On Feb 10, 2007, at 8:35 PM, Carl Witty wrote:
>
> I have a design question about interval arithmetic comparisons.

[...]

I don't quite know the answer to these questions, but I reckon one  
thing: for consistency, you want interval comparison to match up with  
comparison of reals and comparison of p-adic integers. What I mean  
is: a real number (i.e. an IEEE double, or an MPFR real, or whatever)  
is only ever given up to some finite precision, and therefore really  
it's just an interval. Similarly, a p-adic integer is only given up  
to congruence mod some power of p, and so also corresponds to an  
"interval" (really a ball). But p-adics are easier, since two balls  
overlap if and only if one is contained in the other, which is not at  
all true for real intervals. In particular the "overlap test" for p- 
adic balls gives a transitive relation, which seems to be the main  
ingredient missing in the real interval case.

Do you have any other examples of undesirable behaviour apart from  
the business with the leading term of a polynomial? My feeling is  
that it's the code in the polynomials that needs to change, rather  
than anything else.

David



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