Survey of Computational Physics

Rubin H. Landau is professor of physics and director of the computational physics program at Oregon State University. Manuel Jose Paez is professor of physics at Universidad de Antioquia in Colombia. Cristian C. Bordeianu, a PhD candidate at University of Bucharest, is vice principal at Technological High School 1 in Suceava, Romania.

• Preface xxiii CHAPTER 1: Computational Science Basics 1 1.1 Computational Physics and Science 1 1.2 How to Read and Use This Book 3 1.3 Making Computers Obey; Languages (Theory) 6 1.4 Programming Warmup 8 1.4.1 Structured Program Design 10 1.4.2 Shells, Editors, and Execution 11 1.4.3 Java I/O, Scanner Class with printf 12 1.4.4 I/O Redirection 12 1.4.5 Command-Line Input 13 1.4.6 I/O Exceptions: FileCatchThrow.java 14 1.4.7 Automatic Code Documentation 16 1.5 Computer Number Representations (Theory) 17 1.5.1 IEEE Floating-Point Numbers 18 1.5.2 Over/Underflows Exercises 24 1.5.3 Machine Precision (Model) 25 1.5.4 Determine Your Machine Precision 27 1.6 Problem: Summing Series 27 1.6.1 Numerical Summation (Method) 28 1.6.2 Implementation and Assessment 29 CHAPTER 2: Errors & Uncertainties in Computations 30 2.1 Types of Errors (Theory) 30 2.1.1 Model for Disaster: Subtractive Cancellation 32 2.1.2 Subtractive Cancellation Exercises 33 2.1.3 Round-off Error in a Single Step 34 2.1.4 Round-off Error Accumulation After Many Steps 35 2.2 Errors in Spherical Bessel Functions (Problem) 36 2.2.1 Numerical Recursion Relations (Method) 36 2.2.2 Implementation and Assessment: Recursion Relations 38 2.3 Experimental Error Investigation (Problem) 39 2.3.1 Error Assessment 43 CHAPTER 3: Visualization Tools 45 3.1 Data Visualization 45 3.2 PtPlot: 2-D Graphs Within Java 46 3.3 Grace/ACE: Superb 2-D Graphs for Unix/Linux 51 3.3.1 Grace Basics 51 3.4 Gnuplot: Reliable 2-D and 3-D Plots 56 3.4.1 Gnuplot Input Data Format 58 3.4.2 Printing Plots 59 3.4.3 Gnuplot Surface (3-D) Plots 60 3.4.4 Gnuplot Vector Fields 62 3.4.5 Animations from a Plotting Program (Gnuplot) 64 3.5 OpenDX for Dicing and Slicing 65 3.6 Texturing and 3-D Imaging 65 CHAPTER 4: Object-Oriented Programs: Impedance & Batons 67 4.1 Unit I. Basic Objects: Complex Impedance 67 4.2 Complex Numbers (Math) 67 4.3 Resistance Becomes Impedance (Theory) 70 4.4 Abstract Data Structures, Objects (CS) 70 4.4.1 Object Declaration and Construction 72 4.4.2 Implementation in Java 73 4.4.3 Static and Nonstatic Methods 76 4.4.4 Nonstatic Methods 77 4.5 Complex Currents (Solution) 79 4.6 OOP Worked Examples 80 4.6.1 OOP Beats 80 4.6.2 OOP Planet 82 4.7 Unit II. Advanced Objects: Baton Projectiles 85 4.8 Trajectory of a Thrown Baton (Problem) 86 4.8.1 Combined Translation and Rotation (Theory) 86 4.9 OOP Design Concepts (CS) 89 4.9.1 Including Multiple Classes 90 4.9.2 Ball and Path Class Implementation 92 4.9.3 Composition, Objects Within Objects 93 4.9.4 Baton Class Implementation 94 4.9.5 Composition Exercise 95 4.9.6 Calculating the Baton's Energy (Extension) 96 4.9.7 Examples of Inheritance and Object Hierarchies 98 4.9.8 Baton