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

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  • Subject: [mg106921] Re: [mg106656] Re: [mg106882] Re: More /.{I->-1} craziness. Schools
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Thu, 28 Jan 2010 02:44:48 -0500 (EST)
  • References: <hjbvc0$2tp$1@smc.vnet.net> <hjeqh1$g3c$1@smc.vnet.net> <hjh877$r4r$1@smc.vnet.net> <201001261133.GAA00712@smc.vnet.net> <201001270644.BAA04729@smc.vnet.net> <op.u67uiib3tgfoz2@bobbys-imac.local>

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



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