Article

Univariate polynomial real root isolation: continued fractions revisited

Authors:

Elias P. Tsigaridas,

Ioannis Z. EmirisAuthors Info & Claims

ESA'06: Proceedings of the 14th conference on Annual European Symposium - Volume 14

Pages 817 - 828

https://doi.org/10.1007/11841036_72

Published: 11 September 2006 Publication History

Abstract

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( d ⁴ T ²) 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.

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Cited By

Emiris IPan VTsigaridas E(2010)Algebraic and numerical algorithmsAlgorithms and theory of computation handbook10.5555/1882757.1882774(17-17)Online publication date: 1-Feb-2010
https://dl.acm.org/doi/10.5555/1882757.1882774
Xu MChen LZeng ZLi Z(2010)Real root isolation of multi-exponential polynomials with applicationProceedings of the 4th international conference on Algorithms and Computation10.1007/978-3-642-11440-3_24(263-268)Online publication date: 10-Feb-2010
https://dl.acm.org/doi/10.1007/978-3-642-11440-3_24
Ştefănescu D(2008)Computation of dominant real roots of polynomialsProgramming and Computing Software10.1134/S036176880802003534:2(69-74)Online publication date: 1-Mar-2008
https://dl.acm.org/doi/10.1134/S0361768808020035
Show More Cited By

Index Terms

Univariate polynomial real root isolation: continued fractions revisited
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations in finite fields
      2. Computations on polynomials

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Reviews

Reviewer: James Harold Davenport

As stated in this paper: “Real root isolation of univariate integer polynomials is a well known problem and various algorithms exist for it.” What more can this paper add__?__ The answer is a reconsideration of the continued fraction (CF) algorithm, originally attributed to Vincent in 1836. It has fallen out of favor, despite the fact that continued fractions have been known, since Euler, to be the best approximations. This reconsideration is based on better theoretical bounds for the behavior of roots, my own work, and better bounds for the intermediate quantities. The authors compare their implementation (CF) with the standard real solver (RS) [1] and the numerical solver Aberth [2]. CF is comparable with RS, except in special test cases where CF is much faster. The Aberth method, a nonguaranteed method, is generally the fastest when properly working. Online Computing Reviews Service

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Published In

cover image Guide Proceedings

ESA'06: Proceedings of the 14th conference on Annual European Symposium - Volume 14

September 2006

840 pages

ISBN:3540388753

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 11 September 2006

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Cited By

Emiris IPan VTsigaridas E(2010)Algebraic and numerical algorithmsAlgorithms and theory of computation handbook10.5555/1882757.1882774(17-17)Online publication date: 1-Feb-2010
https://dl.acm.org/doi/10.5555/1882757.1882774
Xu MChen LZeng ZLi Z(2010)Real root isolation of multi-exponential polynomials with applicationProceedings of the 4th international conference on Algorithms and Computation10.1007/978-3-642-11440-3_24(263-268)Online publication date: 10-Feb-2010
https://dl.acm.org/doi/10.1007/978-3-642-11440-3_24
Ştefănescu D(2008)Computation of dominant real roots of polynomialsProgramming and Computing Software10.1134/S036176880802003534:2(69-74)Online publication date: 1-Mar-2008
https://dl.acm.org/doi/10.1134/S0361768808020035
Sharma V(2008)Complexity of real root isolation using continued fractionsTheoretical Computer Science10.1016/j.tcs.2008.09.017409:2(292-310)Online publication date: 10-Dec-2008
https://dl.acm.org/doi/10.1016/j.tcs.2008.09.017
Tsigaridas EEmiris I(2008)On the complexity of real root isolation using continued fractionsTheoretical Computer Science10.1016/j.tcs.2007.10.010392:1-3(158-173)Online publication date: 20-Feb-2008
https://dl.acm.org/doi/10.1016/j.tcs.2007.10.010
Ştefănescu D(2007)Bounds for real roots and applications to orthogonal polynomialsProceedings of the 10th international conference on Computer Algebra in Scientific Computing10.5555/2396194.2396224(377-391)Online publication date: 16-Sep-2007
https://dl.acm.org/doi/10.5555/2396194.2396224
Akritas AStrzeboński AVigklas P(2007)Advances on the continued fractions method using better estimations of positive root boundsProceedings of the 10th international conference on Computer Algebra in Scientific Computing10.5555/2396194.2396197(24-30)Online publication date: 16-Sep-2007
https://dl.acm.org/doi/10.5555/2396194.2396197
Sharma VWang D(2007)Complexity of real root isolation using continued fractionsProceedings of the 2007 international symposium on Symbolic and algebraic computation10.1145/1277548.1277594(339-346)Online publication date: 29-Jul-2007
https://dl.acm.org/doi/10.1145/1277548.1277594

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