  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, their extensions to three parameter setting seem to
be challenging. 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, such a relation does not seem to exist when there
are three or more parameters. 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
also 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. The proposed approaches are suitable
especially for problems where the computation of several eigenvalues is
required with high accuracy. Matlab implementations of both methods have been
made available in the package MultiParEig.
