Applied Numerical Methods Using MATLAB

APPLIED NUMERICAL METHODS USING MATLAB®......Page 3
 CONTENTS......Page 9
 Preface......Page 15
  1.1 Basic Operations of MATLAB......Page 17
   1.1.2 Input/Output of Data Through Files......Page 18
   1.1.3 Input/Output of Data Using Keyboard......Page 20
   1.1.4 2-D Graphic Input/Output......Page 21
   1.1.6 Mathematical Functions......Page 26
   1.1.7 Operations on Vectors and Matrices......Page 31
   1.1.8 Random Number Generators......Page 38
   1.1.9 Flow Control......Page 40
  1.2 Computer Errors Versus Human Mistakes......Page 43
   1.2.1 IEEE 64-bit Floating-Point Number Representation......Page 44
   1.2.2 Various Kinds of Computing Errors......Page 47
   1.2.4 Error Propagation......Page 49
   1.2.5 Tips for Avoiding Large Errors......Page 50
   1.3.1 Nested Computing for Computational Efficiency......Page 53
   1.3.2 Vector Operation Versus Loop Iteration......Page 55
   1.3.4 To Avoid Runtime Error......Page 56
   1.3.5 Parameter Sharing via Global Variables......Page 60
   1.3.6 Parameter Passing Through Varargin......Page 61
  Problems......Page 62
 2 System of Linear Equations......Page 87
   2.1.2 The Underdetermined Case (M < N): Minimum-Norm Solution......Page 88
   2.1.3 The Overdetermined Case (M > N): Least-Squares Error Solution......Page 91
   2.1.4 RLSE (Recursive Least-Squares Estimation)......Page 92
   2.2.1 Gauss Elimination......Page 95
   2.2.2 Partial Pivoting......Page 97
   2.2.3 Gauss–Jordan Elimination......Page 105
   2.4.1 LU Decomposition (Factorization): Triangularization......Page 108
   2.4.2 Other Decomposition (Factorization): Cholesky, QR, and SVD......Page 113
   2.5.1 Jacobi Iteration......Page 114
   2.5.2 Gauss–Seidel Iteration......Page 116
   2.5.3 The Convergence of Jacobi and Gauss–Seidel Iterations......Page 119
  Problems......Page 120
  3.1 Interpolation by Lagrange Polynomial......Page 133
  3.2 Interpolation by Newton Polynomial......Page 135
  3.3 Approximation by Chebyshev Polynomial......Page 140
  3.4 Pade Approximation by Rational Function......Page 145
  3.5 Interpolation by Cubic Spline......Page 149
  3.6 Hermite Interpolating Polynomial......Page 155
  3.7 Two-dimensional Interpolation......Page 157
  3.8 Curve Fitting......Page 159
   3.8.1 Straight Line Fit: A Polynomial Function of First Degree......Page 160
   3.8.2 Polynomial Curve Fit: A Polynomial Function of Higher Degree......Page 161
   3.8.3 Exponential Curve Fit and Other Functions......Page 165
  3.9 Fourier Transform......Page 166
   3.9.1 FFT Versus DFT......Page 167
   3.9.2 Physical Meaning of DFT......Page 168
   3.9.3 Interpolation by Using DFS......Page 171
  Problems......Page 173
  4.1 Iterative Method Toward Fixed Point......Page 195
  4.2 Bisection Method......Page 199
  4.3 False Position or Regula Falsi Method......Page 201
  4.4 Newton(–Raphson) Method......Page 202
  4.5 Secant Method......Page 205
  4.6 Newton Method for a System of Nonlinear Equations......Page 207
  4.7 Symbolic Solution for Equations......Page 209
  4.8 A Real-World Problem......Page 210
  Problems......Page 213
  5.1 Difference Approximation for First Derivative......Page 225
  5.2 Approximation Error of First Derivative......Page 227
  5.3 Difference Approximation for Second and Higher Derivative......Page 232
  5.4 Interpolating Polynomial and Numerical Differential......Page 236
