Parti GENERALITIES
1 Introduction 3
1.1 The Nature ofComputational Science 3
1.1.1 How Computational Scientists Do It 4
1.2 Aims of"This Book 5
1.3 Using this Book with the Disk and Web 6
2 Computing Software Basics 9
2.1 Problem 1: Making Computers Obey 9
2.2 Theory: Computer Languages 9
2.3 Implementation: Programming Concepts 11
2.4 Implementation: Fortran,area.f 12
2.5 Implementation: C,area.c 13
2.6 Implementation: Shells,Editors,and Programs 13
2.7 Theory: Program Design 14
2.8 Method: Structured Programming 16
2.9 Method: Programming Hints 18
2.10 Problem 2: Limited Range of Numbers 20
2.11 Theory: Number Representation 20
2.12 Method: Fixed and Floating 21
2.13 Implementation: Over/Underflows,over.f(.c) 23
2.14 Model: Machine Precision 23
2.15 Implementation: limit.f(.c) 24
2.16 Problem 3: Complex Numbers, Inverse Functions 25
2.17 Theory: Complex Numbers 25
2.18 Implementation: complex.c(.0 27
2.19 Exploration: Quantum Complex Energies 28
2.20 Problem 4: Summing Series 28
2.21 Method: Numeric 29
2.22 Implementation: Pseudocode 29
2.23 Implementation: Good Algorithm,exp-good.f(.c) 29
2.24 Implementation: Bad Algorithm,exp-bad.f(.c) 30
2.25 Assessment 30
$ Errors and Uncertainties in Computations 31
3.1 Problem: Living with Errors 31
3.2 Theory: Types of errors 32
3.3 Model: Subtractive Cancellation 33
3.4 Assessment: Cancellation Experiment 34
3.5 Model: Multiplicative Errors 36
3.6 Problem 1: Errors in Spherical Bessel Functions 37
3.7 Method: Numeric Recursion Relations 38
3.8 Implementation: Bessel.f(.c) 40
3.9 Assessment 40
3.10 Problem 2: Errors in Algorithms 40
3.11 Model: Errors in Algorithms 41
3.11.1 Total Error 41
3.12 Method: Optimizing with Known Error Behavior 42
3.13 Method: Empirical Error Analysis 43
3.14 Assessment: Experiment 44
Integration 47
4.1 Problem: Integrating a Spectrum 47CONTENTS fx
4.2 Model: Quadrature, Summing Boxes 47
4.3 Method: Trapezoid Rule 50
4.4 Method: Simpson's Rule 51
4.5 Assessment: Integration Error, Analytic 52
4.6 Method: Gaussian Quadrature 55
4.6.1 Scaling Integration Points 56
4.7 Implementation: integ.f(.c) 57
4.8 Assessment: Empirical Error Estimate 58
4.9 Assessment: Experimentation 59
4.10 Method: Romberg Extrapolation 59
4.10.1 Other Closed Newton-Cotes Formulas 60
Part II APPLICATIONS
5 Data Fitting 63
5.1 Problem: Fitting an Experimental Spectrum 63
5.2 Theory: Curve Fitting 64
5.3 Method: Lagrange Interpolation 65
5.3.1 Example 66
5.4 Implementation: LaGrange.f(.c) 66
5.5 Assessment: Interpolating a Resonant Spectrum 67
5.6 Assessment: Exploration 68
5.7 Method: Cubic Splines 68
5.7.1 Cubic Spline Boundary Conditions 69
5.7.2 Exploration: Cubic Spline Quadrature 70
5.8 Implementation: spline.f 71
5.9 Assessment: Spline Fit ofCross Section 71
5.10 Problem: Fitting Exponential Decay 71
5.11 Model: Exponential Decay 71
5.12 Theory: Probability Theory 73
5.13 Method: Least-Squares Fitting 74
5.14 Theory: Goodness of Fit 76
5.15 Implementation: Least-Squares Fits, fit.f(.c) 77
5.16 Assessment: Fitting Exponential Decay 78
5.17 Assessment: Fitting Heat Flow 79
5.18 Implementation: Linear Quadratic Fits 80
5.19 Assessment: Quadratic Fit 81
5.20 Method: Nonlinear Least-Squares Fitting 81* CONTENTS
5.21 Assessment: Nonlinear Fitting 82
6 Deterministic Randomness 83
6.1 Problem: Deterministic Randomness 83
6.2 Theory: Random Sequences 83
6.3 Method: Pseudo-Random-Number Generators 84
6.4 Assessment: Random Sequences 86
6.5 Implementation: Simple and Not
random.f(.c); call.f(.c) 87
6.6 Assessment: Randomness and Uniformity 87
6.7 Assessment: Tests of Randomness, Uniformity 88
