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amgf-0.8.py
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amgf-0.8.py
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r"""
Let $F(x) = \sum_{\nu \in \NN^d} F_{\nu} x^\nu$ be a multivariate power series with complex coefficients that converges in a neighborhood of the origin. Assume that $F = G/H$ for some functions $G$ and $H$ holomorphic in a neighborhood of the origin.
Assume also that $H$ is a polynomial.
This Python module for use within `Sage <http://www.sagemath.org>`_ computes asymptotics for the coefficients $F_{r \alpha}$ as $r \to \infty$ with $r \alpha \in \NN^d$ for $\alpha$ in a permissible subset of $d$-tuples of positive reals.
More specifically, it computes arbitrary terms of the asymptotic expansion for $F_{r \alpha}$ when the asymptotics are controlled by a strictly minimal multiple point of the alegbraic variety $H = 0$.
The algorithms and formulas implemented here come from [RaWi2008a]_
and [RaWi2012]_.
.. [AiYu1983] I.A. Aizenberg and A.P. Yuzhakov, "Integral representations and residues in multidimensional complex analysis", Translations of Mathematical Monographs, 58. American Mathematical Society, Providence, RI, 1983. x+283 pp. ISBN: 0-8218-4511-X.
.. [Raic2012] Alexander Raichev, "Leinartas's partial fraction decomposition", `<http://arxiv.org/abs/1206.4740>`_.
.. [RaWi2008a] Alexander Raichev and Mark C. Wilson, "Asymptotics of coefficients of multivariate generating functions: improvements for smooth points", Electronic Journal of Combinatorics, Vol. 15 (2008), R89, `<http://arxiv.org/pdf/0803.2914.pdf>`_.
.. [RaWi2012] Alexander Raichev and Mark C. Wilson, "Asymptotics of coefficients of multivariate generating functions: improvements for smooth points", To appear in 2012 in the Online Journal of Analytic Combinatorics, `<http://arxiv.org/pdf/1009.5715.pdf>`_.
AUTHORS:
- Alexander Raichev (2008-10-01): Initial version
- Alexander Raichev (2010-09-28): Corrected many functions
- Alexander Raichev (2010-12-15): Updated documentation
- Alexander Raichev (2011-03-09): Fixed a division by zero bug in relative_error()
- Alexander Raichev (2011-04-26): Rewrote in object-oriented style
- Alexander Raichev (2011-05-06): Fixed bug in cohomologous_integrand() and fixed _crit_cone_combo() to work in SR
- Alexander Raichev (2012-08-06): Major rewrite. Created class FFPD and moved functions around.
- Alexander Raichev (2012-10-03): Fixed whitespace errors, added examples to those six functions missing them (which i overlooked), changed package name to a more descriptive title, made asymptotics methods work for univariate functions.
EXAMPLES::
sage: from sage.combinat.asymptotics_multivariate_generating_functions import *
A univariate smooth point example::
sage: R.<x> = PolynomialRing(QQ)
sage: H = (x - 1/2)^3
sage: Hfac = H.factor()
sage: G = -1/(x + 3)/Hfac.unit()
sage: F = FFPD(G, Hfac)
sage: print F
(-1/(x + 3), [(x - 1/2, 3)])
sage: alpha = [1]
sage: decomp = F.asymptotic_decomposition(alpha)
sage: print decomp
[(0, []), (-1/2*(x^2 + 6*x + 9)*r^2/(x^5 + 9*x^4 + 27*x^3 + 27*x^2) - 1/2*(5*x^2 + 24*x + 27)*r/(x^5 + 9*x^4 + 27*x^3 + 27*x^2) - 3*(x^2 + 3*x + 3)/(x^5 + 9*x^4 + 27*x^3 + 27*x^2), [(x - 1/2, 1)])]
sage: F1 = decomp[1]
sage: p = {x: 1/2}
sage: asy = F1.asymptotics(p, alpha, 3)
sage: print asy
(8/343*(49*r^2 + 161*r + 114)*2^r, 2, 8/7*r^2 + 184/49*r + 912/343)
sage: print F.relative_error(asy[0], alpha, [1, 2, 4, 8, 16], asy[1])
Calculating errors table in the form
exponent, scaled Maclaurin coefficient, scaled asymptotic values, relative errors...
[((1,), 7.555555556, [7.556851312], [-0.0001714971672]), ((2,), 14.74074074, [14.74052478], [0.00001465051901]), ((4,), 35.96502058, [35.96501458], [1.667911934e-7]), ((8,), 105.8425656, [105.8425656], [4.399565380e-11]), ((16,), 355.3119534, [355.3119534], [0.0000000000])]
Another smooth point example (Example 5.4 of [RaWi2008a]_)::
sage: R.<x,y> = PolynomialRing(QQ)
sage: q = 1/2
sage: qq = q.denominator()
sage: H = 1 - q*x + q*x*y - x^2*y
sage: Hfac = H.factor()
sage: G = (1 - q*x)/Hfac.unit()
sage: F = FFPD(G, Hfac)
sage: alpha = list(qq*vector([2, 1 - q]))
sage: print alpha
[4, 1]
sage: I = F.smooth_critical_ideal(alpha)
sage: print I
Ideal (y^2 - 2*y + 1, x + 1/4*y - 5/4) of Multivariate Polynomial Ring
in x, y over Rational Field
sage: s = solve(I.gens(), [SR(x) for x in R.gens()], solution_dict=true)
sage: print s
[{y: 1, x: 1}]
sage: p = s[0]
sage: asy = F.asymptotics(p, alpha, 1) # long time
Creating auxiliary functions...
