motion 35

[Ulrich Kulisch <ulrich.kulisch@xxxxxxx>]

Dear P1788 members.

 

We are supposed to develop a standard for interval arithmetic. All kinds of intervals have already been invented or developed. Real intervals, set-based intervals, standard intervals, classical intervals, common intervals, conventional intervals, wrap around intervals, Kaucher and modal intervals, and so on. Frequently only the context tells you what kind of interval is meant.

 

For me a ‘real interval’ is a closed and connected set of real numbers, an element of the set \overline{IR}. Let me argue a little about this. Zero finding is a central task of mathematics. In conventional numerical analysis Newton's method is the key method for computing zeros of nonlinear functions. It is well known that under certain natural assumptions on the function the method converges quadratically to the solution if the initial value of the iteration is already close enough to the zero. However, Newton's method may well fail to find the solution in finite as well as in infinite precision arithmetic. This may happen, for instance, if the initial value of the iteration is not close enough to the solution. It is one of the main achievements of interval mathematics that the method has been extended so that it can be used to enclose all (single) zeros of a function in a given domain. Newton's method reaches its final elegance and strength in form of the extended interval Newton method. The method is globally convergent. It is locally quadratically convergent and it never fails, not even in rounded arithmetic. The key operation to achieve these fascinating properties is division by an interval which contains zero. Newton’s method uses this operation to separate different zeros. The calculus of \overline[{IR} is needed for an elegant formulation of the extended interval Newton method in closed form.

 

It certainly is a laudable goal keeping the standard open for extensions. However, Motion 35 seems to me degrading the concept of ‘real interval’ to the set IR and turns arithmetic of \overline{IR}\IR into a flavour. I am not against introduction of flavours. But can’t we keep the standard open for extensions without introducing a new concept. Reading the text of motion 35 gives me the impression that our time limit is shifted to infinity.

 

In my opinion our main effort should be finishing the standard until the end of this year more or less with what agreement has already been reached. This does not mean that the group P1788 finishes working. We still could go on developing flavours.

 

I am not against modal or Kaucher arithmetic. I studied Edgar Kaucher’s thesis again and again during recent weeks and I am sure that it will take quite some time to reach general agreement on this topic. I am afraid we do not have this much time for being successful with the standard.

 

Best wishes

Ulrich

 

-- 
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