with a Lead Weight (Application) 99 4.9.9 Encapsulation to Protect Classes 100 4.9.10 Encapsulation Exercise 101 4.9.11 Complex Object Interface (Extension) 102 4.9.12 Polymorphism, Variable Multityping 104 4.10 Supplementary Exercises 105 4.11 OOP Example: Superposition of Motions 105 4.12 Newton's Laws of Motion (Theory) 106 4.13 OOP Class Structure (Method) 106 4.14 Java Implementation 107 CHAPTER 5: Monte Carlo Simulations (Nonthermal) 109 5.1 Unit I. Deterministic Randomness 109 5.2 Random Sequences (Theory) 109 5.2.1 Random-Number Generation (Algorithm) 110 5.2.2 Implementation: Random Sequence 113 5.2.3 Assessing Randomness and Uniformity 114 5.3 Unit II. Monte Carlo Applications 116 5.4 A Random Walk (Problem) 116 5.4.1 Random-Walk Simulation 116 5.4.2 Implementation: Random Walk 117 5.5 Radioactive Decay (Problem) 119 5.5.1 Discrete Decay (Model) 119 5.5.2 Continuous Decay (Model) 120 5.5.3 Decay Simulation 121 5.6 Decay Implementation and Visualization 122 CHAPTER 6: Integration 123 6.1 Integrating a Spectrum (Problem) 123 6.2 Quadrature as Box Counting (Math) 123 6.2.1 Algorithm: Trapezoid Rule 125 6.2.2 Algorithm: Simpson's Rule 126 6.2.3 Integration Error (Analytic Assessment) 128 6.2.4 Algorithm: Gaussian Quadrature 130 6.2.5 Integration Implementation and Error Assessment 132 6.3 Experimentation 135 6.4 Higher-Order Rules (Algorithm) 135 6.5 Monte Carlo Integration by Stone Throwing 136 6.5.1 Stone Throwing Implementation 136 6.5.2 Integration by Mean Value (Math) 137 6.6 High-Dimensional Integration (Problem) 138 6.6.1 Multidimensional Monte Carlo 139 6.6.2 Error in Multidimensional Integration (Assessment) 139 6.6.3 Implementation: 10-D Monte Carlo Integration 139 6.7 Integrating Rapidly Varying Functions (Problem) 140 6.7.1 Variance Reduction (Method) 140 6.7.2 Importance Sampling (Method) 140 6.7.3 Von Neumann Rejection (Method) 141 6.7.4 Simple Gaussian Distribution 141 6.8 Nonuniform Assessment 142 6.8.1 Implementation: Nonuniform Randomness 142 CHAPTER 7: Differentiation & Searching 146 7.1 Unit I. Numerical Differentiation 146 7.2 Forward Difference (Algorithm) 147 7.3 Central Difference (Algorithm) 148 7.4 Extrapolated Difference (Method) 149 7.5 Error Analysis (Assessment) 149 7.6 Second Derivatives (Problem) 151 7.6.1 Second-Derivative Assessment 151 7.7 Unit II. Trial-and-Error Searching 151 7.8 Quantum States in a Square Well (Problem) 152 7.9 Trial-and-Error Roots via the Bisection Algorithm 152 7.9.1 Bisection Algorithm Implementation 153 7.10 Newton-Raphson Searching (A Faster Algorithm) 154 7.10.1 Newton-Raphson Algorithm with Backtracking 156 7.10.2 Newton-Raphson Algorithm Implementation 157 CHAPTER 8: Solving Systems of Equations with Matrices; Data Fitting 158 8.1 Unit I. Systems of Equations and Matrix Computing 158 8.2 Two Masses on a String 159 8.2.1 Statics (Theory) 160 8.2.2 Multidimensional Newton-Raphson Searching 160 8.3 Classes of Matrix Problems (Math) 163 8.3.1 Practical Aspects of Matrix Computing 165 8.3.2 Implementation: Scientific Libraries, World Wide Web 168 8.3.3 JAMA: Java