New and generalized convergence conditions for the Newton-Kantorovich method

• Autorzy: Argyros I. K.
• Języki publikacji: EN

Abstrakty

EN

We present new semilocal convergence theorems for Newton methods in a Banach space. Using earlier general conditions we find more precise error estimates on the distances involved using the majorant principle. Moreover we provide a better information on the location of the solution. In the special case of Newton's method under Lipschitz conditions we show that the famous Newton-Kantorovich hypothesis having gone unchallenged for a long time can be weakened under the same hypotheses/computational cost.

Słowa kluczowe

EN

Newton's method; Banach space; majorant method; Newton-Kantorovich theorem/hypothesis; Frechet-derivative

PL

metoda Newtona; przestrzeń Banacha; metoda majoranta; twierdzenie/hipoteza Newton-Kantorovich; pochodna Frechet'a

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2003
• Tom: Vol. 9, nr 2
• Strony: 287--299
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Argyros I. K.

• Department of Mathematics Cameron University Lawton, OK 73505 USA,

Bibliografia

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• [2] Argyros, I. K., Newton methods on Banach spaces with a convergence structure and applications, Comput. Math. Appl. 40(1) (2000), 37-48.
• [3] Argyros, I. K., Advances in the Efficiency of Computational Methods and Applications, World Scientific Publ. Co., River Edge, NJ, 2000.
• [4] Argyros, I. K., Szidarovszky, F., The Theory and Applications of Iteration Methods, CRC Press, Boca Raton, FL, 1993.
• [5] De Pascale, E., Zabrejko, P. P., New convergence criterion for the Newton-Kantorovich method and some applications to nonlinear integral equations, Rend. Sem. Mat. Univ. Padova 100 (1998), 211-230.
• [6] Gutierrez, J. M., A new semilocal convergence theorem for Newton’s method, J. Cornput. Appl. Math. 79 (1997), 131-145.
• [7] Kantorovich, L. V., Akilov, G. P., Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1964.
• [8] Potra, F. A., On Q-order and R-order of convergence, J. Optim. Theory Appl. 63(3) (1989), 415-431.
• [9] Yamamoto, T., A convergence theorem for Newton-like methods in Banach space, Numer. Math. 51 (1987), 545-557.
• [10] Ypma, T. J., Local convergence of inexact Newton methods, SIAM J. Numer. Anal. 21(3) (1984), 583-590.
• [11] Zabrejko, P. P., Nguen, D. F., The majorant method in the theory of Newton-Kantorovich approximations and the Ptak error estimates, Numer. Funct. Anal. Optim. 9 (1987), 671-684.