DefinitionDefinition: AI-adaptedProof: 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.

Verified point and uniform linear residual certificate data

Definition

Let $n \geq 1$ , let $A$ be a nonempty set of matrices in $C^{n \times n}$ , and let $b$ be a nonempty subset of $C^{n}$ . A uniform residual certificate consists of a point matrix $R \in C^{n \times n}$ , a point vector $\hat{x} \in C^{n}$ , and numbers $q, r \geq 0$ such that

∥ I - R A ∥_{2} \leq q, ∥ R (b - A \hat{x}) ∥_{2} \leq r

for every $A \in A$ and $b \in b$ . Here $∥ \cdot ∥_{2}$ is the operator norm induced by the Euclidean vector norm, as in The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||. The certificate is usable when $q < 1$ ; in that case its reported error radius is $r / (1 - q)$ . The point-data case is $A = {A}$ and $b = {b}$ .

This definition specifies mathematical upper bounds; it does not specify how a computation establishes them.

Depends on

The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||

Used by

Dependency tree · two levels

2 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

Siegfried M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica 19 (2010), §10.5 (standard reference, not scraped)
Nicholas J. Higham, Numerical Linear Algebra and Matrix Analysis, MIMS EPrint 2015.103, §1 (standard reference, not scraped)