Geometric interpretation of some Cauchy related methods
type: Item
instance of: scholarly article
title: Geometric interpretation of some Cauchy related methods
zbMATH Open document ID: 1235.65027
author: Alain Berlinet
author: Christophe Roland
published in: Numerische Mathematik
publication date: 2011-11-23T00:00:00Z
review text: For solving systems of linear equations \(Qz = b\) with a symmetric and positive definite matrix \(Q\), a class of iterative methods of the type \(x_{n+1} = x_n + 2 \alpha_n r_n + \alpha_n^2 Q r_n\) with \(r_n = Qx_n - b\) is considered. The real number \(\alpha_n\) is computed from \(x_0, x_1, \dots x_n\), \(Q\), and \(b\). This class includes the Cauchy-Barzilai-Borwein (CBB) method, presented by \textit{M.~Raydan} and \textit{B.~F.~Svaiter} [Comput. Optim. Appl. 21, No. 2, 155--167 (2002; Zbl 0988.90049)] which combines the Cauchy method (steepest descent method) and the Barzilai-Borwein method [see \textit{J.~Barzilai} and \textit{J.~M.~Borwein}, IMA J. Numer. Anal. 8, No. 1, 141--148 (1988; Zbl 0638.65055)]. A geometric interpretation of this class of iterative methods with the help of Bézier parabolas is given. This is used to localize the iterates of the Cauchy, the CBB, and the RBB methods in the plane of the Bézier parabola, where RBB is a combination of the classical Richardson and the Barzilai-Borwein methods. This localization gives some insight into the baviour of these methods. Finally, numerical examples are presented to compare the behaviour of the different methods.
reviewed by: Michael Jung
Mathematics Subject Classification ID: 65F10
zbMATH DE Number: 5977579
zbMATH Keywords: systems of linear equations
zbMATH Keywords: iterative solvers
zbMATH Keywords: steepest descent method
zbMATH Keywords: Barzilai-Borwein method
zbMATH Keywords: geometric interpretation of iterative methods
zbMATH Keywords: Richardson method
zbMATH Keywords: Cauchy-Barzilai-Borwein method
zbMATH Keywords: Bézier parabolas
zbMATH Keywords: numerical examples
MaRDI profile type: Publication
full work available at URL: s00211-011-0383-2
OpenAlex ID: W2062821291
cites work: Acceleration schemes with application to the EM algorithm
cites work: Two-Point Step Size Gradient Methods
cites work: Hybrid procedures for solving linear systems
cites work: Gradient Method with Retards and Generalizations
cites work: Q5543004
cites work: On the Barzilai and Borwein choice of steplength for the gradient method
cites work: Relaxed steepest descent and Cauchy-Barzilai-Borwein method
cites work: New iterative schemes for nonlinear fixed point problems, with applications to problems with bifurcations and incomplete-data problems
cites work: Squared polynomial extrapolation methods with cycling: an application to the positron emission tomography problem
DOI: 10.1007/S00211-011-0383-2
Recommended article: Geometric methods in rational interpolation theory
Recommended article: Q4676524
Recommended article: Q3754877
Recommended article: Q4407970
Recommended article: Geometric methods in partial differential equations
Recommended article: Q4947448
Recommended article: Q5321587
Recommended article: The Geometric Basis of Numerical Methods
Recommended article: Q4396651
Recommended article: Linear Methods in Geometric Function Theory
description: scientific article; zbMATH DE number 5977579