An introduction to numerical analysis / Kendall E. Atkinson.

Por: Atkinson, Kendall EEditor: New York : Wiley, c1978Descripción: xiii, 587 p. : il. ; 24 cmISBN: 0471029859Tema(s): Numerical analysisOtra clasificación: 65-01
Contenidos:
 ONE
THE SOURCES AND PROPAGATION OF ERRORS [3]
1.1 Mathematical Preliminaries [3]
1.2 Computer Representation of Numbers [12]
1.3 Sources of Error [14]
1.4 Propagation of Errors [19]
1.5 Stability in Numerical Analysis [26]
Discussion of the Literature [32]
Problems [34]
 TWO
ROOTFINDING FOR NONLINEAR EQUATIONS [39]
2.1 Simple Enclosure Methods [42]
2.2 The Secant Method [48]
2.3 Newton’s Method [52]
2.4 A General Theory for One-Point Iteration Methods [58]
2.5 Aitken Extrapolation for Linearly Convergent Sequences [65]
2.6 Error Tests [68]
2.7 The Numerical Evaluation of Multiple Roots [71]
8.8 Brent’s Rootfinding Algorithm [75]
2.9 Roots of Polynomials [78]
2.10 Muller’s Method [85]
2.11 Nonlinear Systems of Equations [88]
2.12 Newton’s Method for Nonlinear Systems [92]
Discussion of the Literature [95]
Problems [97]
 
 THREE
INTERPOLATION THEORY 107 [107]
3.1 Polynomial Interpolation Theory
3.2 Newton Divided Differences [114]
3.3 Finite Differences and Table-Oriented Interpolation Formulas [123]
3.4 Errors in Data and Forward Differences [129]
3.5 Further Results on Interpolation Error [132]
3.6 Hermite Interpolation [137]
3.7 Piecewise Polynomial Interpolation [141]
Discussion of the Literature [150]
Problems [153]
 FOUR
APPROXIMATION OF FUNCTIONS [161]
4.1 The Weierstrass Theorem and Taylor’s Theorem [162]
4.2 The Minimax Approximation Problem [165]
4.3 The Least Squares Approximation Problem [168]
4.4 Orthogonal Polynomials [171]
4.5 The Least Squares Approximation Problem (continued) [180]
4.6 Economization of Taylor Series [186]
4.7 Minimax Approximations [190]
4.8 Near-Minimax Approximations [194]
4.9 The Remes Algorithm [201]
Discussion of the Literature [203]
Problems [205]
 FIVE
NUMERICAL INTEGRATION [213]
5.1 The Trapezoidal Rule and Simpson’s Rule [215]
5.2 Newton-Cotes Integration Formulas [225]
5.3 Gaussian Quadrature [231]
5.4 Patterson’s Method [243]
5.5 Asymptotic Error Formulas and Their Application [248]
5.6 Adaptive Numerical Integration [263]
5.7 Singular Integrals [268]
Discussion of the Literature [277]
Problems [279]
 SIX
NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS [289]
6.1 Existence, Uniqueness, and Stability Theory [292]
6.2 Euler’s Method [300]
6.3 Multistep Methods [314]
6.4 The Midpoint Method [320]
6.5 The Trapezoidal Method [325]
6.6 A Low-Order Predictor-Corrector Algorithm [330]
6.7 Derivation of Higher-Order Multistep Methods [338]
6.8 Convergence and Stability Theory for Multistep Methods [350]
6.9 Single-Step and Runge-Kutta Methods [366]
Discussion of Literature [380]
Problems [384]
 SEVEN
LINEAR ALGEBRA [393]
7.1 Vector Spaces, Matrices, and Linear Systems [393]
7.2 Eigenvalues and Canonical Forms for Matrices [401]
7.3 Vector and Matrix Norms [412]
7.4 Convergence and Perturbation Theorems [421]
Discussion of Literature [427]
Problems [428]
 EIGHT
NUMERICAL SOLUTION OF SYSTEMS OF LINEAR EQUATIONS [435]
8.1 Gaussian Elimination [436]
8.2 Pivoting and Scaling in Gaussian Elimination [444]
8.3 Variants of Gaussian Elimination [450]
8.4 Error Analysis [457]
8.5 The Residual Correction Method [467]
8.6 Iteration Methods [471]
8.7 Error Prediction and Acceleration [478]
8.8 The Numerical Solution of Poisson’s Equation [482]
Discussion of Literature [487]
Problems [490]
 NINE
THE MATRIX EIGENVALUE PROBLEM [499]
9.1 Eigenvalue Location, Error, and Stability Results [500]
9.2 The Power Method [514]
9.3 Orthogonal Transformations Using Householder Matrices [521]
9.4 The Eigenvalues of a Symmetric Tridiagonal Matrix [532]
9.5 The QR Method [539]
9.6 The Calculation of Eigenvectors and Inverse Iteration [548]
Discussion of Literature [553]
Problems [551]
ANSWERS TO SELECTED EXERCISES [561]
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Incluye referencias bibliográficas e índice.

