Open Access
October 2009 Iterative Refinement for Ill-Conditioned Linear Systems
Shin'ichi Oishi, Takeshi Ogita, Siegfried M. Rump
Japan J. Indust. Appl. Math. 26(2-3): 465-476 (October 2009).


This paper treats a linear equation \begin{equation*} Av=b, \end{equation*} where $A \in \mathbb{F}^{n\times n}$ and $b \in \mathbb{F}^n$. Here, $\mathbb{F}$ is a set of floating point numbers. Let $\mathbf{u}$ be the unit round-off of the working precision and $\kappa(A)=\|A\|_{\infty}\|A^{-1}\|_{\infty}$ be the condition number of the problem. In this paper, ill-conditioned problems with \begin{equation*} 1 < \mathbf{u}\kappa(A) < \infty \end{equation*} are considered and an iterative refinement algorithm for the problems is proposed. In this paper, the forward and backward stability will be shown for this iterative refinement algorithm.


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Shin'ichi Oishi. Takeshi Ogita. Siegfried M. Rump. "Iterative Refinement for Ill-Conditioned Linear Systems." Japan J. Indust. Appl. Math. 26 (2-3) 465 - 476, October 2009.


Published: October 2009
First available in Project Euclid: 1 February 2010

zbMATH: 1188.65053
MathSciNet: MR2589485

Keywords: ill-conditioned linear systems , iterative refinement , verified numerical computation

Rights: Copyright © 2009 The Japan Society for Industrial and Applied Mathematics

Vol.26 • No. 2-3 • October 2009
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