Oscillatory integrals Re: Maxima asks about sign of integration(!) variable

John:

Regarding oscillatory integrals ... are you trying to use quadpack on them?
There may be a better way if you approach them symbolically.

See

http://www.cs.berkeley.edu/~fateman/papers/levin.pdf

There is also code (in Mathematica, may be readable source) for
Levin's method, in its NIntegrate suite of routines.  As far as
I can tell, Mathematica is pretty good in this area, though optimal
settings of various parameters like "working precision" is up for
grabs.

Tell me what you think!

RJF



On 9/25/2013 2:45 PM, John Lapeyre wrote:
> On 09/25/2013 09:26 PM, Richard Fateman wrote:
>
> > In Mathematica, the answer came out
> > 3/16 (65697 + 127 I Sqrt[3])
> > or 12318.2 + 41.2445 I
> >
> > Doing NIntegrate in Mathematica provided
> >
> > 12318.1 + 41.2326 I
> >
>
> My nintegrate function, which calls quadpack, seems to agree more
> with Mathematica's exact answer than it's numerical answer. I've
> seen that more than once.
>
> (%i5) nintegrate(sqrt(1+(4*x^(1/3)-1/16*x^((-1)/3))^2),[x,-8,512]);
>
> (%o5) [41.24445772554166*%i+12318.18747666083,1.5204793641032666e-5,2268,
>        "no problems"]
>
> I have been trying a bit to extend it to do oscillatory integrals
> as well, eg. cos(x^2) --> cos(u)/sqrt(u). But changevar is broken,
> ... so it will take a while.
>
> -- John
>