## Table of contents for Applied numerical methods using MATLAB / Won Y. Yang, Wenwu Cao, Tae S. Chung.

Bibliographic record and links to related information available from the Library of Congress catalog.

Note: Contents data are machine generated based on pre-publication provided by the publisher. Contents may have variations from the printed book or be incomplete or contain other coding.

```Table of Contents
PREFACE vii
CHAPTER 1: MATLAB USAGE & COMPUTATIONAL ERRORS 1
1.1 BASIC OPERATIONS OF MATLAB	1
1.1-1 Input/Output of Data from MATLAB Command Window	2
1.1-2 Input/Output of Data through Files	2
1.1-3 Input/Output of Data using Keyboard	3
1.1-4 2-D Graphic Input/Output	5
1.1-5 3-D Graphic Output	10
1.1-6 Mathematical Functions	11
1.1-7 Operations on Vectors and Matrices	13
1.1-8 Random Number Generators	21
1.1-9 Flow Control	23
1.2 COMPUTER ERRORS VS. HUMAN MISTAKES	27
1.2-1 IEEE 64-bit Floating-Point Number Representation	27
1.2-2 Various Kinds of Computing Errors	30
1.2-3 Absolute/Relative Computing Errors	32
1.2-4 Error Propagation	32
1.2-5 Tips for Avoiding Large Errors	33
1.3 TOWARD GOOD PROGRAM	36
1.3-1 Nested Computing for Computational Efficiency	37
1.3-2 Vector Operation vs. Loop Iteration	38
1.3-3 Iterative Routine vs. Nested Routine	39
1.3-4 To Avoid Runtime Error	39
1.3-5 Parameter Sharing via Global Variables	42
1.3-6 Parameter Passing through VARARGIN	43
1.3.7 Adaptive Input Argument List	44
Problems ..........................................................................................................47
CHAPTER 2: SYSTEM OF LINEAR EQUATIONS 67
2.1 SOLUTION FOR A SYSTEM OF LINEAR EQUATIONS	68
2.1.1 The Nonsingular Case of M=N	68
2.1.2 The Underdetermined Case - Minimum Solution	68
2.1.3 The Overdetermined Case - Least-Squares Error Solution	71
2.1.4 RLSE (Recursive Least Square Estimation)	72
2.2 SOLVING A SYSTEM OF LINEAR EQUATIONS	75
2.2.1 Gauss Elimination	75
2.2.2 Partial Pivoting	76
2.2.3 Gauss-Jordan Elimination	83
2.3 INVERSE MATRIX	86
2.4 DECOMPOSITION (FACTORIZATION)	86
2.4.1 LU Decomposition (Factorization) - Triangularization	86
2.4.2 Other Decomposition (Factorization) - Cholesky, QR and SVD 	90
2.5 ITERATIVE METHODS TO SOLVE EQUATIONS	92
2.5.1 Jacobi Iteration	92
2.5.2 Gauss-Seidel Iteration	94
2.5.3 Convergence of Jacobi and Gauss-Seidel Iteration	97
Problems ..........................................................................................................99
CHAPTER 3: INTERPOLATION AND CURVE FITTING 109
3.1 INTERPOLATION BY LAGRANGE POLYNOMIAL	109
3.2 INTERPOLATION BY NEWTON POLYNOMIAL	111
3.3 APPROXIMATION BY CHEBYSHEV POLYNOMIAL	115
3.4 PADE APPROXIMATION BY RATIONAL FUNCTION	120
3.5 INTERPOLATION BY CUBIC SPLINE	123
3.6 HERMITE INTERPOLATING POLYNOMIAL	128
3.7 2-DIMENSIONAL INTERPOLATION	130
3.8 CURVE FITTING	132
3.8.1 Straight Line Fit - a polynomial function of 1st degree	133
3.8.2 Polynomial Curve Fit - a polynomial function of higher degree	134
3.8.3 Exponential Curve Fit and Other Functions	138
3.9 FOURIER TRANSFORM	139
3.9.1 FFT vs. DFT	139
3.9.2 Physical Meaning of DFT	141
3.9.3 Interpolation by using DFS	143
Problems ........................................................................................................147

CHAPTER 4: NONLINEAR EQUATIONS 165
4.1 ITERATIVE METHOD TOWARD FIXED POINT	165
4.2 BISECTION METHOD	168
4.3 FALSE POSITION OR REGULA FALSI METHOD	170
4.4 NEWTON(-RAPHSON) METHOD	171
4.5 SECANT METHOD	174
4.6 NEWTON METHOD FOR A SYSTEM OF NONLINEAR EQUATIONS	175
4.7 SYMBOLIC SOLUTION FOR EQUATIONS	178
4.8 A REAL-WORLD PROBLEM	179
