Error-free computer solution of certain system of linear equations
type: Item
instance of: scholarly article
title: Error-free computer solution of certain system of linear equations
zbMATH Open document ID: 0623.65034
DOI: 10.1016/0377-0427(87)90002-1
author: Stephen F. Chang
author: W. J. Kennedy
published in: Journal of Computational and Applied Mathematics
publication date: 1987-01-01T00:00:00Z
review text: The authors develop a procedure which generates the exact solution for the system \(Ax=b\), where A is an integral nonsingular matrix and b is an integral vector, by improving the initial floating-point approximation to the solution. This procedure is based on an easily programmed method developed earlier by \textit{O. Aberth} [ibid. 4, 285-288 (1978; Zbl 0419.65024)]. This procedure first computes the approximate floating-point solution \(x^*\) by using an available linear equation solving algorithm. Then it extracts the exact solution x from \(x^*\) if the error in the approximation \(x^*\) is sufficiently small. An a posteriori upper bound for the error of \(x^*\) is derived when Gaussian elimination with partial pivoting is used. As a byproduct of the Gaussian elimination process, a computable upper bound for \(| \det (A)|\) is obtained, which is an alternative to using Hadamard's inequality.
Mathematics Subject Classification ID: 65F05
Mathematics Subject Classification ID: 15B36
Mathematics Subject Classification ID: 65F40
zbMATH DE Number: 4011775
zbMATH Keywords: error-free computer solution
zbMATH Keywords: computable error bounds
zbMATH Keywords: exact solution
zbMATH Keywords: integral nonsingular matrix
zbMATH Keywords: floating-point approximation
zbMATH Keywords: Gaussian elimination with partial pivoting
zbMATH Keywords: Hadamard's inequality
reviewed by: Prem Narain
MaRDI profile type: Publication
full work available at URL: 0377-0427(87)90002-1
OpenAlex ID: W2013302193
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description: scientific article; zbMATH DE number 4011775