<p>Hueso et al. (2015) introduced a new family of iterative methods for solving non-linear systems. However, the convergence analysis is based on Taylor series expansion, which requires the existence of derivatives of the involved operator up to the fifth and seventh orders, respectively, for the method with at least fourth-order convergence and the method with sixth-order convergence. In this paper, we obtain at least fourth- and sixth-order convergence for the respective methods by assuming derivatives only up to the third order. We also provide the semi-local convergence analysis (which is not given in Hueso et al. (2015)) in a more general Banach space form. Moreover, our semi-local and local convergence analyses are based on the same set of assumptions, unlike existing studies, where the authors typically use one set of assumptions for semi-local convergence analysis and another set of assumptions for local convergence analysis. Numerical examples and dynamics of the methods are also provided in this study.</p>