A convex combination approach for mean-based variants of Newton's method

[EN] Several authors have designed variants of Newton¿s method for solving nonlinear equations by using different means. This technique involves a symmetry in the corresponding fixed-point operator. In this paper, some known results about mean-based variants of Newton¿s method (MBN) are re-analyzed from the point of view of convex combinations. A new test is developed to study the order of convergence of general MBN. Furthermore, a generalization of the Lehmer mean is proposed and discussed. Numerical tests are provided to support the theoretical results obtained and to compare the different methods employed. Some dynamical planes of the analyzed methods on several equations are presented, revealing the great difference between the MBN when it comes to determining the set of starting points that ensure convergence and observing their symmetry in the complex plane.This research was partially funded by Spanish Ministerio de Ciencia, Innovacion y Universidades PGC2018-095896-B-C22 and by Generalitat Valenciana PROMETEO/2016/089 (Spain).Cordero Barbero, A.; Franceschi, J.; Torregrosa Sánchez, JR.; Zagati, AC. (2019). A convex combination approach for mean-based variants of Newton's method. Symmetry (Basel). 11(9):1-16. https://doi.org/10.3390/sym11091106S116119Weerakoon, 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)00100-2Özban, A. . (2004). Some new variants of Newton’s method. Applied Mathematics Letters, 17(6), 677-682. doi:10.1016/s0893-9659(04)90104-8Zhou, X. (2007). A class of Newton’s methods with third-order convergence. Applied Mathematics Letters, 20(9), 1026-1030. doi:10.1016/j.aml.2006.09.010Singh, M. K., & Singh, A. K. (2017). A New-Mean Type Variant of Newton´s Method for Simple and Multiple Roots. International Journal of Mathematics Trends and Technology, 49(3), 174-177. doi:10.14445/22315373/ijmtt-v49p524Verma, K. L. (2016). On the centroidal mean Newton’s method for simple and multiple roots of nonlinear equations. International Journal of Computing Science and Mathematics, 7(2), 126. doi:10.1504/ijcsm.2016.076403Cordero, 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.2007.01.062Chicharro, 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:10.1155/2013/78015

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