26 February 2014 01:50:15 PM SPARSE_GRID_HW_PRB C++ version Test the SPARSE_GRID_HW library. CCL_TEST: CCL_ORDER + CC Clenshaw Curtis Linear (CCL) quadrature over [0,1]: Level Nodes Estimate Error 1 1 0.193334 0.00977527 2 3 0.191473 5.68167e-05 3 5 0.191462 3.95965e-08 4 7 0.191462 1.17826e-11 5 9 0.191462 6.52348e-15 CCL_SPARSE_TEST: CCL_ORDER + CC Sparse Clenshaw Curtis Linear quadrature over [0,1]. D Level Nodes SG error MC error 10 2 21 0.00390989 0.0258782 10 3 221 6.45374e-05 0.00789979 10 4 1581 1.23817e-07 0.0028176 10 5 8761 1.00775e-08 0.00122981 10 6 40425 8.84205e-11 0.000568074 10 7 162385 2.90642e-12 0.000293862 CCS_TEST: CCS_ORDER + CC Clenshaw Curtis Slow quadrature over [0,1]: Level Nodes Estimate Error 1 1 0.193334 0.00977527 2 3 0.191473 5.68167e-05 3 5 0.191462 3.95965e-08 4 9 0.191462 6.52348e-15 5 9 0.191462 6.52348e-15 CCS_SPARSE_TEST: CCS_ORDER + CC Sparse Clenshaw Curtis Slow quadrature over [0,1]. D Level Nodes SG error MC error 10 2 21 0.00390989 0.0258782 10 3 221 6.45374e-05 0.00789979 10 4 1581 1.23688e-07 0.0028176 10 5 8721 1.00886e-08 0.00125521 10 6 39665 8.86452e-11 0.000575534 10 7 155105 1.10943e-12 0.000297318 CCE_TEST: CCE_ORDER + CC Clenshaw Curtis Exponential quadrature over [0,1]: Level Nodes Estimate Error 1 1 0.193334 0.00977527 2 3 0.191473 5.68167e-05 3 5 0.191462 3.95965e-08 4 9 0.191462 6.52348e-15 5 17 0.191462 2.89932e-16 CCE_SPARSE_TEST: CCE_ORDER + CC Sparse Clenshaw Curtis Exponential quadrature over [0,1]. D Level Nodes SG error MC error 10 2 21 0.00390989 0.0258782 10 3 221 6.45374e-05 0.00789979 10 4 1581 1.23688e-07 0.0028176 10 5 8801 1.00888e-08 0.00122157 10 6 41265 8.87196e-11 0.000563751 10 7 171425 2.86464e-12 0.000279777 GET_SEQ_TEST GET_SEQ returns all D-dimensional vectors that sum to NORM. D = 3 NORM = 6 The compositions Col: 0 1 2 Row 0: 4 1 1 1: 3 2 1 2: 3 1 2 3: 2 3 1 4: 2 2 2 5: 2 1 3 6: 1 4 1 7: 1 3 2 8: 1 2 3 9: 1 1 4 GQN_TEST: Gauss-Hermite quadrature over (-oo,+oo): Level Nodes Estimate Error 1 1 0 1 2 2 1 0.933333 3 3 9 0.4 4 4 15 4.73695e-16 5 5 15 2.36848e-16 GQN_SPARSE_TEST: GQN sparse grid: Sparse Gaussian quadrature with Hermite weight over (-oo,+oo). D Level Nodes SG error MC error 10 2 21 0.933333 1.23671 10 3 221 0.4 0.498557 10 4 1581 5.82645e-14 0.167225 10 5 8761 1.46135e-13 0.0754892 10 6 40405 2.35971e-12 0.0344189 10 7 162025 1.61097e-11 0.0169912 GQN2_SPARSE_TEST: GQN sparse grid: Gauss-Hermite sparse grids over (-oo,+oo). Use GQN2_ORDER, the growth rule N = 2 * L - 1. J W X Y 0 0.166667 -1.73205 0 1 0.166667 0 -1.73205 2 0.333333 0 0 3 0.166667 0 1.73205 4 0.166667 1.73205 0 J W X Y 0 0.0112574 -2.85697 0 1 0.0277778 -1.73205 -1.73205 2 -0.0555556 -1.73205 0 3 0.0277778 -1.73205 1.73205 4 0.222076 -1.35563 0 5 0.0112574 0 -2.85697 6 -0.0555556 0 -1.73205 7 0.222076 0 -1.35563 8 0.177778 0 0 9 0.222076 0 1.35563 10 -0.0555556 0 1.73205 11 0.0112574 0 2.85697 12 0.222076 1.35563 0 13 0.0277778 1.73205 -1.73205 14 -0.0555556 1.73205 0 15 0.0277778 1.73205 1.73205 16 0.0112574 2.85697 0 J W X Y 0 0.000548269 -3.75044 0 1 0.00187624 -2.85697 -1.73205 2 -0.00375247 -2.85697 0 3 0.00187624 -2.85697 1.73205 4 0.0307571 -2.36676 0 5 0.00187624 -1.73205 -2.85697 6 -0.0277778 -1.73205 -1.73205 7 0.0370127 -1.73205 -1.35563 8 -0.0222222 -1.73205 0 9 0.0370127 -1.73205 1.35563 10 -0.0277778 -1.73205 1.73205 11 0.00187624 -1.73205 2.85697 12 0.0370127 -1.35563 -1.73205 13 -0.0740253 -1.35563 0 14 0.0370127 -1.35563 1.73205 15 0.240123 -1.15441 0 16 0.000548269 0 -3.75044 17 -0.00375247 0 -2.85697 18 0.0307571 0 -2.36676 19 -0.0222222 0 -1.73205 20 -0.0740253 0 -1.35563 21 0.240123 0 -1.15441 22 0.114286 0 0 23 0.240123 0 1.15441 24 -0.0740253 0 1.35563 25 -0.0222222 0 1.73205 26 0.0307571 0 2.36676 27 -0.00375247 0 2.85697 28 0.000548269 0 3.75044 29 0.240123 1.15441 0 30 0.0370127 1.35563 -1.73205 31 -0.0740253 1.35563 0 32 0.0370127 1.35563 1.73205 33 0.00187624 1.73205 -2.85697 34 -0.0277778 1.73205 -1.73205 35 0.0370127 1.73205 -1.35563 36 -0.0222222 1.73205 0 37 0.0370127 1.73205 1.35563 38 -0.0277778 1.73205 1.73205 39 0.00187624 1.73205 2.85697 40 0.0307571 2.36676 0 41 0.00187624 2.85697 -1.73205 42 -0.00375247 2.85697 0 43 0.00187624 2.85697 1.73205 44 0.000548269 3.75044 0 GQU_TEST: Gauss-Legendre quadrature over [0,1]: Level Nodes Estimate Error 1 1 0.193334 0.00977527 2 2 0.191455 3.7965e-05 3 3 0.191462 9.46584e-08 4 4 0.191462 1.74249e-10 5 5 0.191462 2.54416e-13 GQU_SPARSE_TEST: GQU sparse grid: Sparse Gaussian unweighted quadrature over [0,1]. D Level Nodes SG error MC error 10 2 21 0.00494443 0.0258782 10 3 221 0.000155187 0.00789979 10 4 1581 3.57517e-06 0.0028176 10 5 8761 6.46636e-08 0.00122981 10 6 40405 9.57024e-10 0.000581704 10 7 162025 1.75903e-11 0.000287111 KPN_TEST: Kronrod-Patterson-Hermite quadrature over (-oo,+oo): Level Nodes Estimate Error 1 1 0 1 2 3 9 0.4 3 3 9 0.4 4 7 15 4.73695e-16 5 9 15 2.36848e-16 KPN_SPARSE_TEST: KPN sparse grid: Sparse Kronrod-Patterson quadrature with Hermite weight over (-oo,+oo). D Level Nodes SG error MC error 10 