Preface to first edition ....................................... xi
Preface ...................................................... xiii
Acknowledgments ................................................ xv
1 Introduction ................................................. 1
1.1 Computation and science ................................. 1
1.2 The emergence of modern computers ....................... 4
1.3 Computer algorithms and languages ....................... 7
Exercises ................................................... 14
2 Approximation of a function ................................. 16
2.1 Interpolation .......................................... 16
2.2 Least-squares approximation ............................ 24
2.3 The Millikan experiment ................................ 27
2.4 Spline approximation ................................... 30
2.5 Random-number generators ............................... 37
Exercises ................................................... 44
3 Numerical calculus .......................................... 49
3.1 Numerical differentiation .............................. 49
3.2 Numerical integration .................................. 56
3.3 Roots of an equation ................................... 62
3.4 Extremes of a function ................................. 66
3.5 Classical scattering ................................... 70
Exercises ................................................... 76
4 Ordinary differential equations ............................. 80
4.1 Initial-value problems ................................. 81
4.2 The Euler and Picard methods ........................... 81
4.3 Predictor-corrector methods ............................ 83
4.4 The Runge-Kutta method ................................. 88
4.5 Chaotic dynamics of a driven pendulum .................. 90
4.6 Boundary-value and eigenvalue problems ................. 94
4.7 The shooting method .................................... 96
4.8 Linear equations and the Sturm-Liouville problem ....... 99
4.9 The one-dimensional Schrödinger equation .............. 105
Exercises .................................................. 115
5 Numerical methods for matrices ............................. 119
5.1 Matrices in physics ................................... 119
5.2 Basic matrix operations ............................... 123
5.3 Linear equation systems ............................... 125
5.4 Zeros and extremes of multivariable functions ......... 133
5.5 Eigenvalue problems ................................... 138
5.6 The Faddeev-Leverrier method .......................... 147
5.7 Complex zeros of a polynomial ......................... 149
5.8 Electronic structures of atoms ........................ 153
5.9 The Lanczos algorithm and the many-body problem ....... 156
5.10 Random matrices ....................................... 158
Exercises .................................................. 160
6 Spectral analysis .......................................... 164
6.1 Fourier analysis and orthogonal functions ............. 165
6.2 Discrete Fourier transform ............................ 166
6.3 Fast Fourier transform ................................ 169
6.4 Power spectrum of a driven pendulum ................... 173
6.5 Fourier transform in higher dimensions ................ 174
6.6 Wavelet analysis ...................................... 175
6.7 Discrete wavelet transform ............................ 180
6.8 Special functions ..................................... 187
6.9 Gaussian quadratures .................................. 191
Exercises .................................................. 193
7 Partial differential equations ............................. 197
7.1 Partial differential equations in physics ............. 197
7.2 Separation of variables ............................... 198
7.3 Discretization of the equation ........................ 204
7.4 The matrix method for difference equations ............ 206
7.5 The relaxation method ................................. 209
7.6 Groundwater dynamics .................................. 213
7.7 Initial-value problems ................................ 216
7.8 Temperature field of a nuclear waste rod .............. 219
Exercises .................................................. 222
8 Molecular dynamics simulations ............................. 226
8.1 General behavior of a classical system ............... 226
8.2 Basic methods for many-body systems ................... 228
8.3 The Verlet algorithm .................................. 232
8.4 Structure of atomic clusters .......................... 236
8.5 The Gear predictor-corrector method ................... 239
8.6 Constant pressure, temperature, and bond length ....... 241
8.7 Structure and dynamics of real materials .............. 246
8.8 Ab initio molecular dynamics .......................... 250
Exercises .................................................. 254
9 Modeling continuous systems ................................ 256
9.1 Hydrodynamic equations ................................ 256
9.2 The basic finite element method ....................... 258
9.3 The Ritz variational method ........................... 262
9.4 Higher-dimensional systems ............................ 266
9.5 The finite element method for nonlinear equations ..... 269
9.6 The particle-in-cell method ........................... 271
9.7 Hydrodynamics and magnetohydrodynamics ................ 276
9.8 The lattice Boltzmann method .......................... 279
Exercises .................................................. 282
10 Monte Carlo simulations .................................... 285
10.1 Sampling and integration .............................. 285
10 2 The Metropolis algorithm .............................. 287
10.3 Applications in statistical physics ................... 292
10.4 Critical slowing down and block algorithms ............ 297
10.5 Variational quantum Monte Carlo simulations ........... 299
10.6 Green's function Monte Carlo simulations .............. 303
10.7 Two-dimensional electron gas .......................... 307
10.8 Path-integral Monte Carlo simulations ................. 313
10.9 Quantum lattice models ................................ 315
Exercises .................................................. 320
11 Genetic algorithm and programming .......................... 323
11.1 Basic elements of a genetic algorithm ................. 324
11.2 The Thomson problem ................................... 332
11.3 Continuous genetic algorithm .......................... 335
11.4 Other applications .................................... 338
11.5 Genetic programming ................................... 342
Exercises .................................................. 345
12 Numerical renormalization .................................. 347
12.1 The scaling concept ................................... 347
12.2 Renormalization transform ............................. 350
12.3 Critical phenomena: the Ising model ................... 352
12.4 Renormalization with Monte Carlo simulation ........... 355
12.5 Crossover: the Kondo problem .......................... 357
12.6 Quantum lattice renormalization ....................... 360
12.7 Density matrix renormalization ........................ 364
Exercises .................................................. 367
References .................................................... 369
Index ......................................................... 381
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