Skip to main content
SpringerLink
Log in

Nonstationary vs. stationary iterative processes

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, we define s-nonstationary iterative process and obtain its properties. We prove, that for any one-point iterative process without memory, there exists an s-nonstationary process of the same order, but of higher efficiency by the criteria of Traub and Ostrowski. We supply constructions of s-nonstationary processes for Newton’s, Halley’s, and Chebyshev’s methods, obtain their properties and, for some of them, also their geometric interpretation. The algorithms we present can be transformed into computer programs in a straightforward manner. Additionally, we illustrate numerical examples, as demonstrations for the methods we present.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Amat, S., Busquier, S.: Third-order iterative methods under Kantorovich conditions. J. Math. Anal. Appl. 336(1), 243–261 (2007)

    Article  MathSciNet  Google Scholar 

  2. Amat, S., Busquier, S., Plaza, S.: Dynamics of a family of third-order iterative methods that do not require using second derivatives. Appl. Math. Comput. 154, 735–746 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Cordero, A., Torregrosa, J.R.: A class of Steffensen type methods with optimal order of convergence. Appl. Math. Comput. 217, 7653–7659 (2011)

    MathSciNet  MATH  Google Scholar 

  4. Grau, M., Diaz-Barrero, D.-B.J.L.: An improvement to Ostrowski root-finding method. Appl. Math. Comput. 173, 450–456 (2006)

    MathSciNet  MATH  Google Scholar 

  5. Gutierrez, J.M., Hernandez, M.A.: An acceleration of Newton’s method: super-Halley method. Appl. Math. Comput. 117, 223–239 (2001)

    MathSciNet  MATH  Google Scholar 

  6. Halley, E.: A new exact and easy method for finding the roots of equations generally and without any previous reduction. Phil. Roy. Soc. London 18, 136–147 (1964)

    Google Scholar 

  7. Householder, A.S.: The Numerical Treatment of a Single Nonlinear Equation. McGraw-Hill, New York (1970)

    MATH  Google Scholar 

  8. Jain, P., Sethi, K.: Aitken type methods with high efficiency. Tran. A. Razmadze Math. Instit. 172, 223–237 (2018)

    Article  MathSciNet  Google Scholar 

  9. Jain, P., Chand, P.B., Sethi, K.: Efficient numerical methods of Aitken type and their dynamics. Eurasian Math. J. 9(3), 58–72 (2018)

    Article  MathSciNet  Google Scholar 

  10. Kantorovich, L.V.: Functional analysis and applied mathematics [Russian]. Uspehi Mathem. Nauk 3, 89–185 (1948)

    MATH  Google Scholar 

  11. Kantorovich, L.V.: On Newton’s method for functional analysis [Russian]. Doklady Akad. Nauk SSSR 59, 1237–1240 (1948)

    MathSciNet  MATH  Google Scholar 

  12. Kogan, T.I.: Generalization of the method of chords for an algebraic or transcendental equation. Tashkent. Gos. Univ. Naučn. Trudy (Russian) Vyp. 276, 53–55 (1966)

    MathSciNet  Google Scholar 

  13. Kogan, T.I.: Construction of iterative processes of high orders. ž.Vyčisl. Mat. i Mat. Fiz. (Russian) 7, 423–424 (1967)

    Google Scholar 

  14. Kogan, T.I.: On one modification of the method of chords for nonlinear functional equation. Tashkent Sib. Mat. Z. 19(3), 710 (1978)

    Google Scholar 

  15. Kogan, T.I., Sapir, L., Sapir, A.: A nonstationary iterative second-order method for solving nonlinear equations. Appl. Math. Comput. 188, 75–82 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Kogan, T.I., Sapir, L., Sapir, A., Sapir, A.: The Fibonacci family of iterative processes for solving nonlinear equations. Appl. Numer. Math. 110, 148–158 (2016)

