On Recurrence Relations of Multi-dimensional Sequences

TL;DR

This paper introduces a novel algorithm for efficiently computing linear recurrence relations of multi-dimensional sequences by leveraging dual module computations, improving over existing matrix-based methods.

Contribution

It proposes a new approach converting the problem into dual module computations, with complexity analysis and experimental validation showing significant speedups.

Findings

The algorithm effectively reduces matrix sizes in recurrence relation computation.

Experimental results demonstrate faster performance compared to traditional methods.

Complexity bounds are established for the proposed algorithm.

Abstract

In this paper, we present a new algorithm for computing the linear recurrence relations of multi-dimensional sequences. Existing algorithms for computing these relations arise in computational algebra and include constructing structured matrices and computing their kernels. The challenging problem is to reduce the size of the corresponding matrices. In this paper, we show how to convert the problem of computing recurrence relations of multi-dimensional sequences into computing the orthogonal of certain ideals as subvector spaces of the dual module of polynomials. We propose an algorithm using efficient dual module computation algorithms. We present a complexity bound for this algorithm, carry on experiments using Maple implementation, and discuss the cases when using this algorithm is much faster than the existing approaches.