Structured eigenvalue backward errors for rational matrix polynomials with symmetry structures

TL;DR

This paper develops formulas for structured backward errors of eigenvalues in rational matrix polynomials with various symmetry structures, highlighting differences between structure-preserving and arbitrary perturbations.

Contribution

It introduces computable formulas for structured backward errors considering multiple symmetry types in rational matrix polynomials.

Findings

Backward errors differ significantly between structure-preserving and arbitrary perturbations.

Formulas are applicable to various symmetry structures including Hermitian, skew-Hermitian, and palindromic.

Numerical experiments validate the theoretical formulas and demonstrate their practical relevance.

Abstract

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, $*$-even, $*$-odd, and $*$-palindromic structures. Numerical experiments show that the backward errors with respect to structure-preserving and arbitrary perturbations are significantly different.