Tue Oct 19 11:53:20 2021 hyperball_monte_carlo_test(): Python version: 3.6.9 Test hyperball_monte_carlo(). hyperball_monte_carlo_test01 Python version: 3.6.9 Use the Monte Carlo method to estimate integrals over the interior of the unit hyperball in M dimensions. Spatial dimension M = 3 N 1 X^2 Y^2 Z^2 X^4 X^2Y^2 Z^4 1 4.18879 2.49898 0.338345 1.14767 1.49086 0.201853 0.314447 2 4.18879 0.305424 1.77158 0.555055 0.0279207 0.117252 0.130648 4 4.18879 0.482968 0.658135 0.817041 0.110411 0.0529315 0.355448 8 4.18879 0.527317 0.747068 1.36386 0.110086 0.0679667 0.748351 16 4.18879 0.681437 0.959758 0.96551 0.240868 0.0823136 0.426471 32 4.18879 1.00963 0.899314 0.614385 0.512992 0.146394 0.210533 64 4.18879 0.937762 0.685779 0.810133 0.418991 0.11626 0.338407 128 4.18879 0.886977 1.03702 0.755431 0.411456 0.13572 0.284696 256 4.18879 0.702135 0.879147 0.815996 0.265251 0.0982451 0.326915 512 4.18879 0.829268 0.842567 0.842794 0.350688 0.12884 0.364796 1024 4.18879 0.815747 0.848893 0.838192 0.335603 0.122059 0.369132 2048 4.18879 0.806212 0.830958 0.846212 0.338451 0.110286 0.366833 4096 4.18879 0.837241 0.850127 0.821198 0.353683 0.121969 0.3466 8192 4.18879 0.838225 0.847643 0.830691 0.359474 0.119968 0.354039 16384 4.18879 0.832542 0.834516 0.832517 0.35413 0.119049 0.354309 32768 4.18879 0.830118 0.841329 0.835174 0.351848 0.119709 0.360042 65536 4.18879 0.831612 0.836304 0.836691 0.354867 0.118994 0.358479 Exact 4.18879 0.837758 0.837758 0.837758 0.359039 0.11968 0.359039 hyperball_monte_carlo_test01 Normal end of execution. hyperball_monte_carlo_test02 Python version: 3.6.9 Use the Monte Carlo method to estimate integrals over the interior of the unit hyperball in M dimensions. Spatial dimension M = 6 N 1 U V^2 V^2W^2 X^4 Y^2Z^2 Z^6 1 5.16771 0.183113 3.26276 0.0216035 2.40961e-05 2.53043e-05 0.00425145 2 5.16771 0.699144 0.0643996 0.0131291 0.0987737 0.00442907 0.172229 4 5.16771 0.711235 0.880608 0.0474521 0.717279 0.0194042 0.0388302 8 5.16771 -0.829426 0.60075 0.05034 0.220361 0.00413613 0.0134181 16 5.16771 -0.488251 0.63409 0.0380013 0.409426 0.00132766 0.0374404 32 5.16771 0.0700833 0.905066 0.0910041 0.178156 0.00295018 0.0675552 64 5.16771 0.243562 0.691267 0.0834463 0.182202 0.00395012 0.0609396 128 5.16771 0.335089 0.666435 0.0902041 0.194046 0.00406604 0.0904223 256 5.16771 0.0623681 0.61263 0.0510995 0.225771 0.00544565 0.093126 512 5.16771 -0.123977 0.636324 0.0625051 0.195932 0.00592631 0.0732874 1024 5.16771 -0.110746 0.646326 0.0696397 0.188978 0.00594064 0.0815842 2048 5.16771 -0.0220412 0.638509 0.0634906 0.198579 0.00539482 0.076892 4096 5.16771 0.0118139 0.633049 0.0625548 0.196586 0.00514268 0.0756478 8192 5.16771 -0.0187385 0.646154 0.064835 0.194015 0.00566376 0.079228 16384 5.16771 0.0180001 0.633276 0.0634938 0.194949 0.00534793 0.0831334 32768 5.16771 -0.0120724 0.645623 0.0639331 0.19648 0.00534836 0.0802457 65536 5.16771 -0.000681546 0.646934 0.0648712 0.189046 0.00539298 0.0812117 Exact 5.16771 0 0.645964 0.0645964 0.193789 0.00538303 0.0807455 hyperball_monte_carlo_test02 Use the Monte Carlo method to estimate integrals hyperball01_monomial_integral_test Python version: 3.6.9 hyperball01_monomial_integral computes the integral of a monomial over the interior of the unit hyperball in M dimensions. Compare with a Monte Carlo estimate. Spatial dimension M = 3 Number of sample points used is 4192 If any exponent is odd, the integral is zero. We will restrict this test to randomly chosen even exponents. Ex Ey Ez MC-Estimate Exact Error 0 8 0 0.127651 0.126933 0.00072 2 2 4 0.00366693 0.00362666 4e-05 4 2 4 0.000820723 0.000836921 1.6e-05 6 8 6 8.87367e-07 9.388e-07 5.1e-08 0 6 4 0.00449421 0.00418461 0.00031 6 0 0 0.198408 0.199466 0.0011 6 2 4 0.000271516 0.000278974 7.5e-06 8 8 8 6.34559e-08 6.81499e-08 4.7e-09 6 8 6 8.87367e-07 9.388e-07 5.1e-08 4 0 0 0.355304 0.359039 0.0037 0 0 2 0.853279 0.837758 0.016 8 0 4 0.00200689 0.00195282 5.4e-05 4 0 0 0.355304 0.359039 0.0037 0 6 0 0.20168 0.199466 0.0022 0 8 6 0.000628323 0.000574358 5.4e-05 4 8 6 4.3118e-06 4.31848e-06 6.7e-09 4 8 6 4.3118e-06 4.31848e-06 6.7e-09 0 4 2 0.0416762 0.0398932 0.0018 0 0 6 0.199554 0.199466 8.8e-05 4 0 8 0.00206645 0.00195282 0.00011 hyperball01_monomial_integral_test Normal end of execution. hyperball01_sample_test Python version: 3.6.9 hyperball01_sample samples the unit hyperball. Sample points in the unit hyperball. Row: 0 1 2 Col 0 : 0.0281771 0.580197 -0.349252 1 : 0.536712 -0.107119 0.649386 2 : 0.0615639 0.0374416 0.276139 3 : 0.661454 -0.122071 0.69254 4 : -0.351402 0.125613 -0.625333 5 : -0.0120536 0.239983 -0.206623 6 : -0.192901 0.1732 0.318189 7 : -0.914321 -0.126092 0.384568 8 : -0.0704035 0.015807 0.945947 9 : -0.480485 0.463615 -0.522886 hyperball01_sample_test Normal end of execution. hyperball01_volume_test Python version: 3.6.9 hyperball01_volume returns the volume of the unit hyperball in M dimensions. M Volume 1 2 2 3.14159 3 4.18879 4 4.9348 5 5.26379 6 5.16771 7 4.72477 8 4.05871 9 3.29851 10 2.55016 hyperball01_volume_test Normal end of execution. hyperball_monte_carlo_test(): Normal end of execution. Tue Oct 19 11:53:24 2021