius=v*y)
@property
def vradius(self):
"""
This Ellipse property is an alias for the Circle's radius.
Whereas hradius, major and minor can use Ellipse's conventions,
the vradius does not exist for a circle. It is always a positive
value in order that the Circle, like Polygons, will have an
area that can be positive or negative as determined by the sign
of the hradius.
Examples
========
>>> from sympy import Point, Circle
>>> c1 = Circle(Point(3, 4), 6)
>>> c1.vradius
6
"""
return abs(self.radius)
from .polygon import Polygon
|
d96f485b0e19b48a15e267c17a2cc13b63a390bb3a43aa92283b4882f3beb92f | """Transform a string with Python-like source code into SymPy expression. """
from tokenize import (generate_tokens, untokenize, TokenError,
NUMBER, STRING, NAME, OP, ENDMARKER, ERRORTOKEN, NEWLINE)
from keyword import iskeyword
import ast
import unicodedata
from io import StringIO
from sympy.assumptions.ask import AssumptionKeys
from sympy.core.compatibility import iterable
from sympy.core.basic import Basic
from sympy.core import Symbol
from sympy.core.function import arity
from sympy.utilities.misc import filldedent, func_name
def _token_splittable(token):
"""
Predicate for whether a token name can be split into multiple tokens.
A token is splittable if it does not contain an underscore character and
it is not the name of a Greek letter. This is used to implicitly convert
expressions like 'xyz' into 'x*y*z'.
"""
if '_' in token:
return False
else:
try:
return not unicodedata.lookup('GREEK SMALL LETTER ' + token)
except KeyError:
pass
if len(token) > 1:
return True
return False
def _token_callable(token, local_dict, global_dict, nextToken=None):
"""
Predicate for whether a token name represents a callable function.
Essentially wraps ``callable``, but looks up the token name in the
locals and globals.
"""
func = local_dict.get(token[1])
if not func:
func = global_dict.get(token[1])
return callable(func) and not isinstance(func, Symbol)
def _add_factorial_tokens(name, result):
if result == [] or result[-1][1] == '(':
raise TokenError()
beginning = [(NAME, name), (OP, '(')]
end = [(OP, ')')]
diff = 0
length = len(result)
for index, token in enumerate(result[::-1]):
toknum, tokval = token
i = length - index - 1
if tokval == ')':
diff += 1
elif tokval == '(':
diff -= 1
if diff == 0:
if i - 1 >= 0 and result[i - 1][0] == NAME:
return result[:i - 1] + beginning + result[i - 1:] + end
else:
return result[:i] + beginning + result[i:] + end
return result
class AppliedFunction:
"""
A group of tokens representing a function and its arguments.
`exponent` is for handling the shorthand sin^2, ln^2, etc.
"""
def __init__(self, function, args, exponent=None):
if exponent is None:
exponent = []
self.function = function
self.args = args
self.exponent = exponent
self.items = ['function', 'args', 'exponent']
def expand(self):
"""Return a list of tokens representing the function"""
result = []
result.append(self.function)
result.extend(self.args)
return result
def __getitem__(self, index):
return getattr(self, self.items[index])
def __repr__(self):
return "AppliedFunction(%s, %s, %s)" % (self.function, self.args,
self.exponent)
class ParenthesisGroup(list):
"""List of tokens representing an expression in parentheses."""
pass
def _flatten(result):
result2 = []
for tok in result:
if isinstance(tok, AppliedFunction):
result2.extend(tok.expand())
else:
result2.append(tok)
return result2
def _group_parentheses(recursor):
def _inner(tokens, local_dict, global_dict):
"""Group tokens between parentheses with ParenthesisGroup.
Also processes those tokens recursively.
"""
result = []
stacks = []
stacklevel = 0
for token in tokens:
if token[0] == OP:
if token[1] == '(':
stacks.append(ParenthesisGroup([]))
stacklevel += 1
elif token[1] == ')':
stacks[-1].append(token)
stack = stacks.pop()
if len(stacks) > 0:
# We don't recurse here since the upper-level stack
# would reprocess these tokens
stacks[-1].extend(stack)
else:
# Recurse here to handle nested parentheses
# Strip off the outer parentheses to avoid an infinite loop
inner = stack[1:-1]
inner = recursor(inner,
local_dict,
global_dict)
parenGroup = [stack[0]] + inner + [stack[-1]]
result.append(ParenthesisGroup(parenGroup))
stacklevel -= 1
continue
if stacklevel:
stacks[-1].append(token)
else:
result.append(token)
if stacklevel:
raise TokenError("Mismatched parentheses")
return result
return _inner
def _apply_functions(tokens, local_dict, global_dict):
"""Convert a NAME token + ParenthesisGroup into an AppliedFunction.
