Оглавление
Won Y. Yang. Applied Numerical Methods Using MATLAB
Table of Contents
List of Tables
List of Illustrations
Guide
Pages
Applied Numerical Methods Using MATLAB®
Preface
Acknowledgments
About the Companion Website
1 MATLAB Usage and Computational Errors. Chapter Outline
1.1 Basic Operations of MATLAB
1.1.1 Input/Output of Data from MATLAB Command Window
1.1.2 Input/Output of Data Through Files
1.1.3 Input/Output of Data Using Keyboard
1.1.4 Two‐Dimensional (2D) Graphic Input/Output
1.1.5 Three Dimensional (3D) Graphic Output
1.1.6 Mathematical Functions
1.1.7 Operations on Vectors and Matrices
1.1.8 Random Number Generators
1.1.8.1 Random Number Having Uniform Distribution
1.1.8.2 Random Number with Normal (Gaussian) Distribution
1.1.9 Flow Control. 1.1.9.1if‐endandswitch‐case‐endStatements
1.1.9.2for index=i_0:increment:i_last‐endLoop
1.1.9.3whileLoop
1.2 Computer Errors vs. Human Mistakes
1.2.1 IEEE 64‐bit Floating‐Point Number Representation
1.2.2 Various Kinds of Computing Errors
1.2.3 Absolute/Relative Computing Errors
1.2.4 Error Propagation
1.2.5 Tips for Avoiding Large Errors
1.3 Toward Good Program
1.3.1 Nested Computing for Computational Efficiency
1.3.2 Vector Operation vs. Loop Iteration
1.3.3 Iterative Routine vs. Recursive Routine
1.3.4 To Avoid Runtime Error
1.3.5 Parameter Sharing via GLOBAL Variables
1.3.6 Parameter Passing Through VARARGIN
1.3.7 Adaptive Input Argument List
Problems
2 System of Linear Equations. Chapter Outline
2.1 Solution for a System of Linear Equations
2.1.1 The Nonsingular Case (M = N)
2.1.2 The Underdetermined Case (M<N): Minimum‐norm Solution
2.1.3 The Overdetermined Case (M>N): Least‐squares Error Solution
2.1.4 Recursive Least‐Squares Estimation (RLSE)
2.2 Solving a System of Linear Equations
2.2.1 Gauss(ian) Elimination
2.2.2 Partial Pivoting
2.2.3 Gauss‐Jordan Elimination
2.3 Inverse Matrix
2.4 Decomposition (Factorization)
2.4.1 LU Decomposition (Factorization) – Triangularization
2.4.2 Other Decomposition (Factorization) – Cholesky, QR and SVD
2.5 Iterative Methods to Solve Equations
2.5.1 Jacobi Iteration
2.5.2 Gauss‐Seidel Iteration
2.5.3 The Convergence of Jacobi and Gauss‐Seidel Iterations
Problems
3 Interpolation and Curve Fitting. Chapter Outline
3.1 Interpolation by Lagrange Polynomial
3.2 Interpolation by Newton Polynomial
3.3 Approximation by Chebyshev Polynomial
3.4 Pade Approximation by Rational Function
3.5 Interpolation by Cubic Spline
3.6 Hermite Interpolating Polynomial
3.7 Two‐Dimensional Interpolation
3.8 Curve Fitting
3.8.1 Straight‐Line Fit – A Polynomial Function of Degree 1
3.8.2 Polynomial Curve Fit – A Polynomial Function of Higher Degree
3.8.3 Exponential Curve Fit and Other Functions
3.9 Fourier Transform
3.9.1 FFT vs. DFT
3.9.2 Physical Meaning of DFT
3.9.3 Interpolation by Using DFS
Problems
Note
4 Nonlinear Equations. Chapter Outline
4.1 Iterative Method toward Fixed Point
4.2 Bisection Method
4.3 False Position or Regula Falsi Method
4.4 Newton(‐Raphson) Method
4.5 Secant Method
4.6 Newton Method for a System of Nonlinear Equations
4.7 Bairstow's Method for a Polynomial Equation
4.8 Symbolic Solution for Equations
4.9 Real‐World Problems
Problems
5 Numerical Differentiation/Integration. Chapter Outline
5.1 Difference Approximation for the First Derivative
Richardson's Extrapolation
5.2 Approximation Error of the First Derivative
5.3 Difference Approximation for Second and Higher Derivative
5.4 Interpolating Polynomial and Numerical Differential
5.5 Numerical Integration and Quadrature
5.6 Trapezoidal Method and Simpson Method
5.7 Recursive Rule and Romberg Integration
5.8 Adaptive Quadrature
5.9 Gauss Quadrature
5.9.1 Gauss‐Legendre Integration
5.9.2 Gauss‐Hermite Integration
5.9.3 Gauss‐Laguerre Integration
