Newton's method and generation of a determinantal family of iteration functions (Q1975681)

Iterative methods of Newton type can be regarded as fixed point iterations, with appropriate iteration functions. For example, the Newton's method is defined by the iteration function \(G(x)=x-\frac{f(x)}{f'(x)}.\) The authors introduce a family \({B_{m}(x)}_{m=2}^{\infty}\) of iteration functions for constructing high-order methods for approximation of roots of a polynomial f(x), and prove that this class is equivalent to the family \({G_{m}(x)}_{m=2}^{\infty}\) of iteration functions introduced by \textit{J. Gerlach} [SIAM Rev. 36, No. 2, 272-276 (1994; Zbl 0814.65046)].