scholarly journals Extended ball convergence for a seventh order derivative free class of algorithms for nonlinear equations

2021 ◽  
Vol 56 (1) ◽  
pp. 72-82
Author(s):  
I.K. Argyros ◽  
D. Sharma ◽  
C.I. Argyros ◽  
S.K. Parhi ◽  
S.K. Sunanda ◽  
...  
Keyword(s):  
Convergence Analysis ◽  
Nonlinear Equations ◽  
Complex Polynomial ◽  
Taylor Formula ◽  
Polynomial Equations ◽  
First Derivative ◽  
Derivative Free ◽  
Uniqueness Of The Solution ◽  
Seventh Order ◽  
Ball Convergence

In the earlier work, expensive Taylor formula and conditions on derivatives up to the eighthorder have been utilized to establish the convergence of a derivative free class of seventh orderiterative algorithms. Moreover, no error distances or results on uniqueness of the solution weregiven. In this study, extended ball convergence analysis is derived for this class by imposingconditions on the first derivative. Additionally, we offer error distances and convergence radiustogether with the region of uniqueness for the solution. Therefore, we enlarge the practicalutility of these algorithms. Also, convergence regions of a specific member of this class are displayedfor solving complex polynomial equations. At the end, standard numerical applicationsare provided to illustrate the efficacy of our theoretical findings.

scholarly journals Ball Convergence of a Parametric Efficient Family of Iterative Methods for Solving Nonlinear Equations

Foundations ◽  
2021 ◽  
Vol 1 (1) ◽  
pp. 23-31
Author(s):  
Samundra Regmi ◽  
Christopher I. Argyros ◽  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Linear Convergence ◽  
Error Tolerance ◽  
First Derivative ◽  
Uniqueness Of The Solution ◽  
Ball Convergence ◽  
Space Case ◽  
Local Results ◽  
Lipschitz Conditions ◽  
Theoretical Results ◽  
General Banach Space

The goal is to extend the applicability of Newton-Traub-like methods in cases not covered in earlier articles requiring the usage of derivatives up to order seven that do not appear in the methods. The price we pay by using conditions on the first derivative that actually appear in the method is that we show only linear convergence. To find the convergence order is not our intention, however, since this is already known in the case where the spaces coincide with the multidimensional Euclidean space. Note that the order is rediscovered by using ACOC or COC, which require only the first derivative. Moreover, in earlier studies using Taylor series, no computable error distances were available based on generalized Lipschitz conditions. Therefore, we do not know, for example, in advance, how many iterates are needed to achieve a predetermined error tolerance. Furthermore, no uniqueness of the solution results is available in the aforementioned studies, but we also provide such results. Our technique can be used to extend the applicability of other methods in an analogous way, since it is so general. Finally note that local results of this type are important, since they demonstrate the difficulty in choosing initial points. Our approach also extends the applicability of this family of methods from the multi-dimensional Euclidean to the more general Banach space case. Numerical examples complement the theoretical results.

scholarly journals A Family of Fifth and Sixth Convergence Order Methods for Nonlinear Models

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 715
Author(s):  
Ioannis K. Argyros ◽  
Debasis Sharma ◽  
Christopher I. Argyros ◽  
Sanjaya Kumar Parhi ◽  
Shanta Kumari Sunanda
Keyword(s):  
Numerical Experiments ◽  
Divided Difference ◽  
Nonlinear Models ◽  
Convergence Order ◽  
Dynamical Analysis ◽  
Not Given ◽  
Convergence Error ◽  
First Derivative ◽  
Derivative Free ◽  
Uniqueness Of The Solution

We study the local convergence of a family of fifth and sixth convergence order derivative free methods for solving Banach space valued nonlinear models. Earlier results used hypotheses up to the seventh derivative to show convergence. However, we only use the first divided difference of order one as well as the first derivative in our analysis. We also provide computable radius of convergence, error estimates, and uniqueness of the solution results not given in earlier studies. Hence, we expand the applicability of these methods. The dynamical analysis of the discussed family is also presented. Numerical experiments complete this article.

