Considered herein is a Jacobian-free Newton method for the numerical solution of nonlinear equations where the Jacobian is approximated using the complex-step derivative approximation. We demonstrate that this method converges for complex-step values sufficiently small and not necessarily tiny. Notably, in the case of scalar equations the convergence rate becomes quadratic as the complex-step tends to zero. On the other hand, in the case of systems of equations the rate is quadratic for any appropriately small value of the complex-step and not just in the limit to zero. This assertion is substantiated through numerical experiments. Furthermore, we demonstrate the method's seamless applicability in solving nonlinear systems that arise in the context of differential equations, employing it as a Jacobian-free Newton-Krylov method.

翻译：本文研究了一种免雅可比矩阵的牛顿法，用于求解非线性方程的数值解，其中雅可比矩阵采用复步长导数近似。我们证明了当复步长取值充分小（并非必须极小）时，该方法收敛。值得注意的是，对于标量方程，随着复步长趋近于零，收敛速度达到二次收敛；而对于方程组情形，只要复步长取值适当小（无需趋近于零），收敛速度即为二次。该论断通过数值实验得到验证。此外，我们展示了该方法在求解微分方程中出现的非线性系统时的无缝适用性，将其作为免雅可比矩阵的牛顿-克雷洛夫方法使用。