A note on the growth factor in Gaussian elimination for Higham matrices

TL;DR

This paper improves the understanding of the growth factor in Gaussian elimination for Higham matrices by establishing a tighter bound of 2, thus confirming Higham's conjecture.

Contribution

It provides a new bound on the growth factor for generalized Higham matrices, strengthening previous results and proving Higham's conjecture.

Findings

Growth factor bound is strengthened to 2

Proves Higham's conjecture for generalized matrices

Uses condition numbers of B and C matrices

Abstract

The Higham matrix is a complex symmetric matrix A=B+iC, where both B and C are real, symmetric and positive definite and $\mathrm{i}=\sqrt{-1}$ is the imaginary unit. For any Higham matrix A, Ikramov et al. showed that the growth factor in Gaussian elimination is less than 3. In this paper, based on the previous results, a new bound of the growth factor is obtained by using the maximum of the condition numbers of matrixes B and C for the generalized Higham matrix A, which strengthens this bound to 2 and proves the Higham's conjecture.