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Introduction To Numerical Analysis 2ND Editionby F B Hildebrand
Synopses & ReviewsPublisher 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 wellknown, 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. FiniteDifference Interpolation 5. Operations with Finite Differences 6. Numerical Solution of Differential Equations 7. LeastSquares 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 predictioncorrection 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:Wellknown, 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 ContentsPreface
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 SecondDegree 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 Uniformspacing Interpolation 3.5 NewtonCotes Integration Formulas 3.6 Composite Integration Formulas 3.7 Use of Integration Formulas 3.8 Richardson Extrapolation. Romberg Integration 3.9 Asympotic Behavior of NewtonCotes Formulas 3.10 Weighting Functions. Filon Integration 3.11 Differentiation Formulas 3.12 Supplementary References Problems 4 FiniteDifference Interpolation 4.1 Introduction 4.2 Difference Notations 4.3 Newton Forward and Backwarddifference 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 CentralDifference Integration Formulas 5.7 Subtabulation 5.8 Summation and Integration. The EulerMaclaurin 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 OpenType Formulas 6.6 Methods Based on ClosedType Formulas. PredictionCorrection Methods 6.7 The Special Case F = Ay 6.8 PropagatedError Bounds 6.9 Application to Equations of Higher Order. Sets of Equations 6.10 Special Secondorder Equations 6.11 Change of Interval 6.12 Use of Higher Derivatives 6.13 A Simple RungeKutta Method 6.14 RungeKutta Methods of Higher Order 6.15 BoundaryValue Problems 6.16 Linear Characteristicvalue Problems 6.17 Selection of a Method 6.18 Supplementary References Problems 7 LeastSquares Polynomial Approximation 7.1 Introduction 7.2 The Principle of Least Squares 7.3 LeastSquares 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: FivePoint LeastSquares 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 LegendreGauss Quadrature 8.6 LaguerreGauss Quadrature 8.7 HermiteGauss Quadrature 8.8 ChebyshevGauss Quadrature 8.9 JacobiGauss Quadrature 8.10 Formulas with Assigned Abscissas 8.11 Radau Quadrature 8.12 Lobatto Quadrature 8.13 Convergence of Gaussianquadrature 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 ContinuedFraction 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 NewtonRaphson 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 Rootsquaring 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 What Our Readers Are SayingBe the first to add a comment for a chance to win!Product Details
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