Numerical methods for eigenvalues of singular polynomial eigenvalue problems
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929720647483392 |
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| author | Hochstenbach, Michiel E. Mehl, Christian Plestenjak, Bor |
| author_facet | Hochstenbach, Michiel E. Mehl, Christian Plestenjak, Bor |
| contents | Recently, three numerical methods for the computation of eigenvalues of singular matrix pencils, based on a rank-completing perturbation, a rank-projection, or an augmentation were developed. We show that all three approaches can be generalized to treat singular polynomial eigenvalue problems. The common denominator of all three approaches is a transformation of a singular into a regular matrix polynomial whose eigenvalues are a disjoint union of the eigenvalues of the singular polynomial, called true eigenvalues, and additional fake eigenvalues. The true eigenvalues can then be separated from the fake eigenvalues using information on the corresponding left and right eigenvectors. We illustrate the approaches on several interesting applications, including bivariate polynomial systems and ZGV points. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_16832 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Numerical methods for eigenvalues of singular polynomial eigenvalue problems Hochstenbach, Michiel E. Mehl, Christian Plestenjak, Bor Numerical Analysis 65F15, 15A18, 15A22, 47A55, 65F22 Recently, three numerical methods for the computation of eigenvalues of singular matrix pencils, based on a rank-completing perturbation, a rank-projection, or an augmentation were developed. We show that all three approaches can be generalized to treat singular polynomial eigenvalue problems. The common denominator of all three approaches is a transformation of a singular into a regular matrix polynomial whose eigenvalues are a disjoint union of the eigenvalues of the singular polynomial, called true eigenvalues, and additional fake eigenvalues. The true eigenvalues can then be separated from the fake eigenvalues using information on the corresponding left and right eigenvectors. We illustrate the approaches on several interesting applications, including bivariate polynomial systems and ZGV points. |
| title | Numerical methods for eigenvalues of singular polynomial eigenvalue problems |
| topic | Numerical Analysis 65F15, 15A18, 15A22, 47A55, 65F22 |
| url | https://arxiv.org/abs/2406.16832 |