Journal of Mathematical Chemistry

, Volume 56, Issue 7, pp 2117–2131 | Cite as

Ball convergence of a sixth-order Newton-like method based on means under weak conditions

  • Á. A. Magreñán
  • I. K. Argyros
  • J. J. RainerEmail author
  • J. A. Sicilia
Original Paper


We study the local convergence of a Newton-like method of convergence order six to approximate a locally unique solution of a nonlinear equation. Earlier studies show convergence under hypotheses on the seventh derivative or even higher. The convergence in this study is shown under hypotheses on the first derivative although only the first derivative appears in these methods. Hence, the applicability of the method is expanded. Finally, we solve the problem of the fractional conversion in the ammonia process showing the applicability of the theoretical results.


Newton-like method Local convergence Stolarky means Gini means Efficiency index 

Mathematics Subject Classification

65D10 65D99 65G99 90C30 



The research of this author was supported by Universidad Internacional de La Rioja (UNIR,, under the Plan Propio de Investigación, Desarrollo e Innovación 4 [2017–2019]. Research group: Modelación matemática aplicada a la ingeniería(MOMAIN), by Programa de Apoyo a la investigación de la fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 19374/PI714 and by Ministerio de Ciencia y Tecnología MTM2014-52016-C2-01-P.


  1. 1.
    S. Amat, S. Busquier, S. Plaza, Dynamics of the King and Jarratt iterations. Aequ. Math. 69(3), 212–223 (2005)CrossRefGoogle Scholar
  2. 2.
    S. Amat, M.A. Hernández, N. Romero, A modified Chebyshev’s iterative method with at least sixth order of convergence. Appl. Math. Comput. 206(1), 164–174 (2008)Google Scholar
  3. 3.
    S. Amat, S. Busquier, S. Plaza, Chaotic dynamics of a third-order Newton-type method. J. Math. Anal. Appl. 366(1), 24–32 (2010)CrossRefGoogle Scholar
  4. 4.
    I.K. Argyros, Convergence and Application of Newton-Type Iterations (Springer, Berlin, 2008)Google Scholar
  5. 5.
    I.K. Argyros, S. Hilout, Numerical Methods in Nonlinear Analysis (World Scientific, Singapore, 2013)Google Scholar
  6. 6.
    I.K. Argyros, Á.A. Magreñán, Iterative Methods and Their Dynamics with Applications (CRC Press, London, 2017)CrossRefGoogle Scholar
  7. 7.
    I.K. Argyros, S. George, Á.A. Magreñán, Local convergence for multi-point-parametric Chebyshev–Halley-type methods of high convergence order. J. Comput. Appl. Math. 282, 215–224 (2015)CrossRefGoogle Scholar
  8. 8.
    D.D. Bruns, J.E. Bailey, Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chem. Eng. Sci. 32, 257–264 (1977)CrossRefGoogle Scholar
  9. 9.
    D. Budzko, A. Cordero, J.R. Torregrosa, A new family of iterative methods widening areas of convergence. Appl. Math. Comput. 252, 405–417 (2015)Google Scholar
  10. 10.
    V. Candela, A. Marquina, Recurrence relations for rational cubic methods I: the Halley method. Computing 44, 169–184 (1990)CrossRefGoogle Scholar
  11. 11.
    V. Candela, A. Marquina, Recurrence relations for rational cubic methods II: the Chebyshev method. Computing 45(4), 355–367 (1990)CrossRefGoogle Scholar
  12. 12.
    C. Chun, Some improvements of Jarratt’s method with sixth-order convergence. Appl. Math. Comput. 190(2), 1432–1437 (1990)Google Scholar
  13. 13.
    A. Cordero, J.R. Torregrosa, Á.A. Magreñán, C. Quemada, Stability study of eighth-order iterative methods for solving nonlinear equations. J. Comput. Appl. Math. 291(9960), 348–357 (2016)CrossRefGoogle Scholar
  14. 14.
    J.A. Ezquerro, M.A. Hernández, Recurrence relations for Chebyshev-type methods. Appl. Math. Optim. 41(2), 227–236 (2000)CrossRefGoogle Scholar
  15. 15.
    J.A. Ezquerro, M.A. Hernández, On the R-order of the Halley method. J. Math. Anal. Appl. 303, 591–601 (2005)CrossRefGoogle Scholar
