A General Approach to Isolating Roots of a Bitstream Polynomial

  • Michael Sagraloff


We describe a new approach to isolate the roots (either real or complex) of a square-free polynomial F with real coefficients. It is assumed that each coefficient of F can be approximated to any specified error bound and refer to such coefficients as bitstream coefficients. The presented method is exact, complete and deterministic. Compared to previous approaches (Eigenwillig in Real root isolation for exact and approximate polynomials using Descartes’ rule of signs, PhD thesis, Universität des Saarlandes, 2008; Eigenwillig et al. in CASC, LNCS, 2005; Mehlhorn and Sagraloff in J. Symb. Comput. 46(1):70–90, 2011) we improve in two aspects. Firstly, our approach can be combined with any existing subdivision method for isolating the roots of a polynomial with rational coefficients. Secondly, the approximation demand on the coefficients and the bit complexity of our approach is considerably smaller. In particular, we can replace the worst-case quantity σ(F) by the average-case quantity \({\prod_{i=1}^n\sqrt[n] {\sigma_i}}\) , where σ i denotes the minimal distance of the i -th root ξ i of F to any other root of F, σ(F) := min i σ i , and n = deg F. For polynomials with integer coefficients, our method matches the best bounds known for existing practical algorithms that perform exact operations on the input coefficients.


Real polynomial Root isolation Bitstream coefficients Root perturbation bounds Adaptive precision management 