Note that ParenthesisGroups, if not applied to any function, are
converted back into lists of tokens.
"""
result = []
symbol = None
for tok in tokens:
if tok[0] == NAME:
symbol = tok
result.append(tok)
elif isinstance(tok, ParenthesisGroup):
if symbol and _token_callable(symbol, local_dict, global_dict):
result[-1] = AppliedFunction(symbol, tok)
symbol = None
else:
result.extend(tok)
else:
symbol = None
result.append(tok)
return result
def _implicit_multiplication(tokens, local_dict, global_dict):
"""Implicitly adds '*' tokens.
Cases:
- Two AppliedFunctions next to each other ("sin(x)cos(x)")
- AppliedFunction next to an open parenthesis ("sin x (cos x + 1)")
- A close parenthesis next to an AppliedFunction ("(x+2)sin x")\
- A close parenthesis next to an open parenthesis ("(x+2)(x+3)")
- AppliedFunction next to an implicitly applied function ("sin(x)cos x")
"""
result = []
skip = False
for tok, nextTok in zip(tokens, tokens[1:]):
result.append(tok)
if skip:
skip = False
continue
if tok[0] == OP and tok[1] == '.' and nextTok[0] == NAME:
# Dotted name. Do not do implicit multiplication
skip = True
continue
if (isinstance(tok, AppliedFunction) and
isinstance(nextTok, AppliedFunction)):
result.append((OP, '*'))
elif (isinstance(tok, AppliedFunction) and
nextTok[0] == OP and nextTok[1] == '('):
# Applied function followed by an open parenthesis
if tok.function[1] == "Function":
result[-1].function = (result[-1].function[0], 'Symbol')
result.append((OP, '*'))
elif (tok[0] == OP and tok[1] == ')' and
isinstance(nextTok, AppliedFunction)):
# Close parenthesis followed by an applied function
result.append((OP, '*'))
elif (tok[0] == OP and tok[1] == ')' and
nextTok[0] == NAME):
# Close parenthesis followed by an implicitly applied function
result.append((OP, '*'))
elif (tok[0] == nextTok[0] == OP
and tok[1] == ')' and nextTok[1] == '('):
# Close parenthesis followed by an open parenthesis
result.append((OP, '*'))
elif (isinstance(tok, AppliedFunction) and nextTok[0] == NAME):
# Applied function followed by implicitly applied function
result.append((OP, '*'))
elif (tok[0] == NAME and
not _token_callable(tok, local_dict, global_dict) and
nextTok[0] == OP and nextTok[1] == '('):
# Constant followed by parenthesis
result.append((OP, '*'))
elif (tok[0] == NAME and
not _token_callable(tok, local_dict, global_dict) and
nextTok[0] == NAME and
not _token_callable(nextTok, local_dict, global_dict)):
# Constant followed by constant
result.append((OP, '*'))
elif (tok[0] == NAME and
not _token_callable(tok, local_dict, global_dict) and
(isinstance(nextTok, AppliedFunction) or nextTok[0] == NAME)):
# Constant followed by (implicitly applied) function
result.append((OP, '*'))
if tokens:
result.append(tokens[-1])
return result
def _implicit_application(tokens, local_dict, global_dict):
"""Adds parentheses as needed after functions."""
result = []
appendParen = 0 # number of closing parentheses to add
skip = 0 # number of tokens to delay before adding a ')' (to
# capture **, ^, etc.)
exponentSkip = False # skipping tokens before inserting parentheses to
# work with function exponentiation
for tok, nextTok in zip(tokens, tokens[1:]):
result.append(tok)
if (tok[0] == NAME and nextTok[0] not in [OP, ENDMARKER, NEWLINE]):
if _token_callable(tok, local_dict, global_dict, nextTok):
result.append((OP, '('))
appendParen += 1
# name followed by exponent - function exponentiation
elif (tok[0] == NAME and nextTok[0] == OP and nextTok[1] == '**'):
if _token_callable(tok, local_dict, global_dict):
exponentSkip = True
elif exponentSkip:
# if the last token added was an applied function (i.e. the
# power of the function exponent) OR a multiplication (as
# implicit multiplication would have added an extraneous
# multiplication)
if (isinstance(tok, AppliedFunction)
or (tok[0] == OP and tok[1] == '*')):
# don't add anything if the next token is a multiplication
# or if there's already a parenthesis (if parenthesis, still
# stop skipping tokens)
if not (nextTok[0] == OP and nextTok[1] == '*'):
if not(nextTok[0] == OP and nextTok[1] == '('):
result.append((OP, '('))
appendParen += 1
exponentSkip = False
elif appendParen:
if nextTok[0] == OP and nextTok[1] in ('^', '**', '*'):
skip = 1
continue
if skip:
skip -= 1
continue
result.append((OP, ')'))
appendParen -= 1
if tokens:
result.append(tokens[-1])
if appendParen:
result.extend([(OP, ')')] * appendParen)
return result
def function_exponentiation(tokens, local_dict, global_dict):
"""Allows functions to be exponentiated, e.g. ``cos**2(x)``.
