Nested-basis H² matrix-vector multiplication

Input: An H² representation $\tilde{A}$ with row bases $U_{\tau }$, column bases $V_{\sigma }$, transfer matrices from Nested H² cluster bases and interaction lists, and a vector $x\in {\mathbb{F}}^J$.

Procedure:

1. Initialize an upward coefficient vector $z_{\sigma }$ for each column cluster. At each leaf $\sigma$, set $z_{\sigma }=V_{\sigma }^{\ast }x{\mathrm{\mid }}_{\sigma }$. At each internal cluster $\sigma$ with children ${\sigma }_1,{\sigma }_2$, set $z_{\sigma }={\sum }_{j=1}^2(E_{{\sigma }_j}^V{)}^{\ast }{z}_{{\sigma }_j}$.
2. Initialize a downward coefficient vector $w_{\tau }=0$ for each row cluster and an output vector $y=0$. For each admissible leaf $(\tau ,\sigma )$, add $S_{\tau \sigma }{z}_{\sigma }$ to $w_{\tau }$. For each near leaf $(\tau ,\sigma )$, add $D_{\tau \sigma }x{\mathrm{\mid }}_{\sigma }$ to $y{\mathrm{\mid }}_{\tau }$.
3. Traverse the row tree from root to leaves. For each parent $\tau$ and child ${\tau }_i$, add $E_{{\tau }_i}^U{w}_{\tau }$ to $w_{{\tau }_i}$. At each row leaf $\tau$, add $U_{\tau }{w}_{\tau }$ to $y{\mathrm{\mid }}_{\tau }$. Return $y$.

The upward and downward traversals follow the finite cluster trees; block and cluster contributions are accumulated in exact arithmetic.