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Re: Unbounded intervals

Jurgen wrote:

Nate,
  I really do not understand the discussion about unbounded intervals. I
thought they were accepted with motion 3

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;

Are you denying that we all support interval arithmetic in order to be
able to prove mathematical facts on a computer?

No.


or where is the misunderstanding

How do you use interval arithmetic to find all the zeros of a function on
the domain [MAXREAL,+Inf]?




| so for this reason alone, truly unbounded intervals are never
| necessary in numeric models or computations

but they are helpful and convenient
in the interval Newton method, e.g. unbounded intervals are generated by
division or reverse multiplication
Since they are used in a partition of the reals they can not be replaced
by some family of overflow thresholds

Why?




   -- 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.

I share Vincent's comments


   -- 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.


again I agree with Vincent


I disagree with both of you, for reasons already stated (it is clear to me
that the midpoint Interval Newton algorithm is undefined at Level 1 on input
from unbounded interval... and Vincent's argument is that unbounded
intervals are essential to make Level 1 definitions of algorithms simple).

Nate