Need help with ideas to make NIntegrate a little faster for multiple variables

*To*: mathgroup at smc.vnet.net*Subject*: [mg23031] Need help with ideas to make NIntegrate a little faster for multiple variables*From*: "Don Taylor" <psu04033 at odin.pdx.edu>*Date*: Thu, 13 Apr 2000 02:43:21 -0400 (EDT)*Organization*: Portland State University*Sender*: owner-wri-mathgroup at wolfram.com

I'm finding probabilities of a Gaussian ball that has been sliced by some hyperplanes. Calculations are taking a LONG time. Would anyone have any advice on how I might find solutions to problems like the following in a more reasonable amount of time that it seems to take in the current form? NIntegrate[ Exp[-x^2/2-y^2/2-z^2/2 -r1^2/(2*(1/8)^2)-r2^2/(2*(1/8)^2) -r3^2/(2*(1/8)^2)-r4^2/(2*(1/8)^2)]/ ((2*Pi)^(3/2)*(2*Pi*(1/8)^2)^2), {x,-5,5},{y,-5,5},{z,-5,5}, {r1,-10,y-x},{r2,z-y,10},{r3,y-x,10},{r4,z-y,10}] The surface is about as smooth as could be expected for a problem. The real bounds of integration are infinity but I would be able to use results that have 4 good digits in an answer and thus I have been using bounds of +/-5 or +/-10 instead of Infinity. But that doesn't seem to be able to give results in a few hours (on a 360 Mhz machine with lots of free memory and nothing else soaking up cycles) I have tried setting NumPoints->10000000 which gives perhaps 10 points per dimension plus some spares for the algorithm to use as it sees fit. But Version 3 claims that it sees an error when NumPoints is set and thus reverts to not using the compiled version I think. The calculations continue, but slowly, in either case. Does anyone have any ideas about what I might do to get answers with a handful of good digits in a plausible amount of time? I believe I have reduced the problem down to the simplest possible form. But I am still left with sixty four such integrals to evaluate, to at least give me eight good points to plot on a graph. (Would upgrading to version 4 make a dramatic change in this particular situation?) Many Thanks Don