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The m-node Gaussian quadrature rule for a positive measure
Definition
Let be moment data as in Positive-measure Gaussian quadrature moment data. Let be its monic degree- orthogonal polynomial and write its ordered roots as , as established in The Gauss orthogonal polynomial has simple roots inside the interval. For , define the Lagrange cardinal polynomial using the convention of Cardinal-polynomial interpolation. Define the Gaussian quadrature weights and the m-node Gaussian rule by for any real-valued function defined at the nodes. This definition asserts no accuracy; well-definedness of the nodes, cardinal polynomials, weights, and finite sum is proved in The Gaussian nodes, cardinal polynomials and weights are well-defined ↗.
Depends on
Used by
- Assemble a weighted Galerkin mass matrix from Gaussian nodes Algorithm
- Tensor-product Gaussian quadrature Definition
- The Gauss–Jacobi quadrature rule Definition
- A signed Simpson-Smolyak weight makes a nonnegative chaos coefficient estimate negative Example
- A two-node Gaussian Nyström rule solves a rank-two separable equation exactly Example
- Gaussian and Simpson mass assembly give distinct exact and approximate checks Example
- The two-node Gauss–Legendre rule is exact through degree three Example
- Two Gaussian weights are the squared norms of the two cardinal polynomials Example
- Two-node Gauss collocation gives the exact endpoint for a nonstiff cubic IVP Example
- Two-node Gauss is exact through degree three but not for x to the fourth Example
- A Gaussian collocation defect has a high-order weighted endpoint bound Lemma
- Gauss–Jacobi nodes and weights are well-defined Lemma
- One-node Gauss data are midpoint data Lemma
- The Gaussian nodes, cardinal polynomials and weights are well-defined Lemma
- A tensor product of two positive Gaussian rules has a derivative error bound Theorem
- Gaussian quadrature has the Hermite mean-value remainder Theorem
- Gaussian quadrature is exact through degree 2m−1 and has positive weights Theorem
- Gaussian weights equal the total mass times a Jacobi eigenvector's first-coordinate square Theorem
- M-node Gauss collocation has local defect order 2M Theorem
Dependency tree · two levels
33 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)