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Re: Computer arithmetic problems worth exploring

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On 5/11/2011 12:46 AM, nmm1@cam.ac.uk wrote:
> In article<0fbd4179-1128-4ae0-94cf-91fcc25f6401@glegroupsg2000goo.googlegroups.com>,
> Paul A. Clayton<comp.arch@googlegroups.com>  wrote:
>>
>>> Note that I have no objection to signed infinitesimals ('zeroes'),
>>> provided that they are not conflated with true or approximate
>>> zeroes, and it is THAT that causes the trouble - NOT the actual
>>> existence of signed infinitesimals.
>>
>> What is the difference between an approximate zero and an
>> infinitesimal?  (It does seem to me that infinitesimals
>> should be treated like infinities and that having a true
>> zero is somewhat important.)  Is an approximate zero
>> infinitesimal of unknown/unspecified sign?  How would such
>> be used to match the intended semantics or help debugging?
>
> All of this predates 'computer science', and is mostly grounded
> in solid mathematics.  Some is grounded in the slightly less solid
> mathematics of the numerical analytic properties of floating-point.
>
> In the field of real numbers, there are positive numbers, negative
> numbers and true zero.  This can be extended by adding infinities
> and infinitesimals.  The latter have values of zero but a known
> sign, and are inverses of the infinities with the same sign.  That
> means that there are 7 possibilities for 'zeroes', corresponding
> to which of those the value can include. The most useful are:
>
>                                -     0     +
> True zero                     No    Yes   No
> Approximate zero              Yes   Yes   Yes
>
> The remainder are:
>                                -     0     +
> Positive infinitesimal        No    No    Yes
> Non-negative infinitesimal    No    Yes   Yes
> Negative infinitesimal        Yes   No    No
> Non-positive infinitesimal    Yes   Yes   No
> Indeterminate infinitesimal   Yes   No    Yes
>
> And, of course:
>                                -     0     +
> Non-zero                      No    No    No
>
> Numerical analysis started to consider finite arithmetics way
> back when, and covered both fixed- and floating-point.  In both,
> but especially in the latter, it was clear that the zero was not
> a true zero but an approximate zero, and the model chosen was
> that it was the one above.  In cases when the sign was known, it
> was typically remarked on and handled specially.
>
> When modern computing started, the design of the floating-point
> was dominated by numerical analysts, who knew the existing theory
> and mapped it to hardware and software.  Considerably later, the
> subject of 'computer science' invented itself and, for complex
> sociological reasons, more-or-less eliminated numerical analysis
> from its canon.  And that is where we are today, and why.
>
>
> Regards,
> Nick Maclaren.


Are you aware of - rather, can you tell us of - any computers that have 
support for all of the zeroes and infinitesimals that you list above?

I'm not aware of any hardware implementations.  So, I'd be interested in 
any software implementations.

Similarly, for infinities: -infinity, +infinity, and the infinity of 
unknown sign. (What is 1/true zero in this scheme?)

---

While we are at it, I have been thinking about complex numbers more and 
more.  What are the infinitesimals for complex numbers?  Intervals for 
the angle, and/or -0,0,+0, -inf,inf,+inf for the radius?

(For that matter, infinitesimals for the angles matter.  But around and 
cut point, not just 0.)

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Re: Computer arithmetic problems worth exploring "Paul A. Clayton" <paaronclayton@gmail.com> - 2011-05-10 19:24 -0700
  Re: Computer arithmetic problems worth exploring nmm1@cam.ac.uk - 2011-05-11 08:46 +0100
    Re: Computer arithmetic problems worth exploring "Andy \"Krazy\" Glew" <andy@SPAM.comp-arch.net> - 2011-05-11 13:11 -0700
      Re: Computer arithmetic problems worth exploring nmm1@cam.ac.uk - 2011-05-11 21:51 +0100
        Re: Computer arithmetic problems worth exploring "Andy \"Krazy\" Glew" <andy@SPAM.comp-arch.net> - 2011-05-12 20:12 -0700
          Re: Computer arithmetic problems worth exploring Quadibloc <jsavard@ecn.ab.ca> - 2011-05-12 20:53 -0700
          Re: Computer arithmetic problems worth exploring Jason Riedy <jason@lovesgoodfood.com> - 2011-05-13 00:03 -0400
      Re: Computer arithmetic problems worth exploring nmm1@cam.ac.uk - 2011-05-12 00:07 +0100
        Re: Computer arithmetic problems worth exploring "Andy \"Krazy\" Glew" <andy@SPAM.comp-arch.net> - 2011-05-12 20:14 -0700

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