In this paper, a derivative-free one-point iterative technique is proposed, with memory for finding multiple roots of practical problems, such as van der Waals and continuous stirred tank reactor problems, whose multiplicity is unknown in the literature. The new technique has an order of convergence of 1.84 and requires two function evaluations. It can be used as a seed to produce higher-order methods with similar properties, and it increases the efficiency of a similar procedure without memory due to Schröder. After studying its order of convergence, its stability is checked by applying it to the considered problems and comparing with the technique of the same nature for finding multiple roots. The geometrical behavior of the numerical results of the techniques is also studied.

In this paper, we have constructed a family of three-step methods with sixth-order convergence and a novel approach to enhance the convergence order $p$ of iterative methods for systems of nonlinear equations. Additionally, we propose a three-step scheme with convergence order $p+3$ (for $p\ge 3$) and have extended it to a generalized $(m+2)$-step scheme by merely incorporating one additional function evaluation, thus achieving convergence orders up to $p+3m$, $m\in \mathbb{N}$. We also provide a thorough local convergence analysis in Banach spaces, including the convergence radius and uniqueness results, under the assumption of a Lipschitz-continuous Fréchet derivative. Theoretical findings have been validated through numerical experiments. Lastly, the performance of these methods is showcased through the analysis of their basins of attraction and their application to systems of nonlinear equations.