The fourth edition of Numerical Methods Using MATLAB® provides a clear and rigorous introduction to a wide range of numerical methods that have practical applications. The authors’ approach is to integrate MATLAB® with numerical analysis in a way which adds clarity to the numerical analysis and develops familiarity with MATLAB®. MATLAB® graphics and numerical output are used extensively to clarify complex problems and give a deeper understanding of their nature.
The text provides an extensive reference providing numerous useful and important numerical algorithms that are implemented in MATLAB® to help researchers analyze a particular outcome. By using MATLAB® it is possible for the readers to tackle some large and difficult problems and deepen and consolidate their understanding of problem solving using numerical methods. Many worked examples are given together with exercises and solutions to illustrate how numerical methods can be used to study problems that have applications in the biosciences, chaos, optimization and many other fields. The text will be a valuable aid to people working in a wide range of fields, such as engineering, science and economics.
Author(s): George Lindfield, John Penny
Edition: 4
Publisher: Academic Press
Year: 2018
Language: English
Commentary: True PDF
Pages: 608
Cover
Numerical Methods: Using MATLAB
Copyright
Dedication
List of Figures
About the Authors
Preface
Acknowledgment
1 An Introduction to Matlab®
1.1 The Software Package Matlab
1.2 Matrices in Matlab
1.3 Manipulating the Elements of a Matrix
1.4 Transposing Matrices
1.5 Special Matrices
1.6 Generating Matrices and Vectors With Specified Element Values
1.7 Matrix Algebra in Matlab
1.8 Matrix Functions
1.9 Using the Matlab Operator for Matrix Division
1.10 Element-by-Element Operations
1.11 Scalar Operations and Functions
1.12 String Variables
1.13 Input and Output in Matlab
1.14 Matlab Graphics
1.15 Three-Dimensional Graphics
1.16 Implicit Graphics
1.17 Manipulating Graphics - Handle Graphics
1.18 Scripting in Matlab
1.19 User-Defined Functions in Matlab
1.20 Data Structures in Matlab
1.21 Editing Matlab Scripts
1.22 Some Pitfalls in Matlab
1.23 Speeding up Calculations in Matlab
1.24 Live Editor
1.25 Summary
1.26 Problems
2 Linear Equations and Eigensystems
2.1 Introduction
2.2 Linear Equation Systems
2.3 Operators \ and / for Solving Ax = b
2.4 Accuracy of Solutions and Ill-Conditioning
2.5 Elementary Row Operations
2.6 Solution of Ax = b by Gaussian Elimination
2.7 LU Decomposition
2.8 Cholesky Decomposition
2.9 QR Decomposition
2.10 Singular Value Decomposition
2.11 The Pseudo-Inverse
2.12 Over- and Under-Determined Systems
2.13 Iterative Methods
2.14 Sparse Matrices
2.15 The Eigenvalue Problem
2.15.1 Eigenvalue Decomposition
2.15.2 Comparing Eigenvalue and Singular Value Decomposition
2.16 Iterative Methods for Solving the Eigenvalue Problem
2.17 Solution of the General Eigenvalue Problem
2.18 The Google `PageRank' Algorithm
2.19 Summary
2.20 Problems
3 Solution of Non-Linear Equations
3.1 Introduction
3.2 The Nature of Solutions to Non-Linear Equations
3.3 The Bisection Algorithm
3.4 Iterative or Fixed Point Methods
3.5 The Convergence of Iterative Methods
3.6 Ranges for Convergence and Chaotic Behavior
3.7 Newton's Method
3.8 Schroder's Method
3.9 Numerical Problems
3.10 The Matlab Function fzero and Comparative Studies
3.11 Methods for Finding all the Roots of a Polynomial
3.11.1 Bairstow's Method
3.11.2 Laguerre's Method
3.12 Solving Systems of Non-Linear Equations
3.13 Broyden's Method for Solving Non-Linear Equations
3.14 Comparing the Newton and Broyden Methods
3.15 Summary
3.16 Problems
4 Differentiation and Integration
4.1 Introduction
4.2 Numerical Differentiation
4.3 Numerical Integration
4.4 Simpson's Rule
4.5 Newton-Cotes Formulae
4.6 Romberg Integration
4.7 Gaussian Integration
4.8 Infinite Ranges of Integration
4.8.1 Gauss-Laguerre Formula
4.8.2 Gauss-Hermite Formula
4.9 Gauss-Chebyshev Formula