scholarly journals Extended convergence of Gauss-Newton’s method and uniqueness of the solution

2018 ◽  
Vol 34 (2) ◽  
pp. 135-142
Author(s):  
IOANNIS K. ARGYROS ◽  
◽  
YEOL JE CHO ◽  
SANTHOSH GEORGE ◽  
◽  
...  
Keyword(s):  
Least Squares ◽  
Convergence Analysis ◽  
Newton’S Method ◽  
Newton's Method ◽  
Nonlinear Least Squares ◽  
Lipschitz Functions ◽  
New Technique ◽  
Least Squares Problems ◽  
Uniqueness Of The Solution ◽  
Ball Convergence

The aim of this paper is to extend the applicability of the Gauss-Newton’s method for solving nonlinear least squares problems using our new idea of restricted convergence domains. The new technique uses tighter Lipschitz functions than in earlier papers leading to a tighter ball convergence analysis.

scholarly journals Some New Variants of Cauchy's Methods for Solving Nonlinear Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Tianbao Liu ◽  
Hengyan Li
Keyword(s):  
Convergence Analysis ◽  
Nonlinear Equations ◽  
Numerical Results ◽  
Convergence Order ◽  
Second Derivative ◽  
Numerical Examples ◽  
First Derivative ◽  
New Methods

We present and analyze some variants of Cauchy's methods free from second derivative for obtaining simple roots of nonlinear equations. The convergence analysis of the methods is discussed. It is established that the methods have convergence order three. Per iteration the new methods require two function and one first derivative evaluations. Numerical examples show that the new methods are comparable with the well-known existing methods and give better numerical results in many aspects.

scholarly journals Local Comparison between Two Ninth Convergence Order Algorithms for Equations

Algorithms ◽  
2020 ◽  
Vol 13 (6) ◽  
pp. 147
Author(s):  
Samundra Regmi ◽  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Error Estimates ◽  
Nonlinear Equations ◽  
Local Convergence ◽  
Numerical Experiments ◽  
Radius Of Convergence ◽  
Convergence Order ◽  
Convergence Criteria ◽  
First Derivative ◽  
Local Comparison ◽  
Uniqueness Of The Solution

A local convergence comparison is presented between two ninth order algorithms for solving nonlinear equations. In earlier studies derivatives not appearing on the algorithms up to the 10th order were utilized to show convergence. Moreover, no error estimates, radius of convergence or results on the uniqueness of the solution that can be computed were given. The novelty of our study is that we address all these concerns by using only the first derivative which actually appears on these algorithms. That is how to extend the applicability of these algorithms. Our technique provides a direct comparison between these algorithms under the same set of convergence criteria. This technique can be used on other algorithms. Numerical experiments are utilized to test the convergence criteria.

scholarly journals An Optimal Derivative Free Family of Chebyshev–Halley’s Method for Multiple Zeros

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 546
Author(s):  
Ramandeep Behl ◽  
Sonia Bhalla ◽  
Ángel Alberto Magreñán ◽  
Alejandro Moysi
Keyword(s):  
Nonlinear Equation ◽  
Weight Function ◽  
Convergence Analysis ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Higher Order ◽  
Multiple Roots ◽  
Iterative Technique ◽  
Derivative Free

In this manuscript, we introduce the higher-order optimal derivative-free family of Chebyshev–Halley’s iterative technique to solve the nonlinear equation having the multiple roots. The designed scheme makes use of the weight function and one parameter α to achieve the fourth-order of convergence. Initially, the convergence analysis is performed for particular values of multiple roots. Afterward, it concludes in general. Moreover, the effectiveness of the presented methods are certified on some applications of nonlinear equations and compared with the earlier derivative and derivative-free schemes. The obtained results depict better performance than the existing methods.

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