Dear Ulrich,
You write: "In the vast majority of these applications the
data are floating-point numbers or how you call it degenerate
intervals."
My point is that in the vast majority of these applications it
is a mistake for inputs to be degenerate intervals. That is
why it is so important to make extra effort required to enter
degenerate intervals into interval programs, as is the case in
Sun's intrinsic compiler implementation.
If anything, I believe concentrating on the implementation of
CA and EDP have distracted much of the interval research
communities' time, effort and energy from other much more
important tasks that need to be solved to bring computing with
intervals into the mainstream of science and engineering.
Sorry, Ulrich, but that is my firm belief.
With respect and best regards,
Bill
On 8/27/13 3:32 AM, Ulrich Kulisch wrote:
Am 24.08.2013 21:49, schrieb G.
William (Bill) Walster:
Ian,
As far as I can tell the only time when a case can be made that EDP is
essential for interval computations is when all interval inputs are
degenerate and therefore infinitely precise. Otherwise, with interval
bounds on the accuracy of typical measured data, I don't see the
requirement for EDP. I continue to wait to see even one practical
example thereof.
Cheers,
Bill
Bill,
the dot product is a fundamental arithmetic operation in the
vector and matrix spaces as well as in mathematics. Computing
an EDP is simple, error free and fast. If in a particular
application the active part of the long accumulator is
supported by hardware it comes with utmost speed. Not a
conventional computation of the dot product in floating-point
arithmetic nor computing a correctly or otherwise rounded dot
product can reach this speed.
Let me now comment on your mail. I wonder why you care so much
finding a small corner of applications where the EDP might not
be useful.
These are typical applications of interval arithmetic:
Verified solution of systems of linear or nonlinear equations,
guaranteed evaluation of polynomials or of arithmetic
expressions, computing enclosures of the solution of an
ordinary or a partial differential equation, and others. In
the vast majority of these applications the data are
floating-point numbers or how you call it degenerate
intervals. Interval arithmetic allows computing close bounds
for the solution and even to prove the existence of a solution
within the computed bounds.
In these and other cases computing these results follows a
general pattern: First an approximate solution is computed,
then bounds for the expected exact solution are established (in
case of a partial differential equation this may require
computing highly accurate eigen- or singular values, for
instance). Then a mathematical fixed-point theorem
(Banach, Brower, or Schauder, depending on the problem) is
applied which verifies the existence of a solution within
these bounds. Finally, if necessary, an interative refinement
is applied to improve the result. In all these steps an EDP is
repeatedly applied and it often is essential for successs.
This can already be seen in case of computing a verified
solution of a system of linear equations.
Let me now consider problems with non degenerate intervals in
the data. If the bounds are small the computation follows
exactly the pattern described above. There is no reason why
less care in computing the approximate solution, in
establishing bounds for the exact solution or in the
verification step is necessary.
If the bounds for the data grow, the problem may gradually
degenerate into an NP-hard problem.
You probably know that I held patents on the EDP in Europe,
the US, and in Japan between 1981 and 2007. I paid quite some
fees for keeping them alife over all these years. The
XSC-languages Pascal-XSC, ACRITH-XSC of IBM, and C-XSC all
provide CA and the EDP. The verification methods developed in
these languages make heavily use of these tools. See the
language descriptions and the corresponding toolbox volumes.
References are listed in my mail of August 7, 2013.
The patents you obtained in the US a few years ago do not make
use of CA and the EDP. I am absolutely convinced that the
methods which use CA and the EDP are superior to those which
do not use these.
With best regards
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
On 8/27/13 3:32 AM, Ulrich Kulisch wrote:
Am 24.08.2013 21:49, schrieb G.
William (Bill) Walster:
Ian,
As far as I can tell the only time when a case can be made that EDP is
essential for interval computations is when all interval inputs are
degenerate and therefore infinitely precise. Otherwise, with interval
bounds on the accuracy of typical measured data, I don't see the
requirement for EDP. I continue to wait to see even one practical
example thereof.
Cheers,
Bill
Bill,
the dot product is a fundamental arithmetic operation in the
vector and matrix spaces as well as in mathematics. Computing an
EDP is simple, error free and fast. If in a particular application
the active part of the long accumulator is supported by hardware
it comes with utmost speed. Not a conventional computation of the
dot product in floating-point arithmetic nor computing a correctly
or otherwise rounded dot product can reach this speed.
Let me now comment on your mail. I wonder why you care so much
finding a small corner of applications where the EDP might not be
useful.
These are typical applications of interval arithmetic: Verified
solution of systems of linear or nonlinear equations, guaranteed
evaluation of polynomials or of arithmetic expressions, computing
enclosures of the solution of an ordinary or a partial
differential equation, and others. In the vast majority of these
applications the data are floating-point numbers or how you call
it degenerate intervals. Interval arithmetic allows computing
close bounds for the solution and even to prove the existence of a
solution within the computed bounds.
In these and other cases computing these results follows a general
pattern: First an approximate solution is computed, then bounds
for the expected exact solution are established (in case of
a partial differential equation this may require computing
highly accurate eigen- or singular values, for instance).
Then a mathematical fixed-point theorem (Banach, Brower, or
Schauder, depending on the problem) is applied which verifies the
existence of a solution within these bounds. Finally, if
necessary, an interative refinement is applied to improve the
result. In all these steps an EDP is repeatedly applied and it
often is essential for successs. This can already be seen in case
of computing a verified solution of a system of linear equations.
Let me now consider problems with non degenerate intervals in the
data. If the bounds are small the computation follows exactly the
pattern described above. There is no reason why less care in
computing the approximate solution, in establishing bounds for the
exact solution or in the verification step is necessary.
If the bounds for the data grow, the problem may gradually
degenerate into an NP-hard problem.
You probably know that I held patents on the EDP in Europe, the
US, and in Japan between 1981 and 2007. I paid quite some fees for
keeping them alife over all these years. The XSC-languages
Pascal-XSC, ACRITH-XSC of IBM, and C-XSC all provide CA and the
EDP. The verification methods developed in these languages make
heavily use of these tools. See the language descriptions and the
corresponding toolbox volumes. References are listed in my mail of
August 7, 2013.
The patents you obtained in the US a few years ago do not make use
of CA and the EDP. I am absolutely convinced that the methods
which use CA and the EDP are superior to those which do not use
these.
With best regards
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
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