Cluster robust estimates for block gradient-type eigensolvers

Sharp convergence estimates have been derived in recent years for gradient-type eigensolvers for large and sparse symmetric matrices or matrix pairs. An extension of these estimates to the corresponding block iterative methods can be achieved by applying a similar analysis to an embedded vector iteration. Although the resulting estimates are also sharp in the sense that they are not improvable without further assumptions, they cannot reflect the well-known cluster robustness of block eigensolvers. In the present paper, we analyze the cluster robustness of the preconditioned inverse subspace iteration. The main estimate has a weaker assumption and a simpler form compared to some known cluster robust estimates. In addition, it is applicable to further block gradient-type eigensolvers such as the locally optimal block preconditioned conjugate gradient method. The analysis is based on an orthogonal splitting for the block power method and a geometric interpretation of preconditioning. As a by-product, a cluster robust Ritz value estimate for the block power method is improved.