## 1 Introduction  
Numerical analysis is a wide-ranging discipline having close connections with mathematics, computer science, engineering and the applied sciences. One of the most basic and earliest problem of numerical analysis concerns with finding efficiently and accurately the approximate locally unique solution \(x^{*}\) of the equation of the form  
\[F(x)=0, \tag{1}\]  
where \(F\) is a Frechet differentiable operator defined on a convex subset \(D\) of \(X\) with values in \(Y\), where \(X\) and \(Y\) are the Banach spaces. Analytical methods for solving such equations are almost non-existent for obtaining the exact numerical values of the required roots. Therefore, it is only possible to obtain approximate solutions and one has to be satisfied with approximate solutions up to any specified degree of accuracy, by relying on numerical methods which are based on iterative procedures. Therefore, researchers worldwide resort to an iterative method and they have proposed a plethora of iterative methods [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16]. While, using these iterative methods researchers face the problems of slow convergence, non-convergence, divergence, inefficiency or failure (for detail please see [PERSON] [15] and [PERSON] _et al._[13]). The convergence analysis of iterative methods is usually divided into two categories: semi-local and local convergence analysis. The semi-local convergence matter is, based on the information around an initial point, to give criteria ensuring the convergence of iteration procedures. On the other hand, the local convergence is based on the information around a solution, to find estimates of the radii of convergence balls. A very important problem in the study of iterative procedures is the convergence domain. Therefore, it is very important to propose the radius of convergence of the iterative methods. We study the local convergence analysis of three step method defined for each \(n=0,\ 1,\ 2,\ \ldots\) by  
\[y_{n} =x_{n}-F^{\prime}(x_{n})^{-1}F(x_{n}),\] \[z_{n} =\phi_{4}(x_{n},y_{n}), \tag{2}\] \[x_{n+1} =\phi_{8}(x_{n},y_{n},z_{n})\] \[=z_{n}-\left[z_{n},\ x_{n};F\right]^{-1}\left[z_{n},\ y_{n};F \right]\left(2\left[z_{n},\ y_{n};F\right]-\left[z_{n},\ x_{n};F\right]\right)^ {-1}F(z_{n}),\]  
where \(x_{0}\in D\) is an initial point, \([\cdot,\ \cdot;F]:D^{2}\to L(X)\), \(\phi_{4}\) is any two-point optimal fourth-order scheme. The eighth order of convergence of Scheme (2) was shown in [1] when \(X=Y=R\) and \([x,\ y;\ F]=\frac{F(x)-F(y)}{x-y}\) for \(x\
eq y\) and \([x,\ x;\ F]=F^{\prime}(x)\). That is when \([\cdot,\ \cdot;\ F]\) is a divided difference of first order of operator \(F\)[5, 6]. The local convergence was shown using Taylor series expansions and hypotheses reaching up to the fifth order derivative. The hypotheses on the derivatives of \(F\) and \(H\) limit the applicability of Scheme (2). As a motivational example, define function \(F\) on \(\mathbb{X}=\mathbb{Y}=\mathbb{R}\), \(D=[-\frac{1}{\pi},\ \frac{2}{\pi}]\) by  
\[F(x)=\left\{ \begin{array}{ll}x^{3}\log(\pi^{2}x^{2})+x^{5}\sin\left(\frac{1 }{x}\right),&\ x\
eq 0,\\ 0,&\ x=0.\end{array} \right.\]  
Then, we have that  
\[F^{\prime}(x)=2x^{2}-x^{3}\cos\left(\frac{1}{x}\right)+3x^{2}\log(\pi^{2}x^{2} )+5x^{4}\sin\left(\frac{1}{x}\right),\]\[F^{\prime\prime}(x)=-8x^{2}\cos\left(\frac{1}{x}\right)+2x(5+3\log(\pi^{2}x^{2}))+ x(20x^{2}-1)\sin\left(\frac{1}{x}\right)\]  
and  
\[F^{\prime\prime\prime}(x)=\frac{1}{x}\left[(1-36x^{2})\cos\left(\frac{1}{x} \right)+x\left(22+6\log(\pi^{2}x^{2})+(60x^{2}-9)\sin\left(\frac{1}{x}\right) \right)\right].\]  
One can easily find that the function \(F^{\prime\prime\prime}(x)\) is unbounded on \(\mathbb{D}\) at the point \(x=0\). Hence, the results in [1], cannot apply to show the convergence of Scheme (2) or its special cases requiring hypotheses on the fifth derivative of function \(F\) or higher.