A combinatorial construction of high order algorithms for finding polynomial roots of known multiplicity

We construct a family of high order iteration functions for finding polynomial roots of a known multiplicity $s$. This family is a generalization of a fundamental family of high order algorithms for simple roots that dates back to Schröder's 1870 paper. It starts with the well known variant of Newton's method $\hat{B}_{2}(x) = x-s \cdot p(x)/p'(x)$ and the multiple root counterpart of Halley's method derived by Hansen and Patrick. Our approach demonstrates the relevance and power of algebraic combinatorial techniques in studying rational root-finding iteration functions.