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 $\min_{E,f} \| [E, f]\|_F$ subject to $(A+E)x=b+f$, arises in numerous application areas. The solution of 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., called RQI-PCGTLS, is based on Rayleigh quotient iteration coupled with the conjugate gradient method preconditioned via Cholesky factors. We develop a mixed precision variant of this algorithm, called 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 the standard least squares case, for total least squares problems, the constraint on this precision 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兴起，利用其快速精确求解数值线性代数问题再次引起关注。总最小二乘问题的解，即求解$\min_{E,f} \| [E, f]\|_F$，满足约束条件$(A+E)x=b+f$，广泛存在于诸多应用领域。该问题的解需要计算$[A,b]$的最小奇异值及对应的右奇异向量，当$A$为大型稀疏矩阵时极具挑战性。Björck等人针对此类问题提出了高效算法RQI-PCGTLS，该算法基于瑞利商迭代与通过乔列斯基因子预处理的共轭梯度法相结合。我们开发了该算法的混合精度变体RQI-PCGTLS-MP，可同时使用最多三种不同精度。我们假设预处理器的计算采用最低精度，并给出该精度必须满足的理论约束条件以确保稳定性。与标准最小二乘情形不同，对于总最小二乘问题，精度约束不仅取决于矩阵$A$，还与右端项$b$相关。我们针对文献中采用的模型总最小二乘问题进行了大量数值实验，结果表明所提算法能获得与RQI-PCGTLS相当的精度，但可能因低精度使用导致收敛延迟。性能建模显示，根据矩阵规模及瑞利商迭代次数，混合精度方法可实现最高4倍的加速比。