2 21 0.4 1.23671 10 3 201 0.4 0.537103 10 4 1201 2.96059e-15 0.200175 10 5 5301 7.07582e-13 0.0966577 10 6 19485 7.22906e-12 0.0506963 10 7 63405 8.71227e-11 0.0269629 KPU_TEST: Kronrod-Patterson quadrature over [0,1]: Level Nodes Estimate Error 1 1 0.193334 0.00977527 2 3 0.191462 9.46584e-08 3 3 0.191462 9.46584e-08 4 7 0.191462 4.34898e-16 5 7 0.191462 4.34898e-16 KPU_SPARSE_TEST: KPU sparse grid: Sparse Kronrod-Patterson unweighted quadrature over [0,1]. D Level Nodes SG error MC error 10 2 21 0.00452901 0.0258782 10 3 201 0.000118923 0.00840541 10 4 1201 2.09585e-06 0.00327686 10 5 5281 2.68308e-08 0.0016055 10 6 19105 2.67746e-10 0.000821658 10 7 60225 6.42504e-12 0.000475706 NWSPGR_SIZE_TEST: NWSPGR_SIZE returns the size of a sparse grid, based on either: one of the built-in 1D rules, or a family of 1D rules supplied by the user. Kronrod-Patterson, [0,1], Dim 2, Level 3, Symmetric Full 21 Kronrod-Patterson, (-oo,+oo), Dim 2, Level 3, Symmetric Full 21 Gauss-Legendre, [0,1], Dim 2, Level 3, Symmetric Full 14 Gauss Hermite, (-oo,+oo), [0,1], Dim 2, Level 3, Symmetric Full 14 Clenshaw Curtis Exponential, [-1,+1], [0,1], Dim 2, Level 3, Unsymmetric Full 25 Dimension / Level table for Clenshaw Curtis Exponential Dim: 1 2 3 4 5 6 7 8 9 10 Level 1 1 1 1 1 1 1 1 1 1 1 2 3 7 10 13 16 19 22 25 28 31 3 5 25 52 87 131 184 246 317 397 486 4 9 67 195 411 746 1228 1884 2741 3826 5166 5 17 161 609 1573 3376 6430 11222 18319 28369 42101 This function measures the time in seconds required by NWSPGR to compute a sparse grid, based on either: one of the built-in 1D rules, or a family of 1D rules supplied by the user. Kronrod-Patterson, [0,1], Dim 20, Level 5, Symmetric Full 5.69 Kronrod-Patterson, (-oo,+oo), Dim 20, Level 5, Symmetric Full 5.74 Gauss-Legendre, [0,1], Dim 20, Level 5, Symmetric Full 1.25 Gauss Hermite, (-oo,+oo), [0,1], Dim 20, Level 5, Symmetric Full 1.25 Clenshaw Curtis Exponential, [-1,+1], [0,1], Dim 20, Level 5, Unsymmetric Full 6.5 Dimension / Level table for Clenshaw Curtis Exponential Dim: 1 2 3 4 5 6 7 8 9 10 Level 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0.01 0 0.01 0.01 0.02 5 0 0 0 0 0.01 0.02 0.03 0.06 0.1 0.17 Dim: 11 12 13 14 15 16 17 18 19 20 Level 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0.01 0 0 0 0 0 3 0 0 0 0.01 0 0.01 0.01 0.01 0.01 0.01 4 0.03 0.03 0.06 0.06 0.09 0.12 0.15 0.21 0.26 0.33 5 0.28 0.44 0.66 0.94 1.33 1.97 2.66 3.74 5.05 6.57 NWSPGR_TEST: NWSPGR generates a sparse grid, based on either: one of the built-in 1D rules, or