• To: mathgroup at smc.vnet.net
• Subject: [mg89421] Re: A question about N[...]
• From: "wyelen at gmail.com" <wyelen at gmail.com>
• Date: Mon, 9 Jun 2008 02:28:09 -0400 (EDT)
• References: <g2fug6\$2kg\$1@smc.vnet.net>

```On 6=D4=C28=C8=D5, =CF=C2=CE=E72=CA=B133=B7=D6, "wye... at gmail.com" <wye...@g=
mail.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.
>

It seems I have mixed

\$MachinePrecision-digit precision (where \$MachinePrecision = "number
of decimal digits of machine-precision"), which is an arbitrary
precision algorithm

and

MachinePrecision = "machine-number precision", which is a machine
precision algorithm.
This led to a confusion at In[7].

And I haven't realized the precision requirements from the
intermediate steps, which may require more than 16 digits.

I don't know whether it is a bug or not, but at least I see a way to
avoid it :)

```

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