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| Preface | |
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| List of Figures | |
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| An Introduction to Matlab� | |
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| The Matlab Software Package | |
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| Matrices and Matrix Operations in Matlab | |
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| Manipulating the Elements of a Matrix | |
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| Transposing Matrices | |
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| Special Matrices | |
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| Generating Matrices and Vectors with Specified Element Values | |
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| Matrix Functions | |
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| Using the Matlab \ Operator for Matrix Division | |
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| Element-by-Element Operations | |
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| Scalar Operations and Functions | |
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| String Variables | |
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| Input and Output in Matlab | |
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| Matlab Graphics | |
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| Three-Dimensional Graphics | |
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| Manipulating Graphics-Handle Graphics | |
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| Scripting in Matlab | |
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| User-Defined Functions in Matlab | |
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| Data Structures in Matlab | |
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| Editing Matlab Scripts | |
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| Some Pitfalls in Matlab | |
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| Faster Calculations in Matlab | |
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| Problems | |
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| Linear Equations and Eigensystems | |
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| Introduction | |
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| linear Equation Systems | |
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| Operators \ and / for Solving Ax = b | |
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| Accuracy of Solutions and Ill-Conditioning | |
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| Elementary Row Operations | |
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| Solution of Ax = b by Gaussian Elimination | |
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| LU Decomposition | |
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| Cholesky Decomposition | |
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| QR Decomposition | |
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| Singular Value Decomposition | |
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| The Pseudo-Inverse | |
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| Over- and Underdetermined Systems | |
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| Iterative Methods | |
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| Sparse Matrices | |
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| The Eigenvalue Problem | |
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| Iterative Methods for Solving the Eigenvalue Problem | |
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| The Matlab Function eig | |
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| Summary | |
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| Problems | |
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| Solution of Nonlinear Equations | |
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| Introduction | |
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| The Nature of Solutions to Nonlinear Equations | |
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| The Bisection Algorithm | |
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| Iterative or Fixed Point Methods | |
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| The Convergence of Iterative Methods | |
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| Ranges for Convergence and Chaotic Behavior | |
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| Newton's Method | |
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| Schroder's Method | |
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| Numerical Problems | |
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| The Matlab Function fzero and Comparative Studies | |
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| Methods for Finding All the Roots of a Polynomial | |
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| Solving Systems of Nonlinear Equations | |
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| Broyden's Method for Solving Nonlinear Equations | |
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| Comparing the Newton and Broyden Methods | |
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| Summary | |
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| Problems | |
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| Differentiation and Integration | |
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| Introduction | |
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| Numerical Differentiation | |
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| Numerical Integration | |
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| Simpson's Rule | |
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| Newton-Cotes Formulae | |
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| Romberg Integration | |
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| Gaussian Integration | |
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| Infinite Ranges of Integration | |
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| Gauss-Chebyshev Formula | |
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| Gauss-Lobatto Integration | |
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| Filon's Sine and Cosine Formulae | |
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| Problems in the Evaluation of Integrals | |
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| Test Integrals | |
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| Repeated Integrals | |
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| Matlab Functions for Double and Triple Integration | |
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| Summary | |
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| Problems | |
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| Solution of Differential Equations | |
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| Introduction | |
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| Euler's Method | |
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| The Problem of Stability | |
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| The Trapezoidal Method | |
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| Runge-Kutta Methods | |
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| Predictor-Corrector Methods | |
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| Hamming's Method and the Use of Error Estimates | |
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| Error Propagation in Differential Equations | |
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| The Stability of Particular Numerical Methods | |
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| Systems of Simultaneous Differential Equations | |
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| The Lorenz Equations | |
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| The Predator-Prey Problem | |
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| Differential Equations Applied to Neural Networks | |
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| Higher-Order Differential Equations | |
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| Stiff Equations | |
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| Special Techniques | |
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| Extrapolation Techniques | |
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| Summary | |
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| Problems | |
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| Boundary Value Problems | |
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| Classification of Second-Order Partial Differential Equations | |
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| The Shooting Method | |
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| The Finite Difference Method | |
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| Two-Point Boundary Value Problems | |
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| Parabolic Partial Differential Equations | |
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| Hyperbolic Partial Differential Equations | |
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| Elliptic Partial Differential Equations | |
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| Summary | |
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| Problems | |
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| Fitting Functions to Data | |
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| Introduction | |
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| Interpolation Using Polynomials | |
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| Interpolation Using Splines | |
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| Fourier Analysis of Discrete Data | |
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| Multiple Regression: Least Squares Criterion | |
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| Diagnostics for Model Improvement | |
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| Analysis of Residuals | |
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| Polynomial Regression | |
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| Fitting General Functions to Data | |
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| Nonlinear Least Squares Regression | |
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| Transforming Data | |
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| Summary | |
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| Problems | |
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| Optimization Methods | |
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| Introduction | |
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| Linear Programming Problems | |
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| Optimizing Single-Variable Functions | |
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| The Conjugate Gradient Method | |
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| Moller's Scaled Conjugate Gradient Method | |
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| Conjugate Gradient Method for Solving Linear Systems | |
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| Genetic Algorithms | |
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| Continuous Genetic Algorithm | |
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| Simulated Annealing | |
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| Constrained Nonlinear Optimization | |
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| The Sequential Unconstrained Minimization Technique | |
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| Summary | |
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| Problems | |
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| Applications of the Symbolic Toolbox | |
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| Introduction to the Symbolic Toolbox | |
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| Symbolic Variables and Expressions | |
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| Variable-Precision Arithmetic in Symbolic Calculations | |
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| Series Expansion and Summation | |
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| Manipulation of Symbolic Matrices | |
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| Symbolic Methods for the Solution of Equations | |
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| Special Functions | |
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| Symbolic Differentiation | |
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| Symbolic Partial Differentiation | |
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| Symbolic Integration | |
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| Symbolic Solution of Ordinary Differential Equations | |
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| The Laplace Transform | |
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| The Z-Transform | |
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| Fourier Transform Methods | |
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| Linking Symbolic and Numerical Processes | |
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| Summary | |
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| Problems | |
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| Appendices | |
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| Matrix Algebra | |
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| Introduction | |
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| Matrices and Vectors | |
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| Some Special Matrices | |
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| Determinants | |
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| Matrix Operations | |
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| Complex Matrices | |
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| Matrix Properties | |
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| Some Matrix Relationships | |
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| Eigenvalues | |
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| Definition of Norms | |
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| Reduced Row Echelon Form | |
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| Differentiating Matrices | |
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| Square Root of a Matrix | |
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| Error Analysis | |
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| Introduction | |
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| Errors in Arithmetic Operations | |
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| Errors in the Solution of Linear Equation Systems | |
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| Solutions to Selected Problems | |
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| Bibliography | |
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| Index | |