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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