Matrix Library 169 8.3.4 Exercises for Testing Matrix Calls 173 8.3.5 Matrix Solution of the String Problem 175 8.3.6 Explorations 175 8.4 Unit II. Data Fitting 176 8.5 Fitting an Experimental Spectrum (Problem) 176 8.5.1 Lagrange Interpolation (Method) 177 8.5.2 Lagrange Implementation and Assessment 178 8.5.3 Explore Extrapolation 179 8.5.4 Cubic Splines (Method) 179 8.5.5 Spline Fit of Cross Section (Implementation) 182 8.6 Fitting Exponential Decay (Problem) 182 8.6.1 Theory to Fit 182 8.7 Least-Squares Fitting (Method) 184 8.7.1 Least-Squares Fitting: Theory and Implementation 186 8.7.2 Exponential Decay Fit Assessment 188 8.7.3 Exercise: Fitting Heat Flow 189 8.7.4 Linear Quadratic Fit (Extension) 190 8.7.5 Linear Quadratic Fit Assessment 191 8.7.6 Nonlinear Fit of the Breit-Wigner Formula to a Cross Section 191 CHAPTER 9: Differential Equation Applications 194 9.1 Unit I. Free Nonlinear Oscillations 194 9.2 Nonlinear Oscillators (Models) 194 9.3 Types of Differential Equations (Math) 196 9.4 Dynamic Form for ODEs (Theory) 198 9.5 ODE Algorithms 200 9.5.1 Euler's Rule 201 9.5.2 Runge-Kutta Algorithm 202 9.5.3 Adams-Bashforth-Moulton Predictor-Corrector 204 9.5.4 Assessment: rk2 versus rk4 versus rk45 205 9.6 Solution for Nonlinear Oscillations (Assessment) 207 9.6.1 Precision Assessment: Energy Conservation 208 9.7 Extensions: Nonlinear Resonances, Beats, and Friction 209 9.7.1 Friction: Model and Implementation 209 9.7.2 Resonances and Beats: Model and Implementation 210 9.8 Implementation: Inclusion of Time-Dependent Force 211 9.9 Unit II. Binding A Quantum Particle 212 9.10 The Quantum Eigenvalue Problem (Theory) 212 9.10.1 Nucleon in a Box (Model) 213 9.11 Combined Algorithms: Eigenvalues via ODE Solver Plus Search 214 9.11.1 Numerov Algorithm for the Schrodinger ODE 216 9.11.2 Implementation: Eigenvalues via an ODE Solver Plus Bisection Algorithm 218 9.12 Explorations 221 9.13 Unit III. Scattering, Projectiles, and Planetary Orbits 222 9.14 Problem 1: Classical Chaotic Scattering 222 9.14.1 Model and Theory 222 9.14.2 Implementation 224 9.14.3 Assessment 225 9.15 Problem 2: Balls Falling Out of the Sky 225 9.16 Theory: Projectile Motion with Drag 226 9.16.1 Simultaneous Second-Order ODEs 227 9.16.2 Assessment 228 9.17 Problem 3: Planetary Motion 228 9.17.1 Implementation: Planetary Motion 229 CHAPTER 10: Fourier Analysis: Signals and Filters 231 10.1 Unit I. Fourier Analysis of Nonlinear Oscillations 231 10.2 Fourier Series (Math) 232 10.2.1 Example 1: Sawtooth Function 234 10.2.2 Example 2: Half-wave Function 235 10.3 Summation of Fourier Series (Exercise) 235 10.4 Fourier Transforms (Theory) 236 10.4.1 Discrete Fourier Transform Algorithm 237 10.4.2 Aliasing and Anti-aliasing 241 10.4.3 DFT for Fourier Series (Algorithm) 243 10.4.4 Assessments 244 10.4.5 DFT of Nonperiodic Functions (Exploration) 246 10.5 Unit II. Filtering Noisy Signals 246 10.6 Noise Reduction via Autocorrelation (Theory) 246 10.6.1 Autocorrelation Function Exercises 249 10.7 Filtering with Transforms (Theory) 250 10.7.1 Digital Filters: Windowed Sinc