Block CG algorithms revisited

TL;DR

This paper clarifies the relationship between block Lanczos and block conjugate gradient algorithms, introduces practical variants for finite precision computations, and demonstrates their effectiveness through numerical experiments.

Contribution

It establishes a one-to-one correspondence between block Lanczos and BCG algorithms under full rank assumptions and discusses variants suitable for finite precision arithmetic.

Findings

Block Lanczos coefficients can be computed within BCG.

Variants of BCG avoid rank deficiency issues.

Numerical results show the effectiveness of proposed variants.

Abstract

Our goal in this paper is to clarify the relationship between the block Lanczos and the block conjugate gradient (BCG) algorithms. Under the full rank assumption for the block vectors, we show the one-to-one correspondence between the algorithms. This allows, for example, the computation of the block Lanczos coefficients in BCG. The availability of block Jacobi matrices in BCG opens the door for further development, e.g., for error estimation in BCG based on (modified) block Gauss quadrature rules. Driven by the need to get a practical variant of the BCG algorithm well suited for computations in finite precision arithmetic, we also discuss some important variants of BCG due to Dubrulle. These variants avoid the troubles with a possible rank deficiency within the block vectors. We show how to incorporate preconditioning and computation of Lanczos coefficients into these variants. We hope to help clarify which variant of the block conjugate gradient algorithm should be used for computations in finite precision arithmetic. Numerical results illustrate the performance of different variants of BCG on some examples.