# Local Convergence Study for an Iterative Scheme with a High Order of Convergence
## 3 Efficiency Index \((EI)\) and Computational Efficiency Index \((CeI)\)  
Sometimes, evaluating the nonlinear operator \(\mathcal{T}\) or its derivatives involves a high computational cost, rendering other operations carried out in the iterative process insignificant. An important measure of the robustness of an algorithm is the Efficiency Index \((EI)\), introduced by [PERSON] in [16]. It is defined by:  
\[EI=p^{\frac{1}{d}},\]  
where \(p\) stands for the order of convergence of the method and \(d\) denotes the number of functional evaluations per iteration. It is important to remember that the number of functional evaluations of \(\mathcal{T}\) and \(\mathcal{T}^{\prime}\) at each iteration is \(N\) and \(N^{2}\), respectively. Thus, for scheme (1), the number of functional evaluations per iteration is \(d=2N^{2}+N\). Given that order of convergence of the method is \(p=4\), the efficiency index is:  
\[EI=4^{\frac{1}{2N^{2}+N}}.\]  
However, the previous concept does not consider the cost of all operations involved in each iteration. The computational efficiency, which takes these costs into consideration, was introduced by [PERSON] [17]. In the N-dimensional case, several linear systems have to be solved for each iteration, and therefore, it is important consider the number of operations performed. Let us recall that the number of products/quotients (per iteration) in the direct solution of a linear system is:  
\[\frac{N^{3}}{3}+N^{2}-\frac{N}{3}\]  
and the number of products/coefficients (per iteration) in the direct solution of \(M\) linear systems with the same coefficient matrix, using LU factorization, is:  
\[\frac{N^{3}}{3}+MN^{2}-\frac{N}{3},\]  
where \(N\) is the size of the linear systems. The cost increases only in \(N^{2}\) operations for each linear system solved with the same coefficient matrix. Moreover, matrix-vector products correspond to \(N^{2}\) operations. Hence, [PERSON] et al. in [18] defined the Computational Efficiency Index \((CEI)\) as:  
\[CEI=p^{\frac{1}{d+op}},\]  
where \(p\) denotes the order of convergence of the method, \(d\) represents the number of functional evaluations per iteration, and \(op\) stands for the number of products/quotients per iteration. However, it is pointless to add \(d\) functional evaluations with operations and, for that reason, we will apply to \(d\) a correction factor \(\mu\), which transforms the number of evaluations into a number of operations, such that:  
\[CEI=p^{\frac{1}{md+op}}.\]  
Therefore, for method (1), the number of products/quotients per iteration is:  
\[op=\frac{2N^{3}}{3}+7N^{2}-\frac{2N}{3}.\]and:  
\[\mu d+op =\mu\Big{(}2N^{2}+N\Big{)}+\left(\frac{2N^{3}}{3}+7N^{2}-\frac{2N}{3 }\right)=\frac{2N^{3}}{3}+(2\mu+7)N^{2}+\left(\mu-\frac{2}{3}\right)N\] \[=\frac{2N^{3}+3(2\mu+7)N^{2}+(3\mu-2)N}{3}.\]  
Consequently, the Computational Efficiency Index of the method is:  
\[CEI=4^{\frac{3}{2N^{3}+3(2\mu+7)N^{2}+(3\mu-2)N}}.\]  
As previously mentioned, the efficiency indexes are analyzed for other methods and then compared the respective indexes of method (1) (hereinafter referred to as M1). The methods considered are [PERSON] (see [13]):  
\[EI=4^{\frac{1}{2N^{2}+N}}.\]0  
as well as methods proposed by [PERSON] and [PERSON] (see [14]), denoted as CHJ:  
\[EI=4^{\frac{1}{2N^{2}+N}}.\]1  
\[EI=4^{\frac{1}{2N^{2}+N}}.\]2  
and [PERSON] et al. (see [15]):  
\[EI=4^{\frac{1}{2N^{2}+N}}.\]3  
\[EI=4^{\frac{1}{2N^{2}+N}}.\]4  
The comparisons of the Computational Efficiency Indexes for the aforementioned methods are shown in \[EI=4^{\frac{1}{2N^{2}+N}}.\]5 The notation \(LS(\mathcal{T}^{\prime}(x))\) and \(LS(Other)\) stands for the number of linear systems to be solved with the same coefficient matrix \(\mathcal{T}^{\prime}(x)\) as well as with other coefficient matrices, respectively. Moreover, in the case of matrix-vector multiplication (M \(\times\) V) \(N^{2}\) products are obtained.