### 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  that I found.

1. Kovacis algorithm  uses $$\alpha _{\infty }^{\pm }$$ deﬁned as $$\alpha _{\infty }^{\pm }=\frac {1}{2}\pm \frac {1}{2}\sqrt {1+4b}$$ for the case when $$O\left ( \infty \right ) =2$$. 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 $$\infty$$. But we do not need to ﬁnd Laurent series expansion of $$r$$ at $$\infty$$ 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 $$\gcd \left ( s,t\right ) =1$$.
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 $$\frac {1}{\left ( x-c\right ) ^{2}}$$ 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.