On quadratic-like convergence of the means for two methods for simultaneous rootfinding of polynomials
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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  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 classificationAMS(MOS): 65H05 CR: 5.15
Key wordspolynomial zeros Durand-Kerner method simultaneous root finding method factorization of polynomials
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