On the semi-local convergence of a sixth order method in Banach space

Abstract

High convergence order methods are important in computational mathematics, since they generate sequences converging to a solution of a non-linear equation. The derivation of the order requires Taylor series expansions and the existence of derivatives not appearing on the method. Therefore, these results cannot assure the convergence of the method in those cases when such high order derivatives do not exist. But, the method may converge. In this article, a process is introduced by which the semi-local convergence analysis of a sixth order method is obtained using only information from the operators on the method. Numerical examples are included to complement the theory.

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

“…Choose the initial points (5,5) and (1, 0). Then, using the aforementioned divided difference and the method (2), we obtain the solution ξ = (x * 1 , x * 2 ) after three iterations with…”

“…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.…”

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.…”