Structured eigenvalue backward errors for rational matrix polynomials with symmetry structures

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