On a perturbation analysis of Higham squared maximum Gaussian elimination growth matrices

TL;DR

This paper investigates the sensitivity of matrices with maximum growth factor in Gaussian elimination to perturbations, revealing high sensitivity and demonstrating how small changes can reduce growth significantly.

Contribution

It provides a perturbation analysis of Higham's maximum growth matrices, highlighting their high sensitivity and proposing strategies to reduce growth through subtle entry modifications.

Findings

Maximum growth factor matrices are highly sensitive to perturbations.

Small entry changes can significantly reduce the growth factor.

Theoretical and empirical evidence supports the sensitivity analysis.

Abstract

Gaussian elimination is the most popular technique for solving a dense linear system. Large errors in this procedure can occur in floating point arithmetic when the matrix's growth factor is large. In the study of numerical linear algebra, it is often valuable to study and characterize the worst case examples. To this end, in their 1989 paper, Higham and Higham characterized the complete set of real n by n matrices that achieves the maximum growth factor under partial pivoting. Left undone is a sensitivity analysis for these matrices under perturbations. The growth factor of these and nearby matrices is the subject of this work. Through theoretical insights and empirical results, we illustrate the high sensitivity of the growth factor of these matrices to perturbations and show how subtle changes can be strategically applied to matrix entries to significantly reduce the growth.