08-Jan-2022 09:34:45 simplex_gm_rule_test(): MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 Test simplex_gm_rule(). SIMPLEX_GM_RULE_TEST01(): SIMPLEX_UNIT_TO_GENERAL maps points in the unit simplex to a general simplex. Here we consider a simplex in 2D, a triangle. The vertices of the general triangle are: 1.000000 1.000000 3.000000 1.000000 2.000000 5.000000 ( XSI ETA ) ( X Y ) 0.000000 0.000000 1.000000 1.000000 1.000000 0.000000 3.000000 1.000000 0.000000 1.000000 2.000000 5.000000 0.086549 0.041793 1.214890 1.167172 0.031500 0.159331 1.222332 1.637322 0.663975 0.313471 2.641421 2.253886 0.018682 0.965717 2.003081 4.862867 0.044246 0.730574 1.819067 3.922297 0.672411 0.297278 2.642099 2.189111 0.334541 0.059389 1.728472 1.237555 0.934954 0.045886 2.915795 1.183545 0.402907 0.288427 2.094241 2.153708 0.299655 0.135238 1.734549 1.540953 SIMPLEX_GM_RULE_TEST02 SIMPLEX_UNIT_TO_GENERAL maps points in the unit simplex to a general simplex. Here we consider a simplex in 3D, a tetrahedron. The vertices of the general tetrahedron, are: 1.000000 1.000000 1.000000 3.000000 1.000000 1.000000 1.000000 4.000000 1.000000 1.000000 1.000000 5.000000 ( XSI ETA MU ) ( X Y Z ) 0.000000 0.000000 0.000000 1.000000 1.000000 1.000000 1.000000 0.000000 0.000000 3.000000 1.000000 1.000000 0.000000 1.000000 0.000000 1.000000 4.000000 1.000000 0.000000 0.000000 1.000000 1.000000 1.000000 5.000000 0.042684 0.422730 0.157456 1.085367 2.268191 1.629825 0.577356 0.048097 0.090153 2.154712 1.144291 1.360614 0.009806 0.646922 0.158225 1.019613 2.940766 1.632900 0.092537 0.079363 0.580888 1.185075 1.238088 3.323551 0.432356 0.233448 0.183661 1.864712 1.700344 1.734642 0.328994 0.098678 0.108101 1.657988 1.296034 1.432405 0.539741 0.176585 0.010418 2.079481 1.529756 1.041674 0.145379 0.406256 0.077616 1.290758 2.218767 1.310463 0.569472 0.299232 0.096558 2.138943 1.897696 1.386231 0.103284 0.338502 0.325800 1.206567 2.015507 2.303200 SIMPLEX_GM_RULE_TEST03 GM_RULE_SIZE returns N, the number of points associated with a Grundmann-Moeller quadrature rule for the unit simplex of dimension M with rule index RULE and degree of exactness DEGREE = 2*RULE+1. M RULE DEGREE N 2 0 1 1 2 1 3 4 2 2 5 10 2 3 7 20 2 4 9 35 2 5 11 56 3 0 1 1 3 1 3 5 3 2 5 15 3 3 7 35 3 4 9 70 3 5 11 126 5 0 1 1 5 1 3 7 5 2 5 28 5 3 7 84 5 4 9 210 5 5 11 462 10 0 1 1 10 1 3 12 10 2 5 78 10 3 7 364 10 4 9 1365 10 5 11 4368 SIMPLEX_GM_RULE_TEST04 GM_UNIT_RULE_SET determines the weights and abscissas of a Grundmann-Moeller quadrature rule for the M dimensional unit simplex, using a rule of index RULE, which will have degree of exactness 2*RULE+1. Here we use M = 3 RULE = 2 DEGREE = 5 POINT W X Y Z 1 0.050794 0.125000 0.125000 0.125000 2 0.050794 0.375000 0.125000 0.125000 3 0.050794 0.625000 0.125000 0.125000 4 0.050794 0.125000 0.375000 0.125000 5 0.050794 