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Mixed Precision FGMRES-Based Iterative Refinement for Weighted Least Squares
Erin Carson, Eda Oktay
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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 minₓb-Ax₂, where A ∈ℝ^(m×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)×(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), minₓD^(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.
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