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Re: Unbounded intervals

Just reading the on-going discussion (and perhaps missing a
few posts) without reviewing, I'm beginning to lose track
of where this discussion is headed with regard to what we
will eventually specify (or not specify) in the standard,
or how we will word the standard.  I admit the discussion
is interesting, can someone perhaps summarize?  What I have
gathered so far is there is contention about whether there
should be a concept of infinity, or just a concept of
overflow, at level 1.  Suppose we have some symbol, say
\aleph, to represent this (be it infinity or overflow).
Can someone explain (or work through, or review, e.g.
by presenting two contrasting tables) how it will affect
the actual operations, whether we think of \aleph as
infinity or overflow?  P-1788 has already decided that
infinity is not an actual element of an interval, but
the symbol can be used to represent an unbounded interval,
and there are consequences of this decision in the result
of operations.

Best regards,

Baker

On 04/26/2012 07:01 AM, Vincent Lefevre wrote:

On 2012-04-25 09:05:19 -0500, Nate Hayes wrote:

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).


I disagree. If the user asks for the range of 1/[0,1], the math result
is [1,+inf]. This is useful information, at least much more than an
error.


    -- An overflown interval [1,+OVR] := { [1,a] | a>= H_f }, where H_f is


As an unknown bounded interval with an arbitrary bound, the result
for 1/[0,1] would be incorrect, because the computed result must
contain the mathematical result.

You may build a theory where the computed result as a *family* of
intervals, but then you should stop calling that an interval. This
is obviously more complex (as a specification) than unbounded
intervals and I don't see what it would bring.


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";


I don't see how this notion of the "largest representable number" is
retained.


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.


How would you define it at Level 1, as a *real number*?
The only information that you would have is an unbounded interval
containing the midpoint.


    -- 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.


Well, once the one who writes the algorithm choose the member in
question, he has an algorithm. This member can potentially be a
parameter, and it will still be an algorithm. I don't see any problem
with that!




--

---------------------------------------------------------------
Ralph Baker Kearfott,   rbk@xxxxxxxxxxxxx   (337) 482-5346 (fax)
(337) 482-5270 (work)                     (337) 993-1827 (home)
URL: http://interval.louisiana.edu/kearfott.html
Department of Mathematics, University of Louisiana at Lafayette
(Room 217 Maxim D. Doucet Hall, 1403 Johnston Street)
Box 4-1010, Lafayette, LA 70504-1010, USA
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