On the generalization of the Complex Step Method

We generalize the well known Complex Step Method for computing derivatives by introducing a complex step in a strict sense. Exploring different combinations of terms, we derive 52 approximations for computing the first order derivatives and 43 for the second order derivatives. For an appropriate combination of terms and appropriate choice of the step size in the real and imaginary directions, fourth order accuracy can be achieved in a very simple and efficient scheme on a compact stencil. New different ways of computing second order derivatives in one single step are shown. Many of the first order derivative approximations avoid the problem of subtractive cancellation inherent to the classic finite difference approximations for real valued steps, and the superior accuracy and stability of the generalized complex step approximations are demonstrated for an analytic test function.