Convergence rates of individual Ritz values in block preconditioned gradient-type eigensolvers

Abstract

Many popular eigensolvers for large and sparse Hermitian matrices or matrix pairs can be interpreted as accelerated block preconditioned gradient (BPG) iterations for the purpose of analyzing their convergence behavior by composing known estimates. An important feature of the BPG method is the cluster robustness, i.e., that reasonable performance for computing clustered eigenvalues is ensured by a sufficiently large block size. Concise estimates reflecting this feature can easily be derived for exact-inverse (exact shift-inverse) preconditioning. Therein, the BPG method is compatible with an abstract block iteration analyzed by Knyazev [Soviet J. Numer. Anal. Math. Modelling, 2 (1987), pp. 371-396]. An adaptation to more general preconditioning is difficult as some orthogonality properties cannot be preserved. Another analysis by Ovtchinnikov [Linear Algebra Appl., 415 (2006), pp. 140-166] provides sumwise estimates for Ritz values containing elegant convergence factors. However, additional technical terms lead to cumbersome bounds and could cause overestimations in the first steps. We expect to improve the existing results by deriving concise estimates for individual Ritz values. A mid-term goal has been achieved for the BPG iteration with fixed step sizes by the authors in [Math. Comp., 88 (2019), pp. 2737-2765. The present paper deals with the more practical case that the step sizes are implicitly optimized by the Rayleigh-Ritz method.