Kocak's method shows that an unwittingly exaggerated convergence order is in fact 2

Kocak M. C.

APPLIED MATHEMATICS AND COMPUTATION, vol.219, no.19, pp.10053-10058, 2013 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 219 Issue: 19
  • Publication Date: 2013
  • Doi Number: 10.1016/j.amc.2013.03.034
  • Journal Name: APPLIED MATHEMATICS AND COMPUTATION
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.10053-10058
  • Keywords: Non-linear equations, Iterative methods, Newton's method, Convergence acceleration
  • Ankara University Affiliated: No

Abstract

Iterative solution of a nonlinear equation f(x) = 0 usually means a repetitive scheme to locate a fixed point of a related equation x = g(x). Kocak's acceleration method smoothly gears up iterations with the aid of a superior secondary solver gK = x + G(x)(g(x) - x) = (g(x) - m(x) x)/(1 - m(x)) where G(x) = 1/(1 - m(x)) is a gain and m(x) = 1 - 1/G(x) is a straight line slope. The accelerator shows that a previously published article [A. Biazar, A. Amirtemoori, An improvement to the fixed point iterative method, AMC 182 (2006) 567571] unwittingly exaggerated the convergence order of the solver it presented. This solver boils down to an indirect application of Newton's method solving g(x) - x = 0 which means that it is of second order. Hence, their claim that it "increases the order of convergence as much as desired'' is false! The scheme wastes higher derivatives. (C) 2013 Elsevier Inc. All rights reserved.