scipy.special.

roots_sh_legendre#

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

Gauss-Legendre (shifted) quadrature.

Compute the sample points and weights for Gauss-Legendre quadrature. The sample points are the roots of the nth degree shifted Legendre polynomial \(P^*_n(x)\). These sample points and weights correctly integrate polynomials of degree \(2n - 1\) or less over the interval \([0, 1]\) with weight function \(w(x) = 1.0\). See 2.2.11 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

References

[AS]

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

Examples

Compute nodes and weights for a 7th-order shifted Gauss-Legendre quadrature:

>>> from scipy.special import roots_sh_legendre, eval_sh_legendre
>>> x, w = roots_sh_legendre(7)
>>> x
array([0.02544604, 0.12923441, 0.29707742, 0.5, 0.70292258, 0.87076559,
    0.97455396])
>>> w
array([0.06474248, 0.1398527, 0.19091503, 0.20897959, 0.19091503, 0.1398527,
    0.06474248])

Verify that x are the roots of the degree-7 shifted Legendre polynomial:

>>> eval_sh_legendre(7, x)
array([5.55111512e-16, 1.11022302e-16,  3.33066907e-16,  0.00000000e+00,
    -2.22044605e-16, -1.11022302e-16, -1.85962357e-15])

The sum of the weights for shifted Gauss-Legendre quadrature is always 1:

>>> x, w, mu = roots_sh_legendre(10, mu=True)
>>> mu 
1.0  # Sum of weights of shifted Gauss-Legendre quadrature is always 1