Generalized analysis of methods for solving systems of nonlinear equations with point and interval coefficients

In this article we consider problems of interval estimation of a set of solutions to point and interval (partially interval) systems of nonlinear equations. Most developed interval methods are intended only for estimating solutions of point nonlinear systems in some given interval box. And methods for estimating solution sets of nonlinear interval systems are not yet very developed, since the solution sets of such systems geometrically represent a rather complex structure. Here we conducted a general analysis on existing classical interval methods to test their applicability for interval systems. In this case, we chose the methods of Newton and Krawczyk. The results of the analysis show that these and similar other iterative methods are generally not applicable for interval systems due to the limited admissible area. Based on the results of the analysis, a new combined vertex method for outer estimation of solution sets of interval nonlinear systems is proposed, which includes these classical interval methods. Numerical experiments have shown that the proposed method is more efficient and gives more accurate estimates in feasible regions than the direct application of Newton, Krawczyk or Hansen-Sengupta interval methods for interval systems.