## 3 Convergence Analysis  
**Theorem 1**.: _Assume that function \(f:D\subseteq\mathbb{R}\rightarrow\mathbb{R}\) is sufficiently differentiable and has a simple zero \(\alpha\in D\). If an initial guess \(x_{0}\) is sufficiently close to \(\alpha\in D\), then the iterative scheme defined by Equation (9) has optimal convergence of order eight when_  
\[\left\{\begin{array}{l}G(0)=1,G^{\prime}(0)=0,G^{\prime\prime}(0)=4\text{ and }|G^{(3)}(0)|\leq\infty,\beta\in\mathbb{R}\backslash\{0\},\\ H_{00}=1,H_{10}=2,H_{01}=1,H_{20}=\dfrac{G^{(3)}(0)}{3}+6,H_{11}=4,H_{30}=3G^ {(3)}(0)+\dfrac{G^{(4)}(0)}{4},\end{array}\right.\]  
_where \( H_{ij}=\dfrac{1}{i!j!}\dfrac{\partial H(u,v)}{\partial u^{i}v^{j}}|_{(0,0)},\;i=0,1,2,3\text{ and }j=0,1,2,3\)._  
_It satisfies the following error equation_  
\[\begin{split} e_{n+1}&=\dfrac{1}{432}c_{2}\big{(}(-18+G ^{(3)}(0))c_{2}^{2}+6c_{3}\big{)}\Big{[}\big{(}H_{02}(-18+G^{(3)}(0))^{2}-3(64 8+2H_{21}(-18+G^{(3)}(0))-28G^{(3)}(0)\\ &+3G^{(4)}(0))\big{)}c_{2}^{4}+36(-2+G^{(2)}(0))c_{3}^{2}-12c_{2}^{2} \big{(}6\beta f^{\prime}(\alpha)^{3}+(-102+18H_{02}+3H_{21}+G^{(3)}(0)-G^{(2)} (0)G^{(3)}(0))c_{3}\big{)}\\ &-72c_{2}c_{4}\Big{]}e_{n}^{8}+O(e_{n}^{9}),\end{split} \tag{10}\]  
_where \(e_{n}\) and \(c_{k}\) are already defined in Equation (6)._  
Proof.: Using [PERSON]'s series and symbolic computation, we can determine the asymptotic error constant of three-step derivative-free class of methods Equation (9). Furthermore, taking into account that \(f(\alpha)=0,\) we can expand \(f(x_{n})\) about \(x_{n}=\alpha\). Therefore, we get  
\[f(x_{n})=f^{\prime}(\alpha)(e_{n}+c_{2}e_{n}^{2}+c_{3}e_{n}^{3}+c_{4}e_{n}^{4} +c_{5}e_{n}^{5}+c_{6}e_{n}^{6}+c_{7}e  
This implies that the derivative-free class of methods Equation (9) arrive at optimal eighth-order of convergence by choosing the weight functions as follow:  
\[\left\{\begin{array}{l}G(0)=1,G^{\prime}(0)=0,G^{\prime\prime}(0)=4\text{ and }|G^{(3)}(0)|\leq\infty,\beta\in\mathbb{R}\backslash 0,\\ H_{00}=1,H_{10}=2,H_{01}=1,H_{20}=\frac{G^{(3)}(0)}{3}+6,H_{11}=4,H_{30}=3G^{(3 )}(0)+\frac{G^{(4)}(0)}{4}.\end{array}\right. \tag{21}\]  
Finally, using Equations (21) in (20), we get the following error equation  
\[\begin{split} e_{n+1}&=\frac{1}{432}c_{2}\big{(}(-18+G^{( 3)}(0))c_{2}^{2}+6c_{3}\big{)}\Big{[}\big{(}H_{02}(-18+G^{(3)}(0))^{2}-3(648+2 H_{21}(-18+G^{(3)}(0))-28G^{(3)}(0)\\ &+3G^{(4)}(0)))c_{2}^{4}+36(-2+G^{(2)}(0))c_{3}^{2}-12c_{2}^{2}\big{(}6 \beta f^{\prime}(\alpha)^{3}+(-102+18H_{02}+3H_{21}+G^{(3)}(0)-G^{(2)}(0)G^{( 3)}(0))c_{3}\big{)}\\ &-72c_{2}c_{4}\Big{]}e_{n}^{8}+O(e_{n}^{9}).\end{split} \tag{22}\]  
This concludes the proof.  
**Remark 1**.: _It is straightforward to see that all the proposed methods of family Equation (9) require four functional evaluations, viz., \(f(x_{n}),\ f(y_{n}),\ f(w_{n}),\ f(z_{n})\), per full iteration. Therefore, these methods are optimal in the sense of Kung-Traub conjecture and have the efficiency indices \(E=8^{\frac{1}{4}}\approx 1.682\). Furthermore, by choosing appropriate weight functions in family Equation (9), we can develop several new optimal derivative-free families having optimal eighth-order convergence._  
**Remark 2**.: _From the application point of view, when the given problem is complicated, it becomes very difficult to evaluate derivatives. For example, the nonlinear function \(h(x)=(\cot x)e^{x}\sqrt{(1/(2x^{2}\cosh x)}\) (see Figure 1), has a very complicated first derivative. Such shortcomings lead us to investigate new optimal iterative methods which are totally free from derivatives._  
Figure 1: The graph of h(x) and its root.