The Asymptotic Expansion Method via Symbolic Computation

Algorithm 5

Implementation of the algorithm. Steps (3.3) and (3.4).

for (𝚒=2;𝚒<=𝚖;𝚒++){         Calculate, for each 𝜌 such that 2≤𝜌≤𝑀,

    xdx [i][𝟶][𝟶] =

      xdx [𝟷][𝟶][𝟶] *xdx [i−𝟷][𝟶][𝟶];      𝑋(𝜌,0,0)=𝑋(1,0,0)×𝑋(𝜌−1,0,0),

    xdx [𝟶][i][𝟶] =

      xdx [𝟶][𝟷][𝟶] *xdx [𝟶][i−𝟷][0];     𝑋(0,𝜌,0)=𝑋(0,1,0)×𝑋(0,𝜌−1,0).

}

for (i=1; i<=m; i++)          For each 𝜌₁,𝜌₂ such that 1≤𝜌₁,𝜌₂≤𝑀,

  for (j=1; j<=m; j++)            determine the quasipolynomial

    xdx [i][j][𝟶] =

     xdx [i][𝟶][𝟶]*xdx [𝟶][𝚓][𝟶];   𝑋(𝜌₁,𝜌₂,0)=𝑋(𝜌₁,0,0)×𝑋(0,𝜌₂,0).