AlgorithmAlgorithm: 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.

Composite-Simpson assembly of the Chebyshev weighted mass matrix

Definition

Input a polynomial degree $N \geq 0$ and an even integer $M \geq 2$ specifying the number of uniform subintervals of $[0, π]$ . For $0 \leq i, j \leq N$ , form

g_{i j} (θ) = T_{i} (\cos θ) T_{j} (\cos θ)

and return

M_{i j}^{S} : = Q_{S} (g_{i j}; 0, π, M),

where $Q_{S}$ is the composite Simpson procedure from Composite Simpson rule on an even uniform grid. By the Chebyshev multiple-angle identity, $g_{i j} (θ) = \cos (i θ) \cos (j θ)$ , so this is the composite-Simpson approximation of the corresponding arcsine-weighted mass entry. The procedure uses exact real arithmetic and asserts no quadrature or floating-point accuracy beyond a separately proved bound.

Depends on

Used by

Dependency tree · two levels

16 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

Jie Shen and Tao Tang, Spectral and High-Order Methods with Applications, §§3.1–3.3 (standard reference, not scraped)
Art B. Owen, Monte Carlo Theory, Methods and Examples, §7.2 (standard reference, not scraped)