def test_poly_gcdex():
f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
g = x**3 + x**2 - 4*x - 4
assert poly_half_gcdex(f, g, x) == \
(Poly(-x/5+Rational(3,5), x), Poly(x+1, x))
assert poly_half_gcdex(Poly(f, x), Poly(g, x)) == \
(Poly(-x/5+Rational(3,5), x), Poly(x+1, x))
assert poly_gcdex(f, g, x) == \
(Poly(-x/5+Rational(3,5), x), Poly(x**2/5-6*x/5+2, x), Poly(x+1, x))
assert poly_gcdex(Poly(f, x), Poly(g, x)) == \
(Poly(-x/5+Rational(3,5), x), Poly(x**2/5-6*x/5+2, x), Poly(x+1, x))
f = x**4 + 4*x**3 - x + 1
g = x**3 - x + 1
s, t, h = poly_gcdex(f, g, x)
S, T, H = poly_gcdex(g, f, x)
assert s*f + t*g == h
assert S*g + T*f == H
assert poly_gcdex(2*x, x**2-16, x) == \
(Poly(x/32, x), Poly(-Rational(1,16), x), Poly(1, x))
def test_poly_gcdex():
f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
g = x**3 + x**2 - 4*x - 4
assert poly_half_gcdex(f, g, x) == \
(Poly(-x/5+Rational(3,5), x), Poly(x+1, x))
assert poly_half_gcdex(Poly(f, x), Poly(g, x)) == \
(Poly(-x/5+Rational(3,5), x), Poly(x+1, x))
assert poly_gcdex(f, g, x) == \
(Poly(-x/5+Rational(3,5), x), Poly(x**2/5-6*x/5+2, x), Poly(x+1, x))
assert poly_gcdex(Poly(f, x), Poly(g, x)) == \
(Poly(-x/5+Rational(3,5), x), Poly(x**2/5-6*x/5+2, x), Poly(x+1, x))
f = x**4 + 4*x**3 - x + 1
g = x**3 - x + 1
s, t, h = poly_gcdex(f, g, x)
S, T, H = poly_gcdex(g, f, x)
assert s*f + t*g == h
assert S*g + T*f == H
assert poly_gcdex(2*x, x**2-16, x) == \
(Poly(x/32, x), Poly(-Rational(1,16), x), Poly(1, x))
def test_poly_gcdex():
f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
g = x**3 + x**2 - 4*x - 4
assert poly_half_gcdex(f, g, x) == \
(Poly(-x+3, x), Poly(5*x+5, x))
assert poly_half_gcdex(Poly(f, x), Poly(g, x)) == \
(Poly(-x+3, x), Poly(5*x+5, x))
assert poly_gcdex(f, g, x) == \
(Poly(-x+3, x), Poly(x**2-6*x+10, x), Poly(5*x+5, x))
assert poly_gcdex(Poly(f, x), Poly(g, x)) == \
(Poly(-x+3, x), Poly(x**2-6*x+10, x), Poly(5*x+5, x))
f = x**4 + 4*x**3 - x + 1
g = x**3 - x + 1
s, t, h = poly_gcdex(f, g, x)
S, T, H = poly_gcdex(g, f, x)
assert s*f + t*g == h
assert S*g + T*f == H
def test_poly_gcdex():
f = x**4 - 2 * x**3 - 6 * x**2 + 12 * x + 15
g = x**3 + x**2 - 4 * x - 4
assert poly_half_gcdex(f, g, x) == \
(Poly(-x+3, x), Poly(5*x+5, x))
assert poly_half_gcdex(Poly(f, x), Poly(g, x)) == \
(Poly(-x+3, x), Poly(5*x+5, x))
assert poly_gcdex(f, g, x) == \
(Poly(-x+3, x), Poly(x**2-6*x+10, x), Poly(5*x+5, x))
assert poly_gcdex(Poly(f, x), Poly(g, x)) == \
(Poly(-x+3, x), Poly(x**2-6*x+10, x), Poly(5*x+5, x))
f = x**4 + 4 * x**3 - x + 1
g = x**3 - x + 1
s, t, h = poly_gcdex(f, g, x)
S, T, H = poly_gcdex(g, f, x)
assert s * f + t * g == h
assert S * g + T * f == H
def apart(f, z, **flags):
"""Compute partial fraction decomposition of a rational function.
Given a rational function 'f', performing only gcd operations
over the algebraic closue of the initial field of definition,
compute full partial fraction decomposition with fractions
having linear denominators.
For all other kinds of expressions the input is returned in an
unchanged form. Note however, that 'apart' function can thread
over sums and relational operators.
