Convergence of the iteration of Halley's family and Smale operator class in Banach space (Q1267228)

The Newton iteration (for solving the equation \(x= f(x)\)) has been extended by Halley to obtain a convergence order three instead of two. Here this Halley's method is generalized in order that it have a convergence order \(k\). For analyzing the convergence of this algorithm, the author uses a new so-called concept of Smale operator of order \(k\), which is a special case of the infinite order model introduced by Smale to study Newton's method. A proper condition for the convergence and the exact estimation error for the iteration are given.