Approximating the closest structured singular matrix polynomial
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910473369157632 |
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| author | Gnazzo, Miryam Guglielmi, Nicola |
| author_facet | Gnazzo, Miryam Guglielmi, Nicola |
| contents | Consider a matrix polynomial $P \left( λ\right)= A_0 + λA_1 + \ldots + λ^d A_d$, with $A_0,\ldots, A_d$ complex (or real) matrices with a certain structure. In this paper we discuss an iterative method to numerically approximate the closest structured singular matrix polynomial $\widetilde P\left( λ\right)$, using the distance induced by the Frobenius norm. An important peculiarity of the approach we propose is the possibility to include different types of structural constraints. The method also allows us to limit the perturbations to just a few matrices and also to include additional structures, such as the preservation of the sparsity pattern of one or more matrices $A_i$, and also collective-like properties, like a palindromic structure. The iterative method is based on the numerical integration of the gradient system associated with a suitable functional which quantifies the distance to singularity of a matrix polynomial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_06335 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Approximating the closest structured singular matrix polynomial Gnazzo, Miryam Guglielmi, Nicola Numerical Analysis 15A18, 47A56, 65K05 Consider a matrix polynomial $P \left( λ\right)= A_0 + λA_1 + \ldots + λ^d A_d$, with $A_0,\ldots, A_d$ complex (or real) matrices with a certain structure. In this paper we discuss an iterative method to numerically approximate the closest structured singular matrix polynomial $\widetilde P\left( λ\right)$, using the distance induced by the Frobenius norm. An important peculiarity of the approach we propose is the possibility to include different types of structural constraints. The method also allows us to limit the perturbations to just a few matrices and also to include additional structures, such as the preservation of the sparsity pattern of one or more matrices $A_i$, and also collective-like properties, like a palindromic structure. The iterative method is based on the numerical integration of the gradient system associated with a suitable functional which quantifies the distance to singularity of a matrix polynomial. |
| title | Approximating the closest structured singular matrix polynomial |
| topic | Numerical Analysis 15A18, 47A56, 65K05 |
| url | https://arxiv.org/abs/2301.06335 |