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：我们描述了一种用于非线性方程的三精度牛顿法变体。该方法采用双精度计算非线性残差，单精度存储雅可比矩阵，并通过半精度分解的迭代精化求解牛顿步方程。我们将该方法分析为一种不精确牛顿法。分析表明，除条件数极差的雅可比矩阵外，该方法的非线性迭代次数与采用双精度存储和分解雅可比矩阵时相同。对于许多病态情形，可将低精度分解作为GMRES迭代的预处理子，从而恢复非线性迭代的快速收敛。文中给出算例验证结论。