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Am 31.05.2015 um 11:03 schrieb David
Lester:
Am 25.05.2015 um 10:48
schrieb Vincent Lefevre:
On 2015-05-22 08:48:39 +0200, Ulrich Kulisch wrote:
I can only repeat myself: The exact dot product brings speed and
accuracy to interval arithmetic. It is the key operation for wide
acceptance of interval arithmetic.
You have never shown why/how this might be true.
It can be computed in a fraction of the time that any other known
method needs to compute a correctly rounded dot product.
This is not correct. You even have representation problems with
large-range formats such as GNU MPFR and DPE[*], where the difference
between the maximum exponent and the minimum one can about around
2^31, and even 2^63.
[*] https://gforge.inria.fr/projects/dpe/
It also is the key to many other valuable application like variable
precision interval arithmetic.
In what way?
Dear Vincent, please find my answers to the three
questions 1), 2), 3) and 4) in your mail below:
1) You have never shown why/how this might be true.
I have written an entire book about this (published before
P1788 was founded).
See, in particular, chapter 8 (Scalar products and
complete arithmetic) and chapter 9
(Sample applications).
Let me repeat a short paragraph of my mail to John Pryce
of May 19, 2015:
Interval arithmetic carries the potential to replace
floating-point arithmetic by some general
computing tool where results come with highly accurate
guarantees. Two things are definitely
necessary to reach this goal:
A. Fast double precision interval arithmetic and
Why? In particular why limit yourself to doubles? Nick Hingham,
at least, is convinced that for many calculations today, higher
precision than double is needed.
B. An
exact dot product (EDP). (for the data format double
precision)
Again, why only for double precision? What’s so special about
doubles?
You find
these two requirements already in the preface of my book
"Computer Arithmetic and
Validity" (page XII). A standard that just specifies
naive interval arithmetic and ignores questions of
accuracy is incomplete. It is, moreover,
counterproductive since it will just reconfirm old
reservations
against interval arithmetic. A simple and very fast tool
for obtaining high accuracy is needed.
3) This is not correct. You even have representation problems with
large-range formats such as GNU MPFR and DPE[*], where the difference
between the maximum exponent and the minimum one can about around
2^31, and even 2^63.
I have not and don’t know anybody who requested this!!!
Let’s list them then:
Paul Zimmerman, Richard Brent, and Vincent and all those who
work on mpfr.
Klaus Weihrauch, Vasco Bratka, Norbert Muller, and all of us
who work on computable arithmetic.
To adequately handle these arithmetics your accumulator is no
longer bounded by a mere 1,000 bytes;
instead we need (using Vincent’s lower limit above) 4GB!
Regards,
Dave Lester
4) In what way?
See chapter 9 of my book.
Best wishes
Ulrich,
David,
I feel that I still owe you an answer.
Looking at the names of the colleagues you mention in your mail I
wonder whether
we are talking about the same matter. The subject of my mail is interval
arithmetic
and in particular long or variable precision interval
arithmetic. It detects by itself
whether it was successful or not.
An exact dot product for a modest data format (double precision) is
the building
block for the latter. By pipelining it can be computed in the time
the processor
needs to read the data, i.e., its arithmetic does not contribute to
the computing
time. The dot product also is a fundamental operation in all vector
and matrix
spaces as well as for residual refinement or defect correction
techniques. It should
already have been included in IEEE 754 35 years ago.
I have doubts whether a very large exponent range is reasonable if
it drives out
other reasonable tools. A scaling of the problem in extreme cases
might be a
better solution.
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
www.math.kit.edu/ianm2/~kulisch/
KIT - Universität des Landes Baden-Württemberg
und nationales Großforschungszentrum in der
Helmholtz-Gesellschaft
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