    Article  MathSciNet  Google Scholar 

  17. Kogan, T.I., Sapir, L., Sapir, A., Sapir, A.: To the question of efficiency of iterative methods. Appl. Math. Lett. 66, 40–46 (2017)

    Article  MathSciNet  Google Scholar 

  18. Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. ACM 21, 643–651 (1974)

    Article  MathSciNet  Google Scholar 

  19. Kumari, Ch., Parida, P.K.: Local convergence analysis for Chebyshev’s method. J. Appl. Math. Comput. 59, 405–421 (2019). https://doi.org/10.1007/s12190-018-1185-9. Springer

    Article  MathSciNet  MATH  Google Scholar 

  20. McNamee, J.M., Pan, V.Y.: Efficient polynomial root-finders: a survey and new record efficiency estimates. Comp. Math. with Appl. 63, 239–254 (2012)

    Article  Google Scholar 

  21. McNamee, J.M., Pan, V.Y.: Numerical methods for roots of polynomials – Part II. Studies Comp. Math. 16(ch. 7), 1–138 (2013)

    MathSciNet  Google Scholar 

  22. Al Din Ide, N.: A nonstationary Halley’s iterative method by using divided differences formula. Appl. Math. 3, 169–171 (2012). SCIRP

    Article  MathSciNet  Google Scholar 

  23. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution on Nonlinear Equations in Several Variables. Academic Press, New York (1970)

    MATH  Google Scholar 

  24. Ostrowski, A.M.: Solution of Equations and Systems of Equations. Academic Press, New York (1960)

    MATH  Google Scholar 

  25. Ostrowski, A.M.: Solution of Equations in Euclidean and Banach Spaces. Academic Press, New York (1973)

    MATH  Google Scholar 

  26. Phiri, P.A., Makinde, O.D.: A new derivative-free method for solving nonlinear equations. Int. J. Physical Sci. 5(7), 935–939 (2018)

    Google Scholar 

  27. Hernández, M.A., Rubio, M.J.: A new type of recurrence relations for the secant method. Int. J. Comput. Math. 72(4), 477–490 (1999)

    Article  MathSciNet  Google Scholar 

  28. Sergeev, A.S.: On the method of chords [Russian]. Sib. Mat. Z. 2, 282–289 (1961)

    Google Scholar 

  29. Solaiman, O.S., Hashim, I.: Two new efficient sixth order iterative methods for solving nonlinear equations, J King Saud Univ. https://doi.org/10.1016/j.jksus.2018.03.021 (2018)

  30. Solaiman, O.S., Karim, S.A.A., Hashim, I: Optimal fourth and eighth order of convergence derivative-free modifications of King’s method. J. King Saud Univ. https://doi.org/10.1016/j.jksus.2018.12.001 (2018)

  31. Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, Engelwood Cliffs (1964)

    MATH  Google Scholar 

  32. Wall, D.: Fourth-order iterative method free from second derivative for solving nonlinear equations. Math. Comp. 10, 167–168 (1956)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departments of Mathematics and Computer Science, Ben-Gurion University, Beer-Sheva, Israel

    Luba Sapir

  2. Amit Educational Network, Beer-Sheva, Israel

    Tamara Kogan

  3. Department of Computer Science, Ben-Gurion University, Beer-Sheva, Israel

    Ariel Sapir

  4. Department of Computer Science, Sapir Academic College, Sha’ar HaNegev, Yehudah, Israel

    Amir Sapir

  5. The Center for Advanced Studies in Mathematics, Ben-Gurion University, Beer-Sheva, Israel

    Amir Sapir

Authors
  1. Luba Sapir
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tamara Kogan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ariel Sapir
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Amir Sapir
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luba Sapir.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sapir, L., Kogan, T., Sapir, A. et al. Nonstationary vs. stationary iterative processes. Numer Algor 86, 515–535 (2021). https://doi.org/10.1007/s11075-020-00899-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-020-00899-5

Keywords

Search

Navigation