Approximation of the Pseudospectral Abscissa via Eigenvalue Perturbation Theory

TL;DR

This paper introduces a new eigenvalue perturbation theory-based approach for approximating the pseudospectral abscissa, providing accurate estimates for small epsilon and effective initialization for large-scale problems.

Contribution

It develops a novel method using eigenvalue perturbation theory to estimate the pseudospectral abscissa, including formulas with high accuracy and fixed-point iterations suitable for large-scale problems.

Findings

Eigenvalue perturbation estimates are accurate for small epsilon.

The method provides a ${ m O}( ext{epsilon}^3)$ error formula for standard eigenvalues.

Fixed-point iterations converge to the global rightmost pseudospectral point in most cases.

Abstract

Reliable and efficient computation of the pseudospectral abscissa in the large-scale setting is still not settled. Unlike the small-scale setting where there are globally convergent criss-cross algorithms, all algorithms in the large-scale setting proposed to date are at best locally convergent. We first describe how eigenvalue perturbation theory can be put in use to estimate the globally rightmost point in the $\epsilon$-pseudospectrum if $\epsilon$ is small. Our treatment addresses both general nonlinear eigenvalue problems, and the standard eigenvalue problem as a special case. For small $\epsilon$, the estimates by eigenvalue perturbation theory are quite accurate. In the standard eigenvalue case, we even derive a formula with an ${\mathcal O}(\epsilon^3)$ error. For larger $\epsilon$, the estimates can be used to initialize the locally convergent algorithms. We also propose fixed-point iterations built on the the perturbation theory ideas for large $\epsilon$ that are suitable for the large-scale setting. The proposed fixed-point iterations initialized by using eigenvalue perturbation theory converge to the globally rightmost point in the pseudospectrum in a vast majority of the cases that we experiment with.