  Many popular eigensolvers for large and sparse Hermitian matrices or matrix
pairs can be interpreted as accelerated block preconditioned gradient (BPG)
iterations in order to analyze their convergence behavior by composing known
estimates. An important feature of BPG is the cluster robustness, i.e.,
reasonable performance for computing clustered eigenvalues is ensured by a
sufficiently large block size. This feature can easily be explained for
exact-inverse (exact shift-inverse) preconditioning by adapting classical
estimates on nonpreconditioned eigensolvers, whereas the existing results for
more general preconditioning are still improvable. We expect to extend certain
sharp estimates for the corresponding vector iterations to BPG where proper
bounds of convergence rates of individual Ritz values are to be derived. Such
an extension has been achieved for BPG with fixed step sizes in [Math. Comp. 88
(2019), 2737--2765]. The present paper deals with the more practical case that
the step sizes are implicitly optimized by the Rayleigh-Ritz method. Our new
estimates improve some previous ones in view of concise and more flexible
bounds.