ONE --
THE SOURCES AND PROPAGATION OF ERRORS [3] --
1.1 Mathematical Preliminaries [3] --
1.2 Computer Representation of Numbers [12] --
1.3 Sources of Error [14] --
1.4 Propagation of Errors [19] --
1.5 Stability in Numerical Analysis [26] --
Discussion of the Literature [32] --
Problems [34] --
TWO --
ROOTFINDING FOR NONLINEAR EQUATIONS [39] --
2.1 Simple Enclosure Methods [42] --
2.2 The Secant Method [48] --
2.3 Newton’s Method [52] --
2.4 A General Theory for One-Point Iteration Methods [58] --
2.5 Aitken Extrapolation for Linearly Convergent Sequences [65] --
2.6 Error Tests [68] --
2.7 The Numerical Evaluation of Multiple Roots [71] --
8.8 Brent’s Rootfinding Algorithm [75] --
2.9 Roots of Polynomials [78] --
2.10 Muller’s Method [85] --
2.11 Nonlinear Systems of Equations [88] --
2.12 Newton’s Method for Nonlinear Systems [92] --
Discussion of the Literature [95] --
Problems [97] --
--
THREE --
INTERPOLATION THEORY 107 [107] --
3.1 Polynomial Interpolation Theory --
3.2 Newton Divided Differences [114] --
3.3 Finite Differences and Table-Oriented Interpolation Formulas [123] --
3.4 Errors in Data and Forward Differences [129] --
3.5 Further Results on Interpolation Error [132] --
3.6 Hermite Interpolation [137] --
3.7 Piecewise Polynomial Interpolation [141] --
Discussion of the Literature [150] --
Problems [153] --
FOUR --
APPROXIMATION OF FUNCTIONS [161] --
4.1 The Weierstrass Theorem and Taylor’s Theorem [162] --
4.2 The Minimax Approximation Problem [165] --
4.3 The Least Squares Approximation Problem [168] --
4.4 Orthogonal Polynomials [171] --
4.5 The Least Squares Approximation Problem (continued) [180] --
4.6 Economization of Taylor Series [186] --
4.7 Minimax Approximations [190] --
4.8 Near-Minimax Approximations [194] --
4.9 The Remes Algorithm [201] --
Discussion of the Literature [203] --
Problems [205] --
FIVE --
NUMERICAL INTEGRATION [213] --
5.1 The Trapezoidal Rule and Simpson’s Rule [215] --
5.2 Newton-Cotes Integration Formulas [225] --
5.3 Gaussian Quadrature [231] --
5.4 Patterson’s Method [243] --
5.5 Asymptotic Error Formulas and Their Application [248] --
5.6 Adaptive Numerical Integration [263] --
5.7 Singular Integrals [268] --
Discussion of the Literature [277] --
Problems [279] --
SIX --
NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS [289] --
6.1 Existence, Uniqueness, and Stability Theory [292] --
6.2 Euler’s Method [300] --
6.3 Multistep Methods [314] --
6.4 The Midpoint Method [320] --
6.5 The Trapezoidal Method [325] --
6.6 A Low-Order Predictor-Corrector Algorithm [330] --
6.7 Derivation of Higher-Order Multistep Methods [338] --
6.8 Convergence and Stability Theory for Multistep Methods [350] --
6.9 Single-Step and Runge-Kutta Methods [366] --
Discussion of Literature [380] --
Problems [384] --
SEVEN --
LINEAR ALGEBRA [393] --
7.1 Vector Spaces, Matrices, and Linear Systems [393] --
7.2 Eigenvalues and Canonical Forms for Matrices [401] --
7.3 Vector and Matrix Norms [412] --
7.4 Convergence and Perturbation Theorems [421] --
Discussion of Literature [427] --
Problems [428] --
EIGHT --
NUMERICAL SOLUTION OF SYSTEMS OF LINEAR EQUATIONS [435] --
8.1 Gaussian Elimination [436] --
8.2 Pivoting and Scaling in Gaussian Elimination [444] --
8.3 Variants of Gaussian Elimination [450] --
8.4 Error Analysis [457] --
8.5 The Residual Correction Method [467] --
8.6 Iteration Methods [471] --
8.7 Error Prediction and Acceleration [478] --
8.8 The Numerical Solution of Poisson’s Equation [482] --
Discussion of Literature [487] --
Problems [490] --
NINE --
THE MATRIX EIGENVALUE PROBLEM [499] --
9.1 Eigenvalue Location, Error, and Stability Results [500] --
9.2 The Power Method [514] --
9.3 Orthogonal Transformations Using Householder Matrices [521] --
9.4 The Eigenvalues of a Symmetric Tridiagonal Matrix [532] --
9.5 The QR Method [539] --
9.6 The Calculation of Eigenvectors and Inverse Iteration [548] --
Discussion of Literature [553] --
Problems [551] --
ANSWERS TO SELECTED EXERCISES [561] --

MR, 80a:65001

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