TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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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.

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H-matrix block multiplication returns the represented matrix action

Statement

For every H-matrix representation $\tilde{A} \in F^{I \times J}$ on the block-cluster partition of Block-cluster leaves partition the index-pair set and every vector $x \in F^{J}$ , H-matrix block matrix-vector multiplication returns $\tilde{A} x$ exactly.

Facts & Assumptions

Given: The represented H-matrix $\tilde{A}$ , its block-cluster partition, and $x \in F^{J}$ .

[F1]

H-matrix block representation: each block leaf stores exactly its represented dense or factorized submatrix, and the leaf blocks determine $\tilde{A}$ .

[F2]

Block-cluster leaves partition the index-pair set: the terminal block rectangles are disjoint and cover $I \times J$ .

[F3]

H-matrix block matrix-vector multiplication: for each leaf it computes the stored block times the corresponding subvector and adds that contribution to the corresponding output rows.

Proof

1.1givenF1construct

By [F1], $\tilde{A}$ is the matrix obtained by placing its stored block matrices on their index rectangles and putting no entries outside those rectangles.

2.1F2F3step 1.1algebra∎

By [F2], each pair $(i, j) \in I \times J$ lies in exactly one leaf rectangle. By [F3], the algorithm adds the contribution ${\tilde{A}}_{i j} x_{j}$ to output coordinate $i$ from that leaf, and no other leaf includes that entry. Summing over the finite partition gives $y_{i} = \sum_{j \in J} {\tilde{A}}_{i j} x_{j}$ for every $i \in I$ , which is exactly $y = \tilde{A} x$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)