26-Apr-2023 22:05:05

square_exactness_test():
  MATLAB/Octave version 5.2.0
  Test square_exactness().

square_exactness_test01()
  Product Gauss-Legendre rules for the 2D Legendre integral.
  Density function rho(x) = 1.
  Region: -1 <= x <= +1.
  Region: -1 <= y <= +1.
  Level: L
  Exactness: 2*L+1
  Order: N = (L+1)*(L+1)

  Quadrature rule for the 2D Legendre integral.
  Number of points in rule is 1

   D   I       J          Relative Error
   0
       0       0                         1
   1
       1       0                         0
       0       1                         0
   2
       2       0                         1
       1       1                         0
       0       2                         1

  Quadrature rule for the 2D Legendre integral.
  Number of points in rule is 4

   D   I       J          Relative Error
   0
       0       0                         1
   1
       1       0                         0
       0       1                         0
   2
       2       0                         1
       1       1                         0
       0       2                         1
   3
       3       0                         0
       2       1                         0
       1       2                         0
       0       3                         0
   4
       4       0                         1
       3       1                         0
       2       2                         1
       1       3                         0
       0       4                         1

  Quadrature rule for the 2D Legendre integral.
  Number of points in rule is 9

   D   I       J          Relative Error
   0
       0       0                         1
   1
       1       0                         0
       0       1                         0
   2
       2       0                         1
       1       1                         0
       0       2                         1
   3
       3       0                         0
       2       1                         0
       1       2                         0
       0       3                         0
   4
       4       0                         1
       3       1                         0
       2       2                         1
       1       3                         0
       0       4                         1
   5
       5       0                         0
       4       1                         0
       3       2                         0
       2       3                         0
       1       4                         0
       0       5                         0
   6
       6       0                         1
       5       1                         0
       4       2                         1
       3       3                         0
       2       4                         1
       1       5                         0
       0       6                         1

  Quadrature rule for the 2D Legendre integral.
  Number of points in rule is 16

   D   I       J          Relative Error
   0
       0       0                         1
   1
       1       0                         0
       0       1                         0
   2
       2       0                         1
       1       1                         0
       0       2                         1
   3
       3       0                         0
       2       1                         0
       1       2                         0
       0       3                         0
   4
       4       0                         1
       3       1                         0
       2       2                         1
       1       3                         0
       0       4                         1
   5
       5       0                         0
       4       1                         0
       3       2                         0
       2       3                         0
       1       4                         0
       0       5                         0
   6
       6       0                         1
       5       1                         0
       4       2                         1
       3       3                         0
       2       4                         1
       1       5                         0
       0       6                         1
   7
       7       0                         0
       6       1                         0
       5       2                         0
       4       3                         0
       3       4                         0
       2       5                         0
       1       6                         0
       0       7                         0
   8
       8       0                         1
       7       1                         0
       6       2                         1
       5       3                         0
       4       4                         1
       3       5                         0
       2       6                         1
       1       7                         0
       0       8                         1

  Quadrature rule for the 2D Legendre integral.
  Number of points in rule is 25

   D   I       J          Relative Error
   0
       0       0                         1
   1
       1       0                         0
       0       1     6.938893903907228e-18
   2
       2       0                         1
       1       1                         0
       0       2                         1
   3
       3       0                         0
       2       1                         0
       1       2     1.734723475976807e-18
       0       3                         0
   4
       4       0                         1
       3       1                         0
       2       2                         1
       1       3                         0
       0       4                         1
   5
       5       0                         0
       4       1                         0
       3       2     8.673617379884035e-19
       2       3     2.168404344971009e-19
       1       4                         0
       0       5                         0
   6
       6       0                         1
       5       1                         0
       4       2                         1
       3       3                         0
       2       4                         1
       1       5     2.710505431213761e-20
       0       6                         1
   7
       7       0     1.110223024625157e-16
       6       1                         0
       5       2                         0
       4       3                         0
       3       4     2.710505431213761e-20
       2       5                         0
       1       6                         0
       0       7                         0
   8
       8       0                         1
       7       1                         0
       6       2                         1
       5       3     2.710505431213761e-20
       4       4                         1
       3       5                         0
       2       6                         1
       1       7     1.694065894508601e-21
       0       8                         1
   9
       9       0     1.110223024625157e-16
       8       1                         0
       7       2                         0
       6       3                         0
       5       4                         0
       4       5                         0
       3       6     1.694065894508601e-21
       2       7                         0
       1       8     4.235164736271502e-22
       0       9                         0
  10
      10       0                         1
       9       1                         0
       8       2                         1
       7       3                         0
       6       4                         1
       5       5     3.388131789017201e-21
       4       6                         1
       3       7     4.235164736271502e-22
       2       8                         1
       1       9     1.058791184067875e-22
       0      10                         1

