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PART Ⅰ Basic Computations and Visualization 3

1 MATLAB Introduction 3

1.1 Vectors and Matrices 3

1.2 Logic，Loops and Iterations 9

1.3 Iteration：The Newton-Raphson Method 13

1.4 Function Calls，Input/Output Interactions and Debugging 18

1.5 Plotting and Importing/Exporting Data 23

2 Linear Systems 31

2.1 Direct Solution Methods for Ax = b 31

2.2 Iterative Solution Methods for Ax = b 35

2.3 Gradient （Steepest） Descent for Ax = b 39

2.4 Eigenvalues，Eigenvectors and Solvability 44

2.5 Eigenvalues and Eigenvectors for Face Recognition 49

2.6 Nonlinear Systems 56

3 Curve Fitting 61

3.1 Least-Square Fitting Methods 61

3.2 Polynomial Fits and Splines 65

3.3 Data Fitting with MATLAB 69

4 Numerical Differentiation and Integration 77

4.1 Numerical Differentiation 77

4.2 Numerical Integration 83

4.3 Implementation of Differentiation and Integration 87

5 Basic Optimization 93

5.1 Unconstrained Optimization （Derivative-Free Methods） 93

5.2 Unconstrained Optimization （Derivative Methods） 99

5.3 Linear Programming 105

5.4 Simplex Method 110

5.5 Genetic Algorithms 113

6 Visualization 119

6.1 Customizing Plots and Basic 2D Plotting 119

6.2 More 2D and 3D Plotting 125

6.3 Movies and Animations 131

PART Ⅱ Differential and Partial Differential Equations 137

7 Initial and Boundary Value Problems of Differential Equations 137

7.1 Initial Value Problems：Euler，Runge-Kutta and Adams Methods 137

7.2 Error Analysis for Time-Stepping Routines 144

7.3 Advanced Time-Stepping Algorithms 149

7.4 Boundary Value Problems：The Shooting Method 153

7.5 Implementation of Shooting and Convergence Studies 160

7.6 Boundary Value Problems：Direct Solve and Relaxation 164

7.7 Implementing MATLAB for Boundary Value Problems 167

7.8 Linear Operators and Computing Spectra 172

8 Finite Difference Methods 180

8.1 Finite Difference Discretization 180

8.2 Advanced Iterative Solution Methods for Ax = b 186

8.3 Fast Poisson Solvers：The Fourier Transform 186

8.4 Comparison of Solution Techniques for Ax = b：Rules of Thumb 190

8.5 Overcoming Computational Difficulties 195

9 Time and Space Stepping Schemes：Method of Lines 200

9.1 Basic Time-Stepping Schemes 200

9.2 Time-Stepping Schemes：Explicit and Implicit Methods 205

9.3 Stability Analysis 209

9.4 Comparison of Time-Stepping Schemes 213

9.5 Operator Splitting Techniques 216

9.6 Optimizing Computational Performance：Rules of Thumb 219

10 Spectral Methods 225

10.1 Fast Fourier Transforms and Cosine/Sine Transform 225

10.2 Chebychev Polynomials and Transform 229

10.3 Spectral Method Implementation 233

10.4 Pseudo-Spectral Techniques with Filtering 235

10.5 Boundary Conditions and the Chebychev Transform 240

10.6 Implementing the Chebychev Transform 244

10.7 Computing Spectra：The Floquet-Fourier-Hill Method 249

11 Finite Element Methods 256

11.1 Finite Element Basis 256

11.2 Discretizing with Finite Elements and Boundaries 261

11.3 MATLAB for Partial Differential Equations 266

11.4 MATLAB Partial Differential Equations Toolbox 271

PART Ⅲ Computational Methods for Data Analysis 279

12 Statistical Methods and Their Applications 279

12.1 Basic Probability Concepts 279

12.2 Random Variables and Statistical Concepts 286

12.3 Hypothesis Testing and Statistical Significance 294

13 Time-Frequency Analysis：Fourier Transforms and Wavelets 301

13.1 Basics of Fourier Series and the Fourier Transform 301

