  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.
