Tue May 20 21:48:15 2025

legendre_exactness_test():
  python version: 3.10.12
  numpy version:  1.26.4
  Test legendre_exactness().

legendre_exactness():
  Investigate the polynomial exactness of a Gauss-Legendre
  quadrature rule by integrating all monomials up to a given
  degree over the [-1,+1] interval.

User input:
  Quadrature rule X file = "leg_o004_x.txt".
  Quadrature rule W file = "leg_o004_w.txt".
  Quadrature rule R file = "leg_o004_r.txt".
  Maximum degree to check =  10

  The quadrature rule to be tested is
  a Gauss-Legendre rule
  ORDER =  4

  Standard rule:
    Integral ( -1 <= x <= +1 ) f(x) dx
    is to be approximated by
    sum ( 1 <= I <= ORDER ) w(i) * f(x(i)).

  Weights W:

[0.34785485 0.65214515 0.65214515 0.34785485]

  Abscissas X:

[-0.86113631 -0.33998104  0.33998104  0.86113631]

  Region R:

[-1.  1.]

  A Gauss-Legendre rule would be able to exactly
  integrate monomials up to and including 
  degree =  7

  Degree          Error

   0        0.0000000000000002
   1        0.0000000000000000
   2        0.0000000000000000
   3        0.0000000000000000
   4        0.0000000000000001
   5        0.0000000000000000
   6        0.0000000000000000
   7        0.0000000000000000
   8        0.0522448979591839
   9        0.0000000000000000
  10        0.1418075801749273

legendre_exactness_test():
  Normal end of execution.
Tue May 20 21:48:16 2025