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Re: Re: Finding derivatives of a list?

• To: mathgroup at smc.vnet.net
• Subject: [mg40888] Re: [mg40854] Re: [mg40816] Finding derivatives of a list?
• From: Selwyn Hollis <selwynh at earthlink.net>
• Date: Wed, 23 Apr 2003 01:17:33 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

I feel compelled to point out that using interpolation for this purpose 
is, in general, a Bad Idea.

Please have a look at the following, which uses an example attributed 
to Runge. The Do loop plots the difference between f''[x] and the 
second derivative of the InterpolatingFunction based on 21 nodes -1, 
-.9, ..., .9, 1, for InterpolationOrders 3 through 20. You'll notice 
that the difference is quite large in all cases.

runge[x_] := 1/(1 + 20 x^2);
Plot[runge[x], {x, -1, 1}, Frame->True, PlotLabel->"Runge's example"];

pts = Table[{x, runge[x]}, {x, -1., 1, 0.1}];

Do[ interp = Interpolation[pts, InterpolationOrder -> n];
     Plot[Evaluate[{runge''[x] - interp''[x]}], {x, -1, 1},
        PlotRange -> {-10, 10},
        PlotLabel -> "InterpolationOrder --> " <> ToString[n]],
   {n, 3, 20}]

-----
Selwyn Hollis
http://www.math.armstrong.edu/faculty/hollis


On Tuesday, April 22, 2003, at 06:44  AM, Daniel Lichtblau wrote:

> AES/newspost wrote:
>>
>> Specific problem is how to generate a list of values of the second
>> derivative of a relatively smooth function at a set of equally spaced
>> points, when the function itself is known only as a list of numerical
>> values at those same points?
>>
>> --
>> "Power tends to corrupt.  Absolute power corrupts absolutely."
>> Lord Acton (1834-1902)
>> "Dependence on advertising tends to corrupt.  Total dependence on
>> advertising  corrupts totally." (today's equivalent)
>
> Here are some possibilities.
>
> (i) Form an interpolation of relatively high order (say 6 or so). Take
> second derivatives.
>
> (ii) Use finite differences to approximate the second derivatives.
>
> (iii) Use Fourier to get the approximated derivatives. See for example
>
> Wang, Jing. B (2002). Numerical differentiation using Fourier. The
> Mathematica Journal 8:3. 383-388.
>
> I believe there was a small error in the code provided; you might want
> to contact the author at wang at physics.uwa.edu.au
>
>
> Daniel Lichtblau
> Wolfram Research
>
>

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  • Re: Re: Re: Finding derivatives of a list?
    • From: Daniel Lichtblau <danl@wolfram.com>