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Davis P.J., Rabinowitz P. Methods of Numerical Integration

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Davis P.J., Rabinowitz P. Methods of Numerical Integration
2nd Ed. — Academic Press, 1984. — 626 p. — (Computer science and applied mathematics). — ISBN 9781483264288, OCR.
This book describes the theoretical and practical aspects of major methods of numerical integration. Numerical integration is the study of how the numerical value of an integral can be found.
This book contains six chapters and begins with a discussion of the basic principles and limitations of numerical integration. The succeeding chapters present the approximate integration rules and formulas over finite and infinite intervals. These topics are followed by a review of error analysis and estimation, as well as the application of functional analysis to numerical integration. A chapter describes the approximate integration in two or more dimensions. The final chapter looks into the goals and processes of automatic integration, with particular attention to the application of Tschebyscheff polynomials.
Preface to First Edition
Preface to Second Edition
Introduction
Why Numerical Integration?
Formal Differentiation and Integration on Computers
Numerical Integration and Its Appeal in Mathematics
Limitations of Numerical Integration
The Riemann Integral
Improper Integrals
The Riemann Integral in Higher Dimensions
More General Integrals
The Smoothness of Functions and Approximate Integration
Weight Functions
Some Useful Formulas
Orthogonal Polynomials
Short Guide to the Orthogonal Polynomials
Some Sets of Polynomials Orthogonal Over Figures in the Complex Plane
Extrapolation and Speed-Up
Numerical Integration and the Numerical Solution of Integral Equations
Approximate Integration Over a Finite Interval
Primitive Rules
Simpson's Rule
Nonequally Spaced Abscissas
Compound Rules
Integration Formulas of Interpolatory Type
Integration Formulas of Open Type
Integration Rules of Gauss Type
Integration Rules Using Derivative Data
Integration of Periodic Functions
Integration of Rapidly Oscillatory Functions
Contour Integrals
Improper Integrals (Finite Interval)
Indefinite Integration
Approximate Integration Over Infinite Intervals
Change of Variable
Proceeding to the Limit
Truncation of the Infinite Interval
Primitive Rules for the Infinite Interval
Formulas of Interpolatory Type
Gaussian Formulas for the Infinite Interval
Convergence of Formulas of Gauss Type for Singly and Doubly Infinite Intervals
Oscillatory Integrands
The Fourier Transform
The Laplace Transform and Its Numerical Inversion
Error Analysis
Types of Errors
Roundoff Error for a Fixed Integration Rule
Truncation Error
Special Devices
Error Estimates through Differences
Error Estimates through the Theory of Analytic Functions
Application of Functional Analysis to Numerical Integration
Errors for Integrands with Low Continuity
Practical Error Estimation
Approximate Integration in Two or More Dimensions
Introduction
Some Elementary Multiple Integrals Over Standard Regions
Change of Order of Integration
Change of Variables
Decomposition into Elementary Regions
Cartesian Products and Product Rules
Rules Exact for Monomials
Compound Rules
Multiple Integration by Sampling
The Present State of the Art
Automatic Integration
The Goals of Automatic Integration
Some Automatic Integrators
Romberg Integration
Automatic Integration Using Tschebyscheff Polynomials
Automatic Integration in Several Variables
Concluding Remarks
Appendix
On the Practical Evaluation of Integrals
Fortran Programs
Bibliography of Algol, Fortran, and PL/I Procedures
Bibliography of Tables
Bibliography of Books and Articles
Index
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