Stable algorithms for general linear systems by preconditioning the normal equations

TL;DR

This paper investigates the accuracy issues of preconditioned LSQR for nonsymmetric linear systems and demonstrates how iterative refinement and restarting can significantly improve solution precision, supported by backward error analysis.

Contribution

It introduces methods to enhance the accuracy of preconditioned LSQR and compares its stability with GMRES, providing new insights into solving nonsymmetric systems.

Findings

Preconditioned LSQR can produce large errors without refinement.

Iterative refinement and restarting improve LSQR accuracy.

Left-preconditioned LSQR is stable without iterative refinement.

Abstract

This paper studies the solution of nonsymmetric linear systems by preconditioned Krylov methods based on the normal equations, LSQR in particular. On some examples, preconditioned LSQR is seen to produce errors many orders of magnitude larger than classical direct methods; this paper demonstrates that the attainable accuracy of preconditioned LSQR can be greatly improved by applying iterative refinement or restarting when the accuracy stalls. This observation is supported by rigorous backward error analysis. This paper also provides a discussion of the relative merits of GMRES and LSQR for solving nonsymmetric linear systems, demonstrates stability for left-preconditioned LSQR without iterative refinement, and shows that iterative refinement can also improve the accuracy of preconditioned conjugate gradient.