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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 Bounds on, the Eigenvalues 10.1-4 Canonical Forms

10.2 2 The Largest Eigenvalue in Magnitude by the Power Method

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

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 Bounds on, the Eigenvalues 10.1-4 Canonical Forms

10.2 2 The Largest Eigenvalue in Magnitude by the Power Method

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

Library of Congress subject headings for this publication: Numerical analysis