A Primer on Scientific Programming with Python

Hans Petter Langtangen

Simula Research Laboratory
Martin Linges vei 17
1325 Lysaker, Fornebu
Norway
hpl@simula.no

Editors

Timothy J. Barth
Michael Griebel
David E. Keyes
Risto M. Nieminen
Dirk Roose
Tamar Schlick

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Book Details
 Price
 2.00 USD
 Pages
 726 p
 File Size
 6,341 KB
 File Type
 PDF format
 ISBN
 eISBN
 978-3-642-02474-0
 978-3-642-02475-7
 Copyright   
 Springer-Verlag Berlin Heidelberg 2009 

Preface
The aim of this book is to teach computer programming using examples
from mathematics and the natural sciences. We have chosen to use the
Python programming language because it combines remarkable power
with very clean, simple, and compact syntax. Python is easy to learn
and very well suited for an introduction to computer programming.
Python is also quite similar to Matlab and a good language for doing
mathematical computing. It is easy to combine Python with compiled
languages, like Fortran, C, and C++, which are widely used languages
for scientific computations. A seamless integration of Python with Java
is offered by a special version of Python called Jython.

The examples in this book integrate programming with applications
to mathematics, physics, biology, and finance. The reader is expected
to have knowledge of basic one-variable calculus as taught in
mathematics-intensive programs in high schools. It is certainly an advantage
to take a university calculus course in parallel, preferably containing
both classical and numerical aspects of calculus. Although not
strictly required, a background in high school physics makes many of
the examples more meaningful.

Many introductory programming books are quite compact and focus
on listing functionality of a programming language. However, learning
to program is learning how to think as a programmer. This book has
its main focus on the thinking process, or equivalently: programming
as a problem solving technique. That is why most of the pages are
devoted to case studies in programming, where we define a problem and
explain how to create the corresponding program. New constructions
and programming styles (what we could call theory) is also usually
introduced via examples. Special attention is paid to verification of
programs and to finding errors. These topics are very demanding for
mathematical software, because we have approximation errors possibly
mixed with programming errors.

By studying the many examples in the book, I hope readers will learn
how to think right and thereby write programs in a quicker and more
reliable way. Remember, nobody can learn programming by just reading
– one has to solve a large amount of exercises hands on. Therefore,
the book is full of exercises of various types: modifications of existing
examples, completely new problems, or debugging of given programs.
To work with this book, you need to install Python version 2.6. In
Chapter 4 and later chapters, you also need the NumPy and SciTools
packages. To make curve plots with SciTools, you must have a plotting
program, for example, Gnuplot or Matplotlib. There is a web page
associated with this book, http://www.simula.no/intro-programming,
which lists the software you need and explains briefly how to install
it. On this page, you will also find all the files associated with the
program examples in this book. Download book-examples.tar.gz, store
this file in some folder of your choice, and unpack it using WinZip
on Windows or the command tar xzf book-examples.tar.gz on Linux
and Mac. This unpacking yields a folder src with subfolders for the
various chapters in the book.

Contents. Chapter 1 introduces variables, objects, modules, and text
formatting through examples concerning evaluation of mathematical
formulas. Chapter 2 presents fundamental elements of programming:
loops, lists, and functions. This is a comprehensive and important chapter
that should be digested before proceeding. How to read data into
programs and deal with errors in input are the subjects of Chapter 3.
Many of the examples in the first three chapters are strongly related.
Typically, formulas from the first chapter are encapsulated in functions
in the second chapter, and in the third chapter the input to the functions
are fetched from the command line or from a question-answer
dialog with the user, and validity of the data is checked. Chapter 4
introduces arrays and array computing (including vectorization) and
how this is used for plotting y = f(x) curves. After the first four chapters,
the reader should have enough knowledge of programming to solve
mathematical problems by “Matlab-style” programming.
Chapter 5 introduces mathematical modeling, using sequences and
difference equations. We also treat sound as a sequence. No new programming
concepts are introduced in this chapter, the aim being to
consolidate the programming knowledge and apply it to mathematical problems.
Chapter 6 explains how to work with files and text data. Class
programming, including user-defined types for mathematical computations
(with overloaded operators), is introduced in Chapter 7. Chapter
8 deals with random numbers and statistical computing with applications
to games and random walk. Object-oriented programming
(class hierarchies and inheritance) is the subject of Chapter 9. The
key examples here deal with building toolkits for graphics and for numerical
differentiation, integration, and solution of ordinary differential equations.
Appendix A deals with functions on a mesh, numerical differentiation,
and numerical integration. The next appendix gives an introduction
to numerical solution of ordinary differential equations. These two
appendices provide the theoretical background on numerical methods
that are much in use in Chapters 7 and 9. Moreover, the appendices
exemplify basic programming from the first four chapters.
Appendix C shows how a complete project in physics can be solved
by mathematical modeling, numerical methods, and programming elements
from the first four chapters. This project is a good example
on problem solving in computational science, where it is necessary to
integrate physics, mathematics, numerics, and computer science.
Appendix D is devoted to the art of debugging, and in fact problem
solving in general, while Appendix E deals with various more advanced technical topics.

