Newton–Cotes cubature rules over (d+1)(d+1)-pencil lattices

In this paper, Newton–Cotes cubature rules are extended to (d+1)(d+1)-pencil lattices over simplices and simplicial partitions. The closed form of the cubature rules as well as the error term are determined. Further, the basic cubature rules can be combined with an adaptive algorithm over simplicial partitions. The key point of the algorithm is a subdivision step that refines a (d+1)(d+1)-pencil lattice over a simplex to its subsimplices. If the number of function evaluations is crucial, the additional freedom provided by (d+1)(d+1)-pencil lattices may be used to decrease it significantly.