A family of derivative-free methods of seventh-order convergence for solving nonlinear equations is suggested. In the proposed methods, several linear combinations of divided differences are used in order to get a good estimation of the derivative of the given function at the different steps of the iteration. The efficiency indices of the members of this family are equal to 1.6266. Also, numerical examples are used to show the performance of the presented methods, on smooth and nonsmooth equations, and to compare with other derivative-free methods, including some optimal fourth-order ones, in the sense of Kung-Traub’s conjecture.
"A Family of Derivative-Free Methods with High Order of Convergence and Its Application to Nonsmooth Equations." Abstr. Appl. Anal. 2012 (SI12) 1 - 15, 2012. https://doi.org/10.1155/2012/836901