"""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 import (S, Poly, E, pi, I, Matrix, Eq, RootSum, Lambda, Symbol, Dummy, factor, together, sqrt, Expr, Rational) from sympy.testing.pytest import raises, ON_TRAVIS, skip, 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 # XXX: This is XFAIL just because it is slow @XFAIL def test_apart_extension_xfail(): if ON_TRAVIS: skip('Too slow for Travis') 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