Computing derivatives of auxiliary functions...
Computing derivatives of more auxiliary functions...
Computing second order differential operator actions...
sage: print asy # long time
(1/12*2^(2/3)*sqrt(3)*gamma(1/3)/(pi*r^(1/3)), 1,
1/12*2^(2/3)*sqrt(3)*gamma(1/3)/(pi*r^(1/3)))
sage: print F.relative_error(asy[0], alpha, [1, 2, 4, 8, 16], asy[1]) # long time
Calculating errors table in the form
exponent, scaled Maclaurin coefficient, scaled asymptotic values,
relative errors...
[((4, 1), 0.1875000000, [0.1953794675], [-0.04202382689]), ((8, 2),
0.1523437500, [0.1550727862], [-0.01791367323]), ((16, 4), 0.1221771240,
[0.1230813519], [-0.007400959228]), ((32, 8), 0.09739671811,
[0.09768973377], [-0.003008475766]), ((64, 16), 0.07744253816,
[0.07753639308], [-0.001211929722])]
A multiple point example (Example 6.5 of [RaWi2012]_)::
sage: R.<x,y>= PolynomialRing(QQ)
sage: H = (1 - 2*x - y)**2 * (1 - x - 2*y)**2
sage: Hfac = H.factor()
sage: G = 1/Hfac.unit()
sage: F = FFPD(G, Hfac)
sage: print F
(1, [(x + 2*y - 1, 2), (2*x + y - 1, 2)])
sage: I = F.singular_ideal()
sage: print I
Ideal (x - 1/3, y - 1/3) of Multivariate Polynomial Ring in x, y over
Rational Field
sage: p = {x: 1/3, y: 1/3}
sage: print F.is_convenient_multiple_point(p)
(True, 'convenient in variables [x, y]')
sage: alpha = (var('a'), var('b'))
sage: decomp = F.asymptotic_decomposition(alpha); print decomp # long time
[(0, []), (-1/9*(2*a^2*y^2 - 5*a*b*x*y + 2*b^2*x^2)*r^2/(x^2*y^2) +
1/9*(5*(a + b)*x*y - 6*a*y^2 - 6*b*x^2)*r/(x^2*y^2) - 1/9*(4*x^2 - 5*x*y
+ 4*y^2)/(x^2*y^2), [(x + 2*y - 1, 1), (2*x + y - 1, 1)])]
sage: F1 = decomp[1]
sage: print F1.asymptotics(p, alpha, 2) # long time
(-3*((2*a^2 - 5*a*b + 2*b^2)*r^2 + (a + b)*r +
3)*((1/3)^(-b)*(1/3)^(-a))^r, (1/3)^(-b)*(1/3)^(-a), -3*(2*a^2 - 5*a*b +
2*b^2)*r^2 - 3*(a + b)*r - 9)
sage: alpha = [4, 3]
sage: decomp = F.asymptotic_decomposition(alpha)
sage: F1 = decomp[1]
sage: asy = F1.asymptotics(p, alpha, 2) # long time
sage: print asy # long time
(3*(10*r^2 - 7*r - 3)*2187^r, 2187, 30*r^2 - 21*r - 9)
sage: print F.relative_error(asy[0], alpha, [1, 2, 4, 8], asy[1]) # long time
Calculating errors table in the form
exponent, scaled Maclaurin coefficient, scaled asymptotic values,
relative errors...
