An inverse iteration method for eigenvalue problems with eigenvector nonlinearities

TL;DR

This paper introduces a generalized inverse iteration method for solving eigenvalue problems where the matrix depends on the eigenvector and exhibits a specific scale invariance, with convergence analysis and practical applications.

Contribution

It proposes a novel inverse iteration algorithm tailored for eigenvalue problems with eigenvector nonlinearities and analyzes its convergence properties.

Findings

The method converges locally similar to standard inverse iteration.

The algorithm is equivalent to a discretization of an associated ODE.

Numerical simulations demonstrate the method's efficiency and convergence.

Abstract

Consider a symmetric matrix $A(v)\in\RR^{n\times n}$ depending on a vector $v\in\RR^n$ and satisfying the property $A(\alpha v)=A(v)$ for any $\alpha\in\RR\backslash{0}$. We will here study the problem of finding $(\lambda,v)\in\RR\times \RR^n\backslash\{0\}$ such that $(\lambda,v)$ is an eigenpair of the matrix $A(v)$ and we propose a generalization of inverse iteration for eigenvalue problems with this type of eigenvector nonlinearity. The convergence of the proposed method is studied and several convergence properties are shown to be analogous to inverse iteration for standard eigenvalue problems, including local convergence properties. The algorithm is also shown to be equivalent to a particular discretization of an associated ordinary differential equation, if the shift is chosen in a particular way. The algorithm is adapted to a variant of the Schr\"odinger equation known as the Gross-Pitaevskii equation. We use numerical simulations toillustrate the convergence properties, as well as the efficiency of the algorithm and the adaption.