Convergence of Newton’s method under the gamma condition


  • Ioannis K. Argyros Cameron University.



Banach space, Newton’s method, local/semilocal convergence, Newton—Kantorovich theorem, Frechet derivative, majorizing sequence, radius of convergence, gamma condition, analytic operator, espacio de Banach, método de Newton, convergencia local/semilocal.


We provide a semilocal as well as a local convergence analysis of Newton’s method using the gamma condition [1], [10], [11]. Using more precise majorizing sequences than before [4], [8]—[11] and under at least as weak hypotheses, we provide in the semilocal case: finer error bounds on the distances involved and an at least as precise information on the location of the solution; in the local case: a larger radius of convergence.

Author Biography

Ioannis K. Argyros, Cameron University.

Department of Mathematical Sciences.


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[10] Wang, X. H. and Han, D.F., On dominating sequence method in the point estimate and Smale theorem, Sci. Sinica Ser. A, 33, pp 135-144, (1990).

[11] Wang, X. H., Convergence of the iteration of Halley family in weak conditions, Chinese Science Bulletin, 42, pp. 552—555, (1997).



How to Cite

I. K. Argyros, “Convergence of Newton’s method under the gamma condition”, Proyecciones (Antofagasta, On line), vol. 25, no. 3, pp. 293-306, May 2017.




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