[((4, 3), 30.72702332, [0.0000000000], [1.000000000]), ((8, 6),
111.9315678, [69.00000000], [0.3835519207]), ((16, 12), 442.7813138,
[387.0000000], [0.1259793763]), ((32, 24), 1799.879232, [1743.000000],
[0.03160169385])]
"""
#*****************************************************************************
# Copyright (C) 2008 Alexander Raichev <[email protected]>
#
# Distributed under the terms of the GNU General Public License (GPL)
# http://www.gnu.org/licenses/
#*****************************************************************************
from functools import total_ordering
# Sage libraries
from sage.categories.unique_factorization_domains import UniqueFactorizationDomains
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.rings.polynomial.polynomial_ring import is_PolynomialRing
from sage.rings.polynomial.multi_polynomial_ring_generic import is_MPolynomialRing
from sage.symbolic.ring import SR
from sage.geometry.cone import Cone
from sage.calculus.functional import diff
from sage.calculus.functions import jacobian
from sage.calculus.var import function, var
from sage.combinat.cartesian_product import CartesianProduct
from sage.combinat.combinat import stirling_number1
from sage.combinat.permutation import Permutation
from sage.combinat.tuple import UnorderedTuples
from sage.functions.log import exp, log
from sage.functions.other import factorial, gamma, sqrt
from sage.matrix.constructor import matrix
from sage.misc.misc import add
from sage.misc.misc_c import prod
from sage.misc.mrange import cartesian_product_iterator, mrange
from sage.modules.free_module_element import vector
from sage.rings.arith import binomial, xgcd
from sage.rings.all import CC
from sage.rings.fraction_field import FractionField
from sage.rings.integer import Integer
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.rings.rational_field import QQ
from sage.sets.set import Set
from sage.structure.sage_object import SageObject
from sage.symbolic.constants import pi
from sage.symbolic.relation import solve
@total_ordering
class FFPD(object):
r"""
Represents a fraction with factored polynomial denominator (FFPD)
$p/(q_1^{e_1} \cdots q_n^{e_n})$ by storing the parts $p$ and
$[(q_1, e_1), \ldots, (q_n, e_n)]$.
Here $q_1, \ldots, q_n$ are elements of a 0- or multi-variate factorial
polynomial ring $R$ , $q_1, \ldots, q_n$ are distinct irreducible elements
of $R$ , $e_1, \ldots, e_n$ are positive integers, and $p$ is a function
of the indeterminates of $R$ (a Sage Symbolic Expression).
An element $r$ with no polynomial denominator is represented as $[r, (,)]$.
AUTHORS:
- Alexander Raichev (2012-07-26)
"""
def __init__(self, numerator=None, denominator_factored=None,
quotient=None, reduce_=True):
r"""
Create a FFPD instance.
INPUT:
- ``numerator`` - (Optional; default=None) An element $p$ of a
0- or 1-variate factorial polynomial ring $R$.
- ``denominator_factored`` - (Optional; default=None)
A list of the form
$[(q_1, e_1), \ldots, (q_n, e_n)]$ where the $q_1, \ldots, q_n$ are
distinct irreducible elements of $R$ and the $e_i$ are positive
integers.
- ``quotient`` - (Optional; default=None) An element of a field of
fractions of a factorial ring.
- ``reduce_`` - (Optional; default=True) If True, then represent
$p/(q_1^{e_1} \cdots q_n^{e_n})$ in lowest terms.
If False, then won't attempt to divide $p$ by any of the $q_i$.
OUTPUT:
A FFPD instance representing the rational expression
$p/(q_1^{e_1} \cdots q_n^{e_n})$.
To get a non-None output, one of ``numerator`` or ``quotient`` must be
non-None.
EXAMPLES::
sage: from sage.combinat.asymptotics_multivariate_generating_functions import *
sage: R.<x, y> = PolynomialRing(QQ)
sage: df = [x, 1], [y, 1], [x*y+1, 1]
sage: f = FFPD(x, df)
sage: print f
(1, [(y, 1), (x*y + 1, 1)])
sage: ff = FFPD(x, df, reduce_=False)
sage: print ff
(x, [(y, 1), (x, 1), (x*y + 1, 1)])
::
sage: f = FFPD(x + y, [(x + y, 1)])
sage: print f
(1, [])
::
sage: R.<x> = PolynomialRing(QQ)
sage: f = 5*x^3 + 1/x + 1/(x-1) + 1/(3*x^2 + 1)
sage: print FFPD(quotient=f)
(5*x^7 - 5*x^6 + 5/3*x^5 - 5/3*x^4 + 2*x^3 - 2/3*x^2 + 1/3*x - 1/3,
[(x - 1, 1), (x, 1), (x^2 + 1/3, 1)])
::
sage: R.<x, y> = PolynomialRing(QQ)
sage: f = 2*y/(5*(x^3 - 1)*(y + 1))
sage: print FFPD(quotient=f)
(2/5*y, [(y + 1, 1), (x - 1, 1), (x^2 + x + 1, 1)])
::
sage: R.<x, y>= PolynomialRing(QQ)
sage: p = 1/x^2
sage: q = 3*x**2*y
sage: qs = q.factor()
sage: f = FFPD(p/qs.unit(), qs)
sage: print f
(1/(3*x^2), [(y, 1), (x, 2)])
::
sage: R.<x, y> = PolynomialRing(QQ)
sage: f = FFPD(cos(x)*x*y^2, [(x, 2), (y, 1)])
sage: print f
(x*y^2*cos(x), [(y, 1), (x, 2)])
::
sage: R.<x, y> = PolynomialRing(QQ)
sage: G = exp(x + y)
sage: H = (1 - 2*x - y) * (1 - x - 2*y)
sage: a = FFPD(quotient=G/H)
sage: print a
(e^(x + y)/(2*x^2 + 5*x*y + 2*y^2 - 3*x - 3*y + 1), [])
sage: print a._ring
None
sage: b = FFPD(G, H.factor())
sage: print b
(e^(x + y), [(x + 2*y - 1, 1), (2*x + y - 1, 1)])
sage: print b._ring
Multivariate Polynomial Ring in x, y over Rational Field
Singular throws a 'not implemented' error when trying to factor in
a multivariate polynomial ring over an inexact field ::
sage: R.<x, y>= PolynomialRing(CC)
sage: f = (x + 1)/(x*y*(x*y + 1)^2)
sage: FFPD(quotient=f)
Traceback (most recent call last):
...
