Algorithm for forming derivative-free optimal methods (Q457040)

The Newton method is the best known method for solving a nonlinear equation \(f(x)=0\) which is given as \[ x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}, \qquad n=0,1,2,\dots, \qquad |f'(x_n)|\neq 0. \] A scheme for constructing optimal derivative free iterative methods is the main aim of the article. The derivative in Newton's method is approximated as follows \[ f'(x_n)\approx \eta_1 f(x_n)+\eta_2 f(x_n+\alpha f(x_n)) \] and the authors propose the method \[ x_{n+1}=x_n -\frac{\alpha\, (f(x_n))^2}{f(x_n+\alpha f(x_n))-f(x_n)}. \] A generalization of Traub's theorem is obtained. Derivative free optimal iterative methods of orders two, four and eight are constructed. Numerical examples illustrate the proposed methods.