Sharp Ritz Value Estimates for Restarted Krylov Subspace Iterations

Gradient iterations for the Rayleigh quotient are elemental methods for computing the smallest eigenvalues of a pair of symmetric and positive definite matrices. A considerable convergence acceleration can be achieved by preconditioning and by computing Rayleigh-Ritz approximations from subspaces of increasing dimensions. An example of the resulting Krylov subspace eigensolvers is the generalized Davidson method. Krylov subspace iterations can be restarted in order to limit their computer storage requirements. For the restarted Krylov subspace eigensolvers, a Chebyshev type convergence estimate was presented by Knyazev in [Russian J. Numer. Anal. Math. Modelling, 2:371-396, 1987]. This estimate has been generalized to arbitrary eigenvalue intervals in [SIAM J. Matrix Anal. Appl., 37(3):955-975, 2016]. The generalized Ritz value estimate is not sharp as it depends only on three eigenvalues. In the present paper, we extend the latter analysis by generalizing the geometric approach from [SIAM J. Matrix Anal. Appl., 32(2):443-456, 2011] in order to derive a sharp Ritz value estimate for restarted Krylov subspace iterations.