Semilocal convergence of a k-step iterative process and its application for solving a special kind of conservative problems

[EN] In this paper, we analyze the semilocal convergence of k-steps Newton's method with frozen first derivative in Banach spaces. The method reaches order of convergence k + 1. By imposing only the assumption that the Fr,chet derivative satisfies the Lipschitz continuity, we define appropriate recurrence relations for obtaining the domains of convergence and uniqueness. We also define the accessibility regions for this iterative process in order to guarantee the semilocal convergence and perform a complete study of their efficiency. Our final aim is to apply these theoretical results to solve a special kind of conservative systems.Hernández-Verón, MA.; Martínez Molada, E.; Teruel-Ferragud, C. (2017). Semilocal convergence of a k-step iterative process and its application for solving a special kind of conservative problems. Numerical Algorithms. 76(2):309-331. https://doi.org/10.1007/s11075-016-0255-zS309331762Amat, S., Busquier, S., Bermúdez, C., Plaza, S.: On two families of high order Newton type methods. Appl. Math. Comput. 25, 2209–2217 (2012)Argyros, I.K., Hilout, S., Tabatabai, M.A.: Mathematical Modelling with Applications in Biosciences and Engineering. Nova Publishers, New York (2011)Argyros, I.K., George, S.: A unified local convergence for Jarratt-type methods in Banach space under weak conditions. Thai. J. Math. 13, 165–176 (2015)Argyros, I.K., Hilout, S.: On the local convergence of fast two-step Newton-like methods for solving nonlinear equations. J. Comput. Appl. Math. 245, 1–9 (2013)Argyros, I.K., Ezquerro, J.A., Gutiérrez, J.M., Hernández, M.A., Hilout, S.: On the semilocal convergence of efficient Chebyshev–Secant-type methods. J. Comput. Appl. Math. 235, 3195–2206 (2011)Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: Generating optimal derivative free iterative methods for nonlinear equations by using polynomial interpolation. Math. Comput. Mod. 57, 1950–1956 (2013)Ezquerro, J.A., Grau-Sánchez, M., Hernández, M. A., Noguera, M.: Semilocal convergence of secant-like methods for differentiable and nondifferentiable operators equations. J. Math. Anal. Appl. 398(1), 100–112 (2013)Honorato, G., Plaza, S., Romero, N.: Dynamics of a higher-order family of iterative methods. J. Complexity 27(2), 221–229 (2011)Jerome, J.W., Varga, R.S.: Generalizations of Spline Functions and Applications to Nonlinear Boundary Value and Eigenvalue Problems, Theory and Applications of Spline Functions. Academic Press, New York (1969)Kantorovich, L.V., Akilov, G.P.: Functional analysis Pergamon Press. Oxford (1982)Keller, H.B.: Numerical Methods for Two-Point Boundary-Value Problems. Dover Publications, New York (1992)Na, T.Y.: Computational Methods in Engineering Boundary Value Problems. Academic Press, New York (1979)Ortega, J.M.: The Newton-Kantorovich theorem. Amer. Math. Monthly 75, 658–660 (1968)Ostrowski, A.M.: Solutions of Equations in Euclidean and Banach Spaces. Academic Press, New York (1973)Plaza, S., Romero, N.: Attracting cycles for the relaxed Newton’s method. J. Comput. Appl. Math. 235(10), 3238–3244 (2011)Porter, D., Stirling, D.: Integral Equations: A Practical Treatment, From Spectral Theory to Applications. Cambridge University Press, Cambridge (1990)Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall. Englewood Cliffs, New Jersey (1964)Argyros, I.K., George, S.: Extending the applicability of Gauss-Newton method for convex composite optimization on Riemannian manifolds using restricted convergence domains. Journal of Nonlinear Functional Analysis 2016 (2016). Article ID 27Xiao, J.Z., Sun, J., Huang, X.: Approximating common fixed points of asymptotically quasi-nonexpansive mappings by a k+1-step iterative scheme with error terms. J. Comput. Appl. Math 233, 2062–2070 (2010)Qin, X., Dehaish, B.A.B., Cho, S.Y.: Viscosity splitting methods for variational inclusion and fixed point problems in Hilbert spaces. J. Nonlinear Sci. Appl. 9, 2789–2797 (2016

On a Moser-Steffensen type method for nonlinear systems of equations

This paper is devoted to the construction and analysis of a Moser–Steffensen iterative scheme. The method has quadratic convergence without evaluating any derivative nor inverse operator. We present a complete study of the order of convergence for systems of equations, hypotheses ensuring the local convergence, and finally, we focus our attention to its numerical behavior. The conclusion is that the method improves the applicability of both Newton and Steffensen methods having the same order of convergence.Peer ReviewedPostprint (author's final draft

On the local convergence study for an efficient k-step iterative method

[EN] This paper is devoted to a family of Newton-like methods with frozen derivatives used to approximate a locally unique solution of an equation. The methods have high order of convergence but only using first order derivatives. Moreover only one LU decomposition is required in each iteration. In particular, the methods are real alternatives to the classical Newton method. We present a local convergence analysis based on hypotheses only on the first derivative. These types of local results were usually proved based on hypotheses on the derivative of order higher than two although only the first derivative appears in these types of methods (Bermficlez et al., 2012; Petkovic et al., 2013; Traub, 1964). We apply these methods to an equation related to the nonlinear complementarity problem. Finally, we find the most efficient method in the family for this problem and we perform a theoretical and a numerical study for it. (C) 2018 Elsevier B.V. All rights reserved.Research was supported in part by Programa de Apoyo a Ia investigacion de Ia fundacion Seneca-Agencia de Ciencia y Tecnologia de la Region de Murcia 19374/PI/14, by the project of Generalitat Valenciana Prometeo/2016/089 and the projects MTM2015-64382-P (MINECO/FEDER), MTM2014-52016-C2-1-P and MTM2014-52016-C2-2-P of the Spanish Ministry of Science and Innovation.Amat, S.; Argyros, IK.; Busquier Saez, S.; Hernández-Verón, MA.; Martínez Molada, E. (2018). On the local convergence study for an efficient k-step iterative method. Journal of Computational and Applied Mathematics. 343:753-761. https://doi.org/10.1016/j.cam.2018.02.028S75376134

A multistep Steffensen-type method for solving nonlinear systems of equations

[EN] This paper is devoted to the semilocal analysis of a high-order Steffensen-type method with frozen divided differences. The methods are free of bilinear operators and derivatives, which constitutes the main limitation of the classical high-order iterative schemes. Although the methods are more demanding, a semilocal convergence analysis is presented using weaker conditions than the classical Steffensen method.This work was supported supported in part by by Programa de Apoyo a la investigación de la fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 19374/PI/14, by the project of Generalitat Valenciana Prometeo/2016/089 and the projects MTM2015-64382-P (MINECO/FEDER), MTM2014-52016-C2-1-P and MTM2014-52016-C2-2-P of the Spanish Ministry of Science and InnovationAmat, S.; Argyros, IK.; Busquier, S.; Hernández-Verón, MA.; Magreñán, AA.; Martínez Molada, E. (2020). A multistep Steffensen-type method for solving nonlinear systems of equations. Mathematical Methods in the Applied Sciences. 43(13):7518-7536. https://doi.org/10.1002/mma.5599S75187536431