Summary: Dynamics of the King and Jarratt
The purpose of this article is to present results that amount to a de
scription of the conjugacy classes of two forth--order root--finding iterative
methods, the King and the Jarratt iterations, for complex polynomials of de
grees 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 fam
ilies, 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 polynomial having the same degree. For de
gree four, partial results analogous to the ones just described are presented.
Key Words. King and Jarratt iterative methods, dynamics, rational
maps, conjugacy classes.
AMS(MOS) subject classifications. 37N30, 65F10, 30C15.