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Symmetries of the Julia sets of Newton’s method for multiple root. (English) Zbl 1207.65049
The symmetries of the Julia sets of Newton’s method for multiple roots are investigated. The paper consists of two sections. The first section is an introduction where the related works are disscussed and the main results are formulated. In the second section the author proves some auxiliary lemmas and the main theorems. It is shown that the group of symmetries of the Julia set of a polynomial is a subgroup of that of the corresponding standard, multiple and relaxed Newton methods when the nonlinear polynomial is in normal form and the Julia set has a finite group of symmetries. A necessary and sufficient condition for Julia sets of standard, multiple and relaxed Newton methods to be a horizontal line is obtained.
MSC:
65H04Roots of polynomial equations (numerical methods)
12Y05Computational aspects of field theory and polynomials
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