06-Jun-2023 14:45:21 test_int_2d_test(): MATLAB/Octave version 5.2.0 Test test_int_2d(). test_int_2d_test01(): Use a Monte Carlo rule. Repeatedly multiply the number of points by 16. Problem Points Approx Error 1 1 1.137094e+00 5.078403e-01 1 4 1.802090e+00 1.571557e-01 1 16 1.688065e+00 4.313105e-02 1 64 1.536400e+00 1.085343e-01 1 256 1.552961e+00 9.197299e-02 1 1024 1.704838e+00 5.990350e-02 1 4096 1.650839e+00 5.904436e-03 1 16384 1.642791e+00 2.143539e-03 1 65536 1.634529e+00 1.040555e-02 1 262144 1.641655e+00 3.278755e-03 1 1048576 1.646611e+00 1.676446e-03 1 4194304 1.645617e+00 6.833472e-04 1 Exact 1.644934e+00 2 1 4.000430e+00 3.547422e-01 2 4 4.041428e+00 3.137438e-01 2 16 4.729218e+00 3.740454e-01 2 64 4.408236e+00 5.306427e-02 2 256 4.380027e+00 2.485457e-02 2 1024 4.347757e+00 7.415389e-03 2 4096 4.360336e+00 5.164189e-03 2 16384 4.351490e+00 3.682527e-03 2 65536 4.359637e+00 4.464346e-03 2 262144 4.355348e+00 1.753795e-04 2 1048576 4.355749e+00 5.769868e-04 2 4194304 4.355282e+00 1.103084e-04 2 Exact 4.355172e+00 3 1 2.997075e+00 1.271192e-01 3 4 2.691324e+00 4.328708e-01 3 16 2.957552e+00 1.666420e-01 3 64 3.098003e+00 2.619173e-02 3 256 3.145820e+00 2.162529e-02 3 1024 3.102543e+00 2.165143e-02 3 4096 3.093247e+00 3.094743e-02 3 16384 3.130386e+00 6.191781e-03 3 65536 3.122400e+00 1.794760e-03 3 262144 3.123834e+00 3.601408e-04 3 1048576 3.123329e+00 8.648477e-04 3 4194304 3.124908e+00 7.137306e-04 3 Exact 3.124194e+00 4 1 2.911525e+00 3.325172e-01 4 4 1.888833e+00 6.901741e-01 4 16 2.561354e+00 1.765355e-02 4 64 2.642402e+00 6.339421e-02 4 256 2.569328e+00 9.679075e-03 4 1024 2.575522e+00 3.485088e-03 4 4096 2.609795e+00 3.078726e-02 4 16384 2.572626e+00 6.381687e-03 4 65536 2.579264e+00 2.561063e-04 4 262144 2.576325e+00 2.682248e-03 4 1048576 2.578526e+00 4.818079e-04 4 4194304 2.578560e+00 4.470870e-04 4 Exact 2.579008e+00 5 1 9.641600e-01 5.197156e-01 5 4 4.886878e-01 4.424334e-02 5 16 4.445767e-01 1.322255e-04 5 64 4.671966e-01 2.275217e-02 5 256 4.497582e-01 5.313767e-03 5 1024 4.487444e-01 4.299998e-03 5 4096 4.475560e-01 3.111540e-03 5 16384 4.451996e-01 7.551628e-04 5 65536 4.444691e-01 2.464926e-05 5 262144 4.446725e-01 2.280833e-04 5 1048576 4.441180e-01 3.264638e-04 5 4194304 4.442429e-01 2.015849e-04 5 Exact 4.444444e-01 6 1 1.245029e+00 6.179871e-01 6 4 1.130799e+00 7.322174e-01 6 16 2.309045e+00 4.460283e-01 6 64 1.734313e+00 1.287029e-01 6 256 1.847904e+00 1.511244e-02 6 1024 1.887022e+00 2.400611e-02 6 4096 1.875351e+00 1.233460e-02 6 16384 1.845997e+00 1.701923e-02 6 65536 1.851261e+00 1.175507e-02 6 262144 1.861751e+00 1.264780e-03 6 1048576 1.863051e+00 3.451031e-05 6 4194304 1.863602e+00 5.858895e-04 6 Exact 1.863016e+00 7 1 4.041559e-01 1.291775e-01 7 4 4.163261e-01 1.170073e-01 7 16 5.558210e-01 2.248765e-02 7 64 5.063932e-01 2.694017e-02 7 256 5.366460e-01 3.312668e-03 7 1024 5.427785e-01 9.445191e-03 7 4096 5.351402e-01 1.806825e-03 7 16384 5.323119e-01 1.021392e-03 7 65536 