a family of 1D rules supplied by the user. Kronrod-Patterson, [0,1], Dim 2, Level 3 0 0.0771605 * f (0.112702,0.112702 ) 1 0.123457 * f (0.112702,0.5 ) 2 0.0771605 * f (0.112702,0.887298 ) 3 0.123457 * f (0.5,0.112702 ) 4 0.197531 * f (0.5,0.5 ) 5 0.123457 * f (0.5,0.887298 ) 6 0.0771605 * f (0.887298,0.112702 ) 7 0.123457 * f (0.887298,0.5 ) 8 0.0771605 * f (0.887298,0.887298 ) Kronrod-Patterson, (-oo,+oo), Dim 2, Level 3 0 0.0277778 * f (-1.73205,-1.73205 ) 1 0.111111 * f (-1.73205,0 ) 2 0.0277778 * f (-1.73205,1.73205 ) 3 0.111111 * f (0,-1.73205 ) 4 0.444444 * f (0,0 ) 5 0.111111 * f (0,1.73205 ) 6 0.0277778 * f (1.73205,-1.73205 ) 7 0.111111 * f (1.73205,0 ) 8 0.0277778 * f (1.73205,1.73205 ) Gauss-Legendre, [0,1], Dim 2, Level 3 0 0.277778 * f (0.112702,0.5 ) 1 0.25 * f (0.211325,0.211325 ) 2 -0.5 * f (0.211325,0.5 ) 3 0.25 * f (0.211325,0.788675 ) 4 0.277778 * f (0.5,0.112702 ) 5 -0.5 * f (0.5,0.211325 ) 6 0.888889 * f (0.5,0.5 ) 7 -0.5 * f (0.5,0.788675 ) 8 0.277778 * f (0.5,0.887298 ) 9 0.25 * f (0.788675,0.211325 ) 10 -0.5 * f (0.788675,0.5 ) 11 0.25 * f (0.788675,0.788675 ) 12 0.277778 * f (0.887298,0.5 ) Gauss Hermite, (-oo,+oo), Dim 2, Level 3 0 0.166667 * f (-1.73205,0 ) 1 0.25 * f (-1,-1 ) 2 -0.5 * f (-1,0 ) 3 0.25 * f (-1,1 ) 4 0.166667 * f (0,-1.73205 ) 5 -0.5 * f (0,-1 ) 6 1.33333 * f (0,0 ) 7 -0.5 * f (0,1 ) 8 0.166667 * f (0,1.73205 ) 9 0.25 * f (1,-1 ) 10 -0.5 * f (1,0 ) 11 0.25 * f (1,1 ) 12 0.166667 * f (1.73205,0 ) Clenshaw Curtis Exponential, [-1,+1], Dim 2, Level 3 0 0.0277778 * f (0,0 ) 1 -0.0222222 * f (0,0.5 ) 2 0.0277778 * f (0,1 ) 3 0.266667 * f (0.146447,0.5 ) 4 -0.0222222 * f (0.5,0 ) 5 0.266667 * f (0.5,0.146447 ) 6 -0.0888889 * f (0.5,0.5 ) 7 0.266667 * f (0.5,0.853553 ) 8 -0.0222222 * f (0.5,1 ) 9 0.266667 * f (0.853553,0.5 ) 10 0.0277778 * f (1,0 ) 11 -0.0222222 * f (1,0.5 ) 12 0.0277778 * f (1,1 ) ORDER_REPORT For each family of rules, report: L, the level index, RP, the required polynomial precision, AP, the actual polynomial precision, O, the rule order (number of points). GQN family Gauss quadrature, exponential weight, (-oo,+oo) L RP AP O 1 1 1 1 2 3 3 2 3 5 5 3 4 7 7 4 5 9 9 5 6 11 11 6 7 13 13 7 8 15 15 8 9 17 17 9 10 19 19 10 11 21 21 11 12 23 23 12 13 25 25 13 14 27 27 14 15 29 29 15 16 31 31 16 17 33 33 17 18 35 35 18 19 37 37 19 20 39 39 20 21 41 41 21 22 43 43 22 23 45 45 23 24 47 47 24 25 49 49 25 GQU family Gauss quadrature, unit weight, [0,1] L RP AP O 1 1 1 1 2 3 3 2 3 5 5 3 4 7 7 4 5 9 9 5 6 11 11 6 7 13 13 7 8 15 15 8 9 17 