Angle‐Free Cluster‐Robust Ritz Value Bounds for Restarted Block Eigensolvers

Abstract

Convergence rates of block iterations for solving Hermitian eigenvalue problems typically measure the errors of Ritz values approximating eigenvalues. These errors are usually bounded in terms of principal angles between the initial or iterative subspace and the invariant subspace associated with the target eigenvalues. Such bounds cannot be applied repeatedly as necessary for investigating multiple steps of restarted block eigensolvers since the left and right‐hand sides of the bounds use respectively angle‐free and angle‐dependent terms. They must be combined with additional bounds, which could cause an overestimation. Alternative bounds that are completely angle‐free and depend only on the errors of Ritz values have been introduced decades ago but only for a single extreme Ritz value. We revisit and extend these bounds to all Ritz values via repeatable intermediate bounds that enable a priori cluster‐robust bounds for restarted block Krylov subspace eigensolvers.