def splitter(p):
for y in p.atoms(Basic.Symbol):
if not isinstance(derivation(y), Basic.Zero):
c, q = p.as_polynomial(y).as_primitive()
q = q.as_basic()
h = gcd(q, derivation(q), y)
s = quo(h, gcd(q, q.diff(y), y), y)
c_split = splitter(c)
if s.as_polynomial(y).degree() == 0:
return (c_split[0], q * c_split[1])
q_split = splitter(normal(q / s, *V))
return (c_split[0]*q_split[0]*s, c_split[1]*q_split[1])
else:
return (S.One, p)
def risch_norman(f, x, rewrite=False):
"""Computes indefinite integral using extended Risch-Norman algorithm,
also known as parallel Risch. This is a simplified version of full
recursive Risch algorithm. It is designed for integrating various
classes of functions including transcendental elementary or special
functions like Airy, Bessel, Whittaker and Lambert.
The main difference between this algorithm and the recursive one
is that rather than computing a tower of differential extensions
in a recursive way, it handles all cases in one shot. That's why
it is called parallel Risch algorithm. This makes it much faster
than the original approach.
Another benefit is that it doesn't require to rewrite expressions
in terms of complex exponentials. Rather it uses tangents and so
antiderivatives are being found in a more familliar form.
Risch-Norman algorithm can also handle special functions very
easily without any additional effort. Just differentiation
method must be known for a given function.
Note that this algorithm is not a decision procedure. If it
computes an antiderivative for a given integral then it's a
proof that such function exists. However when it fails then
there still may exist an antiderivative and a fallback to
recurrsive Risch algorithm would be necessary.
The question if this algorithm can be made a full featured
decision procedure still remains open.
For more information on the implemented algorithm refer to:
[1] K. Geddes, L.Stefanus, On the Risch-Norman Integration
Method and its Implementation in Maple, Proceedings of
ISSAC'89, ACM Press, 212-217.
[2] J. H. Davenport, On the Parallel Risch Algorithm (I),
Proceedings of EUROCAM'82, LNCS 144, Springer, 144-157.
[3] J. H. Davenport, On the Parallel Risch Algorithm (III):
Use of Tangents, SIGSAM Bulletin 16 (1982), 3-6.
[4] J. H. Davenport, B. M. Trager, On the Parallel Risch
Algorithm (II), ACM Transactions on Mathematical
Software 11 (1985), 356-362.
"""
f = Basic.sympify(f)
if not f.has(x):
return f * x
rewritables = {
(sin, cos, cot) : tan,
(sinh, cosh, coth) : tanh,
}
if rewrite:
for candidates, rule in rewritables.iteritems():
f = f.rewrite(candidates, rule)
else:
for candidates in rewritables.iterkeys():
if f.has(*candidates):
break
else:
rewrite = True
terms = components(f)
for g in set(terms):
h = g.diff(x)
if not isinstance(h, Basic.Zero):
terms |= components(h)
terms = [ g for g in terms if g.has(x) ]
V, in_terms, out_terms = [], [], {}
for i, term in enumerate(terms):
V += [ Symbol('x%s' % i) ]
N = term.count_ops(symbolic=False)
in_terms += [ (N, term, V[-1]) ]
out_terms[V[-1]] = term
in_terms.sort(lambda u, v: int(v[0] - u[0]))
def substitute(expr):
for _, g, symbol in in_terms:
expr = expr.subs(g, symbol)
return expr
diffs = [ substitute(g.diff(x)) for g in terms ]
denoms = [ g.as_numer_denom()[1] for g in diffs ]
denom = reduce(lambda p, q: lcm(p, q, V), denoms)
numers = [ normal(denom * g, *V) for g in diffs ]
def derivation(h):
return Basic.Add(*[ d * h.diff(v) for d, v in zip(numers, V) ])
def deflation(p):
for y in p.atoms(Basic.Symbol):
if not isinstance(derivation(p), Basic.Zero):
c, q = p.as_polynomial(y).as_primitive()
return deflation(c) * gcd(q, q.diff(y))
else:
return p
def splitter(p):
for y in p.atoms(Basic.Symbol):
if not isinstance(derivation(y), Basic.Zero):
c, q = p.as_polynomial(y).as_primitive()
q = q.as_basic()
h = gcd(q, derivation(q), y)
s = quo(h, gcd(q, q.diff(y), y), y)
c_split = splitter(c)
if s.as_polynomial(y).degree() == 0:
return (c_split[0], q * c_split[1])
q_split = splitter(normal(q / s, *V))
return (c_split[0]*q_split[0]*s, c_split[1]*q_split[1])
else:
return (S.One, p)
special = []
for term in terms:
if isinstance(term, Basic.Function):
if isinstance(term, Basic.tan):
special += [ (1 + substitute(term)**2, False) ]
elif isinstance(term.func, tanh):
special += [ (1 + substitute(term), False),
(1 - substitute(term), False) ]
#elif isinstance(term.func, Basic.LambertW):
# special += [ (substitute(term), True) ]
ff = substitute(f)
P, Q = ff.as_numer_denom()
u_split = splitter(denom)
v_split = splitter(Q)
s = u_split[0] * Basic.Mul(*[ g for g, a in special if a ])
a, b, c = [ p.as_polynomial(*V).degree() for p in [s, P, Q] ]
candidate_denom = s * v_split[0] * deflation(v_split[1])
monoms = monomials(V, 1 + a + max(b, c))
linear = False
while True:
coeffs, candidate, factors = [], S.Zero, set()
for i, monomial in enumerate(monoms):
coeffs += [ Symbol('A%s' % i, dummy=True) ]
candidate += coeffs[-1] * monomial
candidate /= candidate_denom
polys = [ v_split[0], v_split[1], u_split[0]] + [ s[0] for s in special ]
for irreducibles in [ factorization(p, linear) for p in polys ]:
factors |= irreducibles
for i, irreducible in enumerate(factors):
if not isinstance(irreducible, Basic.Number):
coeffs += [ Symbol('B%s' % i, dummy=True) ]
candidate += coeffs[-1] * Basic.log(irreducible)
h = together(ff - derivation(candidate) / denom)
numerator = h.as_numer_denom()[0].expand()
if not isinstance(numerator, Basic.Add):
numerator = [numerator]
collected = {}
for term in numerator:
coeff, depend = term.as_independent(*V)
if depend in collected:
collected[depend] += coeff
else:
collected[depend] = coeff
solutions = solve(collected.values(), coeffs)
if solutions is None:
if linear:
break
else:
linear = True
else:
break
if solutions is not None:
antideriv = candidate.subs_dict(solutions)
for C in coeffs:
if C not in solutions:
antideriv = antideriv.subs(C, S.Zero)
antideriv = simplify(antideriv.subs_dict(out_terms)).expand()
if isinstance(antideriv, Basic.Add):
return Basic.Add(*antideriv.as_coeff_factors()[1])
else:
return antideriv
else:
if not rewrite:
return risch_norman(f, x, rewrite=True)
else:
return None
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
