07-Jan-2022 15:21:14 annulus_monte_carlo_test(): MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 Test annulus_monte_carlo(). annulus_area_test() annulus_area() computes the area of an annulus with center = (CX,CY), inner radius R1 and outer radius R2. ( CX CY ) R1 R2 Area ( 3.147237, 4.057919) 0.126987 1.040363 3.349657 ( 1.323592,-4.024596) 0.278498 0.825380 1.896549 ( 4.575068, 4.648885) 0.157613 1.128206 3.920728 ( 4.571669,-0.146244) 0.800280 0.942167 0.776694 (-0.782387, 4.157355) 0.792207 1.751700 7.668187 ( 1.557407,-4.642883) 0.849129 1.783123 7.723623 ( 1.787352, 2.577401) 0.743132 1.135359 2.314711 ( 1.554779,-3.288133) 0.706046 0.737879 0.144401 (-2.230770,-4.538286) 0.097132 0.920590 2.632814 ( 1.948286,-1.829005) 0.950222 0.984668 0.209385 ANNULUS_SAMPLE_TEST ANNULUS_SAMPLE can sample an annulus uniformly. Use it to estimate integrals in the annulus centered at (0,0) with R1 = 0, R2 = 1 N 1 X^2 Y^2 X^4 X^2Y^2 Y^4 X^6 1 3.14159 1.02973 0.168971 0.337518 0.0553841 0.0090881 0.11063 2 3.14159 0.0631809 0.999678 0.00232739 0.0277739 0.373765 8.9311e-05 4 3.14159 0.421631 0.971223 0.0975678 0.110475 0.461996 0.0238664 8 3.14159 0.97872 0.650145 0.589351 0.140491 0.255107 0.405148 16 3.14159 0.363934 1.07071 0.0998 0.106269 0.612519 0.037993 32 3.14159 0.577326 0.707424 0.235573 0.0962249 0.331224 0.13071 64 3.14159 0.769761 0.829046 0.380305 0.120032 0.43729 0.231838 128 3.14159 0.771645 0.780854 0.333579 0.136856 0.369452 0.181976 256 3.14159 0.767249 0.742317 0.359574 0.126794 0.359989 0.206693 512 3.14159 0.816209 0.802139 0.400833 0.146886 0.397044 0.243606 1024 3.14159 0.855343 0.760451 0.44131 0.137448 0.374758 0.279704 2048 3.14159 0.783882 0.789154 0.385651 0.132119 0.396395 0.236265 4096 3.14159 0.79767 0.776177 0.402553 0.133542 0.385264 0.252057 8192 3.14159 0.792577 0.783703 0.398758 0.132733 0.38755 0.250231 16384 3.14159 0.78194 0.779487 0.389671 0.129197 0.391631 0.243035 32768 3.14159 0.782447 0.792183 0.390335 0.131573 0.398717 0.243275 65536 3.14159 0.785123 0.789079 0.393866 0.130289 0.396908 0.247058 Exact 3.14159 0.785398 0.785398 0.392699 0.1309 0.392699 0.245437 ANNULUS_SAMPLE_TEST ANNULUS_SAMPLE can sample an annulus uniformly. Use it to estimate integrals in the annulus centered at (0,0) with R1 = 0.5, R2 = 1 N 1 X^2 Y^2 X^4 X^2Y^2 Y^4 X^6 1 2.35619 0.00200068 1.02275 1.6988e-06 0.000868428 0.44394 1.44248e-09 2 2.35619 0.922409 0.860101 0.69707 0.137605 0.431973 0.535938 4 2.35619 0.985959 0.621387 0.577559 0.164013 0.296587 0.396346 8 2.35619 0.233489 1.10132 0.0324797 0.114016 0.583443 0.00529867 16 2.35619 0.647136 0.751926 0.336215 0.0800122 0.445746 0.205896 32 2.35619 0.741706 0.84347 0.364048 0.175576 0.477218 0.207412 64 2.35619 0.859411 0.670963 0.469669 0.146486 0.33046 0.30335 128 2.35619 0.790699 0.693213 0.393832 0.142711 0.360644 0.227172 256 2.35619 0.766084 0.765532 0.401381 0.135896 0.421018 0.247714 512 2.35619 0.788292 0.710632 0.438622 0.12743 0.376427 0.292438 1024 2.35619 0.709754 0.751789 0.362463 0.127864 0.402657 0.226358 2048 2.35619 0.72665 0.750494 0.377743 0.130155 0.396841 0.235208 4096 2.35619 0.730159 0.734023 0.380322 0.127993 0.382387 0.238448 8192 2.35619 0.739556 0.7275 0.388463 0.127483 0.379438 0.245609 16384 2.35619 0.730364 0.735876 0.381008 0.12806 0.383888 0.240238 32768 2.35619 0.735166 0.737821 0.38695 0.127712 0.388966 0.245672 65536 2.35619 0.736693 0.735457 0.385723 0.129424 0.385093 0.243209 ANNULUS_SAMPLE_TEST ANNULUS_SAMPLE can sample an annulus uniformly. Use it to estimate integrals in the annulus centered at (1,0) with R1 = 0, R2 = 1 N 1 X^2 Y^2 X^4 X^2Y^2 Y^4 X^6 1 3.14159 0.948814 1.30944 0.286558 0.395472 0.545781 0.0865453 2 3.14159 1.12817 1.002 0.438354 0.458114 0.610406 0.181274 4 3.14159 3.50113 1.27643 5.64699 1.13003 0.664062 10.4364 8 3.14159 3.97779 1.00578 8.09477 1.12607 0.581449 19.1259 16 3.14159 4.46714 0.564086 10.8788 0.801435 0.169591 31.5839 32 3.14159 3.68248 1.11465 6.92646 1.07252 0.648629 16.2774 64 3.14159 4.0013 0.82693 8.44608 1.09544 0.488955 21.6876 128 3.14159 4.17825 0.881552 8.798 1.14674 0.44728 22.3316 256 3.14159 3.83561 0.798941 8.1404 0.919707 0.413123 21.0928 512 3.14159 3.71731 0.791993 7.37718 0.914248 0.398573 17.8988 1024 3.14159 3.91875 0.732635 8.31443 0.826414 0.349682 21.4895 2048 3.14159 3.94226 0.759555 8.29295 0.879037 0.37075 21.1409 4096 3.14159 3.87551 0.77575 8.215 0.900762 0.387867 21.1527 8192 3.14159 3.9415 0.788972 8.34678 0.926398 0.397675 21.5075 16384 3.14159 3.93807 0.785658 8.2569 0.928768 0.393093 21.0376 32768 3.14159 3.93043 0.789317 8.27701 0.91352 0.39661 21.2172 65536 3.14159 3.93259 0.786994 8.25497 0.920286 0.39448 21.0697 annulus_monte_carlo_test(): Normal end of execution. 07-Jan-2022 15:21:15