Local and Semilocal Convergence of a Family of Multi-point Weierstrass-Type Root-Finding Methods

Abstract

Weierstrass (Sitzungsber Königl Preuss Akad Wiss Berlin II:1085–1101, 1891) introduced his famous iterative method for numerical finding all zeros of a polynomial simultaneously. Kyurkchiev and Ivanov (Ann Univ Sofia Fac Math Mech 78:132–136, 1984) constructed a family of multi-point root-finding methods which are based on the Weierstrass method. The purpose of this research is threefold: (1) to develop a new simple approach for the study of the local convergence of the multi-point simultaneous iterative methods; (2) to present a new local convergence result for this family which improves in several directions the result of Kyurkchiev and Ivanov; (3) to provide semilocal convergence results for Kyurkchiev–Ivanov’s family of iterative methods.

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Acknowledgements

The authors thank Dr. Maria T. Vasileva for cooperation in the preparation of the numerical results.

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Correspondence to Petko D. Proinov.

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This work was supported by the National Science Fund of the Bulgarian Ministry of Education and Science under Grant DN 12/12.

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Proinov, P.D., Petkova, M.D. Local and Semilocal Convergence of a Family of Multi-point Weierstrass-Type Root-Finding Methods. Mediterr. J. Math. 17, 107 (2020). https://doi.org/10.1007/s00009-020-01545-z

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Mathematics Subject Classification

  • Primary 65H05
  • Secondary 12Y05

Keywords

  • Iterative methods
  • simultaneous methods
  • polynomial zeros
  • local convergence
  • semilocal convergence
  • error estimates