Tolerance-controlled H-matrix block product

Input: H-matrix representations $A\in {\mathbb{F}}^{I\times K}$ and $B\in {\mathbb{F}}^{K\times J}$, a recursively generated output block-cluster partition as in Finite cluster trees and geometric admissibility, a rank cap $k\ge 1$, local tolerances ${\delta }_b\ge 0$ for far output blocks, and for each output block $b=(\tau ,\sigma )$ a finite intermediate-cluster family ${\mathcal{K}}_b$ partitioning $K$ such that the restrictions $A_{\tau \kappa }$ and $B_{\kappa \sigma }$ are available in dense or factorized form for each $\kappa \in {\mathcal{K}}_b$. For a far output block, its factor-sum tree has one exact product term for each $\kappa \in {\mathcal{K}}_b$ and node tolerances totaling at most ${\delta }_b$. This compatibility condition is part of the input.

Procedure: For each output block, use the exact identity $C_b={\sum }_{\kappa \in {\mathcal{K}}_b}{A}_{\tau \kappa }{B}_{\kappa \sigma }$. If the output block is near, form and store this dense sum exactly and report $e_b=0$. For a far output block, convert each product term to factor form: $U{V}^{\ast }\cdot P{Q}^{\ast }=U(V^{\ast }P)Q^{\ast }$ when both factors are low rank; $U{V}^{\ast }\cdot D=U(D^{\ast }V{)}^{\ast }$ and $D\cdot P{Q}^{\ast }=(DP)Q^{\ast }$ when one factor is dense; and $D_1{D}_2=D_1(D_2^{\ast }{)}^{\ast }$ when both are dense. These formulas produce factors with matching inner dimensions. Run Bottom-up low-rank recompression of block contributions on the resulting finite factor-sum tree with cap $k$. If any block returns tolerance-not-met, return that status and block identity; otherwise store the root factors and its certified bound $e_b\le {\delta }_b$.