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    Liu, Gang 2012. Superattracting cycles of the relaxed Newton’s method for entire functions. Applied Mathematics and Computation, Vol. 218, Issue. 24, p. 12008.

    Çilingir, Figen 2007. Mystery of the rational iteration arising from relaxed Newton’s method. Chaos, Solitons & Fractals, Vol. 32, Issue. 2, p. 471.

    Çilingir, Figen 2004. On infinite area for complex exponential function. Chaos, Solitons & Fractals, Vol. 22, Issue. 5, p. 1189.

    Buff, Xavier and Henriksen, Christian 2003. On K nig's root-finding algorithms*. Nonlinearity, Vol. 16, Issue. 3, p. 989.

    Hockett, Kevin 1993. Global dynamical properties of Euler and backward Euler. Ergodic Theory and Dynamical Systems, Vol. 13, Issue. 04,

    Haeseler, F. v. and Peitgen, H. O. 1988. Newton's method and complex dynamical systems. Acta Applicandae Mathematicae, Vol. 13, Issue. 1-2, p. 3.

  • Ergodic Theory and Dynamical Systems, Volume 6, Issue 4
  • December 1986, pp. 561-569

Multiple attractors in Newton's method

  • Mike Hurley (a1)
  • DOI:
  • Published online: 01 September 2008

For each d ≥ 2 there exists a polynomial p with real coefficients such that the associated Newton function z–[p(z)/p′(z)] has 2d–2 distinct attracting periodic orbits in the complex plane. According to a theorem of G. Julia, this is the maximal number of attracting orbits that any rational function of degree d can possess.

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[Bla]P. Blanchard . Complex analytic dynamics on the Riemann sphere. Bull. Amer. Math. Soc. 11 (1984), 85141.

[CuGS]J. Curry , L. Garnett & D. Sullivan . On the iteration of a rational function: computer experiments with Newton's method. Comm. Math. Phys. 91 (1983), 267277.

[HM]M. Hurley & C. Martin . Newton's algorithm and chaotic dynamical systems. SIAM J. Math. Anal. 15 (1984), 238252.

[SaU]D. Saari & J. Urenko . Newton's method, circle maps, and chaotic motion. Amer. Math. Monthly91 (1985), 317.

[Sm]S. Smale . The fundamental theorem of algebra and complexity theory. Bull. Amer. Math. Soc. 4 (1981), 136.

[Sul]; D. Sullivan . Conformal dynamical systems, pp. 725752 in Geometric Dynamics, J. Palis edit., (Springer Lect. Notes in Math. 1007). Springer–Verlag, New York, 1983.

[W]S. Wong . Newton's Method and symbolic dynamics. Proc. Amer. Math. Soc. 91 (1984), 245253.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
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