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Re: NumberForm question

  • To: mathgroup at smc.vnet.net
  • Subject: [mg82047] Re: NumberForm question
  • From: DrMajorBob <drmajorbob at bigfoot.com>
  • Date: Wed, 10 Oct 2007 04:30:09 -0400 (EDT)
  • References: <20828645.1191825480672.JavaMail.root@m35> <fefihj$jm0$1@smc.vnet.net> <13608238.1191965108403.JavaMail.root@m35>
  • Reply-to: drmajorbob at bigfoot.com

NumberForm has always been flaky, or its explanation has... and I don't 
care which.

Personally, I never mess with rounding reals or displaying them any  
special way. Either I want the number in all its detailed glory (InputForm  
or RealDigits base 2), or I let it display in the default mode.

Bobby

On Tue, 09 Oct 2007 09:59:06 -0500, Jean-Marc Gulliet  
<jeanmarc.gulliet at gmail.com> wrote:

> DrMajorBob wrote:
>
>> I thought I had an answer: the 2 in your InputForm output is in the  
>> 17th decimal position (counting from the leading decimal as 1st  
>> position), so 16-digit precision leaves it out in the NumberForm output.
>>  $MachineEpsilon
>>  2.22045*10^-16
>>  BUT:
>>  NumberForm[x,{100,100}]
>>   =

>> 1.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
>>   From that, I'd think x is simply 1.0 (contradicting the InputForm).
>>  But no, since:
>>  RealDigits[x, 2]
>>  {{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
>>     0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
>>    0, 0, 0, 0, 0, 0, 0, 1}, 1}
>>  So... you got me, I have no idea why!
> <snip>
>
> If this was not weird enough, we can see that we add three  
> $MachineEpsilon to one, then the latest decimal digit are not displayed  
> or rounded in a consistent way. For instance, compare
> 1.0000000000000010 returned by NumberForm versus
> 1.0000000000000007 returned by InputForm.
>
> It looks like if the Interpretation box used with NumberForm is messing  
> up the displayed number (see below).
>
> In[1]:= x = 1 + 3*$MachineEpsilon;
> NumberForm[x, {16, 16}]
> InputForm[x]
>
> Out[2]//NumberForm= \!\(\*
> TagBox[
> InterpretationBox["\<\"1.0000000000000010\"\>",
> 1.0000000000000007`,
> AutoDelete->True],
> NumberForm[#, {16, 16}]& ]\)
>
> Out[3]//InputForm=
> 1.0000000000000007
>
> In[4]:= Table[{NumberForm[1 + n*$MachineEpsilon, {16, 16}],
>    N[FromDigits[RealDigits[1 + n*$MachineEpsilon, 2], 2], 20]}, {n, 5}]
>
> Out[4]= {{\!\(\*
> TagBox[
> InterpretationBox["\<\"1.0000000000000000\"\>",
> 1.0000000000000002`,
> AutoDelete->True],
> NumberForm[#, {16, 16}]& ]\), 1.0000000000000002220}, {\!\(\*
> TagBox[
> InterpretationBox["\<\"1.0000000000000000\"\>",
> 1.0000000000000004`,
> AutoDelete->True],
> NumberForm[#, {16, 16}]& ]\), 1.0000000000000004441}, {\!\(\*
> TagBox[
> InterpretationBox["\<\"1.0000000000000010\"\>",
> 1.0000000000000007`,
> AutoDelete->True],
> NumberForm[#, {16, 16}]& ]\), 1.0000000000000006661}, {\!\(\*
> TagBox[
> InterpretationBox["\<\"1.0000000000000010\"\>",
> 1.0000000000000009`,
> AutoDelete->True],
> NumberForm[#, {16, 16}]& ]\), 1.0000000000000008882}, {\!\(\*
> TagBox[
> InterpretationBox["\<\"1.0000000000000010\"\>",
> 1.000000000000001,
> AutoDelete->True],
> NumberForm[#, {16, 16}]& ]\), 1.0000000000000011102}}
>
> Regards,



-- 

DrMajorBob at bigfoot.com


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