scipy.special.

roots_laguerre#

scipy.special.roots_laguerre(n, mu=False)[source]#

Gauss-Laguerre quadrature.

Compute the sample points and weights for Gauss-Laguerre quadrature. The sample points are the roots of the nth degree Laguerre polynomial, \(L_n(x)\). These sample points and weights correctly integrate polynomials of degree \(2n - 1\) or less over the interval \([0, \infty]\) with weight function \(w(x) = e^{-x}\). See 22.2.13 in [AS] for details.

Parameters:

nint: quadrature order
mubool, optional: If True, return the sum of the weights, optional.

Returns:

xndarray: Sample points
wndarray: Weights
mufloat: Sum of the weights

See also

scipy.integrate.fixed_quad
numpy.polynomial.laguerre.laggauss

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_laguerre
>>> roots, weights = roots_laguerre(5)
>>> roots
array([ 0.26356032,  1.41340306,  3.59642577,  7.08581001, 12.64080084])
>>> weights
array([5.21755611e-01, 3.98666811e-01, 7.59424497e-02, 3.61175868e-03,
       2.33699724e-05])

Verify that the values in roots are roots of the Laguerre polynomial \(L_5(x)\).

>>> from scipy.special import eval_laguerre
>>> eval_laguerre(5, roots)
array([-5.55111512e-17, -5.55111512e-17,  2.22044605e-16, -1.77635684e-15,
       -5.68434189e-14])

The values of \(L_5(x)\) evaluated at the roots are indeed rather close to zero. The increasing values for larger roots can be explained by the increasing derivative of the Laguerre polynomial at the roots.

Verify that the sum of the weights equals the integral from 0 to \(\infty\) of \(\exp(-x)\) which evaluates to 1. There are two ways to obtain the sum of weights, both resulting in 1 within numerical precision.

>>> sum(weights)
np.float64(0.9999999999999998)
>>> roots, weights, sum_of_weights = roots_laguerre(5, mu=True)
>>> sum_of_weights
np.float64(1.0)

Roots and weights obtained from the Laguerre polynomial \(L_n(x)\) are used in Gauss-Laguerre quadrature where the integral from 0 to \(\infty\) of \(f(x)\exp(-x)\) is evaluated. Roots and weights for order \(n\) are expected to yield the exact result for polynomials \(f(x)\) of a maximal order of \(2n-1\).

>>> f = lambda x: x**4
>>> weights @ f(roots)
np.float64(24.000000000000014)

This result is indeed very close to the exact value of 24.

In general, Gauss-Laguerre quadrature will only yield an approximate value of the integral. Consider the integral from 0 to \(\infty\) of \(\cos(x)\exp(-x)\) which evaluates to 1/2.

>>> import numpy as np
>>> weights @ np.cos(roots)
np.float64(0.5005384852176379)