k-step iterative methods for solving nonlinear systems of equations
One considers a Fréchet differentiable function \(f: C^ n\to C^ n\), supposed to have at least one fixed point. For finding such a point, one uses an iteration of the form \(y_ m=a_ 0f(y_{m-1})+a_ 1y_{m- 1}+...+a_ ky_{m-k},\) \(m\geq k\), \(a_ 0+a_ 1+...+a_ k=1\), \(a_ 0\neq 0\), \(a_ k\neq 0\). Using the linear theory and a theorem of Ostrowski, one proves a convergence theorem under suitable conditions. These results are extended to some nonstationary methods, where the Ostrowski's theorem is replaced by a theorem of Perron and to cyclic systems.
scientific article; zbMATH DE number 3963103
scientific article