Development and adaptation of higher-order iterative methods in R<sup>n</sup> with specific rules

In this article, we propose fourth- and fifth-order two-step iterative methods for solving the systems of nonlinear equations in \(R^n\) with the operations of multiplication and division of vectors. Some of the proposed optimal fourth-order methods are considered as an extension of well-known methods that designed only for solving the nonlinear equations. We also developed \(p\) \((5 \leqslant p \leqslant 8)\)—order three-point iterative methods for solving the systems of nonlinear equations, that contain some known iterations as particular cases. The computational efficiency of the new methods has been calculated and compared. The outcomes of numerical experiments are given to support the theoretical results concerning convergence order and computational efficiency. Comparative analysis demonstrates the superiority of the developed numerical techniques.