AlgorithmAlgorithm: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

H-matrix block matrix-vector multiplication

Definition

Input: An H-matrix representation $\tilde{A}$ on a finite block-cluster partition $P$ as in H-matrix block representation, and a vector $x \in F^{J}$ .

Procedure: Initialize $y \in F^{I}$ to zero. For every far block $b = (τ, σ)$ with factors $U_{b}, V_{b}$ , compute $z_{b} = V_{b}^{*} x ∣_{σ}$ and add $U_{b} z_{b}$ to $y ∣_{τ}$ . For every near block with dense entries $D_{b}$ , compute $D_{b} x ∣_{σ}$ and add it to $y ∣_{τ}$ . Return $y$ . All arithmetic is exact; contributions to a row cluster are summed over its block leaves.

Depends on

H-matrix block representation

Used by

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Steffen Börm, Lars Grasedyck, and Wolfgang Hackbusch, Hierarchical Matrices, MiS Lecture Note 21 (2003; revised June 2006), §1.6 (standard reference, not scraped)
Ralf Hiptmair, AdvNumCSE Lecture Document, Chapter 2, “Local Low-Rank Compression of Non-Local Operators” (ETH Zürich, AT’23), §2.3.5 (standard reference, not scraped)