Randomized Complete Pivoting for Solving Symmetric Indefinite Linear Systems

TL;DR

This paper introduces a randomized complete pivoting algorithm for symmetric indefinite linear systems that achieves stability comparable to complete pivoting while maintaining efficiency similar to existing methods.

Contribution

The paper develops a novel randomized pivoting algorithm that combines efficiency with strong theoretical guarantees on element growth and numerical stability.

Findings

RCP matches efficiency of Bunch-Kaufman and Aasen's algorithms.

RCP has bounded entries and controlled element growth similar to complete pivoting.

Numerical experiments confirm RCP's stability in practice.

Abstract

The Bunch-Kaufman algorithm and Aasen's algorithm are two of the most widely used methods for solving symmetric indefinite linear systems, yet they both are known to suffer from occasional numerical instability due to potentially exponential element growth or unbounded entries in the matrix factorization. In this work, we develop a randomized complete pivoting (RCP) algorithm for solving symmetric indefinite linear systems. RCP is comparable to the Bunch-Kaufman algorithm and Aasen's algorithm in computational efficiency, yet enjoys theoretical element growth and bounded entries in the factorization comparable to that of complete-pivoting, up to a theoretical failure probability that exponentially decays with an oversampling parameter. Our finite precision analysis shows that RCP is as numerically stable as Gaussian elimination with complete pivoting, and RCP has been observed to be numerically stable in our extensive numerical experiments.