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Gaussian quadrature exactly assembles the orthogonal Galerkin mass matrix

Statement

Let $N \geq 0$ , take moment data of order $2 N + 1$ for a finite positive measure $μ$ with support containing at least $N + 2$ points, and use its $(N + 1)$ -node Gaussian rule and orthogonal polynomials $p_{0}, \dots, p_{N}$ . The matrix from Assemble a weighted Galerkin mass matrix from Gaussian nodes has entries $M_{i j}^{G} = \int p_{i} (x) p_{j} (x) d μ (x) = {\begin{cases} 0, & i \neq j, \\ h_{i}, & i = j . \end{cases}$

Facts & Assumptions

Given: $N \geq 0$ , moment data of order $2 N + 1$ , its $(N + 1)$ -node Gaussian rule, and the corresponding orthogonal polynomials.

[F1]

Gaussian quadrature is exact through degree 2m−1 and has positive weights: for an $m$ -node rule from moment data of order $2 m - 1$ , every polynomial of degree at most $2 m - 1$ integrates exactly.

[F2]

Positive-measure Gaussian quadrature moment data and Monic orthogonal polynomials exist uniquely through the quadrature order with $m = N + 1$ : $p_{i}$ is the monic degree- $i$ polynomial orthogonal to every lower-degree polynomial for $0 \leq i \leq N$ . Write $h_{i} : = \int p_{i}^{2} d μ$ ; these integrals are finite since $2 i \leq 2 N < 2 N + 1$ , within the supplied moment order. Only the stated $N + 2$ support points are needed, not infinite support.

[F3]

Assemble a weighted Galerkin mass matrix from Gaussian nodes: $M_{i j}^{G}$ is the Gaussian sum of $p_{i} p_{j}$ over the $N + 1$ nodes.

Proof

1.1givenF1F2F3algebra

By [F2], the product $p_{i} p_{j}$ has degree $i + j \leq 2 N < 2 N + 1 = 2 (N + 1) - 1$ . Thus [F1] applies with $m = N + 1$ and gives $M_{i j}^{G} = \sum_{r = 1}^{N + 1} w_{r} p_{i} (ξ_{r}) p_{j} (ξ_{r}) = \int p_{i} (x) p_{j} (x) d μ (x)$ by [F3].

2.1F2step 1.1algebra∎

If $i \neq j$ , the higher-degree polynomial is orthogonal to the lower-degree one by [F2], so the integral in step 1.1 is zero. If $i = j$ , it is $h_{i}$ by [F2]. This is the displayed diagonal matrix identity.

Depends on

Used by

Gaussian and Simpson mass assembly give distinct exact and approximate checks Example

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Sources

Jie Shen and Tao Tang, Spectral and High-Order Methods with Applications, §§3.1–3.2 (standard reference, not scraped)
Walter Gautschi, The Interplay Between Classical Analysis and (Numerical) Linear Algebra, §2.1 (standard reference, not scraped)