A posteriori superlinear convergence bounds for block conjugate gradient

TL;DR

This paper extends superlinear convergence bounds to block conjugate gradient methods, providing theoretical insights and computational validation on how eigenvalue distribution and block size influence convergence behavior.

Contribution

It introduces a posteriori superlinear convergence bounds for block conjugate gradient methods, expanding previous scalar results to the block case with computational experiments.

Findings

Bounds validate superlinear convergence in block CG

Eigenvalue distribution affects onset of superlinearity

Block size influences convergence behavior

Abstract

In this paper, we extend to the block case, the a posteriori bound showing superlinear convergence of Conjugate Gradients developed in [J. Comput. Applied Math., 48 (1993), pp. 327-341]; that is, we obtain similar bounds, but now for block Conjugate Gradients. We also present a series of computational experiments illustrating the validity of the bound developed here, as well as the bound from [SIAM Review, 47 (2005), pp. 247-272] using angles between subspaces. Using these bounds, we make some observations on the onset of superlinearity, and how this onset depends on the eigenvalue distribution and the block size.