17 9 10 19 19 10 11 21 21 11 12 23 23 12 13 25 25 13 14 27 27 14 15 29 29 15 16 31 31 16 17 33 33 17 18 35 35 18 19 37 37 19 20 39 39 20 21 41 41 21 22 43 43 22 23 45 45 23 24 47 47 24 25 49 49 25 KPN family Gauss-Kronrod-Patterson quadrature, exponential weight, (-oo,+oo) L RP AP O 1 1 1 1 2 3 5 3 3 5 5 3 4 7 7 7 5 9 15 9 6 11 15 9 7 13 15 9 8 15 15 9 9 17 17 17 10 19 29 19 11 21 29 19 12 23 29 19 13 25 29 19 14 27 29 19 15 29 29 19 16 31 31 31 17 33 33 33 18 35 51 35 19 37 51 35 20 39 51 35 21 41 51 35 22 43 51 35 23 45 51 35 24 47 51 35 25 49 51 35 KPU family Gauss-Kronrod-Patterson quadrature, unit weight, [0,1] L RP AP O 1 1 1 1 2 3 5 3 3 5 5 3 4 7 11 7 5 9 11 7 6 11 11 7 7 13 23 15 8 15 23 15 9 17 23 15 10 19 23 15 11 21 23 15 12 23 23 15 13 25 47 31 14 27 47 31 15 29 47 31 16 31 47 31 17 33 47 31 18 35 47 31 19 37 47 31 20 39 47 31 21 41 47 31 22 43 47 31 23 45 47 31 24 47 47 31 25 49 95 63 SYMMETRIC_SPARSE_SIZE_TEST Given a symmetric sparse grid rule represented only by the points with positive values, determine the total number of points in the grid. For dimension DIM, we report R, the number of points in the positive orthant, and R2, the total number of points. DIM R R2 5 6 11 5 21 61 3 23 69 TENSOR_PRODUCT_TEST: Given a sequence of 1D quadrature rules, construct the tensor product rule. A 1D rule over [-1,+1]: 0 1 * f (-1 ) 1 1 * f (1 ) A 2D rule over [-1,+1] x [2.0,3.0]: 0 0.25 * f (-1,2 ) 1 0.25 * f (1,2 ) 2 0.5 * f (-1,2.5 ) 3 0.5 * f (1,2.5 ) 4 0.25 * f (-1,3 ) 5 0.25 * f (1,3 ) A 3D rule over [-1,+1] x [2.0,3.0] x [10.0,15.0]: 0 0.625 * f (-1,2,10 ) 1 0.625 * f (1,2,10 ) 2 1.25 * f (-1,2.5,10 ) 3 1.25 * f (1,2.5,10 ) 4 0.625 * f (-1,3,10 ) 5 0.625 * f (1,3,10 ) 6 0.625 * f (-1,2,15 ) 7 0.625 * f (1,2,15 ) 8 1.25 * f (-1,2.5,15 ) 9 1.25 * f (1,2.5,15 ) 10 0.625 * f (-1,3,15 ) 11 0.625 * f (1,3,15 ) TENSOR_PRODUCT_TEST_CELL: Given a set of 1D quadrature rules stored in a cell array, construct the tensor product rule. A 1D rule over [-1,+1]: 0 1 * f (-1 ) 1 1 * f (1 ) A 1D rule over [-1,+1]: 0 0.25 * f (-1,2 ) 1 0.25 * f (1,2 ) 2 0.5 * f (-1,2.5 ) 3 0.5 * f (1,2.5 ) 4 0.25 * f (-1,3 ) 5 0.25 * f (1,3 ) A 1D rule over [-1,+1]: 0 0.625 * f (-1,2,10 ) 1 0.625 * f (1,2,10 ) 2 1.25 * f (-1,2.5,10 ) 3 1.25 * f (1,2.5,10 ) 4 0.625 * f (-1,3,10 ) 5 0.625 * f (1,3,10 ) 6 0.625 * f (-1,2,15 ) 7 0.625 * f (1,2,15 ) 8 1.25 * f (-1,2.5,15 ) 9 1.25 * f (1,2.5,15 ) 10 0.625 * f (-1,3,15 ) 11 0.625 * f (1,3,15 ) SPARSE_GRID_HW_PRB Normal end of execution. 26 February 2014 02:16:30 PM