
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 


1.31 Significant digits; 1.32 Error in functional Evaluation; 1.33 Norms 

1.4 Roundoff Error 


1.41 The Probabilistic Approach to Roundoff: A Particular Example 

1.5 Computer Arithmetic 


1.51 FixedPoint Arithmetic; 1.52 FloatingPoint Numbers; 1.53 FloatingPoint Arithmetic; 1.54 Overflow and Underflow; 1.55 Single and DoublePrecision Arithmetic 

1.6 Error Analysis 


1.61 Backward Error Analysis 

1.7 Condition and Stability 

Bibliographic Notes; Bibliography; Problems 
Chapter 2. Approximation and Algorithms 

2.1 Approximation 


2.11 Classes of Approximating Functions; 2.12 Types of Approximations; 2.13 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 


3.31 Lagrangian Interpolation at Equal Intervals; 3.32 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 


4.31 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 Inte 

4.8 Particular Gaussian Quadrature Formulas 


4.81 GaussJacobi Quadrature; 4.82 GaussChebyshev Quadrature; 4.83 Singular Integrals 

4.9 Composite Quadrature Formulas 

4.10 NewtonCotes Quadrature Formulas 


4.101 Composite NewtonCotes Formulas; 4.102 Romberg Integration 

4.11 Adaptive Integration 

4.12 Choosing a Quadrature Formula 

4.13 Summation 


4.131 The EulerMaclaurin Sum Formula; 4.132 Summation of Rational Functions; Factorial Functions; 4.133 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 


5.21 The Method of Undetermined Coefficients 

5.3 Truncation Error in Numerical Integration Methods 

5.4 Stability of Numerical Integration Methods 


5.41 Convergence and Stability; 5.42 PropagatedError Bounds and Estimates 

5.5 PredictorCorrector Methods 


5.51 Convergence of the Iterations; 5.52 Predictors and Correctors; 5.53 Error Estimation; 5.54 Stability 

5.6 Starting the Solution and Changing the Interval 


5.61 Analytic Methods; 5.62 A Numerical Method; 5.63 Changing the Interval 

5.7 Using PredictorCorrector Methods 


5.71 VariableOrderVariableStep Methods; 5.72 Some Illustrative Examples 

5.8 RungeKutta Methods 


5.81 Errors in RungeKutta Methods; 5.82 SecondOrder Methods; 5.83 ThirdOrder Methods; 5.84 FourthOrder Methods; 5.85 HigherOrder Methods; 5.86 Practical Error Estimation; 



5.87 Stepsize Strategy; 5.88 Stability; 5.89 Comparison of RungeKutta and PredictorCorrector Methods 

5.9 Other Numerical Integration Methods 


5.91 Methods Based on Higher Derivatives; 5.92 Extrapolation Methods 

5.10 Stiff Equations 

Bibliographic Notes; Bibliography; Problems 
Chapter 6. Functional Approximation: LeastSquares Techniques 

6.1 Introduction 

6.2 The Principle of Least Squares 

6.3 Polynomial LeastSquares Approximations 


6.31 Solution of the Normal Equations; 6.32 Choosing the Degree of the Polyn 

6.4 OrthogonalPolynomial Approximations 

6.5 An Example of the Generation of LeastSquares Approximations 

6.6 The Fourier Approximation 


6.61 The Fast Fourier Transform; 6.62 LeastSquares 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 


7.71 Economization of Power Series; 7.72 Generalization to Rational Functions 

7.8 Chebyshev's Theorem of Minimax Approximations 

7.9 Constructing Minimax Approximations 


7.91 The Second Algorithm of Remes; 7.92 The Differential Correction Algorithm 

Bibliographic Notes; Bibliography; Problems 
Chapter 8. The Solution of Nonlinear Equations 

8.1 Introduction 

8.2 Functional Iteration 


8.21 Computational Efficiency 

8.3 The Secant Method 

8.4 OnePoint Iteration Formulas 

8.5 Multipoint Iteration Formulas 


8.51 Iteration Formulas Using General Inverse Interpolation; 8.52 Derivative Estimated Iteration Formulas 

8.6 Functional Iteration at a Multiple Root 

8.7 Some Computational Aspects of Functional Iteration 


8.71 The delta superscript 2 Process 

8.8 Systems of Nonlinear Equations 

8.9 The Zeros of Polynomials: The Problem 


8.91 Sturm Sequences 

8.10 Classical Methods 


8.101 Bairstow's Method; 8.102 Graeffe's Rootsquaring Method; 8.103 Bernoulli's Method; 8.104 Laguerre's Method 

8.11 The JenkinsTraub Method 

8.12 A Newtonbased Method 

8.13 The Effect of Coefficient Errors on the Roots; Illconditioned Polynomials 

Bibliographic Notes; Bibliography; Problems 
Chapter 9. The Solution of Simultaneous Linear Equations 

9.1 The Basic theorem and the Pr 

9.2 General Remarks 
9.3 Direct Methods 


9.31 Gaussian Elimination; 9.32 Compact forms of Gaussian Elimination; 9.33 The Doolittle, Crout, and Cholesky Algorithms; 9.34 Pivoting and Equilibration 

9.4 Error Analysis 


9.41 RoundoffError Analysis 

9.5 Iterative Refinement 

9.6 Matrix Iterative Methods 

9.7 Stationary Iterative Processes and Related Matters 


9.71 The Jacobi Iteration; 9.72 The GaussSeidel Method; 9.73 Roundoff Error in Iterative Methods; 9.74 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 


10.11 Basic Theorems; 10.12 The characteristic Equation; 10.13 The Location of, and Bo 


10.21 Acceleration of convergence; 10.22 The Inverse Power Method 

10.3 The Eigenvalues and Eigenvectors of Symmetric Matrices 


10.31 The Jacobi Method; 10.32 Givens' Method; 10.33 Householder's Method 

10.4 Methods for Nonsymmetric Matrices 


10.41 Lanczos' Method; 10.42 Supertriangularization; 10.43 JacobiType Methods 

10.5 The LR and QR Algorithms 


10.51 The Simple QR Algorithm; 10.52 The Double QR Algorithm 

10.6 Errors in Computed eigenvalues and Eigenvectors 

Bibliographic Notes; Bibliography; Problems 
Index; Hints and Answers to Problems 