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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.

A fast component restricts the forward-Euler step in a diagonal fast-slow system

Facts & Assumptions

Given: The diagonal system $y_{1}^{'} = - y_{1}$ , $y_{2}^{'} = - 1000 y_{2}$ , and forward Euler with one common step $h > 0$ .

[F1]

Forward Euler has the closed unit disk centered at −1 as its absolute-stability region: a scalar decay equation $y^{'} = - ρ y$ is absolutely stable under forward Euler exactly when $0 < h ρ \leq 2$ .

Example

The slow component alone permits every $0 < h \leq 2$ , but requiring absolute stability for both components gives $0 < h \leq 1 / 500$ . This example exhibits how a fast decay scale can constrain a step selected to track a slower component.

Verification

1.1givenalgebra

Applying forward Euler to each diagonal component gives the multipliers $1 - h$ and $1 - 1000 h$ .

2.1F1step 1.1algebra∎

By [F1], the first component is stable exactly when $0 < h \leq 2$ and the second exactly when $0 < 1000 h \leq 2$ . Their intersection is $0 < h \leq 1 / 500$ .

Depends on

Forward Euler has the closed unit disk centered at −1 as its absolute-stability region

Used by

Nothing in the library uses this result yet.

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

Ernst Hairer and Gerhard Wanner, Introduction à l’Analyse Numérique, Chapter III, §III.9 (standard reference, not scraped)
Masayuki Yano, James Douglass Penn, George Konidaris, and Anthony T. Patera, Math, Numerics, and Programming (for Mechanical Engineers), Unit IV, Chapter 21 (standard reference, not scraped)