BIT Numerical Mathematics

, Volume 49, Issue 2, pp 325–342 | Cite as

New iterations of R-order four with reduced computational cost



From a one-point iterative method of R-order at least three, we construct new two-point iterations to solve nonlinear equations in Banach spaces such that the computational cost is reduced, whereas the R-order of convergence is increased to at least four.


Nonlinear equations in Banach spaces Semilocal convergence Recurrence relations A priori error estimates R-order of convergence 

Mathematics Subject Classification (2000)

45G10 47H17 65J15 


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  1. 1.
    Amat, S., Busquier, S., Gutiérrez, J.M.: An adaptative version of a fourth-order iterative method for quadratic equations. J. Comput. Appl. Math. 191, 259–268 (2006) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Argyros, I.K., Chen, D., Qian, Q.: The Jarratt method in Banach space setting. J. Comput. Appl. Math. 51, 103–106 (1994) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Argyros, I.K., Chen, D., Qian, Q.: An inverse-free Jarratt type approximation in a Banach space. Approx. Theory Appl. 12, 19–30 (1996) MATHMathSciNetGoogle Scholar
  4. 4.
    Bruns, D.D., Bailey, J.E.: Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chem. Eng. Sci. 32, 257–264 (1977) CrossRefGoogle Scholar
  5. 5.
    Ezquerro, J.A., Gutiérrez, J.M., Hernández, M.A., Salanova, M.A.: Chebyshev-like methods and quadratic equations. Rev. Anal. Numér. Théor. Approx. 28, 23–35 (1999) MATHMathSciNetGoogle Scholar
  6. 6.
    Frontini, M., Sormani, E.: Some variant of Newton’s method with third-order convergence. Appl. Math. Comput. 140, 419–426 (2003) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hernández, M.A.: Chebyshev’s approximation algorithms and applications. Comput. Math. Appl. 41, 433–445 (2001) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Homeier, H.H.H.: A modified Newton method for rootfinding with cubic convergence. J. Comput. Appl. Math. 157, 227–230 (2003) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Homeier, H.H.H.: On Newton-type methods with cubic convergence. J. Comput. Appl. Math. 176, 425–432 (2005) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon, Oxford (1982) MATHGoogle Scholar
  11. 11.
    Potra, F.A., Pták, V.: Nondiscrete Induction and Iterative Processes. Pitman, New York (1984) MATHGoogle Scholar
  12. 12.
    Schröeder, E.: Über unendlich viele Algorithmen zur Auflösung der Gleichugen. Math. Ann. 2, 317–365 (1870) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, Englewood Cliffs (1964) MATHGoogle Scholar
  14. 14.
    Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13, 87–93 (2000) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of Mathematics and ComputationUniversity of La RiojaLogroñoSpain

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