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

• Published in: Journal of computational and applied mathematics Vol. 225; no. 1; pp. 105 - 112
• Main Authors: Bi, Weihong, Ren, Hongmin, Wu, Qingbiao
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 01.03.2009; Elsevier
• Subjects: Exact sciences and technology; Global analysis, analysis on manifolds; Iterative methods; King’s methods; Mathematical analysis; Mathematics; Newton’s method; Nonlinear equations; Numerical analysis; Numerical analysis. Scientific computation; Order of convergence; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; 65B99; Order of convergence; Nonlinear equations; Newton’s method; King’s methods; 65H05; Iterative methods; Step method; Iterative method; Iteration; Optimal convergence; King's methods; Non linear equation; Numerical analysis; Applied mathematics; Newton's method; Optimal approximation; Newton method
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2008.07.004

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.