Table Of ContentWalter 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
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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
