scipy.special.

roots_jacobi#

scipy.special.roots_jacobi(n, alpha, beta, mu=False)[source]#

Gauss-Jacobi quadrature.

Compute the sample points and weights for Gauss-Jacobi quadrature. The sample points are the roots of the nth degree Jacobi polynomial, \(P^{\alpha, \beta}_n(x)\). These sample points and weights correctly integrate polynomials of degree \(2n - 1\) or less over the interval \([-1, 1]\) with weight function \(w(x) = (1 - x)^{\alpha} (1 + x)^{\beta}\). See 22.2.1 in [AS] for details.

Parameters:

nint: Quadrature order.
alphafloat: alpha must be > -1
betafloat: beta must be > -1
mubool, optional: If True, return the sum of the weights in addition to sample points and weights.

Returns:

xndarray: Sample points.
wndarray: Weights.
mufloat, optional: Sum of the weights, only returned if mu=True.

See also

scipy.integrate.fixed_quad

References

[AS]

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.

Examples

>>> from scipy.special import roots_jacobi
>>> x, w = roots_jacobi(3, 0.5, 0.5)
>>> x
array([-0.70710678,  0.        ,  0.70710678])
>>> w
array([0.39269908, 0.78539816, 0.39269908])

>>> x, w, mu = roots_jacobi(3, 0.5, 0.5, mu=True)
>>> mu
1.5707963267948966  # Sum of weights, equals pi/2 for alpha = beta = 0.5