Skip to main content
SpringerLink
Log in

On the convergence of Schröder’s method for the simultaneous computation of polynomial zeros of unknown multiplicity

  • Published:
Calcolo Aims and scope Submit manuscript

Abstract

In this paper, we establish a general theorem for iteration functions in a cone normed space over \({{\mathbb {R}}}^n\). Using this theorem together with a general convergence theorem of Proinov (J Complex 33:118–144, 2016), we obtain a local convergence theorem with a priori and a posteriori error estimates as well as a theorem under computationally verifiable initial conditions for the Schröder’s iterative method considered as a method for simultaneous computation of polynomial zeros of unknown multiplicity. Numerical examples which demonstrate the convergence properties of the proposed method are also provided.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. We do not consider the case \(s = 1\) because it is trivial.

References

  1. Batra, P.: Simultaneous point estimates for Newton’s method. BIT 42, 467–476 (2002). doi:10.1023/A:1021968924257

    MathSciNet  MATH  Google Scholar 

  2. Farmer, M., Loizou, G.: An algorithm for the total, partial, factorization of a polynomial. Math. Proc. Camb. Philos. Soc. 82, 427–437 (1977). doi:10.1017/S0305004100054098

    Article  MathSciNet  MATH  Google Scholar 

  3. Gargantini, I.: Parallel square-root iterations for multiple roots. Comput. Math. Appl. 6, 279–288 (1980). doi:10.1016/0898-1221(80)90035-8

    Article  MathSciNet  MATH  Google Scholar 

  4. Iliev, A., Iliev, I.: On a generalization of Weierstrass–Dochev method for simultaneous extraction of all roots of polynomials over an arbitrary Chebyshev system. C. R. Acad. Bulg. Sci. 54, 31–36 (2001)

    MathSciNet  MATH  Google Scholar 

  5. Iliev, A., Semerdzhiev, Kh.: Some generalizations of the Chebyshev method for simultaneous determination of all roots of polynomial equations. Comput. Math. Math. Phys. 39, 1384–1391 (1999)

  6. Ivanov, S.I.: On the convergence of Chebyshev’s method for multiple polynomial zeros. Results Math. 69(1), 93–103 (2016). doi:10.1007/s00025-015-0490-y

    Article  MathSciNet  MATH  Google Scholar 

  7. Ivanov, S.I.: A unified semilocal convergence analysis of a family of iterative algorithms for computing all zeros of a polynomial simultaneously. Numer. Algorithm (2016). doi:10.1007/s11075-016-0237-1

    Google Scholar 

  8. Kalantari, B., Kalantari, I., Zaare-Nahandi, R.: A basic family of iteration functions for polynomial root finding and its characterizations. J. Comput. Appl. Math. 80, 209–226 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kjurkchiev, N.V., Andreev, A.: On Halley-like algorithms with high order of convergence for simultaneous approximation of multiple roots of polynomials. C. R. Acad. Bulg. Sci. 43(9), 29–32 (1990)

    MathSciNet  Google Scholar 

  10. Kyncheva, V.K., Yotov, V.V., Ivanov, S.I.: Convergence of Newton, Halley and Chebyshev iterative methods as methods for simultaneous determination of multiple polynomial zeros. Appl. Numer. Math. 112, 146–154 (2017). doi:10.1016/j.apnum.2016.10.013

    Article  MathSciNet  MATH  Google Scholar 

  11. Kyurkchiev, N.V.: Initial approximations and root finding methods. In: Mathematical Research, vol. 104. Wiley, Berlin (1998)

  12. McNamee, J.M.: Numerical methods for roots of polynomials, part I. In: Chui, C.K., Wuytack, L. (eds.) Studies in Computational Mathematics, vol. 14. Elsevier, Amsterdam (2007)

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Petković, M.: Point estimation of root finding methods. In: Lecture Notes in Mathematics, vol. 1933. Springer, Berlin (2008)

  15. Petković, M., Milovanović, G., Stefanović, L.: Some higher-order methods for the simultaneous approximation of multiple polynomial zeros. Comput. Math. Appl. 12, 951–962 (1986). doi:10.1016/0898-1221(86)90020-9

    Article  MathSciNet  MATH  Google Scholar 

  16. Petković, M., Milošević, D.: Higher order methods for the inclusion of multiple zeros of polynomials. Reliab. Comput. 15, 91–108 (2011)

    MathSciNet  Google Scholar 

  17. Proinov, P.D.: General local convergence theory for a class of iterative processes and its applications to Newton’s method. J. Complex. 25(1), 38–62 (2009). doi:10.1016/j.jco.2008.05.006

    Article  MathSciNet  MATH  Google Scholar 

  18. Proinov, P.D.: New general convergence theory for iterative processes and its applications to Newton–Kantorovich type theorems. J. Complex. 26(1), 3–42 (2010). doi:10.1016/j.jco.2009.05.001

