Dynamics of the King and Jarratt iterations. (English) Zbl 1068.30019

Amat, Sergio; Busquier, Sonia; Plaza, Sergio

Aequationes Math. 69, No. 3, 212-223 (2005).

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 degree 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 polynomial having the same degree. For degree four, partial results analogous to the ones just described are presented.

MSC:

30D05	Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37F10	Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
65F10	Iterative numerical methods for linear systems
30C15	Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
65H05	Numerical computation of solutions to single equations

Keywords:

King’s iterative methods; Jarratt’s iterative method; dynamics; rational maps; conjugacy classes