Finding the nearest $Ω$-stable pencil with Riemannian optimization
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917904247685120 |
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| author | Noferini, Vanni Nyman, Lauri |
| author_facet | Noferini, Vanni Nyman, Lauri |
| contents | This paper considers the problem of finding the nearest $Ω$-stable pencil to a given square pencil $A+xB \in \mathbb{C}^{n \times n}$, where a pencil is called $Ω$-stable if it is regular and all of its eigenvalues belong to the closed set $Ω$. We propose a new method, based on the Schur form of a matrix pair and Riemannian optimization over the manifold $U(n) \times U(n)$, that is, the Cartesian product of the unitary group with itself. While the developed theory holds for any closed set $Ω$, we focus on two cases that are the most common in applications: Hurwitz stability and Schur stability. For these cases, we develop publicly available efficient implementations. Numerical experiments show that the resulting algorithm outperforms existing methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16876 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finding the nearest $Ω$-stable pencil with Riemannian optimization Noferini, Vanni Nyman, Lauri Numerical Analysis This paper considers the problem of finding the nearest $Ω$-stable pencil to a given square pencil $A+xB \in \mathbb{C}^{n \times n}$, where a pencil is called $Ω$-stable if it is regular and all of its eigenvalues belong to the closed set $Ω$. We propose a new method, based on the Schur form of a matrix pair and Riemannian optimization over the manifold $U(n) \times U(n)$, that is, the Cartesian product of the unitary group with itself. While the developed theory holds for any closed set $Ω$, we focus on two cases that are the most common in applications: Hurwitz stability and Schur stability. For these cases, we develop publicly available efficient implementations. Numerical experiments show that the resulting algorithm outperforms existing methods. |
| title | Finding the nearest $Ω$-stable pencil with Riemannian optimization |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2501.16876 |