Most of the examples and exercises in this book are quite compact
and limited. However, many of the exercises are related, and together
they form larger projects in science, for example on Fourier Series (1.13,
2.39, 3.17, 3.18, 3.19, 4.20), Taylor series (2.38, 4.16, 4.18, 5.16, 5.17,
7.31), falling objects (7.25, 9.30, 7.26, 9.31, 9.32, 9.34), oscillatory population
growth (5.21, 5.22, 6.28, 7.41, 7.42), visualization of web data
(6.22, 6.23, 6.24, 6.25), graphics and animation (9.36, 9.37, 9.38, 9.39),
optimization and finance (5.23, 8.42, 8.43), statistics and probability
(3.25, 3.26, 3.27, 8.19, 8.20, 8.21), random walk and statistical physics
(8.33, 8.34, 8.35, 8.36, 8.37, 8.38, 8.39, 8.40), noisy data analysis (8.44,
8.45, 8.46, 8.47, 9.40), numerical methods (5.12, 7.9, 7.22, 9.15, 9.16,
9.26, 9.27, 9.28), building a calculus calculator (9.41, 9.42, 9.43, 9.44),
and creating a toolkit for simulating oscillatory systems (9.45–9.52).
Chapters 1–9 and Appendix C form the core of an introductory firstsemester
course on scientific programming (INF1100) at the University
of Oslo. Normally, each chapter is suited for a 2 × 45 min lecture, but
Chapters 2 and 7 are challenging, and each of them have consumed
two lectures in the course.

Acknowledgments. First, I want to express my thanks to Aslak Tveito
for his enthusiastic role in the initiation of this book project and for
writing Appendices A and B about numerical methods. Without Aslak
there would be no book. Another key contributor is Ilmar Wilbers. His
extensive efforts with assisting the book project and teaching the associated
course (INF1100) at the University of Oslo are greatly appreciated.
Without Ilmar and his solutions to numerous technical problems
the book would never have been completed. Johannes H. Ring also
deserves a special acknowledgment for the development of the Easyviz
graphics tool, which is much used throughout this book, and for his
careful maintenance and support of software associated with this book.
Several people have helped to make substantial improvements of
the text, the exercises, and the associated software infrastructure. The
author is thankful to Ingrid Eide, Tobias Vidarssønn Langhoff, Mathias
Nedrebø, Arve Knudsen, Marit Sandstad, Lars Storjord, Fredrik Heffer
Valdmanis, and Torkil Vederhus for their contributions. Hakon Adler
is greatly acknowledged for his careful reading of various versions of
the manuscript. The professors Fred Espen Bent, Ørnulf Borgan, Geir
Dahl, Knut Mørken, and Geir Pedersen have contributed with many
exciting exercises from various application fields. Great thanks also go
to Jan Olav Langseth for creating the cover image.

