Effective computation of matrix elements between polynomial basis functions

Two methods of evaluating matrix elements of a function in a polynomial basis are considered: the expansion method, where the function is expanded in the basis and the integrals are evaluated analytically, and the numerical method, where the integration is performed directly using numerical quadrature. A reduced grid is proposed for the latter which makes use of the symmetry of the basis. Comparison of the two methods is presented in the context of evaluation of matrix elements in a non-direct product basis. If high accuracy of all matrix elements is required then the expansion method is the best choice. If however the accuracy of high order matrix elements is not important (as in variational ro-vibrational calculations where one is typically interested only in the lowest eigenstates), then the method based on the reduced grid offers sufficient accuracy and is much quicker than the expansion method.