Efficient computation of cubature rules with application to new asymmetric rules on the triangle

This paper presents a new, efficient method for computing cubature rules, based on least-squares minimisation and the use of orthogonal bases. The method, which can be applied for any integration domain, is tested here for the case of asymmetric cubature rules on the triangle showing how the computation of the necessary basis, and its derivatives, can be optimised. The numerical results presented include three new cubature rules with fewer points than known rules of the same degree.