Recent development on mixed precision techniques has largely enhanced the performance of various linear algebra solvers, one of which being the least squares problem $\min_{x}\lVert b-Ax\rVert_{2}$. By transforming the least squares problem into an augmented linear system, 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 the least squares problem -- 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% in average compared to the fixed precision ones from LAPACK on the x86-64 architecture.

翻译：混合精度技术的最新发展极大地提升了各类线性代数求解器的性能，其中之一便是最小二乘问题 $\min_{x}\lVert b-Ax\rVert_{2}$。通过将最小二乘问题转化为增广线性系统，混合精度技术能够将低精度解精化至工作精度。本文针对最小二乘问题的两种变体——带线性等式约束的最小二乘问题（LSE）与广义最小二乘问题（GLS），提出了混合精度迭代精化算法。经典迭代精化与基于GMRES的迭代精化均可应用于这两个问题的增广系统，以提高解的精度。对于条件数适中的问题，在x86-64架构上，我们的算法相较于LAPACK中的固定精度算法，平均可减少约40%的执行时间。