6.8 Problem: A Random Walk 89
6.9 Model: Random Walk Simulation 89
6.10 Method: Numerical Random Walk 90
6.11 Implementation: walk.f(.c) 91
6.12 Assessment: Different Random Walkers 91
7 Monte Carlo Applications 93
7.1 Problem: Radioactive Decay 93
7.2 Theory: Spontaneous Decay 93
7.3 Model: Discrete Decay 94
7.4 Model: Continuous Decay 95
7.5 Method: Decay Simulation 95
7.6 Implementation: decay.f(.c) 97
7.7 Assessment: Decay Visualization 97
7.8 Problem: Measuring by Stone Throwing 97
7.9 Theory: Integration by Rejection 97
7.10 Implementation: Stone Throwing,pond.f(.c) 98
7.n Problem: High-Dimensionallntegration 99
7.12 Method: Integration by Mean Value 99
7.12.1 Multidimensional Monte Carlo 101
7.13 Assessment: Error in N-D Integration 101
7.14 Implementation: 10-D Integration,int_10d.f(.c) 101
7.15 Problem: Integrate a Rapidly Varying Function© 102
7.16 Method: Variance Reduction© 102
7.17 Method: Importance Sampling© 103
7.18 Implementation: Nonuniform Randomness© 103
7.18.1 Inverse Transform Method 103
7.18.2 Uniform Weight Function w 104CONTENTS Xi
7.18.3 Exponential Weight 105
7.18.4 Gaussian(Normal)Distribution 105
7.18.5 Alternate Gaussian Distribution 106
7.19 Method: von Neumann Rejection© 107
7.20 Assessment© 108
Differentiation 109
8.1 Problem 1: Numerical Limits 109
8.2 Method: Numeric 109
8.2.1 Method: Forward Difference 109
8.2.2 Method: Central Difference 111
8.2.3 Method: Extrapolated Difference 111
8.3 Assessment: Error Analysis 112
8.4 Implementation: Differentiation,diff.f(.c) 113
8.5 Assessment: Error Analysis, Numerical 113
8.6 Problem 2: Second Derivatives 114
8.7 Theory: Newton II 114
8.8 Method: Numerical Second Derivatives 114
8.9 Assessment: Numerical Second Derivatives 115
Differential Equations and Oscillations 117
9.1 Problem: A Forced Nonlinear Oscillator 117
9.2 Theory, Physics: Newton's Laws 117
9.3 Model: Nonlinear Oscillator 118
9.4 Theory, Math: Types ofEquations 119
9.4.1 Order 119
9.4.2 Ordinary and Partial 120
9.4.3 Linear and Nonlinear 121
9.4.4 Initial and Boundary Conditions 121
9.5 Theory,^ath, and Physics:
The Dynamical Form for ODEs 122
9.5.1 Second-Order Equation 122
9.6 Implementation: Dynamical Form for Oscillator 123
9.7 Numerical Method: ODE Algorithms 124
9.8 Method(Numerical): Euler's Algorithm 124
9.9 Method(Numerical): Second-Order Runge-Kutta 126
9.10 Method(Numerical): Fourth-Order Runge-Kutta 127
9.11 Implementation: ODE Solver,rk4.f(.c) 127
9.12 Assessment: rk4 and Linear Oscillations 128Xii CONTENTS
9.13 Assessment: rk4 and Nonlinear Oscillations 128
9.14 Exploration: Energy Conservation 129
10 Quantum Eigenvalues; Zero-Finding and Matching 131
10.1 Problem: Binding A Quantum Particle 131
10.2 Theory: Quantum Waves 132
10.3 Model: Particle in a Box 133
10.4 Solution: Semianalytic 133
10.5 Method: Finding Zero via Bisection Algorithm 136
10.6 Method: Eigenvalues from an ODE Solver 136
10.6.1 Matching 138
10.7 Implementation: ODE Eigenvalues,numerov.c 139
10.8 Assessment: Explorations 141
10.9 Extension: Newton's Rule for Finding Roots 141
11 Anharmonic Oscillations 143
11.1 Problem 1: Perturbed Harmonic Oscillator 143
11.2 Theory: Newton II 144
11.3 Implementation: ODE Solver,rk4.f(.c) 144
11.4 Assessment: Amplitude Dependence frequency 145
11.5 Problem 2: Realistic Pendulum 145
11.6 Theory: Newton II for Rotations 146
11.7 Method, Analytic: Elliptic Integrals 147
11.8 Implementation,rk4 for Pendulum 147
11.9 Exploration: Resonance and Beats 148
11.10 Exploration: Phase-Space Plot 149
11.11 Exploration: Damped Oscillator 150
