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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

Abstract

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 

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References

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    P. Fraigniaud,The Durand-Kerner polynomial root-finding method in case of multiple roots. BIT 31 112–113 (1991).Google Scholar
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    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
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Copyright information

© BIT Foundations 1993

Authors and Affiliations

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

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