03 March 2008 09:12:46 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_o8_a0.5_x.txt".
  Quadrature rule W file = "gegen_o8_a0.5_w.txt".
  Quadrature rule R file = "gegen_o8_a0.5_r.txt".
  Maximum degree to check = 18
  Exponent of (1-x^2), ALPHA = 0.5

  Spatial dimension = 1
  Number of points  = 8

  The quadrature rule to be tested is
  a Gauss-Gegenbauer rule
  ORDER = 8
  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] =      0.04083294770910693
  w[ 1] =       0.1442256007956728
  w[ 2] =       0.2617993877991495
  w[ 3] =       0.3385402270935191
  w[ 4] =       0.3385402270935191
  w[ 5] =       0.2617993877991495
  w[ 6] =       0.1442256007956728
  w[ 7] =      0.04083294770910754

  Abscissas X:

  x[ 0] =      -0.9396926207859084
  x[ 1] =       -0.766044443118978
  x[ 2] =                     -0.5
  x[ 3] =      -0.1736481776669303
  x[ 4] =       0.1736481776669303
  x[ 5] =                      0.5
  x[ 6] =        0.766044443118978
  x[ 7] =       0.9396926207859084

  Region R:

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

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

          Error          Degree

     2.827159716856457e-16   0
     6.106226635438361e-16   1
     8.481479150569377e-16   2
     5.204170427930421e-16   3
     9.895059008997603e-16   4
      4.40619762898109e-16   5
     1.583209441439617e-15   6
      3.95516952522712e-16   7
     2.100175789664798e-15   8
     3.469446951953614e-16   9
     5.169663482251825e-15  10
     3.087807787238717e-16  11
     1.261084576731136e-14  12
     2.706168622523819e-16  13
     2.193190568228038e-15  14
     2.393918396847994e-16  15
      0.000699300699301984  16
     2.133709875451473e-16  17
      0.003290826820234406  18

INT_EXACTNESS_GEGENBAUER:
  Normal end of execution.

03 March 2008 09:12:46 AM