On error estimation in the conjugate gradient method and why it works in finite precision computations
type: Item
instance of: scholarly article
title: On error estimation in the conjugate gradient method and why it works in finite precision computations
zbMATH Open document ID: 1026.65027
author: Zdeněk Strakoš
author: Petr Tichý
publication date: 2003-12-16T00:00:00Z
full work available at URL: https://eudml.org/doc/123075
full work available at URL: pp56-80.html
review text: The authors show that the lower bound for the \(A\)-norm of the error based on Gauss quadrature is mathematically equivalent to the original formula of \textit{M. R. Hestenes} and \textit{E. Stiefel} [J. Res. Natl. Bur. Stand. 49, 409-435 (1952; Zbl 0048.09901)]. Existing bounds are compared and the authors demonstrate the necessity of a proper roundoff error analysis with an example of the well-known bound that fails in finite precision arithmetic. The numerical stability of the simplest bound is proved. In addition a lower bound for the Euclidean norm is described. The results are illustrated by numerical examples.
reviewed by: F. Szidarovszky
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F25
Mathematics Subject Classification ID: 65G50
zbMATH DE Number: 1893806
zbMATH Keywords: conjugate gradient method
zbMATH Keywords: Gauss quadrature
zbMATH Keywords: convergence
zbMATH Keywords: error bounds
zbMATH Keywords: finite precision arithmetic
zbMATH Keywords: rounding errors
zbMATH Keywords: loss of orthogonality
zbMATH Keywords: numerical stability
zbMATH Keywords: numerical examples
describes a project that uses: mctoolbox
describes a project that uses: BiCGstab
MaRDI profile type: Publication
published in: ETNA - Electronic Transactions on Numerical Analysis
description: scientific article; zbMATH DE number 1893806