Subspace methods for three‐parameter eigenvalue problems

We propose subspace methods for 3-parameter eigenvalue problems. Such problems arise when separation of variables is applied to separable boundary value problems; a particular example is the Helmholtz equation in ellipsoidal and paraboloidal coordinates. While several subspace methods for 2-parameter eigenvalue problems exist, extensions to 3-parameter setting have not been worked out thoroughly. An inherent difficulty is that, while for 2-parameter eigenvalue problems we can exploit a relation to Sylvester equations to obtain a fast Arnoldi type method, this relation does not seem to extend to three or more parameters in a straightforward way. Instead, we introduce a subspace iteration method with projections onto generalized Krylov subspaces that are constructed from scratch at every iteration using certain Ritz vectors as the initial vectors. Another possibility is a Jacobi-Davidson type method for three or more parameters, which we generalize from its 2-parameter counterpart. For both approaches, we introduce a selection criterion for deflation that is based on the angles between left and right eigenvectors. The Jacobi-Davidson approach is devised to locate eigenvalues close to a prescribed target, yet it often performs well when eigenvalues are sought based on the proximity of one of the components to a prescribed target. The subspace iteration method is devised specifically for the latter task. Matlab implementations of both methods are made available in package MultiParEig and we present extensive numerical experiments which indicate that both approaches are effective in locating eigenvalues in the exterior of the spectrum.