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 total least squares problems, i.e., solving $\min_{E,r} \| [E, r]\|_F$ subject to $(A+E)x=b+r$, arises in numerous applications. Solving this problem requires finding the smallest singular value and corresponding right singular vector of $[A,b]$, which is challenging when $A$ is large and sparse. An efficient algorithm for this case due to Bj\"{o}rck et al. [SIAM J. Matrix Anal. Appl. 22(2), 2000], called RQI-PCGTLS, is based on Rayleigh quotient iteration coupled with the preconditioned conjugate gradient method. We develop a mixed precision variant of this algorithm, RQI-PCGTLS-MP, in which up to three different precisions can be used. We assume that the lowest precision is used in the computation of the preconditioner, and give theoretical constraints on how this precision must be chosen to ensure stability. In contrast to standard least squares, for total least squares, the resulting constraint depends not only on the matrix $A$, but also on the right-hand side $b$. We perform a number of numerical experiments on model total least squares problems used in the literature, which demonstrate that our algorithm can attain the same accuracy as RQI-PCGTLS albeit with a potential convergence delay due to the use of low precision. Performance modeling shows that the mixed precision approach can achieve up to a $4\times$ speedup depending on the size of the matrix and the number of Rayleigh quotient iterations performed.

翻译：随着混合精度硬件的近期出现，其在快速且精确求解数值线性代数问题中的应用重新引起关注。总体最小二乘问题的求解，即求解满足约束条件$(A+E)x=b+r$的$\min_{E,r} \| [E, r]\|_F$问题，广泛出现在各类应用中。解决此问题需要寻找$[A,b]$的最小奇异值及其对应的右奇异向量，当$A$为大规模稀疏矩阵时这一计算极具挑战性。Björck等人[SIAM J. Matrix Anal. Appl. 22(2), 2000]针对该情形提出的高效算法RQI-PCGTLS，基于瑞利商迭代结合预处理共轭梯度法。我们开发了该算法的混合精度变体RQI-PCGTLS-MP，其中最多可混合使用三种不同精度。假设最低精度用于预处理器的计算，并给出该精度必须满足的理论约束条件以确保算法稳定性。与标准最小二乘问题不同，在总体最小二乘问题中，所得约束不仅依赖于矩阵$A$，还依赖于右侧向量$b$。我们针对文献中使用的经典总体最小二乘测试问题开展数值实验，结果表明：尽管因使用低精度可能导致潜在收敛延迟，但本算法可达到与RQI-PCGTLS相同的精度。性能建模分析显示，根据矩阵规模和瑞利商迭代次数，混合精度方法能实现高达4倍的加速比。