15 September 2008

[4S/yim213]Introduction to numerical analysis: 2nd edition

This volume is intended to provide an introductory treatment of the fundamental processesof numerical analysis which is compatible with the expansion of the field brought about by the developmentof the modern high-speed calculating devices,but which also takes into account the fact that very substantial amountsofcomputation will continue to be effected by desk calculators (and by hand or slide rule), and that familiarity with computation on a desk calculatoris a desirable preliminary to largescale computation in any case.

Table Content
Preface
1 Introduction
[Numerical Analysis - Approximation - Errors - Significant Figures - Determinacy of Functions. Error Control - Machine Errors - Random Errors - Recursive Computation - Mathematical Preliminaries]

2 Interpolation with Divided Differences
[Introduction - Linear Interpolation - Divided Differences - Second-Degree Interpolation - Newton's Fundamental Formula - Error Formulas - Iterated Interpolation - Inverse Interpolation - Supplementary - References - Problems]

3 Lagrangian Methods
[Introduction - Lagrange's Interpolation Formula - Numerical Differentiationand Integration - Uniform-spacing Interpolation - Newton-Cotes Integration Formulas - Composite Integration Formulas - Use of Integration Formulas - Richardson Extrapolation. Romberg Integration - Asymptotic Behaviorof Newton-Cotes Formulas - Weighting Functions. Filon Integration - Differentiation Formulas - Supplementary - References - Problems]

4 Finite-Difference Interpolation
[Introduction - Difference Notations - Newton Forward- and Backward-difference Formulas - Gaussian Formulas - Stirling's Formula - Bessel's Formula - Everett's Formulas - Use of Interpolation Formulas - Propagationof Inherent Errors - Throwback Techniques - Interpolation Series - Tablesof Interpolation Coefficients - Supplementary - References - Problems]

5 Operations with Finite Differences
[Introduction - Difference Operators - Differentiation Formulas - Newtonian Integration Formulas - Newtonian Formulas for Repeated Integration - Central-Difference Integration Formulas - Subtabulation - Summationand Integration. The Euler-Maclaurin Sum Formula - Approximate Summation - Error Terms in Integration Formulas - Other Representationsof Error Terms - Supplementary - References - Problems]

6 Numerical Solution of Differential Equations
[Introduction - Formulasof Open Type - Formulasof Closed Type - Startof Solution - Methods Based on Open-Type Formulas - Methods Based on Closed-Type Formulas. Prediction-Correction Methods - The Special CaseF = Ay - Propagated-Error Bounds - Applicationto Equations of Higher Order. Sets of Equations - Special Second-order Equations - Changeof Interval - Use of Higher Derivatives - A Simple Runge-Kutta Method - Runge-Kutta Methods of Higher Order - Boundary-Value Problems - Linear Characteristic-value Problems - Selectionof a Method - Supplementary - References - Problems]

7 Least-Squares Polynomial Approximation
[Introduction - The Principle of Least Squares - Least-Squares Approximation over Discrete Sets of Points - Error Estimation Orthogonal Polynomials - Legendre Approximation - Laguerre Approximation - Hermite Approximation - Chebyshev Approximation - Properties of Orthogonal Polynomials. Recursive Computation - Factorial Power Functions and Summation Formulas - Polynomials Orthogonal over Discrete Sets of Points - Gram Approximation - Example: Five-Point Least-Squares Approximation - Smoothing Formulas - Recursive Computation of Orthogonal Polynomials on Discrete - Sets of Points - Supplementary - References - Problems]

8 Gaussian Quadrature and Related Topics
[Introduction - Hermite Interpolation - Hermite Quadrature - Gaussian Quadrature - Legendre-Gauss Quadrature - Laguerre-Gauss Quadrature - Hermite-Gauss Quadrature - Chebyshev-Gauss Quadrature - Jacobi-Gauss Quadrature - Formulas with Assigned Abscissas - Radau Quadrature - Lobatto Quadrature - Convergenceof Gaussian-quadrature Sequences - Chebyshev Quadrature - Algebraic Derivations - Application to Trigonometric Integral - Supplementary - References - Problems]

9 Approximations of Various Types
[Introduction - Fourier Approximation: Continuous Domain - Fourier Approximation: Discrete Domain - Exponential Approximation - Determination of Constituent Periodicities - Optimum Polynomial Interpolation with Selected Abscissas - Chebyshev Interpolation - Economization of Polynomial Approximations - Uniform (Minimax) Polynomial Approximation - Spline Approximation - Splines withUniform Spacing - Spline Error Estimates - A Special Class of Splines - Approximationby Continued Fractions - Rational Approximations and Continued Fractions - Determination of Convergents of Continued Fractions - Thiele's Continued-Fraction Approximations - Uniformization of Rational Approximations - Supplementary - References - Problems]

10. Numerical Solution of Equations
[Introduction - Sets of Linear Equations - The Gauss Reduction - The Crout Reduction - Intermediate Round off Errors - Determination of the Inverse Matrix - Inherent Errors - Tridiagonal Sets of Equations - Iterative Methods and Relaxation - Iterative Methods for Nonlinear Equations - The Newton-Raphson Method - Iterative Methods of Higher Order - Sets of Nonlinear Equations - Iterated Synthetic Division of Polynomials. Lin's Method - Determinacy of Zeros of Polynomials - Bernoulli's Iteration - Graeffe's Root-squaring Technique - Quadratic Factors. Lin's Quadratic Method - Bairstow Iteration - Supplementary - References - Problems]

Appendixes
A Justification of the Crout Reduction
B Bibliography
C Directoryof Methods
Index

#Introduction to numerical analysis: 2nd edition
#Author Begnaud Francis Hildebrand
#Pages: 669
#Year of Publication: 1987
#ISBN:0-486-65363-3
#Publisher: Dover Publications, Inc. New York, NY, USA

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