Some Efficient Numerical Methods for Solving Nonlinear Equations and Their Dynamics - Sharma, Himani (My University, 2025)
This thesis centers on the formulation and examination of iterative methods aimed at resolving both scalar and systems of nonlinear equations. The main goal is to enhance the order of convergence while analyzing the stability characteristics of the rational functions linked to these methods. The structure of the thesis comprises eight chapters, each focusing on a specific dimension of iterative techniques. The first chapter offers a comprehensive motivation and literature review, highlighting the necessity for effective iterative methods. Chapter 2 presents a novel family of without memory iterative methods, based on a cubically convergent Hansen-Patrick type scheme, which proves effective even when the derivative approaches zero. This methodology is subsequently expanded to incorporate memory, thereby improving convergence. Chapter 3 introduces an iterative family with memory, which enhances the convergence order beyond that of conventional Chebyshev-Halley type methods through the use of self-accelerating parameters. A thorough dynamical analysis identifies stable and efficient members within this family. Chapter 4 investigates an optimal fourth-order iterative family, utilizing complex dynamics to evaluate the stability of various family variants. The analysis of critical and fixed points is conducted to understand convergence behavior, supported by numerical experiments that affirm the theoretical results. Chapter 5 broadens the stability analysis to encompass an optimal mean-based family of iterative methods, employing dynamical tools to examine sensitivity to initial guesses and its implications for solving chemistry-related challenges. Chapter 6 introduces an innovative derivative-free optimal iterative scheme that maintains effectiveness even when traditional methods falter due to diminishing derivatives. An extension incorporating memory further boosts the convergence order. Chapter 7 details an optimal fourth-order iterative method tailored for multiple roots, ensuring robustness even as the derivative nears zero. Lastly, Chapter 8 generalizes the techniques developed to address systems of nonlinear equations, showcasing their efficacy. In its conclusion, the thesis provides a concise summary of the key contributions, focusing on the advancements in iterative methods, their convergence properties, and the stability analysis performed. It also suggests directions for future research, such as the potential for higher-order iterative schemes, their application to large-scale issues, and further investigation into dynamical properties to refine stability evaluations. By combining innovative iterative methods with robust theoretical analysis and practical applications, this thesis makes a significant contribution to the ongoing refinement of efficient numerical techniques for solving nonlinear equations, setting the stage for future innovations in this field.
Keywords: Nonlinear equation; iterative method; Stability; Complex dynamics; Dynamical plane