TypeError: Singular error:
? not implemented
? error occurred in or before STDIN line 17:
`def sage9=factorize(sage8);`
"""
# Attributes are
# self._numerator
# self._denominator_factored
# self._ring
if quotient is not None:
p = quotient.numerator()
q = quotient.denominator()
R = q.parent()
self._numerator = quotient
self._denominator_factored = []
if is_PolynomialRing(R) or is_MPolynomialRing(R):
self._ring = R
if not R(q).is_unit():
# Factor q
try:
df = q.factor()
except NotImplementedError:
# Singular's factor() needs 'proof=False'.
df = q.factor(proof=False)
self._numerator = p/df.unit()
df = sorted([tuple(t) for t in df]) # Sort for consitency.
self._denominator_factored = df
else:
self._ring = None
# Done. No reducing needed, as Sage reduced quotient beforehand.
return
self._numerator = numerator
if denominator_factored:
self._denominator_factored = sorted([tuple(t) for t in
denominator_factored])
self._ring = denominator_factored[0][0].parent()
else:
self._denominator_factored = []
self._ring = None
R = self._ring
if R is not None and numerator in R and reduce_:
# Reduce fraction if possible.
numer = R(self._numerator)
df = self._denominator_factored
new_df = []
for (q, e) in df:
ee = e
quo, rem = numer.quo_rem(q)
while rem == 0 and ee > 0:
ee -= 1
numer = quo
quo, rem = numer.quo_rem(q)
if ee > 0:
new_df.append((q, ee))
self._numerator = numer
self._denominator_factored = new_df
def numerator(self):
r"""
Return the numerator of ``self``.
EXAMPLES::
sage: from sage.combinat.asymptotics_multivariate_generating_functions import *
sage: R.<x,y>= PolynomialRing(QQ)
sage: H = (1 - x - y - x*y)**2*(1-x)
sage: Hfac = H.factor()
sage: G = exp(y)/Hfac.unit()
sage: F = FFPD(G, Hfac)
sage: print F.numerator()
-e^y
"""
return self._numerator
def denominator(self):
r"""
Return the denominator of ``self``.
EXAMPLES::
sage: from sage.combinat.asymptotics_multivariate_generating_functions import *
sage: R.<x,y>= PolynomialRing(QQ)
sage: H = (1 - x - y - x*y)**2*(1-x)
sage: Hfac = H.factor()
sage: G = exp(y)/Hfac.unit()
sage: F = FFPD(G, Hfac)
sage: print F.denominator()
x^3*y^2 + 2*x^3*y + x^2*y^2 + x^3 - 2*x^2*y - x*y^2 - 3*x^2 - 2*x*y
- y^2 + 3*x + 2*y - 1
"""
return prod([q**e for q, e in self.denominator_factored()])
def denominator_factored(self):
r"""
Return the factorization in ``self.ring()`` of the denominator of
``self`` but without the unit part.
EXAMPLES::
sage: from sage.combinat.asymptotics_multivariate_generating_functions import *
sage: R.<x,y>= PolynomialRing(QQ)
sage: H = (1 - x - y - x*y)**2*(1-x)
sage: Hfac = H.factor()
sage: G = exp(y)/Hfac.unit()
sage: F = FFPD(G, Hfac)
sage: print F.denominator_factored()
[(x - 1, 1), (x*y + x + y - 1, 2)]
"""
return self._denominator_factored
def ring(self):
r"""
Return the ring of the denominator of ``self``, which is
None in the case where ``self`` doesn't have a denominator.
EXAMPLES::
sage: from sage.combinat.asymptotics_multivariate_generating_functions import *
sage: R.<x,y>= PolynomialRing(QQ)
sage: H = (1 - x - y - x*y)**2*(1-x)
sage: Hfac = H.factor()
sage: G = exp(y)/Hfac.unit()
sage: F = FFPD(G, Hfac)
sage: print F.ring()
Multivariate Polynomial Ring in x, y over Rational Field
sage: F = FFPD(quotient=G/H)
sage: print F
(e^y/(x^3*y^2 + 2*x^3*y + x^2*y^2 + x^3 - 2*x^2*y - x*y^2 - 3*x^2 -
2*x*y - y^2 + 3*x + 2*y - 1), [])
sage: print F.ring()
None
"""
return self._ring
def dimension(self):
r"""
Return the number of indeterminates of ``self.ring()``.
