Analysis of numerical methods / [by] Eugene Isaacson [and] Herbert Bishop Keller.
Editor: New York : Wiley, [1966]Descripción: xv, 541 p. : il. ; 24 cmTema(s): Numerical analysisOtra clasificación: 65-02Chapter 1 Norms, Arithmetic, and Well-Posed Computations 0. Introduction [1] 1. Norms of Vectors and Matrices [2] 1.1. Convergent Matrices [14] 2. Floating-Point Arithmetic and Rounding Errors [17] 3. Well-Posed Computations [21] Chapter 2 Numerical Solution of Linear Systems and Matrix Inversion 0. Introduction [26] 1. Gaussian Elimination [29] 1.1. Operational Counts [34] 1.2. A Priori Error Estimates; Condition Number [37] 1.3. A Posteriori Error Estimates [46] 2. Variants of Gaussian Elimination [50] 3. Direct Factorization Methods [52] 3.1. Symmetric Matrices (Cholesky Method) [54] 3.2. Tridiagonal or Jacobi Matrices [55] 3.3. Block-Tridiagonal Matrices [58] 4. Iterative Methods [61] 4.1. Jacobi or Simultaneous Iterations [64] 4.2. Gauss-Seidel or Successive Iterations [66] 4.3. Method of Residual Correction [68] 4.4. Positive Definite Systems [70] 4.5. Block Iterations [72] 5. The Acceleration of Iterative Methods [73] 5.1. Practical Application of Acceleration Methods [78] 5.2. Generalizations of the Acceleration Method [80] 6. Matrix Inversion by Higher Order Iterations [82] Chapter 3 Iterative Solutions of Non-Linear Equations 0. Introduction [85] 1. Functional Iteration for a Single Equation [86] 1.1. Error Propagation [91] 1.2. Second and Higher Order Iteration Methods [94] 2. Some Explicit Iteration Procedures [96] 2.1. The Simple Iteration or Chord Method (First Order) [97] 2.2. Newton’s Method (Second Order) [97] 2.3. Method of False Position (Fractional Order) [99] 2.4. Aitken’s δ2-Method (Arbitrary Order) [102] 3. Functional Iteration for a System of Equations [109] 3.1. Some Explicit Iteration Schemes for Systems [113] 3.2. Convergence of Newton’s Method [115] 3.3. A Special Acceleration Procedure for Non-Linear Systems [120] 4. Special Methods for Polynomials [123] 4.1. Evaluation of Polynomials and Their Derivatives [124] 4.2. Sturm Sequences [126] 4.3. Bernoulli’s Method [128] 4.4. Bairstow’s Method [131] Chapter 4 Computation of Eigenvalues and Eigenvectors 0. Introduction [134] 1. Well-Posedness, Error Estimates [135] 1.1. A Posteriori Error Estimates [140] 2. The Power Method [147] 2.1. Acceleration of the Power Method [151] 2.2. Intermediate Eigenvalues and Eigenvectors (Orthogonalization, Deflation, Inverse Iteration) [152] 3. Methods Based on Matrix Transformations [159] Chapter 5 Basic Theory of Polynomial Approximation 0. Introduction [176] 1. Weierstrass’ Approximation Theorem and Bernstein Polynomials [183] 2. The Interpolation Polynomials [187] 2.1. The Pointwise Error in Interpolation Polynomials [189] 2.2. Hermite or Osculating Interpolation [192] 3. Least Squares Approximation [194] 3.1. Construction of Orthonormal Functions [199] 3.2. Weighted Least Squares Approximation [202] 3.3. Some Properties of Orthogonal Polynomials [203] 3.4. Pointwise Convergence of Least Squares Approximations [205] 3.5. Discrete Least Squares Approximation [211] 4. Polynomials of “Best” Approximation [221] 4.1. The Error in the Best Approximation [223] 4.2. Chebyshev Polynomials [226] 5. Trigonometric Approximation [229] 5.1. Trigonometric Interpolation [230] 5.2. Least Squares Trigonometric Approximation, Fourier Series [237] 5.3. “Best” Trigonometric Approximation [240] Chapter 6 Differences, Interpolation Polynomials, and Approximate Differentiation 