Using implicitly filtered RKS for generalised eigenvalue problems (Q1300714)

This paper discusses properties and applications of the rational Krylov sequence (RKS) for generalised eigenvalue problems. The RKS method is considered as a generalisation of the Arnoldi method. It reduces a matrix pencil into a small upper Hessenberg matrix pencil in the rational Krylov subspace. Two approaches and their implementations for implicitly restarting the rational Krylov sequence are discussed. Combining the restarting of the rational Krylov sequence with the implicit filtering of the Krylov subspace, the authors show why the filtering step can fail and propose to solve this problem by deflating a converged eigenvector from the subspace, based on a Schur decomposition.