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.