Scientific Computing, Validated Numerics, Interval Methods PDF
Preview Scientific Computing, Validated Numerics, Interval Methods
SCIENTIFIC 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 http://www.wkap.nl/ W 9 8 7 6 5 4 3 2 1 A C.I.P. record for this book is avai1able from the Library of Congress AH rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any fonn or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permis sion from the Publisher 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 [email protected] 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
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