# Local Convergence Study for an Iterative Scheme with a High Order of Convergence
## 1 Introduction  
A significant issue in Numerical Functional Analysis is problem-solving in science and engineering using mathematical modeling techniques. A specific problem can be modeled, under certain conditions, as a nonlinear equation in a Banach space. Let \(X\), \(Y\) be Banach spaces, \(D\) an open convex subset of \(X\), and \(\mathcal{T}\) a nonlinear operator, \(\mathcal{T}:D\subseteq X\to Y\). The primary focus in studying an iterative method is to find the closest approximation to the solution of the equation \(\mathcal{T}(x)=0\). However, it is crucial to ensure that the selected iterative method indeed converges to this approximation. To achieve both convergence and accurate results for the scheme, we can perform studies aimed at the necessary conditions that must be verified by the solution \(x^{*}\), the initial value \(x_{0}\), or the operator within the framework of the iterative method. In Banach spaces, a convergence study can be approached from two perspectives: considering the neighborhood of the solution (referred to as local convergence analysis) or focusing on the initial estimates and the domain (known as semilocal convergence analysis). When we analyze local convergence (see [1, 2]), conditions are applied on the operators and their respective derivatives at the solution \(x^{*}\). In consequence, we obtain the local convergence ball, centered in the solution and with radius \(r\): \(B(x^{*},r)\). The elements contained within this ball are possible initial estimates which guarantee convergence, meaning that they are suitable initial points for the scheme to work or for which the sequence of iterates converges to the solution of the problem. Additionally, an error bound is established. Local convergence analysis plays a significant role in understanding the behavior, efficiency, and reliability of an iterative method. This analysis provides detailed information about how fast an iterative method converges to a solution given an initial estimate. Moreover, through local convergence analysis we are able to identify the conditions under which an iterative method will converge to a solution, concerning initial estimate proximityand smoothness conditions. This analysis often provides error bounds, showing how the error decreases in each iteration, which helps in determining how accurate the current approximation is. In summary, local convergence analysis helps in choosing the right method, refining existing methods, and understanding their behavior under realistic conditions. Local convergence analysis is essential in science and engineering for guaranteeing the accuracy, efficiency, stability, and adaptability of iterative methods used to solve complex real-world problems. Whether optimizing designs, conducting scientific simulations, solving nonlinear equations, or managing large-scale data, understanding the local convergence properties of an iterative scheme is necessary to achieve reliable and efficient solutions. For example, in Computational Fluid Dynamics (CFD), where simulations of airflow around objects may engage a large number of variables, fast local convergence can save considerable computational cost and make the process feasible. In molecular dynamics, iterative methods are used to study complex systems and the efficiency provided by a local convergence analysis can allow more detailed and expansive simulations. In mechanical engineering, solving the analysis of material deformations often requires iterative methods and local convergence analysis ensures these methods can handle the complexities of the system. In physics and chemistry, where iterative methods are used to solve quantum mechanical equations, local convergence analysis helps ensure accurate energy levels and wave functions are obtained. Computer-Assisted Diagnostic (CAD) systems rely heavily on various computational models and algorithms.