Stability Analysis and Local Convergence of a New Fourth-Order Optimal Jarratt-Type Iterative Scheme

In this work, using the weight function technique, we introduce a new family of fourth-order iterative methods optimal in the sense of Kung and Traub for scalar equations, generalizing Jarratt’s method. Through Taylor series expansions, we confirm that all members of this family achieve fourth-order convergence when derivatives up to the fourth order are bounded. Additionally, a stability analysis is performed on quadratic polynomials using complex discrete dynamics, enabling differentiation among the methods based on their stability. To demonstrate practical applicability, a numerical example illustrates the effectiveness of the proposed family. Extending our findings to Banach spaces, we conduct local convergence analyses on a specific subfamily containing Jarratt’s method, requiring only boundedness of the first derivative. This significantly broadens the method’s applicability to more general spaces and reduces constraints on higher-order derivatives. Finally, additional examples validate the existence and uniqueness of approximate solutions in Banach spaces, provided the initial estimate lies within the locally determined convergence radius obtained using majorizing functions.