Filters 253 10.8 Unit III. Fast Fourier Transform Algorithm 256 10.8.1 Bit Reversal 258 10.9 FFT Implementation 259 10.10 FFT Assessment 263 CHAPTER 11: Wavelet Analysis & Data Compression 264 11.1 Unit I. Wavelet Basics 264 11.2 Wave Packets and Uncertainty Principle (Theory) 266 11.2.1 Wave Packet Assessment 268 11.3 Short-Time Fourier Transforms (Math) 268 11.4 The Wavelet Transform 269 11.4.1 Generating Wavelet Basis Functions 270 11.4.2 Continuous Wavelet Transform Implementation 273 11.5 Unit II. Discrete Wavelet Transform and Multiresolution Analysis 274 11.5.1 Pyramid Scheme Implementation 279 11.5.2 Daubechies Wavelets via Filtering 283 11.5.3 DWT Implementation and Exercise 286 CHAPTER 12: Discrete & Continuous Nonlinear Dynamics 289 12.1 Unit I. Bug Population Dynamics (Discrete) 289 12.2 The Logistic Map (Model) 289 12.3 Properties of Nonlinear Maps (Theory) 291 12.3.1 Fixed Points 291 12.3.2 Period Doubling, Attractors 292 12.4 Mapping Implementation 293 12.5 Bifurcation Diagram (Assessment) 294 12.5.1 Bifurcation Diagram Implementation 295 12.5.2 Visualization Algorithm: Binning 295 12.5.3 Feigenbaum Constants (Exploration) 297 12.6 Random Numbers via Logistic Map (Exploration) 297 12.7 Other Maps (Exploration) 298 12.8 Signals of Chaos: Lyapunov Coefficients 298 12.8.1 Shannon Entropy 299 12.9 Unit I Quiz 300 12.10 Unit II. Pendulums Become Chaotic (Continuous) 302 12.11 Chaotic Pendulum ODE 302 12.11.1 Free Pendulum Oscillations 303 12.11.2 Solution as Elliptic Integrals 304 12.11.3 Implementation and Test: Free Pendulum 305 12.12 Visualization: Phase Space Orbits 305 12.12.1 Chaos in Phase Space 307 12.12.2 Assessment in Phase Space 311 12.13 Exploration: Bifurcations of Chaotic Pendulums 313 12.14 Alternative Problem: The Double Pendulum 315 12.15 Assessment: Fourier/Wavelet Analysis of Chaos 317 12.16 Exploration: Another Type of Phase Space Plot 317 12.17 Further Explorations 318 12.18 Unit III. Coupled Predator-Prey Models 319 12.19 Lotka-Volterra Model 320 12.19.1 LVM with Prey Limit 321 12.19.2 LVM with Predation Efficiency 322 12.19.3 LVM Implementation and Assessment 323 12.19.4 Two Predators, One Prey (Exploration) 324 CHAPTER 13: Fractals & Statistical Growth 326 13.1 Fractional Dimension (Math) 326 13.2 The Sierpnski Gasket (Problem 1) 327 13.2.1 Sierpnski Implementation 328 13.2.2 Assessing Fractal Dimension 328 13.3 Beautiful Plants (Problem 2) 329 13.3.1 Self-affine Connection (Theory) 330 13.3.2 Barnsley's Fern Implementation 331 13.3.3 Self-affinity in Trees Implementation 332 13.4 Ballistic Deposition (Problem 3) 332 13.4.1 Random Deposition Algorithm 332 13.5 Length of the British Coastline (Problem 4) 334 13.5.1 Coastlines as Fractals (Model) 334 13.5.2 Box Counting Algorithm 335 13.5.3 Coastline Implementation and Exercise 336 13.6 Correlated Growth, Forests, and Films (Problem 5) 338 13.6.1 Correlated Ballistic Deposition Algorithm 338 13.7 Globular Cluster (Problem 6) 339 13.7.1 Diffusion-Limited Aggregation Algorithm 339 13.7.2 Fractal Analysis of a DLA (or Pollock) Graph (Assessment) 342 13.8 