24 January 2008 11:58:51 AM

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_o16_a1.0_x.txt".
  Quadrature rule W file = "gen_herm_o16_a1.0_w.txt".
  Quadrature rule R file = "gen_herm_o16_a1.0_r.txt".
  Maximum degree to check = 35
  Power of |X|, ALPHA = 1
  OPTION = 0, integrate |x|^alpha*exp(-x*x)*f(x)

  Spatial dimension = 1
  Number of points  = 16

  The quadrature rule to be tested is
  a generalized Gauss-Hermite rule
  ORDER = 16
  ALPHA = 1

  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[ 0] =    5.240005874357543e-10
  w[ 1] =    4.242873358136269e-07
  w[ 2] =    4.538254386679103e-05
  w[ 3] =     0.001397268117612835
  w[ 4] =      0.01667174613060782
  w[ 5] =      0.08789749331858591
  w[ 6] =       0.2093933904071717
  w[ 7] =       0.1845942946708189
  w[ 8] =       0.1845942946708189
  w[ 9] =       0.2093933904071717
  w[10] =      0.08789749331858591
  w[11] =      0.01667174613060782
  w[12] =     0.001397268117612835
  w[13] =    4.538254386679103e-05
  w[14] =    4.242873358136269e-07
  w[15] =    5.240005874357543e-10

  Abscissas X:

  x[ 0] =       -4.781540728352031
  x[ 1] =       -3.967452411973961
  x[ 2] =       -3.280017684431137
  x[ 3] =       -2.654412440144422
  x[ 4] =       -2.065599227896752
  x[ 5] =       -1.500362166233917
  x[ 6] =      -0.9506323036797034
  x[ 7] =      -0.4126495272081394
  x[ 8] =       0.4126495272081394
  x[ 9] =       0.9506323036797034
  x[10] =        1.500362166233917
  x[11] =        2.065599227896752
  x[12] =        2.654412440144422
  x[13] =        3.280017684431137
  x[14] =        3.967452411973961
  x[15] =        4.781540728352031

  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 = 31

          Error          Degree

     6.661338147750939e-16   0
     8.207652892352808e-18   1
     8.881784197001252e-16   2
     2.533646937060533e-17   3
     4.440892098500626e-16   4
      1.56224639209215e-16   5
                         0   6
     3.481271531897284e-16   7
     5.921189464667501e-16   8
     2.130457268934016e-16   9
     8.289665250534502e-16  10
     5.322131624296844e-15  11
     1.105288700071267e-15  12
     2.586819647376615e-14  13
     1.263187085795734e-15  14
     4.511946372076636e-13  15
     7.218211918832764e-16  16
     9.265477274311706e-12  17
      6.41618837229579e-16  18
      9.00399754755199e-11  19
     3.849713023377474e-16  20
     3.346940502524376e-10  21
     1.119916515891628e-15  22
     3.725290298461914e-09  23
     4.604101231998941e-15  24
     8.940696716308594e-08  25
     1.072056835725322e-15  26
      4.76837158203125e-07  27
     8.751484373267933e-16  28
       1.9073486328125e-05  29
     1.866983332963825e-16  30
            0.000244140625  31
     7.770007770399816e-05  32
               0.001953125  33
     0.0006627359568564141  34
                         0  35

INT_EXACTNESS_GEN_HERMITE:
  Normal end of execution.

24 January 2008 11:58:51 AM