# Applied Numerical Methods Using MATLAB

ISBN: 978-0-471-69833-3
528 pages
May 2005

## Description

In recent years, with the introduction of new media products, there has been a shift in the use of programming languages from FORTRAN or C to MATLAB for implementing numerical methods. This book makes use of the powerful MATLAB software to avoid complex derivations, and to teach the fundamental concepts using the software to solve practical problems. Over the years, many textbooks have been written on the subject of numerical methods. Based on their course experience, the authors use a more practical approach and link every method to real engineering and/or science problems. The main benefit is that engineers don't have to know the mathematical theory in order to apply the numerical methods for solving their real-life problems.

An Instructor's Manual presenting detailed solutions to all the problems in the book is available online.
See More

Preface xiii

1 MATLAB Usage and 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 4

1.1.4 2-D Graphic Input/Output 5

1.1.5 3-D Graphic Output 10

1.1.6 Mathematical Functions 10

1.1.7 Operations on Vectors and Matrices 15

1.1.8 Random Number Generators 22

1.1.9 Flow Control 24

1.2 Computer Errors Versus Human Mistakes 27

1.2.1 IEEE 64-bit Floating-Point Number Representation 28

1.2.2 Various Kinds of Computing Errors 31

1.2.3 Absolute/Relative Computing Errors 33

1.2.4 Error Propagation 33

1.2.5 Tips for Avoiding Large Errors 34

1.3 Toward Good Program 37

1.3.1 Nested Computing for Computational Efficiency 37

1.3.2 Vector Operation Versus Loop Iteration 39

1.3.3 Iterative Routine Versus Nested Routine 40

1.3.4 To Avoid Runtime Error 40

1.3.5 Parameter Sharing via Global Variables 44

1.3.6 Parameter Passing Through Varargin 45

1.3.7 Adaptive Input Argument List 46

Problems 46

2 System of Linear Equations 71

2.1 Solution for a System of Linear Equations 72

2.1.1 The Nonsingular Case (M = N) 72

2.1.2 The Underdetermined Case (M <N): Minimum-Norm Solution 72

2.1.3 The Overdetermined Case (M >N): Least-Squares Error Solution 75

2.1.4 RLSE (Recursive Least-Squares Estimation) 76

2.2 Solving a System of Linear Equations 79

2.2.1 Gauss Elimination 79

2.2.2 Partial Pivoting 81

2.2.3 Gauss–Jordan Elimination 89

2.3 Inverse Matrix 92

2.4 Decomposition (Factorization) 92

2.4.1 LU Decomposition (Factorization): Triangularization 92

2.4.2 Other Decomposition (Factorization): Cholesky, QR, and SVD 97

2.5 Iterative Methods to Solve Equations 98

2.5.1 Jacobi Iteration 98

2.5.2 Gauss–Seidel Iteration 100

2.5.3 The Convergence of Jacobi and Gauss–Seidel Iterations 103

Problems 104

3 Interpolation and Curve Fitting 117

3.1 Interpolation by Lagrange Polynomial 117

3.2 Interpolation by Newton Polynomial 119

3.3 Approximation by Chebyshev Polynomial 124

3.4 Pade Approximation by Rational Function 129

3.5 Interpolation by Cubic Spline 133

3.6 Hermite Interpolating Polynomial 139

3.7 Two-dimensional Interpolation 141

3.8 Curve Fitting 143

3.8.1 Straight Line Fit: A Polynomial Function of First Degree 144

3.8.2 Polynomial Curve Fit: A Polynomial Function of Higher Degree 145

3.8.3 Exponential Curve Fit and Other Functions 149

3.9 Fourier Transform 150

3.9.1 FFT Versus DFT 151

3.9.2 Physical Meaning of DFT 152

3.9.3 Interpolation by Using DFS 155

Problems 157

4 Nonlinear Equations 179

4.1 Iterative Method Toward Fixed Point 179

4.2 Bisection Method 183

4.3 False Position or Regula Falsi Method 185

4.4 Newton(–Raphson) Method  186

4.5 Secant Method 189

4.6 Newton Method for a System of Nonlinear Equations 191

4.7 Symbolic Solution for Equations 193

4.8 A Real-World Problem 194

Problems 197

5 Numerical Differentiation/Integration 209

