07-Jan-2022 19:05:36 disk_integrands_test(): MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 Test disk_integrands(). disk_integrands_test01(): Use a simple Monte Carlo approach to estimate the integral of X^E over the circle of radius 1 centered at the origin. N E Exact Approximate Error 1 2 0.7854 1.7627 9.7730e-01 2 2 0.7854 1.0505 2.6515e-01 4 2 0.7854 1.9311 1.1457e+00 8 2 0.7854 0.4988 2.8664e-01 16 2 0.7854 0.5438 2.4161e-01 32 2 0.7854 0.9285 1.4312e-01 64 2 0.7854 0.6770 1.0836e-01 128 2 0.7854 0.6858 9.9598e-02 256 2 0.7854 0.6697 1.1569e-01 512 2 0.7854 0.7750 1.0376e-02 1024 2 0.7854 0.7782 7.2038e-03 2048 2 0.7854 0.7544 3.0994e-02 4096 2 0.7854 0.7778 7.5990e-03 8192 2 0.7854 0.7756 9.7797e-03 16384 2 0.7854 0.7887 3.2712e-03 32768 2 0.7854 0.7856 1.6651e-04 65536 2 0.7854 0.7860 5.6858e-04 131072 2 0.7854 0.7842 1.2223e-03 262144 2 0.7854 0.7873 1.9444e-03 524288 2 0.7854 0.7862 8.2045e-04 1048576 2 0.7854 0.7848 6.1991e-04 1 4 0.3927 0.0797 3.1298e-01 2 4 0.3927 0.3846 8.1469e-03 4 4 0.3927 0.7216 3.2893e-01 8 4 0.3927 0.4523 5.9597e-02 16 4 0.3927 0.3741 1.8565e-02 32 4 0.3927 0.3066 8.6146e-02 64 4 0.3927 0.4537 6.0964e-02 128 4 0.3927 0.3930 2.7122e-04 256 4 0.3927 0.4354 4.2653e-02 512 4 0.3927 0.3593 3.3413e-02 1024 4 0.3927 0.3752 1.7464e-02 2048 4 0.3927 0.3850 7.6762e-03 4096 4 0.3927 0.4102 1.7530e-02 8192 4 0.3927 0.4032 1.0507e-02 16384 4 0.3927 0.3892 3.5400e-03 32768 4 0.3927 0.3948 2.1055e-03 65536 4 0.3927 0.3915 1.1533e-03 131072 4 0.3927 0.3929 2.2285e-04 262144 4 0.3927 0.3906 2.0800e-03 524288 4 0.3927 0.3924 2.8946e-04 1048576 4 0.3927 0.3929 1.6904e-04 1 6 0.2454 0.0330 2.1246e-01 2 6 0.2454 0.3688 1.2332e-01 4 6 0.2454 0.0058 2.3962e-01 8 6 0.2454 0.4852 2.3979e-01 16 6 0.2454 0.4761 2.3064e-01 32 6 0.2454 0.2565 1.1079e-02 64 6 0.2454 0.2460 6.1264e-04 128 6 0.2454 0.2587 1.3248e-02 256 6 0.2454 0.2317 1.3693e-02 512 6 0.2454 0.2381 7.3527e-03 1024 6 0.2454 0.2628 1.7328e-02 2048 6 0.2454 0.2438 1.6211e-03 4096 6 0.2454 0.2434 2.0144e-03 8192 6 0.2454 0.2520 6.5668e-03 16384 6 0.2454 0.2448 6.4377e-04 32768 6 0.2454 0.2442 1.2470e-03 65536 6 0.2454 0.2490 3.5394e-03 131072 6 0.2454 0.2454 7.0855e-05 262144 6 0.2454 0.2461 6.8633e-04 524288 6 0.2454 0.2460 6.0633e-04 1048576 6 0.2454 0.2444 1.0527e-03 disk_integrands_test02(): Use a simple Monte Carlo approach to estimate the integral of R^E over the disk of radius 1 centered at the origin. N E Exact Approximate Error 1 1 2.0944 1.5855 5.0890e-01 2 1 2.0944 1.9851 1.0928e-01 4 1 2.0944 1.7342 3.6015e-01 8 1 2.0944 1.5047 5.8966e-01 16 1 2.0944 2.3699 2.7552e-01 32 1 2.0944 2.2710 1.7662e-01 64 1 2.0944 2.1470 