Three-step iterative methods with eighth-order convergence for solving nonlinear equations

  • Authors:

  • Weihong Bi;Hongmin Ren;Qingbiao Wu

  • Affiliations:

  • Department of Mathematics, Zhejiang University, Hangzhou 310027, Zhejiang, PR China;Department of Information and Electronics, Hangzhou Radio and TV University, Hangzhou 310012, Zhejiang, PR China;Department of Mathematics, Zhejiang University, Hangzhou 310027, Zhejiang, PR China

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2009

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Abstract

A family of eighth-order iterative methods for the solution of nonlinear equations is presented. The new family of eighth-order methods is based on King's fourth-order methods and the family of sixth-order iteration methods developed by Chun et al. Per iteration the new methods require three evaluations of the function and one evaluation of its first derivative. Therefore this family of methods has the efficiency index which equals 1.682. Kung and Traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2^n^-^1. Thus we provide a new example which agrees with the conjecture of Kung-Traub for n=4. Numerical comparisons are made to show the performance of the presented methods.