Thu Aug 10 18:11:48 2023 disk01_positive_monte_carlo_test(): Python version: 3.8.10 Test disk01_positive_monte_carlo(). disk01_positive_area_test(): disk01_positive_area returns the area of the unit positive disk. disk01_positive_area() = 0.785398 disk01_positive_monomial_integral_test(): disk01_positive_monomial_integral computes monomial integrals over the interior of the unit disk in 2D. Compare with a Monte Carlo value. Number of sample points used is 4192 We will restrict this test to randomly chosen even exponents. Ex Ey MC-Estimate Exact Error 1 3 0.124495 0.0416667 0.083 4 1 0.0993971 0.0285714 0.071 4 0 0.296766 0.0981748 0.2 0 2 0.390784 0.19635 0.19 4 0 0.296766 0.0981748 0.2 2 0 0.394614 0.19635 0.2 0 1 0.497555 0.333333 0.16 4 3 0.0285747 0.00634921 0.022 2 0 0.394614 0.19635 0.2 4 0 0.296766 0.0981748 0.2 2 2 0.0978474 0.0327249 0.065 1 1 0.248857 0.125 0.12 4 0 0.296766 0.0981748 0.2 1 1 0.248857 0.125 0.12 3 3 0.0416306 0.0104167 0.031 0 3 0.331653 0.133333 0.2 1 1 0.248857 0.125 0.12 0 0 0.785398 0.785398 1.1e-16 4 1 0.0993971 0.0285714 0.071 3 2 0.0664584 0.0190476 0.047 disk01_positive_sample_test() Use disk01_positive_sample to estimate integrals in the unit positive disk. Estimate integral of X^0 Y^0 N Estimate Error 1 0.785398 1.11e-16 2 0.785398 1.11e-16 4 0.785398 1.11e-16 8 0.785398 1.11e-16 16 0.785398 1.11e-16 32 0.785398 1.11e-16 64 0.785398 1.11e-16 128 0.785398 1.11e-16 256 0.785398 1.11e-16 512 0.785398 1.11e-16 1024 0.785398 1.11e-16 2048 0.785398 1.11e-16 4096 0.785398 1.11e-16 8192 0.785398 1.11e-16 16384 0.785398 1.11e-16 32768 0.785398 1.11e-16 65536 0.785398 1.11e-16 Exact: 0.785398 0 Estimate integral of X^0 Y^1 N Estimate Error 1 0.660021 3.27e-01 2 0.440017 1.07e-01 4 0.543489 2.10e-01 8 0.541657 2.08e-01 16 0.402428 6.91e-02 32 0.455838 1.23e-01 64 0.507072 1.74e-01 128 0.525215 1.92e-01 256 0.503197 1.70e-01 512 0.493097 1.60e-01 1024 0.490133 1.57e-01 2048 0.506913 1.74e-01 4096 0.505133 1.72e-01 8192 0.503236 1.70e-01 16384 0.498962 1.66e-01 32768 0.500689 1.67e-01 65536 0.500605 1.67e-01 Exact: 0.333333 0 Estimate integral of X^0 Y^2 N Estimate Error 1 0.630217 4.34e-01 2 0.230913 3.46e-02 4 0.518773 3.22e-01 8 0.326814 1.30e-01 16 0.306988 1.11e-01 32 0.379805 1.83e-01 64 0.388919 1.93e-01 128 0.38513 1.89e-01 256 0.382894 1.87e-01 512 0.382783 1.86e-01 1024 0.386276 1.90e-01 2048 0.393891 1.98e-01 4096 0.396912 2.01e-01 8192 0.386509 1.90e-01 16384 0.395894 2.00e-01 32768 0.390443 1.94e-01 65536 0.392235 1.96e-01 Exact: 0.19635 0 Estimate integral of X^0 Y^3 N Estimate Error 1 0.0235719 1.10e-01 2 0.306703 1.73e-01 4 0.0606193 