research-article

Isolating real roots of real polynomials

Authors:

Michael SagraloffAuthors Info & Claims

ISSAC '09: Proceedings of the 2009 international symposium on Symbolic and algebraic computation

Pages 247 - 254

https://doi.org/10.1145/1576702.1576737

Published: 28 July 2009 Publication History

Abstract

We describe a bisection algorithm for root isolation of polynomials with real coefficients. It is assumed that the coefficients can be approximated with arbitrary precision; exact computation in the field of coefficients is not required. We refer to such coefficients as bitstream coefficients. The algorithm is deterministic and has almost the same asymptotic complexity as the randomized algorithm of [12]. We also discuss a partial extension to multiple roots.

References

[1]

O. Aberth. Precise Numerical Analysis. William C. Brown Company Publishers, Dubuque, IA, USA, 1988.

[2]

A. Akritas and A. Strzebonski. A comparative study of two root isolation methods. Nonlinear Analysis: Modelling and Control, 10:297--304, 2005.

[3]

A. G. Akritas. The fastest exact algorithms for the isolation of the real roots of a polynomial equation. Computing, 24(4):299--313, 1980.

[4]

S. Basu, R. Pollack, and M.-F. Roy. Algorithms in Real and Algebraic Geometry. Springer, 2nd edition, 2006.

Digital Library

[5]

E. Berberich, M. Kerber, and M. Sagraloff. Exact geometric-topological analysis of algebraic surfaces. In SoCG, pages 164--173, 2008.

Digital Library

[6]

D. Bini and G. Fiorentino. Design, analysis and implementation of a multiprecision polynomial rootfinder. Numerical Algorithms, 23:127--173, 2000.

[7]

CGAL (Computational Geometry Algorithms Library). www.cgal.org.

[8]

G. Collins, J. Johnson, and W. Krandick. Interval arithmetic in cylindrical algebraic decomposition. J. Symbolic Computation, 34:143--155, 2002.

Digital Library

[9]

G. E. Collins and A. G. Akritas. Polynomial real root isolation using Descartes' rule of signs. In R. D. Jenks, editor, SYMSAC, pages 272--275, Yorktown Heights, NY, 1976. ACM Press.

Digital Library

[10]

A. Eigenwillig. Real Root Isolation for Exact and Approximate Polynomials Using Descartes' Rule of Signs. PhD thesis, Saarland University, 2008.

[11]

A. Eigenwillig, M. Kerber, and N. Wolpert. Fast and exact analysis of real algebraic plane curves. In ISSAC, pages 151--158, 2007.

Digital Library

[12]

A. Eigenwillig, L. Kettner, W. Krandick, K. Mehlhorn, S. Schmitt, and N. Wolpert. An Exact Descartes Algorithm with Approximate Coefficients (Extended Abstract). In CASC, volume 3718 of LNCS, pages 138--149, 2005.

[13]

A. Eigenwillig, V. Sharma, and C. Yap. Almost tight complexity bounds for the Descartes method. In ISSAC, pages 151--158, 2006.

Digital Library

[14]

J. Gerhard. Modular algorithms in symbolic summation and symbolic integration. LNCS, Springer, 3218, 2004.

Digital Library

[15]

J. R. Johnson and W. Krandick. Polynomial real root isolation using approximate arithmetic. In W. K&#252;chlin, editor, ISSAC, pages 225--232. ACM Press, 1997.

Digital Library

[16]

J. R. Johnson, W. Krandick, and A. D. Ruslanov. Architecture-aware classical taylor shift by 1. In ISSAC, pages 200--207, 2005.

Digital Library

[17]

M. Kerber. Analysis of real algebraic plane curves. Master's thesis, Saarland University, 2006. Master's thesis.

[18]

B. Mourrain, F. Rouillier, and M.-F. Roy. Bernstein's basis and real root isolation. Technical report, INRIA, 2004.

[19]

N. Obrechkoff. Zeros of Polynomials. Marina Drinov, Sofia, 2003. Translation of the Bulgarian original.

[20]

N. Obreschkoff. Verteilung und Berechnung der Nullstellen reeller Polynome. VEB Deutscher Verlag der Wissenschaften, 1963.

[21]

V. Pan. Solving a polynomial equation: Some history and recent progress. SIAM Review, 39:187--220, 1997.

Digital Library

[22]

V. Pan. Univariate polynomials: Nearly optimal algorithms for numerical factorization and root finding. J. Symbolic Computation, 33:701--733, 2002.