  Quadrature rule for the 2D Legendre integral.
  Number of points in rule is 36

   D   I       J          Relative Error
   0
       0       0                         1
   1
       1       0     5.551115123125783e-17
       0       1     1.387778780781446e-17
   2
       2       0                         1
       1       1     1.387778780781446e-17
       0       2                         1
   3
       3       0                         0
       2       1                         0
       1       2     3.469446951953614e-18
       0       3     4.336808689942018e-19
   4
       4       0        0.9999999999999999
       3       1     6.938893903907228e-18
       2       2                         1
       1       3                         0
       0       4                         1
   5
       5       0     2.220446049250313e-16
       4       1                         0
       3       2                         0
       2       3                         0
       1       4     5.421010862427522e-20
       0       5     1.355252715606881e-20
   6
       6       0                         1
       5       1     3.469446951953614e-18
       4       2                         1
       3       3     5.421010862427522e-20
       2       4                         1
       1       5     6.776263578034403e-21
       0       6                         1
   7
       7       0                         0
       6       1     3.469446951953614e-18
       5       2                         0
       4       3     5.421010862427522e-20
       3       4     6.776263578034403e-21
       2       5     1.694065894508601e-21
       1       6     8.470329472543003e-22
       0       7     8.470329472543003e-22
   8
       8       0                         1
       7       1                         0
       6       2                         1
       5       3                         0
       4       4                         1
       3       5     1.694065894508601e-21
       2       6                         1
       1       7                         0
       0       8                         1
   9
       9       0                         0
       8       1                         0
       7       2                         0
       6       3     2.710505431213761e-20
       5       4                         0
       4       5     1.694065894508601e-21
       3       6     4.235164736271502e-22
       2       7                         0
       1       8     7.940933880509066e-23
       0       9                         0
  10
      10       0                         1
       9       1     6.938893903907228e-18
       8       2                         1
       7       3     1.694065894508601e-20
       6       4                         1
       5       5     1.694065894508601e-21
       4       6                         1
       3       7     2.646977960169689e-23
       2       8                         1
       1       9     3.308722450212111e-24
       0      10                         1
  11
      11       0                         0
      10       1     1.734723475976807e-18
       9       2     4.336808689942018e-19
       8       3     1.355252715606881e-20
       7       4                         0
       6       5     8.470329472543003e-22
       5       6                         0
       4       7     2.646977960169689e-23
       3       8     3.308722450212111e-24
       2       9                         0
       1      10     8.271806125530277e-25
       0      11                         0
  12
      12       0        0.9999999999999998
      11       1                         0
      10       2                         1
       9       3     2.710505431213761e-20
       8       4                         1
       7       5     4.235164736271502e-22
       6       6                         1
       5       7     3.970466940254533e-23
       4       8                         1
       3       9     2.481541837659083e-24
       2      10                         1
       1      11     4.135903062765138e-25
       0      12                         1

square_exactness_test02():
  Padua rule for the 2D Legendre integral.
  Density function rho(x) = 1.
  Region: -1 <= x <= +1.
  Region: -1 <= y <= +1.
  Level: L
  Exactness: L+1 when L is 0,
             L   otherwise.
  Order: N = (L+1)*(L+2)/2