5.329666e-01 3.667553e-04 7 262144 5.344234e-01 1.090052e-03 7 1048576 5.334336e-01 1.002489e-04 7 4194304 5.332733e-01 5.998571e-05 7 Exact 5.333333e-01 8 1 5.345632e-05 2.666802e+00 8 4 9.522603e-01 1.714595e+00 8 16 2.342914e+00 3.239415e-01 8 64 3.912524e+00 1.245669e+00 8 256 2.310703e+00 3.561523e-01 8 1024 2.617353e+00 4.950292e-02 8 4096 2.695550e+00 2.869380e-02 8 16384 2.666996e+00 1.401225e-04 8 65536 2.641392e+00 2.546378e-02 8 262144 2.677013e+00 1.015690e-02 8 1048576 2.676079e+00 9.223477e-03 8 4194304 2.665668e+00 1.187232e-03 8 Exact 2.666856e+00 test_int_2d_test02(): Apply a product of composite midpoint rules. Repeatedly multiply the number of points by 16. Problem Points Approx Error 1 1 1.333333e+00 3.116007e-01 1 4 1.453480e+00 1.914542e-01 1 16 1.537716e+00 1.072184e-01 1 64 1.587712e+00 5.722188e-02 1 256 1.615219e+00 2.971465e-02 1 1024 1.629748e+00 1.518607e-02 1 4096 1.637245e+00 7.688838e-03 1 16384 1.641062e+00 3.871831e-03 1 65536 1.642990e+00 1.943642e-03 1 262144 1.643960e+00 9.739717e-04 1 1048576 1.644446e+00 4.875785e-04 1 4194304 1.644690e+00 2.439512e-04 1 Exact 1.644934e+00 2 1 4.000000e+00 3.551722e-01 2 4 4.131182e+00 2.239899e-01 2 16 4.247556e+00 1.076158e-01 2 64 4.308696e+00 4.647584e-02 2 256 4.336327e+00 1.884480e-02 2 1024 4.347843e+00 7.329365e-03 2 4096 4.352402e+00 2.770353e-03 2 16384 4.354146e+00 1.026536e-03 2 65536 4.354797e+00 3.751245e-04 2 262144 4.355036e+00 1.357498e-04 2 1048576 4.355123e+00 4.878914e-05 2 4194304 4.355155e+00 1.745056e-05 2 Exact 4.355172e+00 3 1 2.828427e+00 2.957672e-01 3 4 2.991564e+00 1.326305e-01 3 16 3.069680e+00 5.451432e-02 3 64 3.102966e+00 2.122816e-02 3 256 3.116197e+00 7.997223e-03 3 1024 3.121244e+00 2.950642e-03 3 4096 3.123120e+00 1.074022e-03 3 16384 3.123807e+00 3.874281e-04 3 65536 3.124055e+00 1.389026e-04 3 262144 3.124145e+00 4.959099e-05 3 1048576 3.124177e+00 1.765344e-05 3 4194304 3.124188e+00 6.271530e-06 3 Exact 3.124194e+00 4 1 2.309401e+00 2.696065e-01 4 4 2.454879e+00 1.241284e-01 4 16 2.526993e+00 5.201505e-02 4 64 2.558498e+00 2.050949e-02 4 256 2.571220e+00 7.787491e-03 4 1024 2.576120e+00 2.887793e-03 4 4096 2.577953e+00 1.054641e-03 4 16384 2.578626e+00 3.812863e-04 4 65536 2.578871e+00 1.369088e-04 4 262144 2.578959e+00 4.893053e-05 4 1048576 2.578990e+00 1.743104e-05 4 4194304 2.579001e+00 6.195676e-06 4 Exact 2.579008e+00 5 1 5.000000e-01 5.555556e-02 5 4 4.665064e-01 2.206191e-02 5 16 4.528986e-01 8.454132e-03 5 64 4.476040e-01 3.159603e-03 5 256 4.456052e-01 1.160769e-03 5 1024 4.448659e-01 4.214596e-04 5 4096 4.445962e-01 1.517950e-04 5 16384 4.444988e-01 5.436698e-05 5 65536 4.444638e-01 1.939682e-05 5 262144 4.444513e-01 6.901655e-06 5 1048576 4.444469e-01 2.451070e-06 5 4194304 4.444453e-01 8.693267e-07 5 Exact 4.444444e-01 6 1 1.000000e+00 8.630162e-01 6 4 1.000000e+00 8.630162e-01 6 16 1.750000e+00 1.130162e-01 6 64 1.828125e+00 3.489121e-02 6 256 1.854492e+00 8.524020e-03 6 1024 1.860596e+00 2.420504e-03 