  5.5 Numerical Integration and Quadrature......Page 238
  5.6 Trapezoidal Method and Simpson Method......Page 242
  5.7 Recursive Rule and Romberg Integration......Page 244
  5.8 Adaptive Quadrature......Page 247
  5.9 Gauss Quadrature......Page 250
   5.9.1 Gauss–Legendre Integration......Page 251
   5.9.2 Gauss–Hermite Integration......Page 254
   5.9.3 Gauss–Laguerre Integration......Page 255
   5.9.4 Gauss–Chebyshev Integration......Page 256
  5.10 Double Integral......Page 257
  Problems......Page 260
  6.1 Euler’s Method......Page 279
  6.2 Heun’s Method: Trapezoidal Method......Page 282
  6.3 Runge–Kutta Method......Page 283
   6.4.1 Adams–Bashforth–Moulton Method......Page 285
   6.4.2 Hamming Method......Page 289
   6.4.3 Comparison of Methods......Page 290
   6.5.1 State Equation......Page 293
   6.5.2 Discretization of LTI State Equation......Page 297
   6.5.3 High-Order Differential Equation to State Equation......Page 299
   6.5.4 Stiff Equation......Page 300
   6.6.1 Shooting Method......Page 303
   6.6.2 Finite Difference Method......Page 306
  Problems......Page 309
   7.1.1 Golden Search Method......Page 337
   7.1.2 Quadratic Approximation Method......Page 339
   7.1.3 Nelder–Mead Method [W-8]......Page 341
   7.1.4 Steepest Descent Method......Page 344
   7.1.5 Newton Method......Page 346
   7.1.6 Conjugate Gradient Method......Page 348
   7.1.7 Simulated Annealing Method [W-7]......Page 350
   7.1.8 Genetic Algorithm [W-7]......Page 354
   7.2.1 Lagrange Multiplier Method......Page 359
   7.2.2 Penalty Function Method......Page 362
   7.3.1 Unconstrained Optimization......Page 366
   7.3.2 Constrained Optimization......Page 368
   7.3.3 Linear Programming (LP)......Page 371
  Problems......Page 373
  8.1 Eigenvalues and Eigenvectors......Page 387
  8.2 Similarity Transformation and Diagonalization......Page 389
   8.3.1 Scaled Power Method......Page 394
   8.3.3 Shifted Inverse Power Method......Page 396
  8.4 Jacobi Method......Page 397
  8.5 Physical Meaning of Eigenvalues/Eigenvectors......Page 401
  8.6 Eigenvalue Equations......Page 405
  Problems......Page 406
 9 Partial Differential Equations......Page 417
  9.1 Elliptic PDE......Page 418
   9.2.1 The Explicit Forward Euler Method......Page 422
   9.2.2 The Implicit Backward Euler Method......Page 423
   9.2.3 The Crank–Nicholson Method......Page 425
   9.2.4 Two-Dimensional Parabolic PDE......Page 428
  9.3 Hyperbolic PDE......Page 430
   9.3.1 The Explicit Central Difference Method......Page 431
   9.3.2 Two-Dimensional Hyperbolic PDE......Page 433
  9.4 Finite Element Method (FEM) for solving PDE......Page 436
  9.5 GUI of MATLAB for Solving PDEs: PDETOOL......Page 445
   9.5.1 Basic PDEs Solvable by PDETOOL......Page 446
   9.5.2 The Usage of PDETOOL......Page 447
   9.5.3 Examples of Using PDETOOL to Solve PDEs......Page 451
  Problems......Page 460
 Appendix A. Mean Value Theorem......Page 477
 Appendix B. Matrix Operations/Properties......Page 479
 Appendix C. Differentiation with Respect to a Vector......Page 487
 Appendix D. Laplace Transform......Page 489
 Appendix E. Fourier Transform......Page 491
 Appendix F. Useful Formulas......Page 493
 Appendix G. Symbolic Computation......Page 497
 Appendix H. Sparse Matrices......Page 505
 Appendix I. MATLAB......Page 507
 References......Page 513
 Subject Index......Page 515
 Index for MATLAB Routines......Page 519
 Index for Tables......Page 525