Applied Numerical Methods Using MATLAB

Cover......Page 1
APPLIED NUMERICAL METHODS USING MATLAB......Page 4
CONTENTS......Page 10
Preface......Page 16
1.1 Basic Operations of MATLAB......Page 18
1.1.2 Input/Output of Data Through Files......Page 19
1.1.3 Input/Output of Data Using Keyboard......Page 21
1.1.4 2-D Graphic Input/Output......Page 22
1.1.6 Mathematical Functions......Page 27
1.1.7 Operations on Vectors and Matrices......Page 32
1.1.8 Random Number Generators......Page 39
1.1.9 Flow Control......Page 41
1.2 Computer Errors Versus Human Mistakes......Page 44
1.2.1 IEEE 64-bit Floating-Point Number Representation......Page 45
1.2.2 Various Kinds of Computing Errors......Page 48
1.2.4 Error Propagation......Page 50
1.2.5 Tips for Avoiding Large Errors......Page 51
1.3.1 Nested Computing for Computational Efficiency......Page 54
1.3.2 Vector Operation Versus Loop Iteration......Page 56
1.3.4 To Avoid Runtime Error......Page 57
1.3.5 Parameter Sharing via Global Variables......Page 61
1.3.6 Parameter Passing Through Varargin......Page 62
Problems......Page 63
2 System of Linear Equations......Page 88
2.1.2 The Underdetermined Case (M......Page 89
2.1.3 The Overdetermined Case (M > N): Least-Squares Error Solution......Page 92
2.1.4 RLSE (Recursive Least-Squares Estimation)......Page 93
2.2.1 Gauss Elimination......Page 96
2.2.2 Partial Pivoting......Page 98
2.2.3 Gauss-Jordan Elimination......Page 106
2.4.1 LU Decomposition (Factorization): Triangularization......Page 109
2.4.2 Other Decomposition (Factorization): Cholesky, QR, and SVD......Page 114
2.5.1 Jacobi Iteration......Page 115
2.5.2 Gauss-Seidel Iteration......Page 117
2.5.3 The Convergence of Jacobi and Gauss-Seidel Iterations......Page 120
Problems......Page 121
3.1 Interpolation by Lagrange Polynomial......Page 134
3.2 Interpolation by Newton Polynomial......Page 136
3.3 Approximation by Chebyshev Polynomial......Page 141
3.4 Pade Approximation by Rational Function......Page 146
3.5 Interpolation by Cubic Spline......Page 150
3.6 Hermite Interpolating Polynomial......Page 156
3.7 Two-dimensional Interpolation......Page 158
3.8 Curve Fitting......Page 160
3.8.1 Straight Line Fit: A Polynomial Function of First Degree......Page 161
3.8.2 Polynomial Curve Fit: A Polynomial Function of Higher Degree......Page 162
3.8.3 Exponential Curve Fit and Other Functions......Page 166
3.9 Fourier Transform......Page 167
3.9.1 FFT Versus DFT......Page 168
3.9.2 Physical Meaning of DFT......Page 169
3.9.3 Interpolation by Using DFS......Page 172
Problems......Page 174
4.1 Iterative Method Toward Fixed Point......Page 196
4.2 Bisection Method......Page 200
4.3 False Position or Regula Falsi Method......Page 202
4.4 Newton(-Raphson) Method......Page 203
4.5 Secant Method......Page 206
4.6 Newton Method for a System of Nonlinear Equations......Page 208
4.7 Symbolic Solution for Equations......Page 210
4.8 A Real-World Problem......Page 211
Problems......Page 214
5.1 Difference Approximation for First Derivative......Page 226
5.2 Approximation Error of First Derivative......Page 228
5.3 Difference Approximation for Second and Higher Derivative......Page 233
5.4 Interpolating Polynomial and Numerical Differential......Page 237
