Preface xiii
1 Functional Analysis Ideas 1
1.1 Introduction 1
1.2 Some Sets 2
1.3 Some Special Mappings: Metrics, Norms, and Inner Products 4
1.3.1 Metrics and Metric Spaces 6
1.3.2 Norms and Normed Spaces 8
1.3.3 Inner Products and Inner Product Spaces 14
1.4 The Discrete Fourier Series (DFS) 25
Appendix 1.A Complex Arithmetic 28
Appendix 1.B Elementary Logic 31
References 32
Problems 33
2 Number Representations 38
2.1 Introduction 38
2.2 Fixed-Point Representations 38
2.3 Floating-Point Representations 42
2.4 Rounding Effects in Dot Product Computation 48
2.5 Machine Epsilon 53
Appendix 2.A Review of Binary Number Codes 54
References 59
Problems 59
3 Sequences and Series 63
3.1 Introduction 63
3.2 Cauchy Sequences and Complete Spaces 63
3.3 Pointwise Convergence and Uniform Convergence 70
3.4 Fourier Series 73
3.5 Taylor Series 78
3.6 Asymptotic Series 97
3.7 More on the Dirichlet Kernel 103
3.8 Final Remarks 107
Appendix 3.A COordinate Rotation DIgital Computing (CORDIC) 107
3.A.1 Introduction 107
3.A.2 The Concept of a Discrete Basis 108
3.A.3 Rotating Vectors in the Plane 112
3.A.4 Computing Arctangents 114
3.A.5 Final Remarks 115
Appendix 3.B Mathematical Induction 116
Appendix 3.C Catastrophic Cancellation 117
References 119
Problems 120
4 Linear Systems of Equations 127
4.1 Introduction 127
4.2 Least-Squares Approximation and Linear Systems 127
4.3 Least-Squares Approximation and Ill-Conditioned Linear Systems 132
4.4 Condition Numbers 135
4.5 LU Decomposition 148
4.6 Least-Squares Problems and QR Decomposition 161
4.7 Iterative Methods for Linear Systems 176
4.8 Final Remarks 186
Appendix 4.A Hilbert Matrix Inverses 186
Appendix 4.B SVD and Least Squares 191
References 193
Problems 194
5 Orthogonal Polynomials 207
5.1 Introduction 207
5.2 General Properties of Orthogonal Polynomials 207
5.3 Chebyshev Polynomials 218
5.4 Hermite Polynomials 225
5.5 Legendre Polynomials 229
5.6 An Example of Orthogonal Polynomial Least-Squares Approximation 235
5.7 Uniform Approximation 238
References 241
Problems 241
6 Interpolation 251
6.1 Introduction 251
6.2 Lagrange Interpolation 252
6.3 Newton Interpolation 257
6.4 Hermite Interpolation 266
6.5 Spline Interpolation 269
References 284
Problems 285
7 Nonlinear Systems of Equations 290
7.1 Introduction 290
7.2 Bisection Method 292
7.3 Fixed-Point Method 296
7.4 Newton-Raphson Method 305
7.4.1 The Method 305
7.4.2 Rate of Convergence Analysis 309
7.4.3 Breakdown Phenomena 311
7.5 Systems of Nonlinear Equations 312
7.5.1 Fixed-Point Method 312
7.5.2 Newton-Raphson Method 318
7.6 Chaotic Phenomena and a Cryptography Application 323
References 332
Problems 333
8 Unconstrained Optimization 341
8.1 Introduction 341
8.2 Problem Statement and Preliminaries 341
8.3 Line Searches 345
8.4 Newton's Method 353
8.5 Equality Constraints and Lagrange Multipliers 357
Appendix 8.A MATLAB Code for Golden Section Search 362
References 364
Problems 364
9 Numerical Integration and Differentiation 369
9.1 Introduction 369
9.2 Trapezoidal Rule 371
9.3 Simpson's Rule 378
9.4 Gaussian Quadrature 385
9.5 Romberg Integration 393
9.6 Numerical Differentiation 401
References 406
Problems 406
10 Numerical Solution of Ordinary Differential Equations 415
10.1 Introduction 415
10.2 First-Order ODEs 421
10.3 Systems of First-Order ODEs 442
10.4 Multistep Methods for ODEs 455
10.4.1 Adams-Bashforth Methods 459
10.4.2 Adams-Moulton Methods 461
10.4.3 Comments on the Adams Families 462
10.5 Variable-Step-Size (Adaptive) Methods for ODEs 464
10.6 Stiff Systems 467
10.7 Final Remarks 469
Appendix 10.A MATLAB Code for Example 10.8 469
Appendix 10.B MATLAB Code for Example 10.13 470
References 472
Problems 473
11 Numerical Methods for Eigenproblems 480
11.1 Introduction 480
11.2 Review of Eigenvalues and Eigenvectors 480
11.3 The Matrix Exponential 488
11.4 The Power Methods 498
11.5 QR Iterations 508
References 518
Problems 519
12 Numerical Solution of Partial Differential Equations 525
12.1 Introduction 525
12.2 A Brief Overview of Partial Differential Equations 525
12.3 Applications of Hyperbolic PDEs 528
12.3.1 The Vibrating String 528
12.3.2 Plane Electromagnetic Waves 534
12.4 The Finite-Difference (FD) Method 545
12.5 The Finite-Difference Time-Domain (FDTD) Method 550
Appendix 12.A MATLAB Code for Example 12.5 557
References 560
Problems 561
13 An Introduction to MATLAB 565
13.1 Introduction 565
13.2 Startup 565
13.3 Some Basic Operators, Operations, and Functions 566
13.4 Working with Polynomials 571
13.5 Loops 572
13.6 Plotting and M-Files 573
References 577
Index 579
