## 3 Description of a New Method  
Third-order methods free from second derivatives were proposed from quadrature rule for solving systems of nonlinear equations. These methods require one function evaluation and two first order derivatives at two points. One such method is:  
\(3^{rd}CON\) method derived from Closed-Open quadrature formula [10]:  
\[\mathbf{x}^{(k+1)}=\mathbf{G}_{3^{rd}CON}(\mathbf{x}^{(k)})=\mathbf{x}^{(k)} -\mathbf{A}(\mathbf{x}^{(k)})^{-1}\mathbf{F}(\mathbf{x}^{(k)}) \tag{3}\]where  
\[\mathbf{A}(\mathbf{x})=\frac{\mathbf{F}^{\prime}(\mathbf{x})+3\mathbf{F}^{\prime} \left(\mathbf{x}-\frac{2}{3}\mathbf{u}(\mathbf{x})\right)}{4} \tag{4}\]  
This method is also a member of Frontini-Sormani family of third order methods derived from quadrature rules [11]. The convergence analysis of this method using point of attraction theory can be found in [12]. This method is also more efficient than [PERSON]'s method because it does not require the evaluation of a third order tensor of \(n^{3}\) values. Let \(p\) be the order of a method and \(d\) be defined as \(d=d_{0}n+\sum_{j=1}^{q}d_{j}\)\(n^{j+1},\) where \(d_{0}\) and \(d_{j}\) represent the number of times \(\mathbf{F}\) and \(\mathbf{F}^{(j)}\) should be evaluated, respectively. The definition of the logarithm of Informational Efficiency or Efficiency Index for nonlinear systems (\(IE\)) [12] is given by  
\[IE=\frac{\ln p}{d_{0}n+\sum_{j=1}^{q}d_{j}n^{j+1}} \tag{5}\]  
Due to that the efficiency indices for the [PERSON]'s method (\(2^{nd}NM\)) and the third-order method free from second derivatives (\(3^{rd}CON\)) are given by  
\[IE_{2^{nd}NM}=\frac{\ln 2}{n+n^{2}}\quad and\quad IE_{3^{rd}CON}=\frac{\ln 3}{n +2n^{2}} \tag{6}\]  
respectively. We observe that \(IE_{3^{rd}CON}>IE_{2^{nd}NM},\) if \(n=1\). That is, the third order methods free from second derivatives are less efficient than [PERSON]'s method for systems of nonlinear equations. Thus, it is suitable to develop a fourth-order method from the third-order method to improve the efficiency. For the scalar case, we can suggest the following quartical iteration, which is in fact a Jarratt-type iterative method including three functional evaluations to reach the highest possible order four [13]  
\[\left\{ \begin{array}{l}y_{k}=x_{k}-\frac{2}{3}\frac{f(x_{k})}{f^{\prime}(x_ {k})},\\ x_{k+1}=x_{k}-\frac{4f(x_{k})}{f^{\prime}(x_{k})+3f^{\prime}(y_{k})}\ \left[1+\frac{9}{16}(\frac{f^{\prime}(y_{k})}{f^{\prime}(x_{k})}-1)^{2}\right] \end{array} \right. \tag{7}\]  
We here build our efficient high order method according to (7) to reach the highest possible order using a fixed and the smallest possible number of functional evaluations. The improved fourth-order method (\(4^{th}CON\)) to systems of nonlinear equations can be constructed and suggested as follows  
\[\mathbf{x}^{(k+1)}=\mathbf{G}_{4^{th}CON}(\mathbf{x}^{(k)})=\mathbf{x}^{(k)}- \mathbf{H}(\mathbf{x}^{(k)})\mathbf{A}(\mathbf{x}^{(k)})^{-1}\mathbf{F}( \mathbf{x}^{(k)}) \tag{8}\]  
where  
\[\mathbf{H}(\tau(\mathbf{x}))=\mathbf{I}+\frac{9}{16}(\tau(\mathbf{x})- \mathbf{I})^{2},\,\tau(\mathbf{x})=\mathbf{F}^{\prime}(\mathbf{x})^{-1} \mathbf{F}^{\prime}\left(\mathbf{x}-\frac{2}{3}\mathbf{u}(\mathbf{x})\right)\]  
and \(\mathbf{I}\) is the \(n\times n\) identity matrix. We also note that this method is an improvement of the \(3^{th}CON\) method because it can also be written as:  
\[\mathbf{G}_{4^{th}CON}(\mathbf{x}^{(k)})=\mathbf{x}^{(k)}+\mathbf{H}(\mathbf{ x}^{(k)})\left(\mathbf{G}_{3^{th}CON}(\mathbf{x}^{(k)})-\mathbf{x}^{(k)}\right) \tag{9}\]  
We describe the algorithm of \(4^{th}CON\) method, which requires the evaluations of one function with \(n\) values, two Jacobians of \(n^{2}\) values each and the inversion of two matrices as comes next:\(Algorithm\)\(4^{th}CON\)  
\[\left\{