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Walter Gander Jirf Hrebicek Solving Problems in Scientific Computing Using MAPLE andMATLAB® With 97 Figures and 7 Tables Springer-Verlag Berlin Heidelberg NewY ork London Paris Tokyo Hong Kong Barcelona Budapest Walter Gander Institute of Scientific Computing ETHZUrich CH-8092 ZUrich, Switzerland Jiff Hrebicek Institute of Physics and Electronics University of Agriculture and Forestry Brno Zemedelska 1 613 00 Brno, Czech Republic The cover picture shows a plane fitted by least squares to given points (see Chapter 6) Mathematics Subject Classification (1991): 00A35, 08-04, 65Y99, 68Q40, 68N15 ISBN-13: 978-3-540-57329-6 e-ISBN-13: 978-3-642-97533-2 DOl: 10_1007/978-3-642-97533-2 Cataloging-in-Publication Data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication oft his publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. MATLAB® is a registered trademark of The MathWorks Inc. The trademark is being used with the written permission of The MathW orks Inc. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. © Springer-Verlag Berlin Heidelberg 1993 Typesetting: camera-ready copy by the authors 41/3140 - 543 2 10 - Printed on acid-free paper Preface Modern computing tools like MAPLE (a symbolic computation pack age) and MATLAB (a numeric computation and visualization pro gram) make it possible to use the techniques of scientific computing to solve realistic nontrivial problems in a classroom setting. These problems have been traditionally avoided, since the amount of work required to obtain a solution exceeded the classroom time available and the capabilities of the students. Therefore, simplified and lin earized models are often used. This situation has changed, and students can be taught with real-life problems which can be solved by the powerful software tools available. This book is a collection of interesting problems which illustrate some solution techniques in Scientific Computing. The solution technique for each problem is discussed and demonstrated through the use of either MAPLE or MATLAB. The problems are presented in a way such that a reader can easily extend the techniques to even more difficult problems. This book is intended for students of engineering and scientific computing. It is not an introduction to MAPLEl and MATLAB2. It is assumed that the reader is already familiar with these systems. Instead, it teaches problem solving techniques through the use of examples, which are difficult real-life problems. All figures in the book were created either by using graphic com mands of MAPLE and MATLAB or by direct use of xfig on a SUN workstation. Occasionally changes were made by Dr. S. Bartoii in the postscript files to improve the visual quality of the figures. These changes include different font sizes, line types, line thick nesses, as well as additional comments and labels. This book was written as a collaboration between three insti tutes: • the Department of Theoretical Physics and Astrophysics of Masaryk University, Brno, Czech Republic, IB.W. CHAR ET AL., Maple V Language/Reference Manual, Springer Verlag, New York, 1991 2MATLAB High Performance Numeric Computation and Visualization Software,The MathWorks, Inc., 24 Prime Park Way, Natik, MA 01760, 1992 vi • the Institute of Physics of the University of Agriculture and Forestry, Brno, Czech Republic, and • the Institute of Scientific Computing ETH, Zurich, Switzer land. The authors are indebted to the Swiss National Science Founda tion which stimulated this collaboration through a grant from the "Oststaaten-Soforthilfeprogramm". An additional grant from the ETH "Sonderprogramm Ostkontakte" and