A local construction of the Smith normal form of a matrix polynomial

TL;DR

This paper introduces a novel local approach to compute the Smith normal form of a regular matrix polynomial by separately handling irreducible factors and then combining them, differing from traditional methods.

Contribution

It presents a new algorithm that computes local Smith forms for each irreducible factor and merges them, improving upon previous unimodular operation-based algorithms.

Findings

Effective in exact arithmetic for various test cases

Computes local Smith forms for each irreducible factor

Combines local forms into a global Smith form

Abstract

We present an algorithm for computing a Smith form with multipliers of a regular matrix polynomial over a field. This algorithm differs from previous ones in that it computes a local Smith form for each irreducible factor in the determinant separately and then combines them into a global Smith form, whereas other algorithms apply a sequence of unimodular row and column operations to the original matrix. The performance of the algorithm in exact arithmetic is reported for several test cases.