An introduction to numerical analysis for electrical and computer engineers

AN INTRODUCTION TO NUMERICAL ANALYSIS FOR ELECTRICAL AND COMPUTER ENGINEERS......Page 4
CONTENTS......Page 10
Preface......Page 16
1.1 Introduction......Page 20
1.2 Some Sets......Page 21
1.3 Some Special Mappings: Metrics, Norms, and Inner Products......Page 23
1.3.1 Metrics and Metric Spaces......Page 25
1.3.2 Norms and Normed Spaces......Page 27
1.3.3 Inner Products and Inner Product Spaces......Page 33
1.4 The Discrete Fourier Series (DFS)......Page 44
Appendix 1.A Complex Arithmetic......Page 47
Appendix 1.B Elementary Logic......Page 50
References......Page 51
Problems......Page 52
2.2 Fixed-Point Representations......Page 57
2.3 Floating-Point Representations......Page 61
2.4 Rounding Effects in Dot Product Computation......Page 67
2.5 Machine Epsilon......Page 72
Appendix 2.A Review of Binary Number Codes......Page 73
Problems......Page 78
3.2 Cauchy Sequences and Complete Spaces......Page 82
3.3 Pointwise Convergence and Uniform Convergence......Page 89
3.4 Fourier Series......Page 92
3.5 Taylor Series......Page 97
3.6 Asymptotic Series......Page 116
3.7 More on the Dirichlet Kernel......Page 122
3.A.1 Introduction......Page 126
3.A.2 The Concept of a Discrete Basis......Page 127
3.A.3 Rotating Vectors in the Plane......Page 131
3.A.4 Computing Arctangents......Page 133
3.A.5 Final Remarks......Page 134
Appendix 3.B Mathematical Induction......Page 135
Appendix 3.C Catastrophic Cancellation......Page 136
References......Page 138
Problems......Page 139
4.2 Least-Squares Approximation and Linear Systems......Page 146
4.3 Least-Squares Approximation and Ill-Conditioned Linear Systems......Page 151
4.4 Condition Numbers......Page 154
4.5 LU Decomposition......Page 167
4.6 Least-Squares Problems and QR Decomposition......Page 180
4.7 Iterative Methods for Linear Systems......Page 195
Appendix 4.A Hilbert Matrix Inverses......Page 205
Appendix 4.B SVD and Least Squares......Page 210
References......Page 212
Problems......Page 213
5.2 General Properties of Orthogonal Polynomials......Page 226
5.3 Chebyshev Polynomials......Page 237
5.4 Hermite Polynomials......Page 244
5.5 Legendre Polynomials......Page 248
5.6 An Example of Orthogonal Polynomial Least-Squares Approximation......Page 254
5.7 Uniform Approximation......Page 257
Problems......Page 260
6.1 Introduction......Page 270
6.2 Lagrange Interpolation......Page 271
6.3 Newton Interpolation......Page 276
6.4 Hermite Interpolation......Page 285
6.5 Spline Interpolation......Page 288
References......Page 303
Problems......Page 304
7.1 Introduction......Page 309
7.2 Bisection Method......Page 311
7.3 Fixed-Point Method......Page 315
7.4.1 The Method......Page 324
7.4.2 Rate of Convergence Analysis......Page 328
7.4.3 Breakdown Phenomena......Page 330
7.5.1 Fixed-Point Method......Page 331
7.5.2 Newton–Raphson Method......Page 337
7.6 Chaotic Phenomena and a Cryptography Application......Page 342
References......Page 351
Problems......Page 352
8.2 Problem Statement and Preliminaries......Page 360
8.3 Line Searches......Page 364
8.4 Newton’s Method......Page 372
8.5 Equality Constraints and Lagrange Multipliers......Page 376
Appendix 8.A MATLAB Code for Golden Section Search......Page 381
Problems......Page 383
9.1 Introduction......Page 388
9.2 Trapezoidal Rule......Page 390
9.3 Simpson’s Rule......Page 397
9.4 Gaussian Quadrature......Page 404
9.5 Romberg Integration......Page 412
9.6 Numerical Differentiation......Page 420
Problems......Page 425
10.1 Introduction......Page 434
10.2 First-Order ODEs......Page 440
10.3 Systems of First-Order ODEs......Page 461
10.4 Multistep Methods for ODEs......Page 474
10.4.1 Adams–Bashforth Methods......Page 478
10.4.2 Adams–Moulton Methods......Page 480
10.4.3 Comments on the Adams Families......Page 481
10.5 Variable-Step-Size (Adaptive) Methods for ODEs......Page 483
10.6 Stiff Systems......Page 486
Appendix 10.A MATLAB Code for Example 10.8......Page 488
Appendix 10.B MATLAB Code for Example 10.13......Page 489
References......Page 491
Problems......Page 492
11.2 Review of Eigenvalues and Eigenvectors......Page 499
11.3 The Matrix Exponential......Page 507
11.4 The Power Methods......Page 517
11.5 QR Iterations......Page 527
References......Page 537
Problems......Page 538
12.2 A Brief Overview of Partial Differential Equations......Page 544
12.3.1 The Vibrating String......Page 547
12.3.2 Plane Electromagnetic Waves......Page 553
12.4 The Finite-Difference (FD) Method......Page 564
12.5 The Finite-Difference Time-Domain (FDTD) Method......Page 569
Appendix 12.A MATLAB Code for Example 12.5......Page 576
References......Page 579
Problems......Page 580
13.2 Startup......Page 584
13.3 Some Basic Operators, Operations, and Functions......Page 585
13.4 Working with Polynomials......Page 590
13.5 Loops......Page 591
13.6 Plotting and M-Files......Page 592
References......Page 596
Index......Page 598