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Four two-node Gaussian rules give a sixteen-node tensor rule

Example

Take four coordinate moment data sets, each with a two-node Gaussian rule $Q_{j} (f) = w_{j, 1} f (x_{j, 1}) + w_{j, 2} f (x_{j, 2})$ . The tensor rule has node set ${(x_{1, i_{1}}, x_{2, i_{2}}, x_{3, i_{3}}, x_{4, i_{4}}) : i_{j} \in {1, 2}},$ so it has $2^{4} = 16$ distinct nodes. It is exact for every coordinatewise polynomial $P$ with degree at most three in each variable. In particular, $Q_{\otimes} (x_{1}^{3} x_{2}^{2} x_{3} x_{4}^{0}) = \prod_{j = 1}^{4} (\sum_{i = 1}^{2} w_{j, i} x_{j, i}^{α_{j}}) = \prod_{j = 1}^{4} \int x^{α_{j}} d μ_{j} (x), (α_{1}, α_{2}, α_{3}, α_{4}) = (3, 2, 1, 0) .$

Facts & Assumptions

Given: Four order-three moment data sets and their two-node Gaussian rules.

[F1]

Tensor-product Gaussian quadrature: the tensor nodes are all coordinate tuples and the tensor weights are products of the coordinate weights.

[F2]

Coordinatewise product-moment functional for tensor polynomials: the product-moment functional on a coordinate monomial is the product of its coordinate moments.

[F3]

Tensor-product Gaussian quadrature preserves coordinatewise exactness: the tensor rule is exact for the stated coordinatewise degree ranges and has the product number of distinct nodes.

[F4]

Gaussian quadrature is exact through degree 2m−1 and has positive weights: each coordinate rule integrates every degree-at-most-three monomial exactly.

[F5]

The Gauss orthogonal polynomial has simple roots inside the interval: each two-node Gaussian rule has distinct nodes.

Proof

1.1F1construct

Since each coordinate has two distinct nodes by [F5], the sixteen coordinate tuples are distinct: two different tuples differ in a coordinate whose node entries are distinct.

2.1F1F2F3F4step 1.1algebra∎

The node-count statement of [F3] gives $2 \cdot 2 \cdot 2 \cdot 2 = 16$ . For each coordinate degree $α_{j} \leq 3$ , [F4] evaluates the corresponding one-dimensional sum as its moment. Distributing the finite sums in [F1] therefore gives the product of those four moments, which is [F2]. Taking the specified exponents proves the displayed calculation and the general coordinatewise exactness claim.

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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)