Application of the Monte Carlo method to systems of nonlinear algebraic equations

Abstract

It is, in general, a difficult problem to find all the roots of a given system of nonlinear algebraic equations with several unknowns, even though we meet with such demands in many fields of applied sciences. We have several methods for such problems, which have been extended from some powerful method for algebraic equations with a single unknown. Iterative procedures thus developed, are often found unsuccessful, since they do not necessarily converge, given arbitrary starting values. The conventional methods of iteration are guaranteed against divergence, only if the starting value is well enough in the neighborhood of the solution 1. Thus, as long as we have no means to place all the roots even very roughly, we have no working method to find all the roots of a given system of nonlinear algebraic equations. In the present paper, a new scheme of the Monte Carlo technique is proposed in order to find approximate locations of these roots, which are believed to provide successful starting values for the current classical methods of iteration. The results of a test on a digital computer are also illustrated, together with a sketch of the actual computer procedures. Apart from the novelty of the underlying concept, the method has a mer~e~iff that it can treat cases with many variables with no particular increase in complexity. 2. Principle