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.

Clenshaw–Curtis quadrature with the midpoint zero-degree case

Definition

Let $a < b$ , $n \in N$ , $m = (a + b) / 2$ , $d = (b - a) / 2$ , and $f \in C [a, b]$ . For $n \geq 1$ , set $g (t) : = f (m + d t)$ on $[- 1, 1]$ , let $y_{j} = g (\cos (j π / n))$ , and form $a_{0}, \dots, a_{n}$ from the DCT-I coefficients in DCT-I coefficients of the Chebyshev–Lobatto interpolant. With the Chebyshev moments $μ_{k}$ of Exact integrals of Chebyshev polynomials, define $Q_{n, [a, b]} (f) : = d \sum_{k = 0}^{n} a_{k} μ_{k} .$ For $n = 0$ , define instead $Q_{0, [a, b]} (f) : = (b - a) f (m),$ the midpoint rule. This is the explicit endpoint and zero-degree convention for Clenshaw–Curtis quadrature. The sums and coefficients are exact real quantities; finite-precision evaluation is a separate algorithm.

Depends on

Used by

Dependency tree · two levels

36 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

Lloyd N. Trefethen, Is Gauss Quadrature Better than Clenshaw–Curtis?, §2 (standard reference, not scraped)
L. Ridgway Scott, Numerical Analysis, 2nd ed., §13.4.1 (standard reference, not scraped)