A tensor extension of the bisection method

where the involved functions f and g are continuous (not necessarily differentiable) functions. The method is a tensor extension of the bisection method. Classical generalizations of bisection method are based on the topological degree, are called exclusion algorithms and are intended for separating all solutions of a system of equation in a given region [20,11,21,23]. More recent research line starts from different characterization of topological degree (see [10]) and numerical methods are developed for complex functions of complex variable and for analytic functions [14,18,24]. A delicate issue for all these methods is the proper subdivision of the boundary: such kind of investigation is still actual [3,17,22]. Even though the algorithm we introduce is an extension of the bisection method, the tool we use and the results we achieve are quite different from those of the exclusion algorithms. In fact we use topological degree implicitly and we assume that the functions involved in (1) are continuous (not necessarily differentiable) functions, as we follow research line developed in [4]. More specifically the tools we use are similar of those employed in the continuation methods or in the global homotopy methods (as described in [7,2]). The algorithm we introduce inherits its properties from the classical bisection: it does not require the differentiability of the involved functions, it has linear rate of convergence and provides sure convergence. As the classical bisection, and in contrast with exclusion algorithms, if the system of equations has more than one solution in the given region, the method converges to one of them and disregards the others. On the other hand, and as payback, if the system of equations has only one solution in the given region, the method allows to to prove the unicity of the solution. This feature is missing in the aforesaid exclusion algorithms.