Comments on the Basto–Semiao–Calheiros root finding method

In the recent paper [M. Basto, V. Semiao, F.L. Calheiros, A new iterative method to compute nonlinear equations, Appl. Math. Comput. 173 (2006) 468–483] a new cubically convergent iterative method for solving nonlinear equations was proposed. In this paper we give some comments that point to significant similarity of the proposed method to the classic Euler–Chebyshev method and discuss the efficiency of the new method in a comparison procedure involving five other third order methods with similar structure. Five numerical examples (four of them used in the cited paper) showed that the convergence behavior of the new method is equal or worse than competitive tested methods. Some comments on iterative methods involving the third derivative are also exposed.