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Univariate Real Root Isolation over a Single Logarithmic Extension of Real Algebraic Numbers

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Applications of Computer Algebra (ACA 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 198))

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Abstract

We present algorithmic, complexity, and implementation results for the problem of isolating the real roots of a univariate polynomial \(B \in L[x]\), where \(L=\mathbb {Q} [ \lg (\alpha )]\) and \(\alpha \) is a positive real algebraic number. The algorithm approximates the coefficients of B up to a sufficient accuracy and then solves the approximate polynomial. For this we derive worst-case (aggregate) separation bounds. We also estimate the expected number of real roots when we draw the coefficients from a specific distribution and illustrate our results experimentally. A generalization to bivariate polynomial systems is also presented. We implemented the algorithm in \({\mathtt {C}}\) as part of the core library of mathematica for the case \(B \in \mathbb {Z} [ \lg (q)][x]\) where q is positive rational number and we demonstrate its efficiency over various data sets.

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Notes

  1. 1.

    http://members.wolfram.com/adams/LogRootIsolExamples.txt.

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Acknowledgements

Both authors would like to thank an anonymous referee for her, or his, very detailed comments that greatly improved the presentation of the results. Elias Tsigaridas is partially supported by the French National Research Agency (ANR-09-BLAN-0371-01), GeoLMI (ANR 2011 BS03 011 06), HPAC (ANR ANR-11-BS02-013) and an FP7 Marie Curie Career Integration Grant.

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Authors and Affiliations

  1. Wolfram Research Inc., 100 Trade Centre Drive, Champaign, IL, 61820, USA

    Adam Strzeboński

  2. POLSYS Project, INRIA Paris-Rocquencourt UPMC, Univ Paris 06, LIP6, Paris, France

    Elias P. Tsigaridas

Authors
  1. Adam Strzeboński
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  2. Elias P. Tsigaridas
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Correspondence to Elias P. Tsigaridas .

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Editors and Affiliations

  1. Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Ilias S. Kotsireas

  2. Institute of Mathematics, University of Valladolid, Valladolid, Spain

    Edgar Martínez-Moro

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Strzeboński, A., Tsigaridas, E.P. (2017). Univariate Real Root Isolation over a Single Logarithmic Extension of Real Algebraic Numbers. In: Kotsireas, I., Martínez-Moro, E. (eds) Applications of Computer Algebra. ACA 2015. Springer Proceedings in Mathematics & Statistics, vol 198. Springer, Cham. https://doi.org/10.1007/978-3-319-56932-1_27

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