# Source code for diofant.polys.partfrac

"""Algorithms for partial fraction decomposition of rational functions. """

import itertools

from ..core import (Add, Dummy, Function, Integer, Lambda, preorder_traversal,
sympify)
from ..utilities import numbered_symbols
from . import Poly, RootSum, cancel, factor
from .polyerrors import PolynomialError
from .polyoptions import allowed_flags, set_defaults
from .polytools import parallel_poly_from_expr

__all__ = 'apart', 'apart_list', 'assemble_partfrac_list'

[docs]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 ======== 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*y*_a/3 - y/3)/(x - _a))) 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(*[apart(i, x=x, full=full, **_options) for i in 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: nc.append(apart(i, x=x, full=full, **_options)) 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: reps.append((e, apart(e, x=x, full=full, **_options))) pot.skip() # this was handled successfully 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 = Integer(0) 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 = list(itertools.islice(X, n)), q.quo(f) partial.append((coeffs, q, f, i)) symbols.extend(coeffs) dom = Q.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 = [], Integer(0) for (k,), coeff in F.terms(): system.append(coeff - P.coeff_monomial((k,))) from ..solvers import solve solution = solve(system, symbols)[0] 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 ========== * :cite:Bronstein1993partial """ return assemble_partfrac_list(apart_list(P/Q, P.gens[0]))
[docs]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: >>> 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, x - _a), 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, x - _a), 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*t*_a/(4*t - 1) - t/(4*t - 1)), Lambda(_a, x - _a), 1)]) >>> assemble_partfrac_list(pfd) RootSum(t + _w**2 + _w, Lambda(_a, (-2*t*_a/(4*t - 1) - t/(4*t - 1))/(x - _a))) 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, x - _a), 1), (Poly(_w**2 - 1, _w, domain='ZZ'), Lambda(_a, -3*_a - 6), Lambda(_a, x - _a), 2), (Poly(_w + 1, _w, domain='ZZ'), Lambda(_a, -4), Lambda(_a, x - _a), 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 ========== * :cite:Bronstein1993partial """ 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: # pragma: no cover 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 ========== * :cite:Bronstein1993partial """ f, x, U = P/Q, P.gen, [] u = Function('u')(x) a = Dummy('a') Q_c, Q_sqf = Q.sqf_list() if Q_c != 1 and Q_sqf: if Q_sqf[0][1] == 1: Q_sqf[0] = Q_c*Q_sqf[0][0], 1 else: Q_sqf.insert(0, (Poly(Q_c, x), 1)) partial = [] for d, n in Q_sqf: 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(n): P, Q = cancel(H[j]).as_numer_denom() for i in range(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]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: >>> 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, x - _a), 1), (Poly(_w**2 - 1, _w, domain='ZZ'), Lambda(_a, -3*_a - 6), Lambda(_a, x - _a), 2), (Poly(_w + 1, _w, domain='ZZ'), Lambda(_a, -4), Lambda(_a, x - _a), 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, x - _a), 1)]) >>> pfda = assemble_partfrac_list(pfd) >>> pfda RootSum(_w**2 - 2, Lambda(_a, _a/(x - _a)))/2 >>> pfda.doit() -sqrt(2)/(2*(x + sqrt(2))) + sqrt(2)/(2*(x - sqrt(2))) >>> a = Dummy("a") >>> pfd = (1, Poly(0, x), [([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