Re: Re: Numerical Integration

• To: mathgroup at smc.vnet.net
• Subject: [mg71955] Re: [mg71622] Re: Numerical Integration
• From: "Antonio Neves" <aneves at gmail.com>
• Date: Wed, 6 Dec 2006 06:03:32 -0500 (EST)
• References: <ejrn8h\$9a4\$1@smc.vnet.net> <ejuqhi\$huo\$1@smc.vnet.net>

```Dear All,

I have found this discussion very interesting. I have a similar
problem and was wondering if a similar "oneliner" solution for numeric
integrals of associated Legendre polynomial involving intervals that
contains many zeros of LegendreP[n,m,z] is possible.
Any idea how to generate a list of Zeros for LegendreP[n,m,z]?
Haven't found and library as, Needs["NumericalMath`BesselZeros`"]

Many thanks,
Antonio Neves

On 11/25/06, dimitris <dimmechan at yahoo.com> wrote:
>
> Below I also give my final approach
> (\$Version Number->5.2)
> (Pentium 4, CPU 2.80 GHz, 512MB Ram)
>
> Needs["NumericalMath`BesselZeros`"]
> h[x_] := Tan[BesselJ[0, x]]
> Needs["NumericalMath`BesselZeros`"]
> lst[n_] := Prepend[BesselJZeros[0, n], 0];
> int[n_, i_, nopts___] := NIntegrate[h[x], {x, lst[n][[i]], lst[n][[i +
> 1]]}, nopts]
> tab[n_, nopts___] := Table[int[n, i, nopts], {i, 1, n - 1}]
> sumint[n_, nopts___] := SequenceLimit[FoldList[Plus, 0, tab[n, nopts]]]
>
> (Prepend[Timing[sumint[#1, PrecisionGoal -> 32, WorkingPrecision ->
> (6*32)/5]], #1] & ) /@ (2^Range[2, 7])
> {{4, 0.375*Second, 1.3643699115048195797`2.1072099696478674}, {8,
> 2.062000000000012*Second,
>   1.4544903415612941044`4.816479930623699}, {16,
> 7.8279999999999745*Second,
> 1.4545130030320762838486550491`12.34222982222323},
>  {32, 13.297000000000025*Second,
> 1.4545130030326107638283270574106180434`23.480339661790534},
>  {64, 28.546999999999912*Second,
> 1.45451300303261076382832929894487094148`24.98548964011044},
>  {128, 67.82800000000009*Second,
> 1.45451300303261076382832936146487729718`26.189609622766366}}
>
> TableForm[%]
>
> Best Regards
> Dimitris
>
> dimitris wrote:
> > Peter I apologize for the situation.
> > My thanks was for you but somehow I confused the names!
> >
> > Best Regards
> > Dimitris
> >
> > dimitris wrote:
> > > 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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