23-Mar-2019 08:27:12 square_exactness_test MATLAB version Test square_exactness. SQUARE_EXACTNESS_TEST01 Product Gauss-Legendre rules for the 2D Legendre integral. Density function rho(x) = 1. Region: -1 <= x <= +1. Region: -1 <= y <= +1. Level: L Exactness: 2*L+1 Order: N = (L+1)*(L+1) Quadrature rule for the 2D Legendre integral. Number of points in rule is 1 D I J Relative Error 0 0 0 1 1 1 0 0 0 1 0 2 2 0 1 1 1 0 0 2 1 Quadrature rule for the 2D Legendre integral. Number of points in rule is 4 D I J Relative Error 0 0 0 1 1 1 0 0 0 1 0 2 2 0 1 1 1 0 0 2 1 3 3 0 0 2 1 0 1 2 0 0 3 0 4 4 0 1 3 1 0 2 2 1 1 3 0 0 4 1 Quadrature rule for the 2D Legendre integral. Number of points in rule is 9 D I J Relative Error 0 0 0 1 1 1 0 0 0 1 0 2 2 0 1 1 1 0 0 2 1 3 3 0 0 2 1 0 1 2 0 0 3 0 4 4 0 1 3 1 0 2 2 1 1 3 0 0 4 1 5 5 0 0 4 1 0 3 2 0 2 3 0 1 4 0 0 5 0 6 6 0 1 5 1 0 4 2 1 3 3 0 2 4 1 1 5 0 0 6 1 Quadrature rule for the 2D Legendre integral. Number of points in rule is 16 D I J Relative Error 0 0 0 1 1 1 0 0 0 1 0 2 2 0 1 1 1 0 0 2 1 3 3 0 0 2 1 0 1 2 0 0 3 0 4 4 0 1 3 1 0 2 2 1 1 3 0 0 4 1 5 5 0 0 4 1 0 3 2 0 2 3 0 1 4 0 0 5 0 6 6 0 1 5 1 0 4 2 1 3 3 0 2 4 1 1 5 0 0 6 1 7 7 0 0 6 1 0 5 2 0 4 3 0 3 4 0 2 5 0 1 6 0 0 7 0 8 8 0 1 7 1 0 6 2 1 5 3 0 4 4 1 3 5 0 2 6 1 1 7 0 0 8 1 Quadrature rule for the 2D Legendre integral. Number of points in rule is 25 D I J Relative Error 0 0 0 1 1 1 0 1.110223024625157e-16 0 1 6.938893903907228e-18 2 2 0 1 1 1 6.938893903907228e-18 0 2 1 3 3 0 1.110223024625157e-16 2 1 0 1 2 1.301042606982605e-18 0 3 2.710505431213761e-19 4 4 0 1 3 1 0 2 2 1 1 3 2.981555974335137e-19 0 4 1 5 5 0 0 4 1 0 3 2 4.336808689942018e-19 2 3 2.168404344971009e-19 1 4 4.743384504624082e-20 0 5 1.747005453711994e-20 6 6 0 1 5 1 3.469446951953614e-18 4 2 1 3 3 9.486769009248164e-20 2 4 1 1 5 3.970466940254533e-21 0 6 1 7 7 0 0 6 1 0 5 2 4.336808689942018e-19 4 3 6.776263578034403e-20 3 4 2.371692252312041e-20 2 5 1.058791184067875e-21 1 6 2.779326858178173e-22 0 7 1.694065894508601e-21 8 8 0 1 7 1 0 6 2 1 5 3 4.065758146820642e-20 4 4 1 3 5 1.376428539288238e-21 2 6 1 1 7 1.168392615231152e-22 0 8 1 9 9 0 0 8 1 3.469446951953614e-18 7 2 2.168404344971009e-19 6 3 2.710505431213761e-20 5 4 5.082197683525802e-21 4 5 2.117582368135751e-22 3 6 4.599124205794834e-22 2 7 1.139441293791796e-22 1 8 1.596329335256632e-22 0 9 6.208701355549385e-24 10 10 0 0.9999999999999999 9 1 0 8 2 1 7 3 1.355252715606881e-20 6 4 1 5 5 8.470329472543003e-22 4 6 1 3 7 7.734138727370809e-23 2 8 1 1 9 5.2605132666678e-23 0 10 1 Quadrature rule for the 2D Legendre integral. Number of points in rule is 36 D I J Relative Error 0 0 0 1 1 1 0 1.110223024625157e-16 0 1 3.469446951953614e-18 2 2 0 0.9999999999999998 1 1 3.469446951953614e-18 0 2 1 3 3 0 2.220446049250313e-16 2 1 6.938893903907228e-18 1 2 5.637851296924623e-18 0 3 1.524659305057741e-19 4 4 0 1 3 1 3.469446951953614e-18 2 2 1 1 3 1.965116437629977e-19 0 4 1 5 5 0 2.220446049250313e-16 4 1 1.040834085586084e-17 3 2 4.336808689942018e-19 2 3 5.082197683525802e-20 1 4 2.011703249728963e-20 0 5 2.31941443759869e-21 6 6 0 1 5 1 3.469446951953614e-18 4 2 1 3 3 1.01643953670516e-19 2 4 1 1 5 1.204374971877208e-21 0 6 1 7 7 0 0 6 1 0 5 2 1.084202172485504e-19 4 3 1.355252715606881e-20 3 4 6.45862622281404e-21 2 5 1.720535674110298e-22 1 6 1.352647096677339e-21 0 7 1.985265781869944e-22 8 8 0 0.9999999999999994 7 1 3.469446951953614e-18 6 2 1 5 3 4.743384504624082e-20 4 4 1 3 5 6.617444900424221e-24 2 6 1 1 7 4.655895937679821e-22 0 8 1 9 9 0 5.551115123125783e-17 8 1 3.469446951953614e-18 7 2 2.168404344971009e-19 6 3 1.355252715606881e-20 5 4 5.082197683525802e-21 4 5 