Re: A question about N[...]

*To*: mathgroup at smc.vnet.net*Subject*: [mg89415] Re: A question about N[...]*From*: "David Park" <djmpark at comcast.net>*Date*: Mon, 9 Jun 2008 02:27:02 -0400 (EDT)*References*: <g2fug6$2kg$1@smc.vnet.net>

This appears to be a numerical precision problem in evaluating HypergeometricPFQ. We could just take an indefinite integral and evaluate for an upper limit. Integrate[BesselJ[0, 2405/1000*r]^2, r] % /. r -> 20. r HypergeometricPFQ[{1/2, 1/2}, {1, 1, 3/2}, -((231361 r^2)/40000)] 0. Now let's plot this versus r with MachinePrecision. The plot goes quite bad before we reach r = 20. Plot[r HypergeometricPFQ[{1/2, 1/2}, {1, 1, 3/2}, -((231361 r^2)/ 40000)], {r, 0, 30}, PlotRange -> Automatic, WorkingPrecision -> MachinePrecision] Next, let's plot with a much higher precision. Things still go bad, but not until a higher value of r. Plot[r HypergeometricPFQ[{1/2, 1/2}, {1, 1, 3/2}, -((231361 r^2)/ 40000)], {r, 0, 30}, PlotRange -> Automatic, WorkingPrecision -> 60] So, let's use N with the same higher precision. r HypergeometricPFQ[{1/2, 1/2}, {1, 1, 3/2}, -((231361 r^2)/40000)]; % /. r -> 20 N[%, 60] 20 HypergeometricPFQ[{1/2, 1/2}, {1, 1, 3/2}, -(231361/100)] 0.864755185740518759537796371015077138945014301003469172726221 So one has to know the ins and outs of HypergeometricPFQ and why it requires such high precision. -- David Park djmpark at comcast.net http://home.comcast.net/~djmpark/ <wyelen at gmail.com> wrote in message news:g2fug6$2kg$1 at smc.vnet.net... > > Recently I came across a puzzling problem which I believed to be > related to the function N. > > My platform is Mathematica 6.0 for Microsoft Windows (32-bit). When > calculating the following > integral, I got different results from Integrate & NIntegrate: > > In[1]:= Integrate[BesselJ[0, 2.405 * r]^2, {r, 0, 20}] > > Out[1]= 0. > > In[2]:= NIntegrate[BesselJ[0, 2.405 * r]^2, {r, 0, 20}] > > Out[2]= 0.864755 > > Guessing a problem caused by numerical number 2.405, I rewrote it as > an exact number: > > In[3]:= Integrate[BesselJ[0, (2 + 405/1000)*r]^2, {r, 0, > 20}] > > Out[3]= 20*HypergeometricPFQ[{1/2, 1/2}, {1, 1, 3/2}, - > (231361/100)] > > then evaluated the numerical value, which was surprisingly still 0.: > > In[4]:= N[%] > > Out[4]= 0. > > but evaluating with 6-digit precision gave the same result as > NIntegrate: > > In[5]:= N[%%,6] > > Out[5]= 0.864755 > > In help page for N it said "N[expr] is equivalent to > N[expr,MachinePrecision]", but evaluating with a > approximate precision didn't gave 0.: > > In[6]:= N[MachinePrecision] > > Out[6]= 15.9546 > > In[7]:= N[%3,15.9546] > > Out[7]= 0.8647551857405188 > > I wonder is this caused by the function N ,or whether I should just > turn to another OS (say Linux) and things will go well. > > Thanks a lot for your reply! >