Pseudocomposition: A technique to design predictor–corrector methods for systems of nonlinear equations q Alicia Cordero a , Juan R. Torregrosa a,⇑ , María P. Vassileva b a Instituto de Matemática Multidisciplinar, Universidad Politécnica de Valencia, Camino de Vera, s/n, 46022 Valencia, Spain b Instituto Tecnológico de Santo Domingo (INTEC), Avda. Los Próceres, Gala, Santo Domingo, Dominican Republic article info Keywords: Nonlinear systems Iterative methods Jacobian matrix Convergence order Efficiency index abstract A new technique for designing iterative methods for solving nonlinear systems is presented. This procedure, called pseudocomposition, uses a known method as a predictor and the Gaussian quadrature as a corrector. The order of convergence of the resulting scheme depends, among other factors, on the order of the last two steps of the predictor. We also introduce a new iterative algorithm of order six, and apply the mentioned technique to generate a new method of order ten. Finally, some numerical test are shown. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction The search for solutions of nonlinear systems of equations is an old and difficult problem with wide applications in sciences and engineering. The commonly used methods of resolution are iterative. The best known method, for being very simple and effective, is the Newton’s method. Its generalization to a system of equations was proposed by Ostrowski [1] and to Banach spaces by Kantorovic [2]. More recent results in iterative methods on Banach spaces can be found in [3–5]. The extension of the variants of Newton’s method described by Weerakoon and Fernando in [6], by Özban in [7] and Gerlach in [8], to the functions of several variables have been developed in [9,10,12,11]. In [9,10] families of variants of Newton’s method of order three have been designed by using open and close formulas of quadrature, including the families of the methods defined by Frontini et al. in [11]. Using the generic formula of the interpolatory quadrature, in [12] a family of methods is obtained with order of convergence 2d þ 1, under certain conditions, where d is the order up to which the partial derivatives of each coordinate function evaluated in the solution are canceled. Indeed, Darvishi et al. improved in [13] the method from Frontini et al., getting a fourth-order scheme. In addition to multi-step methods based on interpolatory quadrature, other schemes have been developed by using different techniques, as extension to several variables of one-dimensional schemes (see [14]), Adomian decomposition (see [15,16], for example), the one proposed by Darvishi and Barati in [17,18] with super cubic convergence and the methods proposed by Cordero et al. in [19] with order of convergence 4 and 5. Another procedure to develop iterative methods for nonlinear systems is the replacement of the second derivative by some approximation. In [20], Traub presented a family of multi-point methods based on approximating the second derivative that appears in the iterative formula of Chebyshev’s scheme and, more recently, Babajee et al. in [21] designed two Chebyshev-like methods free from second derivatives. The convergence theorems in Sections 2 and 3 are going to be demonstrated by means of the n-dimensional Taylor expan- sion of the functions involved. Let F : D # R n ! R n be sufficiently Frechet differentiable in D. By using the notation introduced 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.04.081 q This research was supported by Ministerio de Ciencia y Tecnología MTM2011-28636-C02-02 and by Vicerrectorado de Investigación, Universitat Politècnica de València PAID-06-2010-2285. ⇑ Corresponding author. E-mail addresses: acordero@mat.upv.es (A. Cordero), jrtorre@mat.upv.es (J.R. Torregrosa), maria.vassilev@gmail.com (M.P. Vassileva). Applied Mathematics and Computation 218 (2012) 11496–11504 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc