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.
@InProceedings{mehlhorn_et_al:DagSemProc.06021.3, author = {Mehlhorn, Kurt and Eigenwillig, Arno and Kettner, Lutz and Krandick, Werner and Schmitt, Susanne and Wolpert, Nicola}, title = {{A Descartes Algorithms for Polynomials with Bit-Stream Coefficients}}, booktitle = {Reliable Implementation of Real Number Algorithms: Theory and Practice}, pages = {1--12}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2006}, volume = {6021}, editor = {Peter Hertling and Christoph M. Hoffmann and Wolfram Luther and Nathalie Revol}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06021.3}, URN = {urn:nbn:de:0030-drops-7157}, doi = {10.4230/DagSemProc.06021.3}, annote = {Keywords: Root Isolation, Interval Arithmetic, Descartes Algorithm} }
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