Wed Oct 8 07:25:45 2025 annulus_integrals_test(): python version: 3.10.12 numpy version: 1.26.4 Test annulus_integrals(). annulus_integrals_test01(): Estimate monomial integrals using Monte Carlo method over the interior of an annulus centered at the origin. Inner radius is 0.0 Outer radius is 1.0 Number of sample points is 4192 If any exponent is odd, the integral is zero. We will restrict this test to randomly chosen even exponents. Ex Ey MC-Estimate Exact Error 0 0 3.14159 3.14159 4.44e-16 0 2 0.819903 0.785398 3.45e-02 0 4 0.416545 0.392699 2.38e-02 0 6 0.262556 0.245437 1.71e-02 0 8 0.184829 0.171806 1.30e-02 2 0 0.754568 0.785398 3.08e-02 2 2 0.133042 0.1309 2.14e-03 2 4 0.0511549 0.0490874 2.07e-03 2 6 0.0258443 0.0245437 1.30e-03 2 8 0.0151521 0.0143172 8.35e-04 4 0 0.371934 0.392699 2.08e-02 4 2 0.0493832 0.0490874 2.96e-04 4 4 0.015327 0.0147262 6.01e-04 4 6 0.00648256 0.00613592 3.47e-04 4 8 0.00325504 0.00306796 1.87e-04 6 0 0.230313 0.245437 1.51e-02 6 2 0.0244716 0.0245437 7.21e-05 6 4 0.00638596 0.00613592 2.50e-04 6 6 0.00233881 0.0021914 1.47e-04 6 8 0.00103511 0.000958738 7.64e-05 8 0 0.15999 0.171806 1.18e-02 8 2 0.0141315 0.0143172 1.86e-04 8 4 0.00317263 0.00306796 1.05e-04 8 6 0.00102522 0.000958738 6.65e-05 8 8 0.000406645 0.000372843 3.38e-05 annulus_integrals_test01(): Estimate monomial integrals using Monte Carlo method over the interior of an annulus centered at the origin. Inner radius is 1.0 Outer radius is 2.0 Number of sample points is 4192 If any exponent is odd, the integral is zero. We will restrict this test to randomly chosen even exponents. Ex Ey MC-Estimate Exact Error 0 0 9.42478 9.42478 1.78e-15 0 2 11.6764 11.781 1.05e-01 0 4 24.3265 24.74 4.13e-01 0 6 61.1127 62.5864 1.47e+00 0 8 170.572 175.757 5.19e+00 2 0 11.8053 11.781 2.44e-02 2 2 8.08123 8.24668 1.65e-01 2 4 12.1064 12.5173 4.11e-01 2 6 24.1153 25.1082 9.93e-01 2 8 56.0614 58.6287 2.57e+00 4 0 25.0614 24.74 3.21e-01 4 2 12.3325 12.5173 1.85e-01 4 4 14.5719 15.0649 4.93e-01 4 6 24.1298 25.1266 9.97e-01 4 8 48.2094 50.2624 2.05e+00 6 0 64.1404 62.5864 1.55e+00 6 2 25.0077 25.1082 1.01e-01 6 4 24.401 25.1266 7.26e-01 6 6 34.4767 35.9017 1.43e+00 6 8 60.2152 62.8309 2.62e+00 8 0 181.999 175.757 6.24e+00 8 2 59.1901 58.6287 5.61e-01 8 4 49.1448 50.2624 1.12e+00 8 6 60.3821 62.8309 2.45e+00 8 8 93.4428 97.7381 4.30e+00 annulus_integrals_test01(): Estimate monomial integrals using Monte Carlo method over the interior of an annulus centered at the origin. Inner radius is 1.0 Outer radius is 2.0 Number of sample points is 67072 If any exponent is odd, the integral is zero. We will restrict this test to randomly chosen even exponents. Ex Ey MC-Estimate Exact Error 0 0 9.42478 9.42478 1.78e-15 0 2 11.7904 11.781 9.39e-03 0 4 24.7631 24.74 2.30e-02 0 6 62.5596 62.5864 2.68e-02 0 8 175.29 175.757 4.68e-01 2 0 11.7276 11.781 5.34e-02 2 2 8.22849 8.24668 1.82e-02 2 4 12.5091 12.5173 8.19e-03 2 6 25.0826 25.1082 2.56e-02 2 8 58.4619 58.6287 1.67e-01 4 0 24.529 24.74 2.11e-01 4 2 12.4533 12.5173 6.39e-02 4 4 15.0464 15.0649 1.85e-02 4 6 25.1363 25.1266 9.67e-03 4 8 50.2314 50.2624 3.10e-02 6 0 61.9066 62.5864 6.80e-01 6 2 24.8959 25.1082 2.12e-01 6 4 25.0646 25.1266 6.20e-02 6 6 35.9979 35.9017 9.62e-02 6 8 63.1183 62.8309 2.87e-01 8 0 173.718 175.757 2.04e+00 8 2 57.9337 58.6287 6.95e-01 8 4 49.976 50.2624 2.86e-01 8 6 62.9879 62.8309 1.57e-01 8 8 98.5351 97.7381 7.97e-01 annulus_integrals_test(): Normal end of execution. Wed Oct 8 07:25:46 2025