An iterative method with cubic convergence for nonlinear equations
A new three step iterative method for solving a nonlinear equation \(f(x)=0\) 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: \[ y_n=x_n-f(x_n)/ f' (x_n),\quad f'(x_n)\neq 0;\quad z_n=-(y_n-x_n)^2\cdot f''(x_n)/2f'(x_n), \] and one corrector step: \[ x_{n+1}=x_n-f(x_n)/f'(x_n)-(y_n+z_n-z_n)^2\cdot f'(x_n)/2\cdot f'(x_n),\quad n=0, 1,2,\dots. \] The authors show that if the function \(f\) is sufficiently differentiable in the open interval, which contain a simple root of the equation \(f(x)=0\) and if \(x_0\) is sufficiently close to this root, then the proposed iterative algorithm has the order of convergence equal to three. Several numerical examples are given to illustrate the efficiency and performance of the new method.
scientific article; zbMATH DE number 5128309