31-Oct-2022 09:30:09 hyperball_integrals_test(): MATLAB/Octave version 4.2.2 Test hyperball_integrals(). hyperball_integrals_test01(): Use the Monte Carlo method to estimate integrals over the interior of the unit hyperball in M dimensions. Spatial dimension M = 3 Number of sample points used is 4192 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 4 0 6 0.00434626 0.00418461 1.62e-04 8 8 0 0.000203537 0.000211605 8.07e-06 0 4 0 0.343366 0.359039 1.57e-02 8 2 6 3.38859e-05 3.02293e-05 3.66e-06 6 4 8 4.68685e-06 4.31848e-06 3.68e-07 4 2 6 0.000295102 0.000278974 1.61e-05 4 0 0 0.367792 0.359039 8.75e-03 0 0 6 0.207105 0.199466 7.64e-03 6 4 4 5.29237e-05 4.92307e-05 3.69e-06 2 6 4 0.000276188 0.000278974 2.79e-06 4 4 2 0.000856273 0.000836921 1.94e-05 2 8 4 0.000113991 0.000114872 8.80e-07 6 6 0 0.00135933 0.00139487 3.55e-05 6 6 6 3.25485e-06 3.08463e-06 1.70e-07 0 0 8 0.132198 0.126933 5.27e-03 8 2 4 0.000127201 0.000114872 1.23e-05 0 0 0 4.18879 4.18879 8.88e-16 0 6 6 0.00131619 0.00139487 7.87e-05 6 4 2 0.000292516 0.000278974 1.35e-05 6 2 6 8.98108e-05 8.20511e-05 7.76e-06 hyperball_integrals_test02(): Use the Monte Carlo method to estimate integrals over the interior of the unit hyperball in M dimensions. Spatial dimension M = 6 Number of sample points used is 4192 If any exponent is odd, the integral is zero. We will restrict this test to randomly chosen even exponents. E1 E2 E3 E4 E5 E6 MC-Estimate Exact Error 4 6 0 0 6 6 2.16592e-09 1.75835e-09 4.08e-10 2 0 0 6 8 0 4.3355e-06 4.77897e-06 4.43e-07 8 0 6 4 8 8 5.57165e-15 5.06519e-15 5.06e-16 2 0 0 2 0 6 0.000341628 0.000360471 1.88e-05 8 6 6 8 4 4 4.31261e-17 5.16856e-17 8.56e-18 4 8 4 6 6 6 4.24945e-16 3.10113e-16 1.15e-16 4 8 2 2 8 2 6.54175e-12 5.98329e-12 5.58e-13 6 0 6 8 6 2 1.98989e-12 1.88549e-12 1.04e-13 8 2 0 0 4 4 1.2038e-07 1.19474e-07 9.06e-10 6 8 6 6 0 8 2.91204e-15 3.61799e-15 7.06e-16 0 8 8 8 2 4 5.90454e-13 5.13272e-13 7.72e-14 0 2 8 2 8 8 5.06244e-12 6.15927e-12 1.10e-12 8 2 8 2 4 8 1.19614e-14 1.35072e-14 1.55e-15 2 0 0 6 4 4 4.71893e-07 4.09626e-07 6.23e-08 8 4 8 8 0 4 2.59112e-14 4.05215e-14 1.46e-14 0 8 2 8 0 6 1.04527e-09 9.57326e-10 8.79e-11 8 0 0 8 0 6 2.57238e-08 2.87198e-08 3.00e-09 2 2 0 4 8 0 1.01409e-06 9.55794e-07 5.83e-08 4 4 4 4 2 4 4.32473e-11 4.22005e-11 1.05e-12 0 6 4 2 0 4 4.61667e-07 4.09626e-07 5.20e-08 hyperball_integrals_test(): Normal end of execution. 31-Oct-2022 09:30:10