Improved cubature formulae of high degrees of exactness for the square

The method of constructing minimal cubature rules with high algebraic degrees of exactness is developed by adapting a powerful algorithm for solving the system of nonlinear equations. As a result, new cubature formulae of degrees 15, 17, 19, 21, and 23 are derived for the square. They lead to lower numbers of knots and/or to better quality with respect to those known previously. The formulae obtained should be considered as the most efficient for the calculation of two-dimensional integrals with a high precision.