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Introduction To Numerical Analysis 2ND Edition

by

Introduction To Numerical Analysis 2ND Edition Cover

 

Synopses & Reviews

Publisher Comments:

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.

Synopsis:

Well-known, respected introduction, updated to integrate concepts and procedures associated with computers. Computation, approximation, interpolation, numerical differentiation and integration, smoothing of data, more. Includes 150 additional problems in this edition.

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

Product Details

ISBN:
9780486653631
Author:
Hildebrand, F. B.
Author:
Hildebrand, Francis Begnaud
Author:
Hildebrand, F. B.
Author:
Mathematics
Publisher:
Dover Publications
Location:
New York :
Subject:
General
Subject:
Calculus
Subject:
Mathematical Analysis
Subject:
Numerical analysis
Subject:
General Mathematics
Subject:
Mathematics-Calculus
Copyright:
Edition Number:
2
Edition Description:
Second Edition
Series:
Dover Books on Mathematics
Series Volume:
99-856
Publication Date:
19870631
Binding:
TRADE PAPER
Language:
English
Illustrations:
Yes
Pages:
704
Dimensions:
8.5 x 5.38 in 1.66 lb

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Related Subjects


Science and Mathematics » Mathematics » Advanced
Science and Mathematics » Mathematics » Analysis General
Science and Mathematics » Mathematics » Calculus » General
Science and Mathematics » Mathematics » General
Science and Mathematics » Mathematics » Numeric Analysis
Science and Mathematics » Physics » General

Introduction To Numerical Analysis 2ND Edition New Trade Paper
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Product details 704 pages Dover Publications - English 9780486653631 Reviews:
"Synopsis" by ,
Well-known, respected introduction, updated to integrate concepts and procedures associated with computers. Computation, approximation, interpolation, numerical differentiation and integration, smoothing of data, more. Includes 150 additional problems in this edition.
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