22 January 2009   9:23:04.630 AM                                                
 
INT_EXACTNESS_LEGENDRE
  FORTRAN90 version
 
  Investigate the polynomial exactness of a Gauss-Legendre
  quadrature rule by integrating weighted 
  monomials up to a given degree over the [-1,+1] interval.
 
INT_EXACTNESS_LEGENDRE: User input:
  Quadrature rule X file = "leg_o8_x.txt".
  Quadrature rule W file = "leg_o8_w.txt".
  Quadrature rule R file = "leg_o8_r.txt".
  Maximum degree to check =       18
 
  Spatial dimension =        1
  Number of points  =        8
 
  The quadrature rule to be tested is
  a Gauss-Legendre rule
  ORDER =        8
 
  Standard rule:
    Integral ( -1 <= x <= +1 ) f(x) dx
    is to be approximated by
    sum ( 1 <= I <= ORDER ) w(i) * f(x(i)).
 
  Weights W:
 
  w( 1) =   0.1012285362903761    
  w( 2) =   0.2223810344533745    
  w( 3) =   0.3137066458778873    
  w( 4) =   0.3626837833783620    
  w( 5) =   0.3626837833783620    
  w( 6) =   0.3137066458778873    
  w( 7) =   0.2223810344533745    
  w( 8) =   0.1012285362903761    
 
  Abscissas X:
 
  x( 1) =  -0.9602898564975364    
  x( 2) =  -0.7966664774136267    
  x( 3) =  -0.5255324099163290    
  x( 4) =  -0.1834346424956498    
  x( 5) =   0.1834346424956498    
  x( 6) =   0.5255324099163290    
  x( 7) =   0.7966664774136267    
  x( 8) =   0.9602898564975364    
 
  Region R:
 
  r( 1) =  -1.0000000000000000    
  r( 2) =   1.0000000000000000    
 
  A Gauss-Legendre rule would be able to exactly
  integrate monomials up to and including degree =       15
 
          Error                      Error           Degree
         (This rule)                (Trapezoid)
 
        0.0000000000000000         0.0000000000000002   0
        0.0000000000000000         0.0000000000000001   1
        0.0000000000000000         0.0408163265306121   2
        0.0000000000000000         0.0000000000000000   3
        0.0000000000000003         0.1349437734277384   4
        0.0000000000000000         0.0000000000000000   5
        0.0000000000000002         0.2780304124981938   6
        0.0000000000000000         0.0000000000000000   7
        0.0000000000000001         0.4628822399940606   8
        0.0000000000000000         0.0000000000000000   9
        0.0000000000000003         0.6807392105352211  10
        0.0000000000000000         0.0000000000000000  11
        0.0000000000000005         0.9228001872701588  12
        0.0000000000000000         0.0000000000000000  13
        0.0000000000000010         1.1814556860179242  14
        0.0000000000000000         0.0000000000000000  15
        0.0003956606287597         1.4508791292288041  16
        0.0000000000000000         0.0000000000000000  17
        0.0018727936427998         1.7270038958037262  18
 
INT_EXACTNESS_LEGENDRE:
  Normal end of execution.
 
22 January 2009   9:23:04.633 AM