Numerical Algorithms

, Volume 57, Issue 3, pp 377–388 | Cite as

Revisit of Jarratt method for solving nonlinear equations

  • Fazlollah Soleymani
Original Paper


In this paper, some sixth-order modifications of Jarratt method for solving single variable nonlinear equations are proposed. Per iteration, they consist of two function and two first derivative evaluations. The convergence analyses of the presented iterative methods are provided theoretically and a comparison with other existing famous iterative methods of different orders is given. Numerical examples include some of the newest and the most efficient optimal eighth-order schemes, such as Petkovic (SIAM J Numer Anal 47:4402–4414, 2010), to put on show the accuracy of the novel methods. Finally, it is also observed that the convergence radii of our schemes are better than the convergence radii of the optimal eighth-order methods.


Nonlinear equations Newton’s method Jarratt method Padé approximant Error equation Simple root Derivative approximation Convergence radius 

Mathematics Subject Classifications (2010)

65H05 41A25 65B99 


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  1. 1.
    Chun, C.: A geometric construction of iterative formulas of order three. Appl. Math. Lett. 23, 512–516 (2010)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Cordero, A., Hueso, J.L., Martinez, E., Torregrosa, J.R.: Efficient three-step iterative methods with sixth order convergence for nonlinear equations. Numer. Algor. 53, 485–495 (2010)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Grau-Sanchez, M., Gutierrez, J.M.: Zero-finders methods derived from Obreshkov’s techniques. Appl. Math. Comput. 215, 2992–3001 (2009)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Gupta, D.K., Parhi, S.K.: An improved class of regula falsi methods of third order for solving nonlinear equations in R. J. Appl. Math. Comput. 33, 35–45 (2010)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Kou, J., Li, Y., Wang, X.: A modification of Newton method with third-order convergence. Appl. Math. Comput. 181, 1106–1111 (2006)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Math. 21, 634–651 (1974)MathSciNetGoogle Scholar
  7. 7.
    Sharma, J.R., Sharma, R.: A new family of modified Ostrowski’s methods with accelerated eighth order convergence. Numer. Algor. 54, 445–458 (2010)MATHCrossRefGoogle Scholar
  8. 8.
    Sharma, J.R., Guha, R.K.: A family of modified Ostrowski methods with accelerated sixth-order convergence. Appl. Math. Comput. 190, 111–115 (2007)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Soleymani, F., Sharifi, M.: On a cubically iterative scheme for solving nonlinear equations. Far East J. Appl. Math. 43, 137–143 (2010)MathSciNetMATHGoogle Scholar
  10. 10.
    Petkovic, M.S.: On a general class of multipoint root-finding methods of high computational efficiency. SIAM J. Numer. Anal. 47, 4402–4414 (2010)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Wang, H., Li, A.: A family of derivative-free methods for nonlinear equations. Rev. Mat. Complut. (2010). doi: 10.1007/s13163-010-0044-5
  12. 12.
    Wang, X., Kou, J., Li, Y.: A variant of Jarratt method with sixth-order convergence. Appl. Math. Comput. 204, 14–19 (2008)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Wang, X., Gu, C., Kou, J.: Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces. Numer. Algor. (2010). doi: 10.1007/s11075-010-9401-1

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Young Researchers Club, Zahedan BranchIslamic Azad UniversityZahedanIran

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