## 1 Introduction  
Since exact solutions for nonlinear equations are rarely available, we usually resort to their numerical solutions. To locate the desired numerical roots, many authors [1; 2; 3; 4; 5; 6; 7; 8; 9] have developed high-order iterative methods including optimal eighth-order ones [10; 11; 12; 13; 14; 15].  
This paper is devoted to devise a class of sixth-order iterative root-finders for nonlinear systems by employing a three-step weighted Jarratt-like method below:  
\[\left\{\begin{array}{l}y_{n}=x_{n}-\gamma\cdot f^{\prime}{(x_{n})}^{-1}f(x_ {n}),\gamma\in\mathbb{R},\\ z_{n}=x_{n}-T_{f}(s)\cdot f^{\prime}{(x_{n})}^{-1}f(x_{n}),\\ x_{n+1}=z_{n}-L_{f}(s)\cdot f^{\prime}{(x_{n})}^{-1}f(z_{n}),\end{array}\right. \tag{1}\]  
where \(s={f^{\prime}{(x_{n})}^{-1}f^{\prime}{(y_{n})}}\), \(\gamma\) is a parameter to be determined later and \(T_{f},L_{f}:\mathbb{C}\rightarrow\mathbb{C}\) are weight functions being analytic [16; 17; 18] in a neighborhood of 1. Note that Scheme (1) uses two functional values as well as two derivatives. We are certainly able to introduce generic weight functions using one derivative and three functional values to develop general optimal eighth-order methods that covers the existing ones for the zero of a given scalar function. However, expanding such approach to a nonlinear system requires different weight functions. For unified analysis to be performed in both scalar and vector functions, we aim to develop a family of Jarratt-like sixth-order iterative methods by maintaining the same form of weight functions with two derivatives as well as two functional values. This extension to nonlinear systems is the main strength of this paper.  
The robustness of the current analysis presented here covers most existing studies on higher-order root-finders using two derivatives and two function values for both scalar and vector equations. The results of Theorem 1 give us not only fairly generic scalar function solvers, but also some advantage of extending to a nonlinear system with any finite dimension. Such an extension is evidently characterized by Theorem 2 to be studied in this analysis.  
Our major aim is not only to design a class of sixth-order methods by fully specifying the algebraic structure of generic weight functions \(T_{f}(s)\) and \(L_{f}(s)\), but also to investigate their basins of attraction behind the extraneous fixed points [19] when applied to polynomials. The last sub-step of (1) in the form of weighted [PERSON]'s method is clearly more convenient in dealing with extraneous fixed points which are the roots of the weight function \(T_{f}(s)+L_{f}(s)\cdot\frac{f(z)}{f(x)}\).  
The extraneous fixed points may lead us to attractive, indifferent, repulsive and chaotic orbits via the related basins of attraction.  
Section 2 investigates the main theorem regarding the convergence behavior with the desired forms of weight functions, while Section 3 deals with special cases of weight functions that can cover many of the existing studies using two derivatives and two functional evaluations. Section 4 discusses the computational and long-term orbit behavior of the proposed iterative methods regarding scalar functions. Section 5 presents numerical experiments in a \(d\)-dimensional Euclidean space by solving a system of nonlinear vector equations \(f:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}\) encountered in a real life with \(d\in\{3,4,9,10\}\). In addition, computational efficiency is addressed with issues related to the accuracy and applicability of the proposed methods. Concluding remarks are stated in Section 6.