Some new efficient multipoint iterative methods for solving nonlinear systems of equations

It is attempted to put forward a new multipoint iterative method of sixth-order convergence for approximating solutions of nonlinear systems of equations. It requires the evaluation of two vector-function and two Jacobian matrices per iteration. Furthermore, we use it as a predictor to derive a general multipoint method. Convergence error analysis, estimating computational complexity, numerical implementation and comparisons are given to verify applicability and validity for the proposed methods.This research was supported by Islamic Azad University - Hamedan Branch, Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and Universitat Politecnica de Valencia SP20120474.Lotfi, T.; Bakhtiari, P.; Cordero Barbero, A.; Mahdiani, K.; Torregrosa Sánchez, JR. (2015). Some new efficient multipoint iterative methods for solving nonlinear systems of equations. International Journal of Computer Mathematics. 92(9):1921-1934. https://doi.org/10.1080/00207160.2014.946412S1921193492

Increasing the order of convergence of iterative schemes for solving nonlinear systems

[EN] A set of multistep iterative methods with increasing order of convergence is presented, for solving systems of nonlinear equations. One of the main advantages of these schemes is to achieve high order of convergence with few Jacobian and functional evaluations, joint with the use of the same matrix of coefficients in the most of the linear systems involved in the process. Indeed, the application of the pseudocomposition technique on these proposed schemes allows us to increase their order of convergence, obtaining new high-order, efficient methods. Finally, some numerical tests are performed in order to check their practical behavior. (C) 2012 Elsevier B.V. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnología MTM2011-28636-C02-02 and FONDOCYT 2011-1-B1-33 República DominicanaCordero Barbero, A.; Torregrosa Sánchez, JR.; Penkova Vassileva, M. (2013). Increasing the order of convergence of iterative schemes for solving nonlinear systems. Journal of Computational and Applied Mathematics. 252:86-94. https://doi.org/10.1016/j.cam.2012.11.024S869425

Some iterative methods for solving a system of nonlinear equations

AbstractIn this paper, we suggest and analyze two new two-step iterative methods for solving the system of nonlinear equations using quadrature formulas. We prove that these new methods have cubic convergence. Several numerical examples are given to illustrate the efficiency and the performance of the new iterative methods. These new iterative methods may be viewed as an extension and generalizations of the existing methods for solving the system of nonlinear equations

Third-order iterative methods with applications to Hammerstein equations: A unified approach

AbstractThe geometrical interpretation of a family of higher order iterative methods for solving nonlinear scalar equations was presented in [S. Amat, S. Busquier, J.M. Gutiérrez, Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157(1) (2003) 197–205]. This family includes, as particular cases, some of the most famous third-order iterative methods: Chebyshev methods, Halley methods, super-Halley methods, C-methods and Newton-type two-step methods. The aim of the present paper is to analyze the convergence of this family for equations defined between two Banach spaces by using a technique developed in [J.A. Ezquerro, M.A. Hernández, Halley’s method for operators with unbounded second derivative. Appl. Numer. Math. 57(3) (2007) 354–360]. This technique allows us to obtain a general semilocal convergence result for these methods, where the usual conditions on the second derivative are relaxed. On the other hand, the main practical difficulty related to the classical third-order iterative methods is the evaluation of bilinear operators, typically second-order Fréchet derivatives. However, in some cases, the second derivative is easy to evaluate. A clear example is provided by the approximation of Hammerstein equations, where it is diagonal by blocks. We finish the paper by applying our methods to some nonlinear integral equations of this type

Efficient high-order methods based on golden ratio for nonlinear system

We derive new iterative methods with order of convergence four or higher, for solving nonlinear systems, by composing iteratively golden ratio methods with a modified Newton's method. We use different efficiency indices in order to compare the new methods with other ones and present several numerical tests which confirm the theoretical results. © 2010 Elsevier Inc. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnologia MTM2010-18539.Cordero Barbero, A.; Hueso Pagoaga, JL.; Martínez Molada, E.; Torregrosa Sánchez, JR. (2011). Efficient high-order methods based on golden ratio for nonlinear system. Applied Mathematics and Computation. 217(9):4548-4556. https://doi.org/10.1016/j.amc.2010.11.006S45484556217

On the Applicability of Two Families of Cubic Techniques for Power Flow Analysis

This work presents a comprehensive analysis of two cubic techniques for Power Flow (PF) studies. In this regard, the families of Weerakoon‐like and Darvishi‐like techniques are considered. Several theoretical findings are presented and posteriorly confirmed by multiple numerical results. Based on the obtained results, the Weerakoon’s technique is considered more reliable than the New‐ ton‐Raphson and Darvishi’s methods. As counterpart, it presents a high computational burden. Re‐ garding this point, the Darvishi’s technique has turned out to be quite efficient and fully competitive with the Newton’s schem