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MathGroup Archive 2006

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Re: Numerical Integration

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71503] Re: Numerical Integration
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Mon, 20 Nov 2006 18:12:06 -0500 (EST)
  • References: <ejrn8h$9a4$1@smc.vnet.net><ejs36f$906$1@smc.vnet.net>

Dear David,

Thanks a lot for your nice solution.

Here is another along the same lines.

h[x_] := Tan[BesselJ[0, x]]

Needs["NumericalMath`BesselZeros`"]

lst = BesselJZeros[0, 10];
lst[[0]] = 0;

f[i_] := NIntegrate[h[x], {x, lst[[i]], lst[[i + 1]]}]

SequenceLimit[FoldList[Plus, 0, Table[f[i], {i, 0, 9}]]]
1.45451


Best Regards
Dimitris

Peter Pein wrote:
> dimitris schrieb:
> > Dear All,
> >
> > I have one question about the numerical integration of one function.
> >
> > $VersionNumber
> > 5.2
> >
> ...
> > h[x_] := Tan[BesselJ[0, x]]
> >
> > Plot[h[x], {x, 0, 40}, PlotPoints -> 100, Axes -> None, Frame -> {True,
> > True, False, False}, PlotStyle -> AbsoluteThickness[2]]
> >
> > Limit[h[x], x -> Infinity]
> > 0
> >
> > I try hard to find any proper settings for getting a numerical
> > estimation of its integral
> > over {0,Infinity} but I can't succeed.
> >
> > Any help will be greatly appreciate.
> >
> > Dimitris
> >
>
> Hi Dimitris,
>
> I tried it this way:
>
> In[1]:=
> Needs["NumericalMath`BesselZeros`"];
> h[x_] := Tan[BesselJ[0, x]];
> t0 = SessionTime[];
> bzlist = NestList[BesselJZerosInterval[0, {1, 2}*Last[#1] + {-1/10, 1/10}] & ,
>     Flatten[{0, BesselJZeros[0, 2]}], 9];
> v0 = (NIntegrate[h[x], Evaluate[Flatten[{x, #1}]]] & ) /@ bzlist;
> SequenceLimit[Rest[FoldList[Plus, 0, v0]]]
> (SessionTime[] - t0)*seconds
> Out[6]=
> 1.4545133229307878
> Out[7]=
> 1.75*seconds
>
> The displayed result (1.45451) does not change any more when increasing the
> number of intervals from 9 to 10 or more.
> 
> Peter


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