# An Optimal Order Method for Multiple Roots in Case of Unknown Multiplicity
## 1 Introduction  
In the present report, we discuss an iterative scheme for calculating the approximate value of a multiple root \(\xi\) of a nonlinear equation \(\psi(x)=0\), in \(\mathcal{R}\). The modified [PERSON]'s scheme [1] is probably the first method for calculating the estimated value of the multiple root when multiplicity \(m\) is available in advance, with a convergence rate twice that in some open set around \(\xi\), under suitable regularity assumptions. For the same case, various schemes are presented in the literature by using numerous modes (one can see [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13] for a few of them).  
For the second case, _i.e._, when the multiplicity \(m\) is not available in advance, for calculating the multiple root of \(\psi(x)=0\), [PERSON] [14] used the relation \(f(x)=\psi(x)/\psi^{\prime}(x)\). In this stage, the problem of finding multiple roots of \(\psi(x)=0\) is equivalent to calculating the simple root of \(f(x)=0\). However, for this substitution, [PERSON]'s method involved the first and second derivatives also. To fend off these evaluations, [PERSON] [15] used finite difference approximation.  
Working with a parallel idea, [PERSON] and [PERSON] [16] developed two iterative methods of order three and four for approximating the multiple roots. The cubic convergence order scheme is as:  
\[x_{n+1}=x_{n}-\frac{f(x_{n})}{g(x_{n})}-\frac{f\left(x_{n}-\frac{f(x_{n})}{g(x _{n})}\right)}{g(x_{n})} \tag{1}\]and the fourth-order method is expressed as:  
\[x_{n+1}=x_{n}-\frac{f(x_{n})}{g(x_{n})}-\frac{f\left(x_{n}-\frac{f (x_{n})}{g(x_{n})}\right)}{g(x_{n})}-\frac{f\left(x_{n}-\frac{f(x_{n})}{g(x_{n} )}-\frac{f\left(x_{n}-\frac{f(x_{n})}{g(x_{n})}\right)}{g(x_{n})}\right)}{g(x_ {n})}\]  
where \(g(x_{n})=\frac{f\left(x_{n}+\beta f(x_{n})\right)-f(x_{n})}{\beta f(x_{n})}\).  
[PERSON] and [PERSON] (2017) also used a similar transformation and presented a uni-parametric scheme of second order for handling similar type problems. In addition, [PERSON] _et al._ (2018) prescribed another transformation given by:  
\[f(x)=\frac{sign(\psi(x))\rho}{sign(\psi(x+sign(\psi(x))|\psi(x)|^{1/m})-\psi(x) )\rho+\psi(x+sign(\psi(x))|\psi(x)|^{1/m})-\psi(x)} \tag{3}\]  
where \(\rho=\psi(x)|\psi(x)|^{1/m}\) and \(m\) represents the multiplicity, and engaged the modified [PERSON]'s method (see (2019, 2020)):  
\[x_{n+1}=x_{n}-h_{n}\frac{f^{2}(x_{n})}{tf^{2}(x_{n})+f(x_{n}+f(x_{n}))-f(x_{n})} \tag{4}\]  
for evaluating the estimated root of \(f(x)=0\), where \(h_{n}(>0)\) is the iteration step size and \(|t|<\infty\). [PERSON] and [PERSON] [PERSON] and [PERSON] (2017) implied a different conversion:  
\[f(x) = \frac{\psi^{2}(x)}{\delta+\psi(x+\psi(x))-\psi(x)},\,\,if\,\,\psi (x)\
eq 0\] (5) \[= 0,\,  
More recently, by using the same transformation, [PERSON] and [PERSON] [PERSON] and [PERSON] (2016) proposed an iterative scheme with a convergence rate of six and expressed as:  
\[y_{n} = x_{n}-\frac{f^{2}(x_{n})}{f(x_{n}+f(x_{n}))-f(x_{n})}\] \[u_{n} = y_{n}-\frac{f(y_{n})f(x_{n})}{f(x_{n}+f(x_{n}))-f(x_{n})} \tag{9}\] \[x_{n+1} = u_{n}-\frac{f(u_{n})f[x_{n},y_{n}]}{f[x_{n},u_{n}]f[y_{n},u_{n}]}\]  
The above mentioned fifth- and sixth-order methods involved four function evaluations \([f(x_{n}),f(x_{n}+f(x_{n})),f(y_{n}),f(u_{n})]\), and by [PERSON]'s [PERSON] and [PERSON] (1999) conjecture, the optimal order should be eight. Motivated by this theory, we are going to present an optimal eighth-order method for multiple roots in case of unknown multiplicity: to our best knowledge, this is the first method of the optimal eighth order. Theoretically, its local rate of convergence is proven and finally supported by numerical testing.