  In this paper, by introducing a class of relaxed filtered Krylov subspaces,
we propose the relaxed filtered Krylov subspace method for computing the
eigenvalues with the largest real parts and the corresponding eigenvectors of
non-symmetric matrices. As by-products, the generalizations of the filtered
Krylov subspace method and the Chebyshev-Davidson method for solving
non-symmetric eigenvalue problems are also presented. We give the convergence
analysis of the complex Chebyshev polynomial, which plays a significant role in
the polynomial acceleration technique. In addition, numerical experiments are
carried out to show the robustness of the relaxed filtered Krylov subspace
method and its great superiority over some state-of-the art iteration methods.
