Re: NIntegrate Precision

*To*: mathgroup at smc.vnet.net*Subject*: [mg30869] Re: NIntegrate Precision*From*: "Allan Hayes" <hay at haystack.demon.co.uk>*Date*: Fri, 21 Sep 2001 04:04:07 -0400 (EDT)*References*: <9oc85h$g35$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Victor, n`p means that the value is only guaranteed to "precision" p That is that it is only guaranteed to lie between n - Abs[n] 10^-p and n + Abs[n] . 10^-p . (note that we are here dealing with relative error). Please note that it is the error that might be introduced during its calculations that Mathematica is reporting on - you are responsible for the input. n` means that Mathematica has been working with machine numbers and has not bothered to compute the precision. Unfortunately you if you ask for Precision[n`] you will get $MachinePrecision, usually 16 as though n` were the same as n`16 -- this is misleading. When you set WorkingPrecision ->20 (anything bigger then $MachinePrecision will do) then Mathematica computes internally to precision 20 or more, keeps track of the precision, and gives you a result n`p, to be interpreted as above. For more see Help Browser>Demos>Numerics Report -- Allan --------------------- Allan Hayes Mathematica Training and Consulting Leicester UK www.haystack.demon.co.uk hay at haystack.demon.co.uk Voice: +44 (0)116 271 4198 Fax: +44 (0)870 164 0565 "Victor Kowalenko" <victork at csse.monash.edu.au> wrote in message news:9oc85h$g35$1 at smc.vnet.net... > 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[-0.0516598116884759770974000991`10.72885,-1.185784385161015463516737 6477`12.0893] > > 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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