Optimal high-order methods for solving nonlinear equations

A class of optimal iterative methods for solving nonlinear equations is extended up to sixteenth-order of convergence. We design them by using the weight function technique, with functions of three variables. Some numerical tests are made in order to confirm the theoretical results and to compare the new methods with other known ones.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and FONDOCYT 2011-1-B1-33 Republica Dominicana.Artidiello Moreno, SDJ.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Penkova Vassileva, M. (2014). Optimal high-order methods for solving nonlinear equations. Journal of Applied Mathematics. 2014. https://doi.org/10.1155/2014/5916382014Kung, H. T., & Traub, J. F. (1974). Optimal Order of One-Point and Multipoint Iteration. Journal of the ACM, 21(4), 643-651. doi:10.1145/321850.321860Artidiello, S., Chicharro, F., Cordero, A., & Torregrosa, J. R. (2013). Local convergence and dynamical analysis of a new family of optimal fourth-order iterative methods. International Journal of Computer Mathematics, 90(10), 2049-2060. doi:10.1080/00207160.2012.748900Chun, C., Lee, M. Y., Neta, B., & Džunić, J. (2012). On optimal fourth-order iterative methods free from second derivative and their dynamics. Applied Mathematics and Computation, 218(11), 6427-6438. doi:10.1016/j.amc.2011.12.013Ik Kim, Y. (2012). A triparametric family of three-step optimal eighth-order methods for solving nonlinear equations. International Journal of Computer Mathematics, 89(8), 1051-1059. doi:10.1080/00207160.2012.673597Khan, Y., Fardi, M., & Sayevand, K. (2012). A new general eighth-order family of iterative methods for solving nonlinear equations. Applied Mathematics Letters, 25(12), 2262-2266. doi:10.1016/j.aml.2012.06.014Džunić, J., & Petković, M. S. (2012). A Family of Three-Point Methods of Ostrowski’s Type for Solving Nonlinear Equations. Journal of Applied Mathematics, 2012, 1-9. doi:10.1155/2012/425867Soleymani, F., Sharifi, M., & Somayeh Mousavi, B. (2011). An Improvement of Ostrowski’s and King’s Techniques with Optimal Convergence Order Eight. Journal of Optimization Theory and Applications, 153(1), 225-236. doi:10.1007/s10957-011-9929-9Thukral, R. (2012). New Sixteenth-Order Derivative-Free Methods for Solving Nonlinear Equations. American Journal of Computational and Applied Mathematics, 2(3), 112-118. doi:10.5923/j.ajcam.20120203.08Sharma, J. R., Guha, R. K., & Gupta, P. (2013). Improved King’s methods with optimal order of convergence based on rational approximations. Applied Mathematics Letters, 26(4), 473-480. doi:10.1016/j.aml.2012.11.011Chun, C. (2008). Some fourth-order iterative methods for solving nonlinear equations. Applied Mathematics and Computation, 195(2), 454-459. doi:10.1016/j.amc.2007.04.105King, R. F. (1973). A Family of Fourth Order Methods for Nonlinear Equations. SIAM Journal on Numerical Analysis, 10(5), 876-879. doi:10.1137/0710072Džunić, J., Petković, M. S., & Petković, L. D. (2011). A family of optimal three-point methods for solving nonlinear equations using two parametric functions. Applied Mathematics and Computation, 217(19), 7612-7619. doi:10.1016/j.amc.2011.02.055Weerakoon, S., & Fernando, T. G. I. (2000). A variant of Newton’s method with accelerated third-order convergence. Applied Mathematics Letters, 13(8), 87-93. doi:10.1016/s0893-9659(00)00100-2Cordero, A., & Torregrosa, J. R. (2007). Variants of Newton’s Method using fifth-order quadrature formulas. Applied Mathematics and Computation, 190(1), 686-698. doi:10.1016/j.amc.2007.01.06

