Dynamics of a family of third-order iterative methods that do not require using second derivatives

Abstract

The purpose of this article is to present results that amount to a description of the conjugacy classes of three third-order root-finding iterative methods that do not require the use of second derivatives for their formulation, for complex polynomials of degrees two, three and four. For degrees two and three, a full description of the conjugacy classes is accomplished, in each case, by a one-parameter family of polynomials. This is done in such a way that, when one applies one of these three root-finding iterative methods to the elements of these parametrized families, a family of iterative methods is obtained, in such a way that its dynamics represents, up to conjugacy, the dynamics of the corresponding iterative root-finding method applied to any complex polynomial having the same degree. For degree four, analogous partial results are obtained.