Numerical Algorithms

, Volume 47, Issue 1, pp 95–107 | Cite as

Simple geometric constructions of quadratically and cubically convergent iterative functions to solve nonlinear equations

Original Paper


In this paper, we derive one-parameter families of Newton, Halley, Chebyshev, Chebyshev-Halley type methods, super-Halley, C-methods, osculating circle and ellipse methods respectively for finding simple zeros of nonlinear equations, permitting f ′ (x) = 0 at some points in the vicinity of the required root. Halley, Chebyshev, super-Halley methods and, as an exceptional case, Newton method are seen as the special cases of the family. All the methods of the family and various others are cubically convergent to simple roots except Newton’s or a family of Newton’s method.


Nonlinear equations Iterative methods One-parameter family Newton’s method Halley’s method Chebyshev’s method super-Halley method 

AMS subject classifications



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ostrowski, A.M.: Solutions of Equations and Systems of Equations. Academic Press, New York (1960)Google Scholar
  2. 2.
    Traub, J.F.: Iterative Methods for Solution of Equations. Prentice-Hall, Englewood Cliffs, NJ (1964)MATHGoogle Scholar
  3. 3.
    Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)MATHGoogle Scholar
  4. 4.
    Halley, E.: A new, exact and easy method for finding the roots of any equations generally, without any previous reduction (Latin). Philos. Trans. R. Soc. Lond. 18, 136–148 (1694)CrossRefGoogle Scholar
  5. 5.
    Gutiérrez, J.M., Hernández, M.A.: A family of Chebyshev–Halley type methods in Banach spaces. Bull. Aust. Math. Soc. 55, 113–130 (1997)MATHCrossRefGoogle Scholar
  6. 6.
    Melman, A.: Geometry and convergence of Euler’s and Halley’s method. SIAM Rev. 39(4), 728–735 (1997)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Salehov, G.S.: On the convergence of the process of tangent hyperbolas. Dokl. Akad. Nauk. SSSR 82, 525–528 (1952)MathSciNetGoogle Scholar
  8. 8.
    Scavo, T.R., Thoo, J.B.: On the geometry of Halley’s method. Am. Math. Mon. 102, 417–426 (1995)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gander, W.: On Halley’s iteration method. Am. Math. Mon. 92, 131–134 (1985)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Amat, S., Busquier, S., Candela, V.F., Potra, F.A.: Convergence of third order iterative methods in Banach spaces, Preprint, Vol. 16, U.P. Caratgena, 2001Google Scholar
  11. 11.
    Gutiérrez, J.M., Hernández, M.A.: An acceleration of Newton’s method: Super-Halley method. Appl. Math. Comput. 117, 223–239 (2001)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Amat, S., Busquier, S., Gutiérrez, J.M.: Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157, 197–205 (2003)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kanwar, V., Singh, Sukhjit, Guha, R.K., Mamta, : On method of osculating circle for solving nonlinear equations. Appl. Math. Comput. 176, 379–382 (2006)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Sharma, J.R.: A family of third-order methods to solve nonlinear equations by quadratic curves approximation. Appl. Math. Comput. 184, 210–215 (2007)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Chun, C.: A one-parameter family of third-order methods to solve nonlinear equations. Appl. Math. Comput. 189, 126–130 (2007)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Jiang, D., Han, D.: Some one-parameter families of third-order methods for solving nonlinear equations. Appl. Math. Comput. DOI 10.1016/j.amc.2007.04.100
  17. 17.
    Ben-Israel, A.: Newton’s method with modified functions. Contemp. Math. 204, 39–50 (1997)MathSciNetGoogle Scholar
  18. 18.
    Kanwar, V., Tomar, S.K.: Modified families of Newton, Halley and Chebyshev methods. Appl. Math. Comput. 192, 20–26 (2007)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Wu, X., Wu, H.W.: On a class of quadratic convergence iteration formulae without derivatives. Appl. Math. Comput. 10(7), 77–80 (2000)CrossRefGoogle Scholar
  20. 20.
    Kanwar, V., Tomar, S.K.: Modified families of multi-point iterative methods for solving nonlinear equations. Numer. Algor 44, 381–389 (2007)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2007

Authors and Affiliations

  1. 1.University Institute of Engineering and TechnologyPanjab UniversityChandigarhIndia
  2. 2.Department of MathematicsSant Longowal Institute of Engineering and TechnologyLongowalIndia
  3. 3.Department of Applied SciencesIndo Global College of EngineeringMohaliIndia

Personalised recommendations