On a quadratically convergent method using divided differences of order one under the gamma condition

• Autorzy: Ioannis Argyros , Hongmin Ren
• Języki publikacji: EN

Abstrakty

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.

Słowa kluczowe

EN

Banach space; local/semilocal convergence; majorizing sequence; Fréchet-derivative; two-point iterative method; Steffensen’s method; gamma condition

Kategorie tematyczne

• 49M15: Newton-type methods
• 65K10: Optimization and variational techniques
• 65G99: None of the above, but in this section
• 65H10: Systems of equations
• 65B05: Extrapolation to the limit, deferred corrections

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2008
• Tom: 6
• Numer: 2
• Strony: 262-271

Daty

wydano

2008-06-01

online

2008-04-15

Twórcy

autor

Ioannis Argyros

• Cameron University

autor

Hongmin Ren

• Hangzhou Radio and TV University

Bibliografia

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• [4] Argyros I.K., On a two-point Newton-like method of convergent order two, Int. J. Comput. Math., 2005, 82, 219–234 http://dx.doi.org/10.1080/00207160412331296661
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• [6] Argyros I.K., On the convergence of the Secant method under the gamma condition, Cent. Eur. J. Math., 2007, 5, 205–214 http://dx.doi.org/10.2478/s11533-007-0007-3
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• [17] Ren H.M., Yang S.J., Wu Q.B., A new semilocal convergence theorem for the Secant method under Hölder continuous divided differences, Appl. Math. Comput., 2006, 182, 41–48 http://dx.doi.org/10.1016/j.amc.2006.03.034
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Identyfikatory

DOI

10.2478/s11533-008-0015-y