First Course in Numerical Analysis

Preface to the Dover Edition; Preface to the Second Edition; Notation Chapter 1. Introduction and Preliminaries 1.1 What Is Numerical Analysis? 1.2 Sources of Error 1.3 Error Definitions and Related Matters 1.3-1 Significant digits; 1.3-2 Error in functional Evaluation; 1.3-3 Norms 1.4 Roundoff Error 1.4-1 The Probabilistic Approach to Roundoff: A Particular Example 1.5 Computer Arithmetic 1.5-1 Fixed-Point Arithmetic; 1.5-2 Floating-Point Numbers; 1.5-3 Floating-Point Arithmetic; 1.5-4 Overflow and Underflow; 1.5-5 Single- and Double-Precision Arithmetic 1.6 Error Analysis 1.6-1 Backward Error Analysis 1.7 Condition and Stability Bibliographic Notes; Bibliography; Problems Chapter 2. Approximation and Algorithms 2.1 Approximation 2.1-1 Classes of Approximating Functions; 2.1-2 Types of Approximations; 2.1-3 The Case for Polynomial Approximation 2.2 Numerical Algorithms 2.3 Functionals and Error Analysis 2.4 The Method of Undetermined Coefficients Bibliographic Notes; Bibliography; Problems Chapter 3. Interpolation 3.1 Introduction 3.2 Lagrangian Interpolation 3.3 Interpolation at Equal Intervals 3.3-1 Lagrangian Interpolation at Equal Intervals; 3.3-2 Finite Differences 3.4 The use of Interpolation Formulas 3.5 Iterated Interpolation 3.6 Inverse Interpolation 3.7 Hermite Interpolation 3.8 Spline Interpolation 3.9 Other Methods of Interpolation; Extrapolation Bibliographic Notes; Bibliography; Problems Chapter 4. Numerical Differentiation, Numerical Quadrature, and Summation 4.1 Numerical Differentiation of Data 4.2 Numerical Differentiation of Functions 4.3 Numerical Quadrature: The General Problem 4.3-1 Numerical Integration of Data 4.4 Guassian Quadrature 4.5 Weight Functions 4.6 Orthogonal Polynomials and Gaussian Quadrature 4.7 Gaussian Quadrature over Infinite Intervals 4.8 Particular Gaussian Quadrature Formulas 4.8-1 Gauss-Jacobi Quadrature; 4.8-2 Gauss-Chebyshev Quadrature; 4.8-3 Singular Integrals 4.9 Composite Quadrature Formulas 4.10 Newton-Cotes Quadrature Formulas 4.10-1 Composite Newton-Cotes Formulas; 4.10-2 Romberg Integration 4.11 Adaptive Integration 4.12 Choosing a Quadrature Formula 4.13 Summation 4.13-1 The Euler-Maclaurin Sum Formula; 4.13-2 Summation of Rational Functions; Factorial Functions; 4.13-3 The Euler Transformation Bibliographic Notes; Bibliography; Problems Chapter 5. The Numerical Solution of Ordinary Differential Equations 5.1 Statement of the Problem 5.2 Numerical Integration Methods 5.2-1 The Method of Undetermined Coefficients 5.3 Truncation Error in Numerical Integration Methods 5.4 Stability of Numerical Integration Methods 5.4-1 Convergence and Stability; 5.4-2 Propagated-Error Bounds and Estimates 5.5 Predictor-Corrector Methods 5.5-1 Convergence of the Iterations; 5.5-2 Predictors and Correctors; 5.5-3 Error Estimation; 5.5-4 Stability 5.6 Starting the Solution and Changing the Interval 5.6-1 Analytic Methods; 5.6-2 A Numerical Method; 5.6-3 Changing the Interval 5.7 Using Predictor-Corrector Methods 5.7-1 Variable-Order--Variable-Step Methods; 5.7-2 Some Illustrative Examples 5.8 Runge-Kutta Methods 5.8-1 Errors in Runge-Kutta Methods; 5.8-2 Second-Order Methods; 5.8-3 Third-Order Methods; 5.8-4 Fourth-Order Methods; 5.8-5 Higher-Order Methods; 5.8-6 Practical Error Estimation; 5.8-7 Step-size Strategy; 5.8-8 Stability; 5.8-9 Comparison of Runge-Kutta and Predictor-Corrector Methods 5.9 Other Numerical Integration Methods 5.9-1 Methods Based on Higher Derivatives; 5.9-2 Extrapolation Methods 5.10 Stiff Equations