EXAMPLES::
sage: from sage.combinat.asymptotics_multivariate_generating_functions import *
sage: R.<x,y>= PolynomialRing(QQ)
sage: H = (1 - x - y - x*y)**2*(1-x)
sage: Hfac = H.factor()
sage: G = exp(y)/Hfac.unit()
sage: F = FFPD(G, Hfac)
sage: print F.dimension()
2
"""
R = self.ring()
if is_PolynomialRing(R) or is_MPolynomialRing(R):
return R.ngens()
else:
return None
def list(self):
r"""
Convert ``self`` into a list for printing.
EXAMPLES::
sage: from sage.combinat.asymptotics_multivariate_generating_functions import *
sage: R.<x,y>= PolynomialRing(QQ)
sage: H = (1 - x - y - x*y)**2*(1-x)
sage: Hfac = H.factor()
sage: G = exp(y)/Hfac.unit()
sage: F = FFPD(G, Hfac)
sage: print F # indirect doctest
(-e^y, [(x - 1, 1), (x*y + x + y - 1, 2)])
"""
return (self.numerator(), self.denominator_factored())
def quotient(self):
r"""
Convert ``self`` into a quotient.
EXAMPLES::
sage: from sage.combinat.asymptotics_multivariate_generating_functions import *
sage: R.<x,y>= PolynomialRing(QQ)
sage: H = (1 - x - y - x*y)**2*(1-x)
sage: Hfac = H.factor()
sage: G = exp(y)/Hfac.unit()
sage: F = FFPD(G, Hfac)
sage: print F
(-e^y, [(x - 1, 1), (x*y + x + y - 1, 2)])
sage: print F.quotient()
-e^y/(x^3*y^2 + 2*x^3*y + x^2*y^2 + x^3 - 2*x^2*y - x*y^2 - 3*x^2 -
2*x*y - y^2 + 3*x + 2*y - 1)
"""
return self.numerator()/self.denominator()
def __str__(self):
r"""
Returns a string representation of ``self``
EXAMPLES::
sage: from sage.combinat.asymptotics_multivariate_generating_functions import *
sage: R.<x,y> = PolynomialRing(QQ)
sage: f = FFPD(x + y, [(y, 1), (x, 1)])
sage: print f
(x + y, [(y, 1), (x, 1)])
"""
return str(self.list())
def __eq__(self, other):
r"""
Two FFPD instances are equal iff they represent the same
fraction.
EXAMPLES::
sage: from sage.combinat.asymptotics_multivariate_generating_functions import *
sage: R.<x, y>= PolynomialRing(QQ)
sage: df = [x, 1], [y, 1], [x*y+1, 1]
sage: f = FFPD(x, df)
sage: ff = FFPD(x, df, reduce_=False)
sage: f == ff
True
sage: g = FFPD(y, df)
sage: g == f
False
::
sage: R.<x, y> = PolynomialRing(QQ)
sage: G = exp(x + y)
sage: H = (1 - 2*x - y) * (1 - x - 2*y)
sage: a = FFPD(quotient=G/H)
sage: b = FFPD(G, H.factor())
sage: bool(a == b)
True
"""
return self.quotient() == other.quotient()
def __ne__(self, other):
r"""
EXAMPLES::
sage: from sage.combinat.asymptotics_multivariate_generating_functions import *
sage: R.<x, y>= PolynomialRing(QQ)
sage: df = [x, 1], [y, 1], [x*y+1, 1]
sage: f = FFPD(x, df)
sage: ff = FFPD(x, df, reduce_=False)
sage: f != ff
False
sage: g = FFPD(y, df)
sage: g != f # indirect doctest
True
"""
return not (self == other)
def __lt__(self, other):
r"""
FFPD A is less than FFPD B iff
(the denominator factorization of A is shorter than that of B) or
(the denominator factorization lengths are equal and
the denominator of A is less than that of B in their ring) or
(the denominator factorization lengths are equal and the
denominators are equal and the numerator of A is less than that of B
in their ring).
EXAMPLES::
sage: from sage.combinat.asymptotics_multivariate_generating_functions import *
sage: R.<x, y>= PolynomialRing(QQ)
sage: df = [x, 1], [y, 1], [x*y+1, 1]
sage: f = FFPD(x, df)
sage: ff = FFPD(x, df, reduce_=False)
sage: g = FFPD(y, df)
sage: h = FFPD(exp(x), df)
sage: i = FFPD(sin(x + 2), df)
sage: print f, ff
(1, [(y, 1), (x*y + 1, 1)]) (x, [(y, 1), (x, 1), (x*y + 1, 1)])
sage: print f < ff
True
sage: print f < g
True
sage: print g < h
True
sage: print h < i
False
"""
sn = self.numerator()
on = other.numerator()
sdf = self.denominator_factored()
odf = other.denominator_factored()
sd = self.denominator()
od = other.denominator()
return bool(len(sdf) < len(odf) or\
(len(sdf) == len(odf) and sd < od) or\
(len(sdf) == len(odf) and sd == od and sn < on))
def univariate_decomposition(self):
r"""
Return the usual univariate partial fraction decomposition
of ``self`` as a FFPDSum instance.
