This paper introduces an interpolationbased order six method for solving nonlinear equations. Our method offers significant improvements in accuracy, stability, and efficiency, making it valuable for computational and applied mathematics. The main goal is to address various nonlinear problems in fields such as physics, engineering, and finance. The paper explains the theoretical foundation and key principles of the method, which uses interpolation points instead of derivatives to approximate solutions. This approach enhances convergence behavior and numerical precision. We also conduct a detailed local and semilocal convergence analysis to evaluate the method’s performance. This analysis provides insights into the convergence region, radii, and error boundaries. It also assesses the method’s effectiveness in scenarios where accurate initial guesses are hard to obtain. Extensive numerical experiments on diverse test problems demonstrate the method’s superior convergence rates and error estimates, confirming its effectiveness and reliability.

01 Mar 2025·International Journal of Computational Methods

On the Implementation of Iterative Methods Without Inverse Updating for Solving Equations in Banach Spaces

Author: Argyros, Ioannis K. ; George, Santhosh

The implementation of iterative methods using inverses to solve equations is a computationally expensive or impossible task in general. This is the case, since the analytical form of the inverse is difficult to find in practice. That is why, we replace the inverse by a sum of linear operators which is well defined. The convergence of the resulting hybrid methods is studied based on majorizing sequences under generalized continuity assumptions on the operators involved and in the setting of a Banach space. It is demonstrated by numerical experimentations that the convergence order as well as the number of iterations required to obtain a predetermined error tolerance when comparing the original to the hybrid method is essentially the same.

01 Dec 2024·MethodsX

A highly accurate family of stable and convergent numerical solvers based on Daftardar–Gejji and Jafari decomposition technique for systems of nonlinear equations

Article

Author: Jafari, Hossein ; Soomro, Amanullah ; Argyros, Ioannis K ; Qureshi, Sania ; Gdawiec, Krzysztof

This study introduces a family of root-solvers for systems of nonlinear equations, leveraging the Daftardar-Gejji and Jafari Decomposition Technique coupled with the midpoint quadrature rule. Despite the existing application of these root solvers to single-variable equations, their extension to systems of nonlinear equations marks a pioneering advancement. Through meticulous derivation, this work not only expands the utility of these root solvers but also presents a comprehensive analysis of their stability and semilocal convergence; two areas of study missing in the existing literature. The convergence of the proposed solvers is rigorously established using Taylor series expansions and the Banach Fixed Point Theorem, providing a solid theoretical foundation for semilocal convergence guarantees. Additionally, a detailed stability analysis further underscores the robustness of these solvers in various computational scenarios. The practical efficacy and applicability of the developed methods are demonstrated through the resolution of five real-world application problems, underscoring their potential in addressing complex nonlinear systems. This research fills a significant gap in the literature by offering a thorough investigation into the stability and convergence of these root solvers when applied to nonlinear systems, setting the stage for further explorations and applications in the field.•The proposed methods combine the Daftardar-Gejji and Jafari Decomposition Technique and the midpoint quadrature rule for solving systems of nonlinear equations.•The numerical results show the superiority of the proposed methods over some other third- and fourth-order convergent methods from the literature.•The proposed methods can be used in various contexts and many real-world applications.

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01 Sep 2023

·www.coatingsworld.com

Barentz CASE Expands Sales, Welcomes Karna Bertucci

Barentz welcomed Karna Bertucci as account manager, covering customers in our Southwest region. With more than 25 years working in the chemicals field, Bertucci is a results-oriented sales professional with broad experience in the coatings, adhesives, sealants, and elastomers (CASE) industry, including many years with Azelis Americas and Buckman Laboratories. She earned a bachelor of science degree in chemistry with a minor in biology at Cameron University and has won various outstanding sales performance awards, such as the prestigious Acorn Award. "It's incredibly exciting for Barentz to announce the addition of a distribution sales professional like Karna, who possesses tremendous knowledge, energy, experience, and relationships," Bill Nicholas, VP, sales (CASE), said. "I knew of her from the great things I'd heard from customers we've shared, so naturally we jumped at the opportunity to bring her on board when she became available, and we couldn't be happier that we did."

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