22 January 2009 9:23:00.380 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_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) = 0.3478548451374538
w( 2) = 0.6521451548625461
w( 3) = 0.6521451548625461
w( 4) = 0.3478548451374538
Abscissas X:
x( 1) = -0.8611363115940526
x( 2) = -0.3399810435848563
x( 3) = 0.3399810435848563
x( 4) = 0.8611363115940526
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 = 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
0.0000000000000001 0.7078189300411522 4
0.0000000000000000 0.0000000000000000 5
0.0000000000000000 1.3397347965249200 6
0.0000000000000000 0.0000000000000000 7
0.0522448979591839 2.0009144947416555 8
0.0000000000000000 0.0000000000000000 9
0.1418075801749273 2.6667908573105947 10
INT_EXACTNESS_LEGENDRE:
Normal end of execution.
22 January 2009 9:23:00.383 AM