Solving two-parameter eigenvalue problems using an alternating method

TL;DR

This paper introduces an alternating method for efficiently computing specific eigenvalues and eigenvectors of two-parameter eigenvalue problems, applicable to right definite cases including certain Helmholtz equations.

Contribution

The paper presents a novel alternating approach that reduces the problem to generalized eigenvalue computations, with convergence proofs for extremal and local convergence for other eigenvalues.

Findings

Method effectively computes selected eigenvalues.

Convergence is proven for extremal eigenvalues.

Empirical results support local convergence for other eigenvalues.

Abstract

We present a new approach to compute selected eigenvalues and eigenvectors of the two-parameter eigenvalue problem. Our method requires computing generalized eigenvalue problems of the same size as the matrices of the initial two-parameter eigenvalue problem. The method is applicable for right definite problems, possibly after performing an affine transformation. This includes a class of Helmholtz equations when separation of variables is applied. We provide a convergence proof for extremal eigenvalues and empirical evidence along with a local convergence proof for other eigenvalues.