Numerical Algorithms

, Volume 55, Issue 1, pp 87–99 | Cite as

A modified Newton-Jarratt’s composition

  • Alicia Cordero
  • José L. Hueso
  • Eulalia Martínez
  • Juan R. Torregrosa
Original Paper


A reduced composition technique has been used on Newton and Jarratt’s methods in order to obtain an optimal relation between convergence order, functional evaluations and number of operations. Following this aim, a family of methods is obtained whose efficiency indices are proved to be better for systems of nonlinear equations.


Nonlinear equations and systems Newton’s method Fixed point iteration Convergence order 


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  1. 1.
    Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cordero, A., Torregrosa, J.R.: On interpolation variants of Newton’s method. In: Proceedings of the 2th International Conference on Approximation Methods and Numerical Modelling in Environment and Natural Resources. Universidad de Granada (Spain), 11–13 July 2007, ISBN: 978-84-338-4782-9.Google Scholar
  3. 3.
    Frontini, M., Sormani, E.: Third-order methods from quadrature formulae for solving systems of nonlinear equations. Appl. Math. Comput. 149, 771–782 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Jarrat, P.: Some fourth order multipoint iterative methods for solving equations. Math. Comput. 20, 434–437 (1966)Google Scholar
  5. 5.
    Gerlach, J.: Accelerated convergence in Newton’s method. SIAM Rev. 36(2), 272–276 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Nedzhibov, G.H.: A family of multi-point iterative methods for solving systems of nonlinear equations. J. Comput. Appl. Math. doi: 10.1016/ (2008)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Ozban, A.Y.: Some new variants of Newton’s method. Appl. Math. Lett. 17, 677–682 (2004)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Ostrowski, A.M.: Solutions of Equations and Systems of Equations. Academic, New York (1966)Google Scholar
  9. 9.
    Romero Alvarez, N., Ezquerro, J.A., Hernandez, M.A.: Aproximación de soluciones de algunas equacuaciones integrales de Hammerstein mediante métodos iterativos tipo Newton. XXI Congreso de ecuaciones diferenciales y aplicaciones, Universidad de Castilla-La Mancha (2009)Google Scholar
  10. 10.
    Traub, J.F.: Iterative Methods for the Solution of Equations. Chelsea, New York (1982)zbMATHGoogle Scholar
  11. 11.
    Wang, X., Kou, J., Li, Y.: Modified Jarratt method with sixth-order convergence. Appl. Math. Lett. 22, 1798–1802 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13(8), 87–93 (2000)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  • Alicia Cordero
    • 1
  • José L. Hueso
    • 1
  • Eulalia Martínez
    • 2
  • Juan R. Torregrosa
    • 1
  1. 1.Instituto de Matemática MultidisciplinarUniversidad Politécnica de ValenciaValenciaSpain
  2. 2.Instituto de Matemática Pura y AplicadaUniversidad Politécnica de ValenciaValenciaSpain

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