    Article  MathSciNet  MATH  Google Scholar 

  19. Proinov, P.D.: A unified theory of cone metric spaces and its applications to the fixed point theory. Fixed Point Theory Appl. 2013, 103 (2013). doi:10.1186/1687-1812-2013-103

    Article  MathSciNet  MATH  Google Scholar 

  20. 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  MATH  Google Scholar 

  21. Proinov, P.D.: A general semilocal convergence theorem for simultaneous methods for polynomial zeros and its applications to Ehrlich’s and Dochev-Byrnev’s methods. Appl. Math. Comput. 284, 102–114 (2016)

    MathSciNet  Google Scholar 

  22. Proinov, P.D.: On the local convergence of Ehrlich method for numerical computation of polynomial zeros. Calcolo 53(3), 413–426 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Proinov, P.D., Cholakov, S.I.: Convergence of Chebyshev-like method for simultaneous approximation of multiple polynomial zeros. C. R. Acad. Bulg. Sci. 67(7), 907–918 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Proinov, P.D., Cholakov, S.I.: Semilocal convergence of Chebyshev-like root-finding method for simultaneous approximation of polynomial zeros. Appl. Math. Comput. 236, 669–682 (2014). doi:10.1016/j.amc.2014.03.092

    MathSciNet  MATH  Google Scholar 

  26. Proinov, P.D., Ivanov, S.I.: Convergence of Schröder’s method for polynomial zeros of unknown multiplicity. C. R. Acad. Bulg. Sci. 66(8), 1073–1080 (2013)

    MATH  Google Scholar 

  27. Proinov, P.D., Ivanov, S.I.: On the convergence of Halley’s method for multiple polynomial zeros. Mediterr. J. Math. 12, 555–572 (2015). doi:10.1007/s00009-014-0400-7

    Article  MathSciNet  MATH  Google Scholar 

  28. Proinov, P.D., Ivanov, S.I.: On the convergence of Halley’s method for simultaneous computation of polynomial zeros. J. Numer. Math. 23(4), 379–394 (2015). doi:10.1515/jnma-2015-0026

    Article  MathSciNet  MATH  Google Scholar 

  29. Proinov, P.D., Petkova, M.D.: Convergence of the two-point Weierstrass root-finding method. Jpn. J. Ind. Appl. Math. 31(2), 279–292 (2014). doi:10.1007/s13160-014-0138-4

    Article  MathSciNet  MATH  Google Scholar 

  30. Proinov, P.D., Vasileva, M.T.: On the convergence of high-order Ehrlich-type iterative methods for approximating all zeros of a polynomial simultaneously. J. Inequal. Appl. 2015:336 (2015). http://link.springer.com/article/10.1186

  31. 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  Google Scholar 

  32. Schröder, E.: Über unendlich viele Algorithmen zur Auflösung der Gleichungen. Math. Ann. 2, 317–365 (1870). doi:10.1007/BF01444024

    Article  MathSciNet  Google Scholar 

  33. Sekuloski, R.: Generalization of Prešic’s iterative method for factorization of a polynomial. Mat Vesnik 9, 257–264 (1972). (in Serbo-Croatian)

    MathSciNet  MATH  Google Scholar 

  34. Semerdzhiev, Kh.: A method for simultaneous finding all roots of algebraic equations with given multiplicity. C. R. Acad. Bulg. Sci. 35, 1057–1060 (1982). (in Bulgarian)

  35. Sendov, Bl., Andreev, A., Kjurkchiev, N.: Numerical solution of polynomial equations. In: Ciarlet, P., Lions, J. (eds.) Handbook of Numerical Analysis, vol. III, pp. 625–778. Elsevier, Amsterdam (1994)

  36. Sun, F., Kosmol, P.: A new simultaneous method of fourth order for finding complex zeros in circular interval arithmetic. J. Comput. Appl. Math. 130, 293–307 (2001). doi:10.1016/S0377-0427(99)00375-1

    Article  MathSciNet  MATH  Google Scholar 

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

Download references

Acknowledgements

The authors express their gratitude to the anonymous referees for the constructive comments and suggestions which have improved the quality of the paper. This research was supported by Project SP15-FFIT-004 of Plovdiv University.

Author information

Authors and Affiliations

  1. Faculty of Physics, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000, Plovdiv, Bulgaria

    Veselina K. Kyncheva, Viktor V. Yotov & Stoil I. Ivanov

Authors
  1. Veselina K. Kyncheva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Viktor V. Yotov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stoil I. Ivanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stoil I. Ivanov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kyncheva, V.K., Yotov, V.V. & Ivanov, S.I. On the convergence of Schröder’s method for the simultaneous computation of polynomial zeros of unknown multiplicity. Calcolo 54, 1199–1212 (2017). https://doi.org/10.1007/s10092-017-0225-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10092-017-0225-4

Keywords

Mathematics Subject Classification

Search

Navigation