RE: NIntegrate Precision

• To: mathgroup at smc.vnet.net
• Subject: [mg30839] RE: NIntegrate Precision
• From: victork at csse.monash.edu.au (Victor Kowalenko)
• Date: Thu, 20 Sep 2001 03:51:34 -0400 (EDT)
• Organization: The Math Forum
• Sender: owner-wri-mathgroup at wolfram.com

I am interested in obtaining very high precision answers to
Mellin-Barnes integrals using at the Compaq Grendel supercomputer at
the Victorian Partnership for Advanced Computing in Australia.
Typically, I am using NIntegrate to evaluate the contour integrals
along the imaginary axis with integrands that are of the following
form:

(-\exp(i\pi/2)/z)^s \Gamma(s-\nu+1/2) \Gamma(s+\nu +1/2)/(\exp(i\pi
s)-\exp(-i\pis)). I create a module that sets s=c+r\exp(i \theta)
and invokes the NIntegrate procedure for r ranging from 0 to \infty.
MinRecursion and MaxRecursion are set equal to 3 and 10 respectively
while WorkingPrecision is set equal to wp. The module calls NINtegrate
twice with \theta equal to -\pi/2 and \pi/2 and subtracts the two
results. The module is set equal to a function with variables z
(typically equal to \exp(49i\pi/100)/2, \nu (initially equal to 1/4),
\nu (equal to 1/3, but will become complex) and wp. When wp is set
equal to 11, I get a result in FullForm of

Complex[-0.051659811688484825, -1.185784385161031]

For wp=16 I get the same result. However, when I put wp equal to 20
I get

Complex[-0.051659811688475977097400099110.72885,-1.185784385161015463516737647712.0893]

I am mystified by what 10.7285 means. In addition the latter value
seems to have less precision than the former values although the
precision is higher. I would have expected the discrepancy to occur
at a high level of precision, although I suspect the last value may be
more accurate than the previous two results. Are you able to shed any
light on this conundrum?

`

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