# An Improved Convergence Analysis of a Multi-Step Method with High-Efficiency Indices
## 1 Introduction  
Numerous problems in science and engineering can be effectively modeled using linear and nonlinear mathematical equations [1, 2, 3]. However, unlike linear equations, finding exact solutions to nonlinear equations is not feasible. Therefore, we have to rely on iterative methods to find approximate solutions to such equations. One of the most prominent iterative methods is [PERSON]'s method [4], which has a quadratic convergence. In recent years, considerable efforts have been made to improve the convergence properties of iterative methods, aiming for higher convergence rate and better computational efficiency [1, 5, 6, 7, 8, 9]. Many of these methods are extensions or modifications of [PERSON]'s method [10, 11, 12] and have been applied to both single-variable and multi-variable nonlinear equations. The order of convergence of an iterative method is a key measure of its efficiency, which indicates how rapidly the method converges to the solution. We say that a sequence \(\{s_{n}\}\) converges to \(s^{*}\) with the order of convergence at least \(p>0\)[13, 14, 15] if there exists a constant \(Q>0\) such that  
\[\|s_{n+1}-s^{*}\|\leq Q\|s_{n}-s^{*}\|^{p}. \tag{1}\]  
The efficiency index (EI) and informational efficiency (IE) are two metrics to evaluate and compare the performance of iterative methods. [PERSON] [16] has introduced EI asand [PERSON] [14] has introduced IE as  
\[IE=\frac{p}{\eta},\]  
where \(p\) is the order of convergence and \(\eta\) is the total number of functions and derivative evaluations per iteration. We consider that the primary goal of the iterative method is to improve the convergence speed while enhancing the accuracy and overall efficiency of the method. In this paper, we focus on the convergence analysis of iterative methods for solving equations of the form  
\[\mathcal{IE}(s)=0, \tag{2}\]  
where \(\mathcal{IE}:\Omega\subset X\to Y\) is a nonlinear operator from Banach space \(X\) into Banach space \(Y\), and \(\Omega\) is a non-empty open convex set. The multi-step method for solving nonlinear systems given by [PERSON] and [PERSON] [17] is defined for \(s_{0}\in\Omega\) and \(n=0,1,2,\dots\) such that  
\[t_{n} = s_{n}-\mathcal{AE}^{\prime}(s_{n})^{-1}\mathcal{AE}(s_{n})\] \[s_{n+1} = t_{n}-2\mathcal{AE}^{\prime}(s_{n})^{-1}\bigg{[}\mathcal{AE}(v_{ n})+\frac{3}{2}\mathcal{AE}(z_{n})-\frac{3}{2}\mathcal{AE}(t_{n})\bigg{]}, \tag{3}\]  
where \(v_{n}=t_{n}+\mathcal{AE}^{\prime}(s_{n})^{-1}\mathcal{E}(t_{n})\) and \(z_{n}=t_{n}-\mathcal{AE}^{\prime}(s_{n})^{-1}\mathcal{AE}(t_{n})\). The iterative method (3) has a convergence order \(5\), which is studied in [17] using Taylor series expansion, and requires the assumption that the operator \(\mathrm{F}\) is at least six times differentiable. These restrictions are the motivations for our study. It is shown in [17] that the method (3) is highly efficient and superior to the earlier methods. A comparison with other methods in terms of efficiency has been given in [17]. If we follow the analysis made in [17], then the method (3) cannot be used to approximate the solution of (2) if \(F\) cannot be differentiated six times. For example, consider \(q:[-2,2]\to\mathbb{R}\) defined as  
\[q(s)=\left\{ \begin{array}{cl}as^{6}\log s^{2}+bs^{7}+cs^{8}&\text{if }s\
eq 0\\ 0&\text{if }s=0,\end{array} \right.\]  
where \(a\
eq 0\) and \(a,b,c\) are real parameters. Notice that \(s^{*}=1\in[-2,2]\) solves the equation \(q(s)=0\). Further, notice the unboundedness of \(q^{(6)}\) on the interval \([-2,2]\) since \(q^{(6)}(s)\) does not exists at \(s=0\). Thus, the method (3) cannot assure the convergence of \(s_{n}\) to the solution \(s^{*}=0\) if we use the analysis in [17]. However, method (3) does converge to \(s^{*}\) if, e.g., \(a=1\), \(b=-2\), and \(c=1\) and the initial guess \(s_{0}=0.95\). We have studied the convergence of the method (3) without using the Taylor series. Thus, our study relaxes the condition that \(F\) has to be six times differentiable and requires \(F\) to be just two times differentiable.