Considered herein is a modified Newton method for the numerical solution of nonlinear equations where the Jacobian is approximated using a complex-step derivative approximation. We show that this method converges for sufficiently small complex-step values, which need not be infinitesimal. Notably, when the individual derivatives in the Jacobian matrix are approximated using the complex-step method, the convergence is linear and becomes quadratic as the complex-step approaches zero. However, when the Jacobian matrix is approximated by the nonlinear complex-step derivative approximation, the convergence rate remains quadratic for any appropriately small complex-step value, not just in the limit as it approaches zero. This claim is supported by numerical experiments. Additionally, we demonstrate the method's robust applicability in solving nonlinear systems arising from differential equations, where it is implemented as a Jacobian-free Newton-Krylov method.

翻译：本文研究了一种改进的牛顿法，用于数值求解非线性方程组，其中雅可比矩阵通过复步导数逼近进行近似。我们证明该方法在复步长充分小时收敛，且该步长无需无穷小。特别值得注意的是，当雅可比矩阵中的各阶导数采用复步法近似时，收敛速度为线性；而当复步长趋近于零时，收敛速度提升为二次。然而，若采用非线性复步导数逼近来近似雅可比矩阵，则对于任意适当小的复步长（不仅限于趋近于零的极限情况），收敛速率始终保持二次。数值实验支持了这一论断。此外，我们通过将本方法实现为无雅可比矩阵的牛顿-克雷洛夫方法，展示了其在求解微分方程导出的非线性方程组中具有鲁棒适用性。