Estudio de métodos tipo secante: convergencia, estabilidad y accesibilidad

This research is part of the study and development of iterative numerical methods for solving nonlinear equations. In particular, it focuses on the analysis and optimization of secant-type methods, with the aim of improving their efficiency and stability compared to traditional techniques. The motivation for this study lies in the importance of solving nonlinear equations in multiple scientific and engineering fields, where exact solutions are often not possible and iterative algorithms must be used. In this context, this thesis develops a new family of iterative methods based on a reformulation of the secant method by introducing control parameters and symmetric divided differences. It is demonstrated that these modifications improve the convergence speed without increasing the computational cost. Through a rigorous theoretical analysis, the local convergence of the proposed methods is studied, establishing conditions under which the iterative scheme reaches a quadratic order of convergence. In addition, the dynamic behavior of these methods is explored using complex analysis tools, identifying the existence of basins of attraction and zones of instability. The development of these methods required the implementation of various numerical simulations in Python, using specialized libraries for solving nonlinear equations. Through these simulations, the performance of the proposed methods was compared with classical schemes, verifying that the new formulations present greater robustness and accessibility. These algorithms have been applied to solve integral equations and problems in which the objective function is not differentiable, demonstrating their usefulness in broader contexts. From a theoretical perspective, the work also addresses the relationship between secant-type methods and iterative schemes based on operators in Banach spaces. It studies how decomposing the operator into a differentiable part and a continuous, non-differentiable part improves the applicability of these algorithms to problems where classical methods present limitations. This approach has made it possible to extend the applicability of secant-type methods to more general equations while maintaining a simple and efficient computational structure. Throughout the development of this thesis, special attention has been paid to the graphical analysis of the behavior of iterative methods, exploring their dynamic properties in the complex plane. Visual representations have been used to understand the distribution of the basins of attraction of the roots and the effects of the introduced parameters on the stability of the iterations. These representations have been key to characterizing the impact of the proposed improvements and have allowed for a visual comparison with other iterative methods. The results obtained show that the optimization of secant-type methods using parametric adjustments and divided-difference techniques allows for more efficient schemes, with larger regions of convergence and improved dynamic behavior. The implementation of these methods in computational environments confirms their practical utility and their potential for use in mathematical and engineering problems where solving nonlinear equations is a recurring challenge.