# On Generalizations of the Newton-Raphson-Simpson Method

## TL;DR

This paper introduces a family of algorithms called NRS(m) that generalize the Newton-Raphson-Simpson method, enabling the evaluation of sums of formal zeros of functions and connecting to hypergeometric series via combinatorial structures.

## Contribution

The paper develops a new class of algorithms NRS(m) that extend the Newton-Raphson-Simpson method and relate to hypergeometric series using novel combinatorial objects.

## Key findings

- NRS(1) recovers the classical Newton-Raphson-Simpson iterations.
- NRS(m) can evaluate certain hypergeometric series.
- The algorithms utilize trees with negative vertex degrees for their construction.

## Abstract

We present generalizations of the Newton-Raphson-Simpson method. Specifically, for a positive integer $m$ and the sequence of coefficients of a Taylor series of a function $f(z)$, we define an algorithm we denote by NRS($m$) which is a way to evaluate, in our terminology, a sum of $m$ formal zeros of $f(z)$. We prove that NRS(1) yields the familiar iterations of the Newton-Raphson-Simpson method. We also prove that NRS($m$) is way to evaluate certain $\mathscr{A}$-hypergeometric series defined by Sturmfels. In order to define these algorithms, we make use of combinatorial objects which we call trees with negative vertex degree.