Solving systems of nonlinear equations using the nonzero value of the topological degree

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Abstract
Two algorithms are described here for the numerical solution of a system of nonlinear equations F(X) = Θ, Θ(0,0,…,0)∈ ℝ, and F is a given continuous mapping of a region 𝒟 in ℝ n into ℝ n . The first algorithm locates at least one root of the sy stem within n -dimensional polyhedron, using the non zero v alue of the topological degree of F at θ relative to the polyhedron; th e second algorithm applies a new generalized bisection method in order to compute an approximate solution to the system. Teh size of the original n -dimensional polyhedron is arbitrary, and the method is globally convergent in a residual sense. These algorithms, in the various function evaluations, only make use of the algebraic sign of F and do not require computations of the topological degree. Moreover, they can be applied to nondifferentiable continuous functions F and do not involve derivatives of F or approximations of such derivatives.

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