Stable iterative refinement algorithms for solving linear systems

TL;DR

This paper introduces a new iterative refinement algorithm with a line search step that guarantees convergence, improving the reliability of solving linear systems on low-precision hardware.

Contribution

A novel IR enhancement with a line search that ensures convergence regardless of initial error magnitude.

Findings

The proposed method guarantees convergence in all cases.

Numerical experiments confirm the theoretical convergence guarantees.

The scheme improves solution accuracy on low-precision hardware.

Abstract

Iterative refinement (IR) is a popular scheme for solving a linear system of equations based on gradually improving the accuracy of an initial approximation. Originally developed to improve upon the accuracy of Gaussian elimination, interest in IR has been revived because of its suitability for execution on fast low-precision hardware such as analog devices and graphics processing units. IR generally converges when the error associated with the solution method is small, but is known to diverge when this error is large. We propose and analyze a novel enhancement to the IR algorithm by adding a line search optimization step that guarantees the algorithm will not diverge. Numerical experiments verify our theoretical results and illustrate the effectiveness of our proposed scheme.