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MathGroup Archive 2005

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Re: Re: inconsistency with Inequality testing and Floor

  • To: mathgroup at smc.vnet.net
  • Subject: [mg60119] Re: [mg60079] Re: inconsistency with Inequality testing and Floor
  • From: Andrzej Kozlowski <andrzej at akikoz.net>
  • Date: Sat, 3 Sep 2005 02:06:07 -0400 (EDT)
  • References: <200508251034.GAA10208@smc.vnet.net> <demmfd$rf1$1@smc.vnet.net> <200509010613.CAA08855@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

On 1 Sep 2005, at 13:13, Richard J. Fateman wrote:

>
>
> Andrzej Kozlowski wrote:
>
>
>
>>
>> You cannot expect inexact numbers, particularly "borderline cases" as
>> in this example, to obey the usual laws of arithmetic.
>>
>>
>
> These inconsistencies are not inherent in computer arithmetic,
> even floating-point arithmetic. They represent choices made by the
> designers of Mathematica.  In particular, choosing to allow two
> distinct numbers to be numerically equal leads to problems. I
> wonder if there is some compensating good reason for this choice
> in Mathematica. I am not aware of any reason good enough to
> justify this.
>
> Calling a bug a feature does not fix it.
>
> RJF
>
>


I can see a number of reasons for keeping numerical equality between  
approximate numbers in Mathematica -the main is that it makes many  
things simpler, in particular in  numerical polynomial algebra (for  
example in numerical Groebner basis). It is not strictly  necessary  
but it is convenient in interval based computations, which is what  
significance arithmetic essentially does. And by the way, while  
writing a review of Hans Stetter's "Numerical Polynomial Algebra", I  
have found that  Mathematica's significance arithmetic with its  
automatic precision tracking performs performs very well when  
combined with   backward error based techniques that Stetter uses; in  
fact quite a lot better than the other well known systems he was  
using. I could even send you some examples of Matheamtica's  
superiority but in view of the countless disputes on this topic that  
you have initiated and that never get anywhere,   I think it would be  
pointless. If you are interested you can read my review when it  
appears in Math Reviews as actually discusses Mathematica's numerics.

Andrzej Kozlowski





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