Examples
========
>>> from sympy.parsing.sympy_parser import (parse_expr,
... standard_transformations, function_exponentiation)
>>> transformations = standard_transformations + (function_exponentiation,)
>>> parse_expr('sin**4(x)', transformations=transformations)
sin(x)**4
"""
result = []
exponent = []
consuming_exponent = False
level = 0
for tok, nextTok in zip(tokens, tokens[1:]):
if tok[0] == NAME and nextTok[0] == OP and nextTok[1] == '**':
if _token_callable(tok, local_dict, global_dict):
consuming_exponent = True
elif consuming_exponent:
if tok[0] == NAME and tok[1] == 'Function':
tok = (NAME, 'Symbol')
exponent.append(tok)
# only want to stop after hitting )
if tok[0] == nextTok[0] == OP and tok[1] == ')' and nextTok[1] == '(':
consuming_exponent = False
# if implicit multiplication was used, we may have )*( instead
if tok[0] == nextTok[0] == OP and tok[1] == '*' and nextTok[1] == '(':
consuming_exponent = False
del exponent[-1]
continue
elif exponent and not consuming_exponent:
if tok[0] == OP:
if tok[1] == '(':
level += 1
elif tok[1] == ')':
level -= 1
if level == 0:
result.append(tok)
result.extend(exponent)
exponent = []
continue
result.append(tok)
if tokens:
result.append(tokens[-1])
if exponent:
result.extend(exponent)
return result
def split_symbols_custom(predicate):
"""Creates a transformation that splits symbol names.
``predicate`` should return True if the symbol name is to be split.
For instance, to retain the default behavior but avoid splitting certain
symbol names, a predicate like this would work:
>>> from sympy.parsing.sympy_parser import (parse_expr, _token_splittable,
... standard_transformations, implicit_multiplication,
... split_symbols_custom)
>>> def can_split(symbol):
... if symbol not in ('list', 'of', 'unsplittable', 'names'):
... return _token_splittable(symbol)
... return False
...
>>> transformation = split_symbols_custom(can_split)
>>> parse_expr('unsplittable', transformations=standard_transformations +
... (transformation, implicit_multiplication))
unsplittable
"""
def _split_symbols(tokens, local_dict, global_dict):
result = []
split = False
split_previous=False
for tok in tokens:
if split_previous:
# throw out closing parenthesis of Symbol that was split
split_previous=False
continue
split_previous=False
if tok[0] == NAME and tok[1] in ['Symbol', 'Function']:
split = True
elif split and tok[0] == NAME:
symbol = tok[1][1:-1]
if predicate(symbol):
tok_type = result[-2][1] # Symbol or Function
del result[-2:] # Get rid of the call to Symbol
i = 0
while i < len(symbol):
char = symbol[i]
if char in local_dict or char in global_dict:
result.extend([(NAME, "%s" % char)])
elif char.isdigit():
char = [char]
for i in range(i + 1, len(symbol)):
if not symbol[i].isdigit():
i -= 1
break
char.append(symbol[i])
char = ''.join(char)
result.extend([(NAME, 'Number'), (OP, '('),
(NAME, "'%s'" % char), (OP, ')')])
else:
use = tok_type if i == len(symbol) else 'Symbol'
result.extend([(NAME, use), (OP, '('),
(NAME, "'%s'" % char), (OP, ')')])
i += 1
# Set split_previous=True so will skip
# the closing parenthesis of the original Symbol
split = False
split_previous = True
continue
else:
split = False
result.append(tok)
return result
return _split_symbols
#: Splits symbol names for implicit multiplication.
#:
#: Intended to let expressions like ``xyz`` be parsed as ``x*y*z``. Does not
#: split Greek character names, so ``theta`` will *not* become
#: ``t*h*e*t*a``. Generally this should be used with
#: ``implicit_multiplication``.
split_symbols = split_symbols_custom(_token_splittable)
def implicit_multiplication(result, local_dict, global_dict):
"""Makes the multiplication operator optional in most cases.
Use this before :func:`implicit_application`, otherwise expressions like
``sin 2x`` will be parsed as ``x * sin(2)`` rather than ``sin(2*x)``.
Examples
========
>>> from sympy.parsing.sympy_parser import (parse_expr,
... standard_transformations, implicit_multiplication)
>>> transformations = standard_transformations + (implicit_multiplication,)
>>> parse_expr('3 x y', transformations=transformations)
3*x*y
"""
# These are interdependent steps, so we don't expose them separately
for step in (_group_parentheses(implicit_multiplication),
_apply_functions,
_implicit_multiplication):
result = step(result, local_dict, global_dict)
result = _flatten(result)
return result
def implicit_application(result, local_dict, global_dict):
"""Makes parentheses optional in some cases for function calls.