Problems ........................................................................................................183
CHAPTER 5: NUMERICAL DIFFERENTIATION/INTEGRATION 193
5.1 DIFFERENCE APPROXIMATION FOR 1ST DERIVATIVE	193
5.2 APPROXIMATION ERROR OF 1ST DERIVATIVE	195
5.3 DIFFERENCE APPROXIMATION FOR 2ND AND HIGHER DERIVATIVE	199
5.4 INTERPOLATING POLYNOMIAL AND NUMERICAL DIFFERENTIAL	203
5.5 NUMERICAL INTEGRATION AND QUADRATURE	204
5.6 TRAPEZOIDAL METHOD AND SIMPSON METHOD	207
5.7 RECURSIVE RULE AND ROMBERG INTEGRATION	208
5.9.1 Gauss-Legendre Integration	214
5.9.2 Gauss-Hermite Integration	217
5.9.2 Gauss-Laguerre Integration	218
5.9.4 Gauss-Chebyshev Integration	219
5.10 DOUBLE INTEGRAL	219
Problems ........................................................................................................223
CHAPTER 6: ORDINARY DIFFERENTIAL EQUATIONS 239
6.1 EULER METHOD	239
6.2 HEUN METHOD	242
6.3 RUNGE-KUTTA METHOD	243
6.4 PREDICTOR-CORRECTOR METHOD	245
6.4.2 Hamming Method	248
6.4.3 Comparison of Methods	249
6.5 VECTOR DIFFERENTIAL EQUATIONS	252
6.5.1 State Equation	252
6.5.2 Discretization of LTI State Equation	255
6.5.3 High-order Differential Equations to State Equations	257
6.5.4 Stiff Equations	258
6.6 BOUNDARY VALUE PROBLEM (BVP)	261
6.6.1 Shooting Method	261
6.6.2 Finite Difference Method	263
Problems ........................................................................................................267
CHAPTER 7: OPTIMIZATION 291
7.1 UNCONSTRAINED OPTIMIZATION	291
7.1.1 Golden Search Method	291
7.1.4 Steepest Descent Method	297
7.1.5 Newton Method	299
7.1.7 Simulated Annealing Method	303
7.1.8 Genetic Algorithm	306
7.2 CONSTRAINED OPTIMIZATION	311
7.2.1 Lagrange Multiplier Method	311
7.2.2 Penalty Function Method	314
7.3 MATLAB BUILT-IN ROUTINES FOR OPTIMIZATION	316
7.3.1 Unconstrained Optimization	317
7.3.2 Constrained Optimization	319
7.3.3 Linear Programing (LP)	321
Problems ........................................................................................................325
CHAPTER 8: MATRICES AND EIGENVALUES 337
8.1 EIGENVALUES AND EIGENVECTORS	337
8.2 SIMILARITY TRANSFORMATION AND DIAGONALIZATION	338
8.3 POWER METHOD	343
8.3.1 Scaled Power Method	343
8.3.2 Inverse Power Method	344
8.3.3 Shifted Inverse Power Method	344
8.4 JACOBI METHOD	346
8.5 PHYSICAL MEANING O F EIGENVALUE/EIGENVECTORS	349
8.6 EIGENVALUE EQUATIONS	352
Problems ........................................................................................................355
CHAPTER 9: PARTIAL DIFFERENTIAL EQUATIONS 363
9.1 ELLIPTIC PDE	364
9.2 PARABOLIC PDE	367
9.2.1 The Explicit Forward Euler Method	368
9.2.2 The Implicit Backward Euler Method	369
9.2.3 The Crank-Nicholson Method	370
9.2.4 Two Dimensional Parabolic PDE	373
9.3 HYPERBOLIC PDE	375
9.3.1 The Explicit Central Difference Method	376
9.3.2 Two Dimensional Hyperbolic PDE	378
9.4 FINITE ELEMENT METHOD (FEM) FOR SOLVING PDE	381
9.5 GUI OF MATLAB FOR SOLVING PDEs - PDETTOOL	390
9.5.1 Basic PEDs Solvable by PDETOOL	390
9.5.2 The Usage of PDETOOL	391
9.5.3 Examples of Using PDETOOL to Solve PDEs	394
Problems ........................................................................................................403
APPENDICIES 417
Appendix A: Mean Value Theorem	417
Appendix B: Matrix Operations/Properties	418
Appendix C: Differentiation w.r.t. A Vector	423
Appendix D: Laplace Trasnform	424
Appendix E: Fourier Trasnform	425
Appendix F: Useful Formulas	427
Appendix G: Symbolic Computation	429
Appendix H: Sparse Matrices	436
Appendix I: MATLAB	438
REFERENCES 442
INDEX 443
```

Library of Congress Subject Headings for this publication:

Numerical analysis -- Data processing.
MATLAB.