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MathGroup Archive 2007

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Re: N function problematic performance (V. 5.2)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg74809] Re: N function problematic performance (V. 5.2)
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Fri, 6 Apr 2007 04:19:11 -0400 (EDT)
  • References: <euvm6e$d7s$1@smc.vnet.net><ev2av0$kcp$1@smc.vnet.net>

But the question about the problematic behavior of the N function

(Warning messages

Infinity::"indet"...
NSum::nsnum :...

and failure to provide a numerical approximation in reasonable timing)

still holds of course!

Dimitris


=CF/=C7 dimitris =DD=E3=F1=E1=F8=E5:
> As I said somewhere else we have Mathematica on one side
> and on the other our mind. We can get the desirable results
> if we bring mind&Mathematica together...
>
> As we see using Integrate on Sin[x - 1]/Sqrt[x*(x - 2)] gives a result in=
 terms
> of MeijerG functions. But then N faces problems to get a numerical
> estimation.
> On the hand applying TrigToExp first gives a result again in terms of Mei=
jerG
> functions which is opposite of the correct result.
>
> However a trivial subsitution results in a very simple result very
> quickly (bear in mind that
> in previous result containg MeijerG functions neither FullSimplify nor
> FunctionExpand
> provides any simplifications).
>
> So...
>
> (*Ins*)
> int = Simplify[(Sin[x - 1]/Sqrt[x*(x - 2)])*Dt[x] /. x -> t + 1] /.
> Dt[t] -> 1
> Plot[int, {t, 1, 10}]
> Timing[Integrate[int, {t, 1, Infinity}]]
> {N[%[[2]],11], NIntegrate[int, {t, 1, Infinity}, Method ->
> Oscillatory,WorkingPrecision->20,PrecisionGoal->10]}
>
> (*Outs*)
> Sin[t]/Sqrt[-1 + t^2]
> Graphics
> {0.672*Second, (1/2)*Pi*BesselJ[0, 1]}
> {1.2019697153, 1.2019697153}
>
> Dimitris
>
> =CF/=C7 dimitris =DD=E3=F1=E1=F8=E5:
> > Consider the following the definite integral
> >
> > h = HoldForm[Integrate[Sin[x - 1]/Sqrt[x*(x - 2)], {x, 2, Infinity}]]
> >
> > Here is a numerical estimation
> >
> >
> > ReleaseHold[h /. Integrate[x___] :> NIntegrate[x, Method ->
> > Oscillatory]]
> > 1.2019697137297456
> >
> > Here is the symbolic result by Mathematica
> >
> > Timing[hh = ReleaseHold[h]]
> >
> > {30.031000000000002*Second, (Pi*(Cos[1]*MeijerG[{{}, {1/4, 3/4}},
> {{0,
> > 1/2, 1/2}, {0}}, 1] -
> >      MeijerG[{{}, {1/4, 3/4}}, {{0, 0, 1/2}, {1/2}}, 1]*Sin[1]))/
> > Sqrt[2]}
> >
> > Application of N function results in unexpected warnings and not in
> > any estimation in reasonable timing...
> >
> > TimeConstrained[N[hh], 120]
> > Infinity::"indet"...
> > NSum::nsnum :...
> > $Aborted
> >
> > What is going here?
> > Did I encouter a known (problematic) situation?
> >
> > BTW,
> >
> > Integrate[TrigToExp[Sin[x - 1]/Sqrt[x*(x - 2)]], {x, 2, Infinity}]
> > ((1/2)*I*Pi^(3/2)*(E^(2*I)*MeijerG[{{}, {1/2, 1/2}}, {{0, 0}, {1/2}},
> > -2*I] - MeijerG[{{}, {1/2, 1/2}}, {{0, 0}, {1/2}}, 2*I]))/
> >   E^I
> >
> > Chop[N[%]]
> > -1.2019697153172064
> >
> > which is opposite of the correct result.



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