Characterizing and approximating eigenvalue sets of symmetric interval matrices

TL;DR

This paper introduces a novel algorithm for approximating eigenvalue sets of symmetric matrices with interval perturbations, guaranteeing exact bounds in many cases, supported by numerical experiments.

Contribution

It presents the first algorithm capable of guaranteeing exact eigenvalue bounds for symmetric interval matrices, with a new characterization of boundary points.

Findings

Algorithm often estimates exact bounds

Numerical experiments validate approach

First guaranteed exactness in eigenvalue set approximation

Abstract

We consider the eigenvalue problem for the case where the input matrix is symmetric and its entries perturb in some given intervals. We present a characterization of some of the exact boundary points, which allows us to introduce an inner approximation algorithm, that in many case estimates exact bounds. To our knowledge, this is the first algorithm that is able to guaran- tee exactness. We illustrate our approach by several examples and numerical experiments.