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Re: analytic integration of InterpolatingFunction compositions
*To*: mathgroup at smc.vnet.net
*Subject*: [mg74087] Re: analytic integration of InterpolatingFunction compositions
*From*: dh <dh at metrohm.ch>
*Date*: Fri, 9 Mar 2007 02:03:14 -0500 (EST)
*References*: <eslsu8$q6c$1@smc.vnet.net>
Hi Roman,
As I saw that you did not get any better answer, you may try
re-interpolation: e.g. to integrate f^2: Choose some points to evaluate
f, calculate f^2 at this points, calculate a new interpolation function
and integrate. E.g.:
f=y/.First[NDSolve[{y'[x]\[Equal]x*y[x]^2,y[0]\[Equal]1},y,{x,0,1}]];
dat=Table[{x,f[x]^2},{x,0,1,0.1}];
f2=Interpolation[dat];
Integrate[f2[x],{x,0,1}]
Daniel
Roman 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.
>
>
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