Efficient Numerical Validation of Solutions of Nonlinear Systems

A new stopping criterion for Newton’s method is derived by combining the properties of the Krawczyk operator and a corollary of the Newton–Kantorovich theorem. When this criterion is satisfied the authors use the last three Newton iterates to compute an interval vector that is very likely to contain a solution of the given nonlinear system. The existence of such a solution is tested using Krawczyk’s operator. Furthermore, each element from this interval vector considered as an approximation to the solution has a relative error that is of the order of the machine precision. Extensive numerical testing has shown that the proposed method has very good practical performance.