7.27e-02 8 0.219125 8.58e-02 16 0.336427 2.03e-01 32 0.374857 2.42e-01 64 0.315493 1.82e-01 128 0.337443 2.04e-01 256 0.312281 1.79e-01 512 0.347664 2.14e-01 1024 0.33249 1.99e-01 2048 0.336414 2.03e-01 4096 0.334285 2.01e-01 8192 0.336799 2.03e-01 16384 0.335462 2.02e-01 32768 0.333547 2.00e-01 65536 0.333857 2.01e-01 Exact: 0.133333 0 Estimate integral of X^0 Y^4 N Estimate Error 1 0.6977 6.00e-01 2 0.111948 1.38e-02 4 0.193507 9.53e-02 8 0.363472 2.65e-01 16 0.331728 2.34e-01 32 0.220133 1.22e-01 64 0.339611 2.41e-01 128 0.317535 2.19e-01 256 0.300352 2.02e-01 512 0.31563 2.17e-01 1024 0.288751 1.91e-01 2048 0.29337 1.95e-01 4096 0.281094 1.83e-01 8192 0.297841 2.00e-01 16384 0.294709 1.97e-01 32768 0.294939 1.97e-01 65536 0.293056 1.95e-01 Exact: 0.0981748 0 Estimate integral of X^1 Y^0 N Estimate Error 1 0.708783 3.75e-01 2 0.729558 3.96e-01 4 0.264466 6.89e-02 8 0.560628 2.27e-01 16 0.512142 1.79e-01 32 0.532915 2.00e-01 64 0.529778 1.96e-01 128 0.520461 1.87e-01 256 0.530808 1.97e-01 512 0.500875 1.68e-01 1024 0.518712 1.85e-01 2048 0.508125 1.75e-01 4096 0.500546 1.67e-01 8192 0.503229 1.70e-01 16384 0.505102 1.72e-01 32768 0.500659 1.67e-01 65536 0.499304 1.66e-01 Exact: 0.333333 0 Estimate integral of X^1 Y^1 N Estimate Error 1 0.392169 2.67e-01 2 0.23477 1.10e-01 4 0.278866 1.54e-01 8 0.217719 9.27e-02 16 0.251013 1.26e-01 32 0.273491 1.48e-01 64 0.248514 1.24e-01 128 0.260288 1.35e-01 256 0.257131 1.32e-01 512 0.246637 1.22e-01 1024 0.255554 1.31e-01 2048 0.244449 1.19e-01 4096 0.248873 1.24e-01 8192 0.250315 1.25e-01 16384 0.249658 1.25e-01 32768 0.250605 1.26e-01 65536 0.249655 1.25e-01 Exact: 0.125 0 Estimate integral of X^1 Y^2 N Estimate Error 1 0.295743 2.29e-01 2 0.272823 2.06e-01 4 0.200029 1.33e-01 8 0.130557 6.39e-02 16 0.0960863 2.94e-02 32 0.153001 8.63e-02 64 0.179715 1.13e-01 128 0.174186 1.08e-01 256 0.173852 1.07e-01 512 0.172714 1.06e-01 1024 0.167896 1.01e-01 2048 0.166424 9.98e-02 4096 0.165644 9.90e-02 8192 0.166788 1.00e-01 16384 0.166963 1.00e-01 32768 0.166372 9.97e-02 65536 0.167417 1.01e-01 Exact: 0.0666667 0 Estimate integral of X^1 Y^3 N Estimate Error 1 0.193262 1.52e-01 2 0.23284 1.91e-01 4 0.0649945 2.33e-02 8 0.153309 1.12e-01 16 0.100293 5.86e-02 32 0.11121 6.95e-02 64 0.131459 8.98e-02 128 0.13033 8.87e-02 256 0.117274 7.56e-02 512 0.114111 7.24e-02 1024 0.12233 8.07e-02 2048 0.12548 8.38e-02 4096 0.125065 8.34e-02 8192 0.126106 8.44e-02 16384 0.125977 8.43e-02 32768 0.125954 8.43e-02 65536 0.124945 8.33e-02 Exact: 0.0416667 0 Estimate integral of X^2 Y^0 N Estimate Error 1 0.780511 5.84e-01 2 0.390398 1.94e-01 4 0.672787 