Linearizability of eigenvector nonlinearities

TL;DR

This paper introduces an exact linearization method for a class of eigenvector nonlinearities, enabling solution computation and analysis of solution count, with practical iterative schemes demonstrated through numerical examples.

Contribution

It presents a novel exact linearization approach for eigenvector nonlinearities using multiparameter problems, facilitating solution enumeration and iterative solution methods.

Findings

Exact linearization via multiparameter problems

Development of inverse and residual inverse iteration schemes

Numerical examples demonstrating method effectiveness

Abstract

We present a method to linearize, without approximation, a specific class of eigenvalue problems with eigenvector nonlinearities (NEPv), where the nonlinearities are expressed by scalar functions that are defined by a quotient of linear functions of the eigenvector. The exact linearization relies on an equivalent multiparameter problem (MEP) that contains the exact solutions of the NEPv. Due to the characterization of MEPs in terms of a generalized eigenvalue problem this provides a direct way to compute all NEPv solutions for small problems, and it opens up the possibility to develop locally convergent iterative methods for larger problems. Moreover, the linear formulation allows us to easily determine the number of solutions of the NEPv. We propose two numerical schemes that exploit the structure of the linearization: inverse iteration and residual inverse iteration. We show how symmetry in the MEP can be used to improve reliability and reduce computational cost of both methods. Two numerical examples verify the theoretical results and a third example shows the potential of a hybrid scheme that is based on a combination of the two proposed methods.