Dear Prof Kulisch, dear colleagues.
let me first clarify an obvious misunderstanding :
in my draft9-2 version of the p1788 document [7] is Marco's,
Stefan's and my contribution to scan2010 .
your book is [4]
and its contents which also was presented in your former books
"Grundlagen des numerischen Rechnens" and
"computer arithmetic in theory and practice" paves my way to
interval arithmetic.
Now, I feel that your way teaching it:
Use the horizontal approach: start with the power set,
(10.2)
round with the hull-operator (semimorphism) to
intervals, (10.3,10.4)
round with outward rounding(semimorphism) to
floatingpoint intervals(12.10.1)(27)
is represented in draft9-2., see 10.2- 10.4 and 12.10
I admit there are some disturbances from multi-precision or
implicit intervals, but I think we have the adequate wording to
keep these types of intervals in the discussion as long as
possible.
Dear Prof Kulisch
Now to complete arithmetic.
In your email, you say :
This immediately leads to the requirement to compute
dot products of two
floating-point vectors with just one rounding (correctly rounded).
and this is required in 12.12.12
Hence, we should be happy that complete arithmetic is included
although its inputs are float vectors.
However, a look into the computing history (before
the electronic age) does not help to define a standard for the
future. perhaps, the dawning of the floating-point age is already
foreseeable ??
I hope some information about this question may already be
discussed at scan2014
your paladin
Jürgen
Am 24.05.2014 16:14, schrieb Ulrich
Kulisch:
Dear
colleagues:
Let me comment a little on [7]. Compared with P1788 the book takes
a more general approach to interval arithmetic. It does not just
consider
arithmetic for intervals over the real numbers. It also defines
and studies
arithmetic for *intervals* in the usual product spaces of
computation like
complex numbers, and for *intervals *of vectors and matrices over
the real
and complex numbers. All these interval operations are defined by
a general
mapping principle (semimorphism) which produces the best possible
answer.
This immediately leads to the requirement to compute dot products
of two
floating-point vectors with just one rounding (correctly rounded).
However, a look into the computing history (before the electronic
age) shows
that the simplest and fastest way for computing a dot product is
to compute
it exactly in fixed point arithmetic^§ . This is an essential mean
to speed up computing.
Complete arithmetic does just this. It is fast and exact vector
processing. *
In addition to this it turns out that the exact dot product is a
most
**general tool for obtaining high accuracy in interval
computations,
*a feature not yet considered in P1788.*
***
There is nothing that can be or needs to be standardized in
complete arithmetic.
Repeating the empty argument "complete arithmetic needs a standard
of its own"
again and again, has distracted the attention of many members of
P1788. It
hurt its development.
In an early stage of the P1788 development complete arithmetic was
listed
very early under basic arithmetic operations. This would give it
the necessary
emphasis. I have no doubts that manufacturers would react and
provide it
well implmented on modern processors. The technology allows this
easily.
For more details see [7].
With best wishes
Ulrich
^§ No intermediate results after multiplications and additions
need to be stored and
read in again for the next operation. No intermediate roundings
and normalizations
have to be performed. No intermediate overflow or underflow can
occur. No error
analysis is necessary. The result is always exact. It is
independent of the order in
which the summands are added. Rounding is only done, if required,
at the very
end of the accumulation.
Am 06.05.2014 01:20, schrieb Jürgen Wolff von Gudenberg:
is now online!
I start the discussion with a few remarks on the bibliography
- drop [1] and [2] they import only notation
- drop [10] or include other motions as well
- include more position papers or drop [5]
-add or replace [7] by
Parallel Detection of Interval Overlapping. Nehmeier, Marco;
Siegel, Stefan; Wolff von Gudenberg, Jürgen K. Jónasson (ed.),
(2012). (Vol. 7134) 127-136.
We should decide whether we list position papers. I suggest to
take those whih appeared also elsewhere
Jürgen
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