Jacobi-Davidson methods for polynomial two-parameter eigenvalue problems

We propose Jacobi–Davidson type methods for polynomial two-parameter eigenvalue problems (PMEP). Such problems can be linearized as singular two-parameter eigenvalue problems, whose matrices are of dimension k(k + 1)n/2, where k is the degree of the polynomial and n is the size of the matrix coefficients in the PMEP. When k2n is relatively small, the problem can be solved numerically by computing the common regular part of the related pair of singular pencils. For large k2n, computing all solutions is not feasible and iterative methods are required. When k is large, we propose to linearize the problem first and then apply Jacobi–Davidson to the obtained singular two-parameter eigenvalue problem. The resulting method may for instance be used for computing zeros of a system of scalar bivariate polynomials close to a given target. On the other hand, when k is small, we can apply a Jacobi–Davidson type approach directly to the original matrices. The original matrices are projected onto a low-dimensional subspace, and the projected polynomial two-parameter eigenvalue problems are solved by a linearization.