13.2 FFT Application：Radar Detection and Filtering 308

13.3 FFT Application：Radar Detection and Averaging 316

13.4 Time-Frequency Analysis：Windowed Fourier Transforms 322

13.5 Time-Frequency Analysis and Wavelets 328

13.6 Multi-Resolution Analysis and the Wavelet Basis 335

13.7 Spectrograms and the Gabor Transform in MATLAB 340

13.8 MATLAB Filter Design and Wavelet Toolboxes 346

14 Image Processing and Analysis 358

14.1 Basic Concepts and Analysis of Images 358

14.2 Linear Filtering for Image Denoising 364

14.3 Diffusion and Image Processing 369

15 Linear Algebra and Singular Value Decomposition 376

15.1 Basics of the Singular Value Decomposition （SVD） 376

15.2 The SVD in Broader Context 381

15.3 Introduction to Principal Component Analysis （PCA） 387

15.4 Principal Components，Diagonalization and SVD 391

15.5 Principal Components and Proper Orthogonal Modes 395

15.6 Robust PCA 403

16 Independent Component Analysis 412

16.1 The Concept of Independent Components 412

16.2 Image Separation Problem 419

16.3 Image Separation and MATLAB 424

17 Image Recognition：Basics of Machine Learning 431

17.1 Recognizing Dogs and Cats 431

17.2 The SVD and Linear Discrimination Analysis 436

17.3 Implementing Cat/Dog Recognition in MATLAB 445

18 Basics of Compressed Sensing 449

18.1 Beyond Least-Square Fitting：The L1 Norm 449

18.2 Signal Reconstruction and Circumventing Nyquist 456

18.3 Data （Image） Reconstruction from Sparse Sampling 464

19 Dimensionality Reduction for Partial Differential Equations 472

19.1 Modal Expansion Techniques for PDEs 472

19.2 PDE Dynamics in the Right （Best） Basis 478

19.3 Global Normal Forms of Bifurcation Structures in PDEs 482

19.4 The POD Method and Symmetries/Invariances 492

19.5 POD Using Robust PCA 499

20 Dynamic Mode Decomposition 506

20.1 Theory of Dynamic Mode Decomposition （DMD） 506

20.2 Dynamics of DMD Versus POD 510

20.3 Applications of DMD 515

21 Data Assimilation Methods 521

21.1 Theory of Data Assimilation 521

21.2 Data Assimilation，Sampling and Kalman Filtering 526

21.3 Data Assimilation for the Lorenz Equation 529

22 Equation-Free Modeling 537

22.1 Multi-Scale Physics：An Equation-Free Approach 537

22.2 Lifting and Restricting in Equation-Free Computing 542

22.3 Equation-Free Space-Time Dynamics 547

23 Complex Dynamical Systems：Combining Dimensionality Reduction，Compressive Sensing and Machine Learning 551

23.1 Combining Data Methods for Complex Systems 551

23.2 Implementing a Dynamical Systems Library 556

23.3 Flow Around a Cylinder：A Prototypical Example 564

PART Ⅳ Scientific Applications 573

24 Applications of Differential Equations and Boundary Value Problems 573

24.1 Neuroscience and the Hodgkin-Huxley Model 573

24.2 Celestial Mechanics and the Three-Body Problem 577

24.3 Atmospheric Motion and the Lorenz Equations 581

24.4 Quantum Mechanics 585

24.5 Electromagnetic Waveguides 588

25 Applications of Partial Differential Equations 590

25.1 The Wave Equation 590

25.2 Mode-Locked Lasers 593

25.3 Bose-Einstein Condensates 600

25.4 Advection-Diffusion and Atmospheric Dynamics 604

25.5 Introduction to Reaction-Diffusion Systems 611

25.6 Steady State Flow Over an Airfoil 616

26 Applications of Data Analysis 620

26.1 Analyzing Music Scores and the Gabor Transform 620

26.2 Image Denoising through Filtering and Diffusion 622

26.3 Oscillating Mass and Dimensionality Reduction 625

26.4 Music Genre Identification 626

References 629

Index of MATLAB Commands 634

Index 636