Structured eigenvalue backward errors for rational matrix functions with symmetry structures

Abstract

We derive computable formulas for the structured backward errors of a complex number \(\lambda \) when considered as an approximate eigenvalue of rational matrix functions that carry a symmetry structure. We consider symmetric, skew-symmetric, Hermitian, skew-Hermitian, \(*\)-palindromic, T-even, T-odd, \(*\)-even, and \(*\)-odd structures. Numerical experiments show that the backward errors with respect to structure-preserving and arbitrary perturbations are significantly different.