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| """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, all_close) | |
| 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_full_floats(): | |
| # https://github.com/sympy/sympy/issues/26648 | |
| f = ( | |
| 6.43369157032015e-9*x**3 + 1.35203404799555e-5*x**2 | |
| + 0.00357538393743079*x + 0.085 | |
| )/( | |
| 4.74334912634438e-11*x**4 + 4.09576274286244e-6*x**3 | |
| + 0.00334241812250921*x**2 + 0.15406018058983*x + 1.0 | |
| ) | |
| expected = ( | |
| 133.599202650992/(x + 85524.0054884464) | |
| + 1.07757928431867/(x + 774.88576677949) | |
| + 0.395006955518971/(x + 40.7977016133126) | |
| + 0.564264854137341/(x + 7.79746609204661) | |
| ) | |
| f_apart = apart(f, full=True).evalf() | |
| # There is a significant floating point error in this operation. | |
| assert all_close(f_apart, expected, rtol=1e-3, atol=1e-5) | |
| 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))) | |
| 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 | |