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On the speed of convergence of Newton's method for complex polynomials
We investigate Newton's method for complex polynomials of arbitrary degree
, normalized so that all their roots are in the unit disk. For each degree
, we give an explicit set of
points with the following universal property: for every normalized polynomial
of degree there are starting points in whose Newton
iterations find all the roots with a low number of iterations: if the roots are
uniformly and independently distributed, we show that with probability at least
the number of iterations for these starting points to reach all
roots with precision is . This is an improvement of an earlier result in
\cite{Schleicher}, where the number of iterations is shown to be in the worst case (allowing multiple roots)
and for
well-separated (so-called -separated) roots.
Our result is almost optimal for this kind of starting points in the sense
that the number of iterations can never be smaller than for fixed
.Comment: 13 pages, 1 figure, to appear in AMS Mathematics of Computatio
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