Mediterranean Journal of Mathematics

, Volume 13, Issue 6, pp 4109–4128 | Cite as

On a Moser–Steffensen Type Method for Nonlinear Systems of Equations

  • S. Amat
  • M. Grau-Sanchez
  • M. A. Hernández-Verón
  • M. J. Rubio


This paper is devoted to the construction and analysis of a Moser–Steffensen iterative scheme. The method has quadratic convergence without evaluating any derivative nor inverse operator. We present a complete study of the order of convergence for systems of equations, hypotheses ensuring the local convergence, and finally, we focus our attention to its numerical behavior. The conclusion is that the method improves the applicability of both Newton and Steffensen methods having the same order of convergence.


Steffensen’s method Moser’s strategy Recurrence relations Local convergence Numerical analysis 

Mathematics Subject Classification

65J1 47H17 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alarcón V., Amat S., Busquier S., López D.J.: A Steffensen’s type method in Banach spaces with applications on boundary-value problems. J. Comput. Appl. Math. 216, 243–250 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Amat S., Ezquerro J.A., Hernández M.A.: Approximation of inverse operators by a new family of high-order iterative methods. Numer. Linear Algebra Appl. 21, 62–644 (2014)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Amat S., Hernández M.A., Rubio M.J.: Improving the applicability of the secant method to solve nonlinear systems of equations. Appl. Math. Comput. 247, 741–752 (2014)MathSciNetMATHGoogle Scholar
  4. 4.
    Argyros I.K.: On Ulm’s method using divided differences of order one. Numer. Algorithms 52, 295–320 (2009)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Argyros I.K.: A new convergence theorem for Steffensen’s method on Banach spaces and applications. Southwest J. Pure Appl. Math. 1, 23–29 (1997)MATHGoogle Scholar
  6. 6.
    Argyros I.K., Magreñán Á.A.: On the convergence of an optimal fourth-order family of methods and its dynamics. Appl. Math. Comput. 252, 336–346 (2015)MathSciNetMATHGoogle Scholar
  7. 7.
    Chicharro F., Cordero A., Gutiérrez J.M., Torregrosa J.R.: Complex dynamics of derivative-free methods for nonlinear equations. Appl. Math. Comput. 219, 7023–7035 (2013)MathSciNetMATHGoogle Scholar
  8. 8.
    Cordero A., Hueso J.L., Martínez E., Torregrosa J.R.: Steffensen type methods for solving nonlinear equations. J. Comput. Appl. Math. 236(12), 3058–3064 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cordero, A., Soleymani, F., Torregrosa, J.R., Shateyi, S.: Basins of attraction for various Steffensen-type methods. J. Appl. Math. vol. 2014, Article ID 539707, p. 17 (2014). doi: 10.1155/2014/539707
  10. 10.
    Dennis, JE., Schnabel, R.B.: Numerical methods for unconstrained optimization and nonlinear equations. SIAM, Philadelphia (1996)Google Scholar
  11. 11.
    Ezquerro J.A., Hernández M.A., Romero N., Velasco A.I.: On Steffensen’s method on Banach spaces. J. Comput. Appl. Math. 249, 9–23 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Golub, G.H., Van Loan, C.F.: Matrix computations. JHU Press (1996)Google Scholar
  13. 13.
    Grau-Sánchez M., Grau A., Noguera M.: Frozen divided differences scheme for solving systems of nonlinear equations. J. Comput. Appl. Math. 235, 1739–1743 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Grau-Sánchez M., Noguera M, Diaz J.L.: On the local convergence of a family of two-step iterative methods for solving nonlinear equations. J. Comput. Appl. Math. 255, 753–764 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hairer, E., Wanner, G.: Solving ordinary differential equations II: stiff and differential algebraic problems. Springer-Verlag, Berlin (1991)Google Scholar
  16. 16.
    Hald O.H.: On a Newton-Moser type method. Numer. Math. 23, 411–425 (1975)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Moser, J.: Stable and random motions in dynamical systems with special emphasis on celestial mechanics, Herman Weil Lectures, Annals of Mathematics Studies, vol. 77. Princeton University Press, Princeton (1973)Google Scholar
  18. 18.
    Ortega J.M., Rheinbold W.C.: Iterative solutions of nonlinear equations in several variables. Academic Press, New York (1970)Google Scholar
  19. 19.
    Petković M.S., Ilić S., Dăunić J.: Derivative free two-point methods with and without memory for solving nonlinear equations. Appl. Math. Comput. 217, 1887–1895 (2010)MathSciNetMATHGoogle Scholar
  20. 20.
    Potra F.A.: A characterisation of the divided differences of an operator which can be represented by Riemann integrals. Anal. Numer. Theory Approx. 9, 251–253 (1980)MathSciNetMATHGoogle Scholar
  21. 21.
    Potra F.A., Pták V.: Nondiscrete induction and iterative processes. Pitman Publishing, Boston (1984)MATHGoogle Scholar
  22. 22.
    Steffensen I.F.: Remarks on iteration. Skand. Aktuarietidskr. 16, 64–72 (1933)MathSciNetMATHGoogle Scholar
  23. 23.
    Tornheim L.: Convergence of multipoint iterative methods. J. ACM. 11, 210–220 (1964)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Traub J.F.: Iterative methods for the solution of equations. Prentice-Hall, Englewood Cliffs (1964)MATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • S. Amat
    • 1
  • M. Grau-Sanchez
    • 2
  • M. A. Hernández-Verón
    • 3
  • M. J. Rubio
    • 3
  1. 1.Department of Applied Mathematics and StatisticsPolytechnique University of CartagenaCartagenaSpain
  2. 2.Department of Applied Mathematics IITechnical University of CataloniaBarcelonaSpain
  3. 3.Department of Mathematics and ComputationUniversity of La RiojaLogroñoSpain

Personalised recommendations