Coefficient-free adaptations of polynomial root-finders

We adapt two celebrated polynomial root-finders. Performing one of them, we involve only the scaled values of the input polynomial c(λ) at the points approximating the roots and recursively updated. Performing another root-finder, we also compute the values of the derivative cʹ(λ) at these points. In neither case do we use the coefficients of c(λ). We also relate our algorithms to approximating the eigenvalues of a matrix.