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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)


Iteration methods for finding all zeros of a polynomial simultaneously

Author: Oliver Aberth
Journal: Math. Comp. 27 (1973), 339-344
MSC: Primary 65H05
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] Immo O. Kerner, Ein Gesamtschrittverfahren zur Berechnung der Nullstellen von Polynomen, Numer. Math. 8 (1966), 290–294 (German). MR 0203931 (34 #3778)
  • [3] Morris Marden, Geometry of polynomials, Second edition. Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972 (37 #1562)
  • [4] Brian T. Smith, Error bounds for zeros of a polynomial based upon Gerschgorin’s theorems, J. Assoc. Comput. Mach. 17 (1970), 661–674. MR 0279998 (43 #5719)

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Additional Information

PII: S 0025-5718(1973)0329236-7
Keywords: Polynomial zeros, iteration algorithm
Article copyright: © Copyright 1973 American Mathematical Society