Distribution of the number of pivots needed using Gaussian elimination with partial pivoting on random matrices

TL;DR

This paper analyzes the distribution of pivot movements in Gaussian elimination with partial pivoting on random matrices, providing theoretical descriptions, comparisons, and new conjectures on spectral properties.

Contribution

It offers the first full distributional analysis of pivot counts in GEPP for specific Haar random ensembles and introduces new ensembles with fixed pivot counts and sparsity.

Findings

Distribution of pivot movements characterized for certain random ensembles

Comparison of pivot distributions across different randomized transformations

Conjecture on universality class of sparse random matrices and their spectral density

Abstract

Gaussian elimination with partial pivoting (GEPP) is a widely used method to solve dense linear systems. Each GEPP step uses a row transposition pivot movement if needed to ensure the leading pivot entry is maximal in magnitude for the leading column of the remaining untriangularized subsystem. We will use theoretical and numerical approaches to study how often this pivot movement is needed. We provide full distributional descriptions for the number of pivot movements needed using GEPP using particular Haar random ensembles, as well as compare these models to other common transformations from randomized numerical linear algebra. Additionally, we introduce new random ensembles with fixed pivot movement counts and fixed sparsity, $\alpha$. Experiments estimating the empirical spectral density (ESD) of these random ensembles leads to a new conjecture on a universality class of random matrices with fixed sparsity whose scaled ESD converges to a measure on the complex unit disk that depends on $\alpha$ and is an interpolation of the uniform measure on the unit disk and the Dirac measure at the origin.