Hybrid accelerated iterative scheme based on the Picard-Ishikawa approach for systems of nonlinear equations.

Systems of nonlinear equations arise in numerous scientific, engineering, and optimization applications, where efficient and reliable numerical methods are required to obtain accurate solutions. Hybridization techniques have recently attracted considerable attention because of their ability to improve the convergence behavior of iterative schemes. Motivated by this development, this paper proposes a novel hybrid accelerated iterative method for solving systems of nonlinear equations by integrating the Picard-Ishikawa process with an accelerated derivative-free framework. The proposed approach employs a diagonal approximation of the Jacobian matrix through an acceleration parameter, thereby reducing computational cost while maintaining numerical accuracy. The novelty of the method lies in combining the three-step Picard-Ishikawa process with an accelerated scheme to enhance convergence performance and robustness. Under standard assumptions, the global convergence of the proposed method is established. Extensive numerical experiments on a collection of benchmark problems demonstrate that the proposed method outperforms several existing approaches in terms of iteration count, computational time, and robustness. These results confirm the effectiveness and suitability of the proposed scheme for solving large-scale systems of nonlinear equations arising in practical applications.