Twice is enough for dangerous eigenvalues
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866909388317392896 |
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| author | Horning, Andrew Nakatsukasa, Yuji |
| author_facet | Horning, Andrew Nakatsukasa, Yuji |
| contents | We analyze the stability of a class of eigensolvers that target interior eigenvalues with rational filters. We show that subspace iteration with a rational filter is robust even when an eigenvalue is near a filter's pole. These dangerous eigenvalues contribute to large round-off errors in the first iteration, but are self-correcting in later iterations. For matrices with orthogonal eigenvectors (e.g., real-symmetric or complex Hermitian), two iterations is enough to reduce round-off errors to the order of the unit-round off. In contrast, Krylov methods accelerated by rational filters with fixed poles typically fail to converge to unit round-off accuracy when an eigenvalue is close to a pole. In the context of Arnoldi with shift-and-invert enhancement, we demonstrate a simple restart strategy that recovers full precision in the target eigenpairs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_09710 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Twice is enough for dangerous eigenvalues Horning, Andrew Nakatsukasa, Yuji Numerical Analysis We analyze the stability of a class of eigensolvers that target interior eigenvalues with rational filters. We show that subspace iteration with a rational filter is robust even when an eigenvalue is near a filter's pole. These dangerous eigenvalues contribute to large round-off errors in the first iteration, but are self-correcting in later iterations. For matrices with orthogonal eigenvectors (e.g., real-symmetric or complex Hermitian), two iterations is enough to reduce round-off errors to the order of the unit-round off. In contrast, Krylov methods accelerated by rational filters with fixed poles typically fail to converge to unit round-off accuracy when an eigenvalue is close to a pole. In the context of Arnoldi with shift-and-invert enhancement, we demonstrate a simple restart strategy that recovers full precision in the target eigenpairs. |
| title | Twice is enough for dangerous eigenvalues |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2010.09710 |