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