Table Of ContentSCIENTIFIC COMPUTING,
VALIDATED NUMERICS,
INTERVAL METHODS
SCIENTIFIC COMPUTING,
VALIDATED NUMERICS,
INTERVAL METHODS
Edited by
Walter Kramer
University of Wuppertal
Wuppertal, Germany
and
Jiirgen Wolff von Gudenberg
University of Wiirzburg
Wiirzburg, Germany
Springer Science+Business Media, LLC
The Publisher makes no warranty of any kind, expressed or implied, with regard to the software
reproduced on the enclosed CD-ROM. The publisher shall not be liable in any event for incidental or
consequential damages or loss in connection with, or arising out of, the furnishings, performance, or use
of the software.
Proceedings of SCAN 2000-The 9th GAMM-IMACS International Symposium on Scientific
Computing, Computer Arithmetic and Validated Numerics, and Interval 2000-The International
Conference on Interval Methods in Science and Engineering, held 19-22 September, 2000, in Karlsruhe,
Gennany
Additional material to this book can be down1oaded from http://extras.springer.com
ISBN 978-1-4419-3376-8 ISBN 978-1-4757-6484-0 (eBook)
DOI 10.1007/978-1-4757-6484-0
©2001 Springer Science+Business Media New York
Originally published by Kluwer Academic/Plenurn Publishers, New York in 2001
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Preface
Scan 2000, the GAMM - IMACS International Symposium on Scientific
Computing, Computer Arithmetic, and Validated Numerics and Interval
2000, the International Conference on Interval Methods in Science and
Engineering were jointly held in Karlsruhe, September 19-22, 2000. The
joint conference continued the series of 7 previous Scan-symposia under the
joint sponsorship of GAMM and IMACS. These conferences have
traditionally covered the numerical and algorithmic aspects of scientific
computing, with a strong emphasis on validation and verification of
computed results as well as on arithmetic, programming, and algorithmic
tools for this purpose. The conference further continued the series of 4
former Interval conferences focusing on interval methods and their
application in science and engineering. The objectives are to propagate
current applications and research as well as to promote a greater
understanding and increased awareness of the subject matters.
The symposium was held in Karlsruhe the European cradle of interval
arithmetic and self-validating numerics and attracted 193 researchers from
33 countries. 12 invited and 153 contributed talks were given. But not only
the quantity was overwhelming we were deeply impressed by the emerging
maturity of our discipline. There were many talks discussing a wide variety
of serious applications stretching all parts of mathematical modelling. New
efficient, publicly available or even commercial tools were proposed or
presented, and also foundations of the theory of intervals and reliable
computations were considerably strengthened.
v
Vl Preface
Hence, a possible subdivision of this book might have been according to
the following headlines:
• strengthen the theory
• improve the tools
• perform applications
Another observation is that the talks were concerned not only with
developing enclosure methods, but also transferred the mechanisms and
design process known from these areas to other fields. And, of course, there
were new applications of well known enclosure methods.
The 31 contributions to this proceedings volume were carefully selected
out of a much larger number of submissions. A thorough referee process has
been installed, and we take the opportunity to thank all the referees for their
detailed comments delivered in time.
The book opens with a historical review and an outline of the coming
perspectives of intervals and reliable computations by G.W. Walster.
In the first section, then, efficient algorithms for elementary functions
and hardware devices are considered. The section is finished by an article
that shows how the principles of enclosure methods can be applied in test
strategies for scientific computations.
The second section deals with the solution of linear systems.
Surprisingly enough that all the authors consider a kind of generalized
arithmetic.
The treatment of polynomial equations and the sharp enclosure of sets,
although difficult tasks in their own, may be considered as two of the basic
ingredients for two traditional application areas: global optimisation and
control theory. Hence, the four topics are presented in this order.
Solutions of differential equations that model dynamic processes are the
topic of the next 4 papers.
The final section is devoted to the treatment of uncertain data, in
particular the relation of interval and stochastic methods is investigated.
