DefinitionDefinition: 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.

Euler–Heun embedded Runge–Kutta pair

Definition

For $y^{'} = f (t, y)$ , take the shared explicit stages $K_{1} = f (t_{n}, y_{n}), K_{2} = f (t_{n} + h, y_{n} + h K_{1}) .$ The Euler member is $Y_{E} = y_{n} + h K_{1}$ and the Heun explicit trapezoidal member is $Y_{H} = y_{n} + \frac{h}{2} (K_{1} + K_{2})$ . Their embedded difference is $Y_{H} - Y_{E} = \frac{h}{2} (K_{2} - K_{1})$ . The local orders are proved separately. In Embedded Runge–Kutta pair and difference estimator notation, the common tableau has $c = (0, 1)$ , $A_{21} = 1$ , high weights $(1 / 2, 1 / 2)$ and low weights $(1, 0)$ .

Depends on

Embedded Runge–Kutta pair and difference estimator

Used by

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

Ernst Hairer, course notes, Chapter III: Équations Différentielles Ordinaires, §III.2 (standard reference, not scraped)
Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, Chapter 6 (standard reference, not scraped)