  Quadrature rule for the 2D Legendre integral.
  Number of points in rule is 1

   D   I       J          Relative Error
   0
       0       0                         0
   1
       1       0                         0
       0       1                         0
   2
       2       0                         1
       1       1                         0
       0       2                         1

  Quadrature rule for the 2D Legendre integral.
  Number of points in rule is 3

   D   I       J          Relative Error
   0
       0       0                         0
   1
       1       0                         0
       0       1                         0
   2
       2       0                         2
       1       1                         0
       0       2        0.5000000000000001

  Quadrature rule for the 2D Legendre integral.
  Number of points in rule is 6

   D   I       J          Relative Error
   0
       0       0     2.220446049250313e-16
   1
       1       0     1.110223024625157e-16
       0       1     4.440892098500626e-16
   2
       2       0     4.996003610813204e-16
       1       1     5.551115123125783e-17
       0       2      3.33066907387547e-16
   3
       3       0     1.110223024625157e-16
       2       1        0.6666666666666665
       1       2     2.775557561562891e-17
       0       3        0.3333333333333338

  Quadrature rule for the 2D Legendre integral.
  Number of points in rule is 10

   D   I       J          Relative Error
   0
       0       0                         0
   1
       1       0                         0
       0       1     6.661338147750939e-16
   2
       2       0     1.665334536937735e-16
       1       1     6.661338147750939e-16
       0       2      3.33066907387547e-16
   3
       3       0                         0
       2       1     7.494005416219807e-16
       1       2     3.608224830031759e-16
       0       3     9.436895709313831e-16
   4
       4       0        0.1666666666666666
       3       1     1.082467449009528e-15
       2       2        0.2499999999999993
       1       3     7.216449660063518e-16
       0       4       0.04166666666666721

  Quadrature rule for the 2D Legendre integral.
  Number of points in rule is 15

   D   I       J          Relative Error
   0
       0       0                         0
   1
       1       0     7.216449660063518e-16
       0       1      3.33066907387547e-16
   2
       2       0     1.665334536937735e-16
       1       1     1.942890293094024e-16
       0       2     1.665334536937735e-16
   3
       3       0      1.97758476261356e-16
       2       1     1.110223024625157e-16
       1       2     3.747002708109903e-16
       0       3     5.551115123125783e-17
   4
       4       0     1.110223024625157e-15
       3       1     3.573530360512223e-16
       2       2     9.992007221626409e-16
       1       3     1.474514954580286e-16
       0       4     2.775557561562891e-16
   5
       5       0     9.367506770274758e-17
       4       1       0.03333333333333316
       3       2     2.983724378680108e-16
       2       3       0.05555555555555577
       1       4     2.133709875451473e-16
       0       5       0.01666666666666702

  Quadrature rule for the 2D Legendre integral.
  Number of points in rule is 21

   D   I       J          Relative Error
   0
       0       0     1.110223024625157e-16
   1
       1       0     8.326672684688674e-17
       0       1     8.326672684688674e-17
   2
       2       0     6.661338147750939e-16
       1       1     1.387778780781446e-16
       0       2     1.665334536937735e-16
   3
       3       0     7.979727989493313e-16
       2       1     5.828670879282072e-16
       1       2     1.387778780781446e-16
       0       3     2.498001805406602e-16
   4
       4       0     1.110223024625157e-15
       3       1     4.718447854656915e-16
       2       2                         0
       1       3     2.289834988289385e-16
       0       4     2.775557561562891e-16
   5
       5       0     9.020562075079397e-16
       4       1      9.71445146547012e-16
       3       2     2.081668171172169e-16
       2       3     2.671474153004283e-16
       1       4      9.71445146547012e-17
       0       5     2.636779683484747e-16
   6
       6       0      0.008333333333334469
       5       1      1.27675647831893e-15
       4       2       0.02083333333333305
       3       3     4.024558464266192e-16
       2       4       0.02083333333333243
       1       5     4.024558464266192e-16
       0       6      0.006249999999999978

square_exactness_test():
  Normal end of execution.

26-Apr-2023 22:05:06