Low Complexity Algorithms for Linear Recurrences - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Conference Papers Year : 2006

Low Complexity Algorithms for Linear Recurrences

Alin Bostan
  • Function : Author
  • PersonId : 831654
Frédéric Chyzak
Thomas Cluzeau
Bruno Salvy

Abstract

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~$O(\sqrt{N}\log^{2}N)$ bit operations; a deterministic one that computes a compact representation of the solution in~$O(N\log^{3}N)$ bit operations. Similar speed-ups are obtained in indefinite and definite hypergeometric summation. We describe the results of an implementation.
Fichier principal
Vignette du fichier
BoChClSa06.pdf (216.94 Ko) Télécharger le fichier
Loading...

Dates and versions

inria-00068922 , version 1 (15-05-2006)

Identifiers

Cite

Alin Bostan, Frédéric Chyzak, Thomas Cluzeau, Bruno Salvy. Low Complexity Algorithms for Linear Recurrences. ISSAC International Symposium on Symbolic and Algebraic Computations, Jul 2006, Genova, Italy, Italy. pp.31-38, ⟨10.1145/1145768.1145781⟩. ⟨inria-00068922⟩

Collections

INRIA INRIA2
99 View
122 Download

Altmetric

Share

Gmail Facebook X LinkedIn More