Conditioning and backward errors for nonlinear eigenvalue problems with eigenvector nonlinearities

TL;DR

This paper derives explicit formulas for eigenvalue condition numbers and backward errors in symmetric nonlinear eigenvalue problems with eigenvector nonlinearities, highlighting the need for careful analysis.

Contribution

It provides new explicit, computable expressions for condition numbers and backward errors specific to symmetric nonlinear eigenvalue problems with eigenvector nonlinearities.

Findings

Explicit formulas enable efficient evaluation of condition numbers and backward errors.

Eigenvector nonlinearities require more careful treatment than linear or eigenvalue nonlinear cases.

Numerical experiments illustrate the importance of specialized analysis for these problems.

Abstract

We consider eigenvalue condition numbers and backward errors for a class of symmetric nonlinear eigenvalue problems with eigenvector nonlinearities. For both of these quantities, we derive explicit and computable expressions that can be evaluated with little computational effort for a given eigenpair, assuming the matrix perturbations are measured by the spectral or Frobenius norm. We also show how symmetric perturbations can be exploited in the analysis. By means of two numerical experiments we demonstrate that problems incorporating eigenvector nonlinearities potentially need to be treated with additional care, when compared to the linear or eigenvalue-nonlinear theory.