  16. 16.
    J.A. Ezquerro, M.A. Hernández, New iterations of R-order four with reduced computational cost. BIT Numer. Math. 49, 325–342 (2009)CrossRefGoogle Scholar
  17. 17.
    J.M. Gutiérrez, M.A. Hernández, Recurrence relations for the super-Halley method. Comput. Math. Appl. 36(7), 1–8 (1998)CrossRefGoogle Scholar
  18. 18.
    J.M. Gutiérrez, Á.A. Magreñán, J.L. Varona, The “gauss-seidelization” of iterative methods for solving nonlinear equations in the complex plane. Appl. Math. Comput. 218(6), 2467–2479 (2011)Google Scholar
  19. 19.
    D. Herceg, D.J. Herceg, Sixth order modifications on Newton’s method based on Stolarsky and Gini means. J. Comput Appl. Math. 267, 244–253 (2014)CrossRefGoogle Scholar
  20. 20.
    M.A. Hernández, Chebyshev’s approximation algorithms and applications. Comput. Math. Appl. 41(3–4), 433–455 (2001)CrossRefGoogle Scholar
  21. 21.
    M.A. Hernández, M.A. Salanova, Sufficient conditions for semilocal convergence of a fourth order multipoint iterative method for solving equations in Banach spaces. Southwest J. Pure Appl. Math 1, 29–40 (1999)Google Scholar
  22. 22.
    P. Jarratt, Some fourth order multipoint methods for solving equations. Math. Comput. 20(95), 434–437 (1966)CrossRefGoogle Scholar
  23. 23.
    J. Kou, Y. Li, An improvement of the Jarratt method. Appl. Math. Comput. 189, 1816–1821 (2007)Google Scholar
  24. 24.
    J. Kou, X. Wang, Semilocal convergence of a modified multi-point Jarratt method in Banach spaces under general continuity conditions. Numer. Algorithms 60, 369–390 (2012)CrossRefGoogle Scholar
  25. 25.
    D. Li, P. Liu, J. Kou, An improvement of the Chebyshev–Halley methods free from second derivative. Appl. Math. Comput. 235, 221–225 (2014)CrossRefGoogle Scholar
  26. 26.
    Á.A. Magreñán, Different anomalies in a Jarratt family of iterative root-finding methods. Appl. Math. Comput. 233, 29–38 (2014)Google Scholar
  27. 27.
    Á.A. Magreñán, A new tool to study real dynamics: the convergence plane. Appl. Math. Comput. 248, 215–224 (2014)Google Scholar
  28. 28.
    Á.A. Magreñán, J.M. Gutiérrez, Real dynamics for damped Newton’s method applied to cubic polynomials. J. Comput. Appl. Math. 275, 527–538 (2015)CrossRefGoogle Scholar
  29. 29.
    Á.A. Magreñán, A. Coredero, J.M. Gutiérrez, J.R. Torregrosa, Real qualitative behavior of a fourth-order family of iterative methods by using the convergence plane. Math. Comput. Simul. 105, 49–61 (2014)CrossRefGoogle Scholar
  30. 30.
    S.K. Parhi, D.K. Gupta, Recurrence relations for a Newton-like method in Banach spaces. J. Comput. Appl. Math. 206(2), 873–887 (2007)CrossRefGoogle Scholar
  31. 31.
    L.B. Rall, Computational Solution of Nonlinear Operator Equations (Robert E. Krieger, New York, 1979)Google Scholar
  32. 32.
    H. Ren, Q. Wu, W. Bi, New variants of Jarratt method with sixth-order convergence. Numer. Algorithms 52(4), 585–603 (2009)CrossRefGoogle Scholar
  33. 33.
    W.C. Rheinboldt, An adaptive continuation process for solving systems of nonlinear equations. Pol. Acad. Sci. Banach Ctr. Publ. 3, 129–142 (1978)Google Scholar
  34. 34.
    J.F. Traub, Iterative Methods for the Solution of Equations. Prentice-Hall Series in Automatic Computation (Prentice-Hall, Englewood Cliffs, 1964)Google Scholar
  35. 35.
    X. Wang, J. Kou, C. Gu, Semilocal convergence of a sixth-order Jarratt method in Banach spaces. Numer. Algorithms 57, 441–456 (2011)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Á. A. Magreñán
    • 1
  • I. K. Argyros
    • 2
  • J. J. Rainer
    • 1
    Email author
  • J. A. Sicilia
    • 1
  1. 1.Escuela de IngenieríaUniversidad Internacional de La RiojaLogroñoSpain
  2. 2.Department of Mathematics SciencesCameron UniversityLawtonUSA

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