  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.
