The construction of cubature formulae by continuation

A cubature formulaQ is an approximation of ann-dimensional integralI. Q is exact for the space spanned by the polynomialsf ₁, ...,f _d if it verifies the system of equations: $$Q[f_i ] = I[f_i ] i = 1,...,d.$$ The unknowns are knots and weights of the cubature formula. We suppose that there are as many unknowns as equations. For searching solutions to this system, we construct a family of systems depending continuously on a parametert: $$Q[f_i (t)] = I[f_i (t)] i = 1,...,d,$$ coinciding with the previous system fort=1 and whose solutions att=0 are easily computed. The solution curves originating from these solutions are followed numerically and may yield a solution fort=1.