Fri May 19 13:32:57 2023

gegenbauer_exactness_test():
  Python version: 3.8.10
  Test gegenbauer_exactness().

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

gegenbauer_exactness(): User input:
  Quadrature rule X file = "gegen_o8_a0.5_x.txt".
  Quadrature rule W file = "gegen_o8_a0.5_w.txt". gegen_o8_a0.5_w.txt
  Quadrature rule R file = "gegen_o8_a0.5_r.txt". gegen_o8_a0.5_r.txt
  Maximum degree to check =  8
  Exponent of (1-x^2), ALPHA =  0.5

  Number of points  =  8

  Testing a Gauss-Gegenbauer rule
  ORDER =  8
  ALPHA =  0.5

  Standard rule:
    Integral ( -1 <= x <= +1 ) (1-x^2)^alpha f(x) dx
    is to be approximated by
    sum ( 1 <= I <= ORDER ) w(i) * f(x(i)).

  Weights W:

[0.04083295 0.1442256  0.26179939 0.33854023 0.33854023 0.26179939
 0.1442256  0.04083295]

  Abscissas X:

[-0.93969262 -0.76604444 -0.5        -0.17364818  0.17364818  0.5
  0.76604444  0.93969262]

  Region R:

[-1.  1.]

  A Gauss-Gegenbauer rule would be able to exactly
  integrate monomials up to and including 
  degree =  15

      Error    Degree

        0.0000000000000000    0
        0.0000000000000006    1
        0.0000000000000007    2
        0.0000000000000005    3
        0.0000000000000007    4
        0.0000000000000004    5
        0.0000000000000016    6
        0.0000000000000004    7
        0.0000000000000021    8

gegenbauer_exactness_test():
  Normal end of execution.
Fri May 19 13:32:57 2023