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Continued fraction expansion of real roots of polynomial systems

Authors:

Angelos Mantzaflaris,

Bernard Mourrain,

Elias TsigaridasAuthors Info & Claims

SNC '09: Proceedings of the 2009 conference on Symbolic numeric computation

Pages 85 - 94

https://doi.org/10.1145/1577190.1577207

Published: 03 August 2009 Publication History

Abstract

We present a new algorithm for isolating the real roots of a system of multivariate polynomials, given in the monomial basis. It is inspired by existing subdivision methods in the Bernstein basis; it can be seen as generalization of the univariate continued fraction algorithm or alternatively as a fully analog of Bernstein subdivision in the monomial basis. The representation of the subdivided domains is done through homographies, which allows us to use only integer arithmetic and to treat efficiently unbounded regions. We use univariate bounding functions, projection and preconditionning techniques to reduce the domain of search. The resulting boxes have optimized rational coordinates, corresponding to the first terms of the continued fraction expansion of the real roots. An extension of Vincent's theorem to multivariate polynomials is proved and used for the termination of the algorithm. New complexity bounds are provided for a simplified version of the algorithm. Examples computed with a preliminary C++ implementation illustrate the approach.

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Cited By

Emiris IMourrain BTsigaridas E(2019)Separation bounds for polynomial systemsJournal of Symbolic Computation10.1016/j.jsc.2019.07.001Online publication date: Jul-2019
https://doi.org/10.1016/j.jsc.2019.07.001
Aizenshtein MBartoň MElber G(2012)Global solutions of well-constrained transcendental systems using expression trees and a single solution testComputer Aided Geometric Design10.1016/j.cagd.2011.07.00229:5(265-279)Online publication date: Jun-2012
https://doi.org/10.1016/j.cagd.2011.07.002
Mantzaflaris AMourrain BSchost ÉEmiris I(2011)Deflation and certified isolation of singular zeros of polynomial systemsProceedings of the 36th international symposium on Symbolic and algebraic computation10.1145/1993886.1993925(249-256)Online publication date: 8-Jun-2011
https://dl.acm.org/doi/10.1145/1993886.1993925
Show More Cited By

Index Terms

Continued fraction expansion of real roots of polynomial systems
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Algebraic algorithms

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Information

Published In

cover image ACM Conferences

SNC '09: Proceedings of the 2009 conference on Symbolic numeric computation

August 2009

210 pages

ISBN:9781605586649

DOI:10.1145/1577190

Editors:
Hiroshi Kai
Ehime University, Japan
,
Hiroshi Sekigawa
NTT Communication Science Laboratories, Japan
,
General Chairs:
Tateaki Sasaki
University of Tsukuba, Japan
,
Kiyoshi Shirayanagi
Tokai University, Japan
,
Program Chair:
Ilias Kotsireas
Wilfrid Laurier University, Canada

Copyright © 2009 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 03 August 2009

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SNC '09: Symbolic Numeric Computation

August 3 - 5, 2009

Kyoto, Japan

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Cited By

Emiris IMourrain BTsigaridas E(2019)Separation bounds for polynomial systemsJournal of Symbolic Computation10.1016/j.jsc.2019.07.001Online publication date: Jul-2019
https://doi.org/10.1016/j.jsc.2019.07.001
Aizenshtein MBartoň MElber G(2012)Global solutions of well-constrained transcendental systems using expression trees and a single solution testComputer Aided Geometric Design10.1016/j.cagd.2011.07.00229:5(265-279)Online publication date: Jun-2012
https://doi.org/10.1016/j.cagd.2011.07.002
Mantzaflaris AMourrain BSchost ÉEmiris I(2011)Deflation and certified isolation of singular zeros of polynomial systemsProceedings of the 36th international symposium on Symbolic and algebraic computation10.1145/1993886.1993925(249-256)Online publication date: 8-Jun-2011
https://dl.acm.org/doi/10.1145/1993886.1993925
Mantzaflaris AMourrain BTsigaridas E(2011)On continued fraction expansion of real roots of polynomial systems, complexity and condition numbersTheoretical Computer Science10.1016/j.tcs.2011.01.009412:22(2312-2330)Online publication date: 1-May-2011
https://dl.acm.org/doi/10.1016/j.tcs.2011.01.009
Hodorog MSchicho J(2011)A Symbolic-Numeric Algorithm for Genus ComputationNumerical and Symbolic Scientific Computing10.1007/978-3-7091-0794-2_4(65-94)Online publication date: 12-Oct-2011
https://doi.org/10.1007/978-3-7091-0794-2_4
Emiris IMourrain BTsigaridas EKoepf W(2010)The DMM boundProceedings of the 2010 International Symposium on Symbolic and Algebraic Computation10.1145/1837934.1837981(243-250)Online publication date: 25-Jul-2010
https://dl.acm.org/doi/10.1145/1837934.1837981

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