09 February 2008 05:12:59 PM

INT_EXACTNESS_JACOBI
  C++ version

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

INT_EXACTNESS_JACOBI: User input:
  Quadrature rule X file = "jac_o8_a0.5_b1.5_x.txt".
  Quadrature rule W file = "jac_o8_a0.5_b1.5_w.txt".
  Quadrature rule R file = "jac_o8_a0.5_b1.5_r.txt".
  Maximum degree to check = 18
  Exponent of (1-x), ALPHA = 0.5
  Exponent of (1+x), BETA  = 1.5

  Spatial dimension = 1
  Number of points  = 8

  The quadrature rule to be tested is
  a Gauss-Jacobi rule
  ORDER = 8
  ALPHA = 0.5
  BETA  = 1.5

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

  Weights W:

  w[ 0] =     0.007943251383316998
  w[ 1] =      0.05574150057932281
  w[ 2] =       0.1640573457854426
  w[ 3] =       0.3008492695346398
  w[ 4] =       0.3883180543538824
  w[ 5] =       0.3606436566318294
  w[ 6] =       0.2248513392666373
  w[ 7] =       0.0683919092594677

  Abscissas X:

  x[ 0] =      -0.8900098006603341
  x[ 1] =      -0.6866356906720188
  x[ 2] =      -0.4095019972429185
  x[ 3] =     -0.08860534544266939
  x[ 4] =       0.2412867334092741
  x[ 5] =       0.5444273641737976
  x[ 6] =       0.7879673764819101
  x[ 7] =       0.9455158043974036

  Region R:

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

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

          Error          Degree

     2.273036412352594e-13   0
     2.274449992211022e-13   1
     2.270209252635738e-13   2
     2.263141353343597e-13   3
     2.263141353343597e-13   4
      2.26285863737191e-13   5
     2.262858637371911e-13   6
     2.256881213970557e-13   7
     2.258496733808761e-13   8
     2.259573747034229e-13   9
     2.231571403172038e-13  10
     2.263098517590307e-13  11
     2.163034197914925e-13  12
     2.264047494278482e-13  13
     2.260673354942748e-13  14
     2.208711609172432e-13  15
     0.0003885003887300939  16
     0.0004088141345338994  17
      0.001921059655487914  18

INT_EXACTNESS_JACOBI:
  Normal end of execution.

09 February 2008 05:12:59 PM