Approximation of the Pseudospectral Abscissa via Eigenvalue Perturbation Theory
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910990377943040 |
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| author | Ahmed, Waqar Mengi, Emre |
| author_facet | Ahmed, Waqar Mengi, Emre |
| contents | Reliable and efficient computation of the pseudospectral abscissa in the large-scale setting is still not settled. Unlike the small-scale setting where there are globally convergent criss-cross algorithms, all algorithms in the large-scale setting proposed to date are at best locally convergent. We first describe how eigenvalue perturbation theory can be put in use to estimate the globally rightmost point in the $ε$-pseudospectrum if $ε$ is small. Our treatment addresses both general nonlinear eigenvalue problems, and the standard eigenvalue problem as a special case. For small $ε$, the estimates by eigenvalue perturbation theory are quite accurate. In the standard eigenvalue case, we even derive a formula with an ${\mathcal O}(ε^3)$ error. For larger $ε$, the estimates can be used to initialize the locally convergent algorithms. We also propose fixed-point iterations built on the the perturbation theory ideas for large $ε$ that are suitable for the large-scale setting. The proposed fixed-point iterations initialized by using eigenvalue perturbation theory converge to the globally rightmost point in the pseudospectrum in a vast majority of the cases that we experiment with. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_05535 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approximation of the Pseudospectral Abscissa via Eigenvalue Perturbation Theory Ahmed, Waqar Mengi, Emre Numerical Analysis 65F15, 93D09, 65H17, 30C15 Reliable and efficient computation of the pseudospectral abscissa in the large-scale setting is still not settled. Unlike the small-scale setting where there are globally convergent criss-cross algorithms, all algorithms in the large-scale setting proposed to date are at best locally convergent. We first describe how eigenvalue perturbation theory can be put in use to estimate the globally rightmost point in the $ε$-pseudospectrum if $ε$ is small. Our treatment addresses both general nonlinear eigenvalue problems, and the standard eigenvalue problem as a special case. For small $ε$, the estimates by eigenvalue perturbation theory are quite accurate. In the standard eigenvalue case, we even derive a formula with an ${\mathcal O}(ε^3)$ error. For larger $ε$, the estimates can be used to initialize the locally convergent algorithms. We also propose fixed-point iterations built on the the perturbation theory ideas for large $ε$ that are suitable for the large-scale setting. The proposed fixed-point iterations initialized by using eigenvalue perturbation theory converge to the globally rightmost point in the pseudospectrum in a vast majority of the cases that we experiment with. |
| title | Approximation of the Pseudospectral Abscissa via Eigenvalue Perturbation Theory |
| topic | Numerical Analysis 65F15, 93D09, 65H17, 30C15 |
| url | https://arxiv.org/abs/2506.05535 |