24 January 2008 12:00:53 PM

INT_EXACTNESS_GEN_HERMITE
  C++ version

  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.

INT_EXACTNESS_GEN_HERMITE: 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
  Power of |X|, ALPHA = 1
  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

  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[ 0] =        1.058214197948879
  w[ 1] =       0.8524080381127395
  w[ 2] =       0.7750492008314336
  w[ 3] =       0.7331317124710707
  w[ 4] =       0.7331317124710707
  w[ 5] =       0.7750492008314336
  w[ 6] =       0.8524080381127395
  w[ 7] =        1.058214197948879

  Abscissas X:

  x[ 0] =       -3.065137992375079
  x[ 1] =       -2.129934340988268
  x[ 2] =       -1.321272530993643
  x[ 3] =      -0.5679328213965031
  x[ 4] =       0.5679328213965031
  x[ 5] =        1.321272530993643
  x[ 6] =        2.129934340988268
  x[ 7] =        3.065137992375079

  Region R:

  r[ 0] =                   -1e+30
  r[ 1] =                    1e+30

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

          Error          Degree

     4.440892098500626e-16   0
     2.547875105340935e-17   1
     4.440892098500626e-16   2
     6.852157730108388e-17   3
     6.661338147750939e-16   4
     8.326672684688674e-17   5
     1.036208156316813e-15   6
     2.220446049250313e-16   7
     8.881784197001252e-16   8
     8.881784197001252e-16   9
     9.473903143468002e-16  10
                         0  11
     7.894919286223335e-16  12
                         0  13
     7.218211918832764e-16  14
                         0  15
       0.01428571428571392  16
                         0  17
       0.06507936507936508  18

INT_EXACTNESS_GEN_HERMITE:
  Normal end of execution.

24 January 2008 12:00:53 PM