Closed-form expressions for the nite di erence approximations of rst and higher derivatives based on Taylor series

Numerical di erentiation formulas based on interpolating polynomials, operators and lozenge diagrams can be simpli ed to one of the nite di erence approximations based on Taylor series. In this paper, we have presented closed-form expressions of these approximations of arbitrary order for rst and higher derivatives. A comparison of the three types of approximations is given with an ideal digital di erentiator by comparing their frequency responses. The comparison reveals that the central di erence approximations can be used as digital di erentiators, because they do not introduce any phase distortion and their amplitude response is closer to that of an ideal di erentiator. It is also observed that central di erence approximations are in fact the same as maximally at digital di erentiators. In the appendix, a computer program, written in MATHEMATICA is presented, which can give the approximation of any order to the derivative of a function at a certain mesh point. c © 1999 Elsevier Science B.V. All rights reserved.