  A new approach is presented to compute the seismic normal modes of a fully
heterogeneous, rotating planet. Special care is taken to separate out the
essential spectrum in the presence of a fluid outer core. The relevant
elastic-gravitational system of equations, including the Coriolis force, is
subjected to a mixed finite-element method, while self-gravitation is accounted
for with the fast multipole method (FMM). To solve the resulting quadratic
eigenvalue problem (QEP), the approach utilizes extended Lanczos vectors
forming a subspace computed from a non-rotating planet -- with the shape of
boundaries of a rotating planet and accounting for the centrifugal potential --
to reduce the dimension of the original problem significantly. The subspace is
guaranteed to be contained in the space of functions to which the seismic
normal modes belong. The reduced system can further be solved with a standard
eigensolver. The computational accuracy is illustrated using all the modes with
relative small meshes and also tested against standard perturbation
calculations relative to a standard Earth model. The algorithm and code are
used to compute the point spectra of eigenfrequencies in several Mars models
studying the effects of heterogeneity on a large range of scales.
