[std-interval] Interval comparison operators

[Fernando Cacciola]

Hello Lee,

For now just a question and a quick comment:

Can you justify your claim that:

given a non-degenerate interval X, the expression ceq(X,X) will return false


Maybe I'm just totally missing a point but your Prinicpal Model doesn't look 
to me like a usefull model of interval numbers. Aren't "bounds" the central 
notion of anything called an "interval"?? This models looks to me like the 
"arithmetic expansions" described by Priest here:

http://citeseer.ist.psu.edu/priest91algorithms.html

In such an expansion, roughly, a "result" is given explicitely as a 
"principal value" and a sequence of error terms.
The comparison semantics that you suggest are in line with those concepts, 
but not so much IMO with interval artihmetic.

IOW, the comparison semantics that take into account the "accumulated 
roundoff" (those you cite from Knuth) are well suited for a model which is 
centered around the fact that a result is always a real-point-value but 
which is represented as machine-point-value which differs from the actual 
value by some error. That's why these comparisons are essentiually 
point-based.
But IIUC the very purpose of interval arithmetic when applied to certified 
computations in the presence of roundoff is precisely to provide an 
alternative view centered not on the "principal" value but the error 
bounds...
If you want a number conceptually centered around an "actual value" which 
can't be exactly represented due to roundoff, OK, but I woudn't call that an 
interval.

Fernando Cacciola
SciSoft
http://fcacciola.50webs.com/