Structured eigenvalue backward errors of Rosenbrock systems and related $\mu$-value problems

Anshul Prajapati; Punit Sharma

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arXiv · 2405.11974

Structured eigenvalue backward errors of Rosenbrock systems and related $\mu$-value problems

Anshul Prajapati·Punit Sharma

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Abstract

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 $\lambda\in \mathbb C$. 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 $\mu$-values of a rectangular matrix under rectangular-block-diagonal perturbations. For the reformulated $\mu$-value problem, we provide an explicit expression using partial isometric matrices and also obtain a computable upper bound, which is equal to the $\mu$-value when the pertrubation matrix has no more than three blocks at the diagonal. The results are illustrated through numerical experiments.

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BibTeX RIS

Anshul Prajapati, Punit Sharma. 2024-05-20. Structured eigenvalue backward errors of Rosenbrock systems and related $\mu$-value problems. https://arxiv.org/abs/2405.11974

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