4.76e-01 8 0.347347 1.51e-01 16 0.353377 1.57e-01 32 0.499681 3.03e-01 64 0.360443 1.64e-01 128 0.382284 1.86e-01 256 0.450838 2.54e-01 512 0.395454 1.99e-01 1024 0.400463 2.04e-01 2048 0.395563 1.99e-01 4096 0.395671 1.99e-01 8192 0.390606 1.94e-01 16384 0.39463 1.98e-01 32768 0.392428 1.96e-01 65536 0.392281 1.96e-01 Exact: 0.19635 0 Estimate integral of X^2 Y^1 N Estimate Error 1 0.291734 2.25e-01 2 0.0386014 2.81e-02 4 0.124351 5.77e-02 8 0.168553 1.02e-01 16 0.149616 8.29e-02 32 0.168854 1.02e-01 64 0.15503 8.84e-02 128 0.154595 8.79e-02 256 0.171853 1.05e-01 512 0.165696 9.90e-02 1024 0.167355 1.01e-01 2048 0.163521 9.69e-02 4096 0.168103 1.01e-01 8192 0.165678 9.90e-02 16384 0.16686 1.00e-01 32768 0.166373 9.97e-02 65536 0.166754 1.00e-01 Exact: 0.0666667 0 Estimate integral of X^2 Y^2 N Estimate Error 1 0.164685 1.32e-01 2 0.117415 8.47e-02 4 0.110102 7.74e-02 8 0.100499 6.78e-02 16 0.0756007 4.29e-02 32 0.122047 8.93e-02 64 0.108484 7.58e-02 128 0.0999264 6.72e-02 256 0.0942987 6.16e-02 512 0.0954029 6.27e-02 1024 0.0964181 6.37e-02 2048 0.0997103 6.70e-02 4096 0.0981155 6.54e-02 8192 0.0980344 6.53e-02 16384 0.0978319 6.51e-02 32768 0.0981349 6.54e-02 65536 0.0980256 6.53e-02 Exact: 0.0327249 0 Estimate integral of X^3 Y^0 N Estimate Error 1 0.292132 1.59e-01 2 0.614681 4.81e-01 4 0.326747 1.93e-01 8 0.466615 3.33e-01 16 0.329722 1.96e-01 32 0.346652 2.13e-01 64 0.288715 1.55e-01 128 0.372699 2.39e-01 256 0.330587 1.97e-01 512 0.33363 2.00e-01 1024 0.330689 1.97e-01 2048 0.339223 2.06e-01 4096 0.336661 2.03e-01 8192 0.328741 1.95e-01 16384 0.33199 1.99e-01 32768 0.334117 2.01e-01 65536 0.334158 2.01e-01 Exact: 0.133333 0 Estimate integral of X^3 Y^1 N Estimate Error 1 0.000155187 4.15e-02 2 0.153769 1.12e-01 4 0.0655632 2.39e-02 8 0.153273 1.12e-01 16 0.129916 8.82e-02 32 0.106151 6.45e-02 64 0.137069 9.54e-02 128 0.128093 8.64e-02 256 0.132918 9.13e-02 512 0.127152 8.55e-02 1024 0.120972 7.93e-02 2048 0.124673 8.30e-02 4096 0.126674 8.50e-02 8192 0.123657 8.20e-02 16384 0.12387 8.22e-02 32768 0.124368 8.27e-02 65536 0.125438 8.38e-02 Exact: 0.0416667 0 Estimate integral of X^4 Y^0 N Estimate Error 1 0.206728 1.09e-01 2 0.0666314 3.15e-02 4 0.0495651 4.86e-02 8 0.271282 1.73e-01 16 0.330456 2.32e-01 32 0.269798 1.72e-01 64 0.283917 1.86e-01 128 0.272954 1.75e-01 256 0.305647 2.07e-01 512 0.299418 2.01e-01 1024 0.294686 1.97e-01 2048 0.29275 1.95e-01 4096 0.294601 1.96e-01 8192 0.290315 1.92e-01 16384 0.294554 1.96e-01 32768 0.291533 1.93e-01 65536 0.29402 1.96e-01 Exact: 0.0981748 0 disk01_positive_monte_carlo_test(): Normal end of execution. Thu Aug 10 18:11:50 2023