On backward errors of structured polynomial eigenproblems solved by structure preserving linearizations

TL;DR

This paper derives explicit formulas for structured backward errors of approximate eigenvalues in structured matrix polynomials, analyzes the impact of structure-preserving linearizations, and compares structured and unstructured pseudospectra.

Contribution

It provides new explicit expressions for structured backward errors and identifies linearizations that minimally affect these errors.

Findings

Explicit formulas for structured backward errors.

Identification of structure-preserving linearizations with minimal error impact.

Partial equivalence between structured and unstructured pseudospectra.

Abstract

First, we derive explicit computable expressions of structured backward errors of approximate eigenelements of structured matrix polynomials including symmetric, skew-symmetric, Hermitian, skew-Hermitian, even and odd polynomials. We also determine minimal structured perturbations for which approximate eigenelements are exact eigenelements of the perturbed polynomials. Next, we analyze the effect of structure preserving linearizations of structured matrix polynomials on the structured backward errors of approximate eigenelements. We identify structure preserving linearizations which have almost no adverse effect on the structured backward errors of approximate eigenelements of the polynomials. Finally, we analyze structured pseudospectra of a structured matrix polynomial and establish a partial equality between unstructured and structured pseudospectra.