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Re: High precision NIntegrate problem. Please help!
*To*: mathgroup at smc.vnet.net
*Subject*: [mg60328] Re: High precision NIntegrate problem. Please help!
*From*: Paul Abbott <paul at physics.uwa.edu.au>
*Date*: Tue, 13 Sep 2005 06:07:07 -0400 (EDT)
*Organization*: The University of Western Australia
*References*: <dfueda$1tu$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
In article <dfueda$1tu$1 at smc.vnet.net>, a_noether_theorem at yahoo.com
wrote:
> I'm trying to numerically evaluate a particularly nasty integral, and
> no matter what I do, I can't get Mathematica to give me an answer without
> complaining, and the answer seems to change as I change the precision
> options, and continues to change up to arbitrary levels of precision.
> I could really use some help.
>
> I've tried every combination of options I could, but nothing works.
>
> In case you care what the function is, it's the following. Otherwise,
> skip what's between the lines.
Actually, what's between the lines is most important. Sometimes,
numerical integration is the only way to proceed, but if symbolic
integration is possible, then it's likely to be more useful.
And the correct approach to using NIntegrate often depends on the form
of the integrand.
> -------------------------------------------------------------
> I 1st evaluate the following sum:
>
> 1/( I omega_n - e0) 1/( I omega_n - e1) 1/( I omega_n - e2) 1/( I
> omega_n - e3)
>
> from n=-Infinity to Infinity, where omega_n = (2 n + 1) Pi/beta
>
> The sum can be done analytically by contour integration and Mathematica can
> also do the sum. Anyhoo, it leads to a longish expression that I need
> to integrate. Here,
>
> e0 = t1 Cos[x] + t2 Cos[y] - mu
> e1 = t1 Cos[x+X1] + t2 Cos[y+Y1] - mu
> e2 = t1 Cos[x+X2] + t2 Cos[y+Y2] - mu
> e3 = t1 Cos[x+X3] + t2 Cos[y+Y3] - mu
>
> and t1, t2, mu, beta, X1.., Y1... are just some numbers I choose. This
> is a Feyman diagram calculation, if anyone cares. I evaluate this
> with a few {Xi,Yi} sets.
> ----------------------------------------------------------------------
You talk about integration -- but you don't mention what you are
integrating (x and y I presume), or the integration range.
Also, can you compute the integral of the summand symbolically, and then
perform the doubly-infinite summation?
Many computations involving Feynman diagrams can be found in closed form
and solved using tricks such as parametric differentiation, and these
should be investigated prior to using NIntegrate.
Cheers,
Paul
_______________________________________________________________________
Paul Abbott Phone: 61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
The University of Western Australia (CRICOS Provider No 00126G)
AUSTRALIA http://physics.uwa.edu.au/~paul
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