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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
|
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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On the convergence of the secant method under the gamma condition

63%
Argyros I.
Open Mathematics
|
2007
|
tom 5
|
nr 2
205-214
EN
We provide sufficient convergence conditions for the Secant method of approximating a locally unique solution of an operator equation in a Banach space. The main hypothesis is the gamma condition first introduced in [10] for the study of Newton’s method. Our sufficient convergence condition reduces to the one obtained in [10] for Newton’s method. A numerical example is also provided.
3
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A refined Newton’s mesh independence principle for a class of optimal shape design problems

63%
Argyros I.
Open Mathematics
|
2006
|
tom 4
|
nr 4
562-572
EN
Shape optimization is described by finding the geometry of a structure which is optimal in the sense of a minimized cost function with respect to certain constraints. A Newton’s mesh independence principle was very efficiently used to solve a certain class of optimal design problems in [6]. Here motivated by optimization considerations we show that under the same computational cost an even finer mesh independence principle can be given.
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