Summary: Dynamics of a family of third­order
iterative methods that do not require
using second derivatives
The purpose of this article is to present results that amount to a de-
scription of the conjugacy classes of three third­order root­finding iterative
methods that do not require the use of second derivatives for their formu-
lation, 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 one­parameter family of polynomials. This is done in such a
way that, when one applies one of these three root­finding 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 root­finding method applied to
any complex polynomial having the same degree. For degree four, analogous
partial results are obtained.
Key Words. Iterative methods, dynamics, rational maps, conjugacy
classes.
AMS(MOS) subject classifications. 37N30, 65F10, 30C15.
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Source: Amat, Sergio - Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena

Collections: Mathematics