This book and the associated course are parts of a comprehensive
reform at the University of Oslo, called Computers in Science Edu-
cation. The goal of the reform is to integrate computer programming
and simulation in all bachelor courses in natural science where mathematical
models are used. The present book lays the foundation for the
modern computerized problem solving technique to be applied in later
courses. It has been extremely inspiring to work with the driving forces
behind this reform, in particular the professors Morten Hjorth–Jensen,
Anders Malthe–Sørenssen, Knut Mørken, and Arnt Inge Vistnes.
The excellent assistance from the Springer and Le-TeX teams, consisting
of Martin Peters, Thanh-Ha Le Thi, Ruth Allewelt, Peggy
Glauch-Ruge, Nadja Kroke, and Thomas Schmidt, is highly appreciated
and ensured a smooth and rapid production of this book.
Oslo, May 2009 
Hans Petter Langtangen

Table of Contents
1 Computing with Formulas . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The First Programming Encounter: A Formula . . . . . . . 1
1.1.1 Using a Program as a Calculator . . . . . . . . . . . . . 2
1.1.2 About Programs and Programming . . . . . . . . . . . 2
1.1.3 Tools for Writing Programs . . . . . . . . . . . . . . . . . . 3
1.1.4 Using Idle to Write the Program. . . . . . . . . . . . . . 4
1.1.5 How to Run the Program . . . . . . . . . . . . . . . . . . . . 7
1.1.6 Verifying the Result . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.7 Using Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.8 Names of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.9 Reserved Words in Python . . . . . . . . . . . . . . . . . . . 10
1.1.10 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.11 Formatting Text and Numbers . . . . . . . . . . . . . . . 11
1.2 Computer Science Glossary . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Another Formula: Celsius-Fahrenheit Conversion . . . . . . 18
1.3.1 Potential Error: Integer Division . . . . . . . . . . . . . . 19
1.3.2 Objects in Python . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3.3 Avoiding Integer Division . . . . . . . . . . . . . . . . . . . . 21
1.3.4 Arithmetic Operators and Precedence . . . . . . . . . 21
1.4 Evaluating Standard Mathematical Functions . . . . . . . . . 22
1.4.1 Example: Using the Square Root Function . . . . . 22
1.4.2 Example: Using More Mathematical Functions . 25
1.4.3 A First Glimpse of Round-Off Errors . . . . . . . . . . 25
1.5 Interactive Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.5.1 Calculating with Formulas in the Interactive Shell . . . .. . . . . 27
1.5.2 Type Conversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.5.3 IPython . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.6 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.6.1 Complex Arithmetics in Python . . . . . . . . . . . . . . 32
1.6.2 Complex Functions in Python . . . . . . . . . . . . . . . . 32
1.6.3 Unified Treatment of Complex and Real Functions . . . . . . . . . . . . 33
1.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.7.1 Chapter Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.7.2 Summarizing Example: Trajectory of a Ball . . . . 38
1.7.3 About Typesetting Conventions in This Book . . 39
1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2 Basic Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.1 Loops and Lists for Tabular Data . . . . . . . . . . . . . . . . . . . 51
2.1.1 A Naive Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.1.2 While Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.1.3 Boolean Expressions . . . . . . . . . . . . . . . . . . . . . . . . 54
2.1.4 Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.1.5 For Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.1.6 Alternative Implementations with Lists and Loops . . . . . . . . . . . . . . . 60
2.1.7 Nested Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.1.8 Printing Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.1.9 Extracting Sublists . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.1.10 Traversing Nested Lists . . . . . . . . . . . . . . . . . . . . . . 68
2.1.11 Tuples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.2.1 Functions of One Variable . . . . . . . . . . . . . . . . . . . 71
2.2.2 Local and Global Variables . . . . . . . . . . . . . . . . . . . 73
