Krylov-Schur-Type restarts for the two-sided arnoldi method

We consider the two-sided Arnoldi method and propose a two-sided Krylov-Schurtype restarting method. We discuss the restart for standard Rayleigh-Ritz extraction as well as harmonic Rayleigh-Ritz extraction. Additionally, we provide error bounds for Ritz values and Ritz vectors in the context of oblique projections and present generalizations of, e.g., the Bauer-Fike theorem and Saad's theorem. Applications of the two-sided Krylov-Schur method include the simultaneous computation of left and right eigenvectors and the computation of eigenvalue condition numbers. We demonstrate how the method can be used to -nd the least sensitive eigenvalues of a nonnormal matrix and how to approximate pseudospectra by using left and right shift-invariant subspaces. The results demonstrate that significant improvements in quality can be obtained over approximations with the (one-sided) Krylov-Schur method.