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Mini Shell
Mini Shell
"""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
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