Numerical Integration: Adaptive Methods

Abstract

In chapter 8 we discussed some basic numerical integration rules and developed two simple routines for the mathlib library. These routines have the major disadvantage that no indicator of numerical error is provided. Also, in gaussint, the user is required to select a particular gaussian rule, which presupposes some understanding of how the rule works. Ideally we would like to have a library routine which requires only the information necessary to specify the problem (the integrand, the limits of integration and the error tolerance). Given this information, the routine should compute the integral to a specified accuracy, determining the stepsize and other parameters automatically, or warn the user if the accuracy cannot be achieved. Routines of this type are called automatic integrators.

They know not well the subtle ways I keep, and pass, and turn again.

Ralph Waldo Emerson, Brahma