# Two Efficient Derivative-Free Iterative Methods for Solving Nonlinear Systems
## 3 Computational Efficiency  
The classical efficiency index \(E=\rho^{1/c}\) (see [9]) is the most used index, but not the only one. We find that the iterative methods with the same classical efficiency index (\(E\)) have the different properties in actual applications. The reason is that the number of functional evaluations of iterative method is not the only influence factor in evaluating the efficiency of the iterative method. The number of matrix products, scalar products, decomposition LU of matrix, and the resolution of the triangular linear systems also play an important role in evaluating the real efficiency of iterative method. In this paper, the computational efficiency index (\(CEI\)) [10] is used to compare the efficiency of the iterative methods. Some discussions on the \(CEI\) can be found in [4; 5; 6; 7]. The \(CEI\) of the iterative methods \(\psi_{i}\) (\(i=1,2,\cdots,5\)) is given by  
\[CEI_{i}(\mu,\;m)=\rho_{i}^{\frac{1}{c_{i}(\mu,\;m)}},\;i=1,2,3,4,5 \tag{28}\]  
where \(\rho_{i}\) is the order of convergence of the method and \(C_{i}(\mu,m)\) is the computational cost of method. The \(C_{i}(\mu,m)\) is given by  
\[C_{i}(\mu,\;m)=a_{i}(m)\mu+p_{i}(m) \tag{29}\]  
where \(a_{i}(m)\) denotes the number of evaluations of scalar functions used in the evaluations of Equation (29) between products (and divisions) and evaluations of functions is required, see [5; 10]. We must add \(m\) products for multiplication of a vector by a scalar and \(m^{2}\) products for matrix-vector multiplication. To compute an inverse linear operator, we need \((m^{3}-m)/3\) products and divisions in the LU decomposition and \(m^{2}\) products and divisions for solving two triangular linear systems. If we compute the first-order divided difference then we need \(m(m-1)\) scalar functional evaluations and \(m^{2}\) quotients. The first-order divided difference \([x,y;F]\) of \(F\) is given by  
\[[y,x;F]_{ij}=(F_{i}(y_{1}\cdots,y_{j-1},y_{j},x_{j+1},\cdots,x_{m})-F_{i}(y_{1 }\cdots,y_{j-1},x_{j},x_{j+1},\cdots,x_{m}))/(y_{j}-x_{j})\]  
where \(1\leq i,j\leq m\), \(x=(x_{1},\cdots x_{j-1},x_{j},x_{j+1},\cdots x_{m})\) and \(y=(y_{1},\cdots y_{j-1},y_{j},y_{j+1},\cdots y_{m})\) (see [9]). Based on Equations (28) and (29), Table 1 shows the computational cost of different methods. From Table 1, we can see that our methods \(\psi_{i}\)\((i=4,5)\) need less number of LU decomposition than methods \(\psi_{2}\) and \(\psi_{3}\). The computational cost of the fourth-order methods show the following order:  
\[C_{4}<C_{2},\text{ for }m\geq 2 \tag{30}\]  
We use the following expressions [10] to compare the CEI of different methods  
\[R_{i,j}=\frac{\ln CEL_{i}}{\ln CEL_{j}}=\frac{\ln(\rho_{i})C_{j}(\mu,m)}{\ln( \rho_{j})C_{i}(\mu,m)},\ i,j=1,2,3,4,5 \tag{31}\]  
For \(R_{i,j}>1\) the iterative method \(M_{i}\) is more efficient than \(M_{j}\). Using the \(CEI\) of the iterative methods,we obtain the following theorem:  
**Theorem 2**. : 1. _For the fourth-order method, we have \(CEI_{4}>CEI_{2}\) for all \(m\geq 2\) and \(\mu>0\). 2. For the sixth-order method, we have \(CEI_{5}>CEI_{3}\) for all \(m\geq 11\) and \(\mu>0\)._  
Proof. : 1. From Table 1, we note that the methods \(\psi_{i}\)\((i=2,4)\) have the same order \(\rho_{2}=\rho_{4}=4\). Based on Equations (29) and (30), we get that \(CEI_{4}>CEI_{2}\) for all \(m\geq 2\) and \(\mu>0\). 2. The methods \(\psi_{i}\)\((i=3,5)\) have the same order and the same functional evaluations. The relation between \(\psi_{5}\) and \(\psi_{3}\) can be given by  
\[R_{5,3}=\frac{\ln(\rho_{5})C_{3}(\mu,m)}{\ln(\rho_{3})C_{5}(\mu,m)}=\frac{m(2 m+3)\mu+2(m^{3}-m)/3+6m^{2}}{m(2m+3)\mu+(m^{3}-m)/3+9m^{2}+4m} \tag{32}\]  
Subtracting the denominator from the numerator of Equation (32), we have  
\[\frac{1}{3}m(m^{2}-9m-13) \tag{33}\]  
The Equation (33) is positive for \(m\geq 10.2662\). Thus, we obtain that \(CEI_{5}>CEI_{3}\) for all \(m\geq 11\) and \(\mu>0\).