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On the dynamics of a third order Newton's approximation method

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Gheondea, Aurelian

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We provide an answer to a question raised by S. Amat, S. Busquier, S. Plaza on the qualitative analysis of the dynamics of a certain third order Newton type approximation function M-f, by proving that for functions f twice continuously differentiable and such that both f and its derivative do not have multiple roots, with at least four roots and infinite limits of opposite signs at +/-infinity, M-f has periodic points of any prime period and that the set of points a at which the approximation sequence (M-f(n)(a))(n is an element of N) does not converge is uncountable. In addition, we observe that in their Scaling Theorem analyticity can be replaced with differentiability.

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Wiley-V C H Verlag Gmbh

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Mathematics

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Mathematische Nachrichten

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10.1002/mana.201500470

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