Low Complexity Algorithms for Linear Recurrences

Alin Bostan (1) , Frédéric Chyzak (1) , Thomas Cluzeau (2) , Bruno Salvy (1)

1 ALGO - Algorithms
2 CAFE - Computer algebra and functional equations

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.

Domaines: Calcul formel [cs.SC]

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⟩