On Mon, 10 Nov 2008 23:02:00 -0100, I wrote a detailed critique of
the opposing points of view of Siegfried Rump and Arnold Neumaier.
Unfortunately I was misled by a typo in Siegfried's example, from
which I concluded that his approach was inconsistent, compared to
Arnold's. So the situation is not clear-cut after all. Here are
some replies to questions Siegfried asked here.
On Thu, 13 Nov 2008 10:30:01 +0100 Siegfried M. Rump wrote:
(*) The Interval->Float functions inf() and sup() that return the
bounds can return Inf -- and if this is fed into an interval
constructor as a value (instead of as a bound), an exception
should arise. Arnold's proposal covers this; the result of a
violation is simply Empty.
This means that if we want to compute an upper bound for the diameter
of an interval, then in Arnold's proposal (using INTLAB notation)
diam = sup( intval(X.sup) - X.inf )
is empty for X = infsup(3,inf) with an exception raised.
I suppose the underlying IA runtime would have diam() and mid()
primitives -- the question is, what would they return. If they
return a float, this would indeed by Inf if either end is unbounded.
Perhaps they should also have a form that returns an interval, which
would be [realmax-lower, Inf] for diam([lower, inf]), and (presumably)
the possible range of midpoints, [(realmax+lower)/2, Inf], for the
mid"point" of that same source interval. Arnold may have more to say
about this.