08-Oct-2025 20:03:55 test_int_2d_test(): MATLAB/Octave version 6.4.0 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 1.025564e+00 6.193701e-01 1 4 2.130111e+00 4.851769e-01 1 16 2.501130e+00 8.561954e-01 1 64 1.535281e+00 1.096529e-01 1 256 1.555334e+00 8.960013e-02 1 1024 1.625532e+00 1.940220e-02 1 4096 1.672829e+00 2.789539e-02 1 16384 1.638783e+00 6.151551e-03 1 65536 1.634500e+00 1.043362e-02 1 262144 1.646420e+00 1.486331e-03 1 1048576 1.642624e+00 2.310088e-03 1 4194304 1.646146e+00 1.212177e-03 1 Exact 1.644934e+00 2 1 4.100406e+00 2.547663e-01 2 4 4.244537e+00 1.106356e-01 2 16 4.261347e+00 9.382474e-02 2 64 4.269221e+00 8.595093e-02 2 256 4.342111e+00 1.306071e-02 2 1024 4.341361e+00 1.381129e-02 2 4096 4.361561e+00 6.388484e-03 2 16384 4.361364e+00 6.191750e-03 2 65536 4.349463e+00 5.709471e-03 2 262144 4.353643e+00 1.528845e-03 2 1048576 4.354342e+00 8.299910e-04 2 4194304 4.355801e+00 6.291595e-04 2 Exact 4.355172e+00 3 1 6.077057e+00 2.952862e+00 3 4 4.049169e+00 9.249748e-01 3 16 3.143994e+00 1.979922e-02 3 64 3.106199e+00 1.799521e-02 3 256 3.019799e+00 1.043952e-01 3 1024 3.083396e+00 4.079799e-02 3 4096 3.093333e+00 3.086124e-02 3 16384 3.122031e+00 2.163805e-03 3 65536 3.122134e+00 2.060692e-03 3 262144 3.123708e+00 4.862200e-04 3 1048576 3.121763e+00 2.431082e-03 3 4194304 3.123901e+00 2.933879e-04 3 Exact 3.124194e+00 4 1 3.287784e+00 7.087761e-01 4 4 2.470815e+00 1.081926e-01 4 16 2.552082e+00 2.692550e-02 4 64 2.687593e+00 1.085858e-01 4 256 2.599547e+00 2.053962e-02 4 1024 2.548528e+00 3.047945e-02 4 4096 2.549244e+00 2.976355e-02 4 16384 2.584813e+00 5.805506e-03 4 65536 2.583821e+00 4.813439e-03 4 262144 2.575801e+00 3.206635e-03 4 1048576 2.578140e+00 8.679854e-04 4 4194304 2.579421e+00 4.136345e-04 4 Exact 2.579008e+00 5 1 8.557337e-01 4.112892e-01 5 4 4.711541e-01 2.670966e-02 5 16 4.164137e-01 2.803076e-02 5 64 4.818401e-01 3.739565e-02 5 256 4.395834e-01 4.861080e-03 5 1024 4.479534e-01 3.508918e-03 5 4096 4.456002e-01 1.155742e-03 5 16384 4.449426e-01 4.982018e-04 5 65536 4.427626e-01 1.681861e-03 5 262144 4.442364e-01 2.080032e-04 5 1048576 4.443906e-01 5.382125e-05 5 4194304 4.444944e-01 4.997562e-05 5 Exact 4.444444e-01 6 1 7.383188e-03 1.855633e+00 6 4 9.594040e-01 9.036122e-01 6 16 1.263317e+00 5.996996e-01 6 64 1.758110e+00 1.049059e-01 6 256 1.913977e+00 5.096042e-02 6 1024 1.853470e+00 9.545714e-03 6 4096 1.844438e+00 1.857813e-02 6 16384 1.852402e+00 1.061471e-02 6 65536 1.863004e+00 1.205699e-05 6 262144 1.858667e+00 4.348916e-03 6 1048576 1.863229e+00 2.127757e-04 6 4194304 1.862322e+00 6.944772e-04 6 Exact 1.863016e+00 7 1 6.730651e-01 1.397318e-01 7 4 4.289298e-01 1.044036e-01 7 16 5.854190e-01 5.208563e-02 7 64 5.730771e-01 3.974376e-02 7 256 5.340047e-01 6.713254e-04 7 1024 5.326121e-01 7.212172e-04 7 4096 5.343421e-01 1.008756e-03 7 16384 5.312343e-01 2.099040e-03 7 65536 5.337415e-01 4.081623e-04 7 262144 5.330982e-01 2.351598e-04 7 1048576 5.332479e-01 8.546526e-05 7 4194304 5.332972e-01 3.616281e-05 7 Exact 5.333333e-01 8 1 7.236730e-07 2.666855e+00 8 4 2.814061e-01 2.385450e+00 8 16 1.286546e+00 1.380310e+00 8 64 1.539331e+00 1.127524e+00 8 256 3.220072e+00 5.532158e-01 8 1024 2.452476e+00 2.143794e-01 8 4096 2.769452e+00 1.025961e-01 8 16384 2.596378e+00 7.047792e-02 8 65536 2.690708e+00 2.385210e-02 8 262144 2.677897e+00 1.104133e-02 8 1048576 2.667077e+00 2.209939e-04 8 4194304 2.669642e+00 2.786042e-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 4.44089e-16 8 3969 2.66686 4.44089e-16 8 16129 2.66686 1.33227e-15 8 65025 2.66686 1.33227e-15 8 Exact 2.66686 test_int_2d_test(): Normal end of execution. 08-Oct-2025 20:09:42