A general approach to isolating roots of a bitstream polynomial. (English) Zbl 1229.65077

Summary: 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 [A. Eigenwillig, Real root isolation for exact and approximate polynomials using Descartes’ rule of signs. PhD thesis. Universität des Saarlandes (2008); A. Eigenwillig et al., Lect. Notes Comput. Sci. 3718, 138–149 (2005; Zbl 1169.65315); K. Mehlhorn and the author, J. Symb. Comput. 46, No. 1, 70–90 (2011; Zbl 1207.65048)] 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 \(\sigma (F)\) by the average-case quantity \(\prod_{i=1} ^n\root n\of{\sigma_i}\), where \(\sigma_i\) denotes the minimal distance of the \(i\)-th root \(\xi_i\) of \(F\) to any other root of \(F\), \(\sigma (F) := \min_i\sigma_i\), 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.


65H04 Numerical computation of roots of polynomial equations
65Y20 Complexity and performance of numerical algorithms
26C10 Real polynomials: location of zeros


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