Re: Integration

[Charles Greathouse <charles.greathouse@case.edu>]

Bill, is there a copy of the thesis online, or would we need to visit
Université Bordeaux 1 to read it?

I found a copy of "Intégration numérique par la méthode double-exponentielle"
http://hal.inria.fr/hal-00491561/PDF/integration.pdf
but I don't know if this is his thesis, if they are substantially the
same, or if they're different.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Mon, Nov 8, 2010 at 4:53 AM, Bill Allombert
<Bill.Allombert@math.u-bordeaux1.fr> wrote:
> On Mon, Oct 11, 2010 at 02:54:48PM +0200, Sumaia Saad-Eddin wrote:
>> Dear all,
>>
>>   here is a simple script I use:
>> ----------------------------------------------------
> {calF3b(n, y, borne=0)=
>   local(res);
>   if(borne == 0,
>      res = intnum(u = y, [[1], I],
>                   (n*(n-1)*log(u/y)^(n-2)
>                    -n*log(u/y)^(n-1)
>                    +2*(-n*log(u/y)^(n-1)+log(u/y)^n))*(-cos(u))/u^3
>                  )*2/factorial(n),
>      res = intnum(u = y, borne,
>                   (n*(n-1)*log(u/y)^(n-2)
>                    -n*log(u/y)^(n-1)
>                    +2*(-n*log(u/y)^(n-1)+log(u/y)^n))*(-cos(u))/u^3
>                  )*2/factorial(n));
>   return(res);
> }
>> Can anyone explain me why these results are so
>> different, or give me a pointer to some litterature?
>
> intnum uses the double-exponential integration method that has strong regularity requirement,
> but is often much faster than intnumromb.
>
> The correct integration formula to use is:
> intnum(u = y,borne, expr ) = intnum(u = y,[[1],I], expr ) - intnum(borne, [[1],I], expr )
>
> This avoid the exponential growth of the cosinus when the imaginary part get large
> (after the change of variable).
>
> I suggest reading Pascal Molin Ph.D. thesis (Bordeaux, 2010) for more detail.
>
> Cheers,
> Bill
>