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  1. 3 Open Mathematics

Autorzy help

  1. 3 Argyros I.

  2. 1 Ren H.

Lata help

  1. 1 2008

  2. 1 2007

  3. 1 2006

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On the convergence of the secant method under the gamma condition

100%
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.
2
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A refined Newton’s mesh independence principle for a class of optimal shape design problems

100%
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.
3
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On a quadratically convergent method using divided differences of order one under the gamma condition

63%
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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