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Introductory Computational Physics
Computers are one of the most important tools available to physicists,
whether for calculating and displaying results, simulating experiments, or
solving complex systems of equations.
Introducing students to computational physics, this textbook shows how to
use computers to solve mathematical problems in physics and teaches
students about choosing different numerical approaches. It also introduces
students to many of the programs and packages available. The book relies
solely on free software: the operating system chosen is Linux, which comes
with an excellent C++ compiler, and the graphical interface is the ROOT
package available for free from CERN.
This up-to-date, broad scope textbook is suitable for undergraduates starting
on computational physics courses. It includes exercises and many examples
of programs. Online resources at www.cambridge.org/9780521828627
feature additional reference information, solutions, and updates on new
techniques, software and hardware used in physics.
Andi Klein is a Technical Staff member at Los Alamos National
Laboratory, New Mexico. He gained his Ph.D. from the University of
Basel, Switzerland. He held the position of Professor of Physics at Old
Dominion University, Virginia, from 1990 to 2002, where he taught courses
in computational physics.
Alexander Godunov is Assistant Professor at the Department of Physics,
OldDominionUniversity,Virginia.HegainedhisPh.D.fromMoscowState
University, Russia and has held research positions at Tulane University,
Louisiana, and visiting positions at research centers in France and Russia.
Introductory Computational
Physics
Andi Klein and Alexander Godunov
Los Alamos National Laboratory
and Old Dominion University
cambridge university press
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Cambridge University Press
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Published in the United States of America by Cambridge University Press, New York
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© Cambridge University Press 2006
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Contents
Preface page ix
1 Introduction 1
1.1 The need for computers in science 1
1.2 What is computational physics? 1
1.3 Linux and C++ 2
2 Basics 5
2.1 Basic computer hardware 5
2.2 Software 7
2.3 How does it work? 9
3 Short introduction to Linux 11
3.1 Getting started and logging in 11
3.2 Getting help 12
3.3 The filesystem, or where is everything? 12
3.4 Moving around in your system 13
3.5 Listing your directory 14
3.6 Creating your own files 15
3.7 Doing some work 17
3.8 Good programming 19
3.9 Machine representation and precision 20
3.10 Exercises 23
4 Interpolation 25
4.1 Lagrange interpolation 27
4.2 Neville’s algorithm 29
4.3 Linear interpolation 30
4.4 Polynomial interpolation 31
v
vi Contents
4.5 Cubic spline 33
4.6 Rational function interpolation 34
4.7 Exercises 35
5 Taking derivatives 37
5.1 General discussion of derivatives with computers 37
5.2 Forward difference 38
5.3 Central difference and higher order methods 38
5.4 Higher order derivatives 40
5.5 Exercises 40
6 Numerical integration 41
6.1 Introduction to numerical integration 41
6.2 The simplest integration methods 42
6.3 More advanced integration 44
6.4 Exercises 49
7 Solution of nonlinear equations 51
7.1 Bisection method 51
7.2 Newton’s method 52
7.3 Method of secants 52
7.4 Brute force method 53
7.5 Exercises 53
8 Differential equations 55
8.1 Introduction 55
8.2 A brush up on differential equations 55
8.3 Introduction to the simple and modified Euler methods 57
8.4 The simple Euler method 58
8.5 The modified Euler method 62
8.6 Runge–Kutta method 65
8.7 Adaptive step size Runge–Kutta 70
8.8 The damped oscillator 72
8.9 Exercises 81
9 Matrices 83
9.1 Linear systems of equations 83
9.2 Gaussian elimination 84
9.3 Standard libraries 86
Contents vii
9.4 Eigenvalue problem 86
9.5 Exercises 88
10 Random processes and Monte Carlo simulation 89
10.1 Random processes in science 89
10.2 Random number generators 90
10.3 The random walk 92
10.4 Random numbers for nonuniform distributions 97
10.5 Monte Carlo integration 101
10.6 Exercises 103
References 105
AppendixATheROOTsystem 107
A.1 What is ROOT 107
A.2 The ROOT basics 107
A.3 The first steps 108
A.4 Lab ROOT 113
A.5 Exercises 115
AppendixBFreescientificlibraries 117
B.1 LAPACK 117
B.2 SLATEC 118
B.3 Where to obtain ROOT 118
AppendixCFORTRANandC++ 119
C.1 Calling FORTRAN from C++ 120
AppendixDProgramlistings 121
D.1 Simple Euler 121
D.2 Runge–Kutta program 123
D.3 Random walk in two dimensions 131
D.4 Acceptance and rejection method with sin(x) distribution 134
Index 137
