Skip to main content
`Ctrl`+`K`
scipy.special.

# roots_legendre

scipy.special.roots_legendre(n, mu=False)

  Gauss-Legendre quadrature.

  Compute the sample points and weights for Gauss-Legendre quadrature [GL]. The sample points are the roots of the nth degree Legendre polynomial \(P_n(x)\). These sample points and weights correctly integrate polynomials of degree \(2n - 1\) or less over the interval \([-1, 1]\) with weight function \(w(x) = 1\). See 2.2.10 in [AS] for more details.

  Parameters:
    n int

      quadrature order.

    mu bool, optional

      If True, return the sum of the weights, optional.

  Returns:
    x ndarray

      Sample points.

    w ndarray

      Weights.

    mu float

      Sum of the weights.

  See also

  `scipy.integrate.fixed_quad`
  `numpy.polynomial.legendre.leggauss`

  References

  [AS]

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

  [GL] (1,2)

  Gauss-Legendre quadrature, Wikipedia, https://en.wikipedia.org/wiki/Gauss%E2%80%93Legendre_quadrature

  Examples

      >>> import numpy as np
      >>> from scipy.special import roots_legendre, eval_legendre
      >>> roots, weights = roots_legendre(9)

  `roots` holds the roots, and `weights` holds the weights for Gauss-Legendre quadrature.

      >>> roots
      array([-0.96816024, -0.83603111, -0.61337143, -0.32425342,  0.        ,
              0.32425342,  0.61337143,  0.83603111,  0.96816024])
      >>> weights
      array([0.08127439, 0.18064816, 0.2606107 , 0.31234708, 0.33023936,
             0.31234708, 0.2606107 , 0.18064816, 0.08127439])

  Verify that we have the roots by evaluating the degree 9 Legendre polynomial at `roots`. All the values are approximately zero:

      >>> eval_legendre(9, roots)
      array([-8.88178420e-16, -2.22044605e-16,  1.11022302e-16,  1.11022302e-16,
              0.00000000e+00, -5.55111512e-17, -1.94289029e-16,  1.38777878e-16,
             -8.32667268e-17])

  Here we’ll show how the above values can be used to estimate the integral from 1 to 2 of f(t) = t + 1/t with Gauss-Legendre quadrature [GL]. First define the function and the integration limits.

      >>> def f(t):
      ...    return t + 1/t
      ...
      >>> a = 1
      >>> b = 2

  We’ll use `integral(f(t), t=a, t=b)` to denote the definite integral of f from t=a to t=b. The sample points in `roots` are from the interval [-1, 1], so we’ll rewrite the integral with the simple change of variable:

      x = 2/(b - a) * t - (a + b)/(b - a)

  with inverse:

      t = (b - a)/2 * x + (a + b)/2

  Then:

      integral(f(t), a, b) =
          (b - a)/2 * integral(f((b-a)/2*x + (a+b)/2), x=-1, x=1)

  We can approximate the latter integral with the values returned by `roots_legendre`.

  Map the roots computed above from [-1, 1] to [a, b].

      >>> t = (b - a)/2 * roots + (a + b)/2

  Approximate the integral as the weighted sum of the function values.

      >>> (b - a)/2 * f(t).dot(weights)
      2.1931471805599276

  Compare that to the exact result, which is 3/2 + log(2):

      >>> 1.5 + np.log(2)
      2.1931471805599454