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 \(\alpha _{\infty }^{\pm }\) 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.