scholarly journals On the Semi-Local Convergence of a Traub-Type Method for Solving Equations

Foundations ◽  
2022 ◽  
Vol 2 (1) ◽  
pp. 114-127
Author(s):  
Samundra Regmi ◽  
Christopher I. Argyros ◽  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Local Convergence ◽  
Type Method ◽  
Lipschitz Constants ◽  
Solving Equations ◽  
Original Domain ◽  
Theoretical Results

The celebrated Traub’s method involving Banach space-defined operators is extended. The main feature in this study involves the determination of a subset of the original domain that also contains the Traub iterates. In the smaller domain, the Lipschitz constants are smaller too. Hence, a finer analysis is developed without the usage of additional conditions. This methodology applies to other methods. The examples justify the theoretical results.

scholarly journals Convergence analysis for a fast class of multi-step Chebyshev-Halley-type methods under weak conditions

2021 ◽  
Vol 4 (1) ◽  
pp. 34-43
Author(s):  
Samundra Regmi ◽  
◽  
Ioannis K. Argyros ◽  
Santhosh George ◽  
◽  
...  
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Computational Cost ◽  
Convergence Domain ◽  
Convergence Region ◽  
Type Method ◽  
Precise Information ◽  
Numerical Examples ◽  
Lipschitz Constants ◽  
Theoretical Results

In this study a convergence analysis for a fast multi-step Chebyshe-Halley-type method for solving nonlinear equations involving Banach space valued operator is presented. We introduce a more precise convergence region containing the iterates leading to tighter Lipschitz constants and functions. This way advantages are obtained in both the local as well as the semi-local convergence case under the same computational cost such as: extended convergence domain, tighter error bounds on the distances involved and a more precise information on the location of the solution. The new technique can be used to extend the applicability of other iterative methods. The numerical examples further validate the theoretical results.

On the Ostrowski Method for Solving Equations

2021 ◽  
Vol 2 ◽  
pp. 3
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George ◽  
Christopher I. Argyros
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Local Convergence ◽  
Numerical Experiments ◽  
Convergence Domain ◽  
Lipschitz Constants ◽  
Solving Equations ◽  
Ostrowski’S Method

In this paper, we revisited the Ostrowski's method for solving Banach space valued equations. We developed a technique  to determine a subset of the original convergence domain and using this new Lipschitz constants derived. These constants are at least as tight as the earlier ones leading to a finer convergence analysis in both the semi-local and the local convergence case. These techniques are very general, so they can be used to extend the applicability of other methods without additional hypotheses. Numerical experiments complete this study.

scholarly journals On the Semi-Local Convergence of an Ostrowski-Type Method for Solving Equations

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2281
Author(s):  
Christopher I. Argyros ◽  
Ioannis K. Argyros ◽  
Janak Joshi ◽  
Samundra Regmi ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Local Convergence ◽  
Crucial Role ◽  
Quantum Physics ◽  
General Nature ◽  
Type Method ◽  
Physical Systems ◽  
Solving Equations ◽  
Abstract Spaces

Symmetries play a crucial role in the dynamics of physical systems. As an example, microworld and quantum physics problems are modeled on principles of symmetry. These problems are then formulated as equations defined on suitable abstract spaces. Then, these equations can be solved using iterative methods. In this article, an Ostrowski-type method for solving equations in Banach space is extended. This is achieved by finding a stricter set than before containing the iterates. The convergence analysis becomes finer. Due to the general nature of our technique, it can be utilized to enlarge the utilization of other methods. Examples finish the paper.

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.

scholarly journals Local Convergence for Multi-Step High Order Solvers under Weak Conditions

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 179
Author(s):  
Ramandeep Behl ◽  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Nonlinear Equations ◽  
Local Convergence ◽  
Nonlinear Problems ◽  
Fréchet Derivative ◽  
High Order ◽  
Order Derivative ◽  
First Order ◽  
Lipschitz Constants ◽  
Convergence Study

Our aim in this article is to suggest an extended local convergence study for a class of multi-step solvers for nonlinear equations valued in a Banach space. In comparison to previous studies, where they adopt hypotheses up to 7th Fŕechet-derivative, we restrict the hypotheses to only first-order derivative of considered operators and Lipschitz constants. Hence, we enlarge the suitability region of these solvers along with computable radii of convergence. In the end of this study, we choose a variety of numerical problems which illustrate that our works are applicable but not earlier to solve nonlinear problems.

scholarly journals Unified Local Convergence for Newton’s Method and Uniqueness of the Solution of Equations under Generalized Conditions in a Banach Space

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 463 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Ángel Alberto Magreñán ◽  
Lara Orcos ◽  
Íñigo Sarría
Keyword(s):  
Banach Space ◽  
Newton’S Method ◽  
Newton's Method ◽  
Convergence Region ◽  
Precise Information ◽  
Solution Of Equations ◽  
Lipschitz Constants ◽  
Uniqueness Of The Solution ◽  
Generalized Newton ◽  
Theoretical Results

Under the hypotheses that a function and its Fréchet derivative satisfy some generalized Newton–Mysovskii conditions, precise estimates on the radii of the convergence balls of Newton’s method, and of the uniqueness ball for the solution of the equations, are given for Banach space-valued operators. Some of the existing results are improved with the advantages of larger convergence region, tighter error estimates on the distances involved, and at-least-as-precise information on the location of the solution. These advantages are obtained using the same functions and Lipschitz constants as in earlier studies. Numerical examples are used to test the theoretical results.

scholarly journals On an improved convergence analysis of a two-step Gauss-Newton type method under generalized Lipschitz conditions

2020 ◽  
Vol 36 (3) ◽  
pp. 365-372
Author(s):  
I. K. ARGYROS ◽  
R. P. IAKYMCHUK ◽  
S. M. SHAKHNO ◽  
H. P. YARMOLA
Keyword(s):  
Convergence Analysis ◽  
Computational Effort ◽  
Convergence Criteria ◽  
Type Method ◽  
Precise Information ◽  
Special Cases ◽  
Original Domain ◽  
Local Convergence Analysis ◽  
Lipschitz Conditions ◽  
Theoretical Results

We present a local convergence analysis of a two-step Gauss-Newton method under the generalized and classical Lipschitz conditions for the first- and second-order derivatives. In contrast to earlier works, we use our new idea using a center average Lipschitz conditions through which, we define a subset of the original domain that also contains the iterates. Then, the remaining average Lipschitz conditions are at least as tight as the corresponding ones in earlier works. This way, we obtain: weaker sufficient convergence criteria, larger radius of convergence, tighter error estimates and more precise information on the location of the solution. These advantages are obtained under the same computational effort, since the new Lipschitz functions are special cases of the ones in earlier works. Finally, we give a numerical example that confirms the theoretical results, and compares favorably to the results from previous works.

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