  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.
