Re: analytic integration of InterpolatingFunction compositions

*To*: mathgroup at smc.vnet.net*Subject*: [mg74065] Re: [mg74046] analytic integration of InterpolatingFunction compositions*From*: "Chris Chiasson" <chris at chiasson.name>*Date*: Thu, 8 Mar 2007 04:37:48 -0500 (EST)*References*: <200703070815.DAA26693@smc.vnet.net>

It is interesting that the result of Integrate on the InterpolatingFunction is just unevaluated (heh, after the smoke of a bunch of burning rule conditions clears) because the documentation for NIntegrateInterpolatingFunction specifically mentions the following: If you simply need to find the integral of an InterpolatingFunction object (as opposed to a function of one), it is better to use Integrate because this gives you the result which is exact for the polynomial approximation used in the InterpolatingFunction object. On 3/7/07, Roman <rschmied at gmail.com> wrote: > Hello all: > > When I have a simple InterpolatingFunction[] object from an NDSolve[] > call, I know I can analytically integrate this by using Integrate[]. > However, what I want to do is analytically integrate compositions of > such InterpolatingFunction[] objects, which Integrate[] cannot handle. > For example, let > > f = y /. First[NDSolve[{y'[x] == x*y[x]^2, y[0] == 1}, y, {x, 0, > 1}]] > > Now I want to integrate f[x]^2: > > NIntegrate[f[x]^2, {x, 0, 1}] > > works fine. But this being an interpolating function, it seems to me > that one could get a much faster and more accurate result by analytic > integration. Unfortunately, > > Integrate[f[x]^2, {x, 0, 1}] > > does not compute. > > In principle one could extract the interpolation grid from f[x] and > set up an analytic integration "by hand", using > NumericalMath`ListIntegrate`, but this quickly becomes nasty, > especially if you integrate products of different > InterpolatingFunction objects like > > NIntegrate[f[x]*g[x], {x, 0, 1}] > > which are both results of NDSolve[] and thus may be using different > grid points. > > Does anyone have any suggestions on how to do these integrals > properly? Or how to coax NIntegrate[] into realizing that it should > use a grid which matches those of the various InterpolatingFunction > objects in its argument? > > Cheers! > Roman. > > > -- http://chris.chiasson.name/

**References**:**analytic integration of InterpolatingFunction compositions***From:*"Roman" <rschmied@gmail.com>