On the speed of convergence of Newton's method for complex polynomials
Abstract
We investigate Newton's method for complex polynomials of arbitrary degree
d
, normalized so that all their roots are in the unit disk. For each degree
d
, we give an explicit set
S
d
of
3.33d
log
2
d(1+o(1))
points with the following universal property: for every normalized polynomial of degree
d
there are
d
starting points in
S
d
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
1−2/d
the number of iterations for these
d
starting points to reach all roots with precision
ε
is
O(
d
2
log
4
d+dlog|logε|)
. This is an improvement of an earlier result in \cite{Schleicher}, where the number of iterations is shown to be
O(
d
4
log
2
d+
d
3
log
2
d|logε|)
in the worst case (allowing multiple roots) and
O(
d
3
log
2
d(logd+logδ)+dlog|logε|)
for well-separated (so-called
δ
-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
O(
d
2
)
for fixed
ε
.