08-Jan-2022 10:26:52 test_int_2d_test(): MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 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 3.816359e+00 2.171425e+00 1 4 4.128758e+00 2.483824e+00 1 16 1.691763e+00 4.682937e-02 1 64 1.661972e+00 1.703808e-02 1 256 1.643713e+00 1.220591e-03 1 1024 1.633152e+00 1.178185e-02 1 4096 1.652032e+00 7.098154e-03 1 16384 1.622305e+00 2.262873e-02 1 65536 1.654883e+00 9.948856e-03 1 262144 1.641527e+00 3.406836e-03 1 1048576 1.641563e+00 3.371541e-03 1 4194304 1.645158e+00 2.243846e-04 1 Exact 1.644934e+00 2 1 4.000023e+00 3.551490e-01 2 4 4.171320e+00 1.838520e-01 2 16 4.234589e+00 1.205827e-01 2 64 4.251388e+00 1.037843e-01 2 256 4.315520e+00 3.965204e-02 2 1024 4.354910e+00 2.619080e-04 2 4096 4.351677e+00 3.494699e-03 2 16384 4.355897e+00 7.244666e-04 2 65536 4.352097e+00 3.074919e-03 2 262144 4.356257e+00 1.084506e-03 2 1048576 4.355681e+00 5.090496e-04 2 4194304 4.354822e+00 3.504536e-04 2 Exact 4.355172e+00 3 1 2.546690e+00 5.775044e-01 3 4 2.879471e+00 2.447231e-01 3 16 3.637996e+00 5.138016e-01 3 64 3.485008e+00 3.608138e-01 3 256 3.205172e+00 8.097772e-02 3 1024 3.118961e+00 5.233014e-03 3 4096 3.138256e+00 1.406182e-02 3 16384 3.112091e+00 1.210367e-02 3 65536 3.125561e+00 1.367022e-03 3 262144 3.126234e+00 2.039701e-03 3 1048576 3.124968e+00 7.732849e-04 3 4194304 3.125227e+00 1.032765e-03 3 Exact 3.124194e+00 4 1 2.636629e+00 5.762190e-02 4 4 1.851330e+00 7.276772e-01 4 16 2.709839e+00 1.308310e-01 4 64 2.656132e+00 7.712431e-02 4 256 2.562187e+00 1.682061e-02 4 1024 2.563053e+00 1.595470e-02 4 4096 2.580591e+00 1.583115e-03 4 16384 2.576056e+00 2.951094e-03 4 65536 2.584135e+00 5.127775e-03 4 262144 2.580762e+00 1.754725e-03 4 1048576 2.580282e+00 1.274510e-03 4 4194304 2.579181e+00 1.729560e-04 4 Exact 2.579008e+00 5 1 5.846969e-01 1.402524e-01 5 4 2.679233e-01 1.765212e-01 5 16 6.059960e-01 1.615516e-01 5 64 3.836714e-01 6.077307e-02 5 256 4.506752e-01 6.230743e-03 5 1024 4.532015e-01 8.757035e-03 5 4096 4.392417e-01 5.202786e-03 5 16384 4.441201e-01 3.243602e-04 5 65536 4.447137e-01 2.692460e-04 5 262144 4.446188e-01 1.743219e-04 5 1048576 4.441027e-01 3.417743e-04 5 4194304 4.441915e-01 2.529137e-04 5 Exact 4.444444e-01 6 1 1.319378e+00 5.436378e-01 6 4 1.411266e+00 4.517500e-01 6 16 1.743472e+00 1.195446e-01 6 64 1.852292e+00 1.072466e-02 6 256 1.916863e+00 5.384667e-02 6 1024 1.854245e+00 8.770771e-03 6 4096 1.878927e+00 1.591089e-02 6 16384 1.865946e+00 2.929804e-03 6 65536 1.869136e+00 6.120190e-03 6 262144 1.858645e+00 4.370773e-03 6 1048576 1.862197e+00 8.187189e-04 6 4194304 1.863529e+00 5.131858e-04 6 Exact 1.863016e+00 7 1 4.100607e-01 1.232726e-01 7 4 5.411195e-01 7.786189e-03 7 16 5.462839e-01 1.295054e-02 7 64 5.369633e-01 3.629971e-03 7 256 5.505856e-01 1.725222e-02 7 1024 5.326911e-01 6.422656e-04 7 4096 5.409599e-01 7.626572e-03 7 16384 5.326605e-01 6.728531e-04 7 65536 5.339703e-01 6.370138e-04 7 262144 5.330631e-01 2.702286e-04 7 1048576 5.334546e-01 1.212654e-04 7 4194304 5.333039e-01 2.942802e-05 7 Exact 5.333333e-01 8 1 1.245957e+01 9.792715e+00 8 4 1.636390e+00 1.030466e+00 8 16 1.442345e+00 1.224511e+00 8 64 2.719139e+00 5.228372e-02 8 256 3.176229e+00 5.093729e-01 8 1024 2.927420e+00 2.605644e-01 8 4096 2.714099e+00 4.724285e-02 8 16384 2.673917e+00 7.060844e-03 8 65536 2.663379e+00 3.477048e-03 8 262144 2.666214e+00 6.418780e-04 8 1048576 2.667185e+00 3.288738e-04 8 4194304 2.669476e+00 2.620195e-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 8.88178e-16 8 3969 2.66686 0 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-Jan-2022 10:26:55