A General Approach to Isolating Roots of a Bitstream Polynomial

Abstract

We describe a new approach to isolate the roots (either real or complex) of a square-free polynomial F with real coefficients. It is assumed that each coefficient of F can be approximated to any specified error bound and refer to such coefficients as bitstream coefficients. The presented method is exact, complete and deterministic. Compared to previous approaches (Eigenwillig in Real root isolation for exact and approximate polynomials using Descartes’ rule of signs, PhD thesis, Universitat des Saarlandes, 2008; Eigenwillig et al. in CASC, LNCS, 2005; Mehlhorn and Sagraloff in J. Symb. Comput. 46(1):70–90, 2011) we improve in two aspects. Firstly, our approach can be combined with any existing subdivision method for isolating the roots of a polynomial with rational coefficients. Secondly, the approximation demand on the coefficients and the bit complexity of our approach is considerably smaller. In particular, we can replace the worst-case quantity σ(F) by the average-case quantity \({\prod_{i=1}^n\sqrt[n] {\sigma_i}}\) , where σi denotes the minimal distance of the i -th root ξi of F to any other root of F, σ(F) := miniσi, and n = deg F. For polynomials with integer coefficients, our method matches the best bounds known for existing practical algorithms that perform exact operations on the input coefficients.