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Articles

An optimal eighth-order class of three-step weighted Newton's methods and their dynamics behind the purely imaginary extraneous fixed points

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Pages 2174-2211 | Received 01 Feb 2017, Accepted 17 Jul 2017, Published online: 29 Aug 2017
 

ABSTRACT

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.

2000 AMS SUBJECT CLASSIFICATIONS:

Disclosure statement

No potential conflict of interest was reported by the authors.

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