Numerical Analysis and Applications

, Volume 7, Issue 1, pp 26–35 | Cite as

New modified optimal families of King’s and Traub-Ostrowski’s methods

  • R. Behl
  • V. Kanwar
  • K. K. Sharma


Based on quadratically convergent Schröder’s method, we derive many new interesting families of fourth-order multipoint iterative methods without memory for obtaining simple roots of nonlinear equations by using the weight function approach. The classical King’s family of fourth-order methods and Traub-Ostrowski’s method are obtained as special cases. According to the Kung-Traub conjecture, these methods have the maximal efficiency index because only three functional values are needed per step. Therefore, the fourth-order family of King’s family and Traub-Ostrowski’smethod are the main findings of the present work. The performance of proposed multipoint methods is compared with their closest competitors, namely, King’s family, Traub-Ostrowski’s method, and Jarratt’s method in a series of numerical experiments. All the methods considered here are found to be effective and comparable to the similar robust methods available in the literature.


nonlinear equations Newton’s method King’s family Traub-Ostrowski’s method Jarratt’s method optimal order of convergence efficiency index 


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  1. 1.
    Jarratt, P., Some Efficient Fourth-Order Multipoint Methods for Solving Equations, BIT, 1969, no. 9, pp. 119–124.Google Scholar
  2. 2.
    Kanwar, V., Behl, Ramandeep, and Sharma, K.K., Simply Constructed Family of Ostrowski’s Method with Optimal Order of Convergence, Comput. Math. Appl., 2011, no. 62, pp. 4021–4027.Google Scholar
  3. 3.
    King, R.F., A Family of Fourth-Order Methods for Nonlinear Equations, SIAM J. Numer. Anal., 1973, no. 10, pp. 876–879.Google Scholar
  4. 4.
    Kung, H.T. and Traub, J.F., Optimal Order of One-Point and Multipoint Iteration, J. Assoc. Comput. Mach., 1974, no. 21, pp. 643–651.Google Scholar
  5. 5.
    Ostrowski, A.M., Solutions of Equations and System of Equations, New York: Academic Press, 1960.Google Scholar
  6. 6.
    Ostrowski, A.M., Solution of Equations in Euclidean and Banach Space, New York: Academic Press, 1973.Google Scholar
  7. 7.
    Schröder, E., Über unendlichviele Algorithm zur Auffosung der Gleichungen, Math. Ann., 1870, no. 2, pp. 317–365.Google Scholar
  8. 8.
    Traub, J.F., Iterative Methods for the Solution of Equations, Englewood Cliffs, NJ: Prentice-Hall, 1964.MATHGoogle Scholar
  9. 9.
    Werner, W., Some Improvement of Classical Methods for the Numerical Solution of Nonlinear Equations, Lect. Notes Math., 1981, no. 878, pp. 426–440.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.School of Mathematics and Computer ApplicationsThapar UniversityPatialaIndia
  2. 2.University Institute of Engineering and TechnologyPanjab UniversityChandigarhIndia
  3. 3.Department of MathematicsSouth Asian UniversityAkbar Bhavan, Chayankya PuriIndia

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