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近期发展，其在快速准确求解数值线性代数问题中的应用重新引起关注。最小二乘问题 $\min_x\|b-Ax\|_2$（其中 $A \in \mathbb{R}^{m\times n}$）的求解广泛应用于多个领域。超定标准最小二乘问题可通过使用Björck提出的迭代精化法在混合精度下求解，该方法将最小二乘问题转换为一个$(m+n)\times(m+n)$的"增广"系统。近期研究表明，基于GMRES的混合精度迭代精化法（称为GMRES-LSIR方法）也可用于此类问题。在实际应用中，我们常遇到超越标准最小二乘的多种类型最小二乘问题，包括加权最小二乘问题 $\min_x\|D^{1/2}(b-Ax)\|_2$，其中 $D^{1/2}$ 为对角权重矩阵。本文讨论一种采用两种不同预条件的混合精度FGMRES-WLSIR算法以求解加权最小二乘问题。