The complex-step Newton method and its convergence

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.