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

Mathematics ◽  
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.

scholarly journals Convergence Analysis and Complex Geometry of an Efficient Derivative-Free Iterative Method

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 919
Author(s):  
Deepak Kumar ◽  
Janak Raj Sharma ◽  
Lorentz Jäntschi
Keyword(s):  
Convergence Analysis ◽  
Complex Geometry ◽  
Basins Of Attraction ◽  
Convergence Domain ◽  
Derivative Free ◽  
Visual Approach ◽  
Higher Derivatives ◽  
Fifth Order ◽  
Local Convergence Analysis ◽  
Theoretical Results

To locate a locally-unique solution of a nonlinear equation, the local convergence analysis of a derivative-free fifth order method is studied in Banach space. This approach provides radius of convergence and error bounds under the hypotheses based on the first Fréchet-derivative only. Such estimates are not introduced in the earlier procedures employing Taylor’s expansion of higher derivatives that may not exist or may be expensive to compute. The convergence domain of the method is also shown by a visual approach, namely basins of attraction. Theoretical results are endorsed via numerical experiments that show the cases where earlier results cannot be applicable.

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

Mathematics ◽  
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.

scholarly journals Convergence of an Iteration of Fifth-Order Using Weaker Conditions on First Order Fréchet Derivative in Banach Spaces

2018 ◽  
Vol 15 (06) ◽  
pp. 1850048
Author(s):  
Sukhjit Singh ◽  
Dharmendra Kumar Gupta ◽  
Randhir Singh ◽  
Mehakpreet Singh ◽  
Eulalia Martinez
Keyword(s):  
Convergence Analysis ◽  
Banach Spaces ◽  
Semilocal Convergence ◽  
Fréchet Derivative ◽  
Frechet Derivative ◽  
First Order ◽  
Weaker Conditions ◽  
Fifth Order ◽  
Derivatives Of ◽  
Text Condition

The convergence analysis both local under weaker Argyros-type conditions and semilocal under [Formula: see text]-condition is established using first order Fréchet derivative for an iteration of fifth order in Banach spaces. This avoids derivatives of higher orders which are either difficult to compute or do not exist at times. The Lipchitz and the Hölder conditions are particular cases of the [Formula: see text]-condition. Examples can be constructed for which the Lipchitz and Hölder conditions fail but the [Formula: see text]-condition holds. Recurrence relations are used for the semilocal convergence analysis. Existence and uniqueness theorems and the error bounds for the solution are provided. Different examples are solved and convergence balls for each of them are obtained. These examples include Hammerstein-type integrals to demonstrate the applicability of our approach.

scholarly journals Convergence Analysis of Weighted-Newton Methods of Optimal Eighth Order in Banach Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 198 ◽  
Author(s):  
Janak Sharma ◽  
Ioannis Argyros ◽  
Sunil Kumar
Keyword(s):  
Banach Spaces ◽  
Order Of Convergence ◽  
Higher Order ◽  
Newton Methods ◽  
Eighth Order ◽  
Good Convergence ◽  
Convergence Error ◽  
Uniqueness Of The Solution ◽  
Derivatives Of ◽  
Theoretical Results

We generalize a family of optimal eighth order weighted-Newton methods to Banach spaces and study their local convergence. In a previous study, the Taylor expansion of higher order derivatives is employed which may not exist or may be very expensive to compute. However, the hypotheses of the present study are based on the first Fréchet-derivative only, thereby the application of methods is expanded. New analysis also provides the radius of convergence, error bounds and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches that use Taylor expansions of derivatives of higher order. Moreover, the order of convergence for the methods is verified by using computational order of convergence or approximate computational order of convergence without using higher order derivatives. Numerical examples are provided to verify the theoretical results and to show the good convergence behavior.

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

2021 ◽  
Vol 5 (2) ◽  
pp. 27
Author(s):  
Debasis Sharma ◽  
Ioannis K. Argyros ◽  
Sanjaya Kumar Parhi ◽  
Shanta Kumari Sunanda
Keyword(s):  
Convergence Analysis ◽  
Banach Spaces ◽  
Fourth Order ◽  
Local Convergence ◽  
Taylor Expansion ◽  
Iterative Algorithms ◽  
Continuity Condition ◽  
Dynamical Analysis ◽  
Local Analysis ◽  
Uniqueness Of The Solution

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.

Semi-local convergence of a Newton-like method for solving equations with a singular derivative

2018 ◽  
Vol 27 (1) ◽  
pp. 01-08
Author(s):  
IOANNIS K. ARGYROS ◽  
◽  
GEORGE SANTHOSH ◽  
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Real Line ◽  
Local Convergence ◽  
Approximate Solutions ◽  
Convergence Domain ◽  
Solving Equations ◽  
Solutions Of Equations ◽  
Local Convergence Analysis ◽  
Special Case

We present a semi-local convergence analysis for a Newton-like method to approximate solutions of equations when the derivative is not necessarily non-singular in a Banach space setting. In the special case when the equation is defined on the real line the convergence domain is improved for this method when compared to earlier results. Numerical results where earlier results cannot apply but the new results can apply to solve nonlinear equations are also presented in this study.

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