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spaces (Shiriaev, 1993). The very powerful methods
of global optimization are impressive applications of
these techniques.
Many other applications require that rigorous
mathematics can be done with the computer using
floating-point arithmetic. As an example, this is es-
sential in simulation runs (eigenfrequencies of a large
generator, fusion reactor, simulation of nuclear explo-
sions) or mathematical modelling where the user has
to distinguish between computational artifacts and
genuine reactions of the model. The model can only
be developed systematically if errors resulting from
the computation can be excluded.
Nowadays computer applications are of immense
variety. Any discussion of where a dot product com-
puted in quadruple or extended precision arithmetic
can be used to substitute for the exact scalar prod-
uct is superfluous. Since the former can fail to pro-
duce a correct answer an error analysis is needed for
all applications. This can be left to the computer.
As the scalar product can always be executed exactly
with moderate technical effort it should indeed al-
ways be executed exactly. An error analysis thus be-
comes irrelevant. Furthermore, the same result is al-
ways obtained on different computer platforms. An
exact scalar product eliminates many rounding errors
in numerical computations. It stabilises these compu-
tations and speeds them up as well. It is the necessary
complement to floating-point arithmetic.
2
REFERENCES
IEEE Floating-Point Arithmetic Standard 754, 2008.
The IFIP WG - IEEE 754R letter, dated September 4, 2007.
The IFIP WG - IEEE P1788 letter, dated September 9,
2009.
U. Kulisch, V. Snyder. The Exact Dot Product as Basic Tool
for Long Interval Arithmetic, passed on Nov. 18, 2009
as official IEEE P1788 document.
U. Kulisch. Computer Arithmetic and Validity – Theory, Im-
plementation, and Applications, de Gruyter, Berlin,
New York, 2008.
U. Kulisch. Implementation and Formalization of Floating-
Point Arithmetics IBM T. J. Watson-Research Cen-
ter, Report Nr. RC 4608, 1 - 50, 1973. Invited talk at
the Caratheodory Symposium, Sept. 1973 in Athens,
2
A disappointing feature is the failure of the numerical
analysts to influence computer hardware and software in the
way they should. It is often said that the use of computers
for scientific work represents a small part of the market and
numerical analysts have resigned themselves to accepting
facilities ”designed” for other purposes and making the best
of them. J. H. Wilkinson: Turing Lecture 1970, J. ACM 18
(1971), 146.
published in: The Greek Mathematical Society, C.
Caratheodory Symposium, 328 - 369, 1973, and in
Computing 14, 323–348, 1975.
U. Kulisch. Grundlagen des Numerischen Rechnens –
Mathematische Begr¨undung der Rechnerarithmetik.
Reihe Informatik, Band 19, Bibliographisches Insti-
tut, Mannheim/Wien/Z¨urich, 1976.
S. M. Rump. Kleine Fehlerschranken bei Matrixproblemen.
Dissertation, Universit¨at Karlsruhe, 1980.
R. Lohner. Interval Arithmetic in Staggered Correction For-
mat. In: E. Adams and U. Kulisch (eds.): Scien-
tific Computing with Automatic Result Verification,
pp. 301–321. Academic Press, (1993).
F. Blomquist, W. Hofschuster, W. Kr¨amer. A Modified Stag-
gered Correction Arithmetic with Enhanced Accuracy
and Very Wide Exponent Range. In: A. Cuyt et al.
(eds.): Numerical Validation in Current Hardware
Architectures, Lecture Notes in Computer Science
LNCS, vol. 5492, Springer-Verlag Berlin Heidelberg,
41-67, 2009.
R. Klatte, U. Kulisch, C. Lawo, M. Rauch, A. Wiethoff,
C–XSC, A C++ Class Library for Extended Scientific
Computing. Springer-Verlag, Berlin/Heidelberg/New
York, 1993. See also: http://www.math.uni-
wuppertal.de/∼xsc/ resp. http://www.xsc.de/.
D. Shiriaev. Fast Automatic Differentiation for Vector Pro-
cessors and Reduction of the Spatial Complexity in
a Source Translation Environment. Dissertation, Uni-
versit¨at Kalrsuhe, 1993.
S. Oishi; K. Tanabe; T. Ogita; S.M. Rump. Conver-
gence of Rump’s method for inverting arbitrarily ill-
conditioned matrices. Journal of Computational and
Applied Mathematics 205, 533-544, 2007.
IBM System/370 RPQ. High Accuracy Arithmetic. SA 22-
7093-0, IBM Deutschland GmbH (Department 3282,
Sch¨onaicher Strasse 220, D-71032 B¨oblingen), 1984.
ACRITH-XSC: IBM High Accuracy Arithmetic, Extended
Scientific Computation. Version 1, Release 1. IBM
Deutschland GmbH (Sch¨onaicher Strasse 220, D-
71032 B¨oblingen), 1990.
1. General Information, GC33-6461-01.
2. Reference, SC33-6462-00.
3. Sample Programs, SC33-6463-00.
4. How To Use, SC33-6464-00.
5. Syntax Diagrams, SC33-6466-00.
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