On the local convergence study for an efficient k-step iterative method

Abstract

This paper is devoted to a family of Newton-like methods with frozen derivatives used to approximate a locally unique solution of an equation. The methods have high order of convergence but only using first order derivatives. Moreover only one LU decomposition is required in each iteration. In particular, the methods are real alternatives to the classical Newton method. We present a local convergence analysis based on hypotheses only on the first derivative. This type of local results were usually proved based on hypotheses on the derivative of order higher than two although only the first derivative appears in these type of methods [2,22,24]. We apply these methods to an equation related to the nonlinear complementarity problem. Finally, we find the most efficient method in the family for this problem and we perform a theoretical and a numerical study for it.

Extensions on a local convergence result by Dennis and Schnabel for Newton's method with applications

“…6,7 This technique can be used to improve the applicability of other methods, since it is so general. 1,[3][4][5]12,13…”

“…In computational sciences and other disciplines, applications appear like Equation (1) (in general) using mathematical modeling. [1][2][3] It is rarely attainable to find solutions for these equations in closed form. Consequently, most methods to obtain the solution are iterative.…”

Extending the Applicability and Convergence Domain of a Fifth-Order Iterative Scheme under Hölder Continuous Derivative in Banach Spaces

“…In Table 3, the radius R is provided. Example 4 [28] Consider the nonlinear integral equation given by . The value of R is presented in Table 4.…”

“…Also, important findings on the local analysis of fourth, fifth, and sixth order schemes are presented in [22][23][24][25][26][27]. More results on the local analysis of iterative procedures have been established in the literature [28][29][30][31][32][33][34][35]. In this paper, we expand the convergence domain as well as the applicability of a fifth-order scheme using Hölder condition on Q' only.…”

Ball comparison between frozen Potra and Schmidt–Schwetlick schemes with dynamical analysis

“…What makes this research study novel is that no additional information is used because the constants are specializations that we can see in References 1‐25. If , then the aforementioned improvements hold for the corresponding to () and () algorithms.…”