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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)

Abstract

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.

Keywords

Real Root Recursive Call Bernstein Polynomial Split Point Recursion Tree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  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)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Lane, J.M., Riesenfeld, R.F.: Bounds on a polynomial. BIT 21, 112–117 (1981)zbMATHCrossRefMathSciNetGoogle 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)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry. Springer, Heidelberg (2003)zbMATHGoogle 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)zbMATHGoogle 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)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Pan, V.Y.: Univariate polynomials: Nearly optimal algorithms for numerical factorization and root finding. J. Symbolic Computation 33, 701–733 (2002)zbMATHCrossRefGoogle Scholar
  14. 14.
    Prautzsch, H., Boehm, W., Paluszny, M.: Bézier and B-Spline Techniques. Springer, Heidelberg (2002)zbMATHGoogle 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)zbMATHCrossRefGoogle Scholar
  18. 18.
    Bini, D.A., Fiorentino, G.: Design, analysis, and implementation of a multiprecision polynomial rootfinder. Numerical Algorithms 23, 127–173 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Yap, C.K.: Fundamental Problems of Algorithmic Algebra. Oxford University Press, Oxford (2000)zbMATHGoogle 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

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