Using an alternating approach to solve two-parameter eigenvalue problems

Two-parameter eigenvalue problems naturally arise when separation of variables is applied to boundary value problems, but also find a variety of other applications. 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 in the initial problem. It is applicable amongst other for a class of Helmholtz equations after separation of variables.