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The dynamical analysis of a uniparametric family of three-point optimal eighth-order multiple-root finders under the Möbius conjugacy map on the Riemann sphere

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Abstract

In this paper, we present dynamical viewpoints under the Möbius conjugacy map on the Riemann sphere for a uniparametric family of optimal eighth-order multiple-root finders. Various conjugacy properties are investigated including the invariance of the fixed point and its multiplier, which enables the family of iterative methods to trace along the consistent orbits in position. The parameter spaces and dynamical planes are studied and illustrated to visualize the periodic components and their geometric properties. Both theoretical and computational analyses along with a numerical algorithm are carried out regarding the bifurcation points of satellite and primitive components.

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Correspondence to Young Ik Kim.

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Lee, M., Kim, Y.I. The dynamical analysis of a uniparametric family of three-point optimal eighth-order multiple-root finders under the Möbius conjugacy map on the Riemann sphere. Numer Algor 83, 1063–1090 (2020). https://doi.org/10.1007/s11075-019-00716-8

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Keywords

  • Parameter space
  • Möbius map
  • Bifurcation point
  • Multiple-root
  • Eighth-order
  • Conjugacy

Mathematics Subject Classification (2010)

  • 65H05
  • 65H99
  • 41A25
  • 65B99