A biparametric family of four-step sixteenth-order root-finding methods with the optimal efficiency index

AbstractA biparametric family of four-step multipoint iterative methods of order sixteen to solve nonlinear equations are developed and their convergence properties are established. The optimal efficiency indices are all found to be 161/5≈1.741101. Numerical examples as well as comparison with existing methods are demonstrated to verify the developed theory

Sixth-Order Two-Point Efficient Family of Super-Halley Type Methods

The main focus of this manuscript is to provide a highly efficient two-point sixth-order family of super-Halley type methods that do not require any second-order derivative evaluation for obtaining simple roots of nonlinear equations, numerically. Each member of the proposed family requires two evaluations of the given function and two evaluations of the first-order derivative per iteration. By using Mathematica-9 with its high precision compatibility, a variety of concrete numerical experiments and relevant results are extensively treated to confirm t he t heoretical d evelopment. From their basins of attraction, it has been observed that the proposed methods have better stability and robustness as compared to the other sixth-order methods available in the literature

On developing a higher-order family of double-Newton methods with a bivariate weighting function

The article of record as published may be found at http://dx.doi.org/10.1016/j.amc.2014.12.130A high-order family of two-point methods costing two derivatives and two functions are developed by introducing a two-variable weighting function in the second step of the classical double-Newton method. Their theoretical and computational properties are fully investigated along with a main theorem describing the order of convergence and the asymptotic error constant as well as proper choices of special cases. A variety of concrete numerical examples and relevant results are extensively treated to verify the underlying theoretical development. In addition, this paper investigates the dynamics of rational iterative maps associated with the proposed method and an existing method based on illustrated description of basins of attraction for various polynomials

Stable high-order iterative methods for solving nonlinear models

[EN] There are several problems of pure and applied science which can be studied in the unified framework of the scalar and vectorial nonlinear equations. In this paper, we propose a sixth-order family of Jarratt type methods for solving nonlinear equations. Further, we extend this family to the multidimensional case preserving the order of convergence. Their theoretical and computational properties are fully investigated along with two main theorems describing the order of convergence. We use complex dynamics techniques in order to select, among the elements of this class of iterative methods, those more stable. This process is made by analyzing the conjugacy class, calculating the fixed and critical points and getting conclusions from parameter and dynamical planes. For the implementation of the proposed schemes for system of nonlinear equations, we consider some applied science problems namely, Van der Pol problem, kinematic syntheses, etc. Further, we compare them with existing sixth-order methods to check the validity of the theoretical results. From the numerical experiments, we find that our proposed schemes perform better than the existing ones. Further, we also consider a variety of nonlinear equations to check the performance of the proposed methods for scalar equations.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P and by Generalitat Valenciana PROMETEO/2016/089.Behl, R.; Cordero Barbero, A.; Motsa, SS.; Torregrosa Sánchez, JR. (2017). Stable high-order iterative methods for solving nonlinear models. Applied Mathematics and Computation. 303:70-88. https://doi.org/10.1016/j.amc.2017.01.029S708830

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/10.1080/00207160.2012.748900S204920609010Bi, 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