SIAM Review, Volume 64, Issue 1, Page 191-211, February 2022.
We investigate the use of reduced precision arithmetic to solve the linear equation for the Newton step. If one neglects the backward error in the linear solve, then well-known convergence theory implies that using single precision in the linear solve has very little negative effect on the nonlinear convergence rate. However, if one considers the effects of backward error, then the usual textbook estimates are very pessimistic and even the state-of-the-art estimates using probabilistic rounding analysis do not fully conform to experiments. We report on experiments with a specific example. We store and factor Jacobians in double, single, and half precision. In the single precision case we observe that the convergence rates for the nonlinear iteration do not degrade as the dimension increases and that the nonlinear iteration statistics are essentially identical to the double precision computation. In half precision we see that the nonlinear convergence rates, while poor, do not degrade as the dimension increases.

中文翻译：

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混合精度中的牛顿法

SIAM 评论，第 64 卷，第 1 期，第 191-211 页，2022 年 2 月。
我们研究了使用降低精度算术来求解牛顿步的线性方程。如果忽略线性求解中的后向误差，那么众所周知的收敛理论意味着在线性求解中使用单精度对非线性收敛速度几乎没有负面影响。然而，如果考虑后向误差的影响，那么通常的教科书估计是非常悲观的，甚至使用概率舍入分析的最先进的估计也不完全符合实验。我们用一个具体的例子报告实验。我们以双精度、单精度和半精度存储和分解雅可比矩阵。在单精度情况下，我们观察到非线性迭代的收敛速度不会随着维度的增加而降低，并且非线性迭代统计与双精度计算基本相同。在半精度中，我们看到非线性收敛速度虽然很差，但不会随着维度的增加而降低。