Some informational requirements for convergence (Q1100851)

Let \(M: R^ 2\to R\) be a \(C^ 1\) function with \(M(0,-)=0\). The author studies iterative methods \(x_{n+1}=x_ n+M(f(x_ n),f'(x_ n))\) for the solution of a polynomial equation. The problem relies on finding such an M that guarantees global convergence of the method for almost all initial iterates and for all polynomial equations. It turns out that such M does not exist. Moreover any finite amount of information is insufficient for global convergence of a smooth iterative method, but for a given polynomial such M exists.