Use this after :func:`implicit_multiplication`, otherwise expressions
like ``sin 2x`` will be parsed as ``x * sin(2)`` rather than
``sin(2*x)``.
Examples
========
>>> from sympy.parsing.sympy_parser import (parse_expr,
... standard_transformations, implicit_application)
>>> transformations = standard_transformations + (implicit_application,)
>>> parse_expr('cot z + csc z', transformations=transformations)
cot(z) + csc(z)
"""
for step in (_group_parentheses(implicit_application),
_apply_functions,
_implicit_application,):
result = step(result, local_dict, global_dict)
result = _flatten(result)
return result
def implicit_multiplication_application(result, local_dict, global_dict):
"""Allows a slightly relaxed syntax.
- Parentheses for single-argument method calls are optional.
- Multiplication is implicit.
- Symbol names can be split (i.e. spaces are not needed between
symbols).
- Functions can be exponentiated.
Examples
========
>>> from sympy.parsing.sympy_parser import (parse_expr,
... standard_transformations, implicit_multiplication_application)
>>> parse_expr("10sin**2 x**2 + 3xyz + tan theta",
... transformations=(standard_transformations +
... (implicit_multiplication_application,)))
3*x*y*z + 10*sin(x**2)**2 + tan(theta)
"""
for step in (split_symbols, implicit_multiplication,
implicit_application, function_exponentiation):
result = step(result, local_dict, global_dict)
return result
def auto_symbol(tokens, local_dict, global_dict):
"""Inserts calls to ``Symbol``/``Function`` for undefined variables."""
result = []
prevTok = (None, None)
tokens.append((None, None)) # so zip traverses all tokens
for tok, nextTok in zip(tokens, tokens[1:]):
tokNum, tokVal = tok
nextTokNum, nextTokVal = nextTok
if tokNum == NAME:
name = tokVal
if (name in ['True', 'False', 'None']
or iskeyword(name)
# Don't convert attribute access
or (prevTok[0] == OP and prevTok[1] == '.')
# Don't convert keyword arguments
or (prevTok[0] == OP and prevTok[1] in ('(', ',')
and nextTokNum == OP and nextTokVal == '=')):
result.append((NAME, name))
continue
elif name in local_dict:
if isinstance(local_dict[name], Symbol) and nextTokVal == '(':
result.extend([(NAME, 'Function'),
(OP, '('),
(NAME, repr(str(local_dict[name]))),
(OP, ')')])
else:
result.append((NAME, name))
continue
elif name in global_dict:
obj = global_dict[name]
if isinstance(obj, (AssumptionKeys, Basic, type)) or callable(obj):
result.append((NAME, name))
continue
result.extend([
(NAME, 'Symbol' if nextTokVal != '(' else 'Function'),
(OP, '('),
(NAME, repr(str(name))),
(OP, ')'),
])
else:
result.append((tokNum, tokVal))
prevTok = (tokNum, tokVal)
return result
def lambda_notation(tokens, local_dict, global_dict):
"""Substitutes "lambda" with its Sympy equivalent Lambda().
However, the conversion doesn't take place if only "lambda"
is passed because that is a syntax error.
"""
result = []
flag = False
toknum, tokval = tokens[0]
tokLen = len(tokens)
if toknum == NAME and tokval == 'lambda':
if tokLen == 2 or tokLen == 3 and tokens[1][0] == NEWLINE:
# In Python 3.6.7+, inputs without a newline get NEWLINE added to
# the tokens
result.extend(tokens)
elif tokLen > 2:
result.extend([
(NAME, 'Lambda'),
(OP, '('),
(OP, '('),
(OP, ')'),
(OP, ')'),
])
for tokNum, tokVal in tokens[1:]:
if tokNum == OP and tokVal == ':':
tokVal = ','
flag = True
if not flag and tokNum == OP and tokVal in ['*', '**']:
raise TokenError("Starred arguments in lambda not supported")
if flag:
result.insert(-1, (tokNum, tokVal))
else:
result.insert(-2, (tokNum, tokVal))
else:
result.extend(tokens)
return result
def factorial_notation(tokens, local_dict, global_dict):
"""Allows standard notation for factorial."""
result = []
nfactorial = 0
for toknum, tokval in tokens:
if toknum == ERRORTOKEN:
op = tokval
if op == '!':
nfactorial += 1
else:
nfactorial = 0
result.append((OP, op))
else:
if nfactorial == 1:
result = _add_factorial_tokens('factorial', result)
elif nfactorial == 2:
result = _add_factorial_tokens('factorial2', result)
elif nfactorial > 2:
raise TokenError
nfactorial = 0
result.append((toknum, tokval))
return result
def convert_xor(tokens, local_dict, global_dict):
"""Treats XOR, ``^``, as exponentiation, ``**``."""
result = []
for toknum, tokval in tokens:
if toknum == OP:
if tokval == '^':
result.append((OP, '**'))
else:
result.append((toknum, tokval))
else:
result.append((toknum, tokval))
return result
def repeated_decimals(tokens, local_dict, global_dict):
"""
Allows 0.2[1] notation to represent the repeated decimal 0.2111... (19/90)
Run this before auto_number.
