A novel family of weighted-Newton optimal eighth order methods with dynamics (Q725379)

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 are known, with order of convergence \(\tau=8\) and optimal in the sense of the Kung-Traub conjecture. The authors construct a new three-stage method, the so-called weighted-Newton method with order of convergence \(\tau=2^3\) and 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. The authors analyze the basins of attraction of the method on the some basic polynomials.