  We describe a subroutine that improves the running time of any subdivision
algorithm for real root isolation. The subroutine first detects clusters of
roots using a result of Ostrowski, and then uses Newton iteration to converge
to them. Near a cluster, we switch to subdivision, and proceed recursively. The
subroutine has the advantage that it is independent of the predicates used to
terminate the subdivision. This gives us an alternative and simpler approach to
recent developments of Sagraloff (2012) and Sagraloff-Mehlhorn (2013), assuming
exact arithmetic.
  The subdivision tree size of our algorithm using predicates based on
Descartes's rule of signs is bounded by $O(n\log n)$, which is better by
$O(n\log L)$ compared to known results. Our analysis differs in two key
aspects. First, we use the general technique of continuous amortization from
Burr-Krahmer-Yap (2009), and second, we use the geometry of clusters of roots
instead of the Davenport-Mahler bound. The analysis naturally extends to other
predicates.
