scipy.special.

roots_genlaguerre#

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

Gauss-generalized Laguerre quadrature.

Compute the sample points and weights for Gauss-generalized Laguerre quadrature. The sample points are the roots of the nth degree generalized Laguerre polynomial, \(L^{\alpha}_n(x)\). These sample points and weights correctly integrate polynomials of degree \(2n - 1\) or less over the interval \([0, \infty]\) with weight function \(w(x) = x^{\alpha} e^{-x}\). See 22.3.9 in [AS] for details.

Parameters:

nint: quadrature order
alphafloat: alpha must be > -1
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

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_genlaguerre
>>> roots, weights = roots_genlaguerre(5, 1.5)
>>> roots
array([ 0.81763176,  2.47233393,  5.11600612,  9.04414651, 15.04988168])
>>> weights
array([3.96031087e-01, 6.94687948e-01, 2.23227600e-01, 1.52629335e-02,
       1.30819389e-04])

Verify that the values in roots are roots of the generalized Laguerre polynomial \(L^{1.5}_5(x)\).

>>> from scipy.special import eval_genlaguerre
>>> eval_genlaguerre(5, 1.5, roots)
array([-3.25585912e-16, -3.25585912e-16,  6.51171825e-16, -1.56281238e-14,
        4.16749968e-14])

The values of \(L^{1.5}_5(x)\) 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 generalized Laguerre polynomial at the roots.

Verify that the sum of the weights equals the integral from 0 to \(\infty\) of \(x^\alpha\exp(-x)\) which for \(\alpha=1.5\) evaluates to \(3\sqrt{\pi}/4\). There are two ways to obtain the sum of weights, both yielding the expected result within numerical precision.

>>> sum(weights)
np.float64(1.3293403881791368)
>>> roots, weights, sum_of_weights = roots_genlaguerre(5, 1.5, mu=True)
>>> sum_of_weights
np.float64(1.329340388179137)
>>> from math import pi, sqrt
>>> 0.75*sqrt(pi)
1.329340388179137

Roots and weights obtained from the generalized Laguerre polynomial \(L^\alpha_n(x)\) are used in Gauss-Laguerre quadrature where the integral from 0 to \(\infty\) of \(f(x)x^\alpha\exp(-x)\) 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(287.88527781504445)

This result agrees with the exact value of \(10395\sqrt{\pi}/64\) within numerical precision.

>>> (10395/64)*sqrt(pi)
287.88527781504433

In general, Gauss-Laguerre quadrature will only yield an approximate value of the integral. Consider the integral from 0 to \(\infty\) of \(\cos(x)x^{1.5}\exp(-x)\).

>>> import numpy as np
>>> weights @ np.cos(roots)
np.float64(-0.20056731778593656)

Because of the small number of nodes, this result is rather imprecise.

>>> from scipy.integrate import quad
>>> import numpy as np
>>> quad(lambda x: np.cos(x) * x**1.5 * np.exp(-x), 0, np.inf)
(-0.2138889584997946, 1.480038372176226e-08)

Here, the first number refers to the result while the second one gives an estimate of the absolute error. A better result can be obtained by means of Gauss-Laguerre quadrature by choosing a larger number of nodes.

>>> roots, weights = roots_genlaguerre(50, 1.5)
>>> weights @ np.cos(roots)
np.float64(-0.21388895849684425)