Re: NumberForm question
- To: mathgroup at smc.vnet.net
- Subject: [mg82049] Re: NumberForm question
- From: DrMajorBob <drmajorbob at bigfoot.com>
- Date: Wed, 10 Oct 2007 04:31:11 -0400 (EDT)
- References: <20828645.1191825480672.JavaMail.root@m35> <fefihj$jm0$1@smc.vnet.net> <13608238.1191965108403.JavaMail.root@m35>
- Reply-to: drmajorbob at bigfoot.com
I didn't follow all that, but here's an annoying "feature":
x=1+3*$MachineEpsilon;
NumberForm[x,{16,16}]
InputForm[x]
1.0000000000000010
1.0000000000000007
InputForm[x]
NumberForm[x,{16,16}]
1.0000000000000010
1.0000000000000007
InputForm[x]
1.0000000000000007
So InputForm displays out of order, compared to the inputs within a cell.
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