A fast numerical algorithm for finding all real solutions to a system of N nonlinear equations in a finite domain

Abstract

A highly recurrent traditional bottleneck in applied mathematics, for which the most popular codes (Mathematica and Matlab) do not offer a solution, is to find all the real solutions of a system of N nonlinear equations in a certain finite domain of the N-dimensional space of variables. We present an algorithm of minimum length and computational weight to solve this problem, resembling a graphical tool of edge detection in an image extended to N dimensions. Once the hypersurfaces (edges) defined by each nonlinear equation have been identified in a single, simultaneous step, the coincidence of the hypersurfaces in the vicinity of all the hyperpoints that constitute the solutions makes the final Newton-Raphson step rapidly convergent to all the solutions with the desired degree of accuracy. As long as N remains smaller than about five, which is often the case for physical systems that depend on fewer than five parameters, this approach demonstrates excellent effectiveness.