Fast computation of the N-th term of a q-holonomic sequence and applications. (March 2023)

• Record Type: Journal Article
• Authors: Bostan, Alin; Yurkevich, Sergey
• Is Part Of: Journal of symbolic computation. Volume 115(2023)
• Page Start: 96
• Page End: 123
• Publication Date: 2023-03
• Subjects: Algorithms -- Complexity -- q-factorial -- q-holonomic sequences -- Polynomial evaluation; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2022.07.008
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: In 1977, Strassen invented a famous baby-step/giant-step algorithm that computes the factorial N ! in arithmetic complexity quasi-linear in N . In 1988, the Chudnovsky brothers generalized Strassen's algorithm to the computation of the N -th term of any holonomic sequence in essentially the same arithmetic complexity. We design q -analogues of these algorithms. We first extend Strassen's algorithm to the computation of the q -factorial of N, then Chudnovskys' algorithm to the computation of the N -th term of any q -holonomic sequence. Both algorithms work in arithmetic complexity quasi-linear in N ; 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 q -differential equations, and the fast evaluation of large classes of polynomials, including a family recently considered by Nogneng and Schost.