An analysis of the properties of the variants of Newton’s method with third order convergence

For the last five years, the variants of the Newton’s method with cubic convergence have become popular iterative methods to find approximate solutions to the roots of non-linear equations. These methods both enjoy cubic convergence at simple roots and do not require the evaluation of second order derivatives. In this paper, we investigate about the relationship between these methods which are in fact based on the approximation of the second order derivative present in the third order limited Taylor expansion. We also prove that they are different forms of the Halley method and are all contractive iterative methods in a common neighbourhood. We extend some of these variants to multivariate cases and prove their respective local cubic convergence from their corresponding linear models.