2.2.3 Multiple Arguments . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.2.4 Multiple Return Values . . . . . . . . . . . . . . . . . . . . . . 77
2.2.5 Functions with No Return Values . . . . . . . . . . . . . 79
2.2.6 Keyword Arguments . . . . . . . . . . . . . . . . . . . . . . . . 80
2.2.7 Doc Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.2.8 Function Input and Output . . . . . . . . . . . . . . . . . . 84
2.2.9 Functions as Arguments to Functions . . . . . . . . . 84
2.2.10 The Main Program . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.2.11 Lambda Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.3 If Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2.4.1 Chapter Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2.4.2 Summarizing Example: Tabulate a Function. . . . 94
2.4.3 How to Find More Python Information . . . . . . . . 98
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3 Input Data and Error Handling . . . . . . . . . . . . . . . . . . . 119
3.1 Asking Questions and Reading Answers . . . . . . . . . . . . . . 120
3.1.1 Reading Keyboard Input . . . . . . . . . . . . . . . . . . . . 120
3.1.2 The Magic “eval” Function . . . . . . . . . . . . . . . . . . . 121
3.1.3 The Magic “exec” Function . . . . . . . . . . . . . . . . . . . 125
3.1.4 Turning String Expressions into Functions . . . . . 126
3.2 Reading from the Command Line . . . . . . . . . . . . . . . . . . . 127
3.2.1 Providing Input on the Command Line . . . . . . . . 127
3.2.2 A Variable Number of Command-Line Arguments .. . . . 128
3.2.3 More on Command-Line Arguments . . . . . . . . . . . 129
3.2.4 Option–Value Pairs on the Command Line . . . . . 130
3.3 Handling Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.3.1 Exception Handling . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.3.2 Raising Exceptions . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.4 A Glimpse of Graphical User Interfaces . . . . . . . . . . . . . . 139
3.5 Making Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.5.1 Example: Compund Interest Formulas . . . . . . . . . 142
3.5.2 Collecting Functions in a Module File . . . . . . . . . 143
3.5.3 Using Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.6.1 Chapter Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.6.2 Summarizing Example: Bisection Root Finding . 152
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
4 Array Computing and Curve Plotting . . . . . . . . . . . . 169
4.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
4.1.1 The Vector Concept . . . . . . . . . . . . . . . . . . . . . . . . . 170
4.1.2 Mathematical Operations on Vectors . . . . . . . . . . 171
4.1.3 Vector Arithmetics and Vector Functions . . . . . . 173
4.2 Arrays in Python Programs . . . . . . . . . . . . . . . . . . . . . . . . 175
4.2.1 Using Lists for Collecting Function Data . . . . . . . 175
4.2.2 Basics of Numerical Python Arrays . . . . . . . . . . . 176
4.2.3 Computing Coordinates and Function Values . . . 177
4.2.4 Vectorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
4.3 Curve Plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
4.3.1 The SciTools and Easyviz Packages . . . . . . . . . . . 180
4.3.2 Plotting a Single Curve . . . . . . . . . . . . . . . . . . . . . . 181
4.3.3 Decorating the Plot . . . . . . . . . . . . . . . . . . . . . . . . . 183
4.3.4 Plotting Multiple Curves . . . . . . . . . . . . . . . . . . . . 183
4.3.5 Controlling Line Styles . . . . . . . . . . . . . . . . . . . . . . 185
4.3.6 Interactive Plotting Sessions . . . . . . . . . . . . . . . . . 189
4.3.7 Making Animations . . . . . . . . . . . . . . . . . . . . . . . . . 190
4.3.8 Advanced Easyviz Topics . . . . . . . . . . . . . . . . . . . . 193
4.3.9 Curves in Pure Text . . . . . . . . . . . . . . . . . . . . . . . . 198
4.4 Plotting Difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
4.4.1 Piecewisely Defined Functions . . . . . . . . . . . . . . . . 199
4.4.2 Rapidly Varying Functions . . . . . . . . . . . . . . . . . . . 205
4.4.3 Vectorizing StringFunction Objects . . . . . . . . . . . 206
4.5 More on Numerical Python Arrays . . . . . . . . . . . . . . . . . . 207
4.5.1 Copying Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
4.5.2 In-Place Arithmetics . . . . . . . . . . . . . . . . . . . . . . . . 207
4.5.3 Allocating Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
