An iteration method with maximal order based on standard information

Abstract

For finding a root of a function f, Halley's iteration family is a higher generalization of Newton's iteration function. In every step, it uses the values of f and its first number of derivatives, called standard information. Based on the standard information, we obtain an iteration method with maximal order of convergence. It is a natural generalization of Halley's iteration family in terms of divided differences. An explicit construction for this method is also obtained. Numerical experiments are given demonstrating the importance of the proposed approach.