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Dear Professor Kulisch,
I am quite surprised from the last paragraph of
your letter. As far as I remember you are the supervisor
of Kaucher's dissertation. But it is never too late to
find out what your good students have been done.
I think that the easiest way to look at Kaucher arithmetic
is when you consider the mid-rad form. When you restrict to
nonnegative radii (rad \in R^+), you have your interval arithmetic.
When you embed the radii in R, then you obtain Kaucher
arithmetic. (Remark: It is not trivial to do the embedding so
that the inclusion property is invariant)
Thus the relation between usual IA and Kaucher IA is the same
as the relation between the arithmetic in R^+ to real arithmetic (in R).
We all compute in R+. but we mention R, because we use the properties
of R. The FP-standard is about computing in R^+, but mentions R.
This is the correct way to proceed. Our standard should mention
Kaucher arithmetic in the same way as The FP-standard mentions
the set R.
If we remain just in R^+, then we are not allowed to use the properties of R.
To show you how this looks like I attach a ``pedagogical'' paper.
The paper shows namely that we compute in R^+, but we make
advantage of the properties of R. We even add real numbers
presenting them in signed-magnitude form, and then perform the
operations over the magnitudes, that we compute in R^+.
Conclusion: To produce a standard for interval arithmetic that does
not mention Kaucher arithmetic will be the same mistake as to
write a floating point standard without mentioning real numbers.
I am glad to see that slowly but surely the participant in this forum
realise this.
Let me recall that mathematicians did not recognise real numbers for
many centuries. It seems that human minds resist agains improper elements.
An interval standard without mentioning Kaucher intervals will be a very
retrograde piece of work -- something good for 16-th century mathematicians.
Best regards,
Svetoslav
On 28 Jun 2012 at 18:02, Ulrich Kulisch wrote:
Date sent: Thu, 28 Jun 2012 18:02:14 +0200
From: Ulrich Kulisch <ulrich.kulisch@xxxxxxx>
To: stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>
Subject: motion 35
>
> 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 f"ur Technologie (KIT)
> Institut f"ur 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 - Universit"at des Landes Baden-W"urttemberg
> und nationales Grossforschungszentrum in der
> Helmholtz-Gesellschaft
Prof. Svetoslav Markov, DSci, PhD,
Dept. "Biomathematics", phone: +359-2-979-2876
Inst. of Mathematics and Informatics, fax: +359-2-971-3649
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