A Descartes Algorithm 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 $q(x)=q_{n^{x^{n}}}+...+q_{0}$ with root separation ρ, coefficients |q _n | ≥ 1 and $|q_{i}|\leq 2^{\tau}$, it needs coefficient approximations to O(n(log(1/ρ) + τ)) bits after the binary point and has an expected cost of O(n ⁴ (log(1/ρ) + τ)²) bit operations.