Wed Oct 8 07:25:46 2025 annulus_monte_carlo_test(): python version: 3.10.12 numpy version: 1.26.4 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 ( 1.666945, 3.557849) 0.528227 1.437347 5.613851 ( 3.132318, 2.167444) 0.466904 0.475344 0.024983 (-2.026508, 1.051260) 0.237508 0.823463 1.953070 ( 0.789158, 1.977272) 0.625883 0.806376 0.812143 (-0.571886,-3.586620) 0.526666 1.151749 3.295997 ( 1.407109, 0.778695) 0.027935 0.657329 1.354972 (-3.711236,-1.165105) 0.126910 0.577763 0.998097 (-3.406165,-1.405080) 0.458707 0.498307 0.119060 (-1.123525, 1.220549) 0.650033 1.614303 6.859454 ( 4.293400,-0.993997) 0.754916 1.261635 3.210160 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 0.00323 3.11452 3.3209e-06 0.00320217 3.08769 3.41436e-09 2 3.14159 0.706027 0.906504 0.283848 0.117281 0.321264 0.120055 4 3.14159 0.891811 0.684507 0.374974 0.138202 0.266441 0.188487 8 3.14159 0.721454 0.81274 0.288133 0.168193 0.34356 0.14187 16 3.14159 0.796449 1.00802 0.448716 0.114759 0.605963 0.311949 32 3.14159 0.851997 0.95206 0.47578 0.125226 0.519143 0.324378 64 3.14159 0.762751 0.800983 0.34452 0.136596 0.417768 0.198478 128 3.14159 0.748058 0.72611 0.361794 0.116999 0.348098 0.218014 256 3.14159 0.726101 0.844784 0.344118 0.116549 0.442182 0.208173 512 3.14159 0.761566 0.788919 0.373708 0.132858 0.386365 0.235375 1024 3.14159 0.82379 0.800758 0.421827 0.133043 0.405381 0.269064 2048 3.14159 0.784679 0.782404 0.389648 0.129534 0.392177 0.244751 4096 3.14159 0.771946 0.795226 0.382328 0.130412 0.399977 0.236323 8192 3.14159 0.785419 0.785461 0.393622 0.131766 0.391611 0.246936 16384 3.14159 0.782572 0.777511 0.38962 0.129372 0.38516 0.242568 32768 3.14159 0.778738 0.786741 0.38725 0.131678 0.392469 0.240998 65536 3.14159 0.784377 0.785626 0.392781 0.130618 0.393885 0.246078 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 0.737132 0.416257 0.230611 0.130226 0.0735382 0.0721462 2 2.35619 0.445585 0.925974 0.161111 0.0561542 0.548054 0.0595329 4 2.35619 0.94207 0.469371 0.410833 0.151476 0.14139 0.19025 8 2.35619 0.724049 0.68215 0.345164 0.167076 0.271606 0.20651 16 2.35619 0.812634 0.677387 0.453659 0.150093 0.319041 0.307581 32 2.35619 0.583734 0.826585 0.22739 0.12745 0.45241 0.104541 64 2.35619 0.71803 0.753397 0.38772 0.121679 0.412031 0.258367 128 2.35619 0.838725 0.747578 0.463703 0.160469 0.393361 0.298522 256 2.35619 0.779475 0.709157 0.41042 0.134323 0.361656 0.256653 512 2.35619 0.756348 0.719738 0.398242 0.136247 0.368123 0.24978 1024 2.35619 0.764155 0.717733 0.405757 0.135174 0.366979 0.256726 2048 2.35619 0.746957 0.720223 0.397818 0.127336 0.373127 0.255042 4096 2.35619 0.727464 0.751703 0.381376 0.129439 0.398523 0.240986 8192 2.35619 0.730761 0.749302 0.383135 0.130127 0.397194 0.242236 16384 2.35619 0.732668 0.739149 0.382578 0.128668 0.388389 0.24015 32768 2.35619 0.733811 0.741123 0.385524 0.12866 0.390985 0.24397 65536 2.35619 0.735659 0.737621 0.386455 0.129035 0.387063 0.244597 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 1.75505 1.27438 0.980461 0.711932 0.516948 0.547735 2 3.14159 5.46465 0.304174 12.0762 0.745048 0.0475916 29.9493 4 3.14159 3.84397 0.747712 8.93603 1.02228 0.306121 24.9412 8 3.14159 4.06174 1.40019 7.5888 1.50517 1.04384 18.2327 16 3.14159 5.07665 0.64618 13.6669 0.973069 0.223708 41.1652 32 3.14159 3.51305 0.871722 6.82396 0.991232 0.438451 16.0343 64 3.14159 4.16697 0.686612 9.22246 0.823763 0.32792 23.766 128 3.14159 3.67232 0.642828 7.78394 0.763878 0.317956 20.368 256 3.14159 4.07589 0.757179 8.74931 0.862009 0.3913 22.2608 512 3.14159 3.92685 0.745142 8.11226 0.895634 0.368471 20.6061 1024 3.14159 3.84785 0.79092 7.94445 0.901214 0.390071 19.976 2048 3.14159 3.9068 0.770108 8.09791 0.874679 0.378874 20.3998 4096 3.14159 3.97488 0.770454 8.35914 0.921048 0.383777 21.2161 8192 3.14159 3.89065 0.77025 8.15744 0.890705 0.381227 20.8669 16384 3.14159 3.9308 0.774344 8.30291 0.905995 0.383673 21.3101 32768 3.14159 3.92354 0.791273 8.21871 0.92377 0.397464 20.927 65536 3.14159 3.94027 0.785289 8.29819 0.919456 0.393807 21.2329 annulus_monte_carlo_test(): Normal end of execution. Wed Oct 8 07:25:48 2025