Nested H² cluster bases and interaction lists

In the H-matrix setting of H-matrix block representation, an H² representation assigns a row basis $U_{\tau }\in {\mathbb{F}}^{\tau \times r_{\tau }}$ to every row cluster $\tau$ and a column basis $V_{\sigma }\in {\mathbb{F}}^{\sigma \times s_{\sigma }}$ to every column cluster $\sigma$, with $0\le r_{\tau },s_{\sigma }\le k$. For every internal row cluster $\tau$ with children ${\tau }_1,{\tau }_2$, it stores transfer matrices $E_{{\tau }_i}^U\in {\mathbb{F}}^{r_{{\tau }_i}\times r_{\tau }}$ satisfying

$U_{\tau }{\mathrm{\mid }}_{{\tau }_i}=U_{{\tau }_i}{E}_{{\tau }_i}^U(i=1,2)\mathrm{.}$

For every internal column cluster $\sigma$ with children ${\sigma }_1,{\sigma }_2$, it stores transfer matrices $E_{{\sigma }_j}^V\in {\mathbb{F}}^{s_{{\sigma }_j}\times s_{\sigma }}$ satisfying $V_{\sigma }{\mathrm{\mid }}_{{\sigma }_j}=V_{{\sigma }_j}{E}_{{\sigma }_j}^V$. These identities are the nested-basis conditions. An admissible block leaf $(\tau ,\sigma )$ is represented by $U_{\tau }{S}_{\tau \sigma }{V}_{\sigma }^{\ast }$ with coupling matrix $S_{\tau \sigma }\in {\mathbb{F}}^{r_{\tau }\times s_{\sigma }}$; near block leaves remain dense.

Let ${\mathcal{P}}_{\mathrm{f}\mathrm{a}\mathrm{r}}$ denote the admissible leaves of the block-cluster partition. The row interaction list of $\tau$ is ${\mathcal{L}}_U(\tau )=\{\sigma :(\tau ,\sigma )\in {\mathcal{P}}_{\mathrm{f}\mathrm{a}\mathrm{r}}\}$, and the column interaction list of $\sigma$ is ${\mathcal{L}}_V(\sigma )=\{\tau :(\tau ,\sigma )\in {\mathcal{P}}_{\mathrm{f}\mathrm{a}\mathrm{r}}\}$. An H² representation may impose a bound on these list lengths or on their total count; any complexity theorem must state which bound it uses.

Storage convention: Store the basis arrays explicitly only at leaf clusters. Store each internal cluster's child-to-parent transfer matrices; the basis of an internal cluster is reconstructed recursively from its children and these transfers.