Angle-free cluster robust Ritz value bounds for restarted block eigensolvers

Abstract: Convergence rates of block iterations for solving eigenvalue problems typically measure errors of Ritz values approximating eigenvalues. The errors of the Ritz values are commonly bounded in terms of principal angles between the initial or iterative subspace and the invariant subspace associated with the target eigenvalues. Such bounds thus cannot be applied repeatedly as needed for restarted block eigensolvers, since the left- and right-hand sides of the bounds use different terms. They must be combined with additional bounds which could cause an overestimation. Alternative repeatable bounds that are angle-free and depend only on the errors of the Ritz values have been pioneered for Hermitian eigenvalue problems in doi:10.1515/rnam.1987.2.5.371 but only for a single extreme Ritz value. We extend this result to all Ritz values and achieve robustness for clustered eigenvalues by utilizing nonconsecutive eigenvalues. Our new bounds cover the restarted block Lanczos method and its modifications with shift-and-invert and deflation, and are numerically advantageous.