0. Introduction [245] 1. Newton’s Interpolation Polynomial and Divided Differences [246] 2. Iterative Linear Interpolation [258] 3. Forward Differences and Equally Spaced Interpolation Points [260] 3.1. Interpolation Polynomials and Remainder Terms for Equidistant Points [264] 3.2. Centered Interpolation Formulae [270] 3.3. Practical Observations on Interpolation [273] 3.4. Divergence of Sequences of Interpolation Polynomials [275] 4. Calculus of Difference Operators [281] 5. Numerical Differentiation [288] 5.1. Differentiation Using Equidistant Points [292] 6. Multivariate Interpolation [294] Chapter 7 Numerical Integration 0. Introduction [300] 1. Interpolator Quadrature [303] 1.1. Newton-Cotes Formulae [308] 1.2. Determination of the Coefficients [315] 2. Roundoff Errors and Uniform Coefficient Formulae [319] 2.1. Determination of Uniform Coefficient Formulae [323] 3. Gaussian Quadrature; Maximum Degree of Precision [327] 4. Weighted Quadrature Formulae [331] 4.1. Gauss-Chebyshev Quadrature [334] 5. Composite Quadrature Formulae [336] 5.1. Periodic Functions and the Trapezoidal Rule [340] 5.2. Convergence for Continuous Functions [341] 6. Singular Integrals; Discontinuous Integrands [346] 6.1. Finite Jump Discontinuities [346] 6.2. Infinite Integrand [346] 6.3. Infinite Integration Limits [350] 7. Multiple Integrals [352] 7.1. The Use of Interpolation Polynomials [354] 7.2. Undetermined Coefficients (and Nodes) [356] 7.3. Separation of Variables [359] 7.4. Composite Formulae for Multiple Integrals [361] Chapter 8 Numerical Solution of Ordinary Differential Equations 0. Introduction [364] 1. Methods Based on Approximating the Derivative: Euler-Cauchy Method [367] 1.1. Improving the Accuracy of the Numerical Solution [372] 1.2. Roundoff Errors [374] 1.3. Centered Difference Method [377] 1.4. A Divergent Method with Higher Order Truncation Error [379] 2. Multistep Methods Based on Quadrature Formulae [384] 2.1. Error Estimates in Predictor-Corrector Methods [388] 2.2. Change of Net Spacing [393] 3. One-Step Methods [395] 3.1. Finite Taylor’s Series [397] 3.2. One-Step Methods Based on Quadrature Formulae [400] 4. Linear Difference Equations [405] 5. Consistency, Convergence, and Stability of Difference Methods [410] 6. Higher Order Equations and Systems [418] 7. Boundary Value and Eigenvalue Problems [421] 7.1. Initial Value or “Shooting” Methods [424] 7.2. Finite Difference Methods [427] 7.3. Eigenvalue Problems [434] Chapter 9 Difference Methods for Partial Differential Equations 0. Introduction [442] 0.1. Conventions of Notation [444] 1. Laplace Equation in a Rectangle [445] 1.1. Matrix Formulation [452] 1.2. An Eigenvalue Problem for the Laplacian Operator [458] 2. Solution of Laplace Difference Equations [463] 2.1. Line or Block Iterations [471] 2.2 Alternating Direction Iterations [475] 3. Wave Equation and an Equivalent System [479] 3.1. Difference Approximations and Domains of Dependence [485] 3.2. Convergence of Difference Solutions [491] 3.3. Difference Methods for a First Order Hyperbolic System [495] 4. Heat Equation [501] 4.1. Implicit Methods [505] 5. General Theory: Consistency, Convergence, and Stability [514] 5.1. Further Consequences of Stability [522] 5.2. The von Neumann Stability Test [523] Bibliography [531] Index [535]
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
|---|---|---|---|---|---|---|---|---|
Libros
|
Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 65 Is73 (Browse shelf) | Available | A-4139 |
Bibliografía: p. 531-533.
