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Re: Evaluating integral with varying upper limit?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg71193] Re: Evaluating integral with varying upper limit?
*From*: "Andrew Moylan" <andrew.j.moylan at gmail.com>
*Date*: Fri, 10 Nov 2006 06:37:51 -0500 (EST)
*References*: <200611060752.CAA08598@smc.vnet.net><eiupn1$fjr$1@smc.vnet.net>
[Chris Chiasson has provided an answer to my question about the best
ways to make available code, notebooks, and packages to users of this
list: see
http://groups-beta.google.com/group/comp.soft-sys.math.mathematica/msg/379ef0c6f644fbaa.]
On Nov 9, 7:43 pm, "Andrew Moylan" <andrew.j.moy... at gmail.com> wrote:
> 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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