Contour Integration for Eigenvector Nonlinearities

TL;DR

This paper introduces a contour integration method to compute all eigenvalues of polynomial eigenvalue problems with eigenvector nonlinearities within a specific complex plane region, extending existing nonlinear eigenvalue techniques.

Contribution

It generalizes contour integration approaches to handle polynomial eigenvalue problems with eigenvector nonlinearities, a previously challenging class of problems.

Findings

Effective computation of eigenvalues within a region of the complex plane.

Applicable to polynomial and rational function systems.

Provides a new tool for solving complex eigenvalue problems.

Abstract

Solving polynomial eigenvalue problems with eigenvector nonlinearities (PEPv) is an interesting computational challenge, outside the reach of the well-developed methods for nonlinear eigenvalue problems. We present a natural generalization of these methods which leads to a contour integration approach for computing all eigenvalues of a PEPv in a compact region of the complex plane. Our methods can be used to solve any suitably generic system of polynomial or rational function equations.