An optimal eighth-order class of three-step weighted Newton's methods and their dynamics behind the purely imaginary extraneous fixed points
Rhee, Min Surp
Kim, Young Ik
MetadataShow full item record
In this paper, we not only develop an optimal class of three-step eighth-order methods with higher order weight functions employed in the second and third sub-steps, but also investigate their dynamics underlying the purely imaginary extraneous fixed points. Their theoretical and computational properties are fully described along with a main theorem stating the order of convergence and the asymptotic error constant as well as extensive studies of special cases with rational weight functions. A number of numerical examples are illustrated to confirm the underlying theoretical development. Besides, to show the convergence behaviour of global character, fully explored is the dynamics of the proposed family of eighth-order methods as well as an existing competitive method with the help of illustrative basins of attraction.
The article of record as published may be found at http://dx.doi.org/10.1080/00207160.2017.1367387
RightsThis publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.