An Optimal Complexity Algorithm for Computing the Topological Degree in Two Dimensions

An algorithm is presented for computing the topological degree for any function from a class F. The class F consists of functions $f:C \to \mathbb{R}^2 $ defined on C, the unit square, which satisfy the Lipschitz condition with constant $K > 0$, and whose infinity norm on the boundary of C is at least $d > 0$. For functions in this class, the algorithm could be useful in solving nonlinear systems of equations and also in other aspects of nonlinear analysis.A worst-case lower bound, $m^ * = 4\lfloor {K / {4d}} \rfloor $, is established on the number of function evaluations necessary to compute the topological degree for any function f from the class F. The parallel information used by our algorithm permits the computation of the degree for every f in F with $m^ * $ function evaluations. The cost of our algorithm is shown to be almost equal to the complexity of the problem.The algorithm has been implemented and tested. Performance information for a set of 10 systems of equations is reported.