$N$-Dimensional curve following for solving numerically systems of nonlinear equations

This paper presents a methodology for finding numerically, by means of curve-following, all real solutions of a general system of $n$ nonlinear equations in $n$ unknowns, within a given $n$-dimensional box. The main idea behind our method is a) to locate all parts of the curves formed by a selected subset of $n-1$ equations of the initial system, b) follow these parts numerically within the given $n$-box and c) during this process, find all their intersection points with the hypersurface that represents the left-out equation of the initial system. With proper handling techniques, both stages (a) and (b) can be done with safety even when using a rapidly - but locally - convergent method such as Newton’s method. Stage © on the other hand is theoretically straightforward and can be implemented by examining sign change and using bisection. However, improvement of performance with automatic step-size adaptation is also feasible. Since the choice of the left-out equation as well as the order of the unknowns, in some problems may affect the algorithm’s performance, a secondary procedure for problem optimization has also been developed. This integrated algorithm has been successfully tested in all classical test-problems that are used in the literature (in total about 130 problems with dimension up to $n=10$) and achieved to locate approximations to all the known real solutions of each test-problem, with a user-provided accuracy. It has also proved to be more than 10 times quicker than Kuiken’s 2D curve-following method, when compared in two-dimensional problems.