  1. 1.
    Akritas A., Strzebonski A.: A comparative study of two root isolation methods. Nonlinear Anal. Model. Control 10, 297–304 (2005)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Akritas A.G.: The fastest exact algorithms for the isolation of the real roots of a polynomial equation. Computing 24(4), 299–313 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Alesina A., Galuzzi M.: A new proof of Vicent’s theorem. L’Enseignement Mathematique 44, 219–256 (1998)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Beberich, E., Emeliyanenko, P., Sagraloff, M.: An elimination method for solving bivariate polynomial systems: eliminating the usual drawbacks. In: ALENEX, pp. 35–47. SIAM, Philadelphia (2011)Google Scholar
  5. 5.
    Berberich E., Kerber M., Sagraloff M.: An efficient algorithm for the stratification and triangulation of an algebraic surface. Comput. Geom. Theory Appl. (CGTA) 43(3), 257–278 (2009)MathSciNetGoogle Scholar
  6. 6.
    Collins G., Johnson J., Krandick W.: Interval arithmetic in cylindrical algebraic decomposition. JSC 34, 143–155 (2002)MathSciNetGoogle Scholar
  7. 7.
    Collins G.E., Akritas A.G.: Polynomial real root isolation using Descartes’ rule of signs. In: Jenks, R.D. (eds) Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation, pp. 272–275. ACM Press, New York (1976)CrossRefGoogle Scholar
  8. 8.
    Du, Z., Sharma, V., Yap, C.: Amortized bounds for root isolation via Sturm sequences. In: SNC, pp. 113–130 (2007)Google Scholar
  9. 9.
    Eigenwillig A.: On multiple roots in Descartes’ rule and their distance to roots of higher derivatives. J. Comput. Appl. Math. 200(1), 226–230 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Eigenwillig, A.: Real root isolation for exact and approximate polynomials using Descartes’ rule of signs. PhD thesis, Universität des Saarlandes (2008)Google Scholar
  11. 11.
    Eigenwillig, A., Kerber, M., Wolpert, N.: Fast and exact geometric analysis of real algebraic plane curves. In: Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation (ISSAC 2007), pp. 151–158 (2007)Google Scholar
  12. 12.
    Eigenwillig, A., Kettner, L., Krandick, W., Mehlhorn, K., Schmitt, S., Wolpert, N.: An exact descartes algorithm with approximate coefficients. In: CASC. LNCS, vol. 3718, pp. 138–149 (2005)Google Scholar
  13. 13.
    Eigenwillig, A., Sharma, V., Yap, C.: Almost tight complexity bounds for the Descartes method. In: ISSAC, pp. 71–78 (2006)Google Scholar
  14. 14.
    Gerhard J.: Modular algorithms in symbolic summation and symbolic integration. LNCS, pp. 3218. Springer, Berlin (2004)Google Scholar
  15. 15.
    Gourdon, X.: Combinatoire, Algorithmique et Géométrie des Polynômes. Thèse, École polytechnique (1996)Google Scholar
  16. 16.
    Hemmer, M., Tsigaridas, E.P., Zafeirakopoulos, Z., Emiris, I.Z., Karavelas, M.I., Mourrain, B.: Experimental evaluation and cross benchmarking of univariate real solvers. In: SNC, pp. 45–54 (2009)Google Scholar
  17. 17.
    Johnson J.: Algorithms for polynomial real root isolation. In: Caviness, B., Johnson, J. (eds) Quantifier Elimination and Cylindrical Algebraic Decomposition, Texts and monographs in Symbolic Computation, pp. 269–299. Springer, Berlin (1998)Google Scholar
  18. 18.
    Johnson J.R., Krandick W.: Polynomial real root isolation using approximate arithmetic. In: Küchlin, W. (eds) ISSAC, pp. 225–232. ACM Press, New York (1997)CrossRefGoogle Scholar
  19. 19.
    Kerber, M.: Geometric Algorithms for Algebraic Curves and Surfaces. PhD thesis, Universität des Saarlandes (2009)Google Scholar
  20. 20.
    Krandick W., Mehlhorn K.: New bounds for the Descartes method. J. Symb. Comput. 41(1), 49–66 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Lickteig T., Roy M.-F.: Sylvester-Habicht sequences and fast Cauchy index computation. J. Symb. Comput. 31, 315–341 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Mehlhorn K., Ray S.: Faster algorithms for computing Hong’s bound on absolute positiveness. J. Symb. Comput. 45((6), 677–683 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Mehlhorn K., Sagraloff M.: A deterministic algorithm for isolating real roots of a real polynomial. J. Symb. Comput. 46(1), 70–90 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Mitchell, D.P.: Robust ray intersection with interval arithmetic. In: Graphics Interface’90, pp. 68–74 (1990)Google Scholar
  25. 25.
    Mourrain, B., Rouillier, F., Roy, M.-F.: The Bernstein basis and real root isolation. In: Goodman, J.E., Pach, J., Welzl, E. (eds.) Combinatorial and Computational Geometry, number 52 in MSRI Publications, pp. 459–478. Cambridge University Press, Cambridge (2005)Google Scholar
  26. 26.
    Pan V.Y.: Sequential and parallel complexity of approximate evaluation of polynomial zeros. Comput. Math. Appl. 14(8), 591–622 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Pan V.Y.: Solving a polynomial equation: some history and recent progress. SIAM Rev. 39(2), 187–220 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Rouillier F., Zimmermann P.: Efficient isolation of [a] polynomial’s real roots. J. Comput. Appl. Math. 162, 33–50 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Sagraloff, M., Yap, C.: A simple but exact and efficient algorithm for complex root isolation. In: International Symposium on Symbolic and Algebraic Computation (ISSAC), pp. 353–360 (2011)Google Scholar
  30. 30.
    Schönhage, A.: The fundamental theorem of algebra in terms of computational complexity, 1982. Manuscript, Department of Mathematics, University of Tübingen. Updated (2004)Google Scholar
  31. 31.
    Schönhage A.: Quasi-GCD computations. J. Complex. 1(1), 118–137 (1985)zbMATHCrossRefGoogle Scholar
  32. 32.
    Sharma V.: Complexity of real root isolation using continued fractions. Theor. Comput. Sci. 409, 292–310 (2008)zbMATHCrossRefGoogle Scholar
  33. 33.
    Smale S.: The fundamental theorem of algebra and complexity theory. Bull. (N.S.) AMS 4(1), 1–36 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Smith B.T.: Error bounds for zeros of a polynomial based upon Gerschgorin’s theorems. J. ACM 17(4), 661–674 (1970)zbMATHCrossRefGoogle Scholar
  35. 35.
    Tsigaridas E.P., Emiris I.Z.: On the complexity of real root isolation using continued fractions. Theor. Comput. Sci. 392(1–3), 158–173 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Vincent A.J.H.: Sur la résolution des equations numériques. Journal de Mathématiques Pures et Appliquées 1, 341–372 (1836)Google Scholar
  37. 37.
    Yap C.K.: Fundamental Problems in Algorithmic Algebra. Oxford University Press, UK (2000)Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany

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