Introduction to Numerical Analysis / Edition 2

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Overview

The ultimate aim of the field of numerical analysis is to provide convenient methods for obtaining useful solutions to mathematical problems and for extracting useful information from available solutions which are not expressed in tractable forms. This well-known, highly respected volume provides an introduction to the fundamental processes of numerical analysis, including substantial grounding in the basic operations of computation, approximation, interpolation, numerical differentiation and integration, and the numerical solution of equations, as well as in applications to such processes as the smoothing of data, the numerical summation of series, and the numerical solution of ordinary differential equations.
Chapter headings include:
l. Introduction
2. Interpolation with Divided Differences
3. Lagrangian Methods
4. Finite-Difference Interpolation
5. Operations with Finite Differences
6. Numerical Solution of Differential Equations
7. Least-Squares Polynomial Approximation
In this revised and updated second edition, Professor Hildebrand (Emeritus, Mathematics, MIT) made a special effort to include more recent significant developments in the field, increasing the focus on concepts and procedures associated with computers. This new material includes discussions of machine errors and recursive calculation, increased emphasis on the midpoint rule and the consideration of Romberg integration and the classical Filon integration; a modified treatment of prediction-correction methods and the addition of Hamming's method, and numerous other important topics.
In addition, reference lists have been expanded and updated, and more than 150 new problems have been added. Widely considered the classic book in the field, Hildebrand's Introduction to Numerical Analysis is aimed at advanced undergraduate and graduate students, or the general reader in search of a strong, clear introduction to the theory and analysis of numbers.

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Product Details

  • ISBN-13: 9780486653631
  • Publisher: Dover Publications
  • Publication date: 6/1/1987
  • Series: Dover Books on Mathematics Series
  • Edition number: 2
  • Pages: 669
  • Sales rank: 628,497
  • Product dimensions: 5.41 (w) x 8.52 (h) x 1.32 (d)

