We introduce a new approach to isolate the real roots of a square-free
polynomial F=βi=0nβAiβxi with real coefficients. It is assumed that
each coefficient of F can be approximated to any specified error bound. The
presented method is exact, complete and deterministic. Due to its similarities
to the Descartes method, we also consider it practical and easy to implement.
Compared to previous approaches, our new method achieves a significantly better
bit complexity. It is further shown that the hardness of isolating the real
roots of F is exclusively determined by the geometry of the roots and not by
the complexity or the size of the coefficients. For the special case where F
has integer coefficients of maximal bitsize Ο, our bound on the bit
complexity writes as O~(n3Ο2) which improves the best bounds
known for existing practical algorithms by a factor of n=degF. The crucial
idea underlying the new approach is to run an approximate version of the
Descartes method, where, in each subdivision step, we only consider
approximations of the intermediate results to a certain precision. We give an
upper bound on the maximal precision that is needed for isolating the roots of
F. For integer polynomials, this bound is by a factor n lower than that of
the precision needed when using exact arithmetic explaining the improved bound
on the bit complexity