Widely Convergent Method for Finding Multiple Solutions of Simultaneous Nonlinear Equations

A new method has been developed for solving a system of nonlinear equations g(x) = 0. This method is based on solving the related system of differential equations dg/dt ± g(x) = 0 wherein the sign is changed whenever the corresponding trajectory x(t) encounters a change in sign of the Jacobian determinant or arrives at a solution point of g(x) = 0. This procedure endows the method with a much wider region of convergence than other methods (occasionally, even global convergence) and enables it to find multiple solutions of g(x) = 0 one after the other. The principal limitations of the method relate to the extraneous singularities of the differential equation. The role of these singularities is illustrated by several examples. In addition, the extension of the method to the problem of finding multiple extrema of a function of N variables is explained and some examples are given.