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| Preface | |
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| Introduction | |
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| Objectives and Approach | |
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| Organization of the Book | |
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| Examples | |
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| Programs | |
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| Problems | |
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| Significant Digits, Precision, Accuracy, Errors, and Number Representation | |
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| Software Packages and Libraries | |
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| The Taylor Series and the Taylor Polynomial | |
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| Basic Tools of Numerical Analysis | |
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| Systems of Linear Algebraic Equations | |
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| Eigenproblems | |
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| Roots of Nonlinear Equations | |
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| Polynomial Approximation and Interpolation | |
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| Numercial Differentiation and Difference Formulas | |
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| Numerical Integration | |
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| Summary | |
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| Systems of Linear Algebraic Equations | |
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| Introduction | |
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| Properties of Matrices and Determinants | |
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| Direct Elimination Methods | |
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| LU Factorization | |
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| Tridiagonal Systems of Equations | |
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| Pitfalls of Elimination Methods | |
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| Iterative Methods | |
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| Programs | |
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| Summary | |
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| Exercise Problems | |
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| Eigenproblems | |
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| Introduction | |
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| Mathematical Characteristics of Eigenproblems | |
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| The Power Method | |
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| The Direct Method | |
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| The QR Method | |
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| Eigenvectors | |
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| Other Methods | |
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| Programs | |
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| Summary | |
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| Exercise Problems | |
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| Nonlinear Equations | |
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| Introduction | |
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| General Features of Root Finding | |
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| Closed Domain (Bracketing) Methods | |
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| Open Domain Methods | |
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| Polynomials | |
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| Pitfalls of Root Finding Methods and Other Methods of Root Finding | |
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| Systems of Nonlinear Equations | |
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| Programs | |
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| Summary | |
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| Exercise Problems | |
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| Polynomial Approximation and Interpolation | |
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| Introduction | |
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| Properties of Polynomials | |
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| Direct Fit Polynomials | |
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| Lagrange Polynomials | |
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| Divided Difference Tables and Divided Difference Polynomials | |
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| Difference Tables and Difference Polynomials | |
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| Inverse Interpolation | |
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| Multivariate Approximation | |
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| Cubic Splines | |
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| Least Squares Approximation | |
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| Programs | |
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| Summary | |
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| Exercise Problems | |
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| Numerical Differentiation and Difference Formulas | |
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| Introduction | |
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| Unequally Spaced Data | |
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| Equally Spaced Data | |
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| Taylor Series Approach | |
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| Difference Formulas | |
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| Error Estimation and Extrapolation | |
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| Programs | |
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| Summary | |
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| Exercise Problems | |
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| Numerical Integration | |
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| Introduction | |
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| Direct Fit Polynomials | |
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| Newton-Cotes Formulas | |
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| Extrapolation and Romberg Integration | |
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| Adaptive Integration | |
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| Gaussian Quadrature | |
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| Multiple Integrals | |
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| Programs | |
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| Summary | |
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| Exercise Problems | |
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| Ordinary Differential Equations | |
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| Introduction | |
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| General Features of Ordinary Differential Equations | |
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| Classification of Ordinary Differential Equations | |
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| Classification of Physical Problems | |
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| Initial-Value Ordinary Differential Equations | |
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| Boundary-Value Ordinary Differential Equations | |
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| Summary | |
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| One-Dimensional Initial-Value Ordinary Differential Equations | |
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| Introduction | |
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| General Features of Initial-Value ODEs | |
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| The Taylor Series Method | |
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| The Finite Difference Method | |
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| The First-Order Euler Methods | |
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| Consistency, Order, Stability, and Convergence | |
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| Single-Point Methods | |
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| Extrapolation Methods | |
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| Multipoint Methods | |
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| Summary of Methods and Results | |
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| Nonlinear Implicit Finite Difference Equations | |
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| Higher-Order Ordinary Differential Equations | |
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| Systems of First-Order Ordinary Differential Equations | |
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| Stiff Ordinary Differential Equations | |
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| Programs | |
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| Summary | |
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| Exercise Problems | |
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| One-Dimensional Boundary-Value Ordinary Differential Equations | |
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| Introduction | |
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| General Features of Boundary-Value ODEs | |
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| The Shooting (Initial-Value) Method | |
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| The Equilibrium (Boundary-Value) Method | |
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| Derivative (and Other) Boundary Conditions | |
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| Higher-Order Equilibrium Methods | |
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| The Equilibrium Method for Nonlinear Boundary-Value Problems | |
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| The Equilibrium Method on Nonuniform Grids | |
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| Eigenproblems | |
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| Programs | |
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| Summary | |
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| Exercise Problems | |
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| Partial Differential Equations | |
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| Introduction | |
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| General Features of Partial Differential Equations | |
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| Classification of Partial Differential Equations | |
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| Classification of Physical Problems | |
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| Elliptic Partial Differential Equations | |
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| Parabolic Partial Differential Equations | |
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| Hyperbolic Partial Differential Equations | |
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| The Convection-Diffusion Equation | |
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| Initial Values and Boundary Conditions | |
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| Well-Posed Problems | |
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| Summary | |
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| Elliptic Partial Differential Equations | |
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| Introduction | |
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| General Features of Elliptic PDEs | |
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| The Finite Difference Method | |
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| Finite Difference Solution of the Laplace Equation | |
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| Consistency, Order, and Convergence | |
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| Iterative Methods of Solution | |
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| Derivative Boundary Conditions | |
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| Finite Difference Solution of the Poisson Equation | |
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| Higher-Order Methods | |
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| Nonrectangular Domains | |
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| Nonlinear Equations and Three-Dimensional Problems | |
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| The Control Volume Method | |
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| Programs | |
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| Summary | |
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| Exercise Problems | |
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| Parabolic Partial Differential Equations | |
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| Introduction | |
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| General Features of Parabolic PDEs | |
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| The Finite Difference Method | |
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| The Forward-Time Centered-Space (FTCS) Method | |
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| Consistency, Order, Stability, and Convergence | |
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| The Richardson and DuFort-Frankel Methods | |
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| Implicit Methods | |
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| Derivative Boundary Conditions | |
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| Nonlinear Equations and Multidimensional Problems | |
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| The Convection-Diffusion Equation | |
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| Asymptotic Steady State Solution to Propagation Problems | |
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| Programs | |
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| Summary | |
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| Exercise Problems | |
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| Hyperbolic Partial Differential Equations | |
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| Introduction | |
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| General Features of Hyperbolic PDEs | |
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| The Finite Difference Method | |
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| The Forward-Time Centered-Space (FTCS) Method and the Lax Method | |
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| Lax-Wendroff Type Methods | |
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| Upwind Methods | |
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| The Backward-Time Centered-Space (BTCS) Method | |
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| Nonlinear Equations and Multidimensional Problems | |
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| The Wave Equation | |
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| Programs | |
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| Summary | |
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| Exercise Problems | |
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| The Finite Element Method | |
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| Introduction | |
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| The Rayleigh-Ritz, Collocation, and Galerkin Methods | |
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| The Finite Element Method for Boundary Value Problems | |
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| The Finite Element Method for the Laplace (Poisson) Equation | |
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| The Finite Element Method for the Diffusion Equation | |
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| Programs | |
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| Summary | |
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| Exercise Problems | |
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| References | |
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| Answers to Selected Problems | |
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| Index | |