# Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative
## 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 problems of numerical analysis are concerned with efficiently and accurately finding the approximate locally unique solution \(x^{*}\) of an equation of the form  
\[F(x)=0, \tag{1}\]  
where \(F\) is a Frechet differentiable operator defined on a convex subset \(\mathbb{D}\) of \(\mathbb{X}\) with values in \(\mathbb{Y}\), where \(\mathbb{X}\) and \(\mathbb{Y}\) are 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, worldwide, researchers resort to an iterative method and a plethora of iterative methods has been proposed [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15]. While using these iterative methods, researchers face the problems of slowconvergence, non-convergence, divergence, inefficiency or failure (for detail please see [PERSON] [14] and [PERSON] et al. [11]). 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 iterative 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 a three step method defined for each \(n=0,\;1,\;2,\;\ldots\) by  
\[ \begin{split} y_{n}&=x_{n}-F^{\prime}(x_{n})^{-1}F(x _{n}),\\ z_{n}&=\phi_{4}(x_{n},y_{n}),\\ x_{n+1}&=\phi_{8}(x_{n},y_{n},z_{n})\\ &=z_{n}-(2[z_{n},\;y_{n};F]-[z_{n},\;x_{n};F])^{-1}\left(F^{ \prime}(x_{n})-[y_{n},\;x_{n},\;F]+[z_{n},\;y_{n};F]\right)F(x_{n})^{-1}F(z_{n }),\end{split}  \tag{2}\]  
where \(x_{0}\in\mathbb{D}\) is an initial point, \([\cdot,\;\cdot,\;F]:\mathbb{D}^{2}\to L(\mathbb{X})\), \(\phi_{4}\) is any two-point optimal fourth-order scheme and \(\phi_{8}(x,y,z):=z-(2[z,y;F]-[z,x;F])^{-1}\left(F^{\prime}(x)-[y,x;F]+[z,y;F] \right)F^{\prime}(x)^{-1}F(z)\). The eighth order of convergence of method Equation (2) was shown in [13] when \(\mathbb{X}=\mathbb{Y}=\mathbb{R}\) 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 \(F\)[4; 5]. The local convergence was shown using Taylor series expansions and hypotheses reaching up to the fifth order derivative. The hypotheses on the derivatives of \(F\) limit the applicability of method Equation (2). As a motivational example, define function \(F\) on \(\mathbb{X}=\mathbb{Y}=\mathbb{R}\), \(\mathbb{D}=[-\frac{1}{\pi},\;\frac{2}{\pi}]\) 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 for \(x\
eq 0\),we have  
\[F^{\prime}(x)=2x^{2}-x^{3}\cos\left(\frac{1}{x}\right)+3x^{2}\log(\pi^{2}x^{2 })+5x^{4}\sin\left(\frac{1}{x}\right),\]  
\[F^{\prime\prime}(x)=-8x^{2}\cos\left(\frac{1}{x}\right)+2x(5+3\log(\pi^{2}x^{ 2}))+x(20x^{2}-1)\sin\left(\frac{1}{x}\right)\]  
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 see that the function \(F^{\prime\prime\prime}\) is unbounded on \(\mathbb{D}\) at the point \(x=0\).