       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

??????  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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