Quadrature nodes used in recursivenodes.

recursivenodes.quadrature.gaussjacobi(n, a=0.0, b=0.0)[source]

The $$n$$-point Gauss-Jacobi quadrature points and weights for weight function $$(1+x)^a (1-x)^b$$ for the interval [-1,1], which exactly integrates polynomials with degree up to $$2n-1$$.

Parameters
• n (int) – The number of points.

• a (float, optional) – the left exponent of the weight function.

• b (float, optional) – the right exponent of the weight function.

Returns

Each of shape ($$n$$,), the points (in ascending order) and weights of the quadrature rule.

Return type

(ndarray, ndarray)

Reference:

[GW69]

recursivenodes.quadrature.lobattogaussjacobi(n, a=0.0, b=0.0)[source]

The $$n$$-point Lobatto-Gauss-Jacobi quadrature points and weights for weight function $$(1+x)^a (1-x)^b$$ for the interval [-1,1], which includes the endpoints $$-1$$ and $$1$$ and exactly integrates polynomials with degree up to $$2n-3$$.

Parameters
• n (int) – The number of points.

• a (float, optional) – the left exponent of the weight function.

• b (float, optional) – the right exponent of the weight function.

Returns

Each of shape ($$n$$,), the points (in ascending order) and weights of the quadrature rule.

Return type

(ndarray, ndarray)

Reference:

[Gol73]

recursivenodes.quadrature.simplexgausslegendre(d, n)[source]

An $$n^d$$-point Gaussian quadrature rule for the biunit simplex that exactly integrates polynomials with degree up to $$2n-1$$.

Parameters
• d (int) – The spatial dimension.

• n (int) – The points per coordinate direction

Returns

The points (shape ($$n, \dots, n, d$$)) and weights (shape ($$n, \dots, n,$$)) of the quadrature rule.

Return type

(ndarray, ndarray)