Re: Numerical Differentiation

*To*: mathgroup at smc.vnet.net*Subject*: [mg4341] Re: Numerical Differentiation*From*: tlm at ameslab.gov (Dr. T. L. Marchioro II)*Date*: Thu, 11 Jul 1996 00:59:18 -0400*Organization*: Iowa State University, Ames, Iowa*Sender*: owner-wri-mathgroup at wolfram.com

Mark Evans wrote: > > If you sample the function at even intervals, to obtain a "data stream," then > you can get the first derivative with a simple digital filtering operation > (discrete convolution using a special kernel). It is also possible to get the > second derivative, but I would not hold my hat for higher-order derivatives. > > If this approach appeals to you, then let me know and I will send you my > Mathematica work on the subject. This work was done under contract with the > specific objective of finding first- and second-order derivatives of a > discrete data stream. The Distributed Approximating Functionals which I discussed in a related post work very well for higher order derivatives as well, although with each increasing order the accuracy does decrease, but only "a bit", i.e., you can often find quite accurate high order derivatives. I would be happy to provide examples if you are interested. > It turns out that the temporal spacing of the data samples (the sampling > frequency) only contributes to the problem by introducing a scale factor. You > are free to select whatever sampling rate you like if you have an analytic > function, but as with most other problems, the more points you include, the > better the answer. ?????? I'm sorry, I must be misunderstanding what you are saying here.... perhaps because you are talking in terms of a data stream instead of an underlying analytic function... BUT!?!? If you sample the function at a rate lower than the Nyquist frequency then you simply do not capture all of its properties, hence the derivatives (and even your representation of the function) will be wrong. I really do not understand what you mean here, but would like to since it's an area I work in when time permits. Regards --- Tom _____________________________________________ Dr. Thomas L. Marchioro II Two-wheeled theoretical physicist Applied Mathematical Sciences 515-294-9779 Ames Laboratory 515-432-9142 (home) Ames, Iowa 50011 tlm at ameslab.gov Project Coordinator: Undergraduate Computational Engineering and Sciences http://uces.ameslab.gov/ ==== [MESSAGE SEPARATOR] ====