5.2588e-02 128 1 2.0944 2.0540 4.0366e-02 256 1 2.0944 2.0870 7.3641e-03 512 1 2.0944 2.0490 4.5440e-02 1024 1 2.0944 2.0850 9.3841e-03 2048 1 2.0944 2.1037 9.2740e-03 4096 1 2.0944 2.0878 6.5793e-03 8192 1 2.0944 2.0887 5.7018e-03 16384 1 2.0944 2.0948 4.3860e-04 32768 1 2.0944 2.0958 1.4114e-03 65536 1 2.0944 2.0930 1.4222e-03 131072 1 2.0944 2.0942 1.9197e-04 262144 1 2.0944 2.0936 7.9640e-04 524288 1 2.0944 2.0934 9.6423e-04 1048576 1 2.0944 2.0934 1.0170e-03 1 3 1.2566 1.5416 2.8493e-01 2 3 1.2566 0.3770 8.7965e-01 4 3 1.2566 2.3324 1.0758e+00 8 3 1.2566 1.5279 2.7130e-01 16 3 1.2566 1.0664 1.9023e-01 32 3 1.2566 1.1415 1.1513e-01 64 3 1.2566 1.2467 9.9489e-03 128 3 1.2566 1.2163 4.0304e-02 256 3 1.2566 1.2807 2.4052e-02 512 3 1.2566 1.2313 2.5381e-02 1024 3 1.2566 1.3045 4.7884e-02 2048 3 1.2566 1.2535 3.1435e-03 4096 3 1.2566 1.2546 2.0172e-03 8192 3 1.2566 1.2609 4.2271e-03 16384 3 1.2566 1.2595 2.8931e-03 32768 3 1.2566 1.2481 8.5291e-03 65536 3 1.2566 1.2595 2.8132e-03 131072 3 1.2566 1.2563 3.4370e-04 262144 3 1.2566 1.2569 3.1263e-04 524288 3 1.2566 1.2578 1.1604e-03 1048576 3 1.2566 1.2574 7.2118e-04 1 5 0.8976 0.5376 3.6004e-01 2 5 0.8976 0.2388 6.5884e-01 4 5 0.8976 1.2665 3.6892e-01 8 5 0.8976 1.1679 2.7033e-01 16 5 0.8976 0.7683 1.2934e-01 32 5 0.8976 0.7204 1.7719e-01 64 5 0.8976 0.8537 4.3914e-02 128 5 0.8976 0.9680 7.0435e-02 256 5 0.8976 0.9021 4.4813e-03 512 5 0.8976 0.9639 6.6302e-02 1024 5 0.8976 0.8605 3.7127e-02 2048 5 0.8976 0.9188 2.1203e-02 4096 5 0.8976 0.8860 1.1592e-02 8192 5 0.8976 0.9150 1.7381e-02 16384 5 0.8976 0.8885 9.0821e-03 32768 5 0.8976 0.8957 1.9290e-03 65536 5 0.8976 0.8964 1.2288e-03 131072 5 0.8976 0.9011 3.5014e-03 262144 5 0.8976 0.8973 3.4452e-04 524288 5 0.8976 0.8962 1.4100e-03 1048576 5 0.8976 0.8984 8.2614e-04 disk_integrands_test03(): Use a simple Monte Carlo approach to estimate the integral of exp(X) over the disk of radius 1 centered at the origin. N Exact Approximate Error 1 3.5510 4.0996 5.4855e-01 2 3.5510 6.5092 2.9582e+00 4 3.5510 3.6177 6.6665e-02 8 3.5510 3.2917 2.5929e-01 16 3.5510 3.2284 3.2263e-01 32 3.5510 3.3637 1.8728e-01 64 3.5510 3.4176 1.3340e-01 128 3.5510 3.2994 2.5156e-01 256 3.5510 3.4142 1.3681e-01 512 3.5510 3.6118 6.0765e-02 1024 3.5510 3.6171 6.6095e-02 2048 3.5510 3.5128 3.8220e-02 4096 3.5510 3.5543 3.2655e-03 8192 3.5510 3.5084 4.2582e-02 16384 3.5510 3.5615 1.0540e-02 32768 3.5510 3.5758 2.4778e-02 65536 3.5510 3.5650 1.4048e-02 131072 3.5510 3.5480 2.9974e-03 262144 3.5510 3.5513 2.9920e-04 524288 3.5510 3.5528 1.8261e-03 1048576 3.5510 3.5517 6.9577e-04 disk_integrands_test(): Normal end of execution. 07-Jan-2022 19:05:37