5 Notation mapping between Saunders/Smith algorithm and original Kovacis algorithm

I have implemented the original Kovacis algorithm using Maple 2021 based on the original paper (1). The following are notation difference between the two algorithms and the implementation by Smith [3] that I found.

1. Kovacis algorithm  uses ${\alpha }_{\mathrm{\infty }}^{\pm }$ defined as ${\alpha }_{\mathrm{\infty }}^{\pm }=\frac{1}{2}\pm \frac{1}{2}\sqrt{1+4b}$ for the case when $O\left(\mathrm{\infty }\right)=2$. Smith algorithm uses $e_0$ for the $\sqrt{1+4b}$ part only. In both algorithms the $b$ value is calculated in the same way. It is the coefficient of $\frac{1}{x^2}$ in the Laurent series expansion of $r$ at $\mathrm{\infty }$. But we do not need to find Laurent series expansion of $r$ at $\mathrm{\infty }$ to find $b$ here. It can be found using $b=\frac{lcoeff\left(s\right)}{lcoeff\left(t\right)}$ where $r=\frac{s}{t}$ and $gcd(s,t)=1$.
2. Smith algorithm finds $e_1,e_2,\cdots$ values for each pole. This is part b of step 1 for poles of order 2, these correspond to only the $\sqrt{1+4b}$ part in Kovacis algorithm (this is part c2 of step1), where there it finds ${\left[\sqrt{r}\right]}_c$ for each pole and ${\alpha }_c^{\pm }=\frac{1}{2}\pm \frac{1}{2}\sqrt{1+4b}$ where $b$ is the coefficient of $\frac{1}{{(x-c)}^2}$ in the partial fraction decomposition of $r$. This $b$ value is also the same for Smith algorithm in its $e^{\prime }s$.

More mappings to be added next.