A fast algorithm for computing the Smith normal form with multipliers for a nonsingular integer matrix

TL;DR

This paper presents a randomized algorithm that efficiently computes the Smith normal form and multipliers of a nonsingular integer matrix, with applications to quickly finding the fractional inverse.

Contribution

It introduces a fast Las Vegas algorithm for computing Smith multipliers with explicit bounds and novel use of the Smith massager, improving efficiency over previous methods.

Findings

Expected running time comparable to matrix multiplication

Explicit bounds on unimodular multiplier entries

Application to fast fractional inverse computation

Abstract

A Las Vegas randomized algorithm is given to compute the Smith multipliers for a nonsingular integer matrix $A$, that is, unimodular matrices $U$ and $V$ such that $AV=US$, with $S$ the Smith normal form of $A$. The expected running time of the algorithm is about the same as required to multiply together two matrices of the same dimension and size of entries as $A$. Explicit bounds are given for the size of the entries in both unimodular multipliers. The main tool used by the algorithm is the Smith massager, a relaxed version of $V$, the unimodular matrix specifying the column operations of the Smith computation. From the perspective of efficiency, the main tools used are fast linear solving and partial linearization of integer matrices. As an application of the Smith with multipliers algorithm, a fast algorithm is given to find the fractional part of the inverse of the input matrix.