The Cherruault decomposition method is proposed as a means to introduce modifications to Newton's iterative scheme, aiming to enhance efficiency and improve convergence order. Regarded as superior, this technique boasts applicability across a variety of numerical problems. It is noteworthy to mention that algorithms due to Weerakoon and Fernando [13], Homeier [7] and Özban [12] are actually attributed to Traub [12].

This note addresses some suggestions for the results presented by Rawat S. et al. [1].

In this paper, we introduce a novel Jungck iteration scheme characterized by second-order convergence. We verify its validity and accuracy using polynomiography. Additionally, we establish that existing well-known iteration schemes for approximating the fixed point of a general mapping exhibit identical convergence rates under specific conditions.

In this paper, we introduce a fourth-order iterative scheme tailored for a parametric matrix family to calculate the inverse square root of matrices. Initially, we will design a fourth-order algorithm for the matrix equation, followed by the development of a coupled scheme to address stability concerns. Moreover, we will conduct a comprehensive analysis of convergence and stability, substantiated by numerical examples.

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