Optimal Newton–Secant like methods without memory for solving nonlinear equations with its dynamics

ABSTRACT

We construct two optimal Newton–Secant like iterative methods for solving nonlinear equations. The proposed classes have convergence order four and eight and cost only three and four function evaluations per iteration, respectively. These methods support the Kung and Traub conjecture and possess a high computational efficiency. The new methods are illustrated by numerical experiments and a comparison with some existing optimal methods. We conclude with an investigation of the basins of attraction of the solutions in the complex plane.

KEYWORDS:

• Multi-point iterative methods
• Newton–Secant method
• Kung and Traub's conjecture

MSC 2010 AMS SUBJECT CLASSIFICATIONS:

• 65H04
• 65H05

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCiD

Mehdi Salimi http://orcid.org/0000-0002-6537-6346