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.

Assemble a weighted Galerkin mass matrix from Gaussian nodes

Definition

Input a degree $N \geq 0$ , the monic orthogonal basis $p_{0}, \dots, p_{N}$ and the $(N + 1)$ -node Gaussian rule for the same positive measure $μ$ from The m-node Gaussian quadrature rule for a positive measure. With Gaussian nodes $ξ_{1}, \dots, ξ_{N + 1}$ and weights $w_{1}, \dots, w_{N + 1}$ , output the matrix

M_{i j}^{G} : = \sum_{r = 1}^{N + 1} w_{r} p_{i} (ξ_{r}) p_{j} (ξ_{r}), 0 \leq i, j \leq N .

The inputs are admissible only when the moment data have order at least $2 N + 1$ and the rule nodes, weights, and basis all use the same measure. The algorithm performs the displayed finite exact-arithmetic sums; no floating-point conditioning claim is made.

Depends on

Used by

Dependency tree · two levels

11 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, §§3.1–3.2 (standard reference, not scraped)
Lloyd N. Trefethen, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, Ch. 8 (standard reference, not scraped)