Efficient Families of Multipoint Iterative Methods for Solving Nonlinear Equations

new efficient families of multipoint iterative methods of fourth and eighth order convergence are constructed for finding simple roots of nonlinear equation Gamma(s) = 0. Both families satisfy the optimality condition of Kung-Traub's conjecture with the family of fourth order methods requiring three function evaluations at each iteration and four evaluations of the functions at each iteration for the family of eighth order methods. Investigation on the theoritical convergence criteria of the families are carried out and fully discussed using the two main theorems which confirm their optimal convergence order. Numerical experiments on test functions are executed by comparing with existing well-known methods of similar nature to demonstrate the effectiveness and good performance of the proposed methods.