Quadrature¶

Gaussian quadrature utilities for use with the Python Active-subspaces Utility Library.

active_subspaces.utils.quadrature.g1d(N, quadtype)¶

One-dimensional Gaussian quadrature rule.

Parameters:

N (int) – number of nodes in the quadrature rule
quadtype (str) – type of quadrature rule {‘Legendre’, ‘Hermite’}

Returns:

x (ndarray) – N-by-1 array of quadrature nodes
w (ndarray) – N-by-1 array of quadrature weights

See also

utils.quadrature.gauss_hermite()

Notes

This computation is inspired by Walter Gautschi’s code at https://www.cs.purdue.edu/archives/2002/wxg/codes/OPQ.html.

active_subspaces.utils.quadrature.gauss_hermite(N)¶

Tensor product Gauss-Hermite quadrature rule.

Parameters:	N (int[]) – number of nodes in each dimension of the quadrature rule
Returns:	x (ndarray) – N-by-1 array of quadrature nodes w (ndarray) – N-by-1 array of quadrature weights

Notes

This computation is inspired by Walter Gautschi’s code at https://www.cs.purdue.edu/archives/2002/wxg/codes/OPQ.html.

active_subspaces.utils.quadrature.gauss_legendre(N)¶

Tensor product Gauss-Legendre quadrature rule.

Parameters:	N (int[]) – number of nodes in each dimension of the quadrature rule
Returns:	x (ndarray) – N-by-1 array of quadrature nodes w (ndarray) – N-by-1 array of quadrature weights

Notes

This computation is inspired by Walter Gautschi’s code at https://www.cs.purdue.edu/archives/2002/wxg/codes/OPQ.html.

active_subspaces.utils.quadrature.gh1d(N)¶

One-dimensional Gauss-Hermite quadrature rule.

Parameters:	N (int) – number of nodes in the quadrature rule
Returns:	x (ndarray) – N-by-1 array of quadrature nodes w (ndarray) – N-by-1 array of quadrature weights

See also

utils.quadrature.gauss_hermite()

Notes

This computation is inspired by Walter Gautschi’s code at https://www.cs.purdue.edu/archives/2002/wxg/codes/OPQ.html.

active_subspaces.utils.quadrature.gl1d(N)¶

One-dimensional Gauss-Legendre quadrature rule.

Parameters:	N (int) – number of nodes in the quadrature rule
Returns:	x (ndarray) – N-by-1 array of quadrature nodes w (ndarray) – N-by-1 array of quadrature weights

See also

utils.quadrature.gauss_legendre()

Notes

This computation is inspired by Walter Gautschi’s code at https://www.cs.purdue.edu/archives/2002/wxg/codes/OPQ.html.

active_subspaces.utils.quadrature.jacobi_matrix(ab)¶

Tri-diagonal Jacobi matrix of recurrence coefficients.

Parameters:	ab (ndarray) – N-by-2 array of recurrence coefficients
Returns:	J – (N-1)-by-(N-1) symmetric, tridiagonal Jacobi matrix associated with the orthogonal polynomials
Return type:	ndarray

See also

utils.quadrature.r_hermite(), utils.quadrature.gauss_hermite()

Notes

This computation is inspired by Walter Gautschi’s code at https://www.cs.purdue.edu/archives/2002/wxg/codes/OPQ.html.

active_subspaces.utils.quadrature.r_hermite(N)¶

Recurrence coefficients for the Hermite orthogonal polynomials.

Parameters:	N (int) – the number of recurrence coefficients
Returns:	ab – an N-by-2 array of the recurrence coefficients
Return type:	ndarray

See also

utils.quadrature.jacobi_matrix(), utils.quadrature.gauss_hermite()

Notes

This computation is inspired by Walter Gautschi’s code at https://www.cs.purdue.edu/archives/2002/wxg/codes/OPQ.html.

active_subspaces.utils.quadrature.r_jacobi(N, l, r, a, b)¶

Recurrence coefficients for the Legendre orthogonal polynomials.

Parameters:	N (int) – the number of recurrence coefficients l (float) – the left endpoint of the interval r (float) – the right endpoint of the interval a (float) – Jacobi weight parameter b (float) – Jacobi weight parameter
Returns:	ab – an N-by-2 array of the recurrence coefficients
Return type:	ndarray

See also

utils.quadrature.jacobi_matrix(), utils.quadrature.gauss_legendre()

Notes

This computation is inspired by Walter Gautschi’s code at https://www.cs.purdue.edu/archives/2002/wxg/codes/OPQ.html.