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,
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