Fractal Structures in a Bifurcation Graph (Problem 7) 343 13.9 Fractals from Cellular Automata 343 13.10 Perlin Noise Adds Realism 345 13.10.1Including Ray Tracing 348 13.11 Quiz 351 CHAPTER 14: High-Performance Computing Hardware, Tuning, and Parallel Computing 352 14.1 Unit I. High-Performance Computers (CS) 352 14.2 Memory Hierarchy 353 14.3 The Central Processing Unit 357 14.4 CPU Design: Reduced Instruction Set Computer 357 14.5 CPU Design: Multiple-Core Processors 358 14.6 CPU Design: Vector Processor 359 14.7 Unit II. Parallel Computing 360 14.8 Parallel Semantics (Theory) 361 14.9 Distributed Memory Programming 363 14.10 Parallel Performance 365 14.10.1 Communication Overhead 367 14.11 Parallelization Strategy 368 14.12 Practical Aspects of Message Passing for MIMD 369 14.12.1 High-Level View of Message Passing 370 14.13 Example of a Supercomputer: IBM Blue Gene/L 372 14.14 Unit III. HPC Program Optimization 374 14.14.1 Programming for Virtual Memory (Method) 376 14.14.2 Optimizing Programs; Java versus Fortran/C 376 14.14.3 Experimental Effects of Hardware on Performance 379 14.14.4 Java versus Fortran/C 380 14.15 Programming for the Data Cache (Method) 385 14.15.1 Exercise 1: Cache Misses 386 14.15.2 Exercise 2: Cache Flow 387 14.15.3 Exercise 3: Large-Matrix Multiplication 388 CHAPTER 15: Thermodynamic Simulations & Feynman Quantum Path Integration 390 15.1 Unit I. Magnets via the Metropolis Algorithm 390 15.2 An Ising Chain (Model) 390 15.3 Statistical Mechanics (Theory) 393 15.3.1 Analytic Solutions 393 15.4 Metropolis Algorithm 394 15.4.1 Metropolis Algorithm Implementation 397 15.4.2 Equilibration, Thermodynamic Properties (Assessment) 397 15.4.3 Beyond Nearest Neighbors and 1-D (Exploration) 400 15.5 Unit II. Magnets via Wang-Landau Sampling 400 15.6 Wang-Landau Sampling 403 15.6.1 WLS Ising Model Implementation 405 15.6.2 WLS Ising Model Assessment 408 15.7 Unit III. Feynman Path Integrals 408 15.8 Feynman's Space-Time Propagation (Theory) 408 15.8.1 Bound-State Wave Function ( Theory) 412 15.8.2 Lattice Path Integration (Algorithm) 413 15.8.3 Lattice Implementation 418 15.8.4 Assessment and Exploration 420 15.9 Exploration: Quantum Bouncer's Paths 421 CHAPTER 16: Simulating Matter with Molecular Dynamics 424 16.1 Molecular Dynamics ( Theory) 424 16.1.1 Connection to Thermodynamic Variables 428 16.1.2 Setting Initial Velocity Distribution 429 16.1.3 Periodic Boundary Conditions and Potential Cutoff 429 16.2 Verlet and Velocity-Verlet Algorithms 431 16.3 1-D Implementation and Exercise 432 16.4 Trajectory Analysis 435 16.5 Quiz 436 CHAPTER 17: PDEs for Electrostatics & Heat Flow 437 17.1 PDE Generalities 437 17.2 Unit I. Electrostatic Potentials 439 17.2.1 Laplace's Elliptic PDE ( Theory) 439 17.3 Fourier Series Solution of a PDE 440 17.3.1 Polynomial Expansion As an Algorithm 442 17.4 Solution: Finite-Difference Method 443 17.4.1 Relaxation and Overrelaxation 445 17.4.2 Lattice PDE Implementation 446 17.5 Assessment via Surface Plot 447 17.6 Alternate Capacitor Problems 448 17.7 Implementation and Assessment 450 