Re: Oscillating method of integration

From: <andreyk1_at_post.tau.ac.il>
Date: Tue, 29 Aug 2006 11:58:24 +0300

Hi Lorenzo,

Your method is very clever (change variables), but it (IntegralMultiple) behaves very strange. I wanted to check whether the integration is stable, so I changed the number of iteration (max_p) from 1e+4 to 1e+7. After max_p = 1e+6 the answer start appearing as "inf"......very unexpected...

Andrew.

Quoting Lorenzo Moneta <Lorenzo.Moneta_at_cern.ch>:

> Hi Andrew,
>
> your integral, as you have written the attached pdf page, it should
> converge.
> When you use IntegralMultiple, have you tried changing the
> integration variable from (0,inf) to (0,1) transforming x -> t with
> x = (1-t)/t ?
>
> Otherwise an alternative for you could be to use Monte Carlo
> integration.
> In ROOT you can do using FOAM to sample randomly your function,
>
> Best Regards,
>
> Lorenzo
> On Aug 26, 2006, at 1:11 PM, andreyk1_at_post.tau.ac.il wrote:
>
> > Hi Lorenzo and George,
> >
> > Well, in order to avoid all misunderstandings, I attached here the
> > exact function I need to calculate. Everything inside the PDF file.
> > I will be glad for every solution you can suggest.
> >
> > Sincerely,
> > Andrew.
> >
> >> Quoting George Japaridze <george.japaridze_at_gmail.com>:
> >
> >> Hi,
> >>
> >> Lorenzo's right, the integral
> >>
> >> \int^{infinity}_{0} dx \int^{\infinity}_{0} dy J_{0}(x*y)
> >>
> >> is equal to one dimensional integral
> >>
> >> 2* \int^{\pi}_{0} dx/sin(x)
> >>
> >> which does not exists as an ordinary function - can be redefined as a
> >> generalized function (distribution).
> >> Depends on a specifics of the problem and are you or are you not
> >> allowed to use regularization.
> >>
> >> Cheers,
> >>
> >> George
> >> On Aug 25, 2006, at 9:50 AM, Lorenzo Moneta wrote:
> >>
> >>> Hi Andrew,
> >>>
> >>> actually I noticed now you want the integral between 0 and inf.
> >>> This is undefined, it is like getting the value of sin(x) for
> >>> x=inf.
> >>>
> >>> Best Regards,
> >>> Lorenzo
> >>>
> >>>
> >>> On Aug 25, 2006, at 12:49 PM, andreyk1_at_post.tau.ac.il wrote:
> >>>
> >>>> Dear Lorenzo,
> >>>>
> >>>> Thanks for solution, but this is not my case. The integral is too
> >>>> complicated to
> >>>> be done analitically. The only way is numerical calculation.
> >>>> Actually I can
> >>>> integrate between BesselJ zeroes and sum up these intervals, but
> >>>> it is not
> >>>> efficient.
> >>>>
> >>>> Andrew.
> >>>>
> >>>> Quoting Lorenzo Moneta <Lorenzo.Moneta_at_cern.ch>:
> >>>>
> >>>>> Hi Andrew,
> >>>>>
> >>>>> I don't know of a method for 2d with oscillatory function.
> >>>>> Hower, you should be able for a Bessel of order 0 to solve the
> >>>>> integral analytically,
> >>>>> using the Bessel J0 definition
> >>>>>
> >>>>> see http://en.wikipedia.org/wiki/Bessel_function
> >>>>>
> >>>>> Cheers,
> >>>>>
> >>>>> Lorenzo
> >>>>> On 24 Aug 2006, at 12:14, andreyk1_at_post.tau.ac.il wrote:
> >>>>>
> >>>>>>
> >>>>>>
> >>>>>> Dear rooters,
> >>>>>>
> >>>>>> I need to integrate over two dimensionl oscillating function
> >>>>>> (TMath::BesselJ(0,x*y), 0<x<inf, 0<y<inf). I use TF1 with
> >>>>>> IntegralMultiple, but
> >>>>>> the answer is not stable, a change in max_points leads to a
> >>>>>> different result. I
> >>>>>> tried to increas the number of max_points to 1e+6 but the
> >>>>>> result is
> >>>>>> still
> >>>>>> unstable. Is there some method of integrating over oscillating
> >>>>>> function with
> >>>>>> arbitry dimensions? (in my case it is dim = 2).
> >>>>>>
> >>>>>>
> >>>>>> Thanks a lot!
> >>>>>>
> >>>>>>
> >>>>>> Andrew
> >>>>>>
> >>>>>> ----------------------------------------------------------------
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> >>>>>>
> >>>>>>
> >>>>>
> >>>>>
> >>>>> +++++++++++++++++++++++++++++++++++++++++++
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> >>>>> at the Tel-Aviv University CC.
> >>>>>
> >>>>
> >>>>
> >>>>
> >>>>
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> >>>>
> >>>
> >>
> >>
> >>
> >>
> >>
> >>
> >>
> >>
> >>
> >>
> >>
> >>
> >> *********************************************************************
> >> ***
> >> *******************
> >> Dr. George Japaridze CTSPS, Clark
> >> Atlanta University
> >>
> >> japar_at_ctsps.cau.edu 404 880 6420
> >> Office
> >> http://www.robotics.cau.edu/people/japar404 226 3847 Cell
> >>
> >>
> >>
> >>
> >>
> >>
> >
> >
> >
> >
> > ----------------------------------------------------------------
> > This message was sent using IMP, the Internet Messaging Program.
> > <diff7.pdf>
>
>
> +++++++++++++++++++++++++++++++++++++++++++
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This message was sent using IMP, the Internet Messaging Program. Received on Tue Aug 29 2006 - 10:58:56 MEST