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

TL;DR

This paper derives simplified formulas for the structured eigenvalue backward error of Rosenbrock system matrices, involving $$-values and providing explicit expressions and bounds, with numerical validation.

Contribution

It introduces new simplified formulas and explicit expressions for the structured eigenvalue backward error of Rosenbrock systems, including bounds and computational methods.

Findings

Derived explicit formulas for structured eigenvalue backward errors.

Provided computable upper bounds matching the $$-value in certain cases.

Validated results through numerical experiments.

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.