89 research outputs found
Efficient Algorithms for Mixed Creative Telescoping
Creative telescoping is a powerful computer algebra paradigm -initiated by
Doron Zeilberger in the 90's- for dealing with definite integrals and sums with
parameters. We address the mixed continuous-discrete case, and focus on the
integration of bivariate hypergeometric-hyperexponential terms. We design a new
creative telescoping algorithm operating on this class of inputs, based on a
Hermite-like reduction procedure. The new algorithm has two nice features: it
is efficient and it delivers, for a suitable representation of the input, a
minimal-order telescoper. Its analysis reveals tight bounds on the sizes of the
telescoper it produces.Comment: To be published in the proceedings of ISSAC'1
Constructing minimal telescopers for rational functions in three discrete variables
We present a new algorithm for constructing minimal telescopers for rational
functions in three discrete variables. This is the first discrete
reduction-based algorithm that goes beyond the bivariate case. The termination
of the algorithm is guaranteed by a known existence criterion of telescopers.
Our approach has the important feature that it avoids the potentially costly
computation of certificates. Computational experiments are also provided so as
to illustrate the efficiency of our approach
Differential Equations for Algebraic Functions
It is classical that univariate algebraic functions satisfy linear
differential equations with polynomial coefficients. Linear recurrences follow
for the coefficients of their power series expansions. We show that the linear
differential equation of minimal order has coefficients whose degree is cubic
in the degree of the function. We also show that there exists a linear
differential equation of order linear in the degree whose coefficients are only
of quadratic degree. Furthermore, we prove the existence of recurrences of
order and degree close to optimal. We study the complexity of computing these
differential equations and recurrences. We deduce a fast algorithm for the
expansion of algebraic series
Symbolic-Numeric Tools for Analytic Combinatorics in Several Variables
Analytic combinatorics studies the asymptotic behaviour of sequences through
the analytic properties of their generating functions. This article provides
effective algorithms required for the study of analytic combinatorics in
several variables, together with their complexity analyses. Given a
multivariate rational function we show how to compute its smooth isolated
critical points, with respect to a polynomial map encoding asymptotic
behaviour, in complexity singly exponential in the degree of its denominator.
We introduce a numerical Kronecker representation for solutions of polynomial
systems with rational coefficients and show that it can be used to decide
several properties (0 coordinate, equal coordinates, sign conditions for real
solutions, and vanishing of a polynomial) in good bit complexity. Among the
critical points, those that are minimal---a property governed by inequalities
on the moduli of the coordinates---typically determine the dominant asymptotics
of the diagonal coefficient sequence. When the Taylor expansion at the origin
has all non-negative coefficients (known as the `combinatorial case') and under
regularity conditions, we utilize this Kronecker representation to determine
probabilistically the minimal critical points in complexity singly exponential
in the degree of the denominator, with good control over the exponent in the
bit complexity estimate. Generically in the combinatorial case, this allows one
to automatically and rigorously determine asymptotics for the diagonal
coefficient sequence. Examples obtained with a preliminary implementation show
the wide applicability of this approach.Comment: As accepted to proceedings of ISSAC 201
- …