Backward Error of Matrix Rational Function

TL;DR

This paper analyzes the backward error of eigenvalues in rational matrix functions, providing explicit formulas and minimal perturbations for improving eigenvalue accuracy in minimal realizations.

Contribution

It introduces explicit backward error formulas and minimal perturbation characterizations for eigenvalues of rational matrix functions with minimal realizations.

Findings

Explicit backward error expressions derived.

Minimal perturbations for exact eigenvalues identified.

Applicable to rational matrix functions in minimal realization.

Abstract

We consider a minimal realization of a rational matrix functions. We perturb the polynomial part and one of the constant matrices from the realization part. We derive explicit computable expressions of backward errors of approximate eigenvalue of rational matrix function. We also determine minimal perturbations for which approximate eigenvalue are exact eigenvalue of the perturbed matrix rational functions.