Multidimensional Quadrature

Abstract

In this thesis original software is presented for the approximate evaluation of multiple integrals over two basic regions of integration; the simplex and the hypercube. The majority of the work is based upon an adaptive approach. One exception is a program which generates a sequence of product rules from the one dimensional Patterson family of formulae and applies these rules iteratively over a hypercube type region of integration. A basic adaptive algorithm for multidimensional quadrature is described and programs for both the simplex and the hypercube based on this are presented. The problems of testing and comparison of quadrature routines are discussed. Two alternative approaches to storing integrand evaluations are presented; the first using a linked list type data structure while the second uses scatter storage techniques. Programs for the hypercube which use the basic adaptive strategy but store the integrand evaluations illustrate the storage techniques. It is suggested that the storing of integrand evaluations is only feasible for particularly "expensive" integrands. A case is presented for the adoption of a global subdivision strategy as opposed to a local subdivision strategy in the construction of multidimensional quadrature algorithms. A global strategy reduces the total number of integrand evaluations used by producing a result which is closer to the required tolerance. One approach to extending the methods to other regions of integration is described and the problems associated with this are considered. Finally the possibilities of using a multiple processor for the evaluation of multiple integrals are discussed.