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Error norm estimates for the block conjugate gradient algorithm

Gérard Meurant, Petr Tichý

TL;DR

This work extends error-norm estimation for the conjugate gradient method to the block setting by deriving Block CG from the block Lanczos process and introducing block Gauss and Gauss-Radau quadrature rules to obtain computable lower and upper bounds on the $A$-norm of the error for each right-hand side. It also analyzes the relations between Lanczos and CG coefficients, presents a concrete error-measurement framework for block systems, and addresses rank-deficiency with DR-BCG as a robust alternative. The numerical experiments demonstrate practical, tight bounds across multiple SPD problems with several right-hand sides, highlighting efficiency gains when solving multiple linear systems simultaneously. The results provide actionable error controls for block linear systems and guide robust algorithm choices in finite-precision computations.

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.

Error norm estimates for the block conjugate gradient algorithm

TL;DR

This work extends error-norm estimation for the conjugate gradient method to the block setting by deriving Block CG from the block Lanczos process and introducing block Gauss and Gauss-Radau quadrature rules to obtain computable lower and upper bounds on the -norm of the error for each right-hand side. It also analyzes the relations between Lanczos and CG coefficients, presents a concrete error-measurement framework for block systems, and addresses rank-deficiency with DR-BCG as a robust alternative. The numerical experiments demonstrate practical, tight bounds across multiple SPD problems with several right-hand sides, highlighting efficiency gains when solving multiple linear systems simultaneously. The results provide actionable error controls for block linear systems and guide robust algorithm choices in finite-precision computations.

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 with a symmetric positive definite matrix , where and are vectors. In this paper, we generalize the most important formulas for estimating the -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 -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.

Paper Structure

This paper contains 10 sections, 7 theorems, 128 equations, 4 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. The block Lanczos algorithm
  3. Block CG from block Lanczos
  4. Relations between the coefficients of the block Lanczos and BCG algorithms
  5. How to measure the error
  6. The block Gauss quadrature rule
  7. The block Gauss-Radau quadrature rule
  8. Rank deficiency
  9. Numerical experiments
  10. Conclusion

Key Result

theorem 1

Consider Algorithm alg-BCG and assume that the block vectors $R_{i}$, $i=0,\dots,k-1$, are of full column rank. Then it holds that and the matrix is symmetric and positive definite.

Figures (4)

  • Figure 1: Poisson equation, standard BCG, first column, $A$-norm of the error and bounds: Gauss (dashed green) and Gauss-Radau (dot-dashed red)
  • Figure 2: bcsstk01, standard BCG, first column, $A$-norm of the error and bounds: Gauss (dashed green) and Gauss-Radau (dot-dashed red)
  • Figure 3: 662_bus, standard BCG, first column, $A$-norm of the error and bounds: Gauss (dashed green) and Gauss-Radau (dot-dashed red)
  • Figure 4: nos7, standard BCG, first column, $A$-norm of the error and bounds: Gauss (dashed green) and Gauss-Radau (dot-dashed red)

Theorems & Definitions (14)

  • theorem 1
  • proof
  • proposition 1
  • proof
  • lemma 1
  • proof
  • lemma 2
  • proof
  • theorem 2
  • proof
  • ...and 4 more