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Iteration methods for finding all zeros of a polynomial simultaneously


Author: Oliver Aberth
Journal: Math. Comp. 27 (1973), 339-344
MSC: Primary 65H05
DOI: https://doi.org/10.1090/S0025-5718-1973-0329236-7
MathSciNet review: 0329236
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Abstract: Durand and Kerner independently have proposed a quadratically convergent iteration method for finding all zeros of a polynomial simultaneously. Here, a new derivation of their iteration equation is given, and a second, cubically convergent iteration method is proposed. A relatively simple procedure for choosing the initial approximations is described, which is applicable to either method.


References [Enhancements On Off] (What's this?)

  • [1] E. Durand, Solutions Numériques des Équations Algébriques. Tome I: Equations du Type $ F(x) = 0$; Racines d'un Polynôme, Masson, Paris, 1960. MR 22 #12714.
  • [2] I. O. Kerner, "Ein Gesamtschrittverfahren zur Berechnung der Nullstellen von Polynomen," Numer. Math., v. 8, 1966, pp. 290-294. MR 34 #3778. MR 0203931 (34:3778)
  • [3] M. Marden, Geometry of Polynomials, 2nd ed., Math. Surveys, no. 3, Amer. Math. Soc., Providence, R.I., 1966. MR 37 #1562. MR 0225972 (37:1562)
  • [4] B. T. Smith, "Error bounds for zeros of a polynomial based upon Gerschgorin's theorems," J. Assoc. Comput. Mach., v. 17, 1970, pp. 661-674. MR 43 #5719. MR 0279998 (43:5719)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1973-0329236-7
Keywords: Polynomial zeros, iteration algorithm
Article copyright: © Copyright 1973 American Mathematical Society

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