Mixed precision iterative refinement for least squares with linear equality constraints and generalized least squares problems

TL;DR

This paper introduces mixed precision iterative refinement algorithms for constrained and generalized least squares problems, significantly improving solution accuracy and reducing computation time on well-conditioned problems.

Contribution

It extends mixed precision iterative refinement techniques to LSE and GLS problems, demonstrating substantial performance gains over fixed precision methods.

Findings

Algorithms reduce execution time by 40% on average.

Both classical and GMRES-based refinements improve accuracy.

Effective for well-conditioned problems.

Abstract

Recent development on mixed precision techniques has largely enhanced the performance of various linear algebra solvers, one of which being the solver for the least squares problem $\min_{x}\lVert b-Ax\rVert_{2}$. By transforming least squares problems into augmented linear systems, mixed precision techniques are capable of refining the lower precision solution to the working precision. In this paper, we propose mixed precision iterative refinement algorithms for two variants of least squares problems -- the least squares problem with linear equality constraints (LSE) and the generalized least squares problem (GLS). Both classical and GMRES-based iterative refinement can be applied to augmented systems of these two problems to improve the accuracy of the solution. For reasonably well-conditioned problems, our algorithms reduce the execution time by a factor of 40% on average compared to the fixed precision ones from LAPACK on the x86-64 architecture.