Iterative Refinement for Ill-Conditioned Linear Systems

Abstract

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.