On the leading and penultimate leading coefficients for NRS(2) applied to a cubic polynomial

TL;DR

This paper analyzes the positivity of specific coefficients in the error terms of NRS(2) when applied to cubic polynomials, providing simplified proofs and extending results to additional coefficients.

Contribution

It offers simplified proofs for the positivity of leading and penultimate coefficients in NRS(2) error terms for cubic polynomials, extending previous work.

Findings

Leading coefficients are positive polynomials in u_1 and u_2.

Penultimate coefficients are also positive polynomials in u_1 and u_2.

Proofs are simplified and extended from prior work.

Abstract

We prove that the leading and penultimate leading coefficients in $u_3$ of the ``error" terms of NRS(2) applied to a cubic polynomial $f(z) =\sum_{i=0}^3 a_i z^i=\prod_{i=1}^3 (1-u_iz)$ with starting point $(-\frac{a_1}{a_2}, -\frac{a_1}{a_2})$ are positive-coefficient polynomials in $u_1$ and $u_2$. Our proof for the leading coefficients simplifies that of \cite{DeFranco} and extends to the penultimate leading coefficients as well.