Inequalities on Upper Bounds for Real Polynomial Roots

  • Doru Ştefănescu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4194)


In this paper we propose two methods for the computation of upper bounds of the real roots of univariate polynomials with real coefficients. Our results apply to polynomials having at least one negative coefficient. The upper bounds of the real roots are expressed as functions of the first positive coefficients and of the two largest absolute values of the negative ones.


Positive Root Real Root Continue Fraction Small Integer Previous Inequality 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akritas, A.G., Strzeboński, A.W.: A comparative study of two real root isolation methods. Nonlin. Anal.: Modell. Control 10, 297–304 (2005)zbMATHGoogle Scholar
  2. 2.
    Emiris, I.Z., Tsigaridas, E.P.: Univariate polynomial real root isolation: Continued fractions revisited (2006),
  3. 3.
    Eve, J.: The evaluation of polynomials. Numer. Math. 6, 17–21 (1964)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Grinstein, L.S.: Upper limits to the real roots of polynomial equations. Amer. Math. Monthly 60, 608–615 (1953)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Herzberger, J.: Construction of bounds for the positive root of a general class of polynomials with applications. In: Inclusion Methods for Nonlinear Problems with Applications in Engineering, Economics and Physiscs (Munich, 2000). Comput. Suppl., vol. 16. Springer, Vienna (2003)Google Scholar
  6. 6.
    Kioustelidis, J.B.: Bounds for positive roots of polynomials. J. Comput. Appl. Math. 16, 241–244 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kjurkchiev, N.: Note on the estimation of the order of convergence of some iterative methods. BIT 32, 525–528 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Mignotte, M., Ştefănescu, D.: Polynomials – An Algorithmic Approach. Springer, Heidelberg (1999)zbMATHGoogle Scholar
  9. 9.
    Nuij, W.: A note on hyperbolic polynomials. Math. Scand. 23, 69–72 (1968)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Rouiller, F., Zimmermann, P.: Efficient isolation of polynomials real roots. J. Comput. Appl. Math. 162, 33–50 (2004)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Ştefănescu, D.: New bounds for the positive roots of polynomials. J. Univ. Comp. Sc. 11, 2125–2131 (2005)zbMATHGoogle Scholar
  12. 12.
    Yap, C.K.: Fundamental Problems of Algorithmic Algebra. Oxford University Press, Oxford (2000)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Doru Ştefănescu
    • 1
  1. 1.University of BucharestRomania

Personalised recommendations