Re: Unbounded intervals
> Date: Wed, 25 Apr 2012 10:25:54 -0500
> From: Ralph Baker Kearfott <rbk@xxxxxxxxxxxxx>
> To: Nate Hayes <nh@xxxxxxxxxxxxxxxxx>
> CC: Vincent Lefevre <vincent@xxxxxxxxxx>,
> stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>
> Subject: Re: Unbounded intervals
>
> Nate, Vincent, P-1788,
>
> Regarding this counterpoint, I think it hinges on what
> each of us means by "level 1," i.e. "mathematical
> intervals" (to my understanding). I always thought
> this was in relation to the real numbers, whereas
> the lower levels corresponded to fitting the ideas
> about reals into the floating point system.
> > Dan: Is that how it was used in 754 discussions?
>
> Baker
Baker,
I must say that 1788 is treating the level structure with
far more intelligent thought than 754 ever did. We spent
most of our time on it trying to decide if a notion was in
level 1 or level 2.
That may be due to the fact that most of our communications
today are via email rather than via spoken word, as much as
anything else. Some amount of thought goes into those words
before one tends to hit send.
That having been said, no, 754 discussions (largely via the
means of the spoken word) did not have much to do with the
notion of an overflow as a way of dealing with infinity.
In spite of the fact that it IS, in fact, the most common
way of arriving at infinity on any computer I have ever had
access to.
You must remember that my involvement with 754 was more
involved with re-upping it than with deciding things like
the meaning of +inf. So, the meaning of +inf was already
well established by the time I came along.
We had no difficulty with 1/Small = +inf, Exp(Huge) = +inf,
Huge^2 = +inf, or Huge + Huge = +inf or any of the more
subtle distinctions that are being made with OVR versus
+inf.
Indeed, this entire discussion would not have taken place
within the 754 world. It would have been as meaningless
there as I believe it will turn out to be here, in the end.
In the end, we will all have to have some symbol Big to
stand for some number beyond our ability to calculate with
any degree of accuracy. We will have to have, 1/Small = Big,
Exp(Huge) = Big, Huge^2 = Big, & Huge + Huge = Big for all
the Small(s) & Huge(s) we can calculate with accurately.
Whether we call Big by the name OVR or +Inf will not matter
to us.
In the end, we will have to use 754's +inf for Big anyway.
The decisions about Big have already been made. So it
will have to behave like that whether we like it or not.
It is the world we live in folks.
Sorry if you don't like it.
Dan
>
> On 04/25/2012 09:05 AM, Nate Hayes wrote:
> > 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
> >
>
>
> --
>
> ---------------------------------------------------------------
> 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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