Gaussian quadrature rules

E697759

numerical integration method quadrature rule

Gaussian quadrature rules are numerical integration methods that approximate definite integrals by optimally choosing evaluation points and weights to achieve exactness for polynomials up to a high degree.

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Observed surface forms (1)

Surface form	Occurrences
Gauss–Legendre Runge–Kutta methods	1

Statements (46)

Predicate	Object
instanceOf	numerical integration method ⓘ quadrature rule ⓘ
advantage	high accuracy with relatively few nodes ⓘ
application	computational engineering ⓘ computational finance ⓘ computational physics ⓘ finite element methods ⓘ probability and statistics ⓘ
appliesTo	integrals with weight functions associated to orthogonal polynomials ⓘ
assumption	integrand is sufficiently smooth ⓘ
constructionMethod	solve moment-matching conditions for polynomials ⓘ use three-term recurrence of orthogonal polynomials ⓘ
contrastWith	Newton–Cotes rules that use equally spaced nodes ⓘ
degreeOfExactness	2n-1 for n nodes ⓘ
disadvantage	less effective for highly oscillatory integrands without adaptation ⓘ nodes depend on integrand weight function and interval ⓘ
errorTerm	proportional to (2n)th derivative of integrand for n-node rule under smoothness assumptions ⓘ
field	numerical analysis ⓘ
generalization	Gauss–Kronrod rules NERFINISHED ⓘ adaptive Gaussian quadrature ⓘ
historicalOrigin	19th century ⓘ
mathematicalBasis	theory of orthogonal polynomials and moment problems ⓘ
namedAfter	Carl Friedrich Gauss NERFINISHED ⓘ
nodeDistribution	nodes are interior to the interval for standard Gauss rules ⓘ
nodeSelectionCriterion	nodes are zeros of an orthogonal polynomial of degree n ⓘ
optimize	choice of evaluation points ⓘ choice of weights ⓘ
property	exactness for polynomials up to maximal possible degree for given number of nodes ⓘ
purpose	approximation of definite integrals ⓘ
relatedTo	Clenshaw–Curtis quadrature NERFINISHED ⓘ Newton–Cotes formulas NERFINISHED ⓘ spectral methods ⓘ
requires	precomputation or tabulation of nodes and weights ⓘ
specialCase	Gauss–Chebyshev quadrature NERFINISHED ⓘ Gauss–Hermite quadrature NERFINISHED ⓘ Gauss–Jacobi quadrature NERFINISHED ⓘ Gauss–Laguerre quadrature NERFINISHED ⓘ Gauss–Legendre quadrature NERFINISHED ⓘ
typicalDomain	integration over a finite interval ⓘ
typicalImplementation	use of eigenvalue problems for Jacobi matrices to compute nodes and weights ⓘ
typicalInterval	[-1,1] after change of variables ⓘ
usedIn	Gaussian process quadrature variants ⓘ high-precision numerical integration ⓘ
uses	orthogonal polynomials ⓘ roots of orthogonal polynomials as nodes ⓘ
weightProperty	weights are positive for standard Gaussian rules ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Jacobi polynomials → usedIn → Gaussian quadrature rules ⓘ

Runge–Kutta methods → hasSubclass → Gaussian quadrature rules ⓘ

this entity surface form: Gauss–Legendre Runge–Kutta methods