Nearly Optimal Refinement of Real Roots of a Univariate Polynomial
Résumé
We assume that a real square-free polynomial $A$ has a degree $d$, a
maximum coefficient bitsize $\tau$ and a real root lying in an
isolating interval and having no nonreal roots nearby (we quantify
this assumption). Then we combine the {\em Double Exponential
Sieve} algorithm (also called the {\em Bisection of the
Exponents}), the bisection, and Newton iteration to decrease the
width of this inclusion interval by a factor of $t=2^{-L}$. The
algorithm has Boolean complexity $O_B~(d^2 \tau + d L )$. This
substantially decreases the known bound $O_B~(d^3 +d^2L)$ and is
optimal up to a polylogarithmic factor. Furthermore we readily
extend our algorithm to support the same upper bound on the
complexity of the refinement of $r$ real roots, for any $r\le d$, by
incorporating the known efficient algorithms for multipoint
polynomial evaluation. The main ingredient for the latter is an
efficient algorithm for (approximate) polynomial division; we
present a variation based on structured matrix computation with
quasi-optimal Boolean complexity.
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