5.5 Numerical Integration and Quadrature......Page 239
5.6 Trapezoidal Method and Simpson Method......Page 243
5.7 Recursive Rule and Romberg Integration......Page 245
5.8 Adaptive Quadrature......Page 248
5.9 Gauss Quadrature......Page 251
5.9.1 Gauss-Legendre Integration......Page 252
5.9.2 Gauss-Hermite Integration......Page 255
5.9.3 Gauss-Laguerre Integration......Page 256
5.9.4 Gauss-Chebyshev Integration......Page 257
5.10 Double Integral......Page 258
Problems......Page 261
6.1 Euler\'s Method......Page 280
6.2 Heun\'s Method: Trapezoidal Method......Page 283
6.3 Runge-Kutta Method......Page 284
6.4.1 Adams-Bashforth-Moulton Method......Page 286
6.4.2 Hamming Method......Page 290
6.4.3 Comparison of Methods......Page 291
6.5.1 State Equation......Page 294
6.5.2 Discretization of LTI State Equation......Page 298
6.5.3 High-Order Differential Equation to State Equation......Page 300
6.5.4 Stiff Equation......Page 301
6.6.1 Shooting Method......Page 304
6.6.2 Finite Difference Method......Page 307
Problems......Page 310
7.1.1 Golden Search Method......Page 338
7.1.2 Quadratic Approximation Method......Page 340
7.1.3 Nelder-Mead Method [W-8]......Page 342
7.1.4 Steepest Descent Method......Page 345
7.1.5 Newton Method......Page 347
7.1.6 Conjugate Gradient Method......Page 349
7.1.7 Simulated Annealing Method [W-7]......Page 351
7.1.8 Genetic Algorithm [W-7]......Page 355
7.2.1 Lagrange Multiplier Method......Page 360
7.2.2 Penalty Function Method......Page 363
7.3.1 Unconstrained Optimization......Page 367
7.3.2 Constrained Optimization......Page 369
7.3.3 Linear Programming (LP)......Page 372
Problems......Page 374
8.1 Eigenvalues and Eigenvectors......Page 388
8.2 Similarity Transformation and Diagonalization......Page 390
8.3.1 Scaled Power Method......Page 395
8.3.3 Shifted Inverse Power Method......Page 397
8.4 Jacobi Method......Page 398
8.5 Physical Meaning of Eigenvalues/Eigenvectors......Page 402
8.6 Eigenvalue Equations......Page 406
Problems......Page 407
9 Partial Differential Equations......Page 418
9.1 Elliptic PDE......Page 419
9.2.1 The Explicit Forward Euler Method......Page 423
9.2.2 The Implicit Backward Euler Method......Page 424
9.2.3 The Crank-Nicholson Method......Page 426
9.2.4 Two-Dimensional Parabolic PDE......Page 429
9.3 Hyperbolic PDE......Page 431
9.3.1 The Explicit Central Difference Method......Page 432
9.3.2 Two-Dimensional Hyperbolic PDE......Page 434
9.4 Finite Element Method (FEM) for solving PDE......Page 437
9.5 GUI of MATLAB for Solving PDEs: PDETOOL......Page 446
9.5.1 Basic PDEs Solvable by PDETOOL......Page 447
9.5.2 The Usage of PDETOOL......Page 448
9.5.3 Examples of Using PDETOOL to Solve PDEs......Page 452
Problems......Page 461
Appendix A. Mean Value Theorem......Page 478
Appendix B. Matrix Operations/Properties......Page 480
Appendix C. Differentiation with Respect to a Vector......Page 488
Appendix D. Laplace Transform......Page 490
Appendix E. Fourier Transform......Page 492
Appendix F. Useful Formulas......Page 494
Appendix G. Symbolic Computation......Page 498
Appendix H. Sparse Matrices......Page 506
Appendix I. MATLAB......Page 508
References......Page 514
Subject Index......Page 516
Index for MATLAB Routines......Page 520
Index for Tables......Page 526