"""Algorithms for partial fraction decomposition of rational functions. """
from __future__ import print_function, division
from sympy.core import S, Add, sympify, Function, Lambda, Dummy
from sympy.core.basic import preorder_traversal
from sympy.core.compatibility import range
from sympy.polys import Poly, RootSum, cancel, factor
from sympy.polys.polyerrors import PolynomialError
from sympy.polys.polyoptions import allowed_flags, set_defaults
from sympy.polys.polytools import parallel_poly_from_expr
from sympy.utilities import numbered_symbols, take, xthreaded, public
[docs]@xthreaded
@public
def apart(f, x=None, full=False, **options):
"""
Compute partial fraction decomposition of a rational function.
Given a rational function ``f``, computes the partial fraction
decomposition of ``f``. Two algorithms are available: One is based on the
undertermined coefficients method, the other is Bronstein's full partial
fraction decomposition algorithm.
The undetermined coefficients method (selected by ``full=False``) uses
polynomial factorization (and therefore accepts the same options as
factor) for the denominator. Per default it works over the rational
numbers, therefore decomposition of denominators with non-rational roots
(e.g. irrational, complex roots) is not supported by default (see options
of factor).
Bronstein's algorithm can be selected by using ``full=True`` and allows a
decomposition of denominators with non-rational roots. A human-readable
result can be obtained via ``doit()`` (see examples below).
Examples
========
>>> from sympy.polys.partfrac import apart
>>> from sympy.abc import x, y
By default, using the undetermined coefficients method:
>>> apart(y/(x + 2)/(x + 1), x)
-y/(x + 2) + y/(x + 1)
The undetermined coefficients method does not provide a result when the
denominators roots are not rational:
>>> apart(y/(x**2 + x + 1), x)
y/(x**2 + x + 1)
You can choose Bronstein's algorithm by setting ``full=True``:
>>> apart(y/(x**2 + x + 1), x, full=True)
RootSum(_w**2 + _w + 1, Lambda(_a, (-2*_a*y/3 - y/3)/(-_a + x)))
Calling ``doit()`` yields a human-readable result:
>>> apart(y/(x**2 + x + 1), x, full=True).doit()
(-y/3 - 2*y*(-1/2 - sqrt(3)*I/2)/3)/(x + 1/2 + sqrt(3)*I/2) + (-y/3 -
2*y*(-1/2 + sqrt(3)*I/2)/3)/(x + 1/2 - sqrt(3)*I/2)
See Also
========
apart_list, assemble_partfrac_list
"""
allowed_flags(options, [])
f = sympify(f)
if f.is_Atom:
return f
else:
P, Q = f.as_numer_denom()
_options = options.copy()
options = set_defaults(options, extension=True)
try:
(P, Q), opt = parallel_poly_from_expr((P, Q), x, **options)
except PolynomialError as msg:
if f.is_commutative:
raise PolynomialError(msg)
# non-commutative
if f.is_Mul:
c, nc = f.args_cnc(split_1=False)
nc = f.func(*nc)
if c:
c = apart(f.func._from_args(c), x=x, full=full, **_options)
return c*nc
else:
return nc
elif f.is_Add:
c = []
nc = []
for i in f.args:
if i.is_commutative:
c.append(i)
else:
try:
nc.append(apart(i, x=x, full=full, **_options))
except NotImplementedError:
nc.append(i)
return apart(f.func(*c), x=x, full=full, **_options) + f.func(*nc)
else:
reps = []
pot = preorder_traversal(f)
next(pot)
for e in pot:
try:
reps.append((e, apart(e, x=x, full=full, **_options)))
pot.skip() # this was handled successfully
except NotImplementedError:
pass
return f.xreplace(dict(reps))
if P.is_multivariate:
fc = f.cancel()
if fc != f:
return apart(fc, x=x, full=full, **_options)
raise NotImplementedError(
"multivariate partial fraction decomposition")
common, P, Q = P.cancel(Q)
poly, P = P.div(Q, auto=True)
P, Q = P.rat_clear_denoms(Q)
if Q.degree() <= 1:
partial = P/Q
else:
if not full:
partial = apart_undetermined_coeffs(P, Q)
else:
partial = apart_full_decomposition(P, Q)
terms = S.Zero
for term in Add.make_args(partial):
if term.has(RootSum):
terms += term
else:
terms += factor(term)
return common*(poly.as_expr() + terms)
def apart_undetermined_coeffs(P, Q):
"""Partial fractions via method of undetermined coefficients. """
X = numbered_symbols(cls=Dummy)
partial, symbols = [], []
_, factors = Q.factor_list()
for f, k in factors:
n, q = f.degree(), Q
for i in range(1, k + 1):
coeffs, q = take(X, n), q.quo(f)
partial.append((coeffs, q, f, i))
symbols.extend(coeffs)
dom = Q.get_domain().inject(*symbols)
F = Poly(0, Q.gen, domain=dom)
for i, (coeffs, q, f, k) in enumerate(partial):
h = Poly(coeffs, Q.gen, domain=dom)
partial[i] = (h, f, k)
q = q.set_domain(dom)
F += h*q
system, result = [], S(0)
for (k,), coeff in F.terms():
system.append(coeff - P.nth(k))
from sympy.solvers import solve
solution = solve(system, symbols)
for h, f, k in partial:
h = h.as_expr().subs(solution)
result += h/f.as_expr()**k
return result
def apart_full_decomposition(P, Q):
"""
Bronstein's full partial fraction decomposition algorithm.
