def sqf(f, var=None, order=None, coeff=None):
"""Square-free decomposition.
Thin wrapper that multiplies the factors of L{factor_.sqf} as
SymPy expressions.
"""
a = factor_.sqf(f, var, order, coeff)
result = S.One
for i, p in enumerate(a):
result *= (p.sympy_expr)**(i+1)
return result
def apart(f, z, domain=None, index=None):
"""Computes full partial fraction decomposition of a univariate
rational function over the algebraic closure of its field of
definition. Although only gcd operations over the initial
field are required, the expansion is returned in a formal
form with linear denominators.
However it is possible to force expansion of the resulting
formal summations, and so factorization over a specified
domain is performed.
To specify the desired behavior of the algorithm use the
'domain' keyword. Setting it to None, which is done be
default, will result in no factorization at all.
Otherwise it can be assigned with one of Z, Q, R, C domain
specifiers and the formal partial fraction expansion will
be rewritten using all possible roots over this domain.
If the resulting expansion contains formal summations, then
for all those a single dummy index variable named 'a' will
be generated. To change this default behavior issue new
name via 'index' keyword.
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.
"""
f = Basic.sympify(f)
if isinstance(f, Basic.Add):
return Add(*[ apart(g) for g in f ])
else:
if f.is_fraction(z):
f = normal(f, z)
else:
return f
P, Q = f.as_numer_denom()
if not Q.has(z):
return f
u = Function('u')(z)
if index is None:
A = Symbol('a', dummy=True)
else:
A = Symbol(index)
partial, r = div(P, Q, z)
f, q, U = r / Q, Q, []
for k, d in enumerate(sqf(q, z)):
n, d = k + 1, d.as_basic()
U += [ u.diff(z, k) ]
h = normal(f * d**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], d.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])
G = gcd(P, d, z)
D = quo(d, G, z)
g, B, _ = ext_gcd(Q, D, z)
b = rem(P * B / g, D, z)
term = b.subs(z, A) / (z - A)**(n-j)
if domain is None:
a = D.diff(z)
if not a.has(z):
partial += term.subs(A, -D.subs(z, 0) / a)
else:
partial += Basic.Sum(term, (A, Basic.RootOf(D, z)))
else:
raise NotImplementedError
return partial