| Julia Sets of Newton Method for Multiple Roots |
| Received:June 22, 2004 |
| Key Words:
standard Newton method relax Newton method multiple roots Newton method Julia set fixed point attraction region.
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| Abstract: |
| In this paper we analyze the theory of Julia sets of Newton method for multiple roots, construct the standard, relax and multiple root's Julia sets. Utilizing the method of experimental mathematics, the authors obtain the following conclusion: (1) The Julia sets of above methods for $f(z)=z^\alpha(z^\beta-1)$ has $\beta$ rotational symmetry and its center is the origin; (2) The multiple root attraction region of these kinds of Julia sets are sensitive to $\alpha$; (3) There is not simple root attraction region in the relax method, because $z^*$, the root of $f(z)$, is neutral or repelling fixed point of the relax Newton transform $F(z)$; (4) $\infty$ is not the fixed point of multiple roots Newton transform $F(z)$, so multiple root's Julia set is multiple and simple roots attraction region; (5) The experimental errors and truncation of coefficients cause the minimal effect to the multiple root method, the more to the relax and the maximal to the standard Newton' method. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2006.02.018 |
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