14-May-2025 08:13:22 annulus_integrals_test(): MATLAB/Octave version 6.4.0 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 Outer radius is 1 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 0.782432 0.785398 2.97e-03 0 4 0.388427 0.392699 4.27e-03 0 6 0.24023 0.245437 5.21e-03 0 8 0.166236 0.171806 5.57e-03 2 0 0.797556 0.785398 1.22e-02 2 2 0.133902 0.1309 3.00e-03 2 4 0.0503587 0.0490874 1.27e-03 2 6 0.0252099 0.0245437 6.66e-04 2 8 0.0146897 0.0143172 3.73e-04 4 0 0.403113 0.392699 1.04e-02 4 2 0.050748 0.0490874 1.66e-03 4 4 0.0153249 0.0147262 5.99e-04 4 6 0.00641847 0.00613592 2.83e-04 4 8 0.00321865 0.00306796 1.51e-04 6 0 0.254245 0.245437 8.81e-03 6 2 0.0254299 0.0245437 8.86e-04 6 4 0.00641942 0.00613592 2.83e-04 6 6 0.0023155 0.0021914 1.24e-04 6 8 0.00102119 0.000958738 6.24e-05 8 0 0.179513 0.171806 7.71e-03 8 2 0.0148056 0.0143172 4.88e-04 8 4 0.00320494 0.00306796 1.37e-04 8 6 0.00101623 0.000958738 5.75e-05 8 8 0.00040087 0.000372843 2.80e-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 Outer radius is 2 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 11.7656 11.781 1.54e-02 0 4 24.5842 24.74 1.56e-01 0 6 61.773 62.5864 8.13e-01 0 8 172.135 175.757 3.62e+00 2 0 11.8526 11.781 7.16e-02 2 2 8.23551 8.24668 1.12e-02 2 4 12.3979 12.5173 1.19e-01 2 6 24.7074 25.1082 4.01e-01 2 8 57.3308 58.6287 1.30e+00 4 0 25.125 24.74 3.85e-01 4 2 12.6013 12.5173 8.40e-02 4 4 14.9946 15.0649 7.04e-02 4 6 24.7985 25.1266 3.28e-01 4 8 49.1919 50.2624 1.07e+00 6 0 64.0202 62.5864 1.43e+00 6 2 25.4653 25.1082 3.57e-01 6 4 25.1217 25.1266 4.93e-03 6 6 35.5438 35.9017 3.58e-01 6 8 61.6748 62.8309 1.16e+00 8 0 180.688 175.757 4.93e+00 8 2 59.9019 58.6287 1.27e+00 8 4 50.5033 50.2624 2.41e-01 8 6 62.408 62.8309 4.23e-01 8 8 96.2981 97.7381 1.44e+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 Outer radius is 2 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 11.7715 11.781 9.47e-03 0 4 24.6848 24.74 5.53e-02 0 6 62.3971 62.5864 1.89e-01 0 8 175.177 175.757 5.80e-01 2 0 11.8005 11.781 1.95e-02 2 2 8.2961 8.24668 4.94e-02 2 4 12.5829 12.5173 6.56e-02 2 6 25.1824 25.1082 7.42e-02 2 8 58.6936 58.6287 6.48e-02 4 0 24.7509 24.74 1.09e-02 4 2 12.6031 12.5173 8.59e-02 4 4 15.1914 15.0649 1.26e-01 4 6 25.275 25.1266 1.48e-01 4 8 50.3776 50.2624 1.15e-01 6 0 62.5685 62.5864 1.79e-02 6 2 25.2679 25.1082 1.60e-01 6 4 25.3811 25.1266 2.55e-01 6 6 36.2141 35.9017 3.12e-01 6 8 63.1249 62.8309 2.94e-01 8 0 175.697 175.757 6.05e-02 8 2 58.965 58.6287 3.36e-01 8 4 50.8222 50.2624 5.60e-01 8 6 63.5311 62.8309 7.00e-01 8 8 98.5007 97.7381 7.63e-01 annulus_integrals_test(): Normal end of execution. 14-May-2025 08:13:22