Re: Motion P1788.1/M004.01
On Tue, May 10, 2016 at 10:46 AM, Ulrich Kulisch <ulrich.kulisch@xxxxxxx> wrote:
> I think interval arithmetic should not be defined over the IEEE 754 binary64
> numbers. This more or less pulls all the IEEE 754 exceptions into interval
> arithmetic. We shoud not bother all users of interval arithmetic with
> constructs which really do not occur and definitly are not needed in
> interval arithmetic.
>
> Interval arithmetic shoud just be defined as a calculus for connected sets
> of real numbers.
I disagree. We don't have native real numbers. We only have finite
(small) sets of rational numbers.
> Since -oo and +oo are not real numbers they cannot be
> elements of a real interval.
Agreed. However, while we CALL them signed infinities, they are not.
In point of fact they are actually signed values for a result that
somewhere in the preceding calculations exceeded the maximum
representable value in the applied FP encoding .
> They just serve as bounds for the description
> of unbounded real intervals.
No. There are no unbounded real intervals. Every single computable
interval over rational FP numbers is bounded. We may lose track of
the bound through representation limitations, but there is such a
bound even if it has an unknown, or possibly unknowable, value.
> This leads to a calculus that is totally free of exceptions. (For proof see
> my book Computer Arithmetic and Validity, in particular Sections 4.11 and
> 4.12, pp. 146 to 151 in the second edition).
Yeah. But much of that is based on invalid assumptions. For example
the assumption that we are dealing with real numbers is provably
false.
Now I am not a mathematician (I leave that to my wife). I deal with
numerical software. My understanding of real analysis is both narrow
and shallow. But one of the primary premises of any kind of real
analysis with which I am familiar is the definition of real numbers.
Our small, finite set of rational numbers does not qualify. So
deriving interval arithmetic over FP numbers from real arithmetic
theory is probably a serious mistake.
Lee Winter
Nashua, New Hampshire
United States of America (RIP)