A note on conjugate gradient convergence. III
type: Item
instance of: scholarly article
title: A note on conjugate gradient convergence. III
zbMATH Open document ID: 1013.65029
author: Shlomo Engelberg
author: Aaron E. Naiman
published in: Numerische Mathematik
publication date: 2003-06-26T00:00:00Z
review text: In this third paper of a series [Part I: ibid. 76, No. 2, 209-230 (1997; Zbl 0905.65047); Part II: ibid. 85, No. 4, 655-683 (2000; reviewed above)], the authors collect the previous results. They show that for some right-hand side (r.h.s.) vectors, one iteration will bring a certain improvement independent of the size of the matrix -- derived from the discretization of a two-point boundary value problem. For other r.h.s. vectors the improvement tends to zero as the size of the system is increased. In the end it is concluded that if the r.h.s. vector projects only on a few eigenvectors, especially at the ends of the spectrum, conjugate gradients brings quick improvements, whereas if the r.h.s. vector projects equally onto many eigenvectors, conjugate gradients can be very slow -- a fact that is already well known.
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 34B05
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 65L12
zbMATH DE Number: 1491750
zbMATH Keywords: conjugate gradient method
zbMATH Keywords: convergence
zbMATH Keywords: two-point boundary value problem
reviewed by: Hermann G. Matthies
MaRDI profile type: Publication
DOI: 10.1007/S002110000139
description: scientific article; zbMATH DE number 1491750