BIT Numerical Mathematics

, Volume 33, Issue 1, pp 63–73 | Cite as

On quadratic-like convergence of the means for two methods for simultaneous rootfinding of polynomials

  • Carsten Carstensen
Part II Numerical Mathematics


Durand-Kerner's method for simultaneous rootfinding of a polynomial is locally second order convergent if all the zeros are simple. If this condition is violated numerical experiences still show linear convergence. For this case of multiple roots, Fraigniaud [4] proves that the means of clustering approximants for a multiple root is a better approximant for the zero and called this Quadratic-Like-Convergence of the Means.

This note gives a new proof and a refinement of this property. The proof is based on the related Grau's method for simultaneous factoring of a polynomial. A similar property of some coefficients of the third order method due to Börsch-Supan, Maehly, Ehrlich, Aberth and others is proved.

Mathematics subject classification

AMS(MOS): 65H05 CR: 5.15 

Key words

polynomial zeros Durand-Kerner method simultaneous root finding method factorization of polynomials 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    O. Aberth,Iteration methods for finding all zeros of a polynomial simultaneously. Math. Comp. 27 339–344 (1973).Google Scholar
  2. 2.
    C. Carstensen,On the Grau's method for simultaneous factorization of polynomials. SIAM J. Numer. Anal. 29, 601–613 (1992).Google Scholar
  3. 3.
    C. Carstensen,On simultaneous factoring of a polynomial. International Journal of Computer Mathematics, accepted (1992).Google Scholar
  4. 4.
    P. Fraigniaud,The Durand-Kerner polynomial root-finding method in case of multiple roots. BIT 31 112–113 (1991).Google Scholar
  5. 5.
    A. A. Grau,The simultaneous Newton improvement of a complete set of approximate factors of a polynomial. SIAM J. Numer. Anal.8, 425–438 (1971).Google Scholar
  6. 6.
    G. Kjellberg,Two observations on Durand-Kerner's root-finding method, BIT 24 556–559 (1984).Google Scholar
  7. 7.
    L. Pasquini, D. Trigiante,A globally convergent method for simultaneously finding polynomial roots. Math. Comp. 44 135–149 (1985).Google Scholar
  8. 8.
    M. S. Petković,Iterative Methods for Simultaneous Inclusion of Polynomial Zeros, Springer Lecture Notes 1387 (1989).Google Scholar

Copyright information

© BIT Foundations 1993

Authors and Affiliations

  • Carsten Carstensen
    • 1
  1. 1.Institut für Angewandte MathematikUniversität HannoverHannover 1Germany

Personalised recommendations