Convergence analysis of a power series based iterative method having seventh order of convergence

C. D. Sreedeep; I. K. Argyros; K. C. Gopika Dinesh; N. Shrestha; M. Argyros

doi:10.30970/ms.64.2.179-193

C. D. Sreedeep Department of Mathematics, Amrita School of Physical sciences, Amritapuri, India
I. K. Argyros Department of Computing and Mathematical Sciences, Cameron University, Lawton, USA
K. C. Gopika Dinesh Department of Mathematics, Amrita School of Physical sciences, Amritapuri, India
N. Shrestha Department of Mathematics, University of Florida, Gainesville, USA
M. Argyros College of Computing and Engineering, Nova Southeastern University, Fort Lauderdale, USA

DOI: https://doi.org/10.30970/ms.64.2.179-193

Keywords: power series iteration, seventh order of convergence, local convergence, semilocal convergence

Abstract

In this paper, we propose a new three-point iterative scheme for solving nonlinear equations, which achieves seventh-order convergence. The method begins with a standard Newton iteration, followed by two weighted-Newton steps constructed using power series expansions.
The present manuscript enhances the order of convergence by integrating divided difference techniques with power series approaches, leading to an efficient and reliable iterative process.
The order of convergence has been established rigorously as seven, and the corresponding error equations are derived to validate the theoretical results. A comprehensive convergence analysis is carried out, encompassing both local and semilocal convergence aspects. The local convergence results are obtained under assumptions involving only the first derivative of the operator, and a computable radius of convergence is derived. Moreover, the uniqueness of the solution within this radius is also discussed in detail. For the semilocal analysis, we employ the majorizing sequence technique, which ensures convergence from a wider range of initial approximations. Extensive numerical experiments are performed to demonstrate the validity and accuracy of the proposed method. The calculated results show excellent agreement with the theoretical predictions, confirming the robustness and efficiency of the new algorithm, particularly when compared in terms of the number of iterations and the approximated computational order of convergence.

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