Stable algorithms for general linear systems by preconditioning the normal equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915182850080768 |
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| author | Epperly, Ethan N. Greenbaum, Anne Nakatsukasa, Yuji |
| author_facet | Epperly, Ethan N. Greenbaum, Anne Nakatsukasa, Yuji |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_17767 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stable algorithms for general linear systems by preconditioning the normal equations Epperly, Ethan N. Greenbaum, Anne Nakatsukasa, Yuji Numerical Analysis 65F08, 65F10 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. |
| title | Stable algorithms for general linear systems by preconditioning the normal equations |
| topic | Numerical Analysis 65F08, 65F10 |
| url | https://arxiv.org/abs/2502.17767 |