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.

Normalized implicit-midpoint collocation residual for a two-point BVP

Definition

For a $d$ -dimensional BVP on $[a, b]$ , an integer $N \geq 1$ , uniform mesh width $h = (b - a) / N$ , mesh nodes $t_{i} = a + i h$ , and node vector $Y = (Y_{0}, \dots, Y_{N}) \in (R^{d})^{N + 1}$ , let $D_{h}$ be the open set of node vectors for which every midpoint pair $(t_{i} + h / 2, (Y_{i} + Y_{i + 1}) / 2)$ lies in the vector-field domain and $(Y_{0}, Y_{N})$ lies in the boundary-map domain. Define $G_{h} : D_{h} \to (R^{d})^{N + 1}$ by

$G_{h, i} (Y) = \frac{Y_{i + 1} - Y_{i}}{h} - f ⁣ (t_{i} + \frac{h}{2}, \frac{Y_{i} + Y_{i + 1}}{2}) (0 \leq i < N), G_{h, N} (Y) = B (Y_{0}, Y_{N}) .$

Use the maximum of the Euclidean block norms for both node and residual vectors. The interior rows are normalized by $h$ ; consequently an unscaled increment row has size $h$ times its normalized row.

Depends on

Square nonlinear two-point boundary-value problems and endpoint residuals

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

T. A. Driscoll and R. J. Braun, Fundamentals of Numerical Computation, Chapter 10 (standard reference, not scraped)