On the growth factor upper bound for Aasen's algorithm

TL;DR

This paper establishes a new, tighter upper bound on the growth factor for Aasen's algorithm in factorizing symmetric indefinite matrices, improving upon previous bounds and analyzing their tightness.

Contribution

It provides a significantly smaller growth factor upper bound for Aasen's algorithm and demonstrates its non-tightness for matrices of size six or more.

Findings

New growth factor upper bound for Aasen's algorithm

The bound is much smaller than Higham's previous bound

The bound is not tight for matrices with dimension ≥ 6

Abstract

Aasen's algorithm factorizes a symmetric indefinite matrix $A$ as $A = P^TLTL^TP$, where $P$ is a permutation matrix, $L$ is unit lower triangular with its first column being the first column of the identity matrix, and $T$ is tridiagonal. In this note, we provide a growth factor upper bound for Aasen's algorithm which is much smaller than that given by Higham. We also show that the upper bound we have given is not tight when the matrix dimension is greater than or equal to $6$.