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