Chapter 1 Norms, Arithmetic, and Well-Posed Computations --
0. Introduction [1] --
1. Norms of Vectors and Matrices [2] --
1.1. Convergent Matrices [14] --
2. Floating-Point Arithmetic and Rounding Errors [17] --
3. Well-Posed Computations [21] --
Chapter 2 Numerical Solution of Linear Systems and Matrix Inversion --
0. Introduction [26] --
1. Gaussian Elimination [29] --
1.1. Operational Counts [34] --
1.2. A Priori Error Estimates; Condition Number [37] --
1.3. A Posteriori Error Estimates [46] --
2. Variants of Gaussian Elimination [50] --
3. Direct Factorization Methods [52] --
3.1. Symmetric Matrices (Cholesky Method) [54] --
3.2. Tridiagonal or Jacobi Matrices [55] --
3.3. Block-Tridiagonal Matrices [58] --
4. Iterative Methods [61] --
4.1. Jacobi or Simultaneous Iterations [64] --
4.2. Gauss-Seidel or Successive Iterations [66] --
4.3. Method of Residual Correction [68] --
4.4. Positive Definite Systems [70] --
4.5. Block Iterations [72] --
5. The Acceleration of Iterative Methods [73] --
5.1. Practical Application of Acceleration Methods [78] --
5.2. Generalizations of the Acceleration Method [80] --
6. Matrix Inversion by Higher Order Iterations [82] --
Chapter 3 Iterative Solutions of Non-Linear Equations --
0. Introduction [85] --
1. Functional Iteration for a Single Equation [86] --
1.1. Error Propagation [91] --
1.2. Second and Higher Order Iteration Methods [94] --
2. Some Explicit Iteration Procedures [96] --
2.1. The Simple Iteration or Chord Method (First Order) [97] --
2.2. Newton’s Method (Second Order) [97] --
2.3. Method of False Position (Fractional Order) [99] --
2.4. Aitken’s δ2-Method (Arbitrary Order) [102] --
3. Functional Iteration for a System of Equations [109] --
3.1. Some Explicit Iteration Schemes for Systems [113] --
3.2. Convergence of Newton’s Method [115] --
3.3. A Special Acceleration Procedure for Non-Linear Systems [120] --
4. Special Methods for Polynomials [123] --
4.1. Evaluation of Polynomials and Their Derivatives [124] --
4.2. Sturm Sequences [126] --
4.3. Bernoulli’s Method [128] --
4.4. Bairstow’s Method [131] --
Chapter 4 Computation of Eigenvalues and Eigenvectors --
0. Introduction [134] --
1. Well-Posedness, Error Estimates [135] --
1.1. A Posteriori Error Estimates [140] --
2. The Power Method [147] --
2.1. Acceleration of the Power Method [151] --
2.2. Intermediate Eigenvalues and Eigenvectors (Orthogonalization, Deflation, Inverse Iteration) [152] --
3. Methods Based on Matrix Transformations [159] --
Chapter 5 Basic Theory of Polynomial Approximation --
0. Introduction [176] --
1. Weierstrass’ Approximation Theorem and Bernstein Polynomials [183] --
2. The Interpolation Polynomials [187] --
2.1. The Pointwise Error in Interpolation Polynomials [189] --
2.2. Hermite or Osculating Interpolation [192] --
3. Least Squares Approximation [194] --
3.1. Construction of Orthonormal Functions [199] --
3.2. Weighted Least Squares Approximation [202] --
3.3. Some Properties of Orthogonal Polynomials [203] --
3.4. Pointwise Convergence of Least Squares Approximations [205] --
3.5. Discrete Least Squares Approximation [211] --
4. Polynomials of “Best” Approximation [221] --
4.1. The Error in the Best Approximation [223] --
4.2. Chebyshev Polynomials [226] --
5. Trigonometric Approximation [229] --
5.1. Trigonometric Interpolation [230] --
5.2. Least Squares Trigonometric Approximation, Fourier Series [237] --
5.3. “Best” Trigonometric Approximation [240] --
Chapter 6 Differences, Interpolation Polynomials, and Approximate Differentiation --
0. Introduction [245] --