4.5.4 Generalized Indexing . . . . . . . . . . . . . . . . . . . . . . . . 209
4.5.5 Testing for the Array Type . . . . . . . . . . . . . . . . . . 210
4.5.6 Equally Spaced Numbers . . . . . . . . . . . . . . . . . . . . 211
4.5.7 Compact Syntax for Array Generation. . . . . . . . . 212
4.5.8 Shape Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . 212
4.6 Higher-Dimensional Arrays . . . . . . . . . . . . . . . . . . . . . . . . . 213
4.6.1 Matrices and Arrays . . . . . . . . . . . . . . . . . . . . . . . . 213
4.6.2 Two-Dimensional Numerical Python Arrays . . . . 214
4.6.3 Array Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
4.6.4 Two-Dimensional Arrays and Functions of Two Variables . .. . . . . . 217
4.6.5 Matrix Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
4.7.1 Chapter Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
4.7.2 Summarizing Example: Animating a Function . . 220
4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
5 Sequences and Difference Equations . . . . . . . . . . . . . . 235
5.1 Mathematical Models Based on Difference Equations . . 236
5.1.1 Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
5.1.2 The Factorial as a Difference Equation . . . . . . . . 239
5.1.3 Fibonacci Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 240
5.1.4 Growth of a Population. . . . . . . . . . . . . . . . . . . . . . 241
5.1.5 Logistic Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
5.1.6 Payback of a Loan . . . . . . . . . . . . . . . . . . . . . . . . . . 244
5.1.7 Taylor Series as a Difference Equation . . . . . . . . . 245
5.1.8 Making a Living from a Fortune . . . . . . . . . . . . . . 246
5.1.9 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
5.1.10 The Inverse of a Function . . . . . . . . . . . . . . . . . . . . 251
5.2 Programming with Sound . . . . . . . . . . . . . . . . . . . . . . . . . . 253
5.2.1 Writing Sound to File . . . . . . . . . . . . . . . . . . . . . . . 253
5.2.2 Reading Sound from File . . . . . . . . . . . . . . . . . . . . 254
5.2.3 Playing Many Notes . . . . . . . . . . . . . . . . . . . . . . . . 255
5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
5.3.1 Chapter Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
5.3.2 Summarizing Example: Music of a Sequence . . . . 257
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
6 Files, Strings, and Dictionaries . . . . . . . . . . . . . . . . . . . . 269
6.1 Reading Data from File . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
6.1.1 Reading a File Line by Line . . . . . . . . . . . . . . . . . . 270
6.1.2 Reading a Mixture of Text and Numbers . . . . . . 273
6.1.3 What Is a File, Really? . . . . . . . . . . . . . . . . . . . . . . 274
6.2 Dictionaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
6.2.1 Making Dictionaries . . . . . . . . . . . . . . . . . . . . . . . . . 278
6.2.2 Dictionary Operations . . . . . . . . . . . . . . . . . . . . . . . 279
6.2.3 Example: Polynomials as Dictionaries . . . . . . . . . 280
6.2.4 Example: File Data in Dictionaries . . . . . . . . . . . . 282
6.2.5 Example: File Data in Nested Dictionaries . . . . . 283
6.2.6 Example: Comparing Stock Prices . . . . . . . . . . . . 287
6.3 Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
6.3.1 Common Operations on Strings . . . . . . . . . . . . . . . 292
6.3.2 Example: Reading Pairs of Numbers . . . . . . . . . . 295
6.3.3 Example: Reading Coordinates . . . . . . . . . . . . . . . 298
6.4 Reading Data from Web Pages . . . . . . . . . . . . . . . . . . . . . . 300
6.4.1 About Web Pages . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
6.4.2 How to Access Web Pages in Programs . . . . . . . . 302
6.4.3 Example: Reading Pure Text Files . . . . . . . . . . . . 302
6.4.4 Example: Extracting Data from an HTML Page 304
6.5 Writing Data to File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
6.5.1 Example: Writing a Table to File . . . . . . . . . . . . . 309
6.5.2 Standard Input and Output as File Objects . . . . 310
6.5.3 Reading and Writing Spreadsheet Files . . . . . . . . 312
6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
6.6.1 Chapter Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
6.6.2 Summarizing Example: A File Database . . . . . . . 319
6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
7 Introduction to Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