5.1 Difference Approximation for First Derivative 209

5.2 Approximation Error of First Derivative 211

5.3 Difference Approximation for Second and Higher Derivative 216

5.4 Interpolating Polynomial and Numerical Differential 220

5.5 Numerical Integration and Quadrature 222

5.6 Trapezoidal Method and Simpson Method 226

5.7 Recursive Rule and Romberg Integration 228

5.9.1 Gauss–Legendre Integration 235

5.9.2 Gauss–Hermite Integration 238

5.9.3 Gauss–Laguerre Integration 239

5.9.4 Gauss–Chebyshev Integration 240

5.10 Double Integral 241

Problems 244

6 Ordinary Differential Equations 263

6.1 Euler’s Method 263

6.2 Heun’s Method: Trapezoidal Method 266

6.3 Runge–Kutta Method 267

6.4 Predictor–Corrector Method 269

6.4.2 Hamming Method 273

6.4.3 Comparison of Methods 274

6.5 Vector Differential Equations 277

6.5.1 State Equation 277

6.5.2 Discretization of LTI State Equation 281

6.5.3 High-Order Differential Equation to State Equation 283

6.5.4 Stiff Equation 284

6.6 Boundary Value Problem (BVP) 287

6.6.1 Shooting Method 287

6.6.2 Finite Difference Method 290

Problems 293

7 Optimization 321

7.1 Unconstrained Optimization [L-2, Chapter 7] 321

7.1.1 Golden Search Method 321

7.1.4 Steepest Descent Method 328

7.1.5 Newton Method 330

7.1.7 Simulated Annealing Method [W-7] 334

7.1.8 Genetic Algorithm [W-7] 338

7.2 Constrained Optimization [L-2, Chapter 10] 343

7.2.1 Lagrange Multiplier Method 343

7.2.2 Penalty Function Method 346

7.3 MATLAB Built-In Routines for Optimization 350

7.3.1 Unconstrained Optimization 350

7.3.2 Constrained Optimization 352

7.3.3 Linear Programming (LP) 355

Problems 357

8 Matrices and Eigenvalues 371

8.1 Eigenvalues and Eigenvectors 371

8.2 Similarity Transformation and Diagonalization 373

8.3 Power Method 378

8.3.1 Scaled Power Method 378

8.3.2 Inverse Power Method 380

8.3.3 Shifted Inverse Power Method 380

8.4 Jacobi Method 381

8.5 Physical Meaning of Eigenvalues/Eigenvectors 385

8.6 Eigenvalue Equations 389

Problems 390

9 Partial Differential Equations 401

9.1 Elliptic PDE 402

9.2 Parabolic PDE 406

9.2.1 The Explicit Forward Euler Method 406

9.2.2 The Implicit Backward Euler Method 407

9.2.3 The Crank–Nicholson Method 409

9.2.4 Two-Dimensional Parabolic PDE 412

9.3 Hyperbolic PDE 414

9.3.1 The Explicit Central Difference Method 415

9.3.2 Two-Dimensional Hyperbolic PDE 417

9.4 Finite Element Method (FEM) for solving PDE 420

9.5 GUI of MATLAB for Solving PDEs: PDETOOL 429

9.5.1 Basic PDEs Solvable by PDETOOL 430

9.5.2 The Usage of PDETOOL 431

9.5.3 Examples of Using PDETOOL to Solve PDEs 435

Problems 444

Appendix A. Mean Value Theorem 461

Appendix B. Matrix Operations/Properties 463

Appendix C. Differentiation with Respect to a Vector 471

Appendix D. Laplace Transform 473

Appendix E. Fourier Transform 475

Appendix F. Useful Formulas 477

Appendix G. Symbolic Computation 481

Appendix H. Sparse Matrices 489

Appendix I. MATLAB 491

References 497

Subject Index 499

Index for MATLAB Routines 503

Index for Tables 509

See More

## Author Information

WON Y. YANG, PhD, is Professor of Electrical Engineering at Chung-Ang University, Korea.

WENWU CAO, PhD, is Professor of Mathematics and Materials Science at The Pennsylvania State University.

TAE-SANG CHUNG, PhD, is Professor of Electrical Engineering at Chung-Ang University, Korea.

JOHN MORRIS, PhD, is Associate Professor of Computer Science and Electrical and Computer Engineering at The University of Auckland, New Zealand.

See More

## Reviews

"For academic libraries, mathematicians, students, and working professionals…highly recommended." (CHOICE, November 2005)
See More

## Related Websites / Extra

Supplementary SoftwareMATLAB program examples on ftp site
See More