04-Feb-2008 11:56:46

INT_EXACTNESS_GEN_HERMITE
  MATLAB 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
  Weighting function exponent ALPHA = 1.000000
  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 = 1.000000

  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) =       0.0000000005240006
  w(2) =       0.0000004242873358
  w(3) =       0.0000453825438668
  w(4) =       0.0013972681176128
  w(5) =       0.0166717461306078
  w(6) =       0.0878974933185859
  w(7) =       0.2093933904071717
  w(8) =       0.1845942946708189
  w(9) =       0.1845942946708189
  w(10) =       0.2093933904071717
  w(11) =       0.0878974933185859
  w(12) =       0.0166717461306078
  w(13) =       0.0013972681176128
  w(14) =       0.0000453825438668
  w(15) =       0.0000004242873358
  w(16) =       0.0000000005240006

  Abscissas X:

  x(1) =      -4.7815407283520308
  x(2) =      -3.9674524119739609
  x(3) =      -3.2800176844311371
  x(4) =      -2.6544124401444220
  x(5) =      -2.0655992278967519
  x(6) =      -1.5003621662339170
  x(7) =      -0.9506323036797034
  x(8) =      -0.4126495272081394
  x(9) =       0.4126495272081394
  x(10) =       0.9506323036797034
  x(11) =       1.5003621662339170
  x(12) =       2.0655992278967519
  x(13) =       2.6544124401444220
  x(14) =       3.2800176844311371
  x(15) =       3.9674524119739609
  x(16) =       4.7815407283520308

  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 = 31

      Error    Degree

        0.0000000000000007    0
        0.0000000000000000    1
        0.0000000000000009    2
        0.0000000000000000    3
        0.0000000000000004    4
        0.0000000000000000    5
        0.0000000000000001    6
        0.0000000000000003    7
        0.0000000000000007    8
        0.0000000000000009    9
        0.0000000000000011   10
        0.0000000000000087   11
        0.0000000000000011   12
        0.0000000000000090   13
        0.0000000000000011   14
        0.0000000000005392   15
        0.0000000000000004   16
        0.0000000000025103   17
        0.0000000000000005   18
        0.0000000000409490   19
        0.0000000000000000   20
        0.0000000000228019   21
        0.0000000000000007   22
        0.0000000003491003   23
        0.0000000000000049   24
        0.0000000376545634   25
        0.0000000000000057   26
        0.0000008772895841   27
        0.0000000000000011   28
        0.0000024154287721   29
        0.0000000000000002   30
        0.0001313187343617   31
        0.0000777000777040   32
        0.0039298262524874   33
        0.0006627359568566   34
        0.0233836077292267   35

INT_EXACTNESS_GEN_HERMITE:
  Normal end of execution.

04-Feb-2008 11:56:46