## 2 Developments of the Methods  
Let \(x_{n+1}=\psi(x_{n})\) define an Iterative Function (I.F.). **Definition 1**. : _[_8_]_ _If the sequence \(\{x_{n}\}\) tends to a limit \(x^{*}\) in such a way that_  
\[\lim_{n\rightarrow\infty}\frac{x_{n+1}-x^{*}}{(x_{n}-x^{*})^{p}}=C\]  
_for \(p\geq 1\), then the order of convergence of the sequence is said to be \(p\), and \(C\) is known as the asymptotic error constant. If \(p=1\), \(p=2\) or \(p=3\), the convergence is said to be linear, quadratic or cubic, respectively._  
_Let \(e_{n}=x_{n}-x^{*}\), then the relation_  
\[e_{n+1}=C\;e_{n}^{p}+O\Big{(}e_{n}^{p+1}\Big{)}=O\Big{(}e_{n}^{p}\Big{)}. \tag{1}\]  
_is called the error equation. The value of \(p\) is called the order of convergence of the method._  
**Definition 2**. : _[_9_]_ _The Efficiency Index is given by_  
\[EI=p^{\frac{1}{4}}, \tag{2}\]  
_where \(d\) is the total number of new function evaluations (the values of \(f\) and its derivatives) per iteration._  
Let \(x_{n+1}\) be determined by new information at \(x_{n},\phi_{1}(x_{n}),...,\phi_{\rm i}(x_{n}),{\rm i}\geq 1\). No old information is reused. Thus,  
\[x_{n+1}=\psi(x_{n},\phi_{1}(x_{n}),...,\phi_{\rm i}(x_{n})). \tag{3}\]  
Then \(\psi\) is called a multipoint I.F without memory. **Kung-Traub Conjecture**[10]  
Let \(\psi\) be an I.F. without memory with \(d\) evaluations. Then  
\[p(\psi)\leq p_{Opt}=2^{d-1}, \tag{4}\]  
where \(p_{opt}\) is the maximum order. The second order Newton (also called Newton-Raphson) I.F. (\(2^{nd}\)_NR_) is given by  
\[\psi_{2^{nd}NR}(x)=x-u(x),\,u(x)=\frac{f(x)}{f^{\prime}(x)}. \tag{5}\]The \(2^{nd}\)_NR_ I.F. is a is 1-point I.F. with \(2\) functions evaluations and it satisfies the [PERSON] conjecture \(d=2\). Thus, \(EI_{2^{nd}NR}=1.414\). The Newton I.F. can be constructed from its local linear model, of the function \(f(x)\), which is the tangent drawn to the function \(f(x)\) at the current point \(x\). The local linear model at \(x\) is  
\[\mathrm{L}(x)=f(x)+f^{\prime}(x)(\psi-x). \tag{6}\]  
This local linear model can be interpreted from the viewpoint of the Newton Theorem:  
\[f(\psi)=f(x)+I_{NT},\text{ where }I_{NT}=\int_{x}^{\psi}f^{\prime}(v)\;dv. \tag{7}\]  
[PERSON] and [PERSON] [11] showed that if the indefinite integral \(I_{NT}\) is approximated by the rectangle: \(I_{NT}\approx f^{\prime}(x)(\psi-x)\), the Newton I.F. is obtained by setting \(\mathrm{L}(x)=0\). [PERSON] _et al._[1] obtained a new linear model  
\[\mathrm{L}_{1}(x)=f(x)+\frac{1}{6}\left(f^{\prime}(x)+4f^{\prime}\left(\frac {x+\psi}{2}\right)+f^{\prime}(\psi)\right)(\psi-x) \tag{8}\]  
by approximating \(I_{NT}\) by [PERSON]'s formula: \(I_{NT}\approx\frac{1}{6}\left(f^{\prime}(x)+4f^{\prime}\left(\frac{x+\psi}{2 }\right)+f^{\prime}(\psi)\right)\). Solving the new linear model, they obtained the implicit I.F. :  
\[\psi(x)=x-\frac{6f(x)}{f^{\prime}(x)+4f^{\prime}\left(\frac{x+\psi(x)}{2} \right)+f^{\prime}(\psi(x))}. \tag{9}\]  
Using the Newton I.F. to estimate \(\psi(x)\) in the first derivative by \(\psi_{2^{nd}NR}(x)\), they obtained the \(3^{rd}3 pVS\) I.F. :  
\[e_{n+1}=C\;e_{n}^{p}+O\Big{(}e_{n}^{p+1}\Big{)}=O\Big{(}e_{n}^{p}\Big{)}. \tag{1}\]0  
The \(3^{rd}3 pVS\) I.F. is a special case of the Frontini-Sormani family of third order I.F.s from quadrature rule [12]. It was extended to systems of equations in [13]. However this I.F. is considered as inefficient. According to [PERSON] conjecture, the optimal order of the I.F. with \(d=4\) is eight. In fact, we can have optimal three-point eighth order I.F.s with \(4\) function evaluations or with \(3\) function and \(1\) first derivative evaluations (see [5,14] and the references therein). However, we can achieve only a maximum of sixth order with I.F.s with \(2\) function and \(2\) first derivative evaluations. The question we now pose: What is the maximum order we can achieve with I.F.s with \(1\) function and \(3\) first derivative evaluations? Let us define \(\tau=\frac{f^{\prime}(\psi_{2^{nd}NR}(x))}{f^{\prime}(x)}\). We improve the \(3^{rd}3 pVS\) I.F.