12 Fourier Analysis of Nonlinear Oscillations 151
12.1 Problem 1: Harmonics in Nonlinear Oscillations 151
12.2 Theory: Fourier Analysis 152
12.2.1 Example 1: Sawtooth Function 154
12.2.2 Example 2: Half-Wave Function 154
12.3 Assessment: Summation of Fourier Series 155
12.4 Theory: Fourier Transforms 156
12.5 Method: Discrete Fourier Transform 157
12.6 Method: DFT for Fourier Series 161
12.7 Implementation: fourier.f(.c), in four.c 162
12.8 Assessment: Simple Analytic Input 162CONTENTS Xiii
12.9 Assessment: Highly Nonlinear Oscillator 163
12.10 Assessment: Nonlinearly Perturbed Oscillator 163
12.11 Exploration: DFT of Nonperiodic Functions 163
12.12 Exploration: Processing Noisy Signals 164
12.13 Model: Autocorrelation Function 164
12.14 Assessment: DFT and Autocorrelation Function 165
12.15 Problem 2: Model Dependence of Data Analysis © 166
12.16 Method: Model-Independent Data Analysis 167
12.17 Assessment 168
13 Unusual Dynamics of Nonlinear Systems 171
13.1 Problem: Variability of Populations 171
13.2 Theory: Nonlinear Dynamics 171
13.3 Model: Nonlinear Growth, The Logistic Map 172
13.3.1 The Logistic Map 173
13.4 Theory: Properties of Nonlinear Maps 174
13.4.1 Fixed Points 174
13.4.2 Period Doubling, Attractors 175
13.5 Implementation: Explicit Mapping 176
13.6 Assessment: Bifurcation Diagram 177
13.7 Implementation: bugs.f(.c) 178
13.8 Exploration: Random Numbers via Logistic Map 179
13.9 Exploration: Feigenbaum Constants 179
13.10 Exploration: Other Maps 180
14 Differential Chaos in Phase Space 181
14.1 Problem: A Pendulum Becomes Chaotic 181
14.2 Theory and Model: The Chaotic Pendulum 182
14.3 Theory: Limit Cycles and Mode-Locking 183
14.4 Implementation 1: Solve ODE,rk4.f(.c) 184
14.5 Visualization: Phase-Space Orbits 184
14.6 Implementation 2: Free Oscillations 187
14.7 Theory: Motion in Phase Space 188
14.8 Implementation 3: Chaotic Pendulum 188
14.9 Assessment: Chaotic Structure in Phase Space 191
14.10 Assessment: Fourier Analysis 191
14.11 Exploration: Pendulum with Vibrating Pivot 192
14.11.1 Implementation: Bifurcation Diagram 192
14.12 Further Explorations 19317 Quantum Scattering via Integral Equations©
17.1 Problem: Quantum Scattering in k Space
17.2 Theory: Lippmann-Schwinger Equation
17.3 Theory(Mathematics): Singular Integrals
xlv CONTENTS
Part III APPLICATIONS
(HIGH PERFORMANCE COMPUTING)
15 Matrix Computing and Subroutine Libraries 197
15.1 Problem 1: Many Simultaneous Linear Equations 197
15.2 Formulation: Linear into Matrix Equation 198
15.3 Problem 2: Simple but Unsolvable Statics 198
15.4 Theory: Statics
15.5 Formulation: Nonlinear Simultaneous Equations 199
15.6 Theory: Matrix Problems
15 6.1 Classes of Matrix Problems 201
15.7 Method: Matrix Computing
15.8 Implementation: Scientific Libraries, WWW
15.9 Implementation: Determining Availability
15.9.1 Determining Contents of a Library
15.9.2 Determining the Needed Routine
15.9.3 Calling lapack from Fortran,lineq.c
15.9.4 Calling lapack from C
15.9.5 Calling lapack Fortran from C
15.9.6 C Compiling Calling Fortran
15.10 Extension: More Netlib Libraries
15.10.1 SLATEC's Common Math Library
15.11 Exercises: Testing Matrix Calls
15.12Implementation: LAPACK Short Contents
15.13 Implementation: Netlib Short Contents
15.14 Implementation: SLATEC Short Contents

6 Bound States in Momentum Space 231
16.1 Problem: Bound States in Nonlocal Potentials
16.2 Theory: k-Space Schrodinger Equation
16.3 Method: Reducing Integral to Linear Equations 233
16.4 Model: The Delta-Shell Potential 23
16.5 Implementation: Binding Energies, bound.c(.0
16.6 Exploration: Wave Function 237