Digital Library

[23]

F. Roullier and P. Zimmermann. Efficient isolation of a polynomial's real roots. J. Computational and Applied Mathematics, pages 33--50, 2004.

Digital Library

[24]

I. J. Schoenberg. Zur Abz&#228;hlung der reellen Wurzeln algebraischer Gleichungen. Mathematische Zeitschrift, pages 546--564, 1934.

[25]

A. Sch&#246;nhage. The fundamental theorem of algebra in terms of computational complexity. http://www.informatik.uni-bonn.de/~schoe/fdmthmrep.ps.gz, 1982.

[26]

A. Sch&#246;nhage. Quasi-GCD computations. Journal of Complexity, 1(1):118--137, 1985.

[27]

V. Sharma. Complexity of real root isolation using continued fractions. Theoretical Computer Science, 409:292--310, 2008.

Digital Library

[28]

E. Tsigaridas and I. Emiris. On the complexity of real root isolation using continued fractions. Theor. Comput. Sci., pages 158--173, 2008.

Digital Library

[29]

A. J. H. Vincent. Sur la r&#233;solution des equations num&#233;riques. Journal de Math&#233;matiques Pures et Appliqu&#233;es, 1:341--372, 1836.

[30]

C. Yap. Fundamental Problems in Algorithmic Algebra. Oxford University Press, 1999.

Digital Library

Cited By

Zsidó J(2014)Theorem of Three Circles in CoqJournal of Automated Reasoning10.1007/s10817-013-9299-053:2(105-127)Online publication date: 1-Aug-2014
https://dl.acm.org/doi/10.1007/s10817-013-9299-0
Yap CSagraloff MSchost ÉEmiris I(2011)A simple but exact and efficient algorithm for complex root isolationProceedings of the 36th international symposium on Symbolic and algebraic computation10.1145/1993886.1993938(353-360)Online publication date: 8-Jun-2011
https://dl.acm.org/doi/10.1145/1993886.1993938
Fasani RSavageau M(2010)Automated construction and analysis of the design space for biochemical systemsBioinformatics10.1093/bioinformatics/btq47926:20(2601-2609)Online publication date: 7-Sep-2010
https://doi.org/10.1093/bioinformatics/btq479
Show More Cited By

Index Terms

Isolating real roots of real polynomials
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations on polynomials

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cover image ACM Conferences

ISSAC '09: Proceedings of the 2009 international symposium on Symbolic and algebraic computation

July 2009

402 pages

ISBN:9781605586090

DOI:10.1145/1576702

General Chairs:
Jeremy Johnson
Drexel University, USA
,
Hyungju Park
Korea Institute for Advanced Study, Korea
,
Program Chair:
Erich Kaltofen
North Carolina State University, USA

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: 28 July 2009

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ISSAC '09: International Symposium on Symbolic and Algebraic Computation

July 28 - 31, 2009

Seoul, Republic of Korea

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Overall Acceptance Rate 395 of 838 submissions, 47%

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Cited By

Zsidó J(2014)Theorem of Three Circles in CoqJournal of Automated Reasoning10.1007/s10817-013-9299-053:2(105-127)Online publication date: 1-Aug-2014
https://dl.acm.org/doi/10.1007/s10817-013-9299-0
Yap CSagraloff MSchost ÉEmiris I(2011)A simple but exact and efficient algorithm for complex root isolationProceedings of the 36th international symposium on Symbolic and algebraic computation10.1145/1993886.1993938(353-360)Online publication date: 8-Jun-2011
https://dl.acm.org/doi/10.1145/1993886.1993938
Fasani RSavageau M(2010)Automated construction and analysis of the design space for biochemical systemsBioinformatics10.1093/bioinformatics/btq47926:20(2601-2609)Online publication date: 7-Sep-2010
https://doi.org/10.1093/bioinformatics/btq479
Emeliyanenko PBerberich ESagraloff M(2009)Visualizing Arcs of Implicit Algebraic Curves, Exactly and FastProceedings of the 5th International Symposium on Advances in Visual Computing: Part I10.1007/978-3-642-10331-5_57(608-619)Online publication date: 26-Nov-2009
https://dl.acm.org/doi/10.1007/978-3-642-10331-5_57

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