A comparison of mixed precision iterative refinement approaches for least-squares problems

TL;DR

This paper compares different mixed precision iterative refinement methods for least-squares problems, analyzing their theoretical guarantees, sensitivities, and extending analysis to two-precision cases, aiding in selecting suitable approaches.

Contribution

It provides a comprehensive comparison of three IR approaches, extends uniform precision analysis to two-precision semi-normal equations, and introduces a new multi-problem solving approach.

Findings

IR methods have different sensitivities to problem conditioning

The analysis extends to two-precision semi-normal equations

A new multi-least-squares approach is proposed

Abstract

Various approaches to iterative refinement (IR) for least-squares problems have been proposed in the literature and it may not be clear which approach is suitable for a given problem. We consider three approaches to IR for least-squares problems when two precisions are used and review their theoretical guarantees, known shortcomings and when the method can be expected to recognize that the correct solution has been found, and extend uniform precision analysis for an IR approach based on the semi-normal equations to the two-precision case. We focus on the situation where it is desired to refine the solution to the working precision level. It is shown that the IR methods exhibit different sensitivities to the conditioning of the problem and the size of the least-squares residual, which should be taken into account when choosing the IR approach. We also discuss a new approach that is based on solving multiple least-squares problems.