Complete pivoting growth of butterfly matrices and butterfly Hadamard matrices

TL;DR

This paper analyzes the growth problem in Gaussian elimination with complete pivoting, introducing a class of butterfly matrices for which exact growth can be computed and presenting a new method to construct Hadamard matrices.

Contribution

It extends exact growth computation to GECP for structured butterfly matrices and introduces a novel approach to generate random Hadamard matrices using butterfly matrices.

Findings

Exact GECP growth can be computed for certain butterfly matrices.

New method for constructing random Hadamard matrices using butterfly matrices.

Advances understanding of growth behavior in structured matrices.

Abstract

The growth problem in Gaussian elimination (GE) remains a foundational question in numerical analysis and numerical linear algebra. Wilkinson resolved the growth problem in GE with partial pivoting (GEPP) in his initial analysis from the 1960s, while he was only able to establish an upper bound for the GE with complete pivoting (GECP) growth problem. The GECP growth problem has seen a spike in recent interest, culminating in improved lower and upper bounds established by Bisain, Edelman, and Urschel in 2023, but still remains far from being fully resolved. Due to the complex dynamics governing the location of GECP pivots, analysis of GECP growth for particular input matrices often estimates the actual growth rather than computes the growth exactly. We present a class of dense random butterfly matrices for which we can compute the exact GECP growth. We extend previous results that established exact growth computations for butterfly matrices when using GEPP and GE with rook pivoting (GERP) to now also include GECP for structured subclasses of inputs. Moreover, we present a new method to construct random Hadamard matrices using butterfly matrices.