Work

A Simple and Fast Algorithm for Computing the $N$-th Term of a Linearly Recurrent Sequence

Year: 2020

Type: preprint

Abstract: We present a simple and fast algorithm for computing the $N$-th term of a given linearly recurrent sequence. Our new algorithm uses $O(\mathsf{M}(d) \log N)$ arithmetic operations, where $d$ is the order of the recurrence, and $\mathsf{M}(d)$ denotes the number of arithmetic operations for computing the product of two polynomials of degree $d$. The state-of-the-art algorithm, due to Charles Fiduccia (1985), has the same arithmetic complexity up to a constant factor. Our algorithm is simpler, faster and obtained by a totally different method. We also discuss several algorithmic applications, notably to polynomial modular exponentiation, powering of matrices and high-order lifting.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2008.08822

Authors: Bostan, Alin, Mori, Ryuhei

Institutions: Tokyo Institute of Technology, Institut national de recherche en sciences et technologies du numérique, Centre Inria de Saclay

Language: English

Topic: Polynomial and algebraic computation

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

Funders: Agence Nationale de la Recherche, Japan Society for the Promotion of Science, Precursory Research for Embryonic Science and Technology

Awards: G3010527862, Operational characterization of quantum nonlocality by Boolean Fourier analysis, Design of exponential-time quantum algorithms, Deciding irrationality and transcendence, Computational Complexity of Minimum Description Size Problems, G8973288803

OpenAlex ID: W3068385242

Cites: 56

Cited by: 1

Related to: 10