7.1 Simple Function Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
7.1.1 Problem: Functions with Parameters . . . . . . . . . . 338
7.1.2 Representing a Function as a Class . . . . . . . . . . . . 340
7.1.3 Another Function Class Example . . . . . . . . . . . . . 346
7.1.4 Alternative Function Class Implementations . . . . 347
7.1.5 Making Classes Without the Class Construct . . . 349
7.2 More Examples on Classes . . . . . . . . . . . . . . . . . . . . . . . . . 352
7.2.1 Bank Accounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
7.2.2 Phone Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
7.2.3 A Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
7.3 Special Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
7.3.1 The Call Special Method . . . . . . . . . . . . . . . . . . . . 357
7.3.2 Example: Automagic Differentiation . . . . . . . . . . . 357
7.3.3 Example: Automagic Integration . . . . . . . . . . . . . . 360
7.3.4 Turning an Instance into a String . . . . . . . . . . . . . 362
7.3.5 Example: Phone Book with Special Methods . . . 363
7.3.6 Adding Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
7.3.7 Example: Class for Polynomials . . . . . . . . . . . . . . . 365
7.3.8 Arithmetic Operations and Other Special Methods . . .  . . . . 369
7.3.9 More on Special Methods for String Conversion. 370
7.4 Example: Solution of Differential Equations . . . . . . . . . . 372
7.4.1 A Function for Solving ODEs . . . . . . . . . . . . . . . . 373
7.4.2 A Class for Solving ODEs . . . . . . . . . . . . . . . . . . . . 374
7.4.3 Verifying the Implementation. . . . . . . . . . . . . . . . . 376
7.4.4 Example: Logistic Growth . . . . . . . . . . . . . . . . . . . 377
7.5 Example: Class for Vectors in the Plane . . . . . . . . . . . . . . 378
7.5.1 Some Mathematical Operations on Vectors . . . . . 378
7.5.2 Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
7.5.3 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
7.6 Example: Class for Complex Numbers . . . . . . . . . . . . . . . 382
7.6.1 Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
7.6.2 Illegal Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
7.6.3 Mixing Complex and Real Numbers . . . . . . . . . . . 384
7.6.4 Special Methods for “Right” Operands . . . . . . . . . 387
7.6.5 Inspecting Instances . . . . . . . . . . . . . . . . . . . . . . . . . 388
7.7 Static Methods and Attributes . . . . . . . . . . . . . . . . . . . . . . 389
7.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
7.8.1 Chapter Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
7.8.2 Summarizing Example: Interval Arithmetics . . . . 392
7.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
8 Random Numbers and Simple Games . . . . . . . . . . . . 417
8.1 Drawing Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 418
8.1.1 The Seed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
8.1.2 Uniformly Distributed Random Numbers . . . . . . 419
8.1.3 Visualizing the Distribution . . . . . . . . . . . . . . . . . . 420
8.1.4 Vectorized Drawing of Random Numbers . . . . . . 421
8.1.5 Computing the Mean and Standard Deviation . . 422
8.1.6 The Gaussian or Normal Distribution . . . . . . . . . 423
8.2 Drawing Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
8.2.1 Random Integer Functions . . . . . . . . . . . . . . . . . . . 425
8.2.2 Example: Throwing a Die . . . . . . . . . . . . . . . . . . . . 426
8.2.3 Drawing a Random Element from a List . . . . . . . 427
8.2.4 Example: Drawing Cards from a Deck . . . . . . . . . 427
8.2.5 Example: Class Implementation of a Deck . . . . . 429
8.3 Computing Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
8.3.1 Principles of Monte Carlo Simulation . . . . . . . . . . 432
8.3.2 Example: Throwing Dice . . . . . . . . . . . . . . . . . . . . . 433
8.3.3 Example: Drawing Balls from a Hat . . . . . . . . . . . 435
8.3.4 Example: Policies for Limiting Population Growth . .  . . . 437
8.4 Simple Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
8.4.1 Guessing a Number . . . . . . . . . . . . . . . . . . . . . . . . . 440
8.4.2 Rolling Two Dice . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
8.5 Monte Carlo Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
8.5.1 Standard Monte Carlo Integration . . . . . . . . . . . . 443
8.5.2 Computing Areas by Throwing Random Points . 446