Assume that ``self`` lies in the field of fractions
of a univariate factorial polynomial ring.
EXAMPLES::
sage: from sage.combinat.asymptotics_multivariate_generating_functions import *
One variable::
sage: R.<x> = PolynomialRing(QQ)
sage: f = 5*x^3 + 1/x + 1/(x-1) + 1/(3*x^2 + 1)
sage: print f
(15*x^7 - 15*x^6 + 5*x^5 - 5*x^4 + 6*x^3 - 2*x^2 + x - 1)/(3*x^4 -
3*x^3 + x^2 - x)
sage: decomp = FFPD(quotient=f).univariate_decomposition()
sage: print decomp
[(5*x^3, []), (1, [(x - 1, 1)]), (1, [(x, 1)]),
(1/3, [(x^2 + 1/3, 1)])]
sage: print decomp.sum().quotient() == f
True
One variable with numerator in symbolic ring::
sage: R.<x> = PolynomialRing(QQ)
sage: f = 5*x^3 + 1/x + 1/(x-1) + exp(x)/(3*x^2 + 1)
sage: print f
e^x/(3*x^2 + 1) + ((5*(x - 1)*x^3 + 2)*x - 1)/((x - 1)*x)
sage: decomp = FFPD(quotient=f).univariate_decomposition()
sage: print decomp
[(e^x/(3*x^2 + 1) + ((5*(x - 1)*x^3 + 2)*x - 1)/((x - 1)*x), [])]
One variable over a finite field::
sage: R.<x> = PolynomialRing(GF(2))
sage: f = 5*x^3 + 1/x + 1/(x-1) + 1/(3*x^2 + 1)
sage: print f
(x^6 + x^4 + 1)/(x^3 + x)
sage: decomp = FFPD(quotient=f).univariate_decomposition()
sage: print decomp
[(x^3, []), (1, [(x, 1)]), (x, [(x + 1, 2)])]
sage: print decomp.sum().quotient() == f
True
One variable over an inexact field::
sage: R.<x> = PolynomialRing(CC)
sage: f = 5*x^3 + 1/x + 1/(x-1) + 1/(3*x^2 + 1)
sage: print f
(15.0000000000000*x^7 - 15.0000000000000*x^6 + 5.00000000000000*x^5
- 5.00000000000000*x^4 + 6.00000000000000*x^3 -
2.00000000000000*x^2 + x - 1.00000000000000)/(3.00000000000000*x^4
- 3.00000000000000*x^3 + x^2 - x)
sage: decomp = FFPD(quotient=f).univariate_decomposition()
sage: print decomp
[(5.00000000000000*x^3, []), (1.00000000000000,
[(x - 1.00000000000000, 1)]), (-0.288675134594813*I,
[(x - 0.577350269189626*I, 1)]), (1.00000000000000, [(x, 1)]),
(0.288675134594813*I, [(x + 0.577350269189626*I, 1)])]
sage: print decomp.sum().quotient() == f # Rounding error coming
False
NOTE::
Let $f = p/q$ be a rational expression where $p$ and $q$ lie in a
univariate factorial polynomial ring $R$.
Let $q_1^{e_1} \cdots q_n^{e_n}$ be the
unique factorization of $q$ in $R$ into irreducible factors.
Then $f$ can be written uniquely as
(*) $p_0 + \sum_{i=1}^{m} p_i/ q_i^{e_i}$,
for some $p_j \in R$.
I call (*) the *usual partial fraction decomposition* of f.
AUTHORS:
- Robert Bradshaw (2007-05-31)
- Alexander Raichev (2012-06-25)
"""
if self.dimension() is None or self.dimension() > 1:
return FFPDSum([self])
R = self.ring()
p = self.numerator()
q = self.denominator()
if p in R:
whole, p = p.quo_rem(q)
else:
whole = p
p = R(1)
df = self.denominator_factored()
decomp = [FFPD(whole, [])]
for (a, m) in df:
numer = p * prod([b**n for (b, n) in df if b != a]).\
inverse_mod(a**m) % (a**m)
# The inverse exists because the product and a**m
# are relatively prime.
decomp.append(FFPD(numer, [(a, m)]))
return FFPDSum(decomp)
def nullstellensatz_certificate(self):
r"""
Let $[(q_1, e_1), \ldots, (q_n, e_n)]$ be the denominator factorization
of ``self``.
Return a list of polynomials $h_1, \ldots, h_m$ in ``self.ring()``
that satisfies $h_1 q_1 + \cdots + h_m q_n = 1$ if it exists.
Otherwise return None.
Only works for multivariate ``self``.