5.955700410381799e-22 3 6 2.452590516219727e-22 2 7 1.112008624258299e-22 1 8 3.258608956802679e-23 0 9 6.58908598197025e-23 10 10 0 0.9999999999999999 9 1 1.387778780781446e-17 8 2 1 7 3 3.388131789017201e-21 6 4 1 5 5 4.532949756790592e-22 4 6 1 3 7 7.284682386721878e-24 2 8 1 1 9 1.132451007984015e-23 0 10 1 11 11 0 1.110223024625157e-16 10 1 3.469446951953614e-18 9 2 5.421010862427522e-20 8 3 1.694065894508601e-21 7 4 2.329340604949326e-21 6 5 4.963083675318166e-23 5 6 8.949060252058068e-23 4 7 3.245714551990298e-23 3 8 1.494317124654816e-24 2 9 1.600412605519801e-24 1 10 5.577216524590542e-25 0 11 1.559745010859278e-25 12 12 0 1 11 1 1.734723475976807e-18 10 2 1 9 3 2.032879073410321e-20 8 4 1 7 5 2.69660879692287e-22 6 6 1 5 7 1.664054747909411e-25 4 8 1 3 9 3.02599312405909e-24 2 10 1 1 11 5.987022862319937e-27 0 12 1 SQUARE_EXACTNESS_TEST02 Padua rule for the 2D Legendre integral. Density function rho(x) = 1. Region: -1 <= x <= +1. Region: -1 <= y <= +1. Level: L Exactness: L+1 when L is 0, L otherwise. Order: N = (L+1)*(L+2)/2 Quadrature rule for the 2D Legendre integral. Number of points in rule is 1 D I J Relative Error 0 0 0 0 1 1 0 0 0 1 0 2 2 0 1 1 1 0 0 2 1 Quadrature rule for the 2D Legendre integral. Number of points in rule is 3 D I J Relative Error 0 0 0 0 1 1 0 0 0 1 0 2 2 0 2 1 1 0 0 2 0.5000000000000001 Quadrature rule for the 2D Legendre integral. Number of points in rule is 6 D I J Relative Error 0 0 0 1.110223024625157e-16 1 1 0 1.110223024625157e-16 0 1 4.440892098500626e-16 2 2 0 4.996003610813204e-16 1 1 5.551115123125783e-17 0 2 3.33066907387547e-16 3 3 0 1.110223024625157e-16 2 1 0.6666666666666665 1 2 2.775557561562891e-17 0 3 0.3333333333333338 Quadrature rule for the 2D Legendre integral. Number of points in rule is 10 D I J Relative Error 0 0 0 1.110223024625157e-16 1 1 0 8.326672684688674e-17 0 1 6.38378239159465e-16 2 2 0 0 1 1 6.800116025829084e-16 0 2 3.33066907387547e-16 3 3 0 2.775557561562891e-17 2 1 7.494005416219807e-16 1 2 4.163336342344337e-16 0 3 9.020562075079397e-16 4 4 0 0.1666666666666668 3 1 1.096345236817342e-15 2 2 0.2499999999999993 1 3 7.216449660063518e-16 0 4 0.04166666666666707 Quadrature rule for the 2D Legendre integral. Number of points in rule is 15 D I J Relative Error 0 0 0 0 1 1 0 8.604228440844963e-16 0 1 3.05311331771918e-16 2 2 0 1.665334536937735e-16 1 1 1.942890293094024e-16 0 2 1.665334536937735e-16 3 3 0 2.498001805406602e-16 2 1 1.249000902703301e-16 1 2 4.163336342344337e-16 0 3 2.081668171172169e-17 4 4 0 1.249000902703301e-15 3 1 3.608224830031759e-16 2 2 1.124100812432971e-15 1 3 1.595945597898663e-16 0 4 1.387778780781446e-16 5 5 0 2.775557561562891e-17 4 1 0.03333333333333315 3 2 3.05311331771918e-16 2 3 0.05555555555555579 1 4 2.012279232133096e-16 0 5 0.01666666666666707 Quadrature rule for the 2D Legendre integral. Number of points in rule is 21 D I J Relative Error 0 0 0 0 1 1 0 8.326672684688674e-17 0 1 7.28583859910259e-17 2 2 0 8.326672684688674e-16 1 1 1.630640067418199e-16 0 2 0 3 3 0 8.326672684688674e-16 2 1 5.93275428784068e-16 1 2 1.786765180256111e-16 0 3 2.445960101127298e-16 4 4 0 9.71445146547012e-16 3 1 4.822531263215524e-16 2 2 1.249000902703301e-16 1 3 2.168404344971009e-16 0 4 2.775557561562891e-16 5 5 0 9.436895709313831e-16 4 1 9.575673587391975e-16 3 2 2.34187669256869e-16 2 3 2.584737979205443e-16 1 4 8.326672684688674e-17 0 5 2.740863092043355e-16 6 6 0 0.008333333333334469 5 1 1.262878690511116e-15 4 2 0.02083333333333284 3 3 4.145989107584569e-16 2 4 0.02083333333333222 1 5 4.198030811863873e-16 0 6 0.006250000000000172 square_exactness_test Normal end of execution. 23-Mar-2019 08:27:12