Weaker convergence criteria for Traub's method (Q2074936)

The Traub's method [\textit{J. F. Traub}, Iterative methods for the solution of equations. Englewood Cliffs: Prentice-Hall, Inc. (1964; Zbl 0121.11204)] for solving the nonlinear equation \[ F(x)=0 \] is defined by the iterative sequence \(\{x_n\}\) given by \[ \begin{cases} y_k=x_k-F'(x_k)^{-1} F(x_k)\\ x_{k+1}=y_k-F'(x_k)^{-1} F(y_k), \,k\geq 0, \end{cases} \] where \(x_0\) is the starting value. It is a two-step third order method which is less expensive than any third order method using the second derivative such as Chebyshev's, Halley's and others. In this paper the author studies the semi-local and local convergence of Traub's method by using mainly the technique of majorizing sequences. Numerical examples to illustrate the theory are also presented.