Re: NIntegrate and speed

*To*: mathgroup at smc.vnet.net*Subject*: [mg116796] Re: NIntegrate and speed*From*: Bob Hanlon <hanlonr at cox.net>*Date*: Mon, 28 Feb 2011 05:01:31 -0500 (EST)

$Version "8.0 for Mac OS X x86 (64-bit) (November 6, 2010)" Clear[R, k] Assuming[Element[{k, R}, Reals], Integrate[Cos[k R Sin[t]], {t, 0, Pi}]] Pi*BesselJ[0, Abs[k]*Abs[R]] R = 8000; Z = 1; rd = 3500; NIntegrate[Exp[-k Abs[Z]]/(1 + (k rd)^2)^1.5 Pi* BesselJ[0, Abs[k] Abs[R]], {k, 0, Infinity}] 0.000424068 I get the same results with version 7.01.0 Bob Hanlon ---- Marco Masi <marco.masi at ymail.com> wrote: ============= I have the following problems with NIntegrate. 1) I would like to make the following double numerical integral converge without errors R = 8000; Z = 1; rd = 3500; NIntegrate[Exp[-k Abs[Z]]/(1 + (k rd)^2)^1.5 (NIntegrate[Cos[k R Sin[\[Theta]]], {\[Theta], 0, \[Pi]}]), {k, 0, \[Infinity]}] It tells non numerical values present and I don't understand why, since it evaluates finally a numerical value? 0.000424067 2) Isn't the second integrand a cylindrical Bessel function of order 0? So, I expected that NIntegrate[Exp[-k Abs[Z]]/(1 + (k rd)^2)^1.5 BesselJZero[0, k R], {k, 0, \[Infinity]}] doing the same job. But it fails to converge and gives 0.00185584- i4.96939*10^-18. Trying with WorkingPrecision didn't make things better. How can this be fixed? 3) The above Nintegrals will go into a loop and should be evaluated as fast as possible. How? With Compile, CompilationTarget -> "C", Paralleization, etc.? Any suggestions? Marco.