Koçak’s method shows that an unwittingly exaggerated convergence order is in fact 2

摘要

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). Koçak’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) 567–571] 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.