From eigenvector nonlinearities with quadratic structure to eigenvalue nonlinearities with algebraic structure

TL;DR

This paper introduces a transformation that converts a class of eigenvalue problems with eigenvector nonlinearities into algebraic eigenvalue problems, enabling the use of adapted NEP solvers for more efficient solutions.

Contribution

The authors develop a novel transformation from NEPv to NEP with algebraic nonlinearities, and adapt existing NEP solvers to handle the resulting polynomial systems effectively.

Findings

The transformation enables solving NEPv via NEP methods.

The adapted solvers show efficiency in a GPE-related problem.

Reproducible simulations demonstrate the approach's effectiveness.

Abstract

Over the past decades, transformations between different classes of eigenvalue problems have played a central role in the development of numerical methods for eigenvalue computations. One of the most well-known and successful examples of this is the companion linearization for polynomial eigenvalue problems. In this paper, we construct a transformation that equivalently reframes a specific type of eigenvalue problem with eigenvector nonlinearities (NEPv) into an eigenvalue problem with eigenvalue nonlinearities (NEP). The NEPv class considered consists of nonlinearities expressed as sums of products of matrices and scalar functions, where the scalar functions depend nonlinearly on the eigenvector. Our transformation defines scalar eigenvalue nonlinearities through a polynomial system, resulting in NEP nonlinearities of algebraic type. We propose methods to solve the polynomial system, one of which involves a multiparameter eigenvalue problem (MEP). We adapt well-established NEP solvers to this setting, with the most effective strategy being a combination of deflation and a locally quadratically convergent iterative method. The efficiency and properties of the approach are illustrated by solving a problem related to a modification of a Gross-Pitaevskii equation (GPE). The simulations are reproducible and publicly available.