midpoint

[Ulrich Kulisch <Ulrich.Kulisch@xxxxxxxxxxx>]

Dear P1788 members:

P1788 considers intervals of \overline{IR}, the set of closed and connected sets of real numbers. In mathematics the midpoint of an interval is well defined for elements of the set IR, the set of nonempty, colsed and bounded real intervals.

During the last 12 months the question was discussed whether and how the midpoint should be defined for elements of the set \overline{IR}\IR. I have great sympathy with Dan's resistance of returning NaN in these cases. First of all, this answer may be wrong. Think of an interval of IR where the lower and the upper bound are greater than Fmax. This interval has a real number as its midpoint. It is just not compubable within the given floating-point system. An answer like "not calculable" or perhaps "indefinite" (a native English speaking colleague may find a better denotation) seems to me being more appropriate. Second: Arithmetic of \overline{IR} is strictly based on mathematical grounds and it is free of exceptions. So elements of IEEE 754 like NaN which are more of a speculative nature should not unnecessarily be introduced into P1788.

Now, the question remains, how should the midpoint of elements of \overline{IR}\IR be defined? I think there is no need at all defining it within P1788. We also don't give a definition of the logarithm for negative numbers. If a user has a reasonable application where he needs a splitting point for an unbounded interval or the empty set we should let him the freedom defining it in a way that is appropriate for his application. We don't need this definition in the standard.

I still remember the old days of interval arithmetic (the 1970s). Arithmetic operations were just defined for closed and bounded real intervals (of IR). No functions or operations were defined for the empty set which considerably simplified the implementation of interval arithmetic. Of course, the empty set could occur in a computation as result of an intersection, for instance, and its appearance could be used to continue computing with another branch of an algorithm. In general, however, the empty set meant an end of this computation.

Then IEEE 754 arithmetic brought the strategy to continue computing even with exceptional numbers like -oo, +oo, -0, +0, or NaNs and we all got used to this strategy. I don't see why we should introduce this strategy into P1788.

Best wishes
Ulrich

-- 
Karlsruher Institut für Technologie (KIT)
Institut für Angewandte und Numerische Mathematik (IANM2)
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Prof. Ulrich Kulisch

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