New complex-step derivative approximations with application to second-order Kalman filtering

This paper presents several extensions of the complex-step approximation to compute numerical derivatives. For first derivatives the complex-step approach does not suffer substraction cancellation 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. In this paper we expand upon the standard complex-step approach to provide a wider range of accuracy for both the first and second derivative approximations. Several formulations are derived using various complex numbers coupled with Richardson extrapolations. The new extensions can allow the use of one step-size to provide optimal accuracy for both derivative approximations. Simulation results are provided to show the performance of the new complex-step approximations on a second-order Kalman filter.