Structured eigenvalue backward errors of Rosenbrock systems and related $μ$-value problems
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| author | Prajapati, Anshul Sharma, Punit |
| author_facet | Prajapati, Anshul Sharma, Punit |
| contents | In this paper, we compute the structured eigenvalue backward error of a Rosenbrock system matrix $S(z)=\left[\begin{array}{cc} A-zI & B \\ C & P(z) \end{array}\right]$ for a given scalar $λ\in \mathbb C$.
We have developed simplified formulas for the structured eigenvalue backward error of the Rosenbrock system matrix, considering both full and partial block perturbations. These formulas involve computing
structured $μ$-values of a rectangular matrix under rectangular-block-diagonal perturbations.
For the reformulated $μ$-value problem, we provide an explicit expression using partial isometric matrices and also obtain a computable upper bound, which is equal to the $μ$-value when the pertrubation matrix has no more than three blocks at the diagonal.
The results are illustrated through numerical experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11974 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Structured eigenvalue backward errors of Rosenbrock systems and related $μ$-value problems Prajapati, Anshul Sharma, Punit Optimization and Control In this paper, we compute the structured eigenvalue backward error of a Rosenbrock system matrix $S(z)=\left[\begin{array}{cc} A-zI & B \\ C & P(z) \end{array}\right]$ for a given scalar $λ\in \mathbb C$. We have developed simplified formulas for the structured eigenvalue backward error of the Rosenbrock system matrix, considering both full and partial block perturbations. These formulas involve computing structured $μ$-values of a rectangular matrix under rectangular-block-diagonal perturbations. For the reformulated $μ$-value problem, we provide an explicit expression using partial isometric matrices and also obtain a computable upper bound, which is equal to the $μ$-value when the pertrubation matrix has no more than three blocks at the diagonal. The results are illustrated through numerical experiments. |
| title | Structured eigenvalue backward errors of Rosenbrock systems and related $μ$-value problems |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2405.11974 |