"""
result = []
def is_digit(s):
return all(i in '0123456789_' for i in s)
# num will running match any DECIMAL [ INTEGER ]
num = []
for toknum, tokval in tokens:
if toknum == NUMBER:
if (not num and '.' in tokval and 'e' not in tokval.lower() and
'j' not in tokval.lower()):
num.append((toknum, tokval))
elif is_digit(tokval)and len(num) == 2:
num.append((toknum, tokval))
elif is_digit(tokval) and len(num) == 3 and is_digit(num[-1][1]):
# Python 2 tokenizes 00123 as '00', '123'
# Python 3 tokenizes 01289 as '012', '89'
num.append((toknum, tokval))
else:
num = []
elif toknum == OP:
if tokval == '[' and len(num) == 1:
num.append((OP, tokval))
elif tokval == ']' and len(num) >= 3:
num.append((OP, tokval))
elif tokval == '.' and not num:
# handle .[1]
num.append((NUMBER, '0.'))
else:
num = []
else:
num = []
result.append((toknum, tokval))
if num and num[-1][1] == ']':
# pre.post[repetend] = a + b/c + d/e where a = pre, b/c = post,
# and d/e = repetend
result = result[:-len(num)]
pre, post = num[0][1].split('.')
repetend = num[2][1]
if len(num) == 5:
repetend += num[3][1]
pre = pre.replace('_', '')
post = post.replace('_', '')
repetend = repetend.replace('_', '')
zeros = '0'*len(post)
post, repetends = [w.lstrip('0') for w in [post, repetend]]
# or else interpreted as octal
a = pre or '0'
b, c = post or '0', '1' + zeros
d, e = repetends, ('9'*len(repetend)) + zeros
seq = [
(OP, '('),
(NAME, 'Integer'),
(OP, '('),
(NUMBER, a),
(OP, ')'),
(OP, '+'),
(NAME, 'Rational'),
(OP, '('),
(NUMBER, b),
(OP, ','),
(NUMBER, c),
(OP, ')'),
(OP, '+'),
(NAME, 'Rational'),
(OP, '('),
(NUMBER, d),
(OP, ','),
(NUMBER, e),
(OP, ')'),
(OP, ')'),
]
result.extend(seq)
num = []
return result
def auto_number(tokens, local_dict, global_dict):
"""
Converts numeric literals to use SymPy equivalents.
Complex numbers use ``I``, integer literals use ``Integer``, and float
literals use ``Float``.
"""
result = []
for toknum, tokval in tokens:
if toknum == NUMBER:
number = tokval
postfix = []
if number.endswith('j') or number.endswith('J'):
number = number[:-1]
postfix = [(OP, '*'), (NAME, 'I')]
if '.' in number or (('e' in number or 'E' in number) and
not (number.startswith('0x') or number.startswith('0X'))):
seq = [(NAME, 'Float'), (OP, '('),
(NUMBER, repr(str(number))), (OP, ')')]
else:
seq = [(NAME, 'Integer'), (OP, '('), (
NUMBER, number), (OP, ')')]
result.extend(seq + postfix)
else:
result.append((toknum, tokval))
return result
def rationalize(tokens, local_dict, global_dict):
"""Converts floats into ``Rational``. Run AFTER ``auto_number``."""
result = []
passed_float = False
for toknum, tokval in tokens:
if toknum == NAME:
if tokval == 'Float':
passed_float = True
tokval = 'Rational'
result.append((toknum, tokval))
elif passed_float == True and toknum == NUMBER:
passed_float = False
result.append((STRING, tokval))
else:
result.append((toknum, tokval))
return result
def _transform_equals_sign(tokens, local_dict, global_dict):
"""Transforms the equals sign ``=`` to instances of Eq.
This is a helper function for `convert_equals_signs`.
Works with expressions containing one equals sign and no
nesting. Expressions like `(1=2)=False` won't work with this
and should be used with `convert_equals_signs`.
Examples: 1=2 to Eq(1,2)
1*2=x to Eq(1*2, x)
This does not deal with function arguments yet.
