       Interpolation

• To: mathgroup at smc.vnet.net
• Subject: [mg32285] Interpolation
• From: Yas <y.tesiram at pgrad.unimelb.edu.au>
• Date: Fri, 11 Jan 2002 04:35:19 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```G'day mathgroup,
I have some general questions on Interpolation as implemented in
Mathematica. Essentially, I have a series of profiles for which I have
no general function. The profile is generated by repetitive
multiplication of rotation matrices and so it is difficult (and also
impractical) to deduce an analytic function. Instead to answer the
questions I have, I have decided to use Interpolation -- a pitiful
state, nevertheless a shade better than extrapolation. The first step
involves creating the Interpolating function,

thing1 = Interpolation[data];

Then I find the first derivative,

thing2 = D[thing1[x], {x, 0, lastdatapoint}]

Next I want to find the points where thing2 = 0, but I have run into
problems with Mathematica complaining about inverse functions etc,
although plots of thing2 versus x look fine.

My primary question is,

1. The profiles that I have, have maxima and minima whose co-ordinates I
want to find, hence the differentiation step. Interpolation does a good
job reproducing the data points and in finding a derivative that can be
plotted but not when asked to Solve for thing2 = 0. How do I go about
finding the accuracy of the Interpolating Function to test whether the
value near thing2 = 0 is well behaved?

And a secondary question is,

2. Is there another efficient method of estimating the co-ordinates of
the minima and maxima by computer of data sets?

The length of each of these data sets is 1000 points and for that reason
I have not pasted it into the email. As a demonstrative example, the
Table of values generated by,

Table[Sin[0.75 + Exp[Pi*t]], {t, 0, 1, 1/1000}];
ListPlot[%]

is closely related.

Any help or comments will be appreciated.

Thanks
Yas

```

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