Mixed Precision FGMRES-Based Iterative Refinement for Weighted Least Squares

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.