Keywords: Newton's method, Kantorovich theorem, majorizing sequences, damped Newtons's method.

This paper begins by introducing the Newton's method, the most known method to solve systems of non-linear equations [1,2,3]. Kantorovich introduced this method to solve functional equations [4] and gives conditions to guarantee the convergence of this method. After that introduction we explain the main framework of our study, the damped Newton's method. We make a study of this method taking into account the Newton-Kantorovich theory (see [4,5,6] for more details). Taking advantage of this theorem we can construct a majorizing sequence that allow us to improve the condition that explains the parameters for the existence of the inverse and the Lipschitz's condition [7], is to said, Kantorovich established that h has to be less or equal than 0.5, but we give another condition taking into account the damping factor. In the next section we make a dynamical study of the damped Newton's method, this study consists of realizing how the fractal dimension [8] varies with different damping factors and linking it with the continuous Newton's method [9]. Finally we provide numerical experiments using our results.

1

B.T. Polyak, "Newton-Kantorovich method and its global convergence", J. Math. Sci., 133, 2006. doi:10.1007/s10958-006-0066-1

2

T. Yamamoto, "Historical developments in convergence analysis for Newton's and Newton-like methods", J. Comput. Appl. Math, 124, 2000. doi:10.1016/S0377-0427(00)00417-9

3

T.J. Ypma, "Historical development of the Newton-Raphson method", SIAM Review, 37, 531-551, 1995. doi:10.1137/1037125

4

L.V. Kantorovich, G.P. Akilov, "Functional analysis", Pergamon Press, Oxford, 1982.

5

J.M. Ortega, W.C. Rheinboldt, "Iterative solution of nonlinear equations in several variables", Academic Press, New York, 1970.

6

A.M. Ostrowski, "Solution of equations and systems of equations", Academic Press, New York, 1966.

7

B.I. Epureanu, H.S. Greenside, "Fractal basins on the attraction associated with a damped Newton's method", SIAM Rev., 40, 102-109, 1998. doi:10.1137/S0036144596310033

8

M. Bochncek, M. Nezádal, O. Zmeškal, "The box-counting: critical study" 4th Conference on Prediction, Synergetic and more., the Faculty of Management, Institute of Information Technologies, Faculty of Technology, Tomas Bata University in Zlin, 18.

9

J.W. Neuberger, "Continuous Newton's Method for polynomials", Math. Intelligencer, 21, 18-23, 1999. doi:10.1007/BF03025411

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