Drawing dynamical and parameters planes of iterative families and methods

The complex dynamical analysis of the parametric fourth-order Kim s iterative family is made on quadratic polynomials, showing the MATLAB codes generated to draw the fractal images necessary to complete the study. The parameter spaces associated with the free critical points have been analyzed, showing the stable (and unstable) regions where the selection of the parameter will provide us the excellent schemes (or dreadful ones).The authors thank the anonymous referees for their valuable comments and for the suggestions to improve the readability of the paper. This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02.Chicharro López, FI.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2013). Drawing dynamical and parameters planes of iterative families and methods. The Scientific World Journal. 2013. https://doi.org/10.1155/2013/780153S2013Douady, A., & Hubbard, J. H. (1985). On the dynamics of polynomial-like mappings. Annales scientifiques de l’École normale supérieure, 18(2), 287-343. doi:10.24033/asens.1491Curry, J. H., Garnett, L., & Sullivan, D. (1983). On the iteration of a rational function: Computer experiments with Newton’s method. Communications in Mathematical Physics, 91(2), 267-277. doi:10.1007/bf01211162Varona, J. L. (2002). Graphic and numerical comparison between iterative methods. The Mathematical Intelligencer, 24(1), 37-46. doi:10.1007/bf03025310Gutiérrez, J. M., Hernández, M. A., & Romero, N. (2010). Dynamics of a new family of iterative processes for quadratic polynomials. Journal of Computational and Applied Mathematics, 233(10), 2688-2695. doi:10.1016/j.cam.2009.11.017Honorato, G., Plaza, S., & Romero, N. (2011). Dynamics of a higher-order family of iterative methods. Journal of Complexity, 27(2), 221-229. doi:10.1016/j.jco.2010.10.005Chicharro, F., Cordero, A., Gutiérrez, J. M., & Torregrosa, J. R. (2013). Complex dynamics of derivative-free methods for nonlinear equations. Applied Mathematics and Computation, 219(12), 7023-7035. doi:10.1016/j.amc.2012.12.075Artidiello, S., Chicharro, F., Cordero, A., & Torregrosa, J. R. (2013). Local convergence and dynamical analysis of a new family of optimal fourth-order iterative methods. International Journal of Computer Mathematics, 90(10), 2049-2060. doi:10.1080/00207160.2012.748900Scott, M., Neta, B., & Chun, C. (2011). Basin attractors for various methods. Applied Mathematics and Computation, 218(6), 2584-2599. doi:10.1016/j.amc.2011.07.076Chun, C., Lee, M. Y., Neta, B., & Džunić, J. (2012). On optimal fourth-order iterative methods free from second derivative and their dynamics. Applied Mathematics and Computation, 218(11), 6427-6438. doi:10.1016/j.amc.2011.12.013Neta, B., Scott, M., & Chun, C. (2012). Basin attractors for various methods for multiple roots. Applied Mathematics and Computation, 218(9), 5043-5066. doi:10.1016/j.amc.2011.10.071Cordero, A., García-Maimó, J., Torregrosa, J. R., Vassileva, M. P., & Vindel, P. (2013). Chaos in King’s iterative family. Applied Mathematics Letters, 26(8), 842-848. doi:10.1016/j.aml.2013.03.012Devaney, R. L. (1999). The Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence. The American Mathematical Monthly, 106(4), 289. doi:10.2307/2589552Ik Kim, Y. (2012). A triparametric family of three-step optimal eighth-order methods for solving nonlinear equations. International Journal of Computer Mathematics, 89(8), 1051-1059. doi:10.1080/00207160.2012.673597Cordero, A., Torregrosa, J. R., & Vindel, P. (2013). Dynamics of a family of Chebyshev–Halley type methods. Applied Mathematics and Computation, 219(16), 8568-8583. doi:10.1016/j.amc.2013.02.04

Study of iterative methods though the Cayley quadratic test

[EN] Many iterative methods for solving nonlinear equations have been developed recently. The main advantage claimed by their authors is the improvement of the order of convergence. In this work, we compare their dynamical behavior on quadratic polynomials with the one of Newton's scheme. This comparison is defined in what we call Cayley Quadratic Test (CQT) which can be used as a first test to check the efficiency of such methods. Moreover we make a brief insight in cubic polynomials. (C) 2014 Elsevier B.V. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02.Babajee, D.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2016). Study of iterative methods though the Cayley quadratic test. Journal of Computational and Applied Mathematics. 291:358-369. https://doi.org/10.1016/j.cam.2014.09.020S35836929

Stability analysis of a family of optimal fourth-order methods for multiple roots

[EN] Complex dynamics tools applied on the rational functions resulting from a parametric family of roots solvers for nonlinear equations provide very useful results that have been stated in the last years. These qualitative properties allow the user to select the most efficient members from the family of iterative schemes, in terms of stability and wideness of the sets of convergent initial guesses. These tools have been widely used in the case of iterative procedures for finding simple roots and only recently are being applied on the case of multiplicity m >1. In this paper, by using weight function procedure, we design a general class of iterative methods for calculating multiple roots that includes some known methods. In this class, conditions on the weight function are not very restrictive, so a large number of different subfamilies can be generated, all of them are optimal with fourth-order of convergence. Their dynamical analysis gives us enough information to select those with better properties and test them on different numerical experiments, showing their numerical properties.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P, Generalitat Valenciana PROMETEO/2016/089 and Schlumberger Foundation-Faculty for Future Program.Zafar, F.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2019). Stability analysis of a family of optimal fourth-order methods for multiple roots. Numerical Algorithms. 81(3):947-981. https://doi.org/10.1007/s11075-018-0577-0S94798181

A new fourth-order family for solving nonlinear problems and its dynamics

In this manuscript, a new parametric class of iterative methods for solving nonlinear systems of equations is proposed. Its fourth-order of convergence is proved and a dynamical analysis on low-degree polynomials is made in order to choose those elements of the family with better conditions of stability. These results are checked by solving the nonlinear system that arises from the partial differential equation of molecular interaction.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-{01, 02} and Universitat Politecnica de Valencia SP20120474.Cordero Barbero, A.; Feng, L.; Magrenan, A.; Torregrosa Sánchez, JR. (2015). A new fourth-order family for solving nonlinear problems and its dynamics. Journal of Mathematical Chemistry. 53(3):893-910. https://doi.org/10.1007/s10910-014-0464-4S893910533R.C. Rach, J.S. Duan, A.M. Wazwaz, Solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. J. 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On the convergence of a higher order family of methods and its dynamics

[EN] In this paper, we present the study of the local convergence of a higher-order family of methods. Moreover, the dynamical behavior of this family of iterative methods applied to quadratic polynomials is studied. Some anomalies are found in this family by means of studying the associated rational function. Parameter spaces are shown and the study of the stability of all the fixed points is presented. (C) 2016 Elsevier B.V. All rights reserved.This research was supported by Universidad Internacional de La Rioja (UNIR, http://www.unir.net), under the Plan Propio de Investigación, Desarrollo e Innovación 3 [2015–2017]. Research group: Modelación matemática aplicada a la ingeniería(MOMAIN), by the grant SENECA 19374/PI/14 and by Ministerio de Ciencia y Tecnología MTM2014-52016-C2-{01,02}-P.Argyros, IK.; Cordero Barbero, A.; Alberto Magreñán, A.; Torregrosa Sánchez, JR. (2017). On the convergence of a higher order family of methods and its dynamics. Journal of Computational and Applied Mathematics. 309:542-562. https://doi.org/10.1016/j.cam.2016.04.022S54256230