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Publisher:

Dover Publications

Publication Date:

2001

Number of Pages:

624

Format:

Paperback

Edition:

2

Price:

21.95

ISBN:

978-0486414546

Category:

Textbook

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

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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 Inte | |||||||

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 Polyn | |||||||

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 Pr | |||||||

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 |

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