EXAMPLES::
sage: from sage.combinat.asymptotics_multivariate_generating_functions import *
sage: R.<x, y> = PolynomialRing(QQ)
sage: G = sin(x)
sage: H = x^2 * (x*y + 1)
sage: f = FFPD(G, H.factor())
sage: L = f.nullstellensatz_certificate()
sage: print L
[y^2, -x*y + 1]
sage: df = f.denominator_factored()
sage: sum([L[i]*df[i][0]**df[i][1] for i in xrange(len(df))]) == 1
True
::
sage: f = 1/(x*y)
sage: L = FFPD(quotient=f).nullstellensatz_certificate()
sage: L is None
True
"""
R = self.ring()
if R is None:
return None
df = self.denominator_factored()
J = R.ideal([q**e for q, e in df])
if R(1) in J:
return R(1).lift(J)
else:
return None
def nullstellensatz_decomposition(self):
r"""
Return a Nullstellensatz decomposition of ``self`` as a FFPDSum
instance.
Recursive.
Only works for multivariate ``self``.
EXAMPLES::
sage: from sage.combinat.asymptotics_multivariate_generating_functions import *
sage: R.<x, y> = PolynomialRing(QQ)
sage: f = 1/(x*(x*y + 1))
sage: decomp = FFPD(quotient=f).nullstellensatz_decomposition()
sage: print decomp
[(0, []), (1, [(x, 1)]), (-y, [(x*y + 1, 1)])]
sage: decomp.sum().quotient() == f
True
sage: for r in decomp:
... L = r.nullstellensatz_certificate()
... L is None
...
True
True
True
::
sage: R.<x, y> = PolynomialRing(QQ)
sage: G = sin(y)
sage: H = x*(x*y + 1)
sage: f = FFPD(G, H.factor())
sage: decomp = f.nullstellensatz_decomposition()
sage: print decomp
[(0, []), (sin(y), [(x, 1)]), (-y*sin(y), [(x*y + 1, 1)])]
sage: bool(decomp.sum().quotient() == G/H)
True
sage: for r in decomp:
... L = r.nullstellensatz_certificate()
... L is None
...
True
True
True
NOTE::
Let $f = p/q$ where $q$ lies in a $d$ -variate polynomial ring $K[X]$ for some field $K$ and $d \ge 1$.
Let $q_1^{e_1} \cdots q_n^{e_n}$ be the
unique factorization of $q$ in $K[X]$ into irreducible factors and
let $V_i$ be the algebraic variety $\{x \in L^d: q_i(x) = 0\}$ of
$q_i$ over the algebraic closure $L$ of $K$.
By [Raic2012]_, $f$ can be written as
(*) $\sum p_A/\prod_{i \in A} q_i^{e_i}$,
where the $p_A$ are products of $p$ and elements in $K[X]$ and
the sum is taken over all subsets
$A \subseteq \{1, \ldots, m\}$ such that
$\cap_{i\in A} T_i \neq \emptyset$.
I call (*) a *Nullstellensatz decomposition* of $f$.
Nullstellensatz decompositions are not unique.
The algorithm used comes from [Raic2012]_.
"""
L = self.nullstellensatz_certificate()
if L is None:
# No decomposing possible.
return FFPDSum([self])
# Otherwise decompose recursively.
decomp = FFPDSum()
p = self.numerator()
df = self.denominator_factored()
m = len(df)
iteration1 = FFPDSum([FFPD(p*L[i],[df[j]
for j in xrange(m) if j != i])
for i in xrange(m) if L[i] != 0])
# Now decompose each FFPD of iteration1.
for r in iteration1:
decomp.extend(r.nullstellensatz_decomposition())
# Simplify and return result.
return decomp.combine_like_terms().whole_and_parts()
def algebraic_dependence_certificate(self):
r"""
Return the ideal $J$ of annihilating polynomials for the set
of polynomials ``[q**e for (q, e) in self.denominator_factored()]``,
which could be the zero ideal.
The ideal $J$ lies in a polynomial ring over the field
``self.ring().base_ring()`` that has
``m = len(self.denominator_factored())`` indeterminates.
Return None if ``self.ring()`` is None.