Bibliographic Notes; Bibliography; Problems Chapter 6. Functional Approximation: Least-Squares Techniques 6.1 Introduction 6.2 The Principle of Least Squares 6.3 Polynomial Least-Squares Approximations 6.3-1 Solution of the Normal Equations; 6.3-2 Choosing the Degree of the Polynomial 6.4 Orthogonal-Polynomial Approximations 6.5 An Example of the Generation of Least-Squares Approximations 6.6 The Fourier Approximation 6.6-1 The Fast Fourier Transform; 6.6-2 Least-Squares Approximations and Trigonometric Interpolation Bibliographic Notes; Bibliography; Problems Chapter 7. Functional Approximation: Minimum Maximum Error Techniques 7.1 General Remarks 7.2 Rational Functions, Polynomials, and Continued Fractions 7.3 Padé Approximations 7.4 An Example 7.5 Chebyshev Polynomials 7.6 Chebyshev Expansions 7.7 Economization of Rational Functions 7.7-1 Economization of Power Series; 7.7-2 Generalization to Rational Functions 7.8 Chebyshev's Theorem of Minimax Approximations 7.9 Constructing Minimax Approximations 7.9-1 The Second Algorithm of Remes; 7.9-2 The Differential Correction Algorithm Bibliographic Notes; Bibliography; Problems Chapter 8. The Solution of Nonlinear Equations 8.1 Introduction 8.2 Functional Iteration 8.2-1 Computational Efficiency 8.3 The Secant Method 8.4 One-Point Iteration Formulas 8.5 Multipoint Iteration Formulas 8.5-1 Iteration Formulas Using General Inverse Interpolation; 8.5-2 Derivative Estimated Iteration Formulas 8.6 Functional Iteration at a Multiple Root 8.7 Some Computational Aspects of Functional Iteration 8.7-1 The delta superscript 2 Process 8.8 Systems of Nonlinear Equations 8.9 The Zeros of Polynomials: The Problem 8.9-1 Sturm Sequences 8.10 Classical Methods 8.10-1 Bairstow's Method; 8.10-2 Graeffe's Root-squaring Method; 8.10-3 Bernoulli's Method; 8.10-4 Laguerre's Method 8.11 The Jenkins-Traub Method 8.12 A Newton-based Method 8.13 The Effect of Coefficient Errors on the Roots; Ill-conditioned Polynomials Bibliographic Notes; Bibliography; Problems Chapter 9. The Solution of Simultaneous Linear Equations 9.1 The Basic theorem and the Problem 9.2 General Remarks 9.3 Direct Methods 9.3-1 Gaussian Elimination; 9.3-2 Compact forms of Gaussian Elimination; 9.3-3 The Doolittle, Crout, and Cholesky Algorithms; 9.3-4 Pivoting and Equilibration 9.4 Error Analysis 9.4-1 Roundoff-Error Analysis 9.5 Iterative Refinement 9.6 Matrix Iterative Methods 9.7 Stationary Iterative Processes and Related Matters 9.7-1 The Jacobi Iteration; 9.7-2 The Gauss-Seidel Method; 9.7-3 Roundoff Error in Iterative Methods; 9.7-4 Acceleration of Stationary Iterative Processes 9.8 Matrix Inversion 9.9 Overdetermined Systems of Linear Equations 9.10 The Simplex Method for Solving Linear Programming Problems 9.11 Miscellaneous topics Bibliographic Notes; Bibliography; Problems Chapter 10. The Calculation of Eigenvalues and Eigenvectors of Matrices 10.1 Basic Relationships 10.1-1 Basic Theorems; 10.1-2 The characteristic Equation; 10.1-3 The Location of, and Bo 10.2-1 Acceleration of convergence; 10.2-2 The Inverse Power Method 10.3 The Eigenvalues and Eigenvectors of Symmetric Matrices 10.3-1 The Jacobi Method; 10.3-2 Givens' Method; 10.3-3 Householder's Method 10.4 Methods for Nonsymmetric Matrices 10.4-1 Lanczos' Method; 10.4-2 Supertriangularization; 10.4-3 Jacobi-Type Methods 10.5 The LR and QR Algorithms 10.5-1 The Simple QR Algorithm; 10.5-2 The Double QR Algorithm 10.6 Errors in Computed eigenvalues and Eigenvectors Bibliographic Notes; Bibliography; Problems Index; Hints and Answers to Problems