227 lines
6.9 KiB
Python
227 lines
6.9 KiB
Python
"""Tests for algorithms for partial fraction decomposition of rational
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functions. """
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from sympy.polys.partfrac import (
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apart_undetermined_coeffs,
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apart,
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apart_list, assemble_partfrac_list
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)
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from sympy.core.expr import Expr
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from sympy.core.function import Lambda
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from sympy.core.numbers import (E, I, Rational, pi)
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from sympy.core.relational import Eq
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from sympy.core.singleton import S
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from sympy.core.symbol import (Dummy, Symbol)
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from sympy.functions.elementary.miscellaneous import sqrt
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from sympy.matrices.dense import Matrix
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from sympy.polys.polytools import (Poly, factor)
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from sympy.polys.rationaltools import together
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from sympy.polys.rootoftools import RootSum
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from sympy.testing.pytest import raises, XFAIL
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from sympy.abc import x, y, a, b, c
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def test_apart():
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assert apart(1) == 1
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assert apart(1, x) == 1
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f, g = (x**2 + 1)/(x + 1), 2/(x + 1) + x - 1
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assert apart(f, full=False) == g
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assert apart(f, full=True) == g
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f, g = 1/(x + 2)/(x + 1), 1/(1 + x) - 1/(2 + x)
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assert apart(f, full=False) == g
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assert apart(f, full=True) == g
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f, g = 1/(x + 1)/(x + 5), -1/(5 + x)/4 + 1/(1 + x)/4
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assert apart(f, full=False) == g
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assert apart(f, full=True) == g
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assert apart((E*x + 2)/(x - pi)*(x - 1), x) == \
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2 - E + E*pi + E*x + (E*pi + 2)*(pi - 1)/(x - pi)
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assert apart(Eq((x**2 + 1)/(x + 1), x), x) == Eq(x - 1 + 2/(x + 1), x)
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assert apart(x/2, y) == x/2
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f, g = (x+y)/(2*x - y), Rational(3, 2)*y/(2*x - y) + S.Half
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assert apart(f, x, full=False) == g
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assert apart(f, x, full=True) == g
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f, g = (x+y)/(2*x - y), 3*x/(2*x - y) - 1
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assert apart(f, y, full=False) == g
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assert apart(f, y, full=True) == g
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raises(NotImplementedError, lambda: apart(1/(x + 1)/(y + 2)))
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def test_apart_matrix():
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M = Matrix(2, 2, lambda i, j: 1/(x + i + 1)/(x + j))
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assert apart(M) == Matrix([
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[1/x - 1/(x + 1), (x + 1)**(-2)],
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[1/(2*x) - (S.Half)/(x + 2), 1/(x + 1) - 1/(x + 2)],
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])
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def test_apart_symbolic():
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f = a*x**4 + (2*b + 2*a*c)*x**3 + (4*b*c - a**2 + a*c**2)*x**2 + \
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(-2*a*b + 2*b*c**2)*x - b**2
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g = a**2*x**4 + (2*a*b + 2*c*a**2)*x**3 + (4*a*b*c + b**2 +
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a**2*c**2)*x**2 + (2*c*b**2 + 2*a*b*c**2)*x + b**2*c**2
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assert apart(f/g, x) == 1/a - 1/(x + c)**2 - b**2/(a*(a*x + b)**2)
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assert apart(1/((x + a)*(x + b)*(x + c)), x) == \
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1/((a - c)*(b - c)*(c + x)) - 1/((a - b)*(b - c)*(b + x)) + \
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1/((a - b)*(a - c)*(a + x))
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def _make_extension_example():
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# https://github.com/sympy/sympy/issues/18531
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from sympy.core import Mul
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def mul2(expr):
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# 2-arg mul hack...
