Ball convergence for a family of eight-order iterative schemes under hypotheses only of the first-order derivative

ABSTRACT

In this paper, we propose a local convergence study of an optimal family of eight-order iteration functions. Lotfi et al. [A new class of three-point methods with optimal convergence order eight and its dynamics, Numer. Algor. 68 (2015), pp. 261–288] used Taylor series expansions, and hypotheses up to the eight or higher-order derivative of the considered function in order to demonstrate the convergence order in their studies. However, the presented scheme did not involve second or higher-order derivative of the considered function. Such conditions restrict the usage of their scheme. Therefore, we extend the suitability of their iteration functions by considering suppositions solely on the first-order derivative. Furthermore, we present the convergence domain of the iteration functions and bounds on the error by using Lipschitz constants. Finally, we discuss a case where the earlier study is not applicable but our results are very useful for the same problem.

KEYWORDS:

• Nonlinear equations
• Newton's method
• Lipschitz constants
• error bounds
• radius of convergence

2010 AMS SUBJECT CLASSIFICATION:

• 65Hxx

Acknowledgments

The authors, therefore, gratefully acknowledge the DSR technical and financial support.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. D-217-130-1439.