• 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.
>
>

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

```

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