Vincent Lefevre wrote:
On 2012-04-23 10:08:24 -0500, Nate Hayes wrote:
We are looking at a model that defines overflow at Level 1 as an abstract
parameterization of Level 2,
Note that in my case,
I'm assuing by this you mean:
Vincent Lefevre wrote:
But if you replace "midpoint" by "any member of
the interval" (or perhaps something more restrictive), I think it is
well-defined at Level 1. Similarly, it is well-defined at Level 2 on
an unbounded input only if some arbitrary value is chosen for the
midpoint on such an interval.
...
I am not really interested in intervals that
overflow. If unbounded intervals occur at Level 2, this is because
they are probably really unbounded intervals at Level 1.
A few points:
-- No computer (that I'm aware of) can numerically prove hardly anything
useful about the domain of a function beyond the underlying numeric limits
of the system; so for this reason alone, truly unbounded intervals are never
necessary in numeric models or computations (you have never answered my
original question from long ago to show a counter-example of this).
-- An overflown interval [1,+OVR] := { [1,a] | a >= H_f }, where H_f is
a parameterization of any would-be Level 2 format, is functionally
equivalent to an unbounded interval but retains a notion of the "largest
representable number"; for this reason it is possible to define
midpoint([1,+OVR]) at Level 1 in the same way P1788 is currently considering
to do so at Level 2.
-- Replacing "midpoint" with "any member of the interval" gives a valid
mathematical definition of the Interval Newton, but such a definition is
also then no longer an algorithm because the exact method of choosing "any
member of the interval" is left undefined.
I think that it is bad to have a notion of overflow at Level 1,
because mathematically at Level 1, there is no overflow. Such a
notion would be, IMHO, artificial.
We are all entitled to our opinions, but I believe for the reasons above it
is neither artificial nor "bad", since overflow at Level 1 in this way
models what actually happens at Level 2 inside a comptuer much more
realistically and allows, for example, an algorithm like Interval Newton to
be defined at Level 1, not just Level 2.
Nate