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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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