Structured eigenvalue backward errors for rational matrix functions with symmetry structures

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