Applied Numerical Methods Using MATLAB

by Yang, Won Y.; Cao, Wenwu; Chung, Tae-Sang; Morris, John

• ISBN13: 9780471698333
• ISBN10: 0471698334
• Format: Hardcover
• Copyright: 2005-05-02
• Publisher: Wiley-Interscience

Summary

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.

Author Biography

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.

Table of Contents

Preface	xiii
MATLAB Usage and Computational Errors	1	(70)
Basic Operations of MATLAB	1	(26)
Input/Output of Data from MATLAB Command Window	2	(1)
Input/Output of Data Through Files	2	(2)
Input/Output of Data Using Keyboard	4	(1)
2-D Graphic Input/Output	5	(5)
3-D Graphic Output	10	(1)
Mathematical Functions	10	(5)
Operations on Vectors and Matrices	15	(7)
Random Number Generators	22	(2)
Flow Control	24	(3)
Computer Errors Versus Human Mistakes	27	(10)
IEEE 64-bit Floating-Point Number Representation	28	(3)
Various Kinds of Computing Errors	31	(2)
Absolute/Relative Computing Errors	33	(1)
Error Propagation	33	(1)
Tips for Avoiding Large Errors	34	(3)
Toward Good Program	37	(34)
Nested Computing for Computational Efficiency	37	(2)
Vector Operation Versus Loop Iteration	39	(1)
Iterative Routine Versus Nested Routine	40	(1)
To Avoid Runtime Error	40	(4)
Parameter Sharing via Global Variables	44	(1)
Parameter Passing Through Varargin	45	(1)
Adaptive Input Argument List	46	(1)
Problems	46	(25)
System of Linear Equations	71	(46)
Solution for a System of Linear Equations	72	(7)
The Nonsingular Case (M = N)	72	(1)
The Underdetermined Case (M < N): Minimum-Norm Solution	72	(3)
The Overdetermined Case (M > N): Least-Squares Error Solution	75	(1)
RLSE (Recursive Least-Squares Estimation)	76	(3)
Solving a System of Linear Equations	79	(13)
Gauss Elimination	79	(2)
Partial Pivoting	81	(8)
Gauss--Jordan Elimination	89	(3)
Inverse Matrix	92	(1)
Decomposition (Factorization)	92	(6)
LU Decomposition (Factorization): Triangularization	92	(5)
Other Decomposition (Factorization): Cholesky, QR, and SVD	97	(1)
Iterative Methods to Solve Equations	98	(19)
Jacobi Iteration	98	(2)
Gauss--Seidel Iteration	100	(3)
The Convergence of Jacobi and Gauss--Seidel Iterations	103	(1)
Problems	104	(13)
Interpolation and Curve Fitting	117	(62)
Interpolation by Lagrange Polynomial	117	(2)
Interpolation by Newton Polynomial	119	(5)
Approximation by Chebyshev Polynomial	124	(5)
Pade Approximation by Rational Function	129	(4)
Interpolation by Cubic Spline	133	(6)
Hermite Interpolating Polynomial	139	(2)
Two-dimensional Interpolation	141	(2)
Curve Fitting	143	(7)
Straight Line Fit: A Polynomial Function of First Degree	144	(1)
Polynomial Curve Fit: A Polynomial Function of Higher Degree	145	(4)
Exponential Curve Fit and Other Functions	149	(1)
Fourier Transform	150	(29)
FFT Versus DFT	151	(1)
Physical Meaning of DFT	152	(3)
Interpolation by Using DFS	155	(2)
Problems	157	(22)
Nonlinear Equations	179	(30)
Iterative Method Toward Fixed Point	179	(4)
Bisection Method	183	(2)
False Position or Regula Falsi Method	185	(1)
Newton(--Raphson) Method	186	(3)
Secant Method	189	(2)
Newton Method for a System of Nonlinear Equations	191	(2)
Symbolic Solution for Equations	193	(1)
A Real-World Problem	194	(15)
Problems	197	(12)
Numerical Differentiation/Integration	209	(54)
