Revision of Motion 13

[Ulrich Kulisch <Ulrich.Kulisch@xxxxxxxxxxx>]

Dear colleagues:

On April 4 my proposal entitled: "Comparison Relations for an Interval
Arithmetic Standard" was proposed as Motion 13 by Bo Einarsson.

To a certain extent selecting the basic coparisons is a matter of taste.
To keep the standard simple the original proposal required only 4
relations as basic comparisons. Many others can be put together using
these 4 relations.

Several very interesting comments were made by a number of colleagues in
recent days. One by Nate Hayes convinced me to revise the motion. In a
mail of April 11 he suggests to extend the 4 comparisons in the proposal
by 3 additional relations. Indeed many derived comparisons can be
expressed simpler by the 7 relations than by the original 4. In
particular Nate Hayes shows in Table 2 of his mail that all certainly
and possibly relations of the Sun interval Fortran can be expressed by
very simple expressions with the 7 suggested relations.

So the revised Motion 13 now contains the 7 comparisons suggested in
Nate's mail as elementary relations for intervals. In the revised Motion
13 they are shown in the same sequence and expressed by the same symbols
as in Nate's mail.

Among the 7 relations are 3 pairs of strict and non-strict relations.
For non of these 3 pairs the two relations are related by what
frequently was called the "algebraic textbook" property.
I have no problem with that. I mentioned already in a mail of April 8
that for vectors and matrices this is also not the case. We usually
define (aij) <= (bij) and (aij) < (bij) if the relation holds for all
components and nobody ever complained that the "algebraic textbook"
property is violated. Intervals can be interpreted as special vectors.
The "algebraic textbook" property is an existence statement. It does not
require that the symbols <= and < only may be used in this relationship.
All 7 relations in the revised motion are clearly defined and easy to
remember.

Alternatively descriptive names could be assigned to the last 3
relations, for instance, "interior", "less than", and "precedes".

With best wishes
Ulrich

--
Karlsruher Institut für Technologie (KIT)
Institut für Angewandte und Numerische Mathematik (IANM2)
D-76128 Karlsruhe, Germany
Prof. Ulrich Kulisch

Telefon: +49 721 608-2680
Fax: +49 721 608-6679
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-Gemeinschaft

Attachment: CompRel.pdf
Description: Adobe PDF document