IEEEP1788

[Ulrich Kulisch <ulrich.kulisch@xxxxxxx>]

 

Dear friends:

 

Please understand that I feel the need to write you again.

 

Interval arithmetic can be seen as arithmetic for connected sets of the real numbers R. Arithmetic for R as well as for subsets F of pure floating-point numbers is well defined. On this base arithmetic for bounded and unbounded intervals of IR and IF easily and clearly can be derived. This leads to well known formulas which can be described on a few pages.

 

If division by an interval which includes zero as an interior point is excluded, interval arithmetic leads to an exception-free, closed calculus, i.e., an operation for two intervals of IR or IF always leads to an interval of IR resp. IF again. As an add-on division by an interval that includes zero as an interior point also can be defined in IR and IF.  It leads to two distinct unbounded real intervals. These can be used to develop the extended interval Newton method which allows computing enclosures of all zeros of a function in a given domain.

 

Operations like oo - oo, oo/oo or 0 · oo, which in IEEE 754 arithmetic are set to NaN, do not occur in the operations of IR and IF. For proof see my book Computer Arithmetic and Validity. The book was published before IEEE P1788 was founded. Also the introduction of -0 or +0 does not make sense in interval arithmetic. There is only one zero in R. If zero is a bound of an interval the other bound clearly tells whether the elements of its interior are positive or negative. 

 

In contrast to the simplicity of arithmetics in IR and IF, IEEE P1788 develops interval arithmetic on the base of IEEE 754 arithmetic with all its exceptions. This is a big mistake. It unnecessarily pulls all the IEEE 754 exceptions into interval arithmetic. It makes interval arithmetic clumsy, difficult to understand and to use. It will prove as a serious hindrance for interval arithmetic to be more widely used in the scientific computing community. At the time of teraflops and petaflops computers, however, this is absolutely necessary.

 

In summary: IEEE 754 arithmetic and interval arithmetic are distinct calculi which strictly must be kept separate.

 

I admire all the work that colleagues have invested into the present draft of IEEE P1788. But it should have been recognized that the mathematical truth is much simpler. I feel that it should not be too difficult to wipe out all the superfluous stuff. It really would be worth doing this.

 

With best wishes

Ulrich

  

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

Telefon: +49 721 608-42680
Fax: +49 721 608-46679
E-Mail: ulrich.kulisch@xxxxxxx
www.kit.edu
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