Note that no factorization of the initial denominator of 'f' is
needed. The final decomposition is formed in terms of a sum of
RootSum instances. By default RootSum tries to compute all its
roots to simplify itself. This behaviour can be however avoided
by seting the keyword flag evaluate=False, which will make this
function return a formal decomposition.
>>> from sympy import *
>>> x,y = symbols('xy')
>>> apart(y/(x+2)/(x+1), x)
y/(1 + x) - y/(2 + x)
>>> apart(1/(1+x**5), x, evaluate=False)
RootSum(Lambda(_a, -1/5/(x - _a)*_a), x**5 + 1, x)
For more information on the implemented algorithm refer to:
[1] M. Bronstein, B. Salvy, Full partial fraction decomposition
of rational functions, in: M. Bronstein, ed., Proceedings
ISSAC '93, ACM Press, Kiev, Ukraine, 1993, pp. 157-160.
"""
if not f.has(z):
return f
f = Poly.cancel(f, z)
P, Q = f.as_numer_denom()
if not Q.has(z):
return f
partial, r = div(P, Q, z)
f, q, U = r / Q, Q, []
u = Function('u')(z)
a = Symbol('a', dummy=True)
for k, d in enumerate(poly_sqf(q, z)):
n, b = k + 1, d.as_basic()
U += [u.diff(z, k)]
h = together(Poly.cancel(f * b**n, z) / u**n)
H, subs = [h], []
for j in range(1, n):
H += [H[-1].diff(z) / j]
for j in range(1, n + 1):
subs += [(U[j - 1], b.diff(z, j) / j)]
for j in range(0, n):
P, Q = together(H[j]).as_numer_denom()
for i in range(0, j + 1):
P = P.subs(*subs[j - i])
Q = Q.subs(*subs[0])
P, Q = Poly(P, z), Poly(Q, z)
G = poly_gcd(P, d)
D = poly_quo(d, G)
B, g = poly_half_gcdex(Q, D)
b = poly_rem(P * poly_quo(B, g), D)
numer = b.as_basic()
denom = (z - a)**(n - j)
expr = numer.subs(z, a) / denom
partial += RootSum(Lambda(a, expr), D, **flags)
return partial
def apart(f, z, **flags):
"""Compute partial fraction decomposition of a rational function.
Given a rational function 'f', performing only gcd operations
over the algebraic closue of the initial field of definition,
compute full partial fraction decomposition with fractions
having linear denominators.
For all other kinds of expressions the input is returned in an
unchanged form. Note however, that 'apart' function can thread
over sums and relational operators.
Note that no factorization of the initial denominator of 'f' is
needed. The final decomposition is formed in terms of a sum of
RootSum instances. By default RootSum tries to compute all its
roots to simplify itself. This behaviour can be however avoided
by seting the keyword flag evaluate=False, which will make this
function return a formal decomposition.
>>> from sympy import *
>>> x,y = symbols('xy')
>>> apart(y/(x+2)/(x+1), x)
y/(1 + x) - y/(2 + x)
>>> apart(1/(1+x**5), x, evaluate=False)
RootSum(Lambda(_a, -1/5/(x - _a)*_a), x**5 + 1, x)
For more information on the implemented algorithm refer to:
[1] M. Bronstein, B. Salvy, Full partial fraction decomposition
of rational functions, in: M. Bronstein, ed., Proceedings
ISSAC '93, ACM Press, Kiev, Ukraine, 1993, pp. 157-160.
"""
if not f.has(z):
return f
f = Poly.cancel(f, z)
P, Q = f.as_numer_denom()
if not Q.has(z):
return f
partial, r = div(P, Q, z)
f, q, U = r / Q, Q, []
u = Function('u')(z)
a = Symbol('a', dummy=True)
for k, d in enumerate(poly_sqf(q, z)):
n, b = k + 1, d.as_basic()
U += [ u.diff(z, k) ]
h = together(Poly.cancel(f*b**n, z) / u**n)
H, subs = [h], []
for j in range(1, n):
H += [ H[-1].diff(z) / j ]
for j in range(1, n+1):
subs += [ (U[j-1], b.diff(z, j) / j) ]
for j in range(0, n):
P, Q = together(H[j]).as_numer_denom()
for i in range(0, j+1):
P = P.subs(*subs[j-i])
Q = Q.subs(*subs[0])
P, Q = Poly(P, z), Poly(Q, z)
G = poly_gcd(P, d)
D = poly_quo(d, G)
B, g = poly_half_gcdex(Q, D)
b = poly_rem(P * poly_quo(B, g), D)
numer = b.as_basic()
denom = (z-a)**(n-j)
expr = numer.subs(z, a) / denom
partial += RootSum(Lambda(a, expr), D, **flags)
return partial
