Convergence criterion and convergence ball of the King-Werner method under the radius Lipschitz condition (Q2470805)

Let \(E\), \(F\) be Banach spaces; and \(f:B[x_0,r]\subset E\to F\) be a nonlinear operator with continuous Fréchet derivative. The convergence of the King-Werner iterative method \[ \begin{aligned} & x_{n+1}=x_n-f'(y_n)^{-1}f(x_n),\\ & z_{n+1}=x_{n+1}-f'(y_n)^{-1}f(x_{n+1}),\\ & y_{n+1}=(1/2)(x_{n+1}+z_{n+1}); n\geq 0\end{aligned} \] [cf. \textit{R. F. King}, ibid. 18, 298--304 (1972; Zbl 0215.27403); \textit{W. Werner}, Numer. Math. 32, 333--342 (1979; Zbl 0431.65040)] towards a solution \(x^*\) of \(f(x)=0\) is analyzed.