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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.10248 |
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| _version_ | 1866915107913596928 |
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| author | He, Yunhui |
| author_facet | He, Yunhui |
| contents | In this work, we analyze the asymptotic convergence factor of minimal residual iteration (MRI) (or GMRES(1)) for solving linear systems $Ax=b$ based on vector-dependent nonlinear eigenvalue problems. The worst-case root-convergence factor is derived for linear systems with $A$ being symmetric or $I-A$ being skew-symmetric. When $A$ is symmetric, the asymptotic convergence factor highly depends on the initial guess. While $M=I-A$ is skew-symmetric, GMRES(1) converges unconditionally and the worst-case root-convergence factor relies solely on the spectral radius of $M$. We also derive the q-linear convergence factor, which is the same as the worst-case root-convergence factor. Numerical experiments are presented to validate our theoretical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10248 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The worst-case root-convergence factor of GMRES(1) He, Yunhui Numerical Analysis 65F10, 15A18 In this work, we analyze the asymptotic convergence factor of minimal residual iteration (MRI) (or GMRES(1)) for solving linear systems $Ax=b$ based on vector-dependent nonlinear eigenvalue problems. The worst-case root-convergence factor is derived for linear systems with $A$ being symmetric or $I-A$ being skew-symmetric. When $A$ is symmetric, the asymptotic convergence factor highly depends on the initial guess. While $M=I-A$ is skew-symmetric, GMRES(1) converges unconditionally and the worst-case root-convergence factor relies solely on the spectral radius of $M$. We also derive the q-linear convergence factor, which is the same as the worst-case root-convergence factor. Numerical experiments are presented to validate our theoretical results. |
| title | The worst-case root-convergence factor of GMRES(1) |
| topic | Numerical Analysis 65F10, 15A18 |
| url | https://arxiv.org/abs/2501.10248 |