Rate of convergence of higher order methods

Methods like the Chebyshev and the Halley method are well known methods for solving nonlinear systems of equations. They are members in the Halley class of methods and all members in this class have local and third order rate of convergence. They are single point iterative methods using the first and second derivatives. Schröderʼs method is another single point method using the first and second derivatives. However, this method is only quadratically convergent. In this paper we derive a unified framework for these methods and show their local convergence and rate of convergence. We also use the same approach to derive inexact methods. The methods in the Halley class require solution of two linear systems of equations for each iteration. However, in the Chebyshev method the coefficient matrices will be the same. Using the unified framework we show how to extend this to all methods in the class. We will illustrate these results with some numerical experiments.