ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Relative error becomes unavailable at an Airy zero

Statement

There is a real $ξ < 0$ with $Ai (ξ) = 0$ . An absolute enclosure remains meaningful there, but the relative certificate in Certified special-function enclosure result cannot be reported by an enclosure containing zero. Near $ξ$ , relative certification requires a radius proportional to the small value $∣ Ai (x) ∣$ and therefore requires increasingly fine absolute enclosures.

Facts & Assumptions

Given: A negative real Airy zero $ξ$ and a real input $x$ near $ξ$ .

[F1]

The Airy Ai function has a negative real zero: some $ξ < 0$ satisfies $Ai (ξ) = 0$ .

[F2]

Airy coefficient recurrence and geometric tail bounds: $Ai$ is entire, hence continuous on the real axis.

[F3]

Certified special-function enclosure result: relative certification requires $0 \notin B$ and $rad (B) / d (B, 0) \leq ε_{r}$ .

[F4]

Soundness of finite Airy enclosures: successful Airy evaluations provide absolute enclosures even when no relative certificate is available.

Proof

1.1F1F3F4

At $x = ξ$ , every enclosure containing the exact value also contains $0$ . Therefore the zero-exclusion condition in [F3] fails, while [F4] still permits an absolute-radius success enclosure.

2.1F1F2F3∎

For any $η > 0$ , continuity in [F2] at the zero [F1] gives a neighborhood of $ξ$ on which $∣ Ai (x) ∣ < η$ . Whenever a relative certificate succeeds for an enclosure $B$ of such a value, [F3] implies $rad (B) \leq ε_{r} d (B, 0) \leq ε_{r} ∣ Ai (x) ∣ < ε_{r} η$ . Thus the required absolute radius tends to zero as the input approaches $ξ$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

DLMF, §9.17(i), numerical evaluation of Airy functions (standard reference, not scraped)
DLMF, §§9.9(i) and 9.17(i) (standard reference, not scraped)
N. M. Temme, “Numerical aspects of special functions,” Acta Numerica 17 (2008), §2 (standard reference, not scraped)