Some theoretical aspects of generalised quadrature methods

Generalised quadrature methods rely on generating quadrature rules for given irregular oscillatory weight functions w(x) commonly belonging to the class C-n[a,b], for some usually small n. If these weight functions are known to satisfy Lw = 0 for a differential operator L, then Lagrange's identity gLw - wMg = Z'(w,g) can be used to generate a quadrature rule by forcing exactness for a set of basis functions. Theorems which give conditions under which the computed quadrature rules will yield results correct to a required precision (usually that of the machine being employed) underpin the practical rule, and finite range integrals with weights such as sin(q(x)) and J(n)(q(x)) have been successfully integrated, for q(x) is an element of C-2[a,b]. Doubly oscillatory weights also become feasible with weights such as J(n)(q(1)(x))J(m)(q2(x)). More recent work has considered multiple quadratures and the special problems which arise with the commonly occurring infinite range integrations. In the latter case, the direct approach results in violations of the conditions of the underlying theorem and requires some modification for success. This approach has enabled several diverse practical problems to be attempted including integrals from financial market predictions, from chemical reactor analysis, from coherent optical imaging and from wave analysis on sloping beaches. (C) 2003 Elsevier Science (USA). All rights reserved.