Dynamics of the King and Jarratt iterations

Summary.

The purpose of this article is to present results that amount to a description of the conjugacy classes of two fourth-order root-finding iterative methods, namely King’s family of iterative methods and Jarratt’s iterative method, 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 two root-finding iterative methods to the elements of these parametrized families, a family of iterative methods is obtained, and its dynamics represents, up to conjugacy, the dynamics of the corresponding iterative root-finding method applied to any complex polyno- mial having the same degree. For degree four, partial results analogous to the ones just described are presented.