Table of Contents

Preface
1 Introduction
  1.1 Numerical Analysis
  1.2 Approximation
  1.3 Errors
  1.4 Significant Figures
  1.5 Determinacy of Functions. Error Control
  1.6 Machine Errors
  1.7 Random Errors
  1.8 Recursive Computation
  1.9 Mathematical Preliminaries
  1.10 Supplementary References
    Problems
2 Interpolation with Divided Differences
  2.1 Introduction
  2.2 Linear Interpolation
  2.3 Divided Differences
  2.4 Second-Degree Interpolation
  2.5 Newton's Fundamental Formula
  2.6 Error Formulas
  2.7 Iterated Interpolation
  2.8 Inverse Interpolation
  2.9 Supplementary References
    Problems
3 Lagrangian Methods
  3.1 Introduction
  3.2 Lagrange's Interpolation Formula
  3.3 Numerical Differentiation and Integration
  3.4 Uniform-spacing Interpolation
  3.5 Newton-Cotes Integration Formulas
  3.6 Composite Integration Formulas
  3.7 Use of Integration Formulas
  3.8 Richardson Extrapolation. Romberg Integration
  3.9 Asympotic Behavior of Newton-Cotes Formulas
  3.10 Weighting Functions. Filon Integration
  3.11 Differentiation Formulas
  3.12 Supplementary References
    Problems
4 Finite-Difference Interpolation
  4.1 Introduction
  4.2 Difference Notations
  4.3 Newton Forward- and Backward-difference Formulas
  4.4 Gaussian Formulas
  4.5 Stirling's Formula
  4.6 Bessel's Formula
  4.7 Everett's Formulas
  4.8 Use of Interpolation Formulas
  4.9 Propogation of Inherent Errors
  4.10 Throwback Techniques
  4.11 Interpolation Series
  4.12 Tables of Interpolation Coefficients
  4.13 Supplementary References
    Problems
5 Operations with Finite Differences
  5.1 Introduction
  5.2 Difference Operators
  5.3 Differentiation Formulas
  5.4 Newtonian Integration Formulas
  5.5 Newtonian Formulas for Repeated Integration
  5.6 Central-Difference Integration Formulas
  5.7 Subtabulation
  5.8 Summation and Integration. The Euler-Maclaurin Sum Formula
  5.9 Approximate Summation
  5.10 Error Terms in Integration Formulas
  5.11 Other Representations of Error Terms
  5.12 Supplementary References
    Problems
6 Numerical Solution of Differential Equations
  6.1 Introduction
  6.2 Formulas of Open Type
  6.3 Formulas of Closed Type
  6.4 Start of Solution
  6.5 Methods Based on Open-Type Formulas
  6.6 Methods Based on Closed-Type Formulas. Prediction-Correction Methods
  6.7 The Special Case F = Ay
  6.8 Propagated-Error Bounds
  6.9 Application to Equations of Higher Order. Sets of Equations
  6.10 Special Second-order Equations
  6.11 Change of Interval
  6.12 Use of Higher Derivatives
  6.13 A Simple Runge-Kutta Method
  6.14 Runge-Kutta Methods of Higher Order
  6.15 Boundary-Value Problems
  6.16 Linear Characteristic-value Problems
  6.17 Selection of a Method
  6.18 Supplementary References
    Problems
7 Least-Squares Polynomial Approximation
  7.1 Introduction
  7.2 The Principle of Least Squares
  7.3 Least-Squares Approximation over Discrete Sets of Points
  7.4 Error Estimation
  7.5 Orthogonal Polynomials
  7.6 Legendre Approximation
  7.7 Laguerre Approximation
  7.8 Hermite Approximation
  7.9 Chebsyshev Approximation
  7.10 Properties of Orthoogonal Polynomials. Recursive Computation
  7.11 Factorial Power Functions and Summation Formulas
  7.12 Polynomials Orthogonal over Discrete Sets of Points
  7.13 Gram Approximation
  7.14 Example: Five-Point Least-Squares Approximation
  7.15 Smoothing Formulas
  7.16 Recursive Computation of Orthogonal Polynomials on Discrete Set of Points
  7.17 Supplementary References
    Problems
8 Gaussian Quadrature and Related Topics
  8.1 Introduction
  8.2 Hermite Interpolation
  8.3 Hermite Quadrature
  8.4 Gaussian Quadrature
  8.5 Legendre-Gauss Quadrature
  8.6 Laguerre-Gauss Quadrature
  8.7 Hermite-Gauss Quadrature
  8.8 Chebyshev-Gauss Quadrature
  8.9 Jacobi-Gauss Quadrature
  8.10 Formulas with Assigned Abscissas
  8.11 Radau Quadrature
  8.12 Lobatto Quadrature
  8.13 Convergence of Gaussian-quadrature Sequences
  8.14 Chebyshev Quadrature
  8.15 Algebraic Derivations
  8.16 Application to Trigonometric Integrals
  8.17 Supplementary References
    Problems
9 Approximations of Various Types
  9.1 Introduction
  9.2 Fourier Approximation: Continuous Domain
  9.3 Fourier Approximation: Discrete Domain
  9.4 Exponential Approximation
  9.5 Determination of Constituent Periodicities
  9.6 Optimum Polynomial Interpolation with Selected Abscissas
  9.7 Chebyshev Interpolation
  9.8 Economization of Polynomial Approximations
  9.9 Uniform (Minimax) Polynomial Approximation
  9.10 Spline Approximation
  9.11 Splines with Uniform Spacing
  9.12 Spline Error Estimates
  9.13 A Special Class of Splines
  9.14 Approximation by Continued Fractions
  9.15 Rational Approximations and Continued Fractions
  9.16 Determination of Convergents of Continued Fractions
  9.17 Thiele's Continued-Fraction Approxmations
  9.18 Uniformization of Rational Approximations
  9.19 Supplementary References
    Problems
10 Numerical Solution of Equations
  10.1 Introduction
  10.2 Sets of Linear Equations
  10.3 The Gauss Reduction
  10.4 The Crout Reduction
  10.5 Intermediate Roudoff Errors
  10.6 Determination of the Inverse Matrix
  10.7 Inherent Errors
  10.8 Tridiagonal Sets of Equations
  10.9 Iterative Methods and Relaxation
  10.10 Iterative Methods for Nonlinear Equations
  10.11 The Newton-Raphson Method
  10.12 Iterative Methods of Higher Order
  10.13 Sets of Nonlinear Equations
  10.14 Iterated Synthetic Division of Polynomials. Lin's Method
  10.15 Determinacy of Zeros of Polynomials
  10.16 Bernoulli's Iteration
  10.17 Graeffe's Root-squaring Technique
  10.18 Quadratic Factors. Lin's Quadratic Method
  10.19 Bairstow Iteration
  10.20 Supplementary References
    Problems
Appendixes
A Justification of the Crout Reduction
B Bibliography
C Directory of Methods
Index
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