The Asymptotic Expansion Method via Symbolic Computation

Algorithm 6

Implementation of the algorithm. Step (4).

for (q=1; q<=order; q++)         For each 𝑞 such that 1≤𝑞≤𝑄, do:

{ 

 U=U*0.;                  Compute the quasipolynomial

     for (k=0; k<=m; k++)  

      for (nu=0; nu<=k; nu++)       𝑈=∑^𝑀_𝜅=0∑_{0≤𝜌≤𝜅}𝑓_{𝜌,𝜅−𝜌}×𝑋(𝜌,𝜅−𝜌,𝑞−1).

        U = U + xdx [nu] [k-nu] [q-1]

           *f [nu] [k-nu];

 xdx [𝟷][𝟶][𝚚] = U.solvePVI         Calculate 𝑋(1,0,𝑞) as the solution to (5.3),

 (a1, a0, t0, 0., 0.);

 xdx [𝟶][𝟷][𝚚]= xdx [𝟷][𝟶][𝚚].der ();              𝑥̈_𝑞+𝑎₁𝑥̇_𝑞+𝑎₀𝑥_𝑞=𝑈,

                         𝑥_𝑞(𝑡₀)=0,𝑥̇_𝑞(𝑡₀)=0,

                      and 𝑋(0,1,𝑞)=𝑑/𝑑𝑡(𝑋(1,0,𝑞)).

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

   for (nu=0; nu<=q; nu++) {

   xdx [i][𝟶][𝚚] = xdx[i][𝟶][𝚚]       𝑋(𝜌,0,𝑞)=∑_{0≤𝑝≤𝑞}𝑋(𝜌−1,0,𝑝) 

       + xdx [i−𝟷][𝟶][𝚗𝚞]                  

      *xdx [𝟷][𝟶][𝚚−𝚗𝚞];           ×𝑋(1,0,𝑞−𝑝),

   xdx [𝟶][i][𝚚] = xdx [𝟶][i][𝚚]      𝑋(0,𝜌,𝑞)=∑_{0≤𝑝≤𝑞}𝑋(0,𝜌−1,𝑝)

     + xdx [𝟶][i−𝟷][𝚗𝚞]

        *xdx [𝟶][𝟷][𝚚−𝚗𝚞];           ×𝑋(0,1,𝑞−𝑝).

  }

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

  for (j=1; j<=m; j++)         compute

   for (nu=0; nu<=q; nu++)

    xdx [i][𝚓][𝚚] =            𝑋(𝜌₁,𝜌₂,𝑞)

     xdx [i][𝚓][𝚚]

     + xdx [i][𝟶][𝚗𝚞]            =∑_{0≤𝑝≤𝑞}𝑋(𝜌₁,0,𝑝)×𝑋(0,𝜌₂,𝑞−𝑝).

         *xdx [𝟶][𝚓][𝚚−nu];

}