On a Path Following Method for Systems of Equations.

TLDR: In this paper, a general predictor-corrector method is described for following the curve, which is a globalization of the classical Davidenko approach and is shown to follow the path to any desired degree of accuracy and is convergent.

Abstract: : The problem of finding one or all solutions to systems of equations, equilibrium, fixed points, or to dynamical systems is considered. In the last few years, a new method has emerged for solving this problem. The idea is to start at a given solution of a simpler problem and to follow a path of solutions as the path parameter (and hence, the problem) is gradually changed. This path is proved to exist via topological approaches and is shown to lead to the 'right' place. In this paper, a general predictor-corrector method is described for following the curve. It is a globalization of the classical Davidenko approach. It is shown that the method can follow the path to any desired degree of accuracy and is convergent.