Re: overflow question
Arnold Neumaier wrote:
On 05/09/2012 09:30 PM, Nate Hayes wrote:
Arnold Neumaier wrote:
On 05/09/2012 05:22 PM, Nate Hayes wrote:
One may even argue that to justify
midpoint([1,+Inf]) = REALMAX
at Level 2 is due to implicit reasoning of overflow.
The right definition of the midpoint of an empty or unbounded interval
is 'undefined' on level 1 and NaN on level 2. Any other definition is
mathematically artificially and hence misleading some users.
That is an argument in favor of my motion, since as I said the motion
does not define the midpoint of an unbounded interval at any level.
But one needs unbounded intervals on level 2 and then doesn't need the
complicated overflow interval sets, as they don't serve any additional
purpose.
First we should be clear it was Alex who just recently suggested
midpoint([1,+Inf]) should be REALMAX at Level 2, but I had said:
Nate Hayes wrote:
P1788 should not define midpoint for unbounded intervals at Level 2
So if X = [1,+Inf], then your position matches my own when you said that the
midpoint of X should be undefined at Level 2, as well, i.e.,
midpoint([1,+Inf]) = undefined (current Level 1)
midpoint([1,+Inf]) = undefined = NAN (current Level 2)
However, this means many interval algorithms involving midpoint are
undefined at both Level 1 and Level 2 in the current model. For example, how
does a branch-and-bound algorithm that bisects on the midpoint even begin to
proceed (at Level 1 or Level 2) when the user provides X = [1,+Inf] as
input?
In our paper and motion,
Z = Omega(X) = Omega([1,+Inf]) = [1,+omega]
is an overflow family. This is different mathematical object than an
unbounded interval, so we may define the midpoint of [1,+omega] as a real
number.
Now all the interval algorithms that are undefined for unbounded intervals
in the current model are defined for overflow in the new model.
We are working on an amendment to the motion text and position paper in
light of recent discussion to make this much more explicitly clear.
Thanks.
Nate