The paper is devoted to the numerical solution of nonlinear equations
. The authors use Newton’s iteration for the direct Newtonian interpolation of the function to construct optimal Steffensen-type methods of second-, fourth- and eighth-order, which use only two, three and four evaluations of the function, respectively. Moreover, they deduce the corresponding error equations and asymptotic convergence constants. The proposed general optimal Steffensen-type family only uses
to achieve the optimal
th order of convergence for solving the simple root of nonlinear functions, and the authors compare this family with Newton’s method, Steffensen’s method, Ren-Wu-Bi’s method, Kung-Traub’s method and Neta-Petković’s method for solving nonlinear equations in numerical examples.