1. Newton’s Interpolation Polynomial and Divided Differences [246] --
2. Iterative Linear Interpolation [258] --
3. Forward Differences and Equally Spaced Interpolation Points [260] --
3.1. Interpolation Polynomials and Remainder Terms for Equidistant Points [264] --
3.2. Centered Interpolation Formulae [270] --
3.3. Practical Observations on Interpolation [273] --
3.4. Divergence of Sequences of Interpolation Polynomials [275] --
4. Calculus of Difference Operators [281] --
5. Numerical Differentiation [288] --
5.1. Differentiation Using Equidistant Points [292] --
6. Multivariate Interpolation [294] --
Chapter 7 Numerical Integration --
0. Introduction [300] --
1. Interpolator Quadrature [303] --
1.1. Newton-Cotes Formulae [308] --
1.2. Determination of the Coefficients [315] --
2. Roundoff Errors and Uniform Coefficient Formulae [319] --
2.1. Determination of Uniform Coefficient Formulae [323] --
3. Gaussian Quadrature; Maximum Degree of Precision [327] --
4. Weighted Quadrature Formulae [331] --
4.1. Gauss-Chebyshev Quadrature [334] --
5. Composite Quadrature Formulae [336] --
5.1. Periodic Functions and the Trapezoidal Rule [340] --
5.2. Convergence for Continuous Functions [341] --
6. Singular Integrals; Discontinuous Integrands [346] --
6.1. Finite Jump Discontinuities [346] --
6.2. Infinite Integrand [346] --
6.3. Infinite Integration Limits [350] --
7. Multiple Integrals [352] --
7.1. The Use of Interpolation Polynomials [354] --
7.2. Undetermined Coefficients (and Nodes) [356] --
7.3. Separation of Variables [359] --
7.4. Composite Formulae for Multiple Integrals [361] --
Chapter 8 Numerical Solution of Ordinary Differential Equations --
0. Introduction [364] --
1. Methods Based on Approximating the Derivative: Euler-Cauchy Method [367] --
1.1. Improving the Accuracy of the Numerical Solution [372] --
1.2. Roundoff Errors [374] --
1.3. Centered Difference Method [377] --
1.4. A Divergent Method with Higher Order Truncation Error [379] --
2. Multistep Methods Based on Quadrature Formulae [384] --
2.1. Error Estimates in Predictor-Corrector Methods [388] --
2.2. Change of Net Spacing [393] --
3. One-Step Methods [395] --
3.1. Finite Taylor’s Series [397] --
3.2. One-Step Methods Based on Quadrature Formulae [400] --
4. Linear Difference Equations [405] --
5. Consistency, Convergence, and Stability of Difference Methods [410] --
6. Higher Order Equations and Systems [418] --
7. Boundary Value and Eigenvalue Problems [421] --
7.1. Initial Value or “Shooting” Methods [424] --
7.2. Finite Difference Methods [427] --
7.3. Eigenvalue Problems [434] --
Chapter 9 Difference Methods for Partial Differential Equations --
0. Introduction [442] --
0.1. Conventions of Notation [444] --
1. Laplace Equation in a Rectangle [445] --
1.1. Matrix Formulation [452] --
1.2. An Eigenvalue Problem for the Laplacian Operator [458] --
2. Solution of Laplace Difference Equations [463] --
2.1. Line or Block Iterations [471] --
2.2 Alternating Direction Iterations [475] --
3. Wave Equation and an Equivalent System [479] --
3.1. Difference Approximations and Domains of Dependence [485] --
3.2. Convergence of Difference Solutions [491] --
3.3. Difference Methods for a First Order Hyperbolic System [495] --
4. Heat Equation [501] --
4.1. Implicit Methods [505] --
5. General Theory: Consistency, Convergence, and Stability [514] --
5.1. Further Consequences of Stability [522] --
5.2. The von Neumann Stability Test [523] --
Bibliography [531] --
Index [535] --
MR, 34 #924
Click on an image to view it in the image viewer
There are no comments on this title.