Structure-preserving deflation of critical eigenvalues in quadratic eigenvalue problems associated with damped mass-spring systems
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866915404079693824 |
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| author | Alam, Rafikul Mehrmann, Volker Truhar, Ninoslav |
| author_facet | Alam, Rafikul Mehrmann, Volker Truhar, Ninoslav |
| contents | For a quadratic matrix polynomial associated with a damped mass-spring system there are three types of critical eigenvalues, the eigenvalues $\infty$ and $0$ and the eigenvalues on the imaginary axis. All these are on the boundary of the set of (robustly) stable eigenvalues. For numerical methods, but also for (robust) stability analysis, it is desirable to deflate such eigenvalues by projecting the matrix polynomial to a lower dimensional subspace before computing the other eigenvalues and eigenvectors. We describe structure-preserving deflation strategies that deflate these eigenvalues via a trimmed structure-preserving linearization. We employ these results for the special case of hyperbolic problems. We also analyze the effect of a (possibly low rank) parametric damping matrix on purely imaginary eigenvalues. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_16024 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structure-preserving deflation of critical eigenvalues in quadratic eigenvalue problems associated with damped mass-spring systems Alam, Rafikul Mehrmann, Volker Truhar, Ninoslav Numerical Analysis Spectral Theory 65F15, 15A57, 15A18, 65F35 For a quadratic matrix polynomial associated with a damped mass-spring system there are three types of critical eigenvalues, the eigenvalues $\infty$ and $0$ and the eigenvalues on the imaginary axis. All these are on the boundary of the set of (robustly) stable eigenvalues. For numerical methods, but also for (robust) stability analysis, it is desirable to deflate such eigenvalues by projecting the matrix polynomial to a lower dimensional subspace before computing the other eigenvalues and eigenvectors. We describe structure-preserving deflation strategies that deflate these eigenvalues via a trimmed structure-preserving linearization. We employ these results for the special case of hyperbolic problems. We also analyze the effect of a (possibly low rank) parametric damping matrix on purely imaginary eigenvalues. |
| title | Structure-preserving deflation of critical eigenvalues in quadratic eigenvalue problems associated with damped mass-spring systems |
| topic | Numerical Analysis Spectral Theory 65F15, 15A57, 15A18, 65F35 |
| url | https://arxiv.org/abs/2507.16024 |