support from the Com puter Science Department of ETH Zurich made it possible for Dr. S. Barton to spend a year in Zurich. He was the communication link between the two groups of authors and without him, the book would not have been produced on time. We would also like to thank Dr. 1. Badoux, Austauschdienst ETH, and Prof. C.A. Zehnder, chair man of the Computer Science Department, for their interest and support. Making our Swiss- and Czech-English understandable and cor rect was a major problem in producing this book. This was ac complished through an internal refereeing and proofreading process which greatly improved the quality of all articles. We had great help from Dr. Kevin Gates, Martha Gonnet, Michael Oettli, Prof. S. Leon, Prof. T. Casavant and Prof. B. Gragg during this process. We thank them all for their efforts to improve our language. Dr. U. von Matt wrote the 1I\TEX style file to generate the layout of our book in order to meet the requirements of the publisher. We are all very thankful for his excellent work. D. Gruntz, our MAPLE expert, gave valuable comments to all the authors and greatly improved the quality of the programs. We wish to thank him for his assistance. The programs were written using MAPLE V Release 2 and MAT LAB 4.1. For MAPLE output we used the ASCII interface instead of the nicer XMAPLE environment. This way it was easier to incor porate MAPLE output in the book. The programs are available in machine readable form. We are thankful to MathW orks for helping us to distribute the software. Included in this book is a postcard addressed to The MathWorks, Inc., 24 Prime Park Way, Natik, MA 01760, USA, with which the reader may order all the programs on diskette. Readers connected to Internet can also obtain the pro grams from neptune. inf . ethz . ch using anonymous ftp. Zurich, September 13, 1993 Walter Gander, Jifi HfebiCek List of Authors Stanislav Barton Institute of Physics and Electronics University of Agriculture and Forestry Brno Zemedelska 1 613 00 Brno, Czech Republic [email protected] Jaroslav Buchar Institute of Physics and Electronics University of Agriculture and Forestry Brno Zemedelska 1 613 00 Brno, Czech Republic Ivan Daler Air Traffic Control Research Department Smetanova 19 602 00 Brno, Czech Republic Walter Gander Institute of Scientific Computing ETH Zurich 8092 Zurich, Switzerland [email protected] Dominik Gruntz Institute of Scientific Computing ETH Zurich 8092 Zurich, Switzerland [email protected] Jurgen Halin Institute of Energy Technology ETH Zurich 8092 Zurich, Switzerland [email protected] viii Jift Hrebicek Institute of Physics and Electronics University of Agriculture and Forestry Brno Zemedelska 1 613 00 Brno, Czech Republic hrebicekGarwen.ics.muni.cs Frantisek Klvana Department of Theoretical Physics and Astrophysics Masaryk University Brno, Faculty of Science Kotlarska2 602 00 Brno, Czech Republic klvanaGsci.muni.cz Urs von Matt Institute of Scientific Computing ETH Ziirich 8092 Ziirich, Switzerland na.vonmattGna-net.ornl.gov Jorg Waldvogel Seminar of Applied Mathematics ETH Ziirich 8092 Ziirich, Switzerland waldvogeGmath.ethz.ch Contents Chapter 1. The Tractrix and Similar Curves 1 1.1 Introduction...... 1 1.2 The Classical Tractrix . 1 1.3 The Child and the Toy . 3 1.4 The Jogger and the Dog 6 1.5 Showing the Movements with MATLAB 11 References. . . . . . . . . . . . . . . . . . . 14 Chapter 2. Trajectory of a Spinning Tennis Ball . 15 2.1 Introduction.... 15 2.2 MAPLE Solution 17 2.3 MATLAB Solution . 21 References . . . . . . . . 23 Chapter 3. The lllumination Problem 25 3.1 Introduction............. 