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return Mul(2, expr, evaluate=False)
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f = ((x**2 + 1)**3/((x - 1)**2*(x + 1)**2*(-x**2 + 2*x + 1)*(x**2 + 2*x - 1)))
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g = (1/mul2(x - sqrt(2) + 1)
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- 1/mul2(x - sqrt(2) - 1)
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+ 1/mul2(x + 1 + sqrt(2))
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- 1/mul2(x - 1 + sqrt(2))
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+ 1/mul2((x + 1)**2)
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+ 1/mul2((x - 1)**2))
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return f, g
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def test_apart_extension():
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f = 2/(x**2 + 1)
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g = I/(x + I) - I/(x - I)
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assert apart(f, extension=I) == g
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assert apart(f, gaussian=True) == g
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f = x/((x - 2)*(x + I))
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assert factor(together(apart(f)).expand()) == f
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f, g = _make_extension_example()
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# XXX: Only works with dotprodsimp. See test_apart_extension_xfail below
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from sympy.matrices import dotprodsimp
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with dotprodsimp(True):
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assert apart(f, x, extension={sqrt(2)}) == g
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def test_apart_extension_xfail():
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f, g = _make_extension_example()
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assert apart(f, x, extension={sqrt(2)}) == g
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def test_apart_full():
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f = 1/(x**2 + 1)
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assert apart(f, full=False) == f
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assert apart(f, full=True).dummy_eq(
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-RootSum(x**2 + 1, Lambda(a, a/(x - a)), auto=False)/2)
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f = 1/(x**3 + x + 1)
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assert apart(f, full=False) == f
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assert apart(f, full=True).dummy_eq(
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RootSum(x**3 + x + 1,
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Lambda(a, (a**2*Rational(6, 31) - a*Rational(9, 31) + Rational(4, 31))/(x - a)), auto=False))
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f = 1/(x**5 + 1)
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assert apart(f, full=False) == \
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(Rational(-1, 5))*((x**3 - 2*x**2 + 3*x - 4)/(x**4 - x**3 + x**2 -
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x + 1)) + (Rational(1, 5))/(x + 1)
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assert apart(f, full=True).dummy_eq(
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-RootSum(x**4 - x**3 + x**2 - x + 1,
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Lambda(a, a/(x - a)), auto=False)/5 + (Rational(1, 5))/(x + 1))
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def test_apart_undetermined_coeffs():
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p = Poly(2*x - 3)
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q = Poly(x**9 - x**8 - x**6 + x**5 - 2*x**2 + 3*x - 1)
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r = (-x**7 - x**6 - x**5 + 4)/(x**8 - x**5 - 2*x + 1) + 1/(x - 1)
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assert apart_undetermined_coeffs(p, q) == r
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p = Poly(1, x, domain='ZZ[a,b]')
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q = Poly((x + a)*(x + b), x, domain='ZZ[a,b]')
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r = 1/((a - b)*(b + x)) - 1/((a - b)*(a + x))
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assert apart_undetermined_coeffs(p, q) == r
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def test_apart_list():
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from sympy.utilities.iterables import numbered_symbols
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def dummy_eq(i, j):
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if type(i) in (list, tuple):
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return all(dummy_eq(i, j) for i, j in zip(i, j))
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return i == j or i.dummy_eq(j)
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w0, w1, w2 = Symbol("w0"), Symbol("w1"), Symbol("w2")
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_a = Dummy("a")
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f = (-2*x - 2*x**2) / (3*x**2 - 6*x)
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got = apart_list(f, x, dummies=numbered_symbols("w"))
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ans = (-1, Poly(Rational(2, 3), x, domain='QQ'),
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[(Poly(w0 - 2, w0, domain='ZZ'), Lambda(_a, 2), Lambda(_a, -_a + x), 1)])
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assert dummy_eq(got, ans)
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got = apart_list(2/(x**2-2), x, dummies=numbered_symbols("w"))
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ans = (1, Poly(0, x, domain='ZZ'), [(Poly(w0**2 - 2, w0, domain='ZZ'),
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Lambda(_a, _a/2),
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Lambda(_a, -_a + x), 1)])
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assert dummy_eq(got, ans)
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f = 36 / (x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2)
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got = apart_list(f, x, dummies=numbered_symbols("w"))
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ans = (1, Poly(0, x, domain='ZZ'),
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[(Poly(w0 - 2, w0, domain='ZZ'), Lambda(_a, 4), Lambda(_a, -_a + x), 1),
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(Poly(w1**2 - 1, w1, domain='ZZ'), Lambda(_a, -3*_a - 6), Lambda(_a, -_a + x), 2),
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(Poly(w2 + 1, w2, domain='ZZ'), Lambda(_a, -4), Lambda(_a, -_a + x), 1)])
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assert dummy_eq(got, ans)
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def test_assemble_partfrac_list():
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f = 36 / (x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2)
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pfd = apart_list(f)
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assert assemble_partfrac_list(pfd) == -4/(x + 1) - 3/(x + 1)**2 - 9/(x - 1)**2 + 4/(x - 2)
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a = Dummy("a")
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pfd = (1, Poly(0, x, domain='ZZ'), [([sqrt(2),-sqrt(2)], Lambda(a, a/2), Lambda(a, -a + x), 1)])
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assert assemble_partfrac_list(pfd) == -1/(sqrt(2)*(x + sqrt(2))) + 1/(sqrt(2)*(x - sqrt(2)))
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@XFAIL
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def test_noncommutative_pseudomultivariate():
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# apart doesn't go inside noncommutative expressions
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class foo(Expr):
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is_commutative=False
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e = x/(x + x*y)
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c = 1/(1 + y)
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assert apart(e + foo(e)) == c + foo(c)
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assert apart(e*foo(e)) == c*foo(c)
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def test_noncommutative():
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class foo(Expr):
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is_commutative=False
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e = x/(x + x*y)
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c = 1/(1 + y)
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assert apart(e + foo()) == c + foo()
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def test_issue_5798():
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assert apart(
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2*x/(x**2 + 1) - (x - 1)/(2*(x**2 + 1)) + 1/(2*(x + 1)) - 2/x) == \
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(3*x + 1)/(x**2 + 1)/2 + 1/(x + 1)/2 - 2/x
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