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Numerical Algorithms

, Volume 54, Issue 4, pp 445–458 | Cite as

A new family of modified Ostrowski’s methods with accelerated eighth order convergence

  • Janak Raj SharmaEmail author
  • Rajni Sharma
Original Paper

Abstract

Based on Ostrowski’s fourth order method, we derive a family of eighth order methods for the solution of nonlinear equations. In terms of computational cost the family requires three evaluations of the function and one evaluation of first derivative. Therefore, the efficiency index of the present methods is 1.682 which is better than the efficiency index 1.587 of Ostrowski’s method. Kung and Traub conjectured that multipoint iteration methods without memory based on n evaluations have optimal order 2n − 1. Thus, the family agrees with Kung–Traub conjecture for the case n = 4. The efficacy of the present methods is tested on a number of numerical examples. It is observed that our methods are competitive with other similar robust methods and very effective in high precision computations.

Keywords

Nonlinear equations Newton’s method Ostrowski’s method Order of convergence Efficiency 

Mathematics Subject Classifications (2000)

65H05 65B99 

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Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  1. 1.Department of MathematicsSant Longowal Institute of Engineering and TechnologySangrurIndia
  2. 2.Department of Applied SciencesD.A.V. Institute of Engineering and TechnologyKabirnagarIndia

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