Third-order modifications of Newton's method based on Stolarsky and Gini means
The authors consider modifications of some third-order variants of the classical Newton's method, namely \(x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}\phi(s(x_n))\), where \(s(x)=\frac{f'(x-\frac{f(x)}{f'(x)})}{f'(x)}\). They approximate \(f'(x_n)\) by the Stolarsky mean [\textit{K. B. Stolarsky}, Am. Math. Mon. 87, 545--548 (1980; Zbl 0455.26008)] and the Gini mean [\textit{K. B. Stolarsky}, Math. Mag. 48, 87--92 (1975; Zbl 0302.26003)], respectively. They prove third-order convergence of such modified Newton's methods. The results of numerical experiments show that their methods produce better results than several well-known algorithms.
scientific article; zbMATH DE number 6156305