"""
result = []
if (OP, "=") in tokens:
result.append((NAME, "Eq"))
result.append((OP, "("))
for index, token in enumerate(tokens):
if token == (OP, "="):
result.append((OP, ","))
continue
result.append(token)
result.append((OP, ")"))
else:
result = tokens
return result
def convert_equals_signs(result, local_dict, global_dict):
""" Transforms all the equals signs ``=`` to instances of Eq.
Parses the equals signs in the expression and replaces them with
appropriate Eq instances.Also works with nested equals signs.
Does not yet play well with function arguments.
For example, the expression `(x=y)` is ambiguous and can be interpreted
as x being an argument to a function and `convert_equals_signs` won't
work for this.
See also
========
convert_equality_operators
Examples
========
>>> from sympy.parsing.sympy_parser import (parse_expr,
... standard_transformations, convert_equals_signs)
>>> parse_expr("1*2=x", transformations=(
... standard_transformations + (convert_equals_signs,)))
Eq(2, x)
>>> parse_expr("(1*2=x)=False", transformations=(
... standard_transformations + (convert_equals_signs,)))
Eq(Eq(2, x), False)
"""
for step in (_group_parentheses(convert_equals_signs),
_apply_functions,
_transform_equals_sign):
result = step(result, local_dict, global_dict)
result = _flatten(result)
return result
#: Standard transformations for :func:`parse_expr`.
#: Inserts calls to :class:`~.Symbol`, :class:`~.Integer`, and other SymPy
#: datatypes and allows the use of standard factorial notation (e.g. ``x!``).
standard_transformations = (lambda_notation, auto_symbol, repeated_decimals, auto_number,
factorial_notation)
def stringify_expr(s, local_dict, global_dict, transformations):
"""
Converts the string ``s`` to Python code, in ``local_dict``
Generally, ``parse_expr`` should be used.
"""
tokens = []
input_code = StringIO(s.strip())
for toknum, tokval, _, _, _ in generate_tokens(input_code.readline):
tokens.append((toknum, tokval))
for transform in transformations:
tokens = transform(tokens, local_dict, global_dict)
return untokenize(tokens)
def eval_expr(code, local_dict, global_dict):
"""
Evaluate Python code generated by ``stringify_expr``.
Generally, ``parse_expr`` should be used.
"""
expr = eval(
code, global_dict, local_dict) # take local objects in preference
return expr
def parse_expr(s, local_dict=None, transformations=standard_transformations,
global_dict=None, evaluate=True):
"""Converts the string ``s`` to a SymPy expression, in ``local_dict``
Parameters
==========
s : str
The string to parse.
local_dict : dict, optional
A dictionary of local variables to use when parsing.
global_dict : dict, optional
A dictionary of global variables. By default, this is initialized
with ``from sympy import *``; provide this parameter to override
this behavior (for instance, to parse ``"Q & S"``).
transformations : tuple, optional
A tuple of transformation functions used to modify the tokens of the
parsed expression before evaluation. The default transformations
convert numeric literals into their SymPy equivalents, convert
undefined variables into SymPy symbols, and allow the use of standard
mathematical factorial notation (e.g. ``x!``).
evaluate : bool, optional
When False, the order of the arguments will remain as they were in the
string and automatic simplification that would normally occur is
suppressed. (see examples)
Examples
========
>>> from sympy.parsing.sympy_parser import parse_expr
>>> parse_expr("1/2")
1/2
>>> type(_)
<class 'sympy.core.numbers.Half'>
>>> from sympy.parsing.sympy_parser import standard_transformations,\\
... implicit_multiplication_application
>>> transformations = (standard_transformations +
... (implicit_multiplication_application,))
>>> parse_expr("2x", transformations=transformations)
2*x
When evaluate=False, some automatic simplifications will not occur:
>>> parse_expr("2**3"), parse_expr("2**3", evaluate=False)
(8, 2**3)
In addition the order of the arguments will not be made canonical.
This feature allows one to tell exactly how the expression was entered:
>>> a = parse_expr('1 + x', evaluate=False)
>>> b = parse_expr('x + 1', evaluate=0)
>>> a == b
False
>>> a.args
(1, x)
>>> b.args
(x, 1)
See Also
========
stringify_expr, eval_expr, standard_transformations,
implicit_multiplication_application
"""
if local_dict is None:
local_dict = {}
elif not isinstance(local_dict, dict):
raise TypeError('expecting local_dict to be a dict')
if global_dict is None:
global_dict = {}
exec('from sympy import *', global_dict)
elif not isinstance(global_dict, dict):
raise TypeError('expecting global_dict to be a dict')
transformations = transformations or ()
if transformations:
if not iterable(transformations):
raise TypeError(
'`transformations` should be a list of functions.')
for _ in transformations:
if not callable(_):
raise TypeError(filldedent('''
expected a function in `transformations`,
not %s''' % func_name(_)))
if arity(_) != 3:
raise TypeError(filldedent('''
a transformation should be function that
takes 3 arguments'''))
code = stringify_expr(s, local_dict, global_dict, transformations)
if not evaluate:
code = compile(evaluateFalse(code), '<string>', 'eval')
try:
return eval_expr(code, local_dict, global_dict)
except Exception as e:
raise e from ValueError(f"Error from parse_expr with transformed code: {code!r}")
def evaluateFalse(s):
"""
Replaces operators with the SymPy equivalent and sets evaluate=False.
