Abstract
In this paper we study the approximation of the phase-portrait of the Newton differential equation u′ = –R/R′ o u by means of its Euler discretization NR,h Here NR,h stands for the relaxed Newton-method for finding a zero of a rational function R with disaetization parameter h. For h suficiently small we prove that each basin of a zero of R corresponding to the differential equation is approximated in the sense of Carathedory by the immediate hasin of attraction with respect to the iteration of NR,h. From that we derive conditions. which ensure the convergence of the Julia set J(NR,h) in the sense of the Hausdorff metric, to the union of all boundaries of basins corresponding to the differential equation and the zeros of R, for h tending to zero. In particular the derived conditions apply to all nonconstant polynomials.
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