17.8 Electric Field Visualization (Exploration) 452 17.9 Laplace Quiz 452 17.10 Unit II. Finite-Element Method 453 17.11 Electric Field from Charge Density (Problem) 454 17.12 Analytic Solution 454 17.13 Finite-Element (Not Difference) Methods 455 17.13.1 Weak Form of PDE 455 17.13.2 Galerkin Spectral Decomposition 456 17.14 FEM Implementation and Exercises 460 17.15 Exploration 463 17.16 Unit III. Heat Flow via Time-Stepping (Leapfrogging) 463 17.17 The Parabolic Heat Equation (Theory) 463 17.17.1 Solution: Analytic Expansion 465 17.17.2 Solution: Time-Stepping 466 17.17.3 Von Neumann Stability Assessment 468 17.17.4 Heat Equation Implementation 470 17.18 Assessment and Visualization 470 17.19 Improved Heat Flow: Crank-Nicolson Method 472 17.19.1 Solution of Tridiagonal Matrix Equations 474 17.19.2 Crank-Nicolson Method Implementation and Assessment 476 CHAPTER 18: PDEWaves: String, Quantum Packet, and E&M 478 18.1 Unit I. Vibrating String 478 18.2 The Hyperbolic Wave Equation (Theory) 478 18.2.1 Solution via Normal-Mode Expansion 480 18.2.2 Algorithm: Time-Stepping 481 18.2.3 Wave Equation Implementation 483 18.2.4 Assessment and Exploration 484 18.3 Waves with Friction (Extension) 486 18.4 Waves for Variable Tension and Density (Extension) 487 18.4.1 Waves on a Catenary 488 18.4.2 Derivation of a Catenary Shape 488 18.4.3 Catenary and Frictional Wave Exercises 490 18.5 Unit II. Quantum Wave Packets 491 18.6 Time-Dependent Schrodinger Equation (Theory) 492 18.6.1 Finite-Difference Algorithm 493 18.6.2 Wave Packet Implementation and Animation 494 18.7 Wave Packets in Other Wells (Exploration) 496 18.8 Algorithm for the 2-D Schrodinger Equation 496 18.9 Unit III. E&M Waves via Finite-Difference Time Domain 499 18.10 Maxwell's Equations 499 18.11 FDTD Algorithm 500 18.11.1 Implementation 503 18.11.2 Assessment 504 18.11.3 Extension: Circularly Polarized EM Waves 506 CHAPTER 19: Solitons & Computational Fluid Dynamics 508 19.1 Unit I. Advection, Shocks, and Russell's Soliton 508 19.2 Theory: Continuity and Advection Equations 509 19.2.1 Advection Implementation 510 19.3 Theory: Shock Waves via Burgers' Equation 510 19.3.1 Algorithm: The Lax-Wendroff Method for Burgers' Equation 511 19.3.2 Implementation and Assessment of Burgers' Shock Equation 513 19.4 Including Dispersion 514 19.5 Shallow-Water Solitons, the KdeV Equation 515 19.5.1 Analytic Soliton Solution 517 19.5.2 Algorithm for KdeV Solitons 518 19.5.3 Implementation: KdeV Solitons 519 19.5.4 Exploration: Solitons in Phase Space and Crossing 520 19.6 Unit II. River Hydrodynamics 521 19.7 Hydrodynamics, the Navier-Stokes Equation (Theory) 521 19.7.1 Boundary Conditions for Parallel Plates 524 19.7.2 Analytic Solution for Parallel Plates 526 19.7.3 Finite-Difference Algorithm and Overrelaxation 527 19.7.4 Successive Overrelaxation Implementation 529 19.8 2-D Flow over a Beam 530 19.9 Theory: Vorticity Form of the Navier-Stokes Equation 530 19.9.1 Finite Differences and the SOR Algorithm 532 19.9.2 Boundary Conditions for a Beam 534 19.9.3 SOR on a Grid Implementation 536 19.9.4 