Abstract perturbed Krylov methods

We introduce the framework of “abstract perturbed Krylov methods”. This is a new and unifying point of view on Krylov subspace methods based solely on the matrix equation and the assumption that the matrix Ck is unreduced Hessenberg. We give polynomial expressions relating the Ritz vectors, quasi-orthogonal residual iterates and quasi-minimal residual iterates to the starting vector q1 and the perturbation term Fk. The properties of these polynomials and similarities between them are analyzed in some detail. The results suggest the interpretation of abstract perturbed Krylov methods as additive overlay of several abstract exact Krylov methods.