Stable algorithms for general linear systems by preconditioning the normal equations

Journal article

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.

Authors: Epperly, E; Greenbaum, A; Nakatsukasa, YN

• Publication status: Accepted
• Peer review status: Peer reviewed
• Publisher: Springer Nature
• Journal: Numerische Mathematik
• Acceptance date: 2026-04-01
• EISSN: 0945-3245
• ISSN: 0029-599X
• Language: English
• Keywords: LSQR; linear system; preconditioning; numerical stability; backward stability