Another note on some quadrature based three-step iterative methods for non-linear equations

The recent paper [H. Ding, Y. Zhang, S. Wang, X. Yang, A note on some quadrature based three-step iterative methods for non-linear equations, Appl. Math. Comput. 215 (1): (2009) 53--57] shows that the Algorithm 2.2 and Algorithm 2.3 in the article [N.A. Mir, T. Zaman, Some quadrature based three-step iterative methods for non-linear equations, Appl. Math. Comput. 193 (2): (2007) 366--373] have twelfth-order and ninth order convergence respectively, not seventh-order as claimed in the later article. In this work; we propose two simple modifications, without increasing the computational cost or functional evaluations, of the Algorithms 2.3 and 2.3. The first modification improves the convergence order of the Algorithm 2.3 from ninth-order to tenth order. In the second modification, we remove evaluation of the second derivative from the Algorithm 2.2 while preserving its twelfth-order convergent nature.