Skip to main content
SpringerLink
Log in

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

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

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.

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

References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  4. Argyros I.K.: On Ulm’s method using divided differences of order one. Numer. Algorithms 52, 295–320 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Dennis, JE., Schnabel, R.B.: Numerical methods for unconstrained optimization and nonlinear equations. SIAM, Philadelphia (1996)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Golub, G.H., Van Loan, C.F.: Matrix computations. JHU Press (1996)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hairer, E., Wanner, G.: Solving ordinary differential equations II: stiff and differential algebraic problems. Springer-Verlag, Berlin (1991)

  16. Hald O.H.: On a Newton-Moser type method. Numer. Math. 23, 411–425 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  18. Ortega J.M., Rheinbold W.C.: Iterative solutions of nonlinear equations in several variables. Academic Press, New York (1970)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  21. Potra F.A., Pták V.: Nondiscrete induction and iterative processes. Pitman Publishing, Boston (1984)

    MATH  Google Scholar 

  22. Steffensen I.F.: Remarks on iteration. Skand. Aktuarietidskr. 16, 64–72 (1933)

    MathSciNet  MATH  Google Scholar 

  23. Tornheim L.: Convergence of multipoint iterative methods. J. ACM. 11, 210–220 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  24. Traub J.F.: Iterative methods for the solution of equations. Prentice-Hall, Englewood Cliffs (1964)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, Polytechnique University of Cartagena, 30230, Cartagena, Spain

    S. Amat

  2. Department of Applied Mathematics II, Technical University of Catalonia, 08034, Barcelona, Spain

    M. Grau-Sanchez

  3. Department of Mathematics and Computation, University of La Rioja, 26004, Logroño, Spain

    M. A. Hernández-Verón & M. J. Rubio

Authors
  1. S. Amat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Grau-Sanchez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. A. Hernández-Verón
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. J. Rubio
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Amat.

Additional information

Research supported by 19374/PI/14 and MTM2015-64382-P.

Research supported by MTM2011-28636-C02-01 of the Spanish Ministry of Science and Innovation. (Spain).

Research supported by MTM2014-52016-C2-1-P of the Spanish Ministry of Science and Innovation. (Spain).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Amat, S., Grau-Sanchez, M., Hernández-Verón, M.A. et al. On a Moser–Steffensen Type Method for Nonlinear Systems of Equations. Mediterr. J. Math. 13, 4109–4128 (2016). https://doi.org/10.1007/s00009-016-0735-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00009-016-0735-3

Mathematics Subject Classification

Keywords

Search

Navigation