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

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.