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MathGroup Archive 2006

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Re: Evaluating integral with varying upper limit?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71120] Re: Evaluating integral with varying upper limit?
  • From: "Andrew Moylan" <andrew.j.moylan at gmail.com>
  • Date: Thu, 9 Nov 2006 03:37:16 -0500 (EST)
  • References: <200611060752.CAA08598@smc.vnet.net><eisf7i$mcg$1@smc.vnet.net>

Hi all,

This is a problem I encounter frequently; I'm surprised it doesn't have
a simple solution in terms of built-in Mathematica functions. The
general problem, for me, is, given some initial value problem {y'(x) ==
y(y(x), x), y(0) == y0} (with y possibly multi-dimensional), give the
FUNCTION y. That is, give a function which, when called, will call
NDSolve over the appropriate interval (but probably no further, unless
the interval is very small), and return the result, while remembering
the results of all previous calls to NDSolve to avoid any unnecessary
recomputation.

I have a written a package called AutoNDSolveInitialValueProblem which
performs this task. I frequently use it to solve exactly the problem
given above, namely, to make an efficient version of f[y_] :=
NIntegrate[bla[x], {x, 0, y}]. Let me know if you'd be interested in
this package.

In another thread I recently asked: what's the right way to submit
packages or general Mathematica notebooks to this newsgroup. If anyone
reading this thread knows the answer I'd be grateful.

Cheers,
Andrew


On Nov 8, 10:32 pm, Carl Woll <c... at wolfram.com> wrote:
> AES wrote:
> >Given a function  f[x]  which happens to be rather messy and not
> >analytically integrable, I want to evaluate the function
>
> >   g[y_] := NIntegrate[f[x], {x, ymin, y} ]
>
> >with ymin fixed and  ymin < y < Infinity.
>
> >I suppose that FunctionInterpolate is the way to go here (???).
>
> >But, are there tricks to tell FunctionInterpolate what I know in
> >advance, namely that f[x] is everywhere positive, and decreases toward
> >zero rapidly enough at large x that g[y] will approach a finite limiting
> >value as y -> Infinity? (which value I'd like to have FI obtain with
> >moderate accuracy -- meaning 3 or 4 significant digits, not 10 or 20)
>
> >Thanks . . .I would recommend using NDSolve to invert the NIntegrate operation,
> although handling the range ymin<y<Infinity is problematic. For example:
>
> In[3]:=
> f[x_]:=BesselJ[2,x]Exp[-x]
> In[4]:=
> g[y_]:=NIntegrate[f[x],{x,1,y}]
>
> Then, using NDSolve we have:
>
> In[5]:=
> NDSolve[{h'[x] == f[x], h[1] == 0}, h, {x, 1, 100}]
> Out[5]=
> {{h -> InterpolatingFunction[{{1., 100.}}, <>]}}
>
> A few checks:
>
> g[10]
> h[10] /. %5[[1]]
> 0.102135
> 0.102135
>
> g[100]
> h[100] /. %5[[1]]
> 0.102141
> 0.102141
>
> g[Infinity]
> 0.102141
>
> If you really need an InterpolationFunction whose range is ymin to
> Infinity, then you can try transforming variables, like y->1/y, but
> getting NDSolve to handle the point at Infinity (or 0 in transformed
> coordinates) becomes difficult.
> 
> Carl Woll
> Wolfram Research


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