Wilkinson's Inertia-Revealing Factorization and Its Application to Sparse Matrices

TL;DR

This paper introduces a new inertia-revealing factorization method for sparse symmetric matrices, demonstrating its applicability, efficiency, and stability through implementation and experiments, offering an alternative to traditional methods.

Contribution

It adapts a classical inertia-revealing factorization scheme to sparse matrices, showing it can be efficiently applied with controlled fill-in and stability.

Findings

Fill-in is bounded by sparse QR factorization

Method is numerically stable

Implementation shows promising performance

Abstract

We propose a new inertia-revealing factorization for sparse symmetric matrices. The factorization scheme and the method for extracting the inertia from it were proposed in the 1960s for dense, banded, or tridiagonal matrices, but they have been abandoned in favor of faster methods. We show that this scheme can be applied to any sparse symmetric matrix and that the fill in the factorization is bounded by the fill in the sparse QR factorization of the same matrix (but is usually much smaller). We describe our serial proof-of-concept implementation, and present experimental results, studying the method's numerical stability and performance.