Gerschgorin's theorem for generalized eigenvalue problems in the Euclidean metric

TL;DR

This paper introduces Gerschgorin-type eigenvalue inclusion sets for generalized eigenvalue problems in the Euclidean metric, offering easier computation and applications in forward error analysis for eigenvalues of diagonalizable pencils.

Contribution

It presents new Gerschgorin-type inclusion sets for generalized eigenvalues defined by Euclidean circles, simplifying computation and enabling error analysis.

Findings

Inclusion sets are easier to compute than previous methods.

Application to forward error analysis of eigenvalues.

Provides bounds for eigenvalues of diagonalizable pencils.

Abstract

We present Gerschgorin-type eigenvalue inclusion sets applicable to generalized eigenvalue problems.Our sets are defined by circles in the complex plane in the standard Euclidean metric, and are easier to compute than known similar results.As one application we use our results to provide a forward error analysis for a computed eigenvalue of a diagonalizable pencil.