In this work, we develop multi-step vectorial iterative schemes for solving nonlinear systems, achieving fourth and sixth-order convergence. The proposed methods are designed to minimize computational costs by employing a single inverse operator and reducing the number of functional evaluations per iteration. Furthermore, we generalize the sixth-order three-step scheme into a \((q+1)\) -step family, increasing the convergence order to \(2q+2\) . While standard local convergence analysis based on Taylor series expansion is common, it limits applicability as it requires the use of higher-order derivatives. To overcome this limitation, our theoretical analysis is conducted in a Banach space setting and relies solely on first-order derivatives. The existence of a unique solution is guaranteed within a specific domain, whose radius of convergence is formally obtained using Lipschitz constants. A detailed computational complexity analysis confirms the superior efficiency of our methods compared to existing approaches. Numerical experiments on different problems demonstrate significantly improved performance, while stability is validated through basins of attraction in the complex plane.
