Tighter bounds of errors of numerical roots

  • Tateaki SasakiEmail author


LetP(z) be a monic univariate polynomial overC, of degreen and having roots ζ1,..., ζ n . Given approximate rootsz 1,...,z n, with ζ i z i (i = 1,...,n), we derive a very tight upper bound of ¦ζ i z i¦, by assuming that ζ i has no close root. The bound formula has a similarity with Smale’s and Smith’s formulas. We also derive a lower bound of ¦ζ i z i¦ and a lower bound of min{ζ j z i¦ ¦ji}.


Error Bound Tight Bound Numerical Root Approximate Root Close Root 
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.


  1. [1]
    L. Blum, F. Cucker, M. Shub and S. Smale, Complexity and Real Computation. Springer-Verlag, New York, 1998.Google Scholar
  2. [2]
    L. Elsner, A remark on simultaneous inclusions of the zeros of a polynomial by Gerschgorin’s theorem. Numer. Math.21 (1973), 425–427.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    D. Inaba and T. Sasaki, Certification of analytic continuation of algebraic function. Proc. CASC 2004 (Computer Algebra in Scientific Computing), V.G. Ganzha, E.W. Mayr and E.V. Vorozhtsov (eds.). Technishe Universität München Press, 2004, 249–260.Google Scholar
  4. [4]
    L. Kantrovich and G. Akilov, Functional Analysis in Normed Spaces (translated from Russian). MacMillan, 1964.Google Scholar
  5. [5]
    A.M. Ostrowski, Solution of Equations in Euclidean and Banach Spaces. Academic Press, New York, 1973.zbMATHGoogle Scholar
  6. [6]
    S. Smale, Newton’s method estimates from data at one point. In R. Ewing, K. Gross, and C. Martin (eds.), The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics. Springer-Verlag, 1986.Google Scholar
  7. [7]
    B.T. Smith, Error bounds for zeros of a polynomial based on Gerschgorin’s theorems. J. ACM,17 (1970), 661–674.zbMATHCrossRefGoogle Scholar
  8. [8]
    T. Sasaki and A. Terui, Computing clustered close-roots of univariate polynomial. Preprint of Univ. Tsukuba (2007).Google Scholar
  9. [9]
    X.H. Wang and D.F. Han, On dominating sequence method in the point estimate and Smale theorem. Sci. China (Series A),33 (1990), 135–144.zbMATHMathSciNetGoogle Scholar

Copyright information

© JJIAM Publishing Committee 2007

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of TsukubaIbarakiJapan

Personalised recommendations