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

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 |