Re: Motion on interval flavors
On 2012-07-06 17:35:24 +0300, Svetoslav Markov wrote:
> Dear prof Kulisch,
>
> > Kaucher or modal intervals are not really intervals.
> > They are abstract entities.
>
> Yes! The sitiation is exactly the same as real numbers.
>
> Let me periphrase your words as follows:
>
> "Real numbers are not really numbers. They are abstract entities."
>
> I hope everybody agrees.
In the context of functions dealt with the IEEE 754 standard, I don't
(see below). I think more people would agree with "complex numbers",
or perhaps "quaternions".
> My question: Why such abstract entities - real numbers -
> are mentioned in the IEEE standard for floating-point numbers?
Because floating-point numbers are a model of real numbers. All the
theory (that's Level 1) is based on real numbers, and functions that
are dealt with (not just arithmetic operations, but also algebraic
functions like sqrt and transcendental functions like exp) make sense
at least on real numbers, for a practical closure of the arithmetic.
Without real numbers, you can't talk about functions like exp.
With just +, -, *, / (not just in the standard, but also for most
applications used in practice, for which the standard has been
designed), I could agree with you, but this is not the case here.
Complex numbers would have been better for the theory, but one would
have gone too far, in particular w.r.t. the existing implementations:
real FP numbers are common, contrary to complex FP numbers.
Going back to interval arithmetic, one has to consider what most
people need and what are the most common implementations (or what
will be implemented in the short term). And I think this restricts
to set-based intervals. Closed and bounded intervals on R would not
be sufficient because some operations could no longer be defined on
all inputs (IEEE 754 introduced NaN for that on FP numbers, but for
intervals, Empty and unbounded intervals fit this purpose in a nice
way).
BTW, an alternate set-based flavor could be the conventional math
intervals (i.e. possibly empty, possibly unbounded) and the closure
of their set complement. However one would again lose some properties
and there would no longer be a canonical hull.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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