### 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.

- 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 \(\gcd \left ( s,t\right ) =1\).
- 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.