Multi-point variants of the Newton-Raphson-Simpson method arising from organizing a formal zero according to a function $\phi$

Abstract

Fix an integer $L \geq 1$, and a function $\phi \colon \mathbb{Z}_{\geq 1} \rightarrow [0,L] \cap \mathbb{Z}$ with $\phi^{-1}(\{0\}) = \{ 1\}$. We define a multi-point variant of the Newton-Raphson-Simpson, which we call the max-phi method, as follows. We use $\phi$ to define the iteration number of a rooted plane tree. Then we construct formal series that are weighted generating functions of rooted plane trees with iteration number at most $N$. Finally we use these formulas to define the max-phi method applied to an arbitrary $L$-differentiable function.