Integration

Abstract

Numerical integration is one of the oldest and most fascinating topics in numerical analysis. Methods of numerical integration date back to Archimedes who tried to calculate the area of a circle, which was long before the integral calculus itself was formulated. Numerical integration is, paradoxically, both simple and difficult. It is simple in that it can often be successfully accomplished by the simplest of methods. It is difficult in that for some problems it may require almost infinite amount of computer time. Unlike differentiation, integration is numerically stable process and in general there is no difficulty in computing an integral to any required accuracy permitted by the computer arithmetic or available data. In contrast to the previous Chapter, most of this Chapter is devoted to the situation where the integrand can be evaluated at any required point. Numerical integration of functions provided in form of table of values is dealt with only briefly in Section 6.1. Numerical integration in one dimension is usually referred to as quadrature. Because of the simplicity of the problem and its practical value, innumerable formulae have been developed for quadrature. Since the computers came into existence, the emphasis in numerical integration has shifted from quadrature rules, to evaluation of multiple integrals, and to development of algorithms for automatic integration.