Rapidly converging iterative processes
The author shows that every rapidly converging iterative process $F(x)$, constructed by a given iterative process $f(x)$, in a certain vicinity of a stationary point, where $F(x)$ and $f(x)$ are assumed to be sufficiently smooth, can be presented in the form (4).
The necessary and sufficient conditions of convergence of an iterative sequence generated by $F(x)$ are found. It is shown that any interval containing a stationary point of the iterative process $f(x)$, in which (8) occurs, is a region of attraction of this stationary point as a stationary point of the process $F(x)$.
Citation Example: Sharkovsky O. M. Rapidly converging iterative processes // Ukr. Mat. Zh. - 1961. - 13, № 2. - pp. 210-215.