Has anyone experimented with this with in code? I think that the theory is adequately understood but to see how messy
this might get and what can be done to keep it simple you would have to try it. In pw.mac, I did not know how messy
diff(signum(x),x) = 2*pwdelta(x) would be until I tried it and in the end I made this feature optional. You can turn it
off and it is turned off by default. I allow the user to turn it on if they want because you cannot get correct results
without it in certain situations. Maybe that would be the way to go. I would like to see this in code, perhaps an
extension to Maxima like pw.mac . . .
Rich
--------------------------------------------------
From: "Viktor T. Toth" <vttoth at vttoth.com>
Sent: Monday, September 13, 2010 7:46 AM
To: "'Stavros Macrakis'" <macrakis at alum.mit.edu>; "'Richard Hennessy'" <rich.hennessy at verizon.net>;
<maxima at math.utexas.edu>
Subject: RE: [Maxima] Extended real arithmetic (was Re: inf - inf = 0 ??)
> Correct me if I am mistaken but I think a basic assumption behind
> simplification is that any unevaluated symbol represents a finite,
> determinate quantity. Hence, you're allowed to simplify 0*x as 0, even
> though 0*inf is definitely not 0, because the value of x, whatever it might
> be (real number, complex number, vector, matrix, operator, quaternion,
> etc.), is not infinite or indeterminate.
>
>
> Viktor
>
>
> -----Original Message-----
> From: maxima-bounces at math.utexas.edu [mailto:maxima-bounces at math.utexas.edu]
> On Behalf Of Stavros Macrakis
> Sent: Sunday, September 12, 2010 6:52 PM
> To: Richard Hennessy; maxima at math.utexas.edu
> Subject: Re: [Maxima] Extended real arithmetic (was Re: inf - inf = 0 ??)
>
> As Fateman has pointed out on this discussion before, Maxima often
> assumes the general case, even when there are special cases which are
> incorrect. For example, Maxima simplifies x/x => 1 without the caveat
> that x # 0.
>
> As for 'limit', it tries to be careful about cases like this, and it
> does assume (sensibly I think) that independent parameters are finite.
> You make it sound very simplistic. Certainly it has bugs and
> limitations, but they're (mostly!) not quite as trivial as you're
> implying.
>
> -s
>
> On 2010-09-12, Richard Hennessy <rich.hennessy at verizon.net> wrote:
>> I get using real number mathematics that 0*x = 0 for all x. I get from
>> extended real arithmetic that und*0 # 0 or inf*0
>> # 0. So I can prove that the answer is both zero and nonzero at the same
>> time depending on the approach. So extended
>> real arithmetic is not self consistent.
>>
>> FWIW.
>>
>> --------------------------------------------------
>> From: "Richard Hennessy" <rich.hennessy at verizon.net>
>> Sent: Saturday, September 11, 2010 11:16 PM
>> To: "Richard Fateman" <fateman at cs.berkeley.edu>
>> Cc: "Maxima List" <maxima at math.utexas.edu>
>> Subject: Re: [Maxima] Extended real arithmetic (was Re: inf - inf = 0 ??)
>>
>>> Rich,
>>>
>>> I have been reading your paper, there is more to this than I thought. I
>>> have not finished reading the paper yet, it
>>> is a lot to digest. I would have been satisfied with not allowing
>>> arithmetic on inf, minf, und or ind. I suppose that
>>> is not good enough.
>>>
>>> Rich
>>>
>>>
>>> --------------------------------------------------
>>> From: "Richard Fateman" <fateman at cs.berkeley.edu>
>>> Sent: Saturday, September 11, 2010 10:07 AM
>>> To: "Leo Butler" <l.butler at ed.ac.uk>
>>> Cc: <maxima at math.utexas.edu>; "Andreas Eder" <andreas_eder at gmx.net>
>>> Subject: Re: [Maxima] Extended real arithmetic (was Re: inf - inf = 0 ??)
>>>
>>>> At the risk of repeating myself, I can refer you to a paper that
>>>> suggests this fits within
>>>> a framework of interval arithmetic.
>>>> e.g. interval(-inf, inf) means --- there is a real number but we
>>>> don't know anything about its value.
>>>> this is perhaps indefinite (IND)
>>>>
>>>> an empty interval (we can choose one to be canonical) that has nothing
>>>> in its interior means
>>>> --- there is no number that this could be, which is perhaps undefined
>>>> (UND)
>>>>
>>>> http://www.cs.berkeley.edu/~fateman/papers/interval.pdf
>>>>
>>>> There are what I consider misuses of intervals regarding limits that are
>>>> present in Mathematica and Maple,
>>>> eg..
>>>>
>>>> limit(sin(x),x,inf) ?=? interval(-1,1).
>>>>
>>>> This leads to either erroneous answers to limit problems and things
>>>> related to limits, or to a substantially
>>>> overloaded notion of "equality". Like any interval that contains 0 is
>>>> equal to 0.
>>>>
>>>> Whether this can be put together in a computer algebra system to be more
>>>> consistent is possible, but
>>>> consistency in general is tough, and certainly not achieved now.
>>>> There are many examples in which, for some procedure f,
>>>>
>>>> f(a) is different from subst(a,x, simplify(f(x)) )
>>>> where x is some arbitrary symbol and a is some particular value.
>>>>
>>>>
>>>> RJF
>>>>
>>>>
>>>>
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>>>>
>>>
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>>
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>
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