Given a univariate rational function ``f``, performing only GCD
operations over the algebraic closure of the initial ground domain
of definition, compute full partial fraction decomposition with
fractions having linear denominators.
Note that no factorization of the initial denominator of ``f`` is
performed. The final decomposition is formed in terms of a sum of
:class:`RootSum` instances.
References
==========
.. [1] [Bronstein93]_
"""
return assemble_partfrac_list(apart_list(P/Q, P.gens[0]))
[docs]@public
def apart_list(f, x=None, dummies=None, **options):
"""
Compute partial fraction decomposition of a rational function
and return the result in structured form.
Given a rational function ``f`` compute the partial fraction decomposition
of ``f``. Only Bronstein's full partial fraction decomposition algorithm
is supported by this method. The return value is highly structured and
perfectly suited for further algorithmic treatment rather than being
human-readable. The function returns a tuple holding three elements:
* The first item is the common coefficient, free of the variable `x` used
for decomposition. (It is an element of the base field `K`.)
* The second item is the polynomial part of the decomposition. This can be
the zero polynomial. (It is an element of `K[x]`.)
* The third part itself is a list of quadruples. Each quadruple
has the following elements in this order:
- The (not necessarily irreducible) polynomial `D` whose roots `w_i` appear
in the linear denominator of a bunch of related fraction terms. (This item
can also be a list of explicit roots. However, at the moment ``apart_list``
never returns a result this way, but the related ``assemble_partfrac_list``
function accepts this format as input.)
- The numerator of the fraction, written as a function of the root `w`
- The linear denominator of the fraction *excluding its power exponent*,
written as a function of the root `w`.
- The power to which the denominator has to be raised.
On can always rebuild a plain expression by using the function ``assemble_partfrac_list``.
Examples
========
A first example:
>>> from sympy.polys.partfrac import apart_list, assemble_partfrac_list
>>> from sympy.abc import x, t
>>> f = (2*x**3 - 2*x) / (x**2 - 2*x + 1)
>>> pfd = apart_list(f)
>>> pfd
(1,
Poly(2*x + 4, x, domain='ZZ'),
[(Poly(_w - 1, _w, domain='ZZ'), Lambda(_a, 4), Lambda(_a, -_a + x), 1)])
>>> assemble_partfrac_list(pfd)
2*x + 4 + 4/(x - 1)
Second example:
>>> f = (-2*x - 2*x**2) / (3*x**2 - 6*x)
>>> pfd = apart_list(f)
>>> pfd
(-1,
Poly(2/3, x, domain='QQ'),
[(Poly(_w - 2, _w, domain='ZZ'), Lambda(_a, 2), Lambda(_a, -_a + x), 1)])
>>> assemble_partfrac_list(pfd)
-2/3 - 2/(x - 2)
Another example, showing symbolic parameters:
>>> pfd = apart_list(t/(x**2 + x + t), x)
>>> pfd
(1,
Poly(0, x, domain='ZZ[t]'),
[(Poly(_w**2 + _w + t, _w, domain='ZZ[t]'),
Lambda(_a, -2*_a*t/(4*t - 1) - t/(4*t - 1)),
Lambda(_a, -_a + x),
1)])
>>> assemble_partfrac_list(pfd)
RootSum(_w**2 + _w + t, Lambda(_a, (-2*_a*t/(4*t - 1) - t/(4*t - 1))/(-_a + x)))
This example is taken from Bronstein's original paper:
>>> f = 36 / (x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2)
>>> pfd = apart_list(f)
>>> pfd
(1,
Poly(0, x, domain='ZZ'),
[(Poly(_w - 2, _w, domain='ZZ'), Lambda(_a, 4), Lambda(_a, -_a + x), 1),
(Poly(_w**2 - 1, _w, domain='ZZ'), Lambda(_a, -3*_a - 6), Lambda(_a, -_a + x), 2),
(Poly(_w + 1, _w, domain='ZZ'), Lambda(_a, -4), Lambda(_a, -_a + x), 1)])
>>> assemble_partfrac_list(pfd)
-4/(x + 1) - 3/(x + 1)**2 - 9/(x - 1)**2 + 4/(x - 2)
See also
========
apart, assemble_partfrac_list
References
==========
.. [1] [Bronstein93]_
"""
allowed_flags(options, [])
f = sympify(f)
if f.is_Atom:
return f
else:
P, Q = f.as_numer_denom()
options = set_defaults(options, extension=True)
(P, Q), opt = parallel_poly_from_expr((P, Q), x, **options)
if P.is_multivariate:
raise NotImplementedError(
"multivariate partial fraction decomposition")
common, P, Q = P.cancel(Q)
poly, P = P.div(Q, auto=True)
P, Q = P.rat_clear_denoms(Q)
polypart = poly
if dummies is None:
def dummies(name):
d = Dummy(name)
while True:
yield d
dummies = dummies("w")
rationalpart = apart_list_full_decomposition(P, Q, dummies)
return (common, polypart, rationalpart)
def apart_list_full_decomposition(P, Q, dummygen):
"""
Bronstein's full partial fraction decomposition algorithm.
