Chapter 6 - Automatic Integration

Abstract

This chapter discusses the automatic integration. The aim of an automatic integration scheme is to relieve the person who has to compute an integral of any need to think. By an automatic integrator, one mean a program for numerical integration Wherein the user inserts the limits of integration, a routine for computing f(x), a tolerance ɛ, and an upper bound N on the number of functional evaluations. An automatic integrator is a program intended to be used “blind,” that is, without any theoretical analysis of the situation either before the problem is placed on the machine or after it comes off the machine. The results of the automatic integration are often utilized or combined in further computation. The attractiveness of an automatic integrator should be plain to anyone who has ever been confronted with the job of carrying out much integration of complicated integrands. Automatic integration schemes can be classified as adaptive or nonadaptive and iterative or noniterative. In an adaptive integration, the points at which the integrand is evaluated are chosen in a way that depends on the nature of the integrand. In a nonadaptive integration, the sequence of integration points is chosen according to a fixed scheme independent of the nature of the integrand; however, the number of such points will usually depend on the integrand. In an iterative scheme, successive approximations to the integral are computed until there is agreement to within the given tolerance. In a noniterative scheme, information from a first approximation is used to generate a second approximation, which is then taken as the final answer.