ExampleConstruction: AI-generatedVerification: 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.

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A quadratic attains the forward truncation bound and exhibits the balanced noisy step

Example

Take $f (t) = t^{2}$ at $x = 1$ . Its exact forward difference has truncation error $h$ . If each call may err by $δ > 0$ , the allowed errors $e_{1 + h} = δ$ and $e_{1} = - δ$ attain the forward bound $h + 2 δ / h$ . This bound is minimized at $h = \sqrt{2 δ}$ .

Facts & Assumptions

Given: The function, point, positive step and evaluation errors above.

[F1]

Forward and central difference stencils with bounded evaluation errors: $D_{h}^{+} f (1) = (f (1 + h) - f (1)) / h$ and the inexact quotient replaces each value by $f (y) + e_{y}$ .

[F2]

The forward and central error bounds have explicit balancing steps: when $M_{2} = 2$ the forward error bound $M_{2} h / 2 + 2 δ / h$ is minimized at $h = \sqrt{4 δ / M_{2}} = \sqrt{2 δ}$ .

Verification

1.1givenF1algebra

Substitution gives $D_{h}^{+} f (1) = ((1 + h)^{2} - 1) / h = 2 + h$ , while $f^{'} (1) = 2$ ; the truncation error is exactly $h = M_{2} h / 2$ because $f^{''} = 2$ .

2.1givenF1step 1.1algebra

With the stated errors, ${\hat{D}}_{h}^{+} f (1) = ((1 + h)^{2} + δ - (1 - δ)) / h = 2 + h + 2 δ / h$ , so the error bound is attained exactly.

3.1givenF2step 2.1algebra∎

Set $δ = 1 / 2$ . Then [F2] gives $h = 1$ ; the truncation and evaluation contributions are each $1$ , so the total error is $2$ , equal to the minimum $2 \sqrt{2 δ}$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, finite differences (standard reference, not scraped)
Nicholas J. Higham, What Is the Complex Step Approximation? (standard reference, not scraped)