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On the Halley method in Banach spaces

100%
Argyros I., Ren H.
Applicationes Mathematicae
|
2012
|
tom 39
|
nr 2
243-255
EN
We provide a semilocal convergence analysis for Halley's method using convex majorants in order to approximate a locally unique solution of a nonlinear operator equation in a Banach space setting. Our results reduce and improve earlier ones in special cases.
2
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Improved ball convergence of Newton's method under general conditions

100%
Argyros I., Ren H.
Applicationes Mathematicae
|
2012
|
tom 39
|
nr 3
365-375
EN
We present ball convergence results for Newton's method in order to approximate a locally unique solution of a nonlinear operator equation in a Banach space setting. Our hypotheses involve very general majorants on the Fréchet derivatives of the operators involved. In the special case of convex majorants our results, compared with earlier ones, have at least as large radius of convergence, no less tight error bounds on the distances involved, and no less precise information on the uniqueness of the solution.
3
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On a quadratically convergent method using divided differences of order one under the gamma condition

100%
Argyros I., Ren H.
Open Mathematics
|
2008
|
tom 6
|
nr 2
262-271
EN
We re-examine a quadratically convergent method using divided differences of order one in order to approximate a locally unique solution of an equation in a Banach space setting [4, 5, 7]. Recently in [4, 5, 7], using Lipschitz conditions, and a Newton-Kantorovich type approach, we provided a local as well as a semilocal convergence analysis for this method which compares favorably to other methods using two function evaluations such as the Steffensen’s method [1, 3, 13]. Here, we provide an analysis of this method under the gamma condition [6, 7, 19, 20]. In particular, we also show the quadratic convergence of this method. Numerical examples further validating the theoretical results are also provided.
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