6 4096 1.862415e+00 6.008938e-04 6 16384 1.862854e+00 1.619652e-04 6 65536 1.862977e+00 3.892630e-05 6 262144 1.863006e+00 1.007300e-05 6 1048576 1.863014e+00 2.532604e-06 6 4194304 1.863016e+00 6.341970e-07 6 Exact 1.863016e+00 7 1 0.000000e+00 5.333333e-01 7 4 3.535534e-01 1.797799e-01 7 16 4.725299e-01 6.080346e-02 7 64 5.126355e-01 2.069782e-02 7 256 5.262356e-01 7.097742e-03 7 1024 5.308825e-01 2.450803e-03 7 4096 5.324821e-01 8.511903e-04 7 16384 5.330363e-01 2.970026e-04 7 65536 5.332293e-01 1.040015e-04 7 262144 5.332968e-01 3.651531e-05 7 1048576 5.333205e-01 1.284579e-05 7 4194304 5.333288e-01 4.525484e-06 7 Exact 5.333333e-01 8 1 2.777249e-01 2.389131e+00 8 4 5.521709e+00 2.854853e+00 8 16 2.825788e+00 1.589318e-01 8 64 2.706977e+00 4.012130e-02 8 256 2.676698e+00 9.842537e-03 8 1024 2.669304e+00 2.448564e-03 8 4096 2.667467e+00 6.113840e-04 8 16384 2.667009e+00 1.527986e-04 8 65536 2.666894e+00 3.819670e-05 8 262144 2.666865e+00 9.548990e-06 8 1048576 2.666858e+00 2.387236e-06 8 4194304 2.666856e+00 5.968083e-07 8 Exact 2.666856e+00 test_int_2d_test03(): Use a product of Gauss-Legendre rules. The 1D rules essentially double in order. Problem Points Approx Error 1 1 1.33333 0.311601 1 9 1.58123 0.0637073 1 49 1.63077 0.0141646 1 225 1.6416 0.00333146 1 961 1.64413 0.000807489 1 3969 1.64474 0.000198756 1 16129 1.64488 4.93032e-05 1 65025 1.64492 1.22778e-05 1 Exact 1.64493 2 1 4 0.355172 2 9 4.30864 0.0465302 2 49 4.3499 0.00526763 2 225 4.35456 0.000611658 2 961 4.3551 7.33017e-05 2 3969 4.35516 8.96068e-06 2 16129 4.35517 1.10734e-06 2 65025 4.35517 1.37619e-07 2 Exact 4.35517 3 1 2.82843 0.295767 3 9 3.10384 0.020355 3 49 3.12224 0.00195471 3 225 3.12398 0.000218752 3 961 3.12417 2.59885e-05 3 3969 3.12419 3.17026e-06 3 16129 3.12419 3.91572e-07 3 65025 3.12419 4.86577e-08 3 Exact 3.12419 4 1 2.3094 0.269606 4 9 2.55776 0.0212471 4 49 2.57694 0.00206741 4 225 2.57878 0.000231912 4 961 2.57898 2.75674e-05 4 3969 2.579 3.36332e-06 4 16129 2.57901 4.15432e-07 4 65025 2.57901 5.1623e-08 4 Exact 2.57901 5 1 0.5 0.0555556 5 9 0.447801 0.00335694 5 49 0.444773 0.000328619 5 225 0.444481 3.69473e-05 5 961 0.444449 4.39436e-06 5 3969 0.444445 5.36199e-07 5 16129 0.444445 6.62327e-08 5 65025 0.444444 8.23038e-09 5 Exact 0.444444 6 1 1 0.863016 6 9 2.06173 0.198712 6 49 1.8629 0.000113944 6 225 1.86838 0.00536441 6 961 1.86276 0.000251951 6 3969 1.86315 0.000137089 6 16129 1.863 1.22869e-05 6 65025 1.86302 5.37063e-06 6 Exact 1.86302 7 1 0 0.533333 7 9 0.443145 0.0901887 7 49 0.504725 0.0286088 7 225 0.523695 0.00963804 7 961 0.530005 0.00332792 7 3969 0.53217 0.00116284 7 16129 0.532925 0.00040872 7 65025 0.533189 0.000144081 7 Exact 0.533333 8 1 0.277725 2.38913 8 9 1.91043 0.756428 8 49 2.6705 0.00364122 8 225 2.66686 1.41248e-10 8 961 2.66686 0 8 3969 2.66686 4.44089e-16 8 16129 2.66686 4.44089e-15 8 65025 2.66686 6.21725e-15 8 Exact 2.66686 test_int_2d_test(): Normal end of execution. 06-Jun-2023 14:59:08