  In this work, we propose a new method, termed as R-CORK, for the numerical
solution of large-scale rational eigenvalue problems, which is based on a
linearization and on a compact decomposition of the rational Krylov subspaces
corresponding to this linearization. R-CORK is an extension of the compact
rational Krylov method (CORK) introduced very recently by Van Beeumen et al. to
solve a family of non-linear eigenvalue problems that can be expressed and
linearized in certain particular ways and which include arbitrary polynomial
eigenvalue problems, but not arbitrary rational eigenvalue problems. The R-CORK
method exploits the structure of the linearized problem by representing the
Krylov vectors in a compact form in order to reduce the cost of storage,
resulting in a method with two levels of orthogonalization. The first level of
orthogonalization works with vectors of the same size that the original
problem, and the second level works with vectors of size much smaller than the
original problem. Since vectors of the size of the linearization are never
stored or orthogonalized, R-CORK is more efficient from the point of views of
memory and orthogonalization than the classical rational Krylov method applied
directly to the linearization. Taking into account that the R-CORK method is
based on a classical rational Krylov method, to implement implicit restarting
is also possible and we show how to do it in a memory efficient way. Finally,
some numerical examples are included in order to show that the R-CORK method
performs satisfactorily in practice.
