Summary

Here are the main steps of the algorithm:

1. Compute a ${\mathbf{C}\{\!\{\partial^{-1}_t\}\!\}}$-basis $\boldsymbol{m}$ of $\mathcal{H}''$.
2. For increasing $k$ compute successively the lattices $\mathcal{H}''_k$ in terms of $\boldsymbol{m}$, and $t$ in terms of $\boldsymbol{m}$ up to order $k$ until $k=k_\infty$ and $\mathcal{H}''_k$ is the saturation $\mathcal{H}''_\infty$ of $\mathcal{H}''$.
3. Compute a ${\mathbf{C}\{\!\{\partial^{-1}_t\}\!\}}$-basis $\boldsymbol{m}'\boldsymbol{m}$ of $\mathcal{H}''_\infty$.
4. Compute $t$ in terms of $\boldsymbol{m}$ up to order $\delta(\boldsymbol{m}')+n+1$.
5. Compute $t$ in terms of $\boldsymbol{m}'\boldsymbol{m}$ up to order $K:=n+1$.
6. Compute $\mathcal{H}''$ in terms of $\boldsymbol{m}'\boldsymbol{m}$.
7. Compute the $V$-filtration on $\mathcal{H}''_\infty/\partial_t^{-K}\mathcal{H}''_\infty$ in terms of $\boldsymbol{m}'\boldsymbol{m}$.
8. Compute the induced $V$-filtration on $\mathcal{H}''/\partial^{-1}_t\mathcal{H}''$ in terms of $\boldsymbol{m}'\boldsymbol{m}$.