Sun May 21 09:35:00 2023

gen_hermite_exactness_test():
  Python version: 3.8.10
  Test gen_hermite_exactness().

gen_hermite_exactness():
  Investigate the polynomial exactness of a generalized Gauss-Hermite
  quadrature rule by integrating exponentially weighted
  monomials up to a given degree over the (-oo,+oo) interval.

User input:
  Quadrature rule X file = "gen_herm_o8_a1.0_x.txt".
  Quadrature rule W file = "gen_herm_o8_a1.0_w.txt".
  Quadrature rule R file = "gen_herm_o8_a1.0_r.txt".
  Maximum degree to check =  18
  Weighting function exponent ALPHA =  1.0
  OPTION = 0, integrate |x|^alpha*exp(-x*x)*f(x).

  The quadrature rule to be tested is
  a generalized Gauss-Hermite rule
  ORDER =  8
  ALPHA =  1.0

  OPTION = 0, standard rule:
    Integral ( -oo < x < +oo ) |x|^alpha exp(-x*x) f(x) dx
    is to be approximated by
    sum ( 1 <= I <= ORDER ) w(i) * f(x(i)).

  Weights W:

[2.69647353e-04 1.94439543e-02 1.78709346e-01 3.01577052e-01
 3.01577052e-01 1.78709346e-01 1.94439543e-02 2.69647353e-04]

  Abscissas X:

[-3.06513799 -2.12993434 -1.32127253 -0.56793282  0.56793282  1.32127253
  2.12993434  3.06513799]

  Region R:

[-1.e+30  1.e+30]

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

  Degree    Error

   0        0.0000000000000007
   1        0.0000000000000001
   2        0.0000000000000009
   3        0.0000000000000000
   4        0.0000000000000007
   5        0.0000000000000001
   6        0.0000000000000003
   7        0.0000000000000003
   8        0.0000000000000003
   9        0.0000000000000044
  10        0.0000000000000009
  11        0.0000000000000071
  12        0.0000000000000014
  13        0.0000000000000000
  14        0.0000000000000018
  15        0.0000000000000000
  16        0.0142857142857165
  17        0.0000000000000000
  18        0.0650793650793677

gen_hermite_exactness():
  Investigate the polynomial exactness of a generalized Gauss-Hermite
  quadrature rule by integrating exponentially weighted
  monomials up to a given degree over the (-oo,+oo) interval.

User input:
  Quadrature rule X file = "gen_herm_o8_a1.0_modified_x.txt".
  Quadrature rule W file = "gen_herm_o8_a1.0_modified_w.txt".
  Quadrature rule R file = "gen_herm_o8_a1.0_modified_r.txt".
  Maximum degree to check =  18
  Weighting function exponent ALPHA =  1.0
  OPTION = 1, integrate                     f(x).

  The quadrature rule to be tested is
  a generalized Gauss-Hermite rule
  ORDER =  8
  ALPHA =  1.0

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

  Weights W:

[1.0582142  0.85240804 0.7750492  0.73313171 0.73313171 0.7750492
 0.85240804 1.0582142 ]

  Abscissas X:

[-3.06513799 -2.12993434 -1.32127253 -0.56793282  0.56793282  1.32127253
  2.12993434  3.06513799]

  Region R:

[-1.e+30  1.e+30]

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

  Degree    Error

   0        0.0000000000000004
   1        0.0000000000000000
   2        0.0000000000000004
   3        0.0000000000000000
   4        0.0000000000000007
   5        0.0000000000000001
   6        0.0000000000000010
   7        0.0000000000000003
   8        0.0000000000000010
   9        0.0000000000000018
  10        0.0000000000000009
  11        0.0000000000000071
  12        0.0000000000000011
  13        0.0000000000000000
  14        0.0000000000000011
  15        0.0000000000000000
  16        0.0142857142857132
  17        0.0000000000000000
  18        0.0650793650793643

gen_hermite_exactness_test():
  Normal end of execution.
Sun May 21 09:35:00 2023