AI 中文总结
该研究基于函数$\phi$定义有根平面树迭代次数，构造对应加权生成函数，提出牛顿-拉夫逊-辛普森法的多点变体max-phi法，用于$L$阶可微函数。
AI 中文摘要
取整数$L \geq 1$，以及函数$\phi \colon \mathbb{Z}_{\geq 1} \rightarrow [0,L] \cap \mathbb{Z}$，满足$\phi^{-1}(\{0\}) = \{ 1\}$。我们定义牛顿-拉夫逊-辛普森法的多点变体，称为max-phi法：用$\phi$定义有根平面树的迭代次数，构造迭代次数不超过$N$的有根平面树的加权生成函数形式级数，最后用这些公式定义max-phi法在任意$L$阶可微函数上的应用。
英文摘要
Fix an integer $L \geq 1$, and a function $ϕ\colon \mathbb{Z}_{\geq 1} \rightarrow [0,L] \cap \mathbb{Z}$ with $ϕ^{-1}(\{0\}) = \{ 1\}$. We define a multi-point variant of the Newton-Raphson-Simpson, which we call the max-phi method, as follows. We use $ϕ$ 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.
