# Local Convergence Study for an Iterative Scheme with a High Order of Convergence
## 1 Introduction
These models often use iterative methods to solve complex mathematical problems (such as nonlinear equations, optimization tasks, and machine learning model training) and local convergence analysis is essential in ensuring the accuracy, efficiency, robustness, and adaptability of iterative methods used in CAD systems. Several studies [2; 3] have explored, for some iterative methods, local convergence under various continuity conditions. For instance, [PERSON] in [4], for the modified [PERSON]'s method for multiple zeros, provides an estimate for the convergence radius, assuming the function meets Holder and center-Holder continuity conditions. This result is further refined by [PERSON] et al. in [5]. [PERSON] et al. in [6] studied third-order methods, and they established local convergence using divided differences and its properties. [PERSON] et al. (see [7]) used [PERSON]'s expansions to establish a new set of bounds in order to ensure convergence. [PERSON] and [PERSON] (see [8]) also employed [PERSON]'s expansions to analyze the local convergence of [PERSON]'s method, but considering that the derivative verifies the center-Holder continuity condition. In addition, [PERSON] et al. [9] examined, for a fifth-order iterative scheme, both the local and semilocal convergence under Holder continuity conditions. In the semilocal convergence analysis, certain conditions are placed both on the operators and their respective derivatives at \(x_{0}\) (the initial estimate), and iterations are then carried out. This leads to results such as the existence of the solution, its uniqueness, a priori error bound, the \(R\)-order of convergence of the scheme, and the ascertainment of the convergence region where the operator \(\mathcal{T}\) is defined. In this work, we will focus on local convergence analysis. Some elements of this study are shown in [10]. This paper presents an iterative method along with its local convergence analysis for solving the nonlinear equation \(\mathcal{T}(x)=0\), which has a solution \(x^{*}\). Consider the (fourth-order) family of iterative schemes studied in [11], defined for \(k=0,1,2,\ldots\) by:  
\[y_{k}=x_{k}-\frac{2}{3}\mathcal{T}^{\prime}(x_{k})^{-1}\mathcal{T}(x_{k})\]  
\[x_{k+1}=x_{k}-\bigg{[}\bigg{(}\frac{5}{8}-\alpha\bigg{)}+\alpha \mathcal{T}^{\prime}(y_{k})^{-1}\mathcal{T}^{\prime}(x_{k})+\frac{\alpha}{3} \mathcal{T}^{\prime}(x_{k})^{-1}\mathcal{T}^{\prime}(y_{k}) \tag{1}\]  
where \(\alpha\in\mathbb{C}\) is a parameter and \(x_{0}\) is the initial estimate. In order to approximate the solution of the nonlinear equation \(\mathcal{T}(x)=0\), it is assumed that the derivative \(\mathcal{T}^{\prime}(x)\) exists in a neighborhood of the solution \(x^{*}\) and that \(\mathcal{T}^{\prime}(x^{*})^{-1}\in\mathcal{L}(Y,X)\), where \(\mathcal{L}(Y,X)\) is the set of bounded linear operators mapping from \(Y\) to \(X\). In the following sections, this paper addresses these topics: a local convergence analysis is presented in Section 2. In Section 3, we undertake a computational efficiency study of the new method and compare it with other existing schemes. Section 4 includes several numerical examples in order to validate the theoretical results. Concluding observations are summarized in Section 5.