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/0521828627 feature additional reference information, solutions, and updates on new techniques, software and hardware used in physics.

Preface

ix

Introduction

1

(4)

The need for computers in science

1

(1)

What is computational physics?

1

(1)

Linux and C++

2

(3)

Basics

5

(6)

Basic computer hardware

5

(2)

Software

7

(2)

How does it work?

9

(2)

Short introduction to Linux

11

(14)

Getting started and logging in

11

(1)

Getting help

12

(1)

The filesystem, or where is everything?

12

(1)

Moving around in your system

13

(1)

Listing your directory

14

(1)

Creating your own files

15

(2)

Doing some work

17

(2)

Good programming

19

(1)

Machine representation and precision

20

(3)

Exercises

23

(2)

Interpolation

25

(12)

Lagrange interpolation

27

(2)

Neville's algorithm

29

(1)

Linear interpolation

30

(1)

Polynomial interpolation

31

(2)

Cubic spline

33

(1)

Rational function interpolation

34

(1)

Exercises

35

(2)

Taking derivatives

37

(4)

General discussion of derivatives with computers

37

(1)

Forward difference

38

(1)

Central difference and higher order methods

38

(2)

Higher order derivatives

40

(1)

Exercises

40

(1)

Numerical integration

41

(10)

Introduction to numerical integration

41

(1)

The simplest integration methods

42

(2)

More advanced integration

44

(5)

Exercises

49

(2)

Solution of nonlinear equations

51

(4)

Bisection method

51

(1)

Newton's method

52

(1)

Method of secants

52

(1)

Brute force method

53

(1)

Exercises

53

(2)

Differential equations

55

(28)

Introduction

55

(1)

A brush up on differential equations

55

(2)

Introduction to the simple and modified Euler methods

57

(1)

The simple Euler method

58

(4)

The modified Euler method

62

(3)

Runge-Kutta method

65

(5)

Adaptive step size Runge-Kutta

70

(2)

The damped oscillator

72

(9)

Exercises

81

(2)

Matrices

83

(6)

Linear systems of equations

83

(1)

Gaussian elimination

84

(2)

Standard libraries

86

(1)

Eigenvalue problem

86

(2)

Exercises

88

(1)

Random processes and Monte Carlo simulation

89

(18)

Random processes in science

89

(1)

Random number generators

90

(2)

The random walk

92

(5)

Random numbers for nonuniform distributions

97

(4)

Monte Carlo integration

101

(2)

Exercises

103

(4)

References

105

(2)

Appendix A The ROOT system

107

(10)

What is ROOT

107

(1)

The ROOT basics

107

(1)

The first steps

108

(5)

Lab ROOT

113

(2)

Exercises

115

(2)

Appendix B Free scientific libraries

117

(2)

LAPACK

117

(1)

SLATEC

118

(1)

Where to obtain ROOT

118

(1)

Appendix C FORTRAN and C++

119

(2)

Calling FORTRAN from C++

120

(1)

Appendix D Program listings

121

(16)

Simple Euler

121

(2)

Runge-Kutta program

123

(8)

Random walk in two dimensions

131

(3)

Acceptance and rejection method with sin(x) distribution

134

(3)

Index

137

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Introductory Computational Physics

Summary

Table of Contents

Supplemental Materials