The bordering method as a preconditioning method
type: Item
instance of: scholarly article
title: The bordering method as a preconditioning method
zbMATH Open document ID: 0913.65027
published in: Moscow University Computational Mathematics and Cybernetics
publication date: 1999-03-21T00:00:00Z
review text: The problem of solving a system of linear algebraic equations \[ Ax = b \tag{1} \] where \(A\) is a symmetric positive definite \((n\times n)\)-matrix is considered. Let \(\{ \lambda_{i} , v^{i} \}\) \((i = 1, 2,\dots, n)\) be the eigenvalues and eigenvectors of \(A\), where \(\lambda_{1} \leq \lambda_{2} \leq \cdots \leq \lambda_{n}\). Let the vector \[ x =\sum_{i=1}^{n}x_{i} v^{i} \tag{2} \] be a solution of the system (1). If the matrix \(A\) of the system (1) is an almost degenerate then its eigenvalues are close to zero. In this case methods of approximation (conjugate gradient method, for example) give a big value for some components of the vector of errors. In the article a method of bordering of matrices (usually used for obtaining inverse matrices) is proposed for obtaining an absolutly degenerate matrix for the system (1) and then the conjugate gradient method is becoming more effective.
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F35
zbMATH DE Number: 1266872
zbMATH Keywords: bordering method
zbMATH Keywords: preconditioning
zbMATH Keywords: degenerate matrices
zbMATH Keywords: symmetric positive definite matrix
zbMATH Keywords: conjugate gradient method
author: Owe Axelsson
author: M. Neytcheva
author: Ben Polman
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1266872