TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge 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.

M-node Gauss collocation has global order 2M

Statement

Assume Countable Choice as required by M-node Gauss collocation has local defect order 2M. Let $M \geq 1$ and suppose the hypotheses of that theorem and Small-step collocation maps are Lipschitz stable hold uniformly on a fixed interval $[0, T]$ . Starting with the exact initial value and using the $M$ -node Gauss collocation endpoint map on a uniform mesh, the maximum grid error is $O (h^{2 M})$ as $h = T / N ↓ 0$ .

Facts & Assumptions

Given: The exact IVP solution, a uniform mesh, the $M$ -node Gauss collocation endpoint map, and the common-tube and branch hypotheses.

[F1]

M-node Gauss collocation has local defect order 2M: the exact-start defect is uniformly $O (h^{2 M + 1})$ .

[F2]

Small-step collocation maps are Lipschitz stable: for sufficiently small $h$ , the endpoint map has a uniform Lipschitz bound $1 + L_{0} h$ .

[F3]

Global error from uniform one-step defect and stability: a uniform exact-start $O (h^{p + 1})$ defect and $(1 + L_{0} h)$ stability on a fixed interval imply global error $O (h^{p})$ from exact initial data.

[A1]

The exact solution, numerical grid states, stages and comparison segments remain in the common tube on every mesh step.

Proof

1.1A1F1F3

By [F1], for a constant independent of the mesh and step index, $∥ Ψ_{h, n} (y (t_{n})) - y (t_{n + 1}) ∥_{2} \leq C h^{2 M + 1}$ . This is the local-defect hypothesis of [F3] with $p = 2 M$ .

2.1A1F2F3step 1.1∎

By [F2] the stability premise of [F3] holds uniformly; the starting value is exact. Applying [F3] with $p = 2 M$ gives $\max_{n} ∥ y_{n} - y (t_{n}) ∥_{2} = O (h^{2 M})$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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

Ernst Hairer, Syvert P. Nørsett, and Gerhard Wanner, Solving Ordinary Differential Equations I, 2nd ed., Chapter II §II.7, Theorems 7.9–7.10 (standard reference, not scraped)
Ernst Hairer and Gerhard Wanner, Introduction à l’Analyse Numérique, Chapter III: Équations Différentielles Ordinaires (standard reference, not scraped)