A Metalgorithm for Adaptive Quadrature

Abstract

The few adaptive quadrature algorithms that have appeared are significantly superior to traditional numerical integration algorithms The concept of metalgorithm is introduced to provide a framework for the systematic study of the range of interesting adaptive quadrature algorithms A principal result is that there are from i to 10 million potentially interesting and distinct algorithms This is followed by a considerable development of metalgorithm analysis. In partmular, theorems about the convergence properties of various classes of algorithms are established which theoretically show the experimentally observed superiority of these algorithms. Roughly, these theorems state. (a) for “well-behaved” integrands adaptive algorithms are just as efficient and effective as traditional algorithms of a “comparable” nature, (b) adaptive algorithms are equally effective for “badly behaved” integrands where traditional ones are ineffective The final part of the paper introduces the concept of a characteristic length and its role is illustrated in an analyms of three concrete realizations of the metalgorithm, including the algorithms CADRE and SQUANK. © 1975, ACM. All rights reserved.