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# numpy.polynomial.hermite.hermgauss

polynomial.hermite.hermgauss(deg)

  Gauss-Hermite quadrature.

  Computes the sample points and weights for Gauss-Hermite quadrature. These sample points and weights will correctly integrate polynomials of degree \(2*deg - 1\) or less over the interval \([-\inf, \inf]\) with the weight function \(f(x) = \exp(-x^2)\).

  Parameters:
    deg int

      Number of sample points and weights. It must be >= 1.

  Returns:
    x ndarray

      1-D ndarray containing the sample points.

    y ndarray

      1-D ndarray containing the weights.

  Notes

  The results have only been tested up to degree 100, higher degrees may be problematic. The weights are determined by using the fact that

  \[w_k = c / (H'_n(x_k) * H_{n-1}(x_k))\]

  where \(c\) is a constant independent of \(k\) and \(x_k\) is the k’th root of \(H_n\), and then scaling the results to get the right value when integrating 1.

  Examples

      >>> from numpy.polynomial.hermite import hermgauss
      >>> hermgauss(2)
      (array([-0.70710678,  0.70710678]), array([0.88622693, 0.88622693]))