Compress exact block LU factors with local tolerances

Input: An exact factorization $A=LU$ of a square matrix $A\in {\mathbb{F}}^{n\times n}$, where $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$ and $n\ge 1$, with $L$ block lower triangular and $U$ block upper triangular on a common ordered cluster partition; an integer rank cap $k\ge 1$; and a nonnegative local Frobenius tolerance for each admissible off-diagonal block of $L$ and $U$. The factorization $A=LU$ is supplied as a hypothesis, not computed by this procedure.

Procedure: Copy every dense nearfield block and every diagonal block exactly. For each strictly lower admissible block $D_b\in {\mathbb{F}}^{m_b\times n_b}$ of $L$ and each strictly upper admissible block $D_b$ of $U$, form the one-pair factor list $(D_b,I_{n_b})$, representing $D_b=D_b{I}_{n_b}^{\ast }$, and pass it to Core-SVD recompression with rank and error tolerances with effective cap $\min (k,m_b,n_b)$ and its given local tolerance. Keep structural zero blocks zero. If any compression reports tolerance-not-met, return that status and the factor/block identity. Otherwise return $\widehat{L},\widehat{U}$ and the local discarded-tail bounds ${ϵ}_b^L,{ϵ}_c^U$ reported by the recompressions. The exact diagonal blocks preserve block triangularity; no claim is made that the compressed factors are exact factors of $A$.