Work

An Improved King–Werner-Type Method Based on Cubic Interpolation: Convergence Analysis and Complex Dynamics

Year: 2025

Type: article

Abstract: In this paper, we study the convergence and complex dynamics of a novel higher-order multipoint iteration scheme to solve nonlinear equations. The approach is based upon utilizing cubic interpolation in the second step of the King–Werner method to improve its convergence order from 2.414 to 3 and the efficiency index from 1.554 to 1.732, which is higher than the efficiency of optimal fourth- and eighth-order iterative schemes. The proposed method is validated through numerical and dynamic experiments concerning the absolute error, approximated computational order, regions of convergence, and CPU time (sec) on the real-world problems, including Kepler’s equation, isentropic supersonic flow, and law of population growth, demonstrating superior performance compared to some existing well-known methods. Commonly, regions of convergence of iterative methods are investigated and compared by plotting attractor basins of iteration schemes in the complex plane on polynomial functions of the type zn−1. However, in this paper, the attractor basins of the proposed method are investigated on diverse nonlinear functions. The proposed scheme creates portraits of basins of attraction faster with wider convergence areas outperforming existing well-known iteration schemes.

Source: Axioms

DOI: 10.3390/axioms14050360

Authors: Moin-ud-Din Junjua, Ibraheem M. Alsulami, Amer Alsulami, Sangeeta Kumari

Institutions: Zhejiang Normal University, Ghazi University, Umm al-Qura University, Taif University, Chandigarh University

Language: English

Topic: Iterative Methods for Nonlinear Equations

Subfield: Numerical Analysis

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $2,896

Funder: Umm Al-Qura University