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

TL;DR

This paper analyzes the convergence rates of individual Ritz values in block preconditioned gradient eigensolvers, providing improved bounds for practical step size optimization methods.

Contribution

It extends existing convergence estimates to more practical BPG methods with Rayleigh-Ritz optimized step sizes, improving the bounds and robustness analysis.

Findings

Improved convergence bounds for Ritz values in BPG eigensolvers.

Enhanced understanding of cluster robustness with block size.

Extension of sharp estimates to practical step size scenarios.

Abstract

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.