Approximated structured pseudospectra

TL;DR

This paper introduces an efficient method for approximating pseudospectra and structured pseudospectra of matrices by analyzing spectra of rank-one perturbations, providing better insights with lower computational costs.

Contribution

A novel approach based on rank-one perturbations inspired by Wilkinson’s sensitivity analysis for computing approximations of pseudospectra and structured pseudospectra.

Findings

Method yields better insights than random perturbations.

Approach reduces computational burden compared to existing methods.

Numerical examples demonstrate effectiveness of the proposed technique.

Abstract

Pseudospectra and structured pseudospectra are important tools for the analysis of matrices. Their computation, however, can be very demanding for all but small matrices. A new approach to compute approximations of pseudospectra and structured pseudospectra, based on determining the spectra of many suitably chosen rank-one or projected rank-one perturbations of the given matrix is proposed. The choice of rank-one or projected rank-one perturbations is inspired by Wilkinson's analysis of eigenvalue sensitivity. Numerical examples illustrate that the proposed approach gives much better insight into the pseudospectra and structured pseudospectra than random or structured random rank-one perturbations with lower computational burden. The latter approach is presently commonly used for the determination of structured pseudospectra.