30 March 2023  10:39:37.514 AM
 
legendre_exactness():
  FORTRAN90 version
  Investigate the polynomial exactness of a Gauss-Legendre
  quadrature rule by integrating 
  monomials up to a given degree over the [-1,+1] interval.
 
LEGENDRE_EXACTNESS: User input:
  Quadrature rule X file = "leg_o4_x.txt".
  Quadrature rule W file = "leg_o4_w.txt".
  Quadrature rule R file = "leg_o4_r.txt".
  Maximum degree to check =       10
 
  Spatial dimension =        1
  Number of points  =        4
 
  The quadrature rule to be tested is
  a Gauss-Legendre rule
  ORDER =        4
 
  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) =   0.3478548451374539    
  w( 2) =   0.6521451548625459    
  w( 3) =   0.6521451548625459    
  w( 4) =   0.3478548451374539    
 
  Abscissas X:
 
  x( 1) =  -0.8611363115940526    
  x( 2) =  -0.3399810435848563    
  x( 3) =   0.3399810435848563    
  x( 4) =   0.8611363115940526    
 
  Region R:
 
  r( 1) =   -1.000000000000000    
  r( 2) =    1.000000000000000    
 
  A Gauss-Legendre rule would be able to exactly
  integrate monomials up to and including degree =        7
 
          Error                      Error           Degree
         (This rule)                (Trapezoid)
 
        0.0000000000000002         0.0000000000000001   0
        0.0000000000000000         0.0000000000000001   1
        0.0000000000000000         0.2222222222222222   2
        0.0000000000000000         0.0000000000000000   3
        0.0000000000000000         0.7078189300411522   4
        0.0000000000000000         0.0000000000000000   5
        0.0000000000000000         1.3397347965249200   6
        0.0000000000000000         0.0000000000000000   7
        0.0522448979591835         2.0009144947416555   8
        0.0000000000000000         0.0000000000000000   9
        0.1418075801749270         2.6667908573105947  10
 
LEGENDRE_EXACTNESS:
  Normal end of execution.
 
30 March 2023  10:39:37.515 AM