Collins und Akritas (1976) have described the Descartes method for isolating the real roots of an integer polynomial in one variable This method recursively subdivides an initial interval until Descartes' Rule of Signs indicates that all roots have been isolated The partial converse of Descartes' Rule by Obreshkoff (1952) in conjunction with the bound of Mahler (1964) and Davenport (1985) leads us to an asymptotically almost tight bound for the resulting subdivision tree It implies directly the best known complexity bounds for the equivalent forms of the Descartes method in the power basis (Collins/Akritas, 1976), the Bernstein basis (Lane/Riesenfeld, 1981) and the scaled Bernstein basis (Johnson, 1991), which are presented here in a unified fashion Without losing correctness of the output, we modify the Descartes method such that it can handle bitstream coefficients, which can be approximated arbitrarily well but cannot be determined exactly We analyze the computing time and precision requirements The method described elsewhere by the author together with Kerber/Wolpert (2007) and Kerber (2008) to determine the arrangement of plane algebraic curves rests in an essential way on variants of the bitstream Descartes algorithm; we analyze a central part of it 
Collins und Akritas (1976) haben das Descartes-Verfahren zur Einschliesung der reellen Nullstellen eines ganzzahligen Polynoms in einer Veranderlichen angegeben Das Verfahren unterteilt rekursiv ein Ausgangsintervall, bis die Descartes'sche Vorzeichenregel anzeigt, dass alle Nullstellen getrennt worden sind Die partielle Umkehrung der Descartes'schen Regel nach Obreschkoff (1952) in Verbindung mit der Schranke von Mahler (1964) und Davenport (1985) fuhrt uns auf eine asymptotisch fast scharfe Schranke fur den sich ergebenden Unterteilungsbaum Daraus folgen direkt die besten bekannten Komplexitatsschranken fur die aquivalenten Formen des Descartes-Verfahrens in der Monom-Basis (Collins/Akritas, 1976), der Bernstein-Basis (Lane/Riesenfeld, 1981) und der skalierten Bernstein-Basis (Johnson, 1991), die hier vereinheitlicht dargestellt werden Ohne dass die Korrektheit der Ausgabe verloren geht, modifizieren wir das Descartes-Verfahren so, dass es mit "Bitstream"-Koeffizienten umgehen kann, die beliebig genau angenahert, aber nicht exakt bestimmt werden konnen Wir analysieren die erforderliche Rechenzeit und Prazision Das vom Verfasser mit Kerber/Wolpert (2007) und Kerber (2008) an anderer Stelle beschriebene Verfahren zur Bestimmung des Arrangements (der Schnittfigur) ebener algebraischer Kurven fust wesentlich auf Varianten des Bitstream-Descartes-Verfahrens; wir analysieren einen zentralen Teil davon

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Real root isolation for exact and approximate polynomials using Descartes' rule of signs

https://hal.inria.fr/inria-00116990/document

On the complexity of real root isolation using continued fractions

Complexity of real root isolation using continued fractions

We present algorithmic, complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real numbers. We improve the previously known bound by a factor of d T, where d is the polynomial degree and T bounds the coefficient bitsize, thus matching the current record complexity for real root isolation by exact methods. Namely, the complexity bound is O˜B(d4T2) using a standard bound on the expected bitsize of the integers in the continued fraction expansion. We show how to compute the multiplicities within the same complexity and extend the algorithm to non square-free polynomials. Finally, we present an efficient open-source C++ implementation in the algebraic library synaps, and illustrate its efficiency as compared to other available software. We use polynomials with coefficient bitsize up to 8000 and degree up to 1000.

/pdf/univariate-polynomial-real-root-isolation-continued-5a6d0mnghs.pdf

Univariate polynomial real root isolation: continued fractions revisited

Finding an upper bound for the positive roots of univariate polynomials is an important step of the continued fractions real root isolation algorithm. The revived interest in this algorithm has highlighted the need for better estimations of upper bounds of positive roots. In this paper we present a new theorem, based on a generalization of a theorem by D. Stefanescu, and describe several implementations of it – including Cauchy's and Kioustelidis' rules as well as two new rules recently developed by us. From the empirical results presented here we see that applying various implementations of our theorem – and taking the minimum of the computed values – greatly improves the estimation of the upper bound and hopefully that will affect the performance of the continued fractions real root isolation method.

J B Kiostelikis

Papers

Bounds for positive roots of polynomials