10-Jan-2023 11:38:12

gen_hermite_exactness_test():
  MATLAB/Octave version 4.2.2
  Test gen_hermite_exactness().
10-Jan-2023 11:38:12

gen_hermite_exactness():
  MATLAB/Octave version 4.2.2

  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.

gen_hermite_exactness: 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 = 1.000000
  OPTION = 0, integrate |x|^alpha*exp(-x*x)*f(x).

  Spatial dimension = 1
  Number of points  = 8

  The quadrature rule to be tested is
  a generalized Gauss-Hermite rule
  ORDER = 8
  ALPHA = 1.000000

  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:

  w(1) =       0.0002696473527807
  w(2) =       0.0194439542575027
  w(3) =       0.1787093462188999
  w(4) =       0.3015770521708171
  w(5) =       0.3015770521708171
  w(6) =       0.1787093462188999
  w(7) =       0.0194439542575027
  w(8) =       0.0002696473527807

  Abscissas X:

  x(1) =      -3.0651379923750790
  x(2) =      -2.1299343409882678
  x(3) =      -1.3212725309936431
  x(4) =      -0.5679328213965031
  x(5) =       0.5679328213965031
  x(6) =       1.3212725309936431
  x(7) =       2.1299343409882678
  x(8) =       3.0651379923750790

  Region R:

  r(1) = -1.000000e+30
  r(2) = 1.000000e+30

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

      Error    Degree

        0.0000000000000007    0
        0.0000000000000001    1
        0.0000000000000009    2
        0.0000000000000000    3
        0.0000000000000007    4
        0.0000000000000001    5
        0.0000000000000003    6
        0.0000000000000003    7
        0.0000000000000003    8
        0.0000000000000044    9
        0.0000000000000009   10
        0.0000000000000071   11
        0.0000000000000013   12
        0.0000000000000000   13
        0.0000000000000016   14
        0.0000000000000000   15
        0.0142857142857163   16
        0.0000000000000000   17
        0.0650793650793675   18

gen_hermite_exactness():
  Normal end of execution.

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10-Jan-2023 11:38:12

gen_hermite_exactness():
  MATLAB/Octave version 4.2.2

  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.

gen_hermite_exactness: 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 = 1.000000
  OPTION = 1, integrate                     f(x).

  Spatial dimension = 1
  Number of points  = 8

  The quadrature rule to be tested is
  a generalized Gauss-Hermite rule
  ORDER = 8
  ALPHA = 1.000000

  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:

  w(1) =       1.0582141979488791
  w(2) =       0.8524080381127395
  w(3) =       0.7750492008314336
  w(4) =       0.7331317124710707
  w(5) =       0.7331317124710707
  w(6) =       0.7750492008314336
  w(7) =       0.8524080381127395
  w(8) =       1.0582141979488791

  Abscissas X:

  x(1) =      -3.0651379923750790
  x(2) =      -2.1299343409882678
  x(3) =      -1.3212725309936431
  x(4) =      -0.5679328213965031
  x(5) =       0.5679328213965031
  x(6) =       1.3212725309936431
  x(7) =       2.1299343409882678
  x(8) =       3.0651379923750790

  Region R:

  r(1) = -1.000000e+30
  r(2) = 1.000000e+30

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

      Error    Degree

        0.0000000000000004    0
        0.0000000000000000    1
        0.0000000000000004    2
        0.0000000000000000    3
        0.0000000000000007    4
        0.0000000000000001    5
        0.0000000000000010    6
        0.0000000000000003    7
        0.0000000000000010    8
        0.0000000000000018    9
        0.0000000000000009   10
        0.0000000000000071   11
        0.0000000000000013   12
        0.0000000000000000   13
        0.0000000000000013   14
        0.0000000000000000   15
        0.0142857142857130   16
        0.0000000000000000   17
        0.0650793650793641   18

gen_hermite_exactness():
  Normal end of execution.

10-Jan-2023 11:38:12

gen_hermite_exactness_test():
  Normal end of execution.

10-Jan-2023 11:38:12