A fmal Dankeschoen to Ulrich Kulisch and the organizing committee of
the symposium. We further thank all those people who helped with the
publication of the proceedings, the authors, the referees, the early subscribers
and the technical staff, in particular Markus Grimmer.
Walter Kramer and Jurgen Woljf v. Gudenberg
Contents
SCAN 2000 Keynote Address The Future of Intervals 1
G. William Walster '
Part I Software-and Hardware-Tools
Variable-Precision Exponential Evaluation 19
Javier Hormigo, Julio Villalba, Michael J. Schulte
Fast computation of some special integrals of mathematical physics 29
Ekatherina A. Karatsuba
Interval Input and Output 41
Eero Hyvonen
A Case for Interval Hardware on Superscalar Processors 53
James E. Stine, Michael J. Schulte
Evaluating the Impact of Accurate Branch Prediction on Interval Software 69
Ahmet Akkas, Michael J. Schulte, James E. Stine
Automatic Test Case Generation using Interval Arithmetic 81
Gunter Schumacher, Armin Bantle
Part II Linear Algebra
On the Hull of the Solution Sets of Interval Linear Equations 91
lana Konickova
Computation of Algebraic Solutions to Interval Systems via Systems of Coordinates 103
Svetoslav Markov
Towards Diagrammatic Analysis of Systems of Interval "Linear Equations" 115
Zenon Kulpa
vii
viii SCIENTIFIC COMPUTING, VALIDATED NUMERICS, INTERVAL METHODS
On the Solution of Parametrised Linear Systems 127
Evgenija D. Popova
Part III Polynomials
Verified solutions of systems of nonlinear polynomial equations 141
Daniela Fausten, Wolfram Luther
Euler-like method with Weierstrass' correction 153
Miodrag S. Petkovic, Dejan V. Vranic
Part IV Set Enclosures
Guaranteed Set Computation with Subpavings 167
Michel Kieffer, Isabelle Braems, Eric Walter, Luc Jaulin
A New Intersection Algorithm for Parametric Surfaces Based on LIEs 179
Katja Buhler, Wilhelm Barth
State estimation using interval constraint propagation 191
Luc Jaulin, Isabelle Braems, Michel Kieffer, Eric Walter
Part V Global Optimization
Interval Methods for Global Optimization Using the Boxing Method 205
Andras Erik Csallner, Rudi Klatte, Dietmar Ratz, Andreas Wiethoff
A Branch-and-Prune Method for Global Optimization 215
Dimitris G. Sotiropoulos and Theodoula N. Grapsa
Simulation of a Controlled Aircraft Elevator under Sensor Uncertainties 227
Jiirgen Heeks, Eberhard P. Hofer, Bernd Tibken, Karin Lunde, Klaus Thorwart
Part VI Control
Traditional parameter estimation versus estimation of guaranteed parameter sets 241
Eberhard P. Hofer, Bernd Tibken, Milan Vlach
Stabilizing Control Design of Nonlinear Process Involving Uncertainties 255
Mikhail Krastanov, Neli Dimitrova
Set Estimation, Computation of Volumes and Data Safety 267
Isabelle Braems, Michel Kieffer, Eric Walter, Luc Jaulin
Contents ix
Part VII ODE and DAE and Applications
Verified High-Order Integration of DAEs and Higher-order ODEs 281
Jens Hoe.fkens, Martin Berz, Kyoko Makino
About a Finite Dimensional Reduction Method for Conservative Dynamical Sys-
tems and its Applications 293
Anatoliy Prykarpatsky, Stanislaw Brzychczy, V. Samoylenko
Verified Determination of Singularities in Chemical Processes 305
Christian H. Bischof, Bruno Lang, Wolfgang Marquardt, Martin Monnigmann
Modeling of Multi body Systems with Interval Arithmetic 317
Christian Horsken, Bolger Traczinski
Part VIII Stochastics and Probability
On the Algebraic Properties of Stochastic Arithmetic. Comparison to Interval
Arithmetic 331
Rene Alt, Svetoslav Markov
Global Random Walk Simulations of Diffusion 343
Calin Vamos, Nicolae Suciu, Harry Vereecken, Olaf Nitzsche, Horst Hardelauf
Interval Computations as a Particular Case of a General Scheme Involving Classes
of Probability Distributions 355
Scott Ferson, Lev Ginzburg, Vladik Kreinovich, Harry Schulte
For reliable and powerful scientific computations 367
