04-Nov-2022 17:06:45 annulus_integrals_test(): MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 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 0.798471 0.785398 1.31e-02 0 4 0.398118 0.392699 5.42e-03 0 6 0.247122 0.245437 1.68e-03 0 8 0.17168 0.171806 1.26e-04 2 0 0.78222 0.785398 3.18e-03 2 2 0.136157 0.1309 5.26e-03 2 4 0.0515752 0.0490874 2.49e-03 2 6 0.0258453 0.0245437 1.30e-03 2 8 0.0151447 0.0143172 8.28e-04 4 0 0.386646 0.392699 6.05e-03 4 2 0.0505345 0.0490874 1.45e-03 4 4 0.0153356 0.0147262 6.09e-04 4 6 0.00634952 0.00613592 2.14e-04 4 8 0.00315988 0.00306796 9.19e-05 6 0 0.239835 0.245437 5.60e-03 6 2 0.0249752 0.0245437 4.31e-04 6 4 0.00637352 0.00613592 2.38e-04 6 6 0.00225925 0.0021914 6.79e-05 6 8 0.000974924 0.000958738 1.62e-05 8 0 0.167322 0.171806 4.48e-03 8 2 0.0143688 0.0143172 5.16e-05 8 4 0.00317567 0.00306796 1.08e-04 8 6 0.000991476 0.000958738 3.27e-05 8 8 0.000379677 0.000372843 6.83e-06 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 11.7764 11.781 4.55e-03 0 4 24.6406 24.74 9.94e-02 0 6 62.142 62.5864 4.44e-01 0 8 174.03 175.757 1.73e+00 2 0 11.7687 11.781 1.22e-02 2 2 8.25802 8.24668 1.13e-02 2 4 12.4568 12.5173 6.05e-02 2 6 24.9042 25.1082 2.04e-01 2 8 58.0968 58.6287 5.32e-01 4 0 24.7835 24.74 4.34e-02 4 2 12.636 12.5173 1.19e-01 4 4 15.0239 15.0649 4.10e-02 4 6 24.8042 25.1266 3.22e-01 4 8 49.2585 50.2624 1.00e+00 6 0 62.9401 62.5864 3.54e-01 6 2 25.645 25.1082 5.37e-01 6 4 25.3004 25.1266 1.74e-01 6 6 35.6088 35.9017 2.93e-01 6 8 61.6274 62.8309 1.20e+00 8 0 177.562 175.757 1.80e+00 8 2 60.6083 58.6287 1.98e+00 8 4 51.2611 50.2624 9.99e-01 8 6 62.8751 62.8309 4.42e-02 8 8 96.3424 97.7381 1.40e+00 r2 = 2 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 11.7946 11.781 1.37e-02 0 4 24.7676 24.74 2.76e-02 0 6 62.574 62.5864 1.24e-02 0 8 175.357 175.757 4.00e-01 2 0 11.8272 11.781 4.62e-02 2 2 8.28172 8.24668 3.50e-02 2 4 12.6148 12.5173 9.75e-02 2 6 25.4036 25.1082 2.95e-01 2 8 59.4617 58.6287 8.33e-01 4 0 24.9377 24.74 1.98e-01 4 2 12.5657 12.5173 4.84e-02 4 4 15.135 15.0649 7.01e-02 4 6 25.3704 25.1266 2.44e-01 4 8 51.0215 50.2624 7.59e-01 6 0 63.3497 62.5864 7.63e-01 6 2 25.2338 25.1082 1.26e-01 6 4 25.1564 25.1266 2.98e-02 6 6 36.0459 35.9017 1.44e-01 6 8 63.4433 62.8309 6.12e-01 8 0 178.592 175.757 2.84e+00 8 2 59.0719 58.6287 4.43e-01 8 4 50.2475 50.2624 1.49e-02 8 6 62.7633 62.8309 6.76e-02 8 8 98.0395 97.7381 3.01e-01 annulus_integrals_test(): Normal end of execution. 04-Nov-2022 17:06:45