Numerical Methods

Dedication

Preface

List of Figures

An Introduction to Matlab^®

1.1 The MATLAB Software Package

1.2 Matrices and Matrix Operations 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 Functions

1.8 Using the MATLAB\ Operator for Matrix Division

1.9 Element-by-Element Operations

1.10 Scalar Operations and Functions

1.11 String Variables

1.12 Input and Output in MATLAB

1.13 MATLAB Graphics

1.14 Three-Dimensional Graphics

1.15 Manipulating Graphics—Handle Graphics

1.16 Scripting in MATLAB

1.17 User-Defined Functions in MATLAB

1.18 Data Structures in MATLAB

1.19 Editing MATLAB Scripts

1.20 Some Pitfalls in MATLAB

1.21 Faster Calculations in MATLAB

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 Underdetermined Systems

2.13 Iterative Methods

2.14 Sparse Matrices

2.15 The Eigenvalue Problem

2.16 Iterative Methods for Solving the Eigenvalue Problem

2.17 The MATLAB Function eig

2.18 Summary

3. Solution of Nonlinear Equations

3.1 Introduction

3.2 The Nature of Solutions to Nonlinear 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

3.12 Solving Systems of Nonlinear Equations

3.13 Broyden’s Method for Solving Nonlinear Equations

3.14 Comparing the Newton and Broyden Methods

3.15 Summary

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

4.9 Gauss–Chebyshev Formula

4.10 Gauss–Lobatto Integration

4.11 Filon’s Sine and Cosine Formulae

4.12 Problems in the Evaluation of Integrals

4.13 Test Integrals

4.14 Repeated Integrals

4.15 MATLAB Functions for Double and Triple Integration

4.16 Summary

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

5.11 The Lorenz Equations

5.12 The Predator–Prey Problem

5.13 Differential Equations Applied to Neural Networks

5.14 Higher-Order Differential Equations

5.15 Stiff Equations

5.16 Special Techniques

5.17 Extrapolation Techniques

5.18 Summary

6. Boundary Value Problems

6.1 Classification of Second-Order Partial 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

7. Fitting Functions to Data

7.1 Introduction

7.2 Interpolation Using Polynomials

7.3 Interpolation Using Splines

7.4 Fourier Analysis of Discrete Data

7.5 Multiple Regression: Least Squares Criterion

7.6 Diagnostics for Model Improvement

7.7 Analysis of Residuals

7.8 Polynomial Regression

7.9 Fitting General Functions to Data

7.10 Nonlinear Least Squares Regression

7.11 Transforming Data

7.12 Summary

8. Optimization Methods

8.1 Introduction

8.2 Linear Programming Problems

8.3 Optimizing Single-Variable Functions

8.4 The Conjugate Gradient Method

8.5 Moller’s Scaled Conjugate Gradient Method

8.6 Conjugate Gradient Method for Solving Linear Systems

8.7 Genetic Algorithms

8.8 Continuous Genetic Algorithm

8.9 Simulated Annealing

8.10 Constrained Nonlinear Optimization

8.11 The Sequential Unconstrained Minimization Technique

8.12 Summary

9. Applications of the Symbolic Toolbox

9.1 Introduction to the Symbolic Toolbox

9.2 Symbolic Variables and Expressions

9.3 Variable-Precision Arithmetic in Symbolic Calculations

9.4 Series Expansion and Summation

9.5 Manipulation of Symbolic Matrices

9.6 Symbolic Methods for the Solution of Equations

9.7 Special Functions

9.8 Symbolic Differentiation

9.9 Symbolic Partial Differentiation

9.10 Symbolic Integration

9.11 Symbolic Solution of Ordinary Differential Equations

9.12 The Laplace Transform

9.13 The Z-Transform

9.14 Fourier Transform Methods

9.15 Linking Symbolic and Numerical Processes

9.16 Summary

Appendices

A: Matrix Algebra

B: Error Analysis

Solutions to Selected Problems

Bibliography

Index