new_contents: # -*- coding: utf-8 -*-
  
  from sympy import Poly, legendre_poly, diff
  from sympy.abc import x
  
  from mpmath import polyroots
  
  def gauss_legendre_points(n):
      """Returns the Gauss-Legendre quadrature points for order *n*
  
      These are defined as the roots of P_n where P_n is the n'th
      Legendre polynomial.
      """
      coeffs = Poly(legendre_poly(n, x)).all_coeffs()
  
      roots = polyroots([float(c) for c in coeffs])
      return [float(r) for r in roots]
  
  def lobatto_points(n):
      """Returns the Lobatto quadrature points for order *n*
  
      These are defined as the roots of P'_(n-1) where P'_(n-1) is the
      first derivative of the n'th - 1 Legendre polynomial plus the
      points -1.0 and +1.0.
      """
      coeffs = Poly(diff(legendre_poly(n-1, x), x)).all_coeffs()
  
      roots = polyroots([float(c) for c in coeffs])
      return [-1.0] + [float(r) for r in roots] + [1.0]
  
subject: Add some polynomial-related functions. These will be useful later.