On weighted three point quadrature rules

Abstract

Weighted three point quadrature rules are obtained in the current work giving explicit a priori bounds on the error. The results are valid for general weight functions. The robustness of the bounds are explored for specific weight functions and for a variety of integrands. A comparison of the current development is made with traditional quadrature rules and it is demonstrated that the current development has some advantages. In particular, this method allows the nodes and weights of an n point rule to be easily obtained, which may be preferential if the region of integration varies. Other explicit error bounds may be obtained in advance, thus making it possible to determine the partition required to achieve a certain error tolerance.