Extended Convergence of Two Multi-Step Iterative Methods

Abstract

Iterative methods which have high convergence order are crucial in computational mathematics since the iterates produce sequences converging to the root of a non-linear equation. A plethora of applications in chemistry and physics require the solution of non-linear equations in abstract spaces iteratively. The derivation of the order of the iterative methods requires expansions using Taylor series formula and higher-order derivatives not present in the method. Thus, these results cannot prove the convergence of the iterative method in these cases when such higher-order derivatives are non-existent. However, these methods may still converge. Our motivation originates from the need to handle these problems. No error estimates are given that are controlled by constants. The process introduced in this paper discusses both the local and the semi-local convergence analysis of two step fifth and multi-step 5+3r order iterative methods obtained using only information from the operators on these methods. Finally, the novelty of our process relates to the fact that the convergence conditions depend only on the functions and operators which are present in the methods. Thus, the applicability is extended to these methods. Numerical applications complement the theory.

Extended Newton-Traub Method of Order Six

“…Evaluating the local and semi-local characteristics of iterative techniques offers valuable insights into convergence traits, error limits, and the area of uniqueness for solutions [10][11][12][13][14]. Numerous research endeavors have concentrated on exploring the local and semi-local convergence of effective iterative approaches, yielding noteworthy outcomes like convergence radii, error approximations, and the broadened applicability of these methods [15][16][17][18][19]. These findings are particularly valuable as they shed light on the intricacies involved in selecting appropriate initial points for the iterative process.…”

Increased and Extended Convergence of a family of three-step methods

“…The analysis of local and semi-local behavior of iterative methods provides valuable insights into convergence properties, error bounds, and the region of uniqueness for solutions. Several studies [15,18] have focused on investigating the local and semi-local convergence of ecient iterative techniques, yielding signicant results such as convergence radii, error estimates, and extended applicability of these methods. These ndings are particularly valuable as they shed light on the intricacies involved in selecting appropriate initial points for the iterative process.…”

Efficient Derivative-Free Class of Seventh Order Method for Non-differentiable Equations

“…Newton's Method is a well-known iterative method for handling non-linear equations. Recently, with advances in Science and Mathematics many new iterative methods of higher order have been discovered for the handling of non-linear equations and are currently being used [1,2,[4][5][6][7][8][10][11][12][13][14][15][16][17][18][19][20][21][22]. The computation of derivatives of second and higher order is a great disadvantage for the iterative systems of higher order and is not suitable for the practical application.…”