Wide stability in a new family of optimal fourth-order iterative methods

"This is the peer reviewed version of the following article: Chicharro, F. I., Cordero, A., Garrido, N., & Torregrosa, J. R. (2019). Wide stability in a new family of optimal fourth-order iterative methods. Computational and Mathematical Methods, 1(2), e1023, which has been published in final form at https://doi.org/10.1002/cmm4.1023. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving."[EN] A new family of two¿steps fourth¿order iterative methods for solving nonlinear equations is introduced based on the weight functions procedure. This family is optimal in the sense of Kung¿Traub conjecture and it is extended to design a class of iterative schemes with four step and seventh order of convergence. We are interested in analyzing the dynamical behavior of different elements of the fourth¿order class. This analysis gives us important information about the stability of these members of the family. The methods are also tested with nonlinear functions and compared with other known schemes. The results show the good features of the introduced class.This research was partially supported by the Ministerio de Ciencia, Innovación y Universidades PGC-2018-095896-B-C22 and by the Generalitat Valenciana PROMETEO/2016/089. The authors also want to thank the anonymous referees for their suggestions and comments that have improved the final version of this paper.Chicharro López, FI.; Cordero Barbero, A.; Garrido-Saez, N.; Torregrosa Sánchez, JR. (2019). Wide stability in a new family of optimal fourth-order iterative methods. Computational and Mathematical Methods. 1(2):1-14. https://doi.org/10.1002/cmm4.102311412Amat, S., & Busquier, S. (Eds.). (2016). Advances in Iterative Methods for Nonlinear Equations. SEMA SIMAI Springer Series. doi:10.1007/978-3-319-39228-8Kung, H. T., & Traub, J. F. (1974). Optimal Order of One-Point and Multipoint Iteration. Journal of the ACM, 21(4), 643-651. doi:10.1145/321850.321860King, R. F. (1973). A Family of Fourth Order Methods for Nonlinear Equations. SIAM Journal on Numerical Analysis, 10(5), 876-879. doi:10.1137/0710072Soleymani, F. (2011). Novel Computational Iterative Methods with Optimal Order for Nonlinear Equations. Advances in Numerical Analysis, 2011, 1-10. doi:10.1155/2011/270903ChicharroFI CorderoA TorregrosaJR.Memory and dynamics for a family of King‐type iterative methods. In: Proceedings of the 17th International Conference on Computational and Mathematical Methods in Science and Engineering (CMMSE);2017;Rota Spain.Blanchard, P. (1984). Complex analytic dynamics on the Riemann sphere. Bulletin of the American Mathematical Society, 11(1), 85-142. doi:10.1090/s0273-0979-1984-15240-6Chicharro, F., Cordero, A., & Torregrosa, J. (2015). Dynamics and Fractal Dimension of Steffensen-Type Methods. Algorithms, 8(2), 271-279. doi:10.3390/a8020271Chicharro, F. I., Cordero, A., & Torregrosa, J. R. (2013). Drawing Dynamical and Parameters Planes of Iterative Families and Methods. The Scientific World Journal, 2013, 1-11. doi:10.1155/2013/780153Varona, J. L. (2002). Graphic and numerical comparison between iterative methods. The Mathematical Intelligencer, 24(1), 37-46. doi:10.1007/bf03025310Amat, S., Busquier, S., & Magreñán, Á. A. (2013). Reducing Chaos and Bifurcations in Newton-Type Methods. Abstract and Applied Analysis, 2013, 1-10. doi:10.1155/2013/72670