4.10 Gauss-Lobatto Integration
4.11 Filon's Sine and Cosine Formulae
4.12 Adaptive Integration
4.13 Problems in the Evaluation of Integrals
4.14 Test Integrals
4.15 Repeated Integrals
4.15.1 Simpson's Rule for Repeated Integrals
4.15.2 Gaussian Integration for Repeated Integrals
4.16 Matlab Functions for Double and Triple Integration
4.17 Summary
4.18 Problems
5 Solution of Differential Equations
5.1 Introduction
5.2 Euler's Method
5.3 The Problem of Stability
5.4 The Trapezoidal Method
5.5 Runge-Kutta Methods
5.6 Predictor-Corrector Methods
5.7 Hamming's Method and the Use of Error Estimates
5.8 Error Propagation in Differential Equations
5.9 The Stability of Particular Numerical Methods
5.10 Systems of Simultaneous Differential Equations
5.10.1 Zeeman Model
5.10.2 The Predator-Prey Problem
5.11 Higher-Order Differential Equations
5.11.1 Conversion to a Set of Simultaneous First-Order Differential Equations
5.11.2 Newmark's Method
5.12 Chaotic Systems
5.12.1 The Lorenz Equations
5.12.2 Duffing's Equation
5.13 Differential Equations Applied to Neural Networks
5.14 Stiff Equations
5.15 Special Techniques
5.16 Extrapolation Techniques
5.17 Simulink
5.18 Summary
5.19 Problems
6 Boundary Value Problems
6.1 Classification of Second-Order Differential Equations
6.2 The Shooting Method
6.3 The Finite Difference Method
6.4 Two-Point Boundary Value Problems
6.5 Parabolic Partial Differential Equations
6.6 Hyperbolic Partial Differential Equations
6.7 Elliptic Partial Differential Equations
6.8 Summary
6.9 Problems
7 Analyzing Data
7.1 Introduction
7.2 Interpolation Using Polynomials
7.3 Interpolation Using Splines
7.4 Multiple Regression: Least Squares Criterion
7.5 Diagnostics for Model Improvement
7.6 Analysis of Residuals
7.7 Polynomial Regression
7.8 Fitting General Functions to Data
7.9 Non-Linear Least Squares Regression
7.10 Transforming Data
7.11 The Kalman Filter
7.12 Principal Component Analysis
7.13 Summary
7.14 Problems
8 Analyzing Data Using Discrete Transforms
8.1 Introduction
8.2 Fourier Analysis of Discrete Data
8.3 The Hilbert Transform
8.4 The Walsh Transforms
8.5 Introduction to Wavelet Analysis
8.6 Discrete Wavelet Transforms
8.6.1 Haar Wavelet
8.6.2 Daubechies Wavelets
8.7 Continuous Wavelet Transforms
8.8 Summary
8.9 Problems
9 Optimization Methods
9.1 Introduction
9.2 Linear Programming Problems
9.3 Optimizing Single-Variable Functions
9.4 The Conjugate Gradient Method
9.5 Moller's Scaled Conjugate Gradient Method
9.6 Conjugate Gradient Method for Solving Linear Systems
9.7 Metaheuristic Methods
9.8 Simulated Annealing
9.9 Evolutionary Algorithm
9.10 Differential Evolution
9.11 Constrained Non-Linear Optimization
9.12 The Sequential Unconstrained Minimization Technique
9.13 Summary
9.14 Problems
10 Applications of the Symbolic Toolbox
10.1 Introduction to the Symbolic Toolbox
10.2 Symbolic Variables and Expressions
10.3 Variable Precision Arithmetic in Symbolic Calculations
10.4 Series Expansion and Summation
10.5 Manipulation of Symbolic Matrices
10.6 Symbolic Methods for the Solution of Equations
10.7 Special Functions
10.8 Symbolic Differentiation
10.9 Symbolic Partial Differentiation
10.10 Symbolic Integration
10.11 Symbolic Solution of Ordinary Differential Equations
10.12 The Laplace Transform
10.13 The Z-Transform
10.14 Fourier Transform Methods
10.15 Linking Symbolic and Numerical Processes
10.16 Summary
10.17 Problems
APPENDIX
A Matrix Algebra
A.1 Introduction
A.2 Matrices and Vectors
A.3 Some Special Matrices
A.4 Determinants
A.5 Matrix Operations
A.6 Complex Matrices
A.7 Matrix Properties
A.8 Some Matrix Relationships
A.9 Eigenvalues
A.10 Definition of Norms
A.11 Reduced Row Echelon Form
A.12 Differentiating Matrices
A.13 Square Root of a Matrix
APPENDIX
B Error Analysis
B.1 Introduction
B.2 Errors in Arithmetic Operations
B.3 Errors in the Solution of Linear Equation Systems
Solutions to Selected Problems
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Bibliography
Index
Back Cover
