With the recent emergence of mixed precision hardware, there has been a renewed interest in its use for solving numerical linear algebra problems fast and accurately. The solution of least squares (LS) problems $\min_x\|b-Ax\|_2$, where $A \in \mathbb{R}^{m\times n}$, arise in numerous application areas. Overdetermined standard least squares problems can be solved by using mixed precision within the iterative refinement method of Bj\"{o}rck, which transforms the least squares problem into an $(m+n)\times(m+n)$ ''augmented'' system. It has recently been shown that mixed precision GMRES-based iterative refinement can also be used, in an approach termed GMRES-LSIR. In practice, we often encounter types of least squares problems beyond standard least squares, including weighted least squares (WLS), $\min_x\|D^{1/2}(b-Ax)\|_2$, where $D^{1/2}$ is a diagonal matrix of weights. In this paper, we discuss a mixed precision FGMRES-WLSIR algorithm for solving WLS problems using two different preconditioners.

翻译：随着混合精度硬件的最新出现，人们对其在快速准确求解数值线性代数问题中的使用重新产生了兴趣。最小二乘问题$\min_x\|b-Ax\|_2$（其中$A \in \mathbb{R}^{m\times n}$）的求解出现在众多应用领域中。过定标准最小二乘问题可以通过在Björck迭代精化方法中使用混合精度来求解，该方法将最小二乘问题转化为$(m+n)\times(m+n)$的“增广”系统。最近研究表明，基于混合精度GMRES的迭代精化也可用于此目的，该方法被称为GMRES-LSIR。在实际应用中，我们常会遇见超出标准最小二乘范围的最小二乘问题类型，包括加权最小二乘问题（WLS）$\min_x\|D^{1/2}(b-Ax)\|_2$，其中$D^{1/2}$为权重对角矩阵。本文讨论了一种使用两种不同预条件子求解WLS问题的混合精度FGMRES-WLSIR算法。