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1
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Local convergence of a multi-step high order method with divided differences under hypotheses on the first derivative

100%
Argyros I. K., George S.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
|
2017
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tom 16
EN
This paper is devoted to the study of a multi-step method with divided differences for solving nonlinear equations in Banach spaces. In earlier studies, hypotheses on the Fréchet derivative up to the sixth order of the operator under consideration is used to prove the convergence of the method. That restricts the applicability of the method. In this paper we extended the applicability of the sixth-order multi-step method by using only hypotheses on the first derivative of the operator involved. Our convergence conditions are weaker than the conditions used in earlier studies. Numerical examples where earlier results cannot be applied to solve equations but our results can be applied are also given in this study.
2
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Local convergence for a multi-point family of super-Halley methods in a Banach space under weak conditions

100%
Argyros I., George S.
Applicationes Mathematicae
|
2015
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tom 42
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nr 2-3
193-203
EN
We present a local multi-point convergence analysis for a family of super-Halley methods of high convergence order in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first and second Fréchet derivative of the operator involved. Earlier studies use hypotheses up to the third Fréchet derivative. Numerical examples are also provided.
3
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Local convergence of two competing third order methods in Banach space

100%
Argyros I., George S.
Applicationes Mathematicae
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2014
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tom 41
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nr 4
341-350
EN
We present a local convergence analysis for two popular third order methods of approximating a solution of a nonlinear equation in a Banach space setting. The convergence ball and error estimates are given for both methods under the same conditions. A comparison is given between the two methods, as well as numerical examples.
4
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On the semilocal convergence of a two-step Newton-like projection method for ill-posed equations

100%
Argyros I., George S.
Applicationes Mathematicae
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2013
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tom 40
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nr 3
367-382
EN
We present new semilocal convergence conditions for a two-step Newton-like projection method of Lavrentiev regularization for solving ill-posed equations in a Hilbert space setting. The new convergence conditions are weaker than in earlier studies. Examples are presented to show that older convergence conditions are not satisfied but the new conditions are satisfied.
5
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Local convergence comparison between two novel sixth order methods for solving equations

100%
George S., Argyros I. K.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
|
2019
|
tom 18
5-19
EN
The aim of this article is to provide the local convergence analysis of two novel competing sixth convergence order methods for solving equations involving Banach space valued operators. Earlier studies have used hypotheses reaching up to the sixth derivative but only the first derivative appears in these methods. These hypotheses limit the applicability of the methods. That is why we are motivated to present convergence analysis based only on the first derivative. Numerical examples where the convergence criteria are tested are provided. It turns out that in these examples the criteria in the earlier works are not satisfied, so these results cannot be used to solve equations but our results can be used.
6
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
Content available

Local convergence comparison between two novel sixth order methods for solving equations

100%
George S., Argyros I. K.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
|
2019
|
tom 18
7
Content available remote

Improved local convergence analysis of inexact Newton-like methods under the majorant condition

100%
Argyros I., George S.
Applicationes Mathematicae
|
2015
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tom 42
|
nr 4
343-357
EN
We present a local convergence analysis of inexact Newton-like methods for solving nonlinear equations. Using more precise majorant conditions than in earlier studies, we provide: a larger radius of convergence; tighter error estimates on the distances involved; and a clearer relationship between the majorant function and the associated least squares problem. Moreover, these advantages are obtained under the same computational cost.
8
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Newton-type iterative methods for nonlinear ill-posed Hammerstein-type equations

81%
Shobha M., Argyros I., George S.
Applicationes Mathematicae
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2014
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tom 41
|
nr 1
107-129
EN
We use a combination of modified Newton method and Tikhonov regularization to obtain a stable approximate solution for nonlinear ill-posed Hammerstein-type operator equations KF(x) = y. It is assumed that the available data is $y^{δ}$ with $||y - y^{δ}|| ≤ δ$, K: Z → Y is a bounded linear operator and F: X → Z is a nonlinear operator where X,Y,Z are Hilbert spaces. Two cases of F are considered: where $F'(x₀)^{-1}$ exists (F'(x₀) is the Fréchet derivative of F at an initial guess x₀) and where F is a monotone operator. The parameter choice using an a priori and an adaptive choice under a general source condition are of optimal order. The computational results provided confirm the reliability and effectiveness of our method.
9
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Local convergence for a family of iterative methods based on decomposition techniques

81%
Argyros I., George S., Erappa S.
Applicationes Mathematicae
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2016
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tom 43
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nr 1
133-143
EN
We present a local convergence analysis for a family of iterative methods obtained by using decomposition techniques. The convergence of these methods was shown before using hypotheses on up to the seventh derivative although only the first derivative appears in these methods. In the present study we expand the applicability of these methods by showing convergence using only the first derivative. Moreover we present a radius of convergence and computable error bounds based only on Lipschitz constants. Numerical examples are also provided.
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