Growth factors of random butterfly matrices and the stability of avoiding pivoting

TL;DR

This paper investigates how preconditioning with random butterfly matrices affects the growth factor in Gaussian elimination, showing they dampen large growth factors more effectively than other methods and analyzing their growth factor distribution.

Contribution

It provides the first theoretical and numerical analysis of the growth factor behavior of butterfly matrices in preconditioning, including the distribution for a subclass of these matrices.

Findings

Butterfly matrices significantly reduce large growth factors.

They cause smaller increases in minimal growth factor systems.

Distribution of growth factors for certain butterfly matrices is fully characterized.

Abstract

Random butterfly matrices were introduced by Parker in 1995 to remove the need for pivoting when using Gaussian elimination. The growing applications of butterfly matrices have often eclipsed the mathematical understanding of how or why butterfly matrices are able to accomplish these given tasks. To help begin to close this gap using theoretical and numerical approaches, we explore the impact on the growth factor of preconditioning a linear system by butterfly matrices. These results are compared to other common methods found in randomized numerical linear algebra. In these experiments, we show preconditioning using butterfly matrices has a more significant dampening impact on large growth factors than other common preconditioners and a smaller increase to minimal growth factor systems. Moreover, we are able to determine the full distribution of the growth factors for a subclass of random butterfly matrices. Previous results by Trefethen and Schreiber relating to the distribution of random growth factors were limited to empirical estimates of the first moment for Ginibre matrices.