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