Given a univariate rational function ``f``, performing only GCD
operations over the algebraic closure of the initial ground domain
of definition, compute full partial fraction decomposition with
fractions having linear denominators.
Note that no factorization of the initial denominator of ``f`` is
performed. The final decomposition is formed in terms of a sum of
:class:`RootSum` instances.
References
==========
.. [1] [Bronstein93]_
"""
f, x, U = P/Q, P.gen, []
u = Function('u')(x)
a = Dummy('a')
partial = []
for d, n in Q.sqf_list_include(all=True):
b = d.as_expr()
U += [ u.diff(x, n - 1) ]
h = cancel(f*b**n) / u**n
H, subs = [h], []
for j in range(1, n):
H += [ H[-1].diff(x) / j ]
for j in range(1, n + 1):
subs += [ (U[j - 1], b.diff(x, j) / j) ]
for j in range(0, n):
P, Q = cancel(H[j]).as_numer_denom()
for i in range(0, j + 1):
P = P.subs(*subs[j - i])
Q = Q.subs(*subs[0])
P = Poly(P, x)
Q = Poly(Q, x)
G = P.gcd(d)
D = d.quo(G)
B, g = Q.half_gcdex(D)
b = (P * B.quo(g)).rem(D)
Dw = D.subs(x, next(dummygen))
numer = Lambda(a, b.as_expr().subs(x, a))
denom = Lambda(a, (x - a))
exponent = n-j
partial.append((Dw, numer, denom, exponent))
return partial
[docs]@public
def assemble_partfrac_list(partial_list):
r"""Reassemble a full partial fraction decomposition
from a structured result obtained by the function ``apart_list``.
Examples
========
This example is taken from Bronstein's original paper:
>>> from sympy.polys.partfrac import apart_list, assemble_partfrac_list
>>> from sympy.abc import x, y
>>> f = 36 / (x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2)
>>> pfd = apart_list(f)
>>> pfd
(1,
Poly(0, x, domain='ZZ'),
[(Poly(_w - 2, _w, domain='ZZ'), Lambda(_a, 4), Lambda(_a, -_a + x), 1),
(Poly(_w**2 - 1, _w, domain='ZZ'), Lambda(_a, -3*_a - 6), Lambda(_a, -_a + x), 2),
(Poly(_w + 1, _w, domain='ZZ'), Lambda(_a, -4), Lambda(_a, -_a + x), 1)])
>>> assemble_partfrac_list(pfd)
-4/(x + 1) - 3/(x + 1)**2 - 9/(x - 1)**2 + 4/(x - 2)
If we happen to know some roots we can provide them easily inside the structure:
>>> pfd = apart_list(2/(x**2-2))
>>> pfd
(1,
Poly(0, x, domain='ZZ'),
[(Poly(_w**2 - 2, _w, domain='ZZ'),
Lambda(_a, _a/2),
Lambda(_a, -_a + x),
1)])
>>> pfda = assemble_partfrac_list(pfd)
>>> pfda
RootSum(_w**2 - 2, Lambda(_a, _a/(-_a + x)))/2
>>> pfda.doit()
-sqrt(2)/(2*(x + sqrt(2))) + sqrt(2)/(2*(x - sqrt(2)))
>>> from sympy import Dummy, Poly, Lambda, sqrt
>>> a = Dummy("a")
>>> pfd = (1, Poly(0, x, domain='ZZ'), [([sqrt(2),-sqrt(2)], Lambda(a, a/2), Lambda(a, -a + x), 1)])
>>> assemble_partfrac_list(pfd)
-sqrt(2)/(2*(x + sqrt(2))) + sqrt(2)/(2*(x - sqrt(2)))
See Also
========
apart, apart_list
"""
# Common factor
common = partial_list[0]
# Polynomial part
polypart = partial_list[1]
pfd = polypart.as_expr()
# Rational parts
for r, nf, df, ex in partial_list[2]:
if isinstance(r, Poly):
# Assemble in case the roots are given implicitly by a polynomials
an, nu = nf.variables, nf.expr
ad, de = df.variables, df.expr
# Hack to make dummies equal because Lambda created new Dummies
de = de.subs(ad[0], an[0])
func = Lambda(an, nu/de**ex)
pfd += RootSum(r, func, auto=False, quadratic=False)
else:
# Assemble in case the roots are given explicitly by a list of algebraic numbers
for root in r:
pfd += nf(root)/df(root)**ex
return common*pfd
