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

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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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References

  1. Aberth, O.: Iteration methods for finding all zeros of a polynomial simultaneously. Math. Comput. 27, 339–344 (1973)

    Article  MathSciNet  Google Scholar 

  2. Abu-Alshaikh, I., Sahin, A.: Two-point iterative methods for solving nonlinear equations. Appl. Math. Comput. 182, 871–878 (2006)

    MathSciNet  MATH  Google Scholar 

  3. Argyros, I.K., González, D.: Unified majorizing sequences for Traub type multipoint iterative procedures. Numer. Algorithms 64, 549–565 (2013)

    Article  MathSciNet  Google Scholar 

  4. Bini, D.A.: Numerical computation of polynomial zeros by means of Aberth’s method. Numer. Algorithms 13, 179–200 (1996)

    Article  MathSciNet  Google Scholar 

  5. Bini, D.A., Fiorentino, G.: Design, analysis, and implementation of a multiprecision polynomial rootfinder. Numer. Algorithms 23, 127–173 (2000)

    Article  MathSciNet  Google Scholar 

  6. Bini, D., Pan, V.Y.: Polynomial and Matrix Computations: Fundamental Algorithms. Springer, Berlin (2012)

    MATH  Google Scholar 

  7. Cauchy, A.-L.: Exercises de Mathématique, IV Année. de Bure Frères, Paris (1829)

  8. Cholakov, S.I., Vasileva, M.T.: A convergence analysis of a fourth-order method for computing all zeros of a polynomial simultaneously. J. Comput. Appl. Math. 321, 270–283 (2017)

    Article  MathSciNet  Google Scholar 

  9. Dochev, K.: Modified Newton method for simultaneous approximation of all roots of a given algebraic equation (in Bulgarian). Phys. Math. J. Bulg. Acad. Sci. 5, 136–139 (1962)

    Google Scholar 

  10. Durand, E.: Solutions Numériques des Équations Algébriques, Tome. 1: Équations du Type \(F(x) = 0\), Racines d’un Polynôme. Masson, Paris (1960)

  11. Ignatova, B., Kyurkchiev, N., Iliev, A.: Multipoint algorithms arising from optimal in the sense of Kung-Traub iterative procedures for numerical solution of nonlinear equations. Gen. Math. Notes 6(2), 45–79 (2011)

    Google Scholar 

  12. Ivanov, S.I.: A unified semilocal convergence analysis of a family of algorithms for computing all zeros of a polynomial simultaneously. Numer. Algorithms 75, 1193–1204 (2017)

    Article  MathSciNet  Google Scholar 

  13. Kanno, S., Kyurkchiev, N.V., Yamamoto, T.: On some methods for the simultaneous determination of polynomial zeros. Japan J. Indust. Appl. Math. 13, 267–288 (1996)

    Article  MathSciNet  Google Scholar 

  14. Kerner, I.O.: Ein Gesamtschrittverfahren zur Berechnung der Nullstellen von Polynomen. Numer. Math. 8, 290–294 (1966)

    Article  MathSciNet  Google Scholar 

  15. Kyurkchiev, N., Iliev, A.: On some multipoint methods arising from optimal in the sense of Kung-Traub algorithms. Biomath 2(1) Art. ID 1305155, 7 pp (2013)

  16. Kyurkchiev, N.V., Ivanov, R.: On some multi-stage schemes with a superlinear rate of convergence (in Russian). Ann. Univ. Sofia Fac. Math. Mech. 78, 132–136 (1984)

    MATH  Google Scholar 

  17. Kürschák, J.: Über Limesbildung und allgemeine Körpertheorie. J. Reine Angew. Math. 132, 211–253 (1913)

    Article  Google Scholar 

  18. McNamee, J.M.: Numerical Methods for Roots of Polynomials, Part I. Studies in Computational Mathematics, vol. 14. Elsevier, Amsterdam (2007)

