A Descartes Algorithms for Polynomials with Bit-Stream Coefficients

Abstract

The Descartes method is an algorithm for isolating the real roots of square-free polynomials with real coefficients. We assume that coefficients are given as (potentially infinite) bit-streams. In other words, coefficients can be approximated to any desired accuracy, but are not known exactly. We show that a variant of the Descartes algorithm can cope with bit-stream coefficients. To isolate the real roots of a square-free real polynomial $q(x) = q_nx^n+ldots+q_0$ with root separation $ ho$, coefficients $abs{q_n}ge1$ and $abs{q_i} le 2^ au$, it needs coefficient approximations to $O(n(log(1/ ho) + au))$ bits after the binary point and has an expected cost of $O(n^4 (log(1/ ho) + au)^2)$ bit operations.