Keywords: damped Newton's method, dynamics, universal Julia set.

This paper begins by introducing the Newton's method, the most well-known method to solve systems of non-linear equations [1,2]. From the continuous Newton's method [3] the damped Newton's method [4] is presented. The dynamics of this method is studied be means of applying the method to polynomials with two different roots. The study is divided in two different parts, first of all polynomials with the same multiplicity for both roots. Afterwards we consider polynomials where the multiplicity of one of the roots is greater than the multiplicity of the other root. In both parts two different Möbius transformations are used that allow new iteration maps to be considered that only depend on the damping factor and on the multiplicity, independently of the roots of the polynomial. Some figures in the paper illustrate the results.

On the one hand when the multiplicities are the same, the damping factor is obtained that introduces free critical points. Moreover the character of the fixed points changes with this damping factor although it does not introduce new fixed points. In this part of the work, the main theoretical result is presented that related to the Julia set, that generalizes the result given by Yang [5]. This result guarantees that when the damping factor is between 0 and 2n then the Julia set associated with Newton's method applied to a polynomial with two different roots is a straight line if and only if the multiplicities are the same. Moreover this straight line is the bisector of the segment joining both roots of the polynomial. Some pictures shows examples of this result.

On the other hand we study the case of polynomials with the multiplicity of one root greater than the multiplicity of the other root. The conclusions drawn from the study are that the damping factor modifies the character of the fixed point and introduces free critical points. Some pictures show the changes of the Julia set in this case. Some conclusions of this part are given, first of all there is no symmetry of the Julia set when the roots have different multiplicities. Furthermore, the Julia set form is modified by the damping factor (from a two rays-like a Mandelbrot-like set) and finally that the convergence of the root with lower multiplicity is even lost for some damping factors used.

1

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

2

K. Kneisl, "Julia sets for the super-Newton method, Cauchy's method and Halley's method", Chaos, 11(2), 359-370, 2001. doi:10.1063/1.1368137

3

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

4

S. Plaza, N. Romero, "Attracting cycles for the relaxed Newton's method", Journal of computational and applied mathematics, 235(10), 3238-3244, 2011. doi:10.1016/j.cam.2011.01.010

5

W. Yang, "Symmetries in the Julia sets of Newton's method for multiple root", Appl. Math. Comput., 217, 2490-2494, 2010. doi:10.1016/j.amc.2010.07.061

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