Some iterative schemes for nonlinear equations (Q865513)

A new three step iterative method for solving nonlinear equations \(f(x)=0\) is introduced based on the following scheme: Let \(x_0\) is an initial guess sufficiently close to simple root of the equation \(f(x)=0\). The iterative step consists from two predictor steps: \[ y_n=x_n-f(x_n)/f'(x_n),\quad f'(x_n)\neq 0;\quad z_n=-f(y_n)/f'(x_n) \] and one corrector step: \(x_{n+1}=x_n-f(x_n)/f' (x_n)-f(y_n+z_n)/f'(x_n)\), \(n=1,2,\dots\). The authors show that if the function \(f\) is sufficiently differentiable, this iterative algorithm has the order of convergence equal to three. In contradiction to other two and three-step methods and methods based on a decomposition technique the using of higher-order derivatives is not necessary in the method proposed. Several numerical examples are given to illustrate the efficiency and performance of the new method.