Local Convergence and Dynamical Analysis of a Third and Fourth Order Class of Equation Solvers

In this article, we suggest the local analysis of a uni-parametric third and fourth order class of iterative algorithms for addressing nonlinear equations in Banach spaces. The proposed local convergence is established using an ω-continuity condition on the first Fréchet derivative. In this way, the utility of the discussed schemes is extended and the application of Taylor expansion in convergence analysis is removed. Furthermore, this study provides radii of convergence balls and the uniqueness of the solution along with the calculable error distances. The dynamical analysis of the discussed family is also presented. Finally, we provide numerical explanations that show the suggested analysis performs well in the situation where the earlier approach cannot be implemented.

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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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Extending the Applicability of a Two-Step Chord-Type Method for Non-Differentiable Operators

Mathematics ◽

10.3390/math7090804 ◽

2019 ◽

Vol 7 (9) ◽

pp. 804

Author(s):

Ioannis K. Argyros ◽

Neha Gupta ◽

J. P. Jaiswal

Keyword(s):

Approximate Solution ◽

Convergence Analysis ◽

Banach Spaces ◽

Recurrence Relation ◽

Convergence Theorem ◽

Local Convergence ◽

Existence And Uniqueness ◽

Numerical Illustration ◽

Type Method ◽

Local Convergence Analysis

The semi-local convergence analysis of a well defined and efficient two-step Chord-type method in Banach spaces is presented in this study. The recurrence relation technique is used under some weak assumptions. The pertinency of the assumed method is extended for nonlinear non-differentiable operators. The convergence theorem is also established to show the existence and uniqueness of the approximate solution. A numerical illustration is quoted to certify the theoretical part which shows that earlier studies fail if the function is non-differentiable.

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Local convergence of two Newton-like methods under Holder continuity condition in Banach spaces

International Journal of Mathematics in Operational Research ◽

10.1504/ijmor.2020.10040267 ◽

2020 ◽

Vol 1 (1) ◽

pp. 1

Author(s):

S.K. Parhi ◽

Debasis Sharma

Keyword(s):

Banach Spaces ◽

Local Convergence ◽

Hölder Continuity ◽

Continuity Condition ◽

Holder Continuity

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A robust semi-local convergence analysis of Newton’s method for cone inclusion problems in Banach spaces under affine invariant majorant condition

Journal of Computational and Applied Mathematics ◽

10.1016/j.cam.2014.11.019 ◽

2015 ◽

Vol 279 ◽

pp. 318-335 ◽

Cited By ~ 14

Author(s):

O.P. Ferreira

Keyword(s):

Convergence Analysis ◽

Banach Spaces ◽

Newton’S Method ◽

Local Convergence ◽

Newton's Method ◽

Affine Invariant ◽

Inclusion Problems ◽

Majorant Condition ◽

Local Convergence Analysis

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A CONTINUATION METHOD AND ITS CONVERGENCE FOR SOLVING NONLINEAR EQUATIONS IN BANACH SPACES

International Journal of Computational Methods ◽

10.1142/s0219876213500217 ◽

2013 ◽

Vol 10 (04) ◽

pp. 1350021 ◽

Cited By ~ 5

Author(s):

M. PRASHANTH ◽

D. K. GUPTA

Keyword(s):

Convergence Analysis ◽

Banach Spaces ◽

Nonlinear Equations ◽

Error Bounds ◽

Newton’S Method ◽

Newton's Method ◽

Continuation Method ◽

Continuity Condition ◽

Majorizing Sequences ◽

Chebyshev’S Method

A continuation method is a parameter based iterative method establishing a continuous connection between two given functions/operators and used for solving nonlinear equations in Banach spaces. The semilocal convergence of a continuation method combining Chebyshev's method and Convex acceleration of Newton's method for solving nonlinear equations in Banach spaces is established in [J. A. Ezquerro, J. M. Gutiérrez and M. A. Hernández [1997] J. Appl. Math. Comput.85: 181–199] using majorizing sequences under the assumption that the second Frechet derivative satisfies the Lipschitz continuity condition. The aim of this paper is to use recurrence relations instead of majorizing sequences to establish the convergence analysis of such a method. This leads to a simpler approach with improved results. An existence–uniqueness theorem is given. Also, a closed form of error bounds is derived in terms of a real parameter α ∈ [0, 1]. Four numerical examples are worked out to demonstrate the efficacy of our convergence analysis. On comparing the existence and uniqueness region and error bounds for the solution obtained by our analysis with those obtained by using majorizing sequences, it is found that our analysis gives better results in three examples, whereas in one example it gives the same results. Further, we have observed that for particular values of the α, our analysis reduces to those for Chebyshev's method (α = 0) and Convex acceleration of Newton's method (α = 1) respectively with improved results.

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Convergence Analysis and Dynamical Nature of an Efficient Iterative Method in Banach Spaces

Mathematics ◽

10.3390/math9192510 ◽

2021 ◽

Vol 9 (19) ◽

pp. 2510

Author(s):

Deepak Kumar ◽

Sunil Kumar ◽

Janak Raj Sharma ◽

Lorentz Jantschi

Keyword(s):

Convergence Analysis ◽

Banach Spaces ◽

Complex Geometry ◽

Basins Of Attraction ◽

New Approach ◽

Convergence Error ◽

Uniqueness Of The Solution ◽

Dynamical Nature ◽

Fifth Order ◽

Local Convergence Analysis

We study the local convergence analysis of a fifth order method and its multi-step version in Banach spaces. The hypotheses used are based on the first Fréchet-derivative only. The new approach provides a computable radius of convergence, error bounds on the distances involved, and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches using Taylor expansions of higher order derivatives, which may not exist or may be very expensive or impossible to compute. Numerical examples are provided to validate the theoretical results. Convergence domains of the methods are also checked through complex geometry shown by drawing basins of attraction. The boundaries of the basins show fractal-like shapes through which the basins are symmetric.

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Local Convergence Analysis of an Efficient Fourth Order Weighted-Newton Method under Weak Conditions

Annals of West University of Timisoara - Mathematics and Computer Science ◽

10.2478/awutm-2018-0002 ◽

2018 ◽

Vol 56 (1) ◽

pp. 23-34

Author(s):

Ioannis K. Argyros ◽

Santhosh George

Keyword(s):

Convergence Analysis ◽

Newton Method ◽

Fourth Order ◽

Local Convergence ◽

Systems Of Nonlinear Equations ◽

Order Method ◽

First Derivative ◽

Fourth Order Method ◽

Order Four ◽

Local Convergence Analysis

Abstract Local convergence analysis of a fourth order method considered by Sharma et. al in [19] for solving systems of nonlinear equations. Using conditions on derivatives upto the order five, they proved that the method is of order four. In this study using conditions only on the first derivative, we prove the convergence of the method in [19]. This way we extended the applicability of the method. Numerical example which do not satisfy earlier conditions but satisfy our conditions are presented in this study.

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