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Re: [IEEE P1788 er subgroup]: My first cut at the Level 1 list...

Hi, Dan,

thank you for your mail.


 SHOULD the basis for our (interval) arithmetic be R or R* (including	-oo & +oo)?


The basis for real interval arithemetic is R*, of course, the same as for floating-point arithmetic. On the computer the lower bound of the result of an interval operation has to be rounded downwards and the upper bound rounded upwards. These roundings are mappings from R* onto F* (floating-point numbers with -oo & +oo).
In floating-point arithmetic an overflow or division of a number not equal 0 by zero delivers -oo or +oo as result. In interval arithmetic the result of an operation is a set. In the first case it is an unbounded real interval and in the second case it is the empty set. 
There are no operations for real intervals which deliver [-oo, -oo] or [+oo, +oo] as result. Both are not real intervals. So in contrast to floating-point arithmetic there is no need in interval arithmetic to define operations like [+oo, +oo] - [+oo, +oo] or [+oo, +oo]/[+oo, +oo] which are set to be NaN in floating-point arithmetic. Further, if A and B are real intervals with 0 in A and 0 in B then A/B = (-oo, +oo) in interval arithmetic. So again there is no need in interval arithmetic to introduce NaN for 0/0 for instance as in floating-point arithmetic. Interval arithmetic can be kept free of exceptions.

Floating-point arithmetic and interval arithmetic are different calculi. Each one has its own rules. Mixing the two causes complicated problems. It would be unwise. In simplicity lies truth!

Best whishes
Ulrich




Dan Zuras Intervals schrieb:

Date: Fri, 27 Mar 2009 12:15:40 +0100
From: Ulrich Kulisch <Ulrich.Kulisch@xxxxxxxxxxx>
To: Dan Zuras Intervals <intervals08@xxxxxxxxxxxxxx>
CC: stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>
Subject: Re: [IEEE P1788 er subgroup]: My first cut at the Level 1 list...


Dear colleagues:

Please see the attached comments.

Best whishes
Ulrich Kulisch

	Hi, Ulrich,


	While Ulrich's argument ASSUMES that the basis for the interval
	domain is R, he is quite correct.  If we choose R then both
	1/([1,1] - 1)^2 & 1/[0,0] have no answer.  Or more precisely,
	those arguments are outside the domain of those functions so
	the answer is [empty].

	But in assuming the answer we are dodging the question which
is: As the distinction between the two seems to be whether or not 
	we include the two elements [-oo,-oo] & [+oo,+oo] in our set
	of intervals, I was exploring what would happen if we made it
	R*.

	There is also the fact, which I failed to mention, that the
	basis for our underlying floating-point arithmetic is R*.

	Still, Ulrich is correct to point out that including the two
	infinity singletons doesn't answer all our questions.  Namely
	we still must decide what to do with odd order singularities
	such as 1/xx.  I don't know the answer.  Maybe choosing R
	instead is the correct answer.

	I thought that if we allow ourselves to use R* in designing
	the underpinnings of intervals we should also allow our users
	the same benifits.

	So I chose R* but, as I said above, I admit that the case is
	a weak one.

	It is not a result I'm married to so I would be happy either
	way we go so long as any inconsistencies are ironed out first.

	OK, ironed out eventually. :-)

	Nice to hear from you Ulrich.  Its been a long time.


				   Dan