Structured eigenvalue backward errors of Rosenbrock systems and related $μ$-value problems
Anshul Prajapati, Punit Sharma · arXiv (Cornell University) · 2024
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.