On the guaranteed convergence of the square-root iteration method
The paper is devoted to the convergence analysis of the so-called Ostrowski-like iterative method for the simultaneous approximation of all simple zeros \(z_{i}\) of a given polynomial \(P\). (This method is an extension of the well-known square-root method \[ z^{m+1}_{i}=z^{m}_{i}-\frac{P(z^{m}_{i})}{\sqrt{P'(z^{m}_{i})-P(z^{m}_{i})P''(z^{m}_{i})}}, \] also known as Ostrowski's method). The main result of the paper (Theorem 1) asserts that for a polynomial of degree \(n\geq3\) with simple zeros, satisfying a certain initial condition, the Ostrowski-like simultaneous method is convergent and its order of convergence is four.
scientific article; zbMATH DE number 2085563