5.9.4 Gauss‐Chebyshev Integration
5.10 Double Integral
5.11 Integration Involving PWL Function
Problems
6 Ordinary Differential Equations. Chapter Outline
6.1 Euler's Method
6.2 Heun's Method – Trapezoidal Method
6.3 Runge‐Kutta Method
6.4 Predictor‐Corrector Method
6.4.1 Adams‐Bashforth‐Moulton Method
Adams‐Bashforth‐Moulton Method with Modification Formulas
6.4.2 Hamming Method
Hamming Method with Modification Formulas
6.4.3 Comparison of Methods
6.5 Vector Differential Equations
6.5.1 State Equation
6.5.2 Discretization of LTI State Equation
6.5.3 High‐order Differential Equation to State Equation
6.5.4 Stiff Equation
6.6 Boundary Value Problem (BVP)
6.6.1 Shooting Method
6.6.2 Finite Difference Method
Problems
7 Optimization. Chapter Outline
7.1 Unconstrained Optimization. 7.1.1 Golden Search Method
Golden Search Procedure
7.1.2 Quadratic Approximation Method
7.1.3 Nelder‐Mead Method
Nelder‐Mead Algorithm
7.1.4 Steepest Descent Method
Steepest Descent Algorithm
7.1.5 Newton Method
7.1.6 Conjugate Gradient Method
Conjugate Gradient Algorithm
7.1.7 Simulated Annealing
7.1.8 Genetic Algorithm. Hybrid Genetic Algorithm
7.2 Constrained Optimization
7.2.1 Lagrange Multiplier Method
7.2.2 Penalty Function Method
7.3 MATLAB Built‐In Functions for Optimization
7.3.1 Unconstrained Optimization
7.3.2 Constrained Optimization
7.3.3 Linear Programming (LP)
7.3.4 Mixed Integer Linear Programming (MILP)
7.4 Neural Network[K‐1]
7.5 Adaptive Filter[Y‐3]
7.6 Recursive Least Square Estimation (RLSE)[Y‐3]
Problems
8 Matrices and Eigenvalues. Chapter Outline
8.1 Eigenvalues and Eigenvectors
8.2 Similarity Transformation and Diagonalization
8.3 Power Method
8.3.1 Scaled Power Method
SCALED POWER METHOD
8.3.2 Inverse Power Method
8.3.3 Shifted Inverse Power Method
8.4 Jacobi Method
8.5 Gram‐Schmidt Orthonormalization and QR Decomposition
8.6 Physical Meaning of Eigenvalues/Eigenvectors
8.7 Differential Equations with Eigenvectors
8.8 DoA Estimation with Eigenvectors[Y-3]
Problems
9 Partial Differential Equations. Chapter Outline
9.1 Elliptic PDE
9.2 Parabolic PDE
9.2.1 The Explicit Forward Euler Method
9.2.2 The Implicit Backward Euler Method
9.2.3 The Crank‐Nicholson Method
9.2.4 Using the MATLAB function ‘pdepe()’
9.2.5 Two‐Dimensional Parabolic PDEs
9.3 Hyperbolic PDES
9.3.1 The Explicit Central Difference Method
9.3.2 Two‐Dimensional Hyperbolic PDEs
9.4 Finite Element Method (FEM) for Solving PDE
9.5 GUI of MATLAB for Solving PDES – PDEtool
9.5.1 Basic PDEs Solvable by PDEtool
9.5.2 The Usage of PDEtool
9.5.3 Examples of Using PDEtool to Solve PDEs
Problems
Appendix A Mean Value Theorem. Theorem A.1 Mean Value Theorem
Theorem A.2 Taylor Series Theorem
Appendix B Matrix Operations/Properties. Chapter Outline
B.1 Addition and Subtraction
B.2 Multiplication
B.3 Determinant
B.4 Eigenvalues and Eigenvectors of a Matrix1
B.5 Inverse Matrix
B.6 Symmetric/Hermitian Matrix
B.7 Orthogonal/Unitary Matrix
B.8 Permutation Matrix
B.9 Rank
B.10 Row Space and Null Space
B.11 Row Echelon Form
B.12 Positive Definiteness
B.13 Scalar (Dot) Product and Vector (Cross) Product
B.14 Matrix Inversion Lemma
Note
Appendix C Differentiation W.R.T. A Vector
Appendix D. Laplace Transform
Appendix E Fourier Transform
Appendix F Useful Formulas
Appendix G Symbolic Computation. Chapter Outline
G.1 How to Declare Symbolic Variables and Handle Symbolic Expressions
G.2 Calculus. G.2.1 Symbolic Summation
G.2.2 Limits
G.2.3 Differentiation
G.2.4 Integration
G.2.5 Taylor Series Expansion
G.3 Linear Algebra
G.4 Solving Algebraic Equations
G.5 Solving Differential Equations
Appendix H Sparse Matrices
Appendix I MATLAB
References
Index
Index for MATLAB Functions
Index for Tables
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