25 3.2 Finding the Minimal Illumination Point on a Road 26 3.3 Varying h2 to Maximize the Illumination 29 3.4 Optimal Illumination 31 3.5 Conclusion. 34 References . . . . . . . . . 35 Chapter 4. Orbits in the Planar Three-Body Problem 37 4.1 Introduction................... 37 4.2 Equations of Motion in Physical Coordinates . 38 4.3 Global Regularization. . . . . . . . . . 42 4.4 The Pythagorean Three-Body Problem 48 4.5 Conclusions 55 References. . . . . . . . . . . . . . . . . . . 57 Chapter 5. The Internal Field in Semiconductors 59 5.1 Introduction.................... 59 5.2 Solving a Nonlinear Poisson Equation Using MAPLE 60 5.3 MATLAB Solution . 66 References . . . . . . . . . . . . . . . . . . . . . . . . .. 67 x CONTENTS Chapter 6. Some Least Squares Problems . . . . . .. 69 6.1 Introduction...................... 69 6.2 Fitting Lines, Rectangles and Squares in the Plane 69 6.3 Fitting Hyperplanes. 81 References . . . . . . . . . . . . . . . . . . . . . . 87 Chapter 7. The Generalized Billiard Problem 89 7.1 Introduction............. 89 7.2 The Generalized Reflection Method 89 7.2.1 Line and Curve Reflection 90 7.2.2 Mathematical Description 91 7.2.3 MAPLE Solution ..... 92 7.3 The Shortest Trajectory Method. 93 7.3.1 MAPLE Solution ..... 94 7.4 Examples . . . . . . . . . . . . . 94 7.4.1 The Circular Billiard Table 94 7.4.2 The Elliptical Billiard Table 98 7.4.3 The Snail Billiard Table 100 7.4.4 The Star Billiard Table. 102 7.5 Conclusions 105 References. . . . . . . . . . . 107 Chapter 8. Mirror Curves . . . . . . . . 109 8.1 The Interesting Waste ....... 109 8.2 The Mirror Curves Created by MAPLE 109 8.3 The Inverse Problem . . . . . . . . . . 110 8.3.1 Outflanking Manoeuvre .... 110 8.3.2 Geometrical Construction of a Point on the Pattern Curve . . 112 8.3.3 MAPLE Solution 113 8.3.4 Analytic Solution 114 8.4 Examples ....... . 114 8.4.1 The Circle as the Mirror Curve 114 8.4.2 The Line as the Mirror Curve 117 8.5 Conclusions 118 References . . . . . . . . . . . . . 120 Chapter 9. Smoothing Filters. 121 9.1 Introduction ....... . 121 9.2 Savitzky-Golay Filter . . . 121 9.2.1 Filter Coefficients . 122 9.2.2 Results.... 125 9.3 Least Squares Filter ... 126 CONTENTS ~ 9.3.1 Lagrange Equations. . . . . . . . . . 127 9.3.2 Zero Finder . . . . . . . . . . . . . 129 9.3.3 Evaluation of the Secular Function 130 9.3.4 MEX-Files.. 132 9.3.5 Results. 136 References . . . . . . . . . 138 Chapter 10. The Radar Problem. 141 10.1 Introduction . . . . . . . . . . 141 10.2 Converting Degrees into Radians 142 10.3 Transformation of Geographical into Geocentric Co- ordinates. . . . . . . 143 10.4 The Transformations 146 10.5 Final Algorithm. . 148 10.6 Practical Example 150 References. . . . . . . . 151 Chapter 11. Conformal Mapping of a Circle . 153 11.1 Introduction. . . 153 11.2 Problem Outline 153 11.3 MAPLE Solution 154 References . . . . . . . 159 Chapter 12. The Spinning Top. 161 12.1 Introduction . . . . . . . . . 161 12.2 Formulation and Basic Analysis of the Solution 163 12.3 The Numerical Solution 168 References . . . . . . . . . . . . . . . . . 171 Chapter 13. The Calibration Problem 173 13.1 Introduction. . . . . . . . . . . . . 173 13.2 The Physical Model Description . . 173 13.3 Approximation by Splitting the Solution 176 13.4 Conclusions 182 References . . . . . . . . . . . . . . 182 Chapter 14. Heat Flow Problems 183 14.1 Introduction. . . . . . . . . . 183 14.2 Heat Flow through a Spherical Wall. 183 14.2.1 A Steady State Heat Flow Model 184 14.2.2 Fourier Model for Steady State . 185 14.2.3 MAPLE Plots . . . . . . . . . . . 186 14.3 Non Stationary Heat Flow through an Agriculture Field. . . . ... . . . 187 14.3.1 MAPLE Plots 191 References . . . . . . . . . 191