The Julia set of Newton’s method for multiple root

Abstract

In this paper, we analyze the Julia set of Newton method for multiple roots, construct the Julia sets of standard, relaxed and multiple roots. Through the experimental mathematics method we drew the following conclusion: (1) the Julia sets of above methods for f ( z ) = z α ( z β − 1) has β time rotation symmetry and its center is origin; (2) the multiple roots domain of attraction of these kinds of Julia sets are sensitive to α ; (3) there is not simple root domain of attraction in relax method, because z *, the root of f ( z ), is neutral or repelling fixed point of F ( z ); (4) ∞ is not the fixed point of F ( z ), so multiple root’s Julia set consists of a great number of attraction domain of multiple and simple roots; (5) the experimental errors and truncation of coefficients cause the minimal effect to multiple root, the more to relax, the maximal to standard Newton’ method.