8.6 Random Walk in One Space Dimension . . . . . . . . . . . . . . 447
8.6.1 Basic Implementation . . . . . . . . . . . . . . . . . . . . . . . 448
8.6.2 Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
8.6.3 Random Walk as a Difference Equation . . . . . . . . 449
8.6.4 Computing Statistics of the Particle Positions . . 450
8.6.5 Vectorized Implementation . . . . . . . . . . . . . . . . . . . 451
8.7 Random Walk in Two Space Dimensions . . . . . . . . . . . . . 453
8.7.1 Basic Implementation . . . . . . . . . . . . . . . . . . . . . . . 453
8.7.2 Vectorized Implementation . . . . . . . . . . . . . . . . . . . 455
8.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
8.8.1 Chapter Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
8.8.2 Summarizing Example: Random Growth . . . . . . . 457
8.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
9 Object-Oriented Programming . . . . . . . . . . . . . . . . . . . . 479
9.1 Inheritance and Class Hierarchies . . . . . . . . . . . . . . . . . . . 479
9.1.1 A Class for Straight Lines . . . . . . . . . . . . . . . . . . . . 480
9.1.2 A First Try on a Class for Parabolas . . . . . . . . . . 481
9.1.3 A Class for Parabolas Using Inheritance . . . . . . . 481
9.1.4 Checking the Class Type . . . . . . . . . . . . . . . . . . . . 483
9.1.5 Attribute versus Inheritance . . . . . . . . . . . . . . . . . . 484
9.1.6 Extending versus Restricting Functionality . . . . . 485
9.1.7 Superclass for Defining an Interface . . . . . . . . . . . 486
9.2 Class Hierarchy for Numerical Differentiation . . . . . . . . . 488
9.2.1 Classes for Differentiation . . . . . . . . . . . . . . . . . . . . 488
9.2.2 A Flexible Main Program . . . . . . . . . . . . . . . . . . . . 491
9.2.3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
9.2.4 Alternative Implementation via Functions . . . . . . 495
9.2.5 Alternative Implementation via Functional Programming . .. . . . 496
9.2.6 Alternative Implementation via a Single Class . . 497
9.3 Class Hierarchy for Numerical Integration . . . . . . . . . . . . 499
9.3.1 Numerical Integration Methods . . . . . . . . . . . . . . . 499
9.3.2 Classes for Integration . . . . . . . . . . . . . . . . . . . . . . . 501
9.3.3 Using the Class Hierarchy. . . . . . . . . . . . . . . . . . . . 504
9.3.4 About Object-Oriented Programming . . . . . . . . . 507
9.4 Class Hierarchy for Numerical Methods for ODEs . . . . . 508
9.4.1 Mathematical Problem . . . . . . . . . . . . . . . . . . . . . . 508
9.4.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . 510
9.4.3 The ODE Solver Class Hierarchy . . . . . . . . . . . . . 511
9.4.4 The Backward Euler Method . . . . . . . . . . . . . . . . . 515
9.4.5 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518
9.4.6 Application 1: u′ = u . . . . . . . . . . . . . . . . . . . . . . . . 518
9.4.7 Application 2: The Logistic Equation . . . . . . . . . . 519
9.4.8 Application 3: An Oscillating System. . . . . . . . . . 521
9.4.9 Application 4: The Trajectory of a Ball . . . . . . . . 523
9.5 Class Hierarchy for Geometric Shapes . . . . . . . . . . . . . . . 525
9.5.1 Using the Class Hierarchy. . . . . . . . . . . . . . . . . . . . 526
9.5.2 Overall Design of the Class Hierarchy . . . . . . . . . 527
9.5.3 The Drawing Tool . . . . . . . . . . . . . . . . . . . . . . . . . . 529
9.5.4 Implementation of Shape Classes . . . . . . . . . . . . . 530
9.5.5 Scaling, Translating, and Rotating a Figure . . . . 534
9.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
9.6.1 Chapter Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
9.6.2 Summarizing Example: Input Data Reader . . . . . 540
9.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
A Discrete Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
A.1 Discrete Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
A.1.1 The Sine Function . . . . . . . . . . . . . . . . . . . . . . . . . . 574
A.1.2 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576
A.1.3 Evaluating the Approximation . . . . . . . . . . . . . . . . 576
A.1.4 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577
A.2 Differentiation Becomes Finite Differences . . . . . . . . . . . . 579
A.2.1 Differentiating the Sine Function. . . . . . . . . . . . . . 580
A.2.2 Differences on a Mesh . . . . . . . . . . . . . . . . . . . . . . . 580
