Many scientific disciplines, from basic sciences to engineering, frequently encounter the challenge of solving systems of nonlinear problems. Addressing this challenge demands the development of accurate and efficient computational methods. In this work, we propose a novel multi-step iterative method that achieves an exceptional convergence order of , where m ≥ 3 denotes the number of iterative steps. This method significantly enhances computational efficiency without compromising accuracy, as it requires only a single evaluation and inversion of the Jacobian matrix per iteration cycle. To further optimize performance, the linear systems arising at every step are solved via LU decomposition, bypassing the computational burden of direct matrix inversion. As a result, the proposed method attains a higher convergence order than existing multi-step methods while maintaining comparable computational costs. Its efficiency makes it particularly well-suited for large-scale problems, where computational overhead is a critical concern. To validate the method’s effectiveness, we conducted comprehensive numerical experiments, assessing its efficiency, accuracy, and the geometry of its basins of attraction. The results consistently aligned with theoretical predictions, demonstrating the method’s superior performance over conventional approaches. Additionally, the method to solve standard nonlinear systems commonly arising in science and engineering is applied. Finally, we extended its application to image processing tasks, where it effectively addressed systems of nonlinear problem. The numerical outcomes underscored the method’s robustness, stability, and potential to outperform traditional iterative methods.
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