ADAPTIVE PROCEDURE FOR EVALUATION OF DOUBLE INTEGRALS

Abstract

Adaptive procedure of numerical integration is wellknown in a case when an integrand is a function in one variable. The problem of numerical integration over multidimensional domains is considered far less often, especially when one has to develop the adaptive procedures. In this paper the problem of evaluation of double integrals with given precision is considered and the adaptive procedure is being built, and its effective recursive realization with object-oriented programming method is described The quadrature formulae of classes PB (positive boundary) and PI (positive inside) are considered, i. e. all the weights are positive and all nodes either lie strictly in the interior of the domain of integration or, in the first case, at least one node lies on the boundary of the domain. Respectively, in the first case one may hope that some of the nodes could be used more than once in the algorithm and in the second one one has the full guarantee that one will receive results within the given precision. The considered problem is actual and it also will allow to solve the equations of mathematical physics with more precision (e. g. using the method of finite elements). The main goal of this paper is to create an effective algorithm of numerical integration in a two-dimensional case which will give acceptable results on a wide enough class of functions. Practically interesting integrals, the examples of which are given in the present paper, have a physical sense and are widely used in a branch of laser refractography.