Structured Eigenvalue Backward Errors of Matrix Pencils and Polynomials with Palindromic Structures

We derive formulas for the backward error of an approximate eigenvalue of a ∗palindromic matrix polynomial with respect to ∗-palindromic perturbations. Such formulas are also obtained for complex T -palindromic pencils and quadratic polynomials. When the T -palindromic polynomial is real, then we derive the backward error of a real number considered as an approximate eigenvalue of the matrix polynomial with respect to real T -palindromic perturbations. In all cases the corresponding minimal structure preserving perturbations are obtained as well. The results are illustrated by numerical experiments. These show that there is significant difference between the backward errors with respect to structure preserving and arbitrary perturbations in many cases.