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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 94
PROCEEDINGS OF THE SEVENTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by:
Paper 40
A Dynamic and Analytical Study of the Damped Newton's Method J.M. Gutiérrez, A.A. Magreñán and N. Romero
Department of Mathematics and Computation, University of La Rioja, Spain Full Bibliographic Reference for this paper
, "A Dynamic and Analytical Study of the Damped Newton's Method", in , (Editors), "Proceedings of the Seventh International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 40, 2010. doi:10.4203/ccp.94.40
Keywords: Newton's method, Kantorovich theorem, majorizing sequences, damped Newtons's method.
Summary
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.
References
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