Error norm estimates for the block conjugate gradient algorithm

TL;DR

This paper extends error norm estimation techniques from the classical conjugate gradient method to the block conjugate gradient algorithm, providing bounds and practical insights for symmetric positive definite systems.

Contribution

It generalizes key error estimation formulas to the block CG algorithm and derives related quadrature rules, enhancing error analysis for block iterative methods.

Findings

Derived bounds for the $A$-norm of the error in block CG

Connected error estimates to block Lanczos algorithm quantities

Numerical experiments validate the bounds' effectiveness

Abstract

In the book [Meurant and Tichy, SIAM, 2024] we discussed the estimation of error norms in the conjugate gradient (CG) algorithm for solving linear systems $Ax=b$ with a symmetric positive definite matrix $A$, where $b$ and $x$ are vectors. In this paper, we generalize the most important formulas for estimating the $A$-norm of the error to the block case. First, we discuss in detail the derivation of various variants of the block CG (BCG) algorithm from the block Lanczos algorithm. We then consider BCG and derive the related block Gauss and block Gauss-Radau quadrature rules. We show how to obtain lower and upper bounds on the $A$-norm of the error of each system, both in terms of the quantities computed in BCG and in terms of the underlying block Lanczos algorithm. Numerical experiments demonstrate the behavior of the bounds in practical computations.