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 B̂2(x) = x − s · 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.