Asymmetric cubature formulas for polynomial integration in the triangle and square

We present five new cubature formula in the triangle and square for exact integration of polynomials. The points were computed numerically with a cardinal function algorithm which does not impose any symmetry requirements on the points. Cubature formula are presented which integrate degrees 10, 11 and 12 in the triangle and degrees 10 and 12 in the square. They have positive weights, contain no points outside the domain, and have fewer points than previously known results.