The solution of linear systems by using the Sherman-Morrison formula
type: Item
instance of: scholarly article
title: The solution of linear systems by using the Sherman-Morrison formula
zbMATH Open document ID: 1111.65027
author: P. Maponi
published in: Linear Algebra and its Applications
publication date: 2007-01-09T00:00:00Z
review text: The paper presents the solution of the linear system \(Ax=b\) (LS) by using the Sherman-Morrison formula (SH), where \(A\) is the nonsingular matrix, \(b\) is the known vector, and \(x\) is the vector to be determined. Consider the matrix \(A=A_o+u_1v_1^T+\dots u_Mv_M^T\), where \(A, A_o\in \mathbb{R}^{N\times N}\), and \(u_j, v_j\in \mathbb{R}^N, j=1,\dots,M\). Suppose \(x=x_l\) solves \(A_lx=b\) and for \(k=l+1,\dots,M\) \(y\in \mathbb{R}^N.\) \(x_l\) solves the LS for \(l=M\). Given \(x_o\) and a nonsingular \(A_o\) results in \(A_l=A_{l-1}+u_lv_l^T\) for \(l>0\). The repeated application of the SH on \(A_l\) leads to the solution \(x=x_M\). The convergence analysis and the computational costs are considered. When some \(A_l\) is singular, the partial pivoting technique of the Gauss elimination method is applied to compute the approximate solution \(x=x_M\). The numerical comparison tests of this method with the restarted generalized minimum residual method and the Gaussian elimination method with partial pivoting are supplied for randomly generated matrices as well as for the LS with Pascal, Cauchy, and Vandermonde matrices. The method is also applied in the preconditioning of large sparse nonsymmetric LS.
Mathematics Subject Classification ID: 65F05
Mathematics Subject Classification ID: 65F35
Mathematics Subject Classification ID: 65F50
zbMATH DE Number: 5083602
zbMATH Keywords: direct methods
zbMATH Keywords: approximate inverse preconditioner
zbMATH Keywords: sparse matrix
zbMATH Keywords: numerical examples
zbMATH Keywords: Pascal matrix
zbMATH Keywords: Vandermonde matrix
zbMATH Keywords: Cauchy matrix
zbMATH Keywords: convergence
zbMATH Keywords: partial pivoting
zbMATH Keywords: Gauss elimination
reviewed by: Lubomír Bakule
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describes a project that uses: NAG
describes a project that uses: LAPACK
describes a project that uses: nag
describes a project that uses: mctoolbox
describes a project that uses: SparseMatrix
MaRDI profile type: Publication
full work available at URL: j.laa.2006.07.007
OpenAlex ID: W2070948799
DOI: 10.1016/J.LAA.2006.07.007
description: scientific article; zbMATH DE number 5083602