Growth factors of orthogonal matrices and local behavior of Gaussian elimination with partial and complete pivoting

TL;DR

This paper investigates the growth factors of orthogonal matrices during Gaussian elimination with partial and complete pivoting, analyzing their behavior and differences to understand why large growth is rare in practice.

Contribution

It provides new insights into the growth factors of orthogonal matrices under GEPP and GECP, including their comparative behavior and local dominance near initial matrices.

Findings

GEPP growth factors for Gaussian matrices are mostly polynomial with high probability

GECP has tighter bounds on growth factors, improving understanding of worst-case scenarios

Small growth differences between GEPP and GECP dominate behavior near initial matrices

Abstract

Gaussian elimination (GE) is the most used dense linear solver. Error analysis of GE with selected pivoting strategies on well-conditioned systems can focus on studying the behavior of growth factors. Although exponential growth is possible with GE with partial pivoting (GEPP), growth tends to stay much smaller in practice. Support for this behavior was provided recently by Huang and Tikhomirov's average-case analysis of GEPP, which showed GEPP growth factors for Gaussian matrices stay at most polynomial with very high probability. GE with complete pivoting (GECP) has also seen a lot of recent interest, with improvements to both lower and upper bounds on worst-case GECP growth provided by Bisain, Edelman and Urschel in 2023. We are interested in studying how GEPP and GECP behave on the same linear systems as well as studying large growth on particular subclasses of matrices, including orthogonal matrices. Moreover, as a means to better address the question of why large growth is rarely encountered, we further study matrices with a large difference in growth between using GEPP and GECP, and we explore how the smaller growth strategy dominates behavior in a small neighborhood of the initial matrix.