03 March 2008 09:12:51 AM

INT_EXACTNESS_GEGENBAUER
  C++ version

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

INT_EXACTNESS_GEGENBAUER: User input:
  Quadrature rule X file = "gegen_o16_a0.5_x.txt".
  Quadrature rule W file = "gegen_o16_a0.5_w.txt".
  Quadrature rule R file = "gegen_o16_a0.5_r.txt".
  Maximum degree to check = 35
  Exponent of (1-x^2), ALPHA = 0.5

  Spatial dimension = 1
  Number of points  = 16

  The quadrature rule to be tested is
  a Gauss-Gegenbauer rule
  ORDER = 16
  ALPHA = 0.5

  Standard rule:
    Integral ( -1 <= x <= +1 ) (1-x^2)^alpha f(x) dx
    is to be approximated by
    sum ( 1 <= I <= ORDER ) w(i) * f(x(i)).

  Weights W:

  w[ 0] =     0.006239551412252181
  w[ 1] =      0.02411551965623589
  w[ 2] =      0.05121365616644176
  w[ 3] =      0.08387420745120895
  w[ 4] =       0.1176861850811666
  w[ 5] =       0.1480830941187536
  w[ 6] =       0.1709596635633561
  w[ 7] =       0.1832262859480332
  w[ 8] =       0.1832262859480331
  w[ 9] =       0.1709596635633561
  w[10] =       0.1480830941187535
  w[11] =       0.1176861850811666
  w[12] =      0.08387420745120895
  w[13] =      0.05121365616644213
  w[14] =      0.02411551965623589
  w[15] =     0.006239551412252181

  Abscissas X:

  x[ 0] =      -0.9829730996839018
  x[ 1] =      -0.9324722294043558
  x[ 2] =      -0.8502171357296141
  x[ 3] =      -0.7390089172206591
  x[ 4] =      -0.6026346363792564
  x[ 5] =      -0.4457383557765383
  x[ 6] =      -0.2736629900720829
  x[ 7] =       -0.092268359463302
  x[ 8] =        0.092268359463302
  x[ 9] =       0.2736629900720829
  x[10] =       0.4457383557765383
  x[11] =       0.6026346363792564
  x[12] =       0.7390089172206591
  x[13] =       0.8502171357296141
  x[14] =       0.9324722294043558
  x[15] =       0.9829730996839018

  Region R:

  r[ 0] =                       -1
  r[ 1] =                        1

  A Gauss-Gegenbauer rule would be able to exactly
  integrate monomials up to and including degree = 31

          Error          Degree

     8.481479150569371e-16   0
     1.925543058334256e-16   1
     1.413579858428229e-16   2
     2.237793284010081e-16   3
     5.654319433712916e-16   4
     1.665334536937735e-16   5
     4.523455546970334e-16   6
     1.205632815803881e-16   7
     1.615519838203691e-16   8
     8.326672684688674e-17   9
     2.369429096032086e-15  10
     5.551115123125783e-17  11
      9.59520873599777e-15  12
     4.597017211338539e-17  13
     1.518362701080949e-15  14
     3.469446951953614e-17  15
      4.85876064345902e-15  16
      2.34187669256869e-17  17
     8.336108947111107e-16  18
     1.734723475976807e-17  19
     4.550387289926831e-15  20
     1.301042606982605e-17  21
     6.933923489412328e-15  22
     9.540979117872439e-18  23
     3.206597028700531e-15  24
     6.071532165918825e-18  25
     1.396651150278459e-15  26
     5.204170427930421e-18  27
     4.433813175487184e-15  28
     3.469446951953614e-18  29
     8.317221956775846e-15  30
     2.168404344971009e-18  31
     2.828239023384806e-08  32
     1.734723475976807e-18  33
     2.468282487081176e-07  34
     8.673617379884035e-19  35

INT_EXACTNESS_GEGENBAUER:
  Normal end of execution.

03 March 2008 09:12:51 AM