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.

翻译：文献中已提出多种针对最小二乘问题的迭代精化方法，对于特定问题应选用何种方法可能并不明确。本文研究采用双精度计算时最小二乘问题的三种迭代精化方法，系统评述其理论保证、已知缺陷、以及方法何时能识别出已获得正确解的条件，并将基于半正规方程的迭代精化方法在均匀精度下的分析框架扩展至双精度情形。研究聚焦于期望将解精化至工作精度水平的情境。分析表明，不同迭代精化方法对问题条件数与最小二乘残差规模的敏感性存在差异，选择迭代精化方法时应将此纳入考量。文中还讨论了一种基于求解多重最小二乘问题的新方法。