6 research outputs found
Low Complexity Algorithms for Linear Recurrences
We consider two kinds of problems: the computation of polynomial and rational
solutions of linear recurrences with coefficients that are polynomials with
integer coefficients; indefinite and definite summation of sequences that are
hypergeometric over the rational numbers. The algorithms for these tasks all
involve as an intermediate quantity an integer (dispersion or root of an
indicial polynomial) that is potentially exponential in the bit size of their
input. Previous algorithms have a bit complexity that is at least quadratic in
. We revisit them and propose variants that exploit the structure of
solutions and avoid expanding polynomials of degree . We give two
algorithms: a probabilistic one that detects the existence or absence of
nonzero polynomial and rational solutions in bit
operations; a deterministic one that computes a compact representation of the
solution in bit operations. Similar speed-ups are obtained in
indefinite and definite hypergeometric summation. We describe the results of an
implementation.Comment: This is the author's version of the work. It is posted here by
permission of ACM for your personal use. Not for redistributio
Fast Computation of the -th Term of a -Holonomic Sequence and Applications
33 pages. Long version of the conference paper Computing the -th term of a -holonomic sequence. Proceedings ISSAC'20, pp. 46â53, ACM Press, 2020 (https://hal.inria.fr/hal-02882885)International audienceIn 1977, Strassen invented a famous baby-step/giant-step algorithm that computes the factorial in arithmetic complexity quasi-linear in . In 1988, the Chudnovsky brothers generalized Strassenâs algorithm to the computation of the -th term of any holonomic sequence in essentially the same arithmetic complexity. We design -analogues of these algorithms. We first extend Strassenâs algorithm to the computation of the -factorial of , then Chudnovskys' algorithm to the computation of the -th term of any -holonomic sequence. Both algorithms work in arithmetic complexity quasi-linear in ; surprisingly, they are simpler than their analogues in the holonomic case. We provide a detailed cost analysis, in both arithmetic and bit complexity models. Moreover, we describe various algorithmic consequences, including the acceleration of polynomial and rational solving of linear -differential equations, and the fast evaluation of large classes of polynomials, including a family recently considered by Nogneng and Schost
Algorithmes rapides pour les polynÎmes, séries formelles et matrices
Notes d'un cours dispensĂ© aux JournĂ©es Nationales du Calcul Formel 2010International audienceLe calcul formel calcule des objets mathĂ©matiques exacts. Ce cours explore deux directions : la calculabilitĂ© et la complexitĂ©. La calculabilitĂ© Ă©tudie les classes d'objets mathĂ©matiques sur lesquelles des rĂ©ponses peuvent ĂȘtre obtenues algorithmiquement. La complexitĂ© donne ensuite des outils pour comparer des algorithmes du point de vue de leur efficacitĂ©. Ce cours passe en revue l'algorithmique efficace sur les objets fondamentaux que sont les entiers, les polynĂŽmes, les matrices, les sĂ©ries et les solutions d'Ă©quations diffĂ©rentielles ou de rĂ©currences linĂ©aires. On y montre que de nombreuses questions portant sur ces objets admettent une rĂ©ponse en complexitĂ© (quasi-)optimale, en insistant sur les principes gĂ©nĂ©raux de conception d'algorithmes efficaces. Ces notes sont dĂ©rivĂ©es du cours " Algorithmes efficaces en calcul formel " du Master Parisien de Recherche en Informatique (2004-2010), co-Ă©crit avec FrĂ©dĂ©ric Chyzak, Marc Giusti, Romain Lebreton, Bruno Salvy et Ăric Schost. Le support de cours complet est disponible Ă l'url https://wikimpri.dptinfo.ens-cachan.fr/doku.php?id=cours:c-2-2
Low Complexity Algorithms for Linear Recurrences
We consider two kinds of problems: the computation of polynomial and rational solutions of linear recurrences with coefficients that are polynomials with integer coefficients; indefinite and definite summation of sequences that are hypergeometric over the rational numbers. The algorithms for these tasks all involve as an intermediate quantity an integer N (dispersion or root of an indicial polynomial) that is potentially exponential in the bit size of their input. Previous algorithms have a bit complexity that is at least quadratic in N. We revisit them and propose variants that exploit the structure of solutions and avoid expanding polynomials of degree N. We give two algorithms: a probabilistic one that detects the existence or absence of nonzero polynomial and rational solutions in O ( â N log 2 N) bit operations; a deterministic one that computes a compact representation of the solution in O(N log 3 N) bit operations. Similar speedups are obtained in indefinite and definite hypergeometric summation. We describe the results of an implementation