Motion 45
It was clear to me from the beginning that I shall vote No on Motion 45.
But the motion was accepted. So reading the text of the motion more
carefully now a number of inconsistencies occur to me. I wonder whether
somebody of the Yes voters can help me staighten these out.
The product of two real numbers is a particular form of a dot product.
Now consider the interval product with a>0 and b<0:
[0, a] × [b, 0] = [a×b, +0] or [0, 0] × [c, d] = [+0, +0]
By motion 45 the zeros in the results have to carry a plus sign. This
looks very strange to me in particular since a×b<0. A signed zero does
not make sense in interval arithmetic. An interval with zero as one
bound has already a direction.
Similar for division by an interval where a bound is -oo or +oo:
If in the result of an interval division one bound is zero, like [a, 0]
or [0, b], the other bound shows immediately whether the elements are
positive or negative. So there is absolutely no need for indicating this
by a minus or plus symbol in front of the zero.
A real interval is defined as a closed and connected set of real
numbers. In case of unbounded intervals -oo and +oo are just used as
bounds. They are themselves not real numbers! So in case of interval
division the rules for computing the bounds of the result do not
strictly follow the pattern of IEEE 754 arithmetic for extended real
numbers. For instance 0×(-oo) = (-oo)×0 = 0×(+oo) = (+oo)×0 = 0 in
interval arithmetic and not NaN like in IEEE 754 arithmetic.
Well defined interval arithmetic over the real numbers (or the subset of
floating-point numbers) including CA and the EDP leads to a calculus
that is free of exceptions. We must not make the mistake defining
interval arithmetic over all IEEE 754 datums!
The development of IEEE P1788 suffers from the fact that many members of
the group take IEEE 754 arithmetic as basic truth for scientific
computing. This may be true for floating-point arithmetic. But it is not
true for interval arithmetic. The two should be separated more clearly.
Within IEEE P1788 we have the unique chance of defining interval
arithmetic including CA and the EDP as a powerful but simple calculus
that is free of exceptions.
What in Motion 45 under 1. is called the "broader aim" actually is a
much narrower aim.
I shall comment on a number of misunderstandings concerning CA and the
EDP in another mail.
With best regards
Ulrich
--
Karlsruher Institut für Technologie (KIT)
Institut für Angewandte und Numerische Mathematik
D-76128 Karlsruhe, Germany
Prof. Ulrich Kulisch
Telefon: +49 721 608-42680
Fax: +49 721 608-46679
E-Mail:ulrich.kulisch@xxxxxxx
www.kit.edu
www.math.kit.edu/ianm2/~kulisch/
KIT - Universität des Landes Baden-Württemberg
und nationales Großforschungszentrum in der
Helmholtz-Gesellschaft