Probabilistic Analysis of Block Wiedemann for Leading Invariant Factors

TL;DR

This paper analyzes the probability that the block Wiedemann algorithm accurately computes leading invariant factors, providing bounds and an algorithm to optimize block size for high probability correctness across different matrices.

Contribution

It introduces a probabilistic analysis of the block Wiedemann algorithm, deriving tight bounds and an algorithm for selecting block size to ensure high probability accuracy.

Findings

High probability of correct leading invariant factors with slightly larger block size.

Provides a method to compute probability bounds for specific matrices.

Improves worst-case probability bounds using partial invariant factor information.

Abstract

We determine the probability, structure dependent, that the block Wiedemann algorithm correctly computes leading invariant factors. This leads to a tight lower bound for the probability, structure independent. We show, using block size slightly larger than $r$, that the leading $r$ invariant factors are computed correctly with high probability over any field. Moreover, an algorithm is provided to compute the probability bound for a given matrix size and thus to select the block size needed to obtain the desired probability. The worst case probability bound is improved, post hoc, by incorporating the partial information about the invariant factors.