  We introduce a new algorithm denoted DSC2 to isolate the real roots of a
univariate square-free polynomial f with integer coefficients. The algorithm
iteratively subdivides an initial interval which is known to contain all real
roots of f. The main novelty of our approach is that we combine Descartes' Rule
of Signs and Newton iteration. More precisely, instead of using a fixed
subdivision strategy such as bisection in each iteration, a Newton step based
on the number of sign variations for an actual interval is considered, and,
only if the Newton step fails, we fall back to bisection. Following this
approach, our analysis shows that, for most iterations, we can achieve
quadratic convergence towards the real roots. In terms of complexity, our
method induces a recursion tree of almost optimal size O(nlog(n tau)), where n
denotes the degree of the polynomial and tau the bitsize of its coefficients.
The latter bound constitutes an improvement by a factor of tau upon all
existing subdivision methods for the task of isolating the real roots. In
addition, we provide a bit complexity analysis showing that DSC2 needs only
\tilde{O}(n^3tau) bit operations to isolate all real roots of f. This matches
the best bound known for this fundamental problem. However, in comparison to
the much more involved algorithms by Pan and Sch\"onhage (for the task of
isolating all complex roots) which achieve the same bit complexity, DSC2
focuses on real root isolation, is very easy to access and easy to implement.
