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

Abstract

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 a significant difference between the backward errors with respect to structure preserving and arbitrary perturbations in many cases.