A family of iterative methods with sixth and seventh order convergence for nonlinear equations

In this paper, we study a new family of iterative methods for solving nonlinear equations with sixth and seventh order convergence. The new methods are obtained by composing known methods of third and fourth order with Newton’s method and using an adequate approximation for the last derivative, which provides high order of convergence and reduces the required number of functional evaluations per step. The new methods attain efficiency indices of 1.5651 and 1.6266, which makes them competitive. We introduce a new efficiency index involving the computational effort as well as the functional evaluations per iteration. We use this new index, in combination with the usual efficiency index, in order to compare the methods described in the paper with other known methods and present several numerical tests.