Convergence of Polynomial Restart Krylov Methods for Eigenvalue Computations

Krylov subspace methods have proved effective for many non-Hermitian eigenvalue problems, yet the analysis of such algorithms is involved. Convergence can be characterized by the angle the approximating subspace forms with a desired invariant subspace, resulting in a geometric framework that is robust to eigenvalue ill-conditioning. This paper describes a new bound on this angle that handles the complexities introduced by non-Hermitian matrices, yet has a simpler derivation than similar previous bounds. The new bound suggests that ill-conditioning of the desired eigenvalues exerts little influence on convergence, while instability of unwanted eigenvalues plays an essential role. Practical considerations restrict the dimension of the approximating Krylov space; to obtain convergence, one refines the vector that generates the subspace by applying a polynomial filter. Such filters dynamically steer a low-dimensional Krylov space toward a desired invariant subspace. We address the design of these filters, and illustrate with examples the subtleties that arise when restarting non-Hermitian iterations.