Variants of Newton’s method for functions of several variables q A. Cordero * , Juan R. Torregrosa Departamento de Matema ´ tica Aplicada, Universidad Polite ´cnica de Valencia, Camino de Vera, s/n, 46071 Valencia, Spain Abstract Some variants of Newton’s Method are developed in this work in order to solve systems of nonlinear equations, based in trapezoidal and midpoint rules of quadrature. We prove the quadratic convergence of one of these methods. Moreover, different numeric tests confirm theoretic results and allow us to compare these variants with Newton’s classical method. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Newton’s method; Fixed point iteration; Convergence order 1. Introduction Let us consider the problem of finding a real zero of a function F : D  R n ! R n , that is, a real solution  x, of the nonlinear equation system F(x) = 0, of n equations with n variables. This solution can be obtained as a fixed point of some function G : R n ! R n by means of the fixed point iteration method x ðkþ1Þ ¼ Gðx ðkÞ Þ; k ¼ 0; 1; ... ; where x (0) is the initial estimation. The best known fixed point method is the classical Newton’s method, given by x ðkþ1Þ ¼ x ðkÞ  J F ðx ðkÞ Þ 1 F ðx ðkÞ Þ; k ¼ 0; 1; ... ; where J F (x (k) ) is the Jacobian matrix of the function F evaluated in x (k) . In the following, we remember the most common notions about the convergence of an iterative method. Definition 1. Let {x (k) } kP0 be a sequence in R n convergent to  x. Then, convergence is called (a) linear, if there exists M,0< M < 1, and k 0 such that kx ðkþ1Þ   xk 6 M kx ðkÞ   xk 8k P k 0 : 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.05.062 q This research was supported by Ministerio de Ciencia y Tecnologı ´a MTM2004-03244 and by Generalitat Valenciana GRUPOS03/062. * Corresponding author. E-mail addresses: acordero@mat.upv.es (A. Cordero), jrtorre@mat.upv.es (J.R. Torregrosa). Applied Mathematics and Computation 183 (2006) 199–208 www.elsevier.com/locate/amc