"""
node = ast.parse(s)
node = EvaluateFalseTransformer().visit(node)
# node is a Module, we want an Expression
node = ast.Expression(node.body[0].value)
return ast.fix_missing_locations(node)
class EvaluateFalseTransformer(ast.NodeTransformer):
operators = {
ast.Add: 'Add',
ast.Mult: 'Mul',
ast.Pow: 'Pow',
ast.Sub: 'Add',
ast.Div: 'Mul',
ast.BitOr: 'Or',
ast.BitAnd: 'And',
ast.BitXor: 'Not',
}
def flatten(self, args, func):
result = []
for arg in args:
if isinstance(arg, ast.Call):
arg_func = arg.func
if isinstance(arg_func, ast.Call):
arg_func = arg_func.func
if arg_func.id == func:
result.extend(self.flatten(arg.args, func))
else:
result.append(arg)
else:
result.append(arg)
return result
def visit_BinOp(self, node):
if node.op.__class__ in self.operators:
sympy_class = self.operators[node.op.__class__]
right = self.visit(node.right)
left = self.visit(node.left)
if isinstance(node.left, ast.UnaryOp) and (isinstance(node.right, ast.UnaryOp) == 0) and sympy_class in ('Mul',):
left, right = right, left
if isinstance(node.op, ast.Sub):
right = ast.Call(
func=ast.Name(id='Mul', ctx=ast.Load()),
args=[ast.UnaryOp(op=ast.USub(), operand=ast.Num(1)), right],
keywords=[ast.keyword(arg='evaluate', value=ast.NameConstant(value=False, ctx=ast.Load()))],
starargs=None,
kwargs=None
)
if isinstance(node.op, ast.Div):
if isinstance(node.left, ast.UnaryOp):
if isinstance(node.right,ast.UnaryOp):
left, right = right, left
left = ast.Call(
func=ast.Name(id='Pow', ctx=ast.Load()),
args=[left, ast.UnaryOp(op=ast.USub(), operand=ast.Num(1))],
keywords=[ast.keyword(arg='evaluate', value=ast.NameConstant(value=False, ctx=ast.Load()))],
starargs=None,
kwargs=None
)
else:
right = ast.Call(
func=ast.Name(id='Pow', ctx=ast.Load()),
args=[right, ast.UnaryOp(op=ast.USub(), operand=ast.Num(1))],
keywords=[ast.keyword(arg='evaluate', value=ast.NameConstant(value=False, ctx=ast.Load()))],
starargs=None,
kwargs=None
)
new_node = ast.Call(
func=ast.Name(id=sympy_class, ctx=ast.Load()),
args=[left, right],
keywords=[ast.keyword(arg='evaluate', value=ast.NameConstant(value=False, ctx=ast.Load()))],
starargs=None,
kwargs=None
)
if sympy_class in ('Add', 'Mul'):
# Denest Add or Mul as appropriate
new_node.args = self.flatten(new_node.args, sympy_class)
return new_node
return node
|
cd40a07dccc2bc7b9ca48c9e220d67e85db65bdabbdfbd70c8ea4237c07b8edf | """
This module implements Pauli algebra by subclassing Symbol. Only algebraic
properties of Pauli matrices are used (we don't use the Matrix class).
See the documentation to the class Pauli for examples.
References
==========
.. [1] https://en.wikipedia.org/wiki/Pauli_matrices
"""
from sympy import Symbol, I, Mul, Pow, Add
from sympy.physics.quantum import TensorProduct
__all__ = ['evaluate_pauli_product']
def delta(i, j):
"""
Returns 1 if ``i == j``, else 0.
This is used in the multiplication of Pauli matrices.
Examples
========
>>> from sympy.physics.paulialgebra import delta
>>> delta(1, 1)
1
>>> delta(2, 3)
0
"""
if i == j:
return 1
else:
return 0
def epsilon(i, j, k):
"""
Return 1 if i,j,k is equal to (1,2,3), (2,3,1), or (3,1,2);
-1 if ``i``,``j``,``k`` is equal to (1,3,2), (3,2,1), or (2,1,3);
else return 0.
This is used in the multiplication of Pauli matrices.
