## 1 Introduction  
This paper addresses the problem of multiple roots \(x^{\ast}\) of nonlinear equation \(f(x)=0\) with unknown multiplicity \(m\) whose multiplicity \(m\) is the highest multiplicity, where \(f:[a,b]\subset R\to R\) is a nonlinear differential function on \([a,b]\). In case the multiplicity \(m\) is given explicitly, there are many iterative methods established via various techniques (see [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15] for more details). If the multiplicity \(m\) is not known explicitly, [PERSON] [16] utilized a simple transformation \(F(x)=f(x)/f^{\prime}(x)\) instead of \(f(x)\) for computing a multiple root of \(f(x)=0\). In this case, the aim of solving a multiple root is reduced to that of solving a simple root of the transformed equation \(f(x)=0\), and thus any iterative method can be used to preserve the original convergence order. However, [PERSON]'s method for this transformed equation requires evaluations of the derivatives \(f^{\prime}(x)\) and \(f^{\prime\prime}(x)\). In order to avoid the evaluations of these derivatives with the multiplicity \(m\) unknown, for multiple roots, [PERSON] [17] proposed the secant method which does not use the function \(F=f/f^{\prime}\), but rather use \(F=\dfrac{f(x)}{\dfrac{f(x-f(x))-f(x)}{(x-f(x))-x}}=\dfrac{-f^{2}(x)}{f(x-f(x)) -f(x)}\). [PERSON] and [PERSON] [18] further used \(F(x)=\dfrac{f^{2}(x)}{f(x)-f(x-f(x))}\) and transformed the problem of solving multiple roots of \(f(x)=0\) into that of solving simple root of \(f(x)=0\). Actually, they established the following iteration formulae:  
\[x_{n+1}=x_{n}-\dfrac{F^{2}(x_{n})}{p\cdot F^{2}(x_{n})+F(x_{n})-F(x_{n}-F(x_{n }))}, \tag{1}\]  
where \(p\in R\), \(|p|<\infty\). So, the sequence \(\{x_{n}\}\) produced by the iteration Formulae (1) is at least quadratically convergent for multiple roots. Moreover, [PERSON] _et al._[19] defined a function  
\[F(x)=\dfrac{sign(f(x))f(x)|f(x)|^{1/m}}{sign(f(x+sign(f(x))|f(x)|^{1/m})-f(x)) f(x)|f(x)|^{1/m}+f(x+sign(f(x))|f(x)|^{1/m})-f(x)}, \tag{2}\]  
where \(m\) is the multiplicity, and employed the modified [PERSON]'s method (see [20, 21])  
\[ \begin{split} x_{n+1}=& x_{n}-h_{n}\dfrac{F^{2}(x_ {n})}{t\cdot F^{2}(x_{n})+F(x_{n}+F(x_{n}))-F(x_{n})}\\ =& x_{n}-h_{n}\dfrac{F(x_{n})}{t\cdot F(x_{n})+(F(x _{n}+F(x_{n}))-F(x_{n}))/(F(x_{n}))}\end{split}  \tag{3}\]  
to compute the approximate solution of the equation \(f(x)=0\), where \(h_{n}(>0)\) is the step size of iteration and \(|t|<\infty\). [PERSON] and [PERSON] [22] suggested another transformation  
\[F(x)= \begin{cases}\dfrac{f^{2}(x)}{\delta+f(x+f(x))-f(x)}&\text{if }f(x)\
eq 0,\\ 0&\text{if }f(x)=0\;.\end{cases}  \tag{4}\]  
where \(\delta=sign(f(x+f(x))-f(x))f^{2}(x)\), and transform the task of solving multiple zeros of \(f\) into that of solving simple zero of \(F\). In this case, they utilized a quadratically convergent derivative free Newton-like iterative method:  
\[x_{n+1}=x_{n}-\dfrac{F^{2}(x_{n})}{p\cdot F^{2}(x_{n})+F(x_{n})-F(x_{n}-F(x_{ n}))} \tag{5}\]  
where the parameter \(p\) should be chosen such that the denominator is the largest in magnitude. [PERSON] [23] suggested a new transformation of \(f(x)\) as  
\[H_{\varepsilon}(x)=\dfrac{\varepsilon f^{2}(x)}{f(x+\varepsilon f(x))-f(x)}, \tag{6}\]took \(\varepsilon\) such that \(\max\limits_{a\leq x\leq b}|\varepsilon f(x)|=\delta\), that is  
\[\varepsilon=\frac{\delta}{\max_{a\leq x\leq b}|f(x)|}=\frac{\delta}{\max\left\{ \left|f(a)\right|,\left|f(b)\right|\right\}},\]  
and proposed an iterative method as follows:  
\[x_{n+1}=x_{n}-\frac{2(x_{n}-x_{n-1})H_{\varepsilon}(x_{n})}{H_{\varepsilon}(2 x_{n}-x_{n-1})-H_{\varepsilon}(x_{n-1})}. \tag{7}\]  
Recently, for the transformed equation \(K(x)=0\) with a simple root, [PERSON] [24] proposed a Steffensen-type iterative formula  
\[p_{k+1}=p_{k}-\frac{\mu K(p_{k})^{2}}{K(p_{k}+\mu K(p_{k}))-K(p_{k})},\quad k \geq 0, \tag{8}\]  
where \(K(x)=K(\varepsilon;x)=\left\{