venv/lib/python3.11/site-packages/sympy/polys/tests/test_partfrac.py

"""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)))


@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