research-article

A simple but exact and efficient algorithm for complex root isolation

Authors:

Michael SagraloffAuthors Info & Claims

ISSAC '11: Proceedings of the 36th international symposium on Symbolic and algebraic computation

Pages 353 - 360

https://doi.org/10.1145/1993886.1993938

Published: 08 June 2011 Publication History

Abstract

We present a new exact subdivision algorithm CEVAL for isolating the complex roots of a square-free polynomial in any given box. It is a generalization of a previous real root isolation algorithm called EVAL. Under suitable conditions, our approach is applicable for general analytic functions. CEVAL is based on the simple Bolzano Principle and is easy to implement exactly. Preliminary experiments have shown its competitiveness.

We further show that, for the "benchmark problem" of isolating all roots of a square-free polynomial with integer coefficients, the asymptotic complexity of both algorithms EVAL and CEVAL matches (up a logarithmic term) that of more sophisticated real root isolation methods which are based on Descartes' Rule of Signs, Continued Fraction or Sturm sequence. In particular, we show that the tree size of EVAL matches that of other algorithms. Our analysis is based on a novel technique called Δ-clusters from which we expect to see further applications.

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Cited By

Kelmendi E(2022)Computing the Density of the Positivity Set for Linear Recurrence SequencesProceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3531130.3532399(1-12)Online publication date: 2-Aug-2022
https://dl.acm.org/doi/10.1145/3531130.3532399
Ergür ATonelli-Cueto JTsigaridas EMaza MZhi L(2022)Beyond Worst-Case Analysis for Root Isolation AlgorithmsProceedings of the 2022 International Symposium on Symbolic and Algebraic Computation10.1145/3476446.3535475(139-148)Online publication date: 4-Jul-2022
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Hormann KKania LYap CChyzak FLabahn G(2021)Novel Range Functions via Taylor Expansions and Recursive Lagrange Interpolation with Application to Real Root IsolationProceedings of the 2021 International Symposium on Symbolic and Algebraic Computation10.1145/3452143.3465532(193-200)Online publication date: 18-Jul-2021
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Index Terms

A simple but exact and efficient algorithm for complex root isolation
1. Mathematics of computing
  1. Mathematical analysis
    1. Nonlinear equations
    2. Numerical analysis
      1. Computations on polynomials

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cover image ACM Conferences

ISSAC '11: Proceedings of the 36th international symposium on Symbolic and algebraic computation

June 2011

372 pages

ISBN:9781450306751

DOI:10.1145/1993886

General Chair:
Éric Schost
The University of Western Ontario, Canada
,
Program Chair:
Ioannis Z. Emiris
University of Athens, Greece

Copyright © 2011 ACM.

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Published: 08 June 2011

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ISSAC '11: International Symposium on Symbolic and Algebraic Computation

June 8 - 11, 2011

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Cited By

Kelmendi E(2022)Computing the Density of the Positivity Set for Linear Recurrence SequencesProceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3531130.3532399(1-12)Online publication date: 2-Aug-2022
https://dl.acm.org/doi/10.1145/3531130.3532399
Ergür ATonelli-Cueto JTsigaridas EMaza MZhi L(2022)Beyond Worst-Case Analysis for Root Isolation AlgorithmsProceedings of the 2022 International Symposium on Symbolic and Algebraic Computation10.1145/3476446.3535475(139-148)Online publication date: 4-Jul-2022
https://dl.acm.org/doi/10.1145/3476446.3535475
Hormann KKania LYap CChyzak FLabahn G(2021)Novel Range Functions via Taylor Expansions and Recursive Lagrange Interpolation with Application to Real Root IsolationProceedings of the 2021 International Symposium on Symbolic and Algebraic Computation10.1145/3452143.3465532(193-200)Online publication date: 18-Jul-2021
https://dl.acm.org/doi/10.1145/3452143.3465532
Li WPaulson LMahboubi AMyreen M(2019)Counting polynomial roots in Isabelle/HOL: a formal proof of the Budan-Fourier theoremProceedings of the 8th ACM SIGPLAN International Conference on Certified Programs and Proofs10.1145/3293880.3294092(52-64)Online publication date: 14-Jan-2019
https://dl.acm.org/doi/10.1145/3293880.3294092
Li WPaulson L(2019)Evaluating Winding Numbers and Counting Complex Roots Through Cauchy Indices in Isabelle/HOLJournal of Automated Reasoning10.1007/s10817-019-09521-3Online publication date: 3-Apr-2019
https://doi.org/10.1007/s10817-019-09521-3
Burr MGao STsigaridas EYap CEl Din M(2017)The Complexity of an Adaptive Subdivision Method for Approximating Real CurvesProceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3087604.3087654(61-68)Online publication date: 23-Jul-2017
https://dl.acm.org/doi/10.1145/3087604.3087654
Bouzidi YCluzeau TMoroz GQuadrat A(2017)Computing effectively stabilizing controllers for a class of nD systemsIFAC-PapersOnLine10.1016/j.ifacol.2017.08.20050:1(1847-1852)Online publication date: Jul-2017
https://doi.org/10.1016/j.ifacol.2017.08.200
Becker RSagraloff MSharma VXu JYap CAbramov SZima EGao X(2016)Complexity Analysis of Root Clustering for a Complex PolynomialProceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2930889.2930939(71-78)Online publication date: 20-Jul-2016
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Kobel ARouillier FSagraloff MAbramov SZima EGao X(2016)Computing Real Roots of Real Polynomials ... and now For Real!Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2930889.2930937(303-310)Online publication date: 20-Jul-2016
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Sagraloff MNagasaka KWinkler FSzanto A(2014)A near-optimal algorithm for computing real roots of sparse polynomialsProceedings of the 39th International Symposium on Symbolic and Algebraic Computation10.1145/2608628.2608632(359-366)Online publication date: 23-Jul-2014
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