## 4 Computational Efficiency of the Algorithms  
The efficiency index of any iterative method is measured using the [PERSON]'s definition [16], \(EI=p^{\frac{1}{2}}\), where \(p\) denotes the order of convergence and \(d\) denotes the number of functional evaluations per iteration. Different algorithms considered here are compared with the proposed algorithms in terms of computational cost. To evaluate Jacobian \(\Phi^{\prime}\) and \(\Phi\), one needs \(n^{2}\) function evaluations (all the elements of matrix \(\Phi^{\prime}\)) and \(n\) scalar functional evaluations (the coordinate terms in \(\Phi\)) respectively. Whereas, any iterative method which solves system of nonlinear equations needs one or more matrix inversion. This indicates few linear systems must be solved. Hence, the number of operations required for solving a linear system of equations dominates when the computational cost of an iterative method is measured. Due to this, [PERSON] et al. [3] introduced the concept of computational efficiency index (CE), where the efficiency index given by [PERSON] is combined with the number of products-quotients required per iteration. Computational efficiency index is defined as \(CE=p^{1/(d+op)}\)where \(op\) is the number of products-quotients per iteration and the details of its calculation is given in the next paragraph. The total cost of computation for a nonlinear system of \(n\) equations with \(n\) unknowns is calculated as follows: For an iterative function, \(n\) scalar functions \(\phi_{i}\) (\(1\leq i\leq n\)) are evaluated and for the computation of divided difference \(n(n-1)\) scalar functions are evaluated, where \(\Phi(x)\) and \(\Phi(y)\) are computed separately. Also, \(n^{2}\) quotients are added for divided difference. For calculating inverse linear operator, \(\frac{n^{2}-n}{3}\) products and divisions in the LU decomposition and \(n^{2}\) products and divisions for solving two triangular linear systems are taken. Moreover, \(n^{2}\) products for multiplying a matrix with a vector or a scalar and \(n\) products for multiplying a vector by a scalar are required. The computational cost and efficiency of the proposed algorithms are compared with methods given by (1) to (6) and the algorithms presented recently by [PERSON] et al. [17] which are given below: A fifth-order three-step method (\(5^{th}SD\)) is given by  
\[x^{(r+1)}=G_{5^{th}SD}(x^{(r)})=z(x^{(r)})-\psi(x_{n},y_{n})\Phi( z^{(r)}),\] \[\textit{where}\;\;z(x^{(r)})=G_{2^{nd}NM}(x^{(r)})-\[PERSON]^{\prime}( x^{(r)})^{-1}\[PERSON](G_{2^{nd}NM}(x^{(r)})),\] \[\psi(x_{n},y_{n})=\Big{(}2I-\[PERSON]^{\prime}(x^{(r)})^{-1}[z_{r},y_{ r};\Phi]\Big{)}\[PERSON]^{\prime}(x^{(r)})^{-1}.\]  
An eighth-order four-step method (\(8^{th}SD\)) is given by  
\[x^{(r+1)}=G_{8^{th}SD}(x^{(r)})=w(x^{(r)})-\psi(x_{n},y_{n})\Phi( w^{(r)}),\] \[\textit{where}\;\;w(x^{(r)})=z(x^{(r)})-\psi(x_{n},y_{n})\Phi(z^{ (r)}),\] \[z(x^{(r)})=G_{2^{nd}NM}(x^{(r)})-\Big{(}3I-2\[PERSON]^{\prime}(x^{(r)} )^{-1}[y_{r},x_{r};\Phi],\] \[\psi(x_{n},y_{n})=\Big{(}2I-\[PERSON]^{\prime}(x^{(r)})^{-1}[z_{r},y_{ r};\Phi]\Big{)}\Phi^{\prime}(x^{(r)})^{-1}.\]  
Table 1 displays the computational cost and computational efficiency (CE) of various methods. To compare the CE of considered iterative methods, we calculate the following ratio [17], where \(C_{method}\) and \(CE_{method}\) denote respectively the computational cost and computational efficiency of the method. \[R_{method1;method2}=\frac{log(CE_{method1})}{log(CE_{method2})}=\frac{C_{ method2}\;log(order\;of\;method1)}{C_{method1}\;log(order\;of\;method2)}. \tag{23}\]