Numerical Computing with MATLAB is a textbook for an introductory course in numerical methods, Matlab, and technical computing. The emphasis is on informed use of mathematical software. We want you learn enough about the mathematical functions in Matlab that you will be able to use them correctly, appreciate their limitations, and modify them when necessary to suit your own needs.

Preface

1 Introduction to MATLAB 1

1.1 The Golden Ratio 2

1.2 Fibonacci Numbers 8

1.3 Fractal Fern 13

1.4 Magic Squares 19

1.5 Cryptography 26

1.6 The 3n+l Sequence 31

1.7 Floating-Point Arithmetic 35

1.8 Further Reading 43

Exercises 43

2 Linear Equations 55

2.1Solving Linear Systems 55

2.2 The MATLAB Backslash Operator 56

2.3 A 3-by-3 Example 56

2.4 Permutation and Triangular Matrices 58

2.5 LU Factorization 59

2.6 Why Is Pivoting Necessary? 61

2.7 lutx, bslashtx, lugui 62

2.8 Effect of Roundoff Errors 65

2.9 Norms and Condition Numbers 68

2.10 Sparse Matrices and Band Matrices 75

2.11 PageRank and Markov Chains 77

2.12 Further Reading 85

Exercises 85

3 Interpolation 97

3.1The Interpolating Polynomial 97

3.2 Piecewise Linear Interpolation 102

3.3 Piecewise Cubic Hermite Interpolation 104

3.4 Shape-Preserving Piecewise Cubic 104

3.5 Cubic Spline 106

3.6 pchiptx, splinetx 110

3.7 interpgui 113

Exercises 115

4 Zeros and Roots 123

4.1 Bisection 123

4.2 Newton's Method 125

4.3 A Perverse Example 127

4.4 Secant Method 128

4.5 Inverse Quadratic Interpolation 129

4.6 Zeroin 130

4.7 fzerotx 130

4.8 feerogui 135

4.9 Value Finding and Reverse Interpolation 137

4.10 Optimization and finintx 138

Exercises 141

5Least Squares 147

5.1 Models and Curve Fitting 147

5.2 Norms 149

5.3 censnsgui 150

5.4 Householder Reflections 152

5.5 The QR Factorization 153

5.6 Pseudoinverse 158

5.7 Rank Deficiency 160

5.8 Separable Least Squares 162

5.9 Further Reading 165

Exercises 165

6 Quadrature 173

6.1Adaptive Quadrature 173

6.2 Basic Quadrature Rules 174

6.3 quadtx, quadgui 177

6.4 Specifying Integrands 179

6.5 Performance 181

6.6 Integrating Discrete Data 183

6.7 Further Reading 185

Exercises 185

7 Ordinary Differential Equations 193

7.1 Integrating DifEerential Equations 193

7.2 Systems of Equations 194

7.3 Linearized Differential Equations 195

7.4 Single-Step Methods 197

7.5 The BS23 Algorithm 200

7.6 ode23tx 202

7.7 Examples 205

7.8 Lorenz Attractor 208

7.9 Stiffness 211

7.10 Events 215

7.11 Multistep Methods 218

7.12 The MATLAB ODE Solvers 219

7.13 Errors 220

7.14 Performance 224

7.15 Further Reading 226

Exercises 226

8 Fourier Analysis 245

8.1Touch-Tone Dialing 245

8.2 Finite Fourier Transform 249

8.3 fftgui 250

8.4 Sunspots 254

8.5 Periodic Time Series 257

8.6 Fast Finite Fourier Transform 257

8.7 ffttx 259

8.8 fftmatrix 260

8.9 Other Fourier Transforms and Series 261

8.10 Further Reading 263

Exercises 263

9 Random Numbers 265

9.1Pseudorandom Numbers 265

9.2 Uniform Distribution 265

9.3 Normal Distribution 269

9.4 randtx, randntx 271

9.5 Twister 273

Exercises 274

10 Eigenvalues and Singular Values 279

10.1 Eigenvalue and Singular Value Decompositions 279

10.2 A Small Example 282

10.3 eigshow 283

10.4 Characteristic Polynomial 285

10.5 Symmetric and Hermitian Matrices 286

10.6 Eigenvalue Sensitivity and Accuracy 287

10.7 Singular Value Sensitivity and Accuracy 293

10.8 Jordan and Schur Forms 294

10.9 The QR Algorithm 295

10.10 eigsvdgui 297

10.11 Principal Components 299

10.12 Circle Generator 304

10.13 Further Reading 308

Exercises 309

11 Partial Differential Equations 317

11.1 Model Problems 317

11.2 Finite Difference Methods 318

11.3 Matrix Representation 320

11.4 Numerical Stability 323

11.5 The L-Shaped Membrane 324

Exercises 329

Bibliography 337

Index 342

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