Difference Approximation for First Derivative	209	(2)
Approximation Error of First Derivative	211	(5)
Difference Approximation for Second and Higher Derivative	216	(4)
Interpolating Polynomial and Numerical Differential	220	(2)
Numerical Integration and Quadrature	222	(4)
Trapezoidal Method and Simpson Method	226	(2)
Recursive Rule and Romberg Integration	228	(3)
Adaptive Quadrature	231	(3)
Gauss Quadrature	234	(7)
Gauss--Legendre Integration	235	(3)
Gauss--Hermite Integration	238	(1)
Gauss--Laguerre Integration	239	(1)
Gauss--Chebyshev Integration	240	(1)
Double Integral	241	(22)
Problems	244	(19)
Ordinary Differential Equations	263	(58)
Euler's Method	263	(3)
Heun's Method: Trapezoidal Method	266	(1)
Runge--Kutta Method	267	(2)
Predictor--Corrector Method	269	(8)
Adams--Bashforth--Moulton Method	269	(4)
Hamming Method	273	(1)
Comparison of Methods	274	(3)
Vector Differential Equations	277	(10)
State Equation	277	(4)
Discretization of LTI State Equation	281	(2)
High-Order Differential Equation to State Equation	283	(1)
Stiff Equation	284	(3)
Boundary Value Problem (BVP)	287	(34)
Shooting Method	287	(3)
Finite Difference Method	290	(3)
Problems	293	(28)
Optimization	321	(50)
Unconstrained Optimization [L-2, Chapter 7]	321	(22)
Golden Search Method	321	(2)
Quadratic Approximation Method	323	(2)
Nelder--Mead Method [W-8]	325	(3)
Steepest Descent Method	328	(2)
Newton Method	330	(2)
Conjugate Gradient Method	332	(2)
Simulated Annealing Method [W-7]	334	(4)
Genetic Algorithm [W-7]	338	(5)
Constrained Optimization [L-2, Chapter 10]	343	(7)
Lagrange Multiplier Method	343	(3)
Penalty Function Method	346	(4)
MATLAB Built-In Routines for Optimization	350	(21)
Unconstrained Optimization	350	(2)
Constrained Optimization	352	(3)
Linear Programming (LP)	355	(2)
Problems	357	(14)
Matrices and Eigenvalues	371	(30)
Eigenvalues and Eigenvectors	371	(2)
Similarity Transformation and Diagonalization	373	(5)
Power Method	378	(3)
Scaled Power Method	378	(2)
Inverse Power Method	380	(1)
Shifted Inverse Power Method	380	(1)
Jacobi Method	381	(4)
Physical Meaning of Eigenvalues/Eigenvectors	385	(4)
Eigenvalue Equations	389	(12)
Problems	390	(11)
Partial Differential Equations	401	(60)
Elliptic PDE	402	(4)
Parabolic PDE	406	(8)
The Explicit Forward Euler Method	406	(1)
The Implicit Backward Euler Method	407	(2)
The Crank--Nicholson Method	409	(3)
Two-Dimensional Parabolic PDE	412	(2)
Hyperbolic PDE	414	(6)
The Explicit Central Difference Method	415	(2)
Two-Dimensional Hyperbolic PDE	417	(3)
Finite Element Method (FEM) for solving PDE	420	(9)
GUI of MATLAB for Solving PDEs: PDETOOL	429	(32)
Basic PDEs Solvable by PDETOOL	430	(1)
The Usage of PDETOOL	431	(4)
Examples of Using PDETOOL to Solve PDEs	435	(9)
Problems	444	(17)
Appendix A. Mean Value Theorem	461	(2)
Appendix B. Matrix Operations/Properties	463	(8)
Appendix C. Differentiation with Respect to a Vector	471	(2)
Appendix D. Laplace Transform	473	(2)
Appendix E. Fourier Transform	475	(2)
Appendix F. Useful Formulas	477	(4)
Appendix G. Symbolic Computation	481	(8)
Appendix H. Sparse Matrices	489	(2)
Appendix I. MATLAB	491	(6)
References	497	(2)
Subject Index	499	(4)
Index for MATLAB Routines	503	(6)
Index for Tables	509