10-Feb-2008 10:50:47

INT_EXACTNESS_JACOBI
  MATLAB version

  Investigate the polynomial exactness of a Gauss-Jacobi
  quadrature rule by integrating all monomials up to a given
  degree over the [-1,+1] interval.

  The rule will be adjusted to the [0,1] hypercube.

INT_EXACTNESS_JACOBI: User input:
  Quadrature rule X file = "jac_o16_a0.5_b1.5_x.txt".
  Quadrature rule W file = "jac_o16_a0.5_b1.5_w.txt".
  Quadrature rule R file = "jac_o16_a0.5_b1.5_r.txt".
  Maximum degree to check = 35
  Exponent of (1-x), ALPHA = 0.500000
  Exponent of (1+x), BETA  = 1.500000

  Spatial dimension = 1
  Number of points  = 16

  The quadrature rule to be tested is
  a Gauss-Jacobi rule
  ORDER = 16
  ALPHA = 0.500000
  BETA  = 1.500000

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

  Weights W:

  w(1) =       0.0003988966638936
  w(2) =       0.0031982402989070
  w(3) =       0.0115995225865755
  w(4) =       0.0286563600804907
  w(5) =       0.0558165755710824
  w(6) =       0.0918391988776129
  w(7) =       0.1324747827148990
  w(8) =       0.1710802448681879
  w(9) =       0.2000637064771614
  w(10) =       0.2127999546344757
  w(11) =       0.2054979111101689
  w(12) =       0.1784862462090924
  w(13) =       0.1365200311515025
  w(14) =       0.0879649206719887
  w(15) =       0.0430169974798177
  w(16) =       0.0113827373986822

  Abscissas X:

  x(1) =      -0.9671984819405669
  x(2) =      -0.9040845839929046
  x(3) =      -0.8119779665780460
  x(4) =      -0.6938290260300457
  x(5) =      -0.5534333596595263
  x(6) =      -0.3953028602480263
  x(7) =      -0.2245197622496786
  x(8) =      -0.0465730782535859
  x(9) =       0.1328178890268011
  x(10) =       0.3078873946321201
  x(11) =       0.4730085776933708
  x(12) =       0.6228743183404623
  x(13) =       0.7526678154351841
  x(14) =       0.8582174038957007
  x(15) =       0.9361306362220502
  x(16) =       0.9839033190008087

  Region R:

  r(1) = -1.000000e+00
  r(2) = 1.000000e+00

  A Gauss-Jacobi rule would be able to exactly
  integrate monomials up to and including 
  degree = 31

      Error    Degree

        0.0000000000002279    0
        0.0000000000002279    1
        0.0000000000002279    2
        0.0000000000002276    3
        0.0000000000002279    4
        0.0000000000002279    5
        0.0000000000002281    6
        0.0000000000002273    7
        0.0000000000002276    8
        0.0000000000002279    9
        0.0000000000002253   10
        0.0000000000002285   11
        0.0000000000002188   12
        0.0000000000002357   13
        0.0000000000002288   14
        0.0000000000002235   15
        0.0000000000002326   16
        0.0000000000002321   17
        0.0000000000002266   18
        0.0000000000002212   19
        0.0000000000002238   20
        0.0000000000002269   21
        0.0000000000002217   22
        0.0000000000002401   23
        0.0000000000002307   24
        0.0000000000002211   25
        0.0000000000002273   26
        0.0000000000002155   27
        0.0000000000002241   28
        0.0000000000002372   29
        0.0000000000002372   30
        0.0000000000002180   31
        0.0000000149732520   32
        0.0000000154003224   33
        0.0000001343179534   34
        0.0000001379400370   35

INT_EXACTNESS_JACOBI:
  Normal end of execution.

10-Feb-2008 10:50:47