Structured Eigenvalue Backward Errors for Rational Matrix Functions with Symmetry Structures

We derive computable formulas for the structured backward errors of a complex number λ 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.