Assessment 538 19.9.5 Exploration 539 CHAPTER 20: Integral Equations in Quantum Mechanics 540 20.1 Unit I. Bound States of Nonlocal Potentials 540 20.2 Momentum-Space Schrodinger Equation (Theory) 541 20.2.1 Integral to Linear Equations (Method) 542 20.2.2 Delta-Shell Potential (Model) 544 20.2.3 Binding Energies Implementation 544 20.2.4 Wave Function (Exploration) 546 20.3 Unit II. Nonlocal Potential Scattering 546 20.4 Lippmann-Schwinger Equation (Theory) 547 20.4.1 Singular Integrals (Math) 548 20.4.2 Numerical Principal Values 549 20.4.3 Reducing Integral Equations to Matrix-Equations (Algorithm) 549 20.4.4 Solution via Inversion or Elimination 551 20.4.5 Scattering Implementation 552 20.4.6 Scattering Wave Function (Exploration) 553 Appendix A: Glossary 555 Appendix B: Installing Packages 562 B.1 Installing Java Developer's Kit 564 B.2 Using Classes and Packages 565 B.2.1 Including Packages 565 Appendix C: OpenDX: Industrial-Strength Data Visualization 568 C.1 Getting DX and Unix Running (for Windows) 569 C.2 Test Drive of DX Visual Programming 569 C.3 DX Tools Summary 576 C.4 DX Data Structure and Storage 577 C.5 Sample Visual Programs 579 C.5.1 Sample 1: Linear Plot 579 C.5.2 Sample 2: Fourier Transform 580 C.5.3 Sample 3: Potential of a 2-D Capacitor 580 C.5.4 Sample 4: Vector Field Plots 581 C.5.5 Sample 5: 3-D Scalar Potentials 582 C.5.6 Sample 6: 3-D Functions, the Hydrogen Atom 585 C.6 Animations with OpenDX 586 C.6.1 Scripted Animation with OpenDX 588 C.6.2 Wave Packet and Slit Animation 591 Appendix D: An MPI Tutorial 593 D.1 Running on a Beowulf 593 D.2 Running MPI 597 D.2.1 MPI under the SGE Queueing System 598 D.2.2 MPI Under the Torque/PBS Queueing System 600 D.2.3 Running Parallel Jobs with Torque 602 D.3 Your First MPI Program: MPIhello.c 604 D.3.1 MPIhello.c Explained 605 D.3.2 Send/Receive Messages: MPImessage2.c 606 D.3.3 Receive More Messages: MPImessage3.c 608 D.3.4 Broadcast Messages 609 D.3.5 Exercise 610 D.4 Parallel Tuning 611 D.5 A String Vibrating in Parallel 614 D.5.1 MPIstring.c Exercise 617 D.6 Deadlock 618 D.6.1 Nonblocking Communication 619 D.6.2 Collective Communication 619 D.7 Bootable Cluster CD 620 D.8 Parallel Computing Exercises 620 D.9 List of MPI Commands 621 Appendix E: Calling LAPACK from C 623 E.1 Calling LAPACK Fortran from C 624 E.2 Compiling C Programs with Fortran Calls 625 Appendix F: Software on the CD 626 Appendix G: Compression via DWT with Thresholding 635 G.1 More on Thresholding 637 G.2 Wavelet Implementation and Assessment 638 Bibliography 641 Index 651

Rubin H. Landau, Winner of the 2008 Undergraduate Computational Engineering and Sciences Awards, The Krell Institute "Landau and Piez, authors of Computational Physics, have teamed up with Bordeianu to create an expanded work on introductory computational physics. Even more comprehensive than the first book, this volume contains up-to-date treatments of many new topics at the forefront of the field... This volume offers everything needed for a graduate or undergraduate computational physics course."--K.D. Fisher, Choice