An optimal eighth-order class of three-step weighted Newton's methods and their dynamics behind the purely imaginary extraneous fixed points. Issue 11 (2nd November 2018)

• Record Type: Journal Article
• Authors: Rhee, Min Surp; Kim, Young Ik; Neta, Beny
• Is Part Of: International journal of computer mathematics. Volume 95:Issue 11(2018)
• Page Start: 2174
• Page End: 2211
• Publication Date: 2018-11-02
• Subjects: Eighth-order convergence -- weight function -- asymptotic error constant -- efficiency index -- purely imaginary extraneous fixed point -- basin of attraction; 65H05 -- 65H99; Computers -- Periodicals; Numerical analysis -- Periodicals; Automation -- Periodicals; 004.0151
• DOI: 10.1080/00207160.2017.1367387
• Languages: English
• ISSNs: 0020-7160

Abstract

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.