Re: A question about N[...]

*To*: mathgroup at smc.vnet.net*Subject*: [mg89432] Re: [mg89395] A question about N[...]*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Mon, 9 Jun 2008 02:30:13 -0400 (EDT)*References*: <200806080632.CAA02634@smc.vnet.net>

On 8 Jun 2008, at 15:32, wyelen at gmail.com wrote: > > 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! > The problem you are seeing with N is a usual problem with machine precision arithmetic. The only way to avoid it is to use arbitrary precision arithmetic, as you have discovered for yourself. However, note that N[expr,MachinePrecision] and N[expr,N[MachinePrecision]] are not the saame thing. a= N[2,MachinePrecision];b=N[2,N[MachinePrecision]]; a = N[2, MachinePrecision]; b = N[2, N[MachinePrecision]]; Precision[a] MachinePrecision Precision[b] 15.9546 a is a machine precision number but b is an arbitrary precision number with (nearly) 16 digits of precision. There is a world of difference between these two quantities. Andrzej Kozlowski

**References**:**A question about N[...]***From:*"wyelen@gmail.com" <wyelen@gmail.com>