Iterative refinement for ill-conditioned linear systems

Abstract

This paper treats a linear equation $$Av = b,$$ where\(A \in {\mathbb{F}}^{n \times n} \) and\(b \in {\mathbb{F}}^n \). Here,\({\mathbb{F}}\) is a set of floating point numbers. Letu be the unit round-off of the working precision and κ(A)=‖A‖∞‖A −1‖∞ be the condition number of the problem. In this paper, ill-conditioned problems with $$1< u\kappa ({\rm A})< \infty $$ 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.