We describe a three precision variant of Newton's method for nonlinear equations. We evaluate the nonlinear residual in double precision, store the Jacobian matrix in single precision, and solve the equation for the Newton step with iterative refinement with a factorization in half precision. We analyze the method as an inexact Newton method. This analysis shows that, except for very poorly conditioned Jacobians, the number of nonlinear iterations needed is the same that one would get if one stored and factored the Jacobian in double precision. In many ill-conditioned cases one can use the low precision factorization as a preconditioner for a GMRES iteration. That approach can recover fast convergence of the nonlinear iteration. We present an example to illustrate the results.

翻译：本文描述了一种用于非线性方程的三精度牛顿法变体。我们使用双精度计算非线性残差，以单精度存储雅可比矩阵，并通过半精度矩阵分解结合迭代精化技术求解牛顿步。我们将其视为非精确牛顿法进行分析。分析表明，除非雅可比矩阵病态程度极高，否则所需的非线性迭代次数与采用双精度存储并分解雅可比矩阵时相同。在许多病态情形下，可将低精度分解作为GMRES迭代的预条件子。该方法能够恢复非线性迭代的快速收敛。我们通过算例验证了上述结论。