Computing the structured pseudospectrum of a Toeplitz matrix and its extreme points

TL;DR

This paper introduces algorithms to compute the structured pseudospectrum of Toeplitz matrices, extending unstructured methods and enabling visualization of extremal points and sections of the pseudospectrum.

Contribution

It presents novel algorithms for structured pseudospectra of Toeplitz matrices, addressing challenges in extending unstructured approaches and enabling new visualizations.

Findings

Algorithms effectively compute structured pseudospectra near extremal points.

The methods show promising convergence properties.

Applications demonstrate utility in illustrative examples.

Abstract

The computation of the structured pseudospectral abscissa and radius (with respect to the Frobenius norm) of a Toeplitz matrix is discussed and two algorithms based on a low rank property to construct extremal perturbations are presented. The algorithms are inspired by those considered in [SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1166-1192] for the unstructured case, but their extension to structured pseudospectra and analysis presents several difficulties. Natural generalizations of the algorithms, allowing to draw significant sections of the structured pseudospectra in proximity of extremal points are also discussed. Since no algorithms are available in the literature to draw such structured pseudospectra, the approach we present seems promising to extend existing software tools (Eigtool, Seigtool) to structured pseudospectra representation for Toeplitz matrices. We discuss local convergence properties of the algorithms and show some applications to a few illustrative examples.