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

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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.