  This paper describes a set of rational filtering algorithms to compute a few
eigenvalues (and associated eigenvectors) of non-Hermitian matrix pencils. Our
interest lies in computing eigenvalues located inside a given disk, and the
proposed algorithms approximate these eigenvalues and associated eigenvectors
by harmonic Rayleigh-Ritz projections on subspaces built by computing range
spaces of rational matrix functions through randomized range finders. These
rational matrix functions are designed so that directions associated with
non-sought eigenvalues are dampened to (approximately) zero. Variants based on
matrix partitionings are introduced to further reduce the overall complexity of
the proposed framework. Compared with existing eigenvalue solvers based on
rational matrix functions, the proposed technique requires no estimation of the
number of eigenvalues located inside the disk. Several theoretical and
practical issues are discussed, and the competitiveness of the proposed
framework is demonstrated via numerical experiments.