EXAMPLES::
sage: from sage.combinat.asymptotics_multivariate_generating_functions import *
sage: R.<x, y> = PolynomialRing(QQ)
sage: f = 1/(x^2 * (x*y + 1) * y^3)
sage: ff = FFPD(quotient=f)
sage: J = ff.algebraic_dependence_certificate()
sage: print J
Ideal (1 - 6*T2 + 15*T2^2 - 20*T2^3 + 15*T2^4 - T0^2*T1^3 -
6*T2^5 + T2^6) of Multivariate Polynomial Ring in
T0, T1, T2 over Rational Field
sage: g = J.gens()[0]
sage: df = ff.denominator_factored()
sage: g(*(q**e for q, e in df)) == 0
True
::
sage: R.<x, y> = PolynomialRing(QQ)
sage: G = exp(x + y)
sage: H = x^2 * (x*y + 1) * y^3
sage: ff = FFPD(G, H.factor())
sage: J = ff.algebraic_dependence_certificate()
sage: print J
Ideal (1 - 6*T2 + 15*T2^2 - 20*T2^3 + 15*T2^4 - T0^2*T1^3 -
6*T2^5 + T2^6) of Multivariate Polynomial Ring in
T0, T1, T2 over Rational Field
sage: g = J.gens()[0]
sage: df = ff.denominator_factored()
sage: g(*(q**e for q, e in df)) == 0
True
::
sage: f = 1/(x^3 * y^2)
sage: J = FFPD(quotient=f).algebraic_dependence_certificate()
sage: print J
Ideal (0) of Multivariate Polynomial Ring in T0, T1 over
Rational Field
::
sage: f = sin(1)/(x^3 * y^2)
sage: J = FFPD(quotient=f).algebraic_dependence_certificate()
sage: print J
None
"""
R = self.ring()
if R is None:
return None
df = self.denominator_factored()
if not df:
return R.ideal() # The zero ideal.
m = len(df)
F = R.base_ring()
Xs = list(R.gens())
d = len(Xs)
# Expand R by 2*m new variables.
S = 'S'
while S in [str(x) for x in Xs]:
S = S + 'S'
Ss = [S + str(i) for i in xrange(m)]
T = 'T'
while T in [str(x) for x in Xs]:
T = T + 'T'
Ts = [T + str(i) for i in xrange(m)]
Vs = [str(x) for x in Xs] + Ss + Ts
RR = PolynomialRing(F, Vs)
Xs = RR.gens()[:d]
Ss = RR.gens()[d: d + m]
Ts = RR.gens()[d + m: d + 2 * m]
# Compute the appropriate elimination ideal.
J = RR.ideal([ Ss[j] - RR(df[j][0]) for j in xrange(m)] +\
[ Ss[j]**df[j][1] - Ts[j] for j in xrange(m)])
J = J.elimination_ideal(Xs + Ss)
# Coerce J into the polynomial ring in the indeteminates Ts[m:].
# I choose the negdeglex order because i find it useful in my work.
RRR = PolynomialRing(F, [str(t) for t in Ts], order ='negdeglex')
return RRR.ideal(J)
def algebraic_dependence_decomposition(self, whole_and_parts=True):
r"""
Return an algebraic dependence decomposition of ``self`` as a FFPDSum
instance.
Recursive.
EXAMPLES::
sage: from sage.combinat.asymptotics_multivariate_generating_functions import *
sage: R.<x, y> = PolynomialRing(QQ)
sage: f = 1/(x^2 * (x*y + 1) * y^3)
sage: ff = FFPD(quotient=f)
sage: decomp = ff.algebraic_dependence_decomposition()
sage: print decomp
[(0, []), (-x, [(x*y + 1, 1)]), (x^2*y^2 - x*y + 1,
[(y, 3), (x, 2)])]
sage: print decomp.sum().quotient() == f
True
sage: for r in decomp:
... J = r.algebraic_dependence_certificate()
... J is None or J == J.ring().ideal() # The zero ideal
...
True
True
True
::
sage: R.<x, y> = PolynomialRing(QQ)
sage: G = sin(x)
sage: H = x^2 * (x*y + 1) * y^3
sage: f = FFPD(G, H.factor())
sage: decomp = f.algebraic_dependence_decomposition()
sage: print decomp
[(0, []), (x^4*y^3*sin(x), [(x*y + 1, 1)]),
(-(x^5*y^5 - x^4*y^4 + x^3*y^3 - x^2*y^2 + x*y - 1)*sin(x),
[(y, 3), (x, 2)])]
sage: bool(decomp.sum().quotient() == G/H)
True
sage: for r in decomp:
... J = r.algebraic_dependence_certificate()
... J is None or J == J.ring().ideal()
...
True
True
True
NOTE::
Let $f = p/q$ where $q$ lies in a $d$ -variate polynomial ring
$K[X]$ for some field $K$.
Let $q_1^{e_1} \cdots q_n^{e_n}$ be the
unique factorization of $q$ in $K[X]$ into irreducible factors and
let $V_i$ be the algebraic variety $\{x\in L^d: q_i(x) = 0\}$ of
$q_i$ over the algebraic closure $L$ of $K$.
By [Raic2012]_, $f$ can be written as
(*) $\sum p_A/\prod_{i \in A} q_i^{b_i}$,
where the $b_i$ are positive integers, each $p_A$ is a products
of $p$ and an element in $K[X]$,
and the sum is taken over all subsets
$A \subseteq \{1, \ldots, m\}$ such that $|A| \le d$ and
$\{q_i : i\in A\}$ is algebraically independent.
I call (*) an *algebraic dependence decomposition* of $f$.
Algebraic dependence decompositions are not unique.
The algorithm used comes from [Raic2012]_.
"""
J = self.algebraic_dependence_certificate()
if not J:
# No decomposing possible.
return FFPDSum([self])
# Otherwise decompose recursively.