Wed Oct  8 08:55:54 2025

sphere_integrals_test():
  python version: 3.10.12
  numpy version:  1.26.4
  Test sphere_integrals().

sphere01_area_test
  sphere01_area returns the volume of the unit sphere.

  sphere01_area() =  12.5664

sphere01_monomial_integral_test():
  sphere01_monomial_integral() returns the integral of a monomial
  over the surface of the unit sphere in 3D.
  Compare with a Monte Carlo estimate.

  If any exponent is odd, the integral is zero.
  We will restrict this test to randomly chosen even exponents.

  Ex  Ey  Ez     MC-Estimate           Exact      Error

   8
   4
   0
       0.0287233       0.0292922     0.00057
   2
   0
   2
         0.84274        0.837758       0.005
   6
   0
   2
         0.20093        0.199466      0.0015
   4
   8
   2
      0.00193104      0.00195282     2.2e-05
   4
   6
   0
       0.0548628       0.0543999     0.00046
   0
   0
   0
         12.5664         12.5664     1.8e-15
   6
   6
   6
     6.53671e-05     6.47772e-05     5.9e-07
   6
   0
   0
         1.79409          1.7952      0.0011
   8
   8
   2
     0.000207497     0.000211605     4.1e-06
   4
   8
   4
     0.000342832     0.000344615     1.8e-06
   2
   0
   2
         0.84274        0.837758       0.005
   8
   6
   8
     6.64913e-06      6.5716e-06     7.8e-08
   2
   0
   0
         4.18538         4.18879      0.0034
   2
   0
   6
        0.202212        0.199466      0.0027
   8
   0
   2
        0.128084        0.126933      0.0012
   4
   8
   6
     9.08308e-05      9.0688e-05     1.4e-07
   0
   4
   8
       0.0284925       0.0292922      0.0008
   0
   4
   8
       0.0284925       0.0292922      0.0008
   8
   6
   4
     9.04521e-05      9.0688e-05     2.4e-07
   6
   0
   0
         1.79409          1.7952      0.0011

sphere01_sample_test():
  sphere01_sample() samples the unit sphere.

  Sample points on the unit sphere.

  Row:       0             1             2       
  Col
      0 :   -0.997124     0.0114407     0.0749144  
      1 :   -0.858213     -0.398626      -0.32337  
      2 :  -0.0917259      0.973477     -0.209594  
      3 :    0.823306     -0.133892      0.551579  
      4 :    0.410162      0.246416     -0.878092  
      5 :   -0.872424      -0.48516     0.0591233  
      6 :    0.823052     -0.564015    -0.0668815  
      7 :   -0.104837     -0.173199      0.979291  
      8 :  -0.0410237     -0.314898     -0.948238  
      9 :   -0.112008     -0.109378      0.987669  

sphere_integrals_test():
  Normal end of execution.
Wed Oct  8 08:55:55 2025