A generalization of two modified iterative schemes for solving nonlinear equations suggested by Traub and Ezquerro et al. is presented. This new technique allows to obtain local order of convergence 2 rho + 2 from rho improving the above mentioned results where have been obtained rho+1 and 2 rho+1 from rho respectively. The efficiency is also improved. New numerical algorithms of third and fourth order are used to check the theoretical result given. They are illustrated with numerical examples. (C) 2014 Elsevier Ltd. All rights reserved.
Grau Sanchez, Miguel; Noguera Batlle, Miguel; Díaz Barrero, José Luis Journal of computational and applied mathematics Vol. 255, p. 753-764 DOI: 10.1016/j.cam.2013.06.043 Date of publication: 2014-01-01 Journal article
A local convergence analysis for a generalized family of two step Secant-like methods with frozen operator for solving nonlinear equations is presented. Unifying earlier methods such as Secant’s, Newton, Chebyshev-like, Steffensen and other new variants the family of iterative schemes is built up, where a profound and clear study of the computational efficiency is also carried out. Numerical examples and an application using multiple precision and a stopping criterion are implemented without using any known root. Finally, a study comparing the order, efficiency and elapsed time of the methods suggested supports the theoretical results claimed.