1. 1. Overview of Numerical Quadrature The numerical evaluation of integrals is one of the oldest problems in mathematics. One can trace its roots back at least to Archimedes. The task is to compute the value of the definite integral of a given function. This is the area under a curve in one dimension or a volume in several dimensions. In addition to being a problem of great practi cal interest it has also lead to the development of mathematics of much beauty and insight. Many portions of approximation theory are directly applicable to integration and results from areas as diverse as orthogo nal polynomials, Fourier series and number theory have had important implications for the evaluation of integrals. We denote the problem addressed here as numerical integration or numerical quadrature. Over the years analysts and engineers have contributed to a growing body of theorems, algorithms and lately, programs, for the solution of this specific problem. Much effort has been devoted to techniques for the analytic evalua tion of integrals. However, most routine integrals in practical scien tific work are incapable of being evaluated in closed form. Even if an expression can be derived for the value of an integral, often this reveals itself only after inordinate amounts of error prone algebraic manipulation. Recently some computer procedures have been developed which can perform analytic integration when it is possible.
Authors and Affiliations
Computer Science Department, University of Leuven, Heverlee, Belgium
Robert Piessens
Computer Science Department, Western Michigan University, Kalamazoo, USA
Elise Doncker-Kapenga
Department of Applied and Numerical Mathematics, Technical University Vienna, Wien, Austria
Christoph W. Überhuber
National Bureau of Standards, USA
David K. Kahaner
Bibliographic Information
Book Title: Quadpack
Book Subtitle: A Subroutine Package for Automatic Integration
Authors: Robert Piessens, Elise Doncker-Kapenga, Christoph W. Überhuber, David K. Kahaner
