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Re: Re: Re: Re: More /.{I->-1} craziness. Schools

  • To: mathgroup at smc.vnet.net
  • Subject: [mg106955] Re: [mg106925] Re: [mg106656] Re: [mg106882] Re: More /.{I->-1} craziness. Schools
  • From: Andrzej Kozlowski <akozlowski at gmail.com>
  • Date: Fri, 29 Jan 2010 07:47:56 -0500 (EST)
  • References: <hjbvc0$2tp$1@smc.vnet.net> <hjeqh1$g3c$1@smc.vnet.net> <201001280745.CAA23668@smc.vnet.net>

The definition of precision in Mathematica is this. Suppose x is a 
number known up to an error of epsilon, that is it can be viewed as 
lying in the interval (x-epsilon/2,x+epsilon/2). Then its precision is

-Log[10,epsilon/x]. Its accuracy is -Log[10,epsilon]. The two are 
related by the equation:
Precision[x] - Accuracy[x] == RealExponent[x]

The interpretation in terms of digits is only approximate. Both accuracy 
and precision can be negative - this depends on the scale of the number 
i.e. RealExponent. A number will have negative accuracy if its absolute 
error is large. It is easy to produce such numbers by cancellation

With[{x = N[10^100, 50] - N[10^100, 50]},
     Accuracy[x]]

-50.301

On the other hand, since

$MinPrecision

0

You won't normally in Mathematica see numbers with negative Precision. 
Precision is the main concept, Accuracy is only used because Precision 
is singular at 0 (remember - its relative error).

It's all perfectly documented so this tired scape goat is not available 
this time.

Andrzej Kozlowski


On 28 Jan 2010, at 08:45, DrMajorBob wrote:

> OK... so numbers are allowed to have NEGATIVE precision?
>
> LESS than zero digits of accuracy? Really?
>
> Whatever for?
>
> Bobby
>
> On Wed, 27 Jan 2010 18:23:32 -0600, Daniel Lichtblau 
<danl at wolfram.com> 
> wrote:
>
>> DrMajorBob wrote:
>>> 0 and 1 are not "fuzzballs", so what interval could be >= 1 and 
also 0.?
>>> Bobby
>>
>> I had in mind the spoiler answer Richard Fateman provided in his 
first 
>> post mentioning this particular tangent, err, example.
>>
>> http://forums.wolfram.com/mathgroup/archive/2010/Jan/msg00638.html
>>
>> At the bottom we find:
>> ---
>> I would especially avoid .nb objects, and most especially on topics 
of
>> numerical analysis, where the design flaws are, in my opinion, so
>> fundamental.  Example (mathematica 7.0):
>> {x >= 1, x > 1, x > 0, x}
>>     evaluates to
>> {True, False, False, 0.}
>>
>>   can you construct x?
>>
>> RJF
>>
>> One possible answer, below....
>>
>> x=0``-.5
>> ---
>>
>> The point is that with Mathematica's version of significance 
arithmetic, 
>> equality, I believe, is effectively treated as having a nontrivial an 

>> intersection (of the implicit intervals defining two numbers). If 
>> neither has any fuzz (i.e. both are exact), then Equal allows for no 

>> fuzz, so this is only a subtlety if at least one of the values is 
>> approximate.
>>
>> One implication is that a "zero" of sufficiently low (as in bad) 
>> accuracy can be regarded as 1, or -1, or Pi, if those values happen 
to 
>> fall within the accuracy (which I refer to as fuzz).
>>
>> The other inequalities follow from the preservation of trichotomy. 
For 
>> explicitly real values we regard that as important. mathematica makes 
no 
>> pretense that Equal is transitive and I do not see any way to do that 

>> and also have useful approximate arithmetic.
>>
>> There has been some amount of communication off-line on this topic, 

>> which is why some of us (well, me, at least) sometimes forget the 
>> examples are not universally obvious to those who have not memorized 
the 
>> enitre thread.
>>
>> Daniel
>>
>>
>>> On Wed, 27 Jan 2010 00:44:22 -0600, Daniel Lichtblau 
<danl at wolfram.com> 
>>> wrote:
>>>
>>>> Richard Fateman wrote:
>>>>> [...]
>>>>> If all of Mathematica functionality were available in the free 
player
>>>>> version, WRI would need to drastically change its business model. 
And
>>>>> even it it were free, we still have behavior like this: (..for 
some
>>>>> values of zero)
>>>>>
>>>>>  {x >== 1, x > 0, x}  evaluates to {True, False, 0.}
>>>>>
>>>>> RJF
>>>>
>>>> Let's take simple intervals, that is, intervals that are segments.
>>>> Define less and greater in the obvious ways, that is, one segment 
lies
>>>> strictly below the other (right endpoint of lesser is less than 
left
>>>> endpoint of larger). Let us further define two intervals to be 
equal
>>>> whenever they have nonempty intersection.
>>>>
>>>> With these definitions, which I think are sensible, the behavior 
you
>>>> describe above is consistent with arithmetic on intervals. As the
>>>> numbers involved, at least some of them, are fuzzballs, this 
strikes me
>>>> as an appropriate behavior.
>>>>
>>>> Daniel Lichtblau
>>>> Wolfram Research
>>
>
>
> --
> DrMajorBob at yahoo.com
>



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