RE: Can Octave solve this kind of numerical analysis problem ???

[Ted Harding]

On 24-Dec-99 Ramesh S Guzar wrote:
> Hi,
> 
> I have a problem which is related to the numerical analysis.
> Hence I thought I would seek your help on this.
> 
> I have a Parameter "Z" which depends on 5 independent variables
> l, m, n , o and p.
> A1, A2 ......An are the values of Z for (l1,m1,n1,o1,p1) .....
> to (ln,mn,nn,on,pn) co-ordinates.
> 
> 
> Is it possible to fit a curve or get an expression for "Z" interms
> of the know points like A1,A2.....An, and interpolate the value
> of "Z" for any (l,m,n,o,p) set ? Can Octave do this ? If not 
> directly, do you have any suggestions on how to use Octave to 
> solve this problem ? 

The short answer is "Yes, you could do this in octave".

The generic method would be: Choose appropriate fitting
and interpolation methods for your class of problem, and
then (which might not necessarily be very hard) implement
them in octave.

Beyond this, not much can be said. Octave itself does not come
with an extensive repertoire of such methods, so you would not
be in the situation which some packages might offer, that you can
point and click on an array of methods icons and finally settle
on the one which seems to give the result you most like the look of.

In fact, one might say that using a package such as octave would
oblige you to carry out the necessary thought and investigation
required in order to solve your problem properly, in terms of
what you are trying to achieve and in terms of the nature of the
problem.

The sort of question you need to consider is:

1. Do you need the fitted function to exactly reproduce the
   given values A1, ... , An at the given points? Or would
   a fit of a simpler function, with discrepancies at these
   points, be acceptable?

2. Are the values A1, ... , An obtained by accurate computation
   from the values of (l, m, n, o, p) or are they the results
   of measurement (with error)? And, in the latter case, do
   the values of (l, m, n, o, p) also contain measurement
   error?

3. What do you know about the form of the function which relates
   the independent variables (l, m, n, o, p) to the value Z?

4. If you know the generic form of this function (to within
   the values of certain constants in its equation), do you
   want to exactly reproduce the whole course of this function
   for all values of (l, m, n, o, p) or only at the given points?

5. If the latter, what criterion of discrepancy do you want to
   adopt at points (l, m, n, o, p) other than the given points?
   (e.g. absolute error, relative error, squared error, ... ).

The various answers to questions such as these influence the
choice of methods to be used for such problems. There is no
unique approach; it depends on the problem itself and on
what you are trying to achieve.

As you say, the problem is related to numerical analysis;
and a good book on numerical analysis would have extensive
discussions of such questions. One of the wisest books on
the whole subject is R W Hamming's "Numerical Methods for
Scientists and Engineers", old though it be.

Ted.

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E-Mail: (Ted Harding) <address@hidden>
Date: 24-Dec-99                                       Time: 12:49:46
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