Examples
========
>>> from sympy.physics.paulialgebra import epsilon
>>> epsilon(1, 2, 3)
1
>>> epsilon(1, 3, 2)
-1
"""
if (i, j, k) in [(1, 2, 3), (2, 3, 1), (3, 1, 2)]:
return 1
elif (i, j, k) in [(1, 3, 2), (3, 2, 1), (2, 1, 3)]:
return -1
else:
return 0
class Pauli(Symbol):
"""
The class representing algebraic properties of Pauli matrices.
Explanation
===========
The symbol used to display the Pauli matrices can be changed with an
optional parameter ``label="sigma"``. Pauli matrices with different
``label`` attributes cannot multiply together.
If the left multiplication of symbol or number with Pauli matrix is needed,
please use parentheses to separate Pauli and symbolic multiplication
(for example: 2*I*(Pauli(3)*Pauli(2))).
Another variant is to use evaluate_pauli_product function to evaluate
the product of Pauli matrices and other symbols (with commutative
multiply rules).
See Also
========
evaluate_pauli_product
Examples
========
>>> from sympy.physics.paulialgebra import Pauli
>>> Pauli(1)
sigma1
>>> Pauli(1)*Pauli(2)
I*sigma3
>>> Pauli(1)*Pauli(1)
1
>>> Pauli(3)**4
1
>>> Pauli(1)*Pauli(2)*Pauli(3)
I
>>> from sympy.physics.paulialgebra import Pauli
>>> Pauli(1, label="tau")
tau1
>>> Pauli(1)*Pauli(2, label="tau")
sigma1*tau2
>>> Pauli(1, label="tau")*Pauli(2, label="tau")
I*tau3
>>> from sympy import I
>>> I*(Pauli(2)*Pauli(3))
-sigma1
>>> from sympy.physics.paulialgebra import evaluate_pauli_product
>>> f = I*Pauli(2)*Pauli(3)
>>> f
I*sigma2*sigma3
>>> evaluate_pauli_product(f)
-sigma1
"""
__slots__ = ("i", "label")
def __new__(cls, i, label="sigma"):
if not i in [1, 2, 3]:
raise IndexError("Invalid Pauli index")
obj = Symbol.__new__(cls, "%s%d" %(label,i), commutative=False, hermitian=True)
obj.i = i
obj.label = label
return obj
def __getnewargs_ex__(self):
return (self.i, self.label), {}
def _hashable_content(self):
return (self.i, self.label)
# FIXME don't work for -I*Pauli(2)*Pauli(3)
def __mul__(self, other):
if isinstance(other, Pauli):
j = self.i
k = other.i
jlab = self.label
klab = other.label
if jlab == klab:
return delta(j, k) \
+ I*epsilon(j, k, 1)*Pauli(1,jlab) \
+ I*epsilon(j, k, 2)*Pauli(2,jlab) \
+ I*epsilon(j, k, 3)*Pauli(3,jlab)
return super().__mul__(other)
def _eval_power(b, e):
if e.is_Integer and e.is_positive:
return super().__pow__(int(e) % 2)
def evaluate_pauli_product(arg):
'''Help function to evaluate Pauli matrices product
with symbolic objects.
Parameters
==========
arg: symbolic expression that contains Paulimatrices
Examples
========
>>> from sympy.physics.paulialgebra import Pauli, evaluate_pauli_product
>>> from sympy import I
>>> evaluate_pauli_product(I*Pauli(1)*Pauli(2))
-sigma3
>>> from sympy.abc import x
>>> evaluate_pauli_product(x**2*Pauli(2)*Pauli(1))
-I*x**2*sigma3
'''
start = arg
end = arg
if isinstance(arg, Pow) and isinstance(arg.args[0], Pauli):
if arg.args[1].is_odd:
return arg.args[0]
else:
return 1
if isinstance(arg, Add):
return Add(*[evaluate_pauli_product(part) for part in arg.args])
if isinstance(arg, TensorProduct):
return TensorProduct(*[evaluate_pauli_product(part) for part in arg.args])
elif not(isinstance(arg, Mul)):
return arg
while ((not(start == end)) | ((start == arg) & (end == arg))):
start = end
tmp = start.as_coeff_mul()
sigma_product = 1
com_product = 1
keeper = 1
for el in tmp[1]:
if isinstance(el, Pauli):
sigma_product *= el
elif not(el.is_commutative):
if isinstance(el, Pow) and isinstance(el.args[0], Pauli):
if el.args[1].is_odd:
sigma_product *= el.args[0]
elif isinstance(el, TensorProduct):
keeper = keeper*sigma_product*\
TensorProduct(
*[evaluate_pauli_product(part) for part in el.args]
)
sigma_product = 1
else:
keeper = keeper*sigma_product*el
sigma_product = 1
else:
com_product *= el
end = (tmp[0]*keeper*sigma_product*com_product)
if end == arg: break
return end
|
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