  19. McNamee, J.M., Pan, V.Y.: Numerical Methods for Roots of Polynomials Part II. Elsevier, Amsterdam (2013)

    Google Scholar 

  20. Pan, V.Y.: Solving a polynomial equation: some history and recent progress. SIAM Rev. 39, 187–220 (1997)

    Article  MathSciNet  Google Scholar 

  21. Prešić, S.B.: Un procéd é itératif pour la factorisation des polynômes. C. R. Acad. Sci. Paris Sèr. A, 262, 862–863 (1966)

  22. Petković, M.: Point Estimation of Root Finding Methods. Lecture Notes in Mathematics, vol. 1933. Springer, Berlin (2008)

  23. Petković, M.S., Neta, B., Petković, L.D., Džunić, J.: Multipoint methods for solving nonlinear equations. Elsevier, Amsterdam (2013)

    MATH  Google Scholar 

  24. Proinov, P.D.: General convergence theorems for iterative processes and applications to the Weierstrass root-finding method. J. Complex. 33, 118–144 (2016)

    Article  MathSciNet  Google Scholar 

  25. Proinov, P.D.: Relationships between different types of initial conditions for simultaneous root finding methods. Appl. Math. Lett. 52, 102–111 (2016)

    Article  MathSciNet  Google Scholar 

  26. Proinov, P.D., Ivanov, S.I.: Convergence analysis of Sakurai–Torii–Sugiura iterative method for simultaneous approximation of polynomial zeros. J. Comput. Appl. Math. 357, 56–70 (2019)

    Article  MathSciNet  Google Scholar 

  27. Proinov, P.D., Petkova, M.D.: A new semilocal convergence theorem for the Weierstrass method for finding zeros of a polynomial simultaneously. J. Complexity 30, 366–380 (2014)

    Article  MathSciNet  Google Scholar 

  28. Proinov, P.D., Petkova, M.D.: Convergence of the two-point Weierstrass root-finding method. Japan J. Indust. Appl. Math. 31, 279–292 (2014)

    Article  MathSciNet  Google Scholar 

  29. Proinov, P.D., M. T. Vasileva, M.T.: On the convergence of high-order Ehrlich-type iterative methods for approximating all zeros of a polynomial simultaneously, J. Inequal. Appl. 2015, Articl ID 336 (2015)

  30. Proinov, P.D., Vasileva, M.T.: On a family of Weierstrass-type root-finding methods with accelerated convergence. Appl. Math. Comput. 273, 957–968 (2016)

    MathSciNet  MATH  Google Scholar 

  31. Sendov, Bl., Andreev, A., Kyurkchiev, N. Numerical Solution of Polynomial Equations. Handb. Numer. Anal. 3, 625–778 (1994)

  32. Smale, S.: The fundamental theorem of algebra and complexity theory. Bull. Am. Math. Soc. 4, 1–36 (1981)

    Article  MathSciNet  Google Scholar 

  33. Stewart, G.W.: On the convergence of multipoint iterations. Numer. Math. 68, 143–147 (1994)

    Article  MathSciNet  Google Scholar 

  34. Tričković, S.B., Petković, M.S.: Multipoint methods for the determination of multiple zeros of a polynomial. Novi Sad J. Math. 29, 221–233 (1999)

    MathSciNet  MATH  Google Scholar 

  35. Weierstrass, K.: Neuer Beweis des Satzes, dass jede ganze rationale Function einer Veränderlichen dargestellt werden kann als ein Product aus linearen Functionen derselben Veränderlichen. Sitzungsber. Königl. Preuss. Akad. Wiss. Berlin II, pp. 1085–1101 (1891)

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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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  1. Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, BG, 4000, Plovdiv, Bulgaria

    Petko D. Proinov & Milena D. Petkova

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  1. Petko D. Proinov
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  2. Milena D. Petkova
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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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  • DOI: https://doi.org/10.1007/s00009-020-01545-z

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