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 find 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.