Fabienne Jezequel, Jean-Marie Chesneaux
Reliable representations of strange attractors 379
Dominique Michelucci
Appendix: The Referees 391
Index
393
Links to some freely available interval software
• C-XSC with Toolbox for Verified Computing (current versions):
http://www.math.uni-wuppertal.derxsc/xsc/download.html
• Interval libraries filib and filib++:
http://www.math.uni-wuppertal.de/wrswt/software.html
• GLOBSOL: http://studsys.mscs.mu.edurglobsol/
• PROFIL/BIAS:
http://www.ti3.tu-harburg.derknueppellprofil/index_e.html
• INTLAB: http://www.ti3.tu-harburg.derrump/intlab/
• INTLIB: ftp://interval.louisiana.edu/pub/intervaLmath/intlib
• PASCAL-XSC with Toolbox for Verified Computing:
http://www.math.uni-wuppertal.derxsc/xsc/download.html
• Sun Forte Fortran!HPC and C++ compilers:
CD in this book, see also: http://www.sun.com/forte/index.html
Further links to interval software are available under:
http://www.cs.utep.edu/interval-comp/intsoft.html
About the enclosed CD
The enclosed CD ROM contains a full set of the latest version of Sun's
ForteTM Developer 6 update 2 compilers and productivity tools which supports
Interval Arithmetic in both Fortran 95 and C++. A 30-day trial set of license
tokens can be obtained at no cost and if you decide to purchase the product,
reinstallation of the software is unnecessary.
Forte Developer 6 software is an outstanding solution for software develop
ment on the Solaris™ Operating Environment for both individuals and teams
of software developers. It is a comprehensive, integrated, development envi
ronment that helps you build high-performance, reliable, scalable, open, appli
cations more rapidly and efficiently with GUI- and CLI-based tools. Interval
Arithmetic is supported as a native data type in Fortran 95 and as a class library
in C++.
SCAN 2000 KEYNOTE ADDRESS
THE FUTURE OF INTERVALS
G. William Walster
Sun Microsystems, Inc.
Menlo Park, CA, USA
bill.walster@eng.sun.com
Abstract The 45 year floating-point-interval, (1955, 1999], is briefly reviewed and con
trasted with the first interval-interval, (1958, 1999]. Tasks are identified that will
close the commercial-funding-feedback-loop and thereby accelerate the transi
tion from floating-point to interval computing in the second interval-interval,
[2000, 2050].
Keywords: interval arithmetic, floating-point arithmetic, history, future, commercial support
1. Overview
In spite of undesirable floating-point-number properties, numerical algo
rithms, as well as, technical and scientific computing applications have been
developed during the floating-point-interval (the years [1955, 1999], see Sec
tion 2). Intervals' properties, on the other hand, have stimulated the discovery
of numerical algorithms in the first interval-interval (the years [1958, 1999],
see Section 3 ). Some of these algorithms have been thought to be impossible.
The superior properties of intervals, together with the increasing require
ments of computer users, logically lead to the conclusion that intervals will
become the dominant technical and scientific computing paradigm in the sec
ond interval-interval (the years [2000, 2050] , see Section 4).
Section 5 highlights the differences between the floating-point-interval and
the first interval-interval. A similarity between the floating-point-interval and
the second interval-interval is also described.
The commercial-funding-feedback-loop, which will accelerate the transition
from floating-point to interval-computing in the second interval-interval, is
described in Section 6. The steps needed to close the commercial-funding
feedback-loop using the SCAN 95 and SCAN 2000 wish-lists are also described.
Section 7 contains conclusions.
Scientific Computing, Validated Numerics, Interval Methods, Edited by Kramer and
Wolff von Gudenberg, Kluwer Academic/Plenum Publishers, New York, 2001 1
