Function Repository Resource:

GaussianQuadratureWeights

Get a numerically sorted list of abscissa-weight pairs for Gaussian quadrature

Contributed by: Paul Abbott (additional contributions by Wolfram Research staff)

ResourceFunction["GaussianQuadratureWeights"][n,{a,b}]

gives a list of the n pairs {x_i,w_i} of the elementary n-point Gaussian formula for quadrature on the interval a to b, where w_i is the weight of the abscissa x_i.

ResourceFunction["GaussianQuadratureWeights"][n,{a,b},prec]

uses the working precision prec.

Details and Options

Gaussian quadrature approximates the value of an integral as a linear combination of values of the integrand evaluated at optimal abscissas x_i:

The abscissas are optimal in the sense that the quadrature formula is exact for all polynomials up to degree 2n-1.

The precision argument acts similarly to the WorkingPrecision option used in many Wolfram Language numeric functions; it is not at all like the PrecisionGoal option. If the n argument is too small to allow for a high-precision result, extra precision in a result will not be meaningful.

Examples

Basic Examples (3)

The abscissas and weights for a 10-point quadrature on a given interval:

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The exact abscissas and weights:

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Use the specified precision:

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Applications (3)

Use the Gaussian quadrature to approximate the area under a curve:

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Plot the curve over a given interval:

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Approximate the area under the curve for a specific n:

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$area[n_] := Module[{a, w}, {a, w} = Transpose@ ResourceFunction["GaussianQuadratureWeights"][n, {-0.5, 1.5}]; w . (f /@ a)]$

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Compare to the output of NIntegrate:

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The difference in Gaussian quadratures of two different orders:

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The abscissas and weights for an n-point quadrature on {-1,1} for n=2 … 5:

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The triangle of polynomials whose roots determine the weights for n up to 8:

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Dataset@Table[
Association[
n -> TraditionalForm@
MinimalPolynomial[
ResourceFunction["GaussianQuadratureWeights"][n, {-1, 1}][[1, 2]], x]], {n, 8}]

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Properties and Relations (3)

For the interval {-1,1}, abscissas for the n-point Gaussian quadrature are the roots of the Legendre polynomial P_n(x):

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The weights at abscissas x_i are 2(1-x_i²)/((n+1)²P_n+1(x_i))²):

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$% === (Table[( 2 (1 - xx[[i]]^2))/((n + 1)^2 LegendreP[n + 1, xx[[i]]]^2), {i, n}] // Simplify)$

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The sum of the weights is exactly 2:

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Resource function GaussianQuadratureError gives an upper bound of the error in the weights used for the approximation:

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The abscissas and weights for the elementary n-point Gaussian quadrature are related to the eigensystem of the n⨯n symmetric tridiagonal Gaussian quadrature matrix:

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