## 1 Introduction  
One of the most basic and earliest problem of numerical analysis concerns with finding efficiently and accurately the simple roots of a nonlinear equation of the form  
\[f(x)=0, \tag{1}\]  
where \(f:D\subseteq\mathbb{R}\rightarrow\mathbb{R}\) is a nonlinear continuous function. Analytical methods for solving such equations are almost non-existent and therefore, it is only possible to obtain approximate solutions by relying on numerical methods based on iterative procedure (see e.g., [1; 2; 3; 4; 5; 6; 7]). [PERSON]'s method [5] is one of the most famous and basic method for solving such equations, which is given by  
\[x_{n+1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})},\qquad n=0,1,2,\ldots. \tag{2}\]It converges quadratically for simple roots and linearly for multiple roots. Multipoint iterative methods for solving nonlinear equation are of great practical importance since they overcome the limitations of one-point methods regarding the convergence order and computational efficiency. According to the [PERSON] conjecture [2], the order of convergence of any multipoint method without memory requiring \(n\) function evaluations per iteration, cannot exceed the bound \(2^{n-1}\), called the optimal order. Thus, the optimal order for a method with three functional evaluations per step would be four. As the order of an iterative method increases, so does the number of functional evaluations per step. Commonly, the efficiency of an iterative method is measured by the efficiency index defined by [PERSON] in [3] as \(p^{1/d}\), where \(p\) is the order of convergence and \(d\) is the number of functional evaluations per step. To improve the order and efficiency of [PERSON]'s method Equation (2), [PERSON] and [PERSON] [4] proposed the following third-order method:  
\[\left\{ \begin{array}{l}y_{n}=x_{n}-\dfrac{f(x_{n})}{f^{\prime}(x_{n})},\\ x_{n+1}=x_{n}-\dfrac{f(x_{n})+f(y_{n})}{f^{\prime}(x_{n})},\ \ n=0,1,2, \ldots.\end{array} \right. \tag{3}\]  
It satisfies the following error equation  
\[e_{n+1}=2c_{2}^{2}e_{n}^{3}+(-9c_{2}^{3}+7c_{2}c_{3})e_{n}^{4}+O(e_{n}^{5}).\]  
However, there are many practical situations in which the calculations of derivatives are expensive or it requires a great deal of time for them to be given or calculated. Therefore, the idea of removing derivatives from the iteration process is very significant. In particular, when the first-order derivative \(f^{\prime}(x_{n})\) in [PERSON]'s method is replaced by forward-difference approximation \(\frac{f(x_{n}+f(x_{n}))-f(x_{n})}{f(x_{n})}\), we get the well-known Steffensen method [6] as follows:  
\[x_{n+1}=x_{n}-\dfrac{f(x_{n})}{f[x_{n},w_{n}]},\]  
where \(w_{n}=x_{n}+f(x_{n})\) and \(f[\cdot,\cdot]\) denotes the first order divided difference. As a matter of fact, both methods maintain quadratic convergence using only two functional evaluations per full step, but [PERSON] method is derivative free, which is very useful in optimization problems. Recently, many higher-order derivative-free methods are built according to the [PERSON]'s method, (_cf._[7,8] and the references cited therein). [PERSON] _et al._ in [9] presented the following fourth-order optimal Steffensen type methods given by  
\[\left\{ \begin{array}{l}y_{n}=x_{n}-\dfrac{f(x_{n})}{f[x_{n},w_{n}]},\ \ w_{n}=x_{n}+\beta f(x_{n}),\\ x_{n+1}=x_{n}-\dfrac{f(x_{n})+f(y_{n})}{f[x_{n},w_{n}]}-\left(\dfrac{2f(x_{n}) +af(y_{n})}{f[x_{n},w_{n}]}\left(\dfrac{f(y_{n})}{f(x_{n})}\right)^{2}\right) \left(1-\dfrac{\beta f[x_{n},w_{n}]}{2+2\beta f[x_{n},w_{n}]}\right),\ \ a\in\mathbb{R},\end{array} \right. \tag{4}\]  
where \(\beta\in\mathbb{R}\backslash\{0\}\). The construction of this family is based on [PERSON]'s method. However, we do not have any higher-order derivative-free modifications of [PERSON]'s method till date. With this aim, we intend to propose a new derivative-free modification of [PERSON]'s method having optimal eighth-order convergence. The construction of the proposed class is based on weight function approach.