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.

Periodic Fourier collocation differentiation matrix

Definition

Input an integer $N \geq 2$ and real data $U_{0}, \dots, U_{N - 1}$ on the grid of Periodic Fourier collocation grid and even-grid Nyquist convention. Set $K = (N - 1) / 2$ when $N$ is odd and $K = N / 2$ when $N$ is even. Form the $(N \times N)$ matrix $(D_{N})_{i j} = {\begin{cases} - \frac{2}{N} \sum_{k = 1}^{K} k \sin (k (x_{i} - x_{j})), & N odd, \\ - \frac{2}{N} \sum_{k = 1}^{K - 1} k \sin (k (x_{i} - x_{j})), & N even, \end{cases} 0 \leq i, j < N .$ The algorithm returns $D_{N}$ and the derivative vector $D_{N} U$ . In the even case it omits the derivative of the Nyquist cosine because $\sin ((N / 2) (x_{i} - x_{j})) = 0$ for grid indices $i, j$ . All operations are in exact real arithmetic; no floating-point stability or error guarantee is asserted.

Depends on

Used by

Dependency tree · two levels

19 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, §2.2 (standard reference, not scraped)
Lloyd N. Trefethen, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, Ch. 7 (standard reference, not scraped)