Alphabeta MathHow 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.
Tensor-product Gaussian quadrature
Definition
Let and, for each coordinate , let be a well-defined Gaussian rule as in The m-node Gaussian quadrature rule for a positive measure and The Gaussian nodes, cardinal polynomials and weights are well-defined. Its tensor-product quadrature rule is Its node set is the Cartesian product of the one-dimensional node sets; the displayed finite sum is its value on any real-valued function defined on that node set.
Depends on
- The m-node Gaussian quadrature rule for a positive measure
- The Gaussian nodes, cardinal polynomials and weights are well-defined
- Finite sums and finite products, by recursion
- The product $g_0 g_1 \cdots g_{n-1}$ of a finite list in a monoid, by recursion, with the empty product ($n = 0$) equal to the identity
Used by
- Tensor Gaussian rules recover polynomial-chaos coefficients exactly Corollary
- A signed Simpson-Smolyak weight makes a nonnegative chaos coefficient estimate negative Example
- Four two-node Gaussian rules give a sixteen-node tensor rule Example
- Tensor-product Gaussian quadrature preserves coordinatewise exactness Theorem
Dependency tree · two levels
27 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
- John A. Gubner, Gaussian Quadrature and the Eigenvalue Problem, §§2–5, 2020 (standard reference, not scraped)
- MATH 373 Lecture Notes, Lecture 14: Gaussian Quadrature, §§10.2–10.5 (standard reference, not scraped)
- Walter Gautschi, The Interplay Between Classical Analysis and (Numerical) Linear Algebra, §2.1 (standard reference, not scraped)