On optimal parameter of Laguerre's family of zero-finding methods

ABSTRACT

A one parameter Laguerre's family of iterative methods for solving nonlinear equations is considered. This family includes the Halley, Ostrowski and Euler methods, most frequently used one-point third-order methods for finding zeros. Investigation of convergence quality of these methods and their ranking is reduced to searching optimal parameter of Laguerre's family, which is the main goal of this paper. Although methods from Laguerre's family have been extensively studied in the literature for more decades, their proper ranking was primarily discussed according to numerical experiments. Regarding that such ranking is not trustworthy even for algebraic polynomials, more reliable comparison study is presented by combining the comparison by numerical examples and the comparison using dynamic study of methods by basins of attraction that enable their graphic visualization. This combined approach has shown that Ostrowski's method possesses the best convergence behaviour for most polynomial equations.

Keywords:

• Laguerre's family
• nonlinear equations
• dynamic study
• basin of attraction
• convergence behaviour

2010 AMS SUBJECT CLASSIFICATION:

• 65H05
• 65D18
• 68W30
• 33F05

Acknowledgments

The authors are grateful to the anonymous referee for useful suggestions that improved the presentation. We also thank the Serbian Ministry of Education and Science, Grant 174022, that supported this work.

Disclosure statement

No potential conflict of interest was reported by the authors.