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On the Quality of Some Root-Bounds

  • Prashant BatraEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9582)

Abstract

Bounds for the maximum modulus of all positive (or all complex) roots of a polynomial are a fundamental building block of algorithms involving algebraic equations. We apply known results to show which are the salient features of the Lagrange (real) root-bound as well as the related bound by Fujiwara. For a polynomial of degree n, we construct a bound of relative overestimation at most 1.72n which overestimates the Cauchy root by a factor of two at most. This can be carried over to the bounds by Kioustelidis and Hong. Giving a very short variant of a recent proof presented by Collins, we sketch a way to further definite, measurable improvement.

Keywords

Maximum modulus of polynomial roots Maximum overestimation Improvements of Lagrange’s bound 

References

  1. 1.
    Batra, P.: A property of the nearly optimal root-bound. J. Comput. Appl. Math. 167(2), 489–491 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Batra, P., Sharma, V.: Bounds on absolute positiveness of multivariate polynomials. J. Symb. Comput. 45(6), 617–628 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Burr, M.A., Krahmer, F.: SqFreeEVAL: an (almost) optimal real-root isolation algorithm. J. Symb. Comput. 47(2), 153–166 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Collins, G.E.: Krandick’s proof of Lagrange’s real root bound claim. J. Symb. Comput. 70, 106–111 (2015)CrossRefzbMATHGoogle Scholar
  5. 5.
    Collins, G.E.: Continued fraction real root isolation using the Hong bound. J. Symb. Comput. 72, 21–54 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Collins, G.E., Krandick, W.: On the computing time of the continued fractions method. J. Symb. Comput. 47(11), 1372–1412 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dieudonné, J.: La théorie analytique des polynômes d’une variable (à coefficients quelconques). Gauthier-Villars, Paris (1938)zbMATHGoogle Scholar
  8. 8.
    Fujiwara, M.: Über die obere Schranke des absoluten Betrages der Wurzeln einer algebraischen Gleichung. Tôhoku Math. J. 10, 167–171 (1916)zbMATHGoogle Scholar
  9. 9.
    Hong, H.: Bounds for absolute positiveness of multivariate polynomials. J. Symb. Comput. 25(5), 571–585 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kioustelidis, J.B.: Bounds for positive roots of polynomials. J. Comput. Appl. Math. 16(2), 241–244 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lagrange, J.-L.: Sur la résolution des équations numériques. In: Mémoires de l’Académie royale des Sciences et Belles-lettres de Berlin, t. XXIII, pp. 539–578 (1769)Google Scholar
  12. 12.
    Marden, M.: Geometry of Polynomials. AMS Mathematical Surveys 3, 2nd edn. AMS, Providence, Rhode Island (1966)zbMATHGoogle Scholar
  13. 13.
    Mehlhorn, K., Ray, S.: Faster algorithms for computing Hong’s bound on absolute positiveness. J. Symb. Comput. 45(6), 677–683 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ostrowski, A.: Solution of Equations in Euclidean and Banach Spaces, 3rd edn. Academic Press, New York (1973)zbMATHGoogle Scholar
  15. 15.
    Rahman, Q.I., Schmeisser, G.: Analytic Theory of Polynomials. Oxford University Press, Oxford (2002)zbMATHGoogle Scholar
  16. 16.
    Runge, C.: Separation und Approximation der Wurzeln. In: Encyklopädie der mathematischen Wissenschaften, vol. 1, pp. 404–448. Verlag Teubner, Leipzig (1899)Google Scholar
  17. 17.
    Sagraloff, M.: On the complexity of the Descartes method when using approximate arithmetic. J. Symb. Comput. 65, 79–110 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sharma, V.: Complexity of real root isolation using continued fractions. Theor. Comput. Sci. 409(2), 292–310 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Specht, W.: Algebraische Gleichungen mit reellen oder komplexen Koeffizienten. In: Enzyklopädie der mathematischen Wissenschaften, Band I, Heft 3, Teil II. B.G. Teubner Verlagsgesellschaft, Stuttgart. Zweite, völlig neubearbeitete Auflage (1958)Google Scholar
  20. 20.
    van der Sluis, A.: Upperbounds for roots of polynomials. Numer. Math. 15, 250–262 (1970)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institute for Reliable ComputingHamburg University of TechnologyHamburgGermany

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