Journal of Mathematical Chemistry

, Volume 55, Issue 7, pp 1505–1520 | Cite as

New improved convergence analysis for Newton-like methods with applications

  • Á. Alberto Magreñán
  • Ioannis K. Argyros
  • Juan Antonio Sicilia
Original Paper


We present a new semilocal convergence analysis for Newton-like methods using restricted convergence domains in a Banach space setting. The main goal of this study is to expand the applicability of these methods in cases not covered in earlier studies. The advantages of our approach include, under the same computational cost as previous studies, a more precise convergence analysis under the same computational cost on the Lipschitz constants involved. Numerical studies including a chemical application are also provided in this study.


Newton-type method Banach space Majorizing sequence Restricted domains Local convergence Semilocal convergence 

Mathematics Subject Classification

65H10 65G99 65B05 65N30 47J25 47J05 



This research was supported by Universidad Internacional de La Rioja (UNIR,, under the Plan Propio de Investigación, Desarrollo e Innovación [2015–2017]. Research group: Modelación matemática aplicada a la ingeniería (MOMAIN), by the the grant SENECA 19374/PI/14 and by Ministerio de Ciencia y Tecnología MTM2014-52016-C2-\(\{01\}\)-P.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    S. Amat, S. Busquier, M. Negra, Adaptive approximation of nonlinear operators. Numer. Funct. Anal. Optim. 25, 397–405 (2004)Google Scholar
  2. 2.
    I.K. Argyros, in Computational Theory of Iterative Methods, Series: Studies in Computational Mathematics, ed. by C.K. Chui, L. Wuytack (Elsevier Publ. Co., New York, 2007)Google Scholar
  3. 3.
    I.K. Argyros, S. Hilout, Weaker conditions for the convergence of Newton’s method. J. Complex. AMS 28, 364–387 (2012)Google Scholar
  4. 4.
    I.K. Argyros, Y.J. Cho, S. Hilout, Numerical Method for Equations and Its Applications (CRC Press, New York, 2012)Google Scholar
  5. 5.
    W.E. Bosarge, P.L. Falb, A multipoint method of third order. J. Optim. Thory Appl. 4, 156–166 (1969)Google Scholar
  6. 6.
    W.E. Bosarge, P.L. Falb, Infinite dimensional multipoint methods and the solution of two point boundary value problems. Numer. Math. 14, 264–286 (1970)Google Scholar
  7. 7.
    E. Cătinaş, On some iterative methods for solving nonlinear equations. ANTA 23(1), 47–53 (1994)Google Scholar
  8. 8.
    E. Cătinaş, The inexact, inexact perturbed, and quasi-Newton methods are equivalent models. Math. Comput. 74(249), 291–301 (2005)Google Scholar
  9. 9.
    S. Chandrasekhar, Radiative Transfer (Dover Publ, New York, 1960)Google Scholar
  10. 10.
    J.E. Dennis, Toward a unified convergence theory for Newtonlike methods, in Nonlinear Functional Analysis and Applications, ed. by L.B. Rall (Academic Press, New York, 1971), pp. 425–472Google Scholar
  11. 11.
    J.A. Ezquerro, M.A. Hernández, M.J. Rubio, Secant-like methods for solving nonlinear integral equations of the Hammerstein type. J. Comput. Appl. Math. 115, 245–254 (2000)Google Scholar
  12. 12.
    J.A. Ezquerro, J.M. Gutiérrez, M.A. Hernández, N. Romero, M.J. Rubio, The Newton method: from Newton to Kantorovich (Spanish). Gac. R. Soc. Mat. Esp. 13(1), 53–76 (2010)Google Scholar
  13. 13.
    V.B. Gopalan, J.D. Seader, Application of interval Newton’s method to chemical engineering problems. Reliab. Comput. 1(3), 215–223 (1995)Google Scholar
  14. 14.
    W.B. Gragg, R.A. Tapia, Optimal error bounds for the Newton–Kantorovich theorem. SIAM J. Numer. Anal. 11, 10–13 (1974)Google Scholar
  15. 15.
    L.V. Kantorovich, G.P. Akilov, Functional Analysis (Pergamon Press, Oxford, 1982)Google Scholar
  16. 16.
    H.J. Kornstaedt, Ein allgemeiner Konvergenzstaz fr verschrfte Newton Verfahrem, ISNM, vol. 28 (Birkhaser, Basel, 1975), pp. 53–69Google Scholar
  17. 17.
    P. Laasonen, Ein überquadratich konvergenter iterativer algorithmus. Ann. Acad. Sci. Fenn. Ser. I(450), 1–10 (1969)Google Scholar
  18. 18.
    A.A. Magreñán, I.K. Argyros, Improved convergence analysis for newton-like methods. Numer. Algorithms, 1–23 (2015)Google Scholar
  19. 19.
    G.J. Miel, The Kantorovich theorem with optimal error bounds. Am. Math. Mon. 86, 212–215 (1979)Google Scholar
  20. 20.
    G.J. Miel, An updated version of the Kantorovich theorem for Newton’s method. Computing 27, 237–244 (1981)Google Scholar
  21. 21.
    L.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic press, New York, 1970)Google Scholar
  22. 22.
    F.A. Potra, On a modified secant method. Rev. Anal. Numer. Theory Approx. 8, 203–214 (1979)Google Scholar
  23. 23.
    F.A. Potra, An application of the induction method of V. Ptak to the study of regula falsi. Apl. Mat. 26, 111–120 (1981)Google Scholar
  24. 24.
    F.A. Potra, On the Convergence of a Class of Newton-Like Methods. Iterative Solution of Nonlinear Systems of Equations, Lecture Notes in Math (Springer, Berlin, 1982), pp. 125–137Google Scholar
  25. 25.
    F.A. Potra, On the a posteriori error estimates for Newton’s method. Beitr. Zur Numer. Math. 12, 125–138 (1984)Google Scholar
  26. 26.
    F.A. Potra, Sharp error bounds for a class of Newton-like methods. Lib. Math. 5, 71–84 (1985)Google Scholar
  27. 27.
    F.A. Potra, V. Pták, Nondiscrete Induction and Iterative Processes, vol. 103, Research Notes in Mathematics (Pitman (Advanced Publishing Program), Boston, 1984)Google Scholar
  28. 28.
    P.D. Proinov, New general convergence theory for iterative processes and its applications to Newton–Kantorovich type theorems. J. Complex. 26, 3–42 (2010)Google Scholar
  29. 29.
    W.C. Rheinboldt, An Adaptive Continuation Process for Solving Systems of Nonlinear Equations, vol. 3 (Polish Academy of Science, Warsaw, 1977)Google Scholar
  30. 30.
    J.W. Schmidt, Untere Fehlerschranken fun Regula-Falsi Verhafren. Period Hung. 9, 241–247 (1978)Google Scholar
  31. 31.
    M. Shacham, An improved memory method for the solution of a nonlinear equation. Chem. Eng. Sci. 44, 1495–1501 (1989)Google Scholar
  32. 32.
    J.F. Traub, Iterative Method for Solutions of Equations (Prentice-Hall, New Jersey, 1964)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  • Á. Alberto Magreñán
    • 1
  • Ioannis K. Argyros
    • 2
  • Juan Antonio Sicilia
    • 1
  1. 1.Universidad Internacional de La Rioja (UNIR)LogroñoSpain
  2. 2.Department of Mathematics Sciences LawtonCameron UniversityLawtonUSA

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