We shall establish a five point formula for evaluating ∫1 − 1 ∫1 − 1 f(x y)dx dy when f(x v) is a symmetric function composed of monomials x m y n , which is exact for m+ n≦5. When f(x y) is not symmetric then the rule is exact for monomials x m y n where m + n≦3. This result will be used 10 construct a family of nine point formulae which arc exact for m +n≦7 when f(x y) is symmetric.

Some integration formulae for symmetric functions of two variables

In this thesis numerical integration in one and two dimensions is
considered. In chapter two transformation methods are considered
primarily for singular integrals and methods of computing the
transformations themselves are derived. The well-known transformation
based on the IMT rule and error function are extended to non-standard
functions. The implementation of these rules and their performances
are demonstrated.
These transformations are then extended to two-dimensions and are
used to develop accurate rules for integrating singular integrals. In
addition to this, a polynomial transformation with the aim of the
reduction in the number of function evaluations is also considered
and the resultant product rule is applied to two-dimensional non-singular
integrals.
Finally, the use of monomials in the construction of integration
rules for non-singular two-dimensional integrals is considered and some
rules developed. In all these situations the rules developed are
tested and compared with existing methods. The results show that the
new rules compare favourably with existing ones.

Some quadrature methods for general and singular integrals in one and two dimensions

In this paper, we showcase some formulae that have been derived using the least squares optimisation method, which produces cubatures that are superior for use with singular integrals. We consider some formulae of standard and well-known structure for two differing weight functions, (i) w(x, y)=1 and (ii) w(x, y)= \sqrt{x^2 + y^2} .

The Construction Of Some Optimal 2D Cubature Formulae

The basis for the construction of optimum quadrature rules has been given in Cohen and Gismalla [1]. In this paper the ideas are extended to the construction of optimal Lobatto type formulae and the construction of optimal rules for where w(x) is a singular symmetric weight function.

Quadrature rules by parameter optimisation II

Given a symmetric rule for the approximate numerical integration of we choose criteria for optimising the parameters in that rule. The method is extended to find integration rules in higher dimensions.

D.A. Gismalla

Papers

The construction of quadrature rules by parameter optimisation