PI-type fully symmetric quadrature rules on the 3-, …, 6-simplexes

We consider fully symmetric quadrature rules with positive weights, and with nodes lying inside the 3,…,6 dimensional simplex (so-called PI-type). PI-type fully symmetric quadrature rules up to 20-th order on the tetrahedron, 16-th order on 4-simplex, 10-th order on 5- and 6-simplexes are presented. The number of nodes of the presented quadrature rules for the corresponding orders does not exceed the known ones, and most of them are new. In the calculation we applied the modified Levenberg-Marquardt methods for solving nonlinear equations with convex constraints. The corresponding programs are implemented in MAPLE-FORTRAN environment, and the weights and nodes are first calculated using a FORTRAN program with an accuracy of 10−25 and refined up to accuracy of 10−50 using a MAPLE program.