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Dear P1788 members,we are all looking back on a long and sontimes dffuse discussion on Kaucher/modal arithmetic. The only outcome of all this so far was an abstract motion on flavors where Kaucher/modal arithmetic is considered as one possibler flavor. The proposal I prepared is an attempt to reduce the requirements for Kaucher /modal arihtmetic to a basic core of operations. I tryed to keep the text simple and understandable and not to overload it with too many details. Of course, the proposal does not define Kaucher/modal arithmetic completely. But I am convinced that accepting this proposal as it is as a motion would be a (perhaps first but) great step ahead for both Kaucher/modal arithmetic and a future standard P 1788 for interval arithmetic.
The arguments made by Michel Hack below are more or less comments on Motion 5.01 which was accepted. There division by an interval of IR that contains zero is defined. It leads to arithmetic of \ovederline{IR} which is free of exceptions.
With best regards UlrichP.S.: I once more attach the entire text of the proposal which contains the motion at the end.
Am 01.10.2012 19:31, schrieb Ralph Baker Kearfott:
P-1788, Does someone wish to take action on this motion? I certainly did not mean to quash it, but I hoped to have its underlying intent be clear and precise. Does someone wish to present formally how the specified set of operations should be considered (required, optional but as stated if implemented, or recommended as stated)? Baker ============================================================= OK. However, I'd like the eventual motion to state clearly what will be required, what will be specified but optional, and what will be recommended. We need unambiguous guidance at this point for the actual standard, to avoid controversy at the last minute, since we are approaching a deadline for submitting it for IEEE review. Baker On 09/26/2012 01:34 PM, Michel Hack wrote:I take Ulrich's paper to be the basis for developing an actual motion, after we have discussed the ideas presented therein. An actual motion to be voted on (as a position paper, at this point) would state what requirements or recommendations (Baker's (a) and (b)) should be included in P1788, and (if that motion passes) there would be a later text motion for John's "Chapter 3" mentioned in Ulrich's reply to Baker. The rules being proposed in Ulrich's document are incomplete at this point. As he himself points out, divisors containing zero, and overflow during outward rounding, need special treatment -- and this is not covered in the motion proper as it now stands. In the rationale leading up to the motion, Ulrich addresses these two issues as follows: (1) Division by an interval containing zero produces two unbounded results. (2) Overflow is not acceptable and the computation must be stopped at that point. This leads to a third issue: the given rules only cover inputs in I*R -- but division by zero will lead (without overflow) to elements of (I*R)bar. For overflow Ulrich recommends that a computation be redone with scaled operands. For division with two results I assume a program would bifurcate. I would know how to handle that at the assembly language level (a condition code would indicate that two results were produced, and a single-threaded program would set one aside and continue with the other, later handling the second path). I can even imagine hardware that automatically spawns another tread and (depending on compute resources) allocates the various threads among available processing elements. What I don't know is how this notion would fit generically into higher-level programming languages. The redo-scaled-after-oveflow could be handled by decorations, thus avoiding the need for exceptions, or for explicit tests after every operation, but the bifurcation needs to happen at the point where it occurs -- unless decorations are used to restart a compound operation after an argument-splitting step that avoids intervals containing zero. So there are a number of ways to deal with these issues -- and this is one reason why some of us feel that standardization of Kaucher intervals might be premature. Michel. P.S. Would it be possible to get a plain-text version (e.g. the TeX source) of the motion? Trying to extract text from a .pdf does not work too well and requires a very large amount of manual repair to get useful results. ---Sent: 2012-09-26 19:08:02 UTC
-- 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
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