Local convergence and dynamical analysis of a new family of optimal fourth-order iterative methods

In this paper, a family of new fourth-order optimal iterative methods for solving nonlinear equations is proposed. The classical King s family of fourth-order schemes is obtained as an special case. We also present results for describing the conjugacy classes and dynamics of some of the presented methods for complex polynomials of different degrees.The authors thank the referees for their valuable comments and for their suggestions to improve the readability of the paper. This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and FONDOCYT Republica Dominicana.Artidiello Moreno, SDJ.; Chicharro López, FI.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (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. https://doi.org/10.1080/00207160.2012.748900204920609010Bi, W., Ren, H., & Wu, Q. (2009). Three-step iterative methods with eighth-order convergence for solving nonlinear equations. Journal of Computational and Applied Mathematics, 225(1), 105-112. doi:10.1016/j.cam.2008.07.004Blanchard, P. (1995). The dynamics of Newton’s method. Proceedings of Symposia in Applied Mathematics, 139-154. doi:10.1090/psapm/049/1315536Chun, 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.2011.12.013Douady, 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.1491DRAKOPOULOS, V. (1999). HOW IS THE DYNAMICS OF KÖNIG ITERATION FUNCTIONS AFFECTED BY THEIR ADDITIONAL FIXED POINTS? Fractals, 07(03), 327-334. doi:10.1142/s0218348x99000323King, R. F. (1973). A Family of Fourth Order Methods for Nonlinear Equations. SIAM Journal on Numerical Analysis, 10(5), 876-879. doi:10.1137/0710072Kneisl, K. (2001). Julia sets for the super-Newton method, Cauchy’s method, and Halley’s method. Chaos: An Interdisciplinary Journal of Nonlinear Science, 11(2), 359-370. doi:10.1063/1.1368137Kung, H. T., & Traub, J. F. (1974). Optimal Order of One-Point and Multipoint Iteration. Journal of the ACM, 21(4), 643-651. doi:10.1145/321850.321860Wang, X., & Liu, L. (2009). Two new families of sixth-order methods for solving non-linear equations. Applied Mathematics and Computation, 213(1), 73-78. doi:10.1016/j.amc.2009.03.00