A Descartes Algorithm for Polynomials with Bit-Stream Coefficients

  • Arno Eigenwillig
  • Lutz Kettner
  • Werner Krandick
  • Kurt Mehlhorn
  • Susanne Schmitt
  • Nicola Wolpert
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3718)


The Descartes method is an algorithm for isolating the real roots of square-free polynomials with real coefficients. We assume that coefficients are given as (potentially infinite) bit-streams. In other words, coefficients can be approximated to any desired accuracy, but are not known exactly. We show that a variant of the Descartes algorithm can cope with bit-stream coefficients. To isolate the real roots of a square-free real polynomial \(q(x)=q_{n^{x^{n}}}+...+q_{0}\) with root separation ρ, coefficients |q n | ≥ 1 and \(|q_{i}|\leq 2^{\tau}\), it needs coefficient approximations to O(n(log(1/ρ) + τ)) bits after the binary point and has an expected cost of O(n 4 (log(1/ρ) + τ)2) bit operations.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Collins, G.E., Akritas, A.G.: Polynomial real root isolation using Descartes’ rule of signs. In: Jenks, R.D. (ed.) Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation, pp. 272–275. ACM Press, New York (1976)CrossRefGoogle Scholar
  2. 2.
    Uspensky, J.: Theory of Equations. McGraw-Hill, New York (1948)Google Scholar
  3. 3.
    Krandick, W.: Isolierung reeller Nullstellen von Polynomen. In: Herzberger, J. (ed.) Wissenschaftliches Rechnen, pp. 105–154. Akademie-Verlag (1995)Google Scholar
  4. 4.
    Rouillier, F., Zimmermann, P.: Efficient isolation of a polynomial’s real roots. J. Computational and Applied Mathematics 162, 33–50 (2004)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Lane, J.M., Riesenfeld, R.F.: Bounds on a polynomial. BIT 21, 112–117 (1981)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Mourrain, B., Vrahatis, M.N., Yakoubsohn, J.C.: On the complexity of isolating real roots and computing with certainty the topological degree. J. Complexity 18, 612–640 (2002)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry. Springer, Heidelberg (2003)MATHGoogle Scholar
  8. 8.
    Mourrain, B., Rouillier, F., Roy, M.F.: Bernstein’s basis and real root isolation. Rapport de recherche 5149, INRIA-Rocquencourt (2004), http://www.inria.fr/rrrt/rr-5149.html—
  9. 9.
    Collins, G.E., Johnson, J.R., Krandick, W.: Interval arithmetic in cylindrical algebraic decomposition. J. Symbolic Computation 34, 143–155 (2002)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Henrici, P.: Applied and Computational Complex Analysis, vol. 1. Wiley, Chichester (1974)MATHGoogle Scholar
  11. 11.
    Weyl, H.: Randbemerkungen zu Hauptproblemen der Mathematik II: Fundamentalsatz der Algebra und Grundlagen der Mathematik. Math. Z. 20, 131–152 (1924)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Pan, V.: Solving a polynomial equation: Some history and recent progress. SIAM Review 39, 187–220 (1997)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Pan, V.Y.: Univariate polynomials: Nearly optimal algorithms for numerical factorization and root finding. J. Symbolic Computation 33, 701–733 (2002)MATHCrossRefGoogle Scholar
  14. 14.
    Prautzsch, H., Boehm, W., Paluszny, M.: Bézier and B-Spline Techniques. Springer, Heidelberg (2002)MATHGoogle Scholar
  15. 15.
    Hoschek, J., Lasser, D.: Fundamentals of computer aided geometric design. A K Peters, Wellesley (1996); Translation of: Grundlagen der geometrischen Datenverarbeitung, Teubner(1989) Google Scholar
  16. 16.
    Krandick, W., Mehlhorn, K.: New bounds for the Descartes method. Technical report, Drexel University, Dept. of Computer Science (2004); to appear in J. Symbolic Computation, http://www.mcs.drexel.edu/page.php?name=reports/DU-CS-04-04.html
  17. 17.
    Smith, B.T.: Error bounds for zeros of a polynomial based upon Gerschgorin’s theorems. J. ACM 17, 661–674 (1970)MATHCrossRefGoogle Scholar
  18. 18.
    Bini, D.A., Fiorentino, G.: Design, analysis, and implementation of a multiprecision polynomial rootfinder. Numerical Algorithms 23, 127–173 (2000)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Yap, C.K.: Fundamental Problems of Algorithmic Algebra. Oxford University Press, Oxford (2000)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Arno Eigenwillig
    • 1
  • Lutz Kettner
    • 1
  • Werner Krandick
    • 2
  • Kurt Mehlhorn
    • 1
  • Susanne Schmitt
    • 1
  • Nicola Wolpert
    • 1
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany
  2. 2.Dept. of Computer ScienceDrexel UniversityPhiladelphiaUSA

Personalised recommendations