A note on "New classes of iterative methods for nonlinear equations" and "Some iterative methods free from second derivatives for nonlinear equations"
Abstract In [M.A. Noor, New classes of iterative methods for nonlinear equations, Appl. Math. Comput., in press; M.A. Noor, Some iterative methods free from second derivatives for nonlinear equations, Appl. Math. Comput., in press], Noor introduced a generalized one parameter Halley’s method x n + 1 = x n - f ( x n ) f ′ 2 ( x n ) f ′ 3 ( x n ) - α f ( x n ) f ″ ( x n ) for solving the nonlinear equation f ( x ) = 0. Noor further showed that for α = 1 2 f ′ ( x n ) , the above method reduces to the Halley’s method [E. Halley, A new exact and easy method for finding the roots of equations generally and without any previous reduction, Philos. Roy. Soc. London 18 (1964) 136–147]. It is interesting to note that for α = f ′ 3 ( x n ) f ( x n ) f ′ ′ ( x n ) , the above method fails. In this note, we point out some major bugs in the results of Noor (in press).