A COMPARISON BETWEEN THREE METHODS OF
ORDER SIX USING SIMILAR INFORMATION

Ioannis K. Argyros, Stepan M. Shakhno, Mykhailo M. Havdiak, Michael I. Argyros

Анотація


This study provides a comparative analysis of three sixth-order iterative methods for solving nonlinear equations. Unlike many existing approaches that rely on higher-order derivatives, this work develops an optimized local convergence analysis based on the operator F and its derivative F′. More significantly, introduce a semi-local convergence analysis for these methods using majorizing sequences, which provides a practical framework
for establishing convergence based on the conditions at the initial guess. Both local and semi-local convergence analysis are performed within the context of Banach spaces. This approach enhances the theoretical robustness and practical applicability of the methods. To validate the theoretical findings, a series of numerical experiments is conducted on various standard benchmark problems, including small- to large-scale nonlinear systems. The performance of the methods is compared against the classical Newton’s method. The findings confirm that the sixth-order methods consistently outperform Newton’s method in terms of the total number of iterations. We analyze the Computational Order of Convergence (COC) and Approximate Computational Order of Convergence (ACOC), which empirically confirm the high convergence order of the proposed methods. Furthermore, we investigate the numerical stability and performance under high-precision requirements, utilizing arbitrary-precision arithmetic to solve problems where standard double precision fails. These results underscore the practical advantages and theoretical robustness of the methods. The methodology presented can be applied to other similar iterative methods.


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DOI: http://dx.doi.org/10.30970/vam.2025.35.13707

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