A multi-point iterative method for solving nonlinear equations with optimal order of convergence (Q1756715)

The article is devoted to the classical problem of constructing efficient and higher-order of convergence iterative algorithms for solving nonlinear equations. In this area of numerical analysis, several iterative families with order of convergence \(\tau=8\) and optimal in the sense of Kung-Traub's conjecture are known. The authors construct a new three-stage method, free from second-order derivatives. The technique used for obtaining the new family includes weight functions technique and Newton interpolation. The numerical procedure has order of convergence \(\tau=2^3\) and the efficiency index \(EI=2^{\frac{3}{4}}\). The new algorithm requires only three function calculations and one derivative calculation at a point, that is, it is optimal in the sense of Kung-Traub considerations. As a result of the convergence theorem, some interesting choices of weight functions are listed. Some comparison of basin of attraction for popular methods are given.