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A new Kantorovich-type theorem for Newton's method

100%
Argyros I. K.
Applicationes Mathematicae
|
1999
|
tom 26
|
nr 2
151-157
EN
A new Kantorovich-type convergence theorem for Newton's method is established for approximating a locally unique solution of an equation F(x)=0 defined on a Banach space. It is assumed that the operator F is twice Fréchet differentiable, and that F', F'' satisfy Lipschitz conditions. Our convergence condition differs from earlier ones and therefore it has theoretical and practical value.
2
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The effect of rounding errors on a certain class of iterative methods

100%
Argyros I. K.
Applicationes Mathematicae
|
2000
|
tom 27
|
nr 3
369-375
EN
In this study we are concerned with the problem of approximating a solution of a nonlinear equation in Banach space using Newton-like methods. Due to rounding errors the sequence of iterates generated on a computer differs from the sequence produced in theory. Using Lipschitz-type hypotheses on the mth Fréchet derivative (m ≥ 2 an integer) instead of the first one, we provide sufficient convergence conditions for the inexact Newton-like method that is actually generated on the computer. Moreover, we show that the ratio of convergence improves under our conditions. Furthermore, we provide a wider choice of initial guesses than before. Finally, a numerical example is provided to show that our results compare favorably with earlier ones.
3
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Local convergence of inexact Newton methods under affine invariant conditions and hypotheses on the second Fréchet derivative

100%
Argyros I. K.
Applicationes Mathematicae
|
1999
|
tom 26
|
nr 4
457-465
EN
We use inexact Newton iterates to approximate a solution of a nonlinear equation in a Banach space. Solving a nonlinear equation using Newton iterates at each stage is very expensive in general. That is why we consider inexact Newton methods, where the Newton equations are solved only approximately, and in some unspecified manner. In earlier works [2], [3], natural assumptions under which the forcing sequences are uniformly less than one were given based on the second Fréchet derivative of the operator involved. This approach showed that the upper error bounds on the distances involved are smaller compared with the corresponding ones using hypotheses on the first Fréchet derivative. However, the conditions on the forcing sequences were not given in affine invariant form. The advantages of using conditions given in affine invariant form were explained in [3], [10]. Here we reproduce all the results obtained in [3] but using affine invariant conditions.
4
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Local convergence of a multi-step high order method with divided differences under hypotheses on the first derivative

64%
Argyros I. K., George S.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
|
2017
|
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.
5
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Local convergence comparison between two novel sixth order methods for solving equations

64%
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

64%
George S., Argyros I. K.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
|
2019
|
tom 18
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