The solution of simultaneous nonlinear equations

In this paper we present an algorithm for the numerical solution of (1.1) which is quadratically convergent and requires only (N2/2 + 3N/2) function evaluations per iterative step as compared with (N2 + N) evaluations for Newton's Method. Brown [1] has formalized the algorithm in terms of an iteration function [1, pp. 8-9] and proved that the method converges locally and that the convergence is quadratic in nature [1, pp. 21-32]. Computer results, obtained by applying an ALGOL procedure based on the method to some specific nonlinear systems, are included and a comparison is made with some of the better recent methods as well as with the classical Newton's Method; these results illustrate the quadratic convergence of the method.