Second-order divided-difference filter using a generalized complex-step approximation

This paper presents a framework for the second-order divided-difference Alter using a generalized complex-step derivative approximation. For first derivatives the complex-step approach does not suffer substraction cancelation errors as in standard numerical finite-difference approaches. Therefore, since an arbitrarily small step size can be chosen, the complex-step method can achieve near analytical accuracy. However, for second derivatives straight implementation of the complex-step approach does suffer from roundoff errors. Therefore, an arbitrarily small step size cannot be chosen. Previous work expanded upon the standard complex-step approach to provide a wider range of accuracy for both the first and second derivative approximations. The new extensions can allow the use of one step size to provide optimal accuracy for both derivative approximations, which are used in the divided-difference filter in order to improve its accuracy. Simulation results are provided to show the performance of the new complex-step approximations on the divided-difference filter.