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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/



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