A.2.3 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582
A.3 Integration Becomes Summation . . . . . . . . . . . . . . . . . . . . 583
A.3.1 Dividing into Subintervals . . . . . . . . . . . . . . . . . . . 584
A.3.2 Integration on Subintervals . . . . . . . . . . . . . . . . . . 585
A.3.3 Adding the Subintervals . . . . . . . . . . . . . . . . . . . . . 586
A.3.4 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
A.4 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
A.4.1 Approximating Functions Close to One Point . . . 589
A.4.2 Approximating the Exponential Function . . . . . . 589
A.4.3 More Accurate Expansions . . . . . . . . . . . . . . . . . . . 590
A.4.4 Accuracy of the Approximation . . . . . . . . . . . . . . . 592
A.4.5 Derivatives Revisited . . . . . . . . . . . . . . . . . . . . . . . . 594
A.4.6 More Accurate Difference Approximations . . . . . 595
A.4.7 Second-Order Derivatives . . . . . . . . . . . . . . . . . . . . 597
A.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
B Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
B.1 The Simplest Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606
B.2 Exponential Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608
B.3 Logistic Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612
B.4 A General Ordinary Differential Equation . . . . . . . . . . . . 614
B.5 A Simple Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
B.6 A Model for the Spread of a Disease . . . . . . . . . . . . . . . . . 619
B.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621
C A Complete Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625
C.1 About the Problem: Motion and Forces in Physics . . . . . 626
C.1.1 The Physical Problem . . . . . . . . . . . . . . . . . . . . . . . 626
C.1.2 The Computational Algorithm . . . . . . . . . . . . . . . 628
C.1.3 Derivation of the Mathematical Model . . . . . . . . . 628
C.1.4 Derivation of the Algorithm . . . . . . . . . . . . . . . . . . 631
C.2 Program Development and Testing . . . . . . . . . . . . . . . . . . 632
C.2.1 Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
C.2.2 Callback Functionality. . . . . . . . . . . . . . . . . . . . . . . 635
C.2.3 Making a Module . . . . . . . . . . . . . . . . . . . . . . . . . . . 636
C.2.4 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637
C.3 Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639
C.3.1 Simultaneous Computation and Plotting . . . . . . . 639
C.3.2 Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 642
C.3.3 Remark on Choosing t . . . . . . . . . . . . . . . . . . . . . 643
C.3.4 Comparing Several Quantities in Subplots . . . . . 644
C.3.5 Comparing Approximate and Exact Solutions . . 645
C.3.6 Evolution of the Error as t Decreases . . . . . . . . 646
C.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649
D Debugging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651
D.1 Using a Debugger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651
D.2 How to Debug . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653
D.2.1 A Recipe for Program Writing and Debugging . . 654
D.2.2 Application of the Recipe . . . . . . . . . . . . . . . . . . . . 656
E Technical Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669
E.1 Different Ways of Running Python Programs . . . . . . . . . 669
E.1.1 Executing Python Programs in IPython . . . . . . . 669
E.1.2 Executing Python Programs on Unix . . . . . . . . . . 669
E.1.3 Executing Python Programs on Windows . . . . . . 671
E.1.4 Executing Python Programs on Macintosh . . . . . 673
E.1.5 Making a Complete Stand-Alone Executable . . . 673
E.2 Integer and Float Division . . . . . . . . . . . . . . . . . . . . . . . . . . 673
E.3 Visualizing a Program with Lumpy . . . . . . . . . . . . . . . . . . 674
E.4 Doing Operating System Tasks in Python . . . . . . . . . . . . 675
E.5 Variable Number of Function Arguments . . . . . . . . . . . . . 678
E.5.1 Variable Number of Positional Arguments . . . . . 679
E.5.2 Variable Number of Keyword Arguments . . . . . . 681
E.6 Evaluating Program Efficiency . . . . . . . . . . . . . . . . . . . . . . 683
E.6.1 Making Time Measurements . . . . . . . . . . . . . . . . . 683
E.6.2 Profiling Python Programs . . . . . . . . . . . . . . . . . . . 685
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689

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