Semilocal convergence of a family of third-order methods in Banach spaces under Hölder continuous second derivative

In this paper, the semilocal convergence for ameliorated super-Halley methods in Banach spaces is considered. Different from the results in [J. M. Gutiérrez and M. A. Hernández, Comput. Math. Appl. 36 (1998) 1–8], these ameliorated methods do not need to compute a second derivative, the computation for inversion is reduced and the $R$-order is also heightened. Under a weaker condition, an existence–uniqueness theorem for the solution is proved.

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Recurrence relations for Super-Halley's method with Hölder continuous second derivative in Banach spaces

Kodai Mathematical Journal ◽

10.2996/kmj/1364562724 ◽

2013 ◽

Vol 36 (1) ◽

pp. 119-136 ◽

Cited By ~ 6

Author(s):

Maroju Prashanth ◽

Dharmendra K. Gupta

Keyword(s):

Banach Spaces ◽

Recurrence Relations ◽

Second Derivative ◽

Hölder Continuous ◽

Halley’S Method ◽

Halley's Method

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SEMILOCAL CONVERGENCE OF A STIRLING-LIKE METHOD IN BANACH SPACES

International Journal of Computational Methods ◽

10.1142/s0219876210002210 ◽

2010 ◽

Vol 07 (02) ◽

pp. 215-228 ◽

Cited By ~ 16

Author(s):

S. K. PARHI ◽

D. K. GUPTA

Keyword(s):

Convergence Analysis ◽

Banach Spaces ◽

Existence And Uniqueness ◽

Recurrence Relations ◽

Lipschitz Continuity ◽

Semilocal Convergence ◽

Continuity Condition ◽

Uniqueness Of Solutions ◽

Third Order ◽

Numerical Examples

The aim of this paper is to establish the semilocal convergence of a third order Stirling–like method employed for solving nonlinear equations in Banach spaces by using the first Fréchet derivative, which satisfies the Lipschitz continuity condition. This makes it possible to avoid the evaluation of higher order Fréchet derivatives which are computationally difficult at times or may not even exist. The recurrence relations are used for convergence analysis. A convergence theorem is given for deriving error bounds and the domains of existence and uniqueness of solutions. The R order of the method is also established to be equal to 3. Finally, two numerical examples are worked out, and the results obtained are compared with the existing results. It is observed that our convergence analysis is more effective.

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Generalizing the local convergence analysis of a class of $k$-step iterative algorithms with Hölder continuous derivative in Banach spaces

Applicationes Mathematicae ◽

10.4064/am2410-7-2020 ◽

2021 ◽

Author(s):

Ioannis K. Argyros ◽

Debasis Sharma ◽

Sanjaya Kumar Parhi

Keyword(s):

Convergence Analysis ◽

Banach Spaces ◽

Local Convergence ◽

Iterative Algorithms ◽

Continuous Derivative ◽

Hölder Continuous ◽

Local Convergence Analysis

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An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior

Open Mathematics ◽

10.1515/math-2019-0122 ◽

2019 ◽

Vol 17 (1) ◽

pp. 1567-1598

Author(s):

Tianbao Liu ◽

Xiwen Qin ◽

Qiuyue Li

Keyword(s):

Nonlinear Equations ◽

Fourth Order ◽

Dynamical Behavior ◽

Basins Of Attraction ◽

Second Derivative ◽

Third Order ◽

Numerical Tests ◽

The Family ◽

Theoretical Results ◽

High Computational Efficiency

Abstract In this paper, we derive and analyze a new one-parameter family of modified Cauchy method free from second derivative for obtaining simple roots of nonlinear equations by using Padé approximant. The convergence analysis of the family is also considered, and the methods have convergence order three. Based on the family of third-order method, in order to increase the order of the convergence, a new optimal fourth-order family of modified Cauchy methods is obtained by using weight function. We also perform some numerical tests and the comparison with existing optimal fourth-order methods to show the high computational efficiency of the proposed scheme, which confirm our theoretical results. The basins of attraction of this optimal fourth-order family and existing fourth-order methods are presented and compared to illustrate some elements of the proposed family have equal or better stable behavior in many aspects. Furthermore, from the fractal graphics, with the increase of the value m of the series in iterative methods, the chaotic behaviors of the methods become more and more complex, which also reflected in some existing fourth-order methods.

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