0.375000 0.375000 0.125000 6 0.050794 0.125000 0.625000 0.125000 7 0.050794 0.125000 0.125000 0.375000 8 0.050794 0.375000 0.125000 0.375000 9 0.050794 0.125000 0.375000 0.375000 10 0.050794 0.125000 0.125000 0.625000 11 -0.096429 0.166667 0.166667 0.166667 12 -0.096429 0.500000 0.166667 0.166667 13 -0.096429 0.166667 0.500000 0.166667 14 -0.096429 0.166667 0.166667 0.500000 15 0.044444 0.250000 0.250000 0.250000 SIMPLEX_GM_RULE_TEST05 GM_UNIT_RULE_SET determines the weights and abscissas of a Grundmann-Moeller quadrature rule for the M dimensional unit simplex, using a rule of index RULE, which will have degree of exactness 2*RULE+1. In this test, we compute various rules, and simply report the number of points, and the sum of weights. M RULE N WEIGHT SUM 2 0 1 0.5000000000000000 2 1 4 0.4999999999999999 2 2 10 0.4999999999999999 2 3 20 0.5000000000000006 2 4 35 0.4999999999999999 2 5 56 0.5000000000000028 3 0 1 0.1666666666666667 3 1 5 0.1666666666666667 3 2 15 0.1666666666666667 3 3 35 0.1666666666666670 3 4 70 0.1666666666666664 3 5 126 0.1666666666666647 5 0 1 0.0083333333333333 5 1 7 0.0083333333333333 5 2 28 0.0083333333333333 5 3 84 0.0083333333333333 5 4 210 0.0083333333333336 5 5 462 0.0083333333333329 10 0 1 0.0000002755731922 10 1 12 0.0000002755731922 10 2 78 0.0000002755731922 10 3 364 0.0000002755731922 10 4 1365 0.0000002755731922 10 5 4368 0.0000002755731922 SIMPLEX_GM_RULE_TEST06 GM_UNIT_RULE_SET determines the weights and abscissas of a Grundmann-Moeller quadrature rule for the M dimensional unit simplex, using a rule of index RULE, which will have degree of exactness 2*RULE+1. In this test, we write a rule to a file. Here we use M = 3 RULE = 2 DEGREE = 5 Wrote rule 2 to "gm2_3d_w.txt" and "gm2_3d_x.txt". SIMPLEX_GM_RULE_TEST07 GM_UNIT_RULE_SET determines the weights and abscissas of a Grundmann-Moeller quadrature rule for the M dimensional unit simplex, using a rule of index RULE, which will have degree of exactness 2*RULE+1. In this test, look at all the monomials up to some maximum degree, choose a few low order rules and determine the quadrature error for each. Here we use M = 5 Rule Order Quad_Error F(X) = X1^0 * X2^0 * X3^0 * X4^0 * X5^0 0 1 0.000000e+00 1 7 2.220446e-16 2 28 2.220446e-16 3 84 2.220446e-16 F(X) = X1^1 * X2^0 * X3^0 * X4^0 * X5^0 0 1 1.110223e-16 1 7 3.330669e-16 2 28 1.110223e-15 3 84 2.220446e-16 F(X) = X1^0 * X2^1 * X3^0 * X4^0 * X5^0 0 1 1.110223e-16 1 7 3.330669e-16 2 28 4.440892e-16 3 84 3.330669e-16 F(X) = X1^0 * X2^0 * X3^1 * X4^0 * X5^0 0 1 1.110223e-16 1 7 3.330669e-16 2 28 2.220446e-16 3 84 1.110223e-16 F(X) = X1^0 * X2^0 * X3^0 * X4^1 * X5^0 0 1 1.110223e-16 1 7 3.330669e-16 2 28 2.220446e-16 3 84 1.110223e-16 F(X) = X1^0 * X2^0 * X3^0 * X4^0 * X5^1 0 1 1.110223e-16 1 7 1.110223e-16 2 28 2.220446e-16 3 84 2.220446e-16 F(X) = X1^2 * X2^0 * X3^0 * X4^0 * X5^0 0 1 4.166667e-01 1 7 0.000000e+00 2 28 4.440892e-16 3 84 2.220446e-16 F(X) = X1^1 * X2^1 * X3^0 * X4^0 * X5^0 0 1 1.666667e-01 1 7 2.220446e-16 2 28 8.881784e-16 3 84 2.220446e-16 F(X) = X1^0 * X2^2 * X3^0 * X4^0 * X5^0 0 1 4.166667e-01 1 7 0.000000e+00 2 28 4.440892e-16 3 84 2.220446e-16 F(X) = X1^1 * X2^0 * X3^1 * X4^0 * X5^0 0 1 1.666667e-01 1 7 2.220446e-16 2 28 8.881784e-16 3 84 6.661338e-16 F(X) = X1^0 * X2^1 * X3^1 * X4^0 * X5^0 0 1 1.666667e-01 1 7 2.220446e-16 2 28 8.881784e-16 3 84 4.440892e-16 F(X) = X1^0 * X2^0 * X3^2 * X4^0 * X5^0 0 1 4.166667e-01 1 7 0.000000e+00 2 28 8.881784e-16 3 84 4.440892e-16 F(X) = X1^1 * X2^0 * X3^0 * X4^1 * X5^0 0 1 1.666667e-01 1 7 2.220446e-16 2 28 8.881784e-16 3 84 0.000000e+00 F(X) = X1^0 * X2^1 * X3^0 * X4^1 * X5^0 0 1 1.666667e-01 1 7 2.220446e-16 2 28 8.881784e-16 3 84 5.551115e-16 F(X) = X1^0 * X2^0 * X3^1 * X4^1 * X5^0 0 1 1.666667e-01 1 7 2.220446e-16 2 28 8.881784e-16 3 84 6.661338e-16 F(X) = X1^0 * X2^0 * X3^0 * X4^2 * X5^0 0 1 4.166667e-01 1 7 0.000000e+00 2 28 8.881784e-16 3 84 8.881784e-16 F(X) = X1^1 * X2^0 * X3^0 * X4^0 * X5^1 0 1 1.666667e-01 1 7 1.110223e-16 2 28 8.881784e-16 3 84 2.220446e-16 F(X) = X1^0 * X2^1 * X3^0 * X4^0 * X5^1 0 1 1.666667e-01 1 7 1.110223e-16 2 28 8.881784e-16 3 84 7.771561e-16 F(X) = X1^0 * X2^0 * X3^1 * X4^0 * X5^1 0 1 1.666667e-01 1 7 1.110223e-16 2 28 6.661338e-16 3 84 1.110223e-16 F(X) = X1^0 * X2^0 * X3^0 * X4^1 * X5^1 0 1 1.666667e-01 1 7 1.110223e-16 2 28 6.661338e-16 3 84 1.110223e-16 F(X) = X1^0 * X2^0 * X3^0 * X4^0 * X5^2 0 1 4.166667e-01 1 7 1.110223e-16 2 28 8.881784e-16 3 84 2.220446e-16 F(X) = X1^3 * X2^0 * X3^0 * X4^0 * X5^0 0 1 7.407407e-01 1 7 2.220446e-16 2 28 2.220446e-16 3 84 4.440892e-16 F(X) = X1^2 * X2^1 * X3^0 * X4^0 * X5^0 0 1 2.222222e-01 1 7 1.110223e-16 2 28 2.220446e-16 3 84 2.220446e-16 F(X) = X1^1 * X2^2 * X3^0 * X4^0 * X5^0 0 1 2.222222e-01 1 7 1.110223e-16 2 28 6.661338e-16 3 84 8.881784e-16 F(X) = X1^0 * X2^3 * X3^0 * X4^0 * X5^0 0 1 7.407407e-01 1 7 2.220446e-16 2 28 2.220446e-16 3 84 1.554312e-15 F(X) = X1^2 * X2^0 * X3^1 * X4^0 * X5^0 0 1 2.222222e-01 1 7 1.110223e-16 2 28 2.220446e-16 3 84 2.220446e-16 F(X) = X1^1 * X2^1 * X3^1 * X4^0 * X5^0 0 1 5.555556e-01 1 7 5.551115e-16 2 28 1.332268e-15 3 84 2.220446e-16 F(X) = X1^0 * X2^2 * X3^1 * X4^0 * X5^0 0 1 2.222222e-01 1 7 1.110223e-16 2 28 2.220446e-16 3 84 7.771561e-16 F(X) = X1^1 * X2^0 * X3^2 * X4^0 * X5^0 0 1 2.222222e-01 1 7 1.110223e-16 2 28 6.661338e-16 3 84 5.551115e-16 F(X) = X1^0 * X2^1 * X3^2 * X4^0 * X5^0 0 1 2.222222e-01 1 7 1.110223e-16 2 28 6.661338e-16 3 84 0.000000e+00 F(X) = X1^0 * X2^0 * X3^3 * X4^0 * X5^0 0 1 7.407407e-01 1 7 0.000000e+00 2 28 2.220446e-16 3 84 6.661338e-16 F(X) = X1^2 * X2^0 * X3^0 * X4^1 * X5^0 0 1 2.222222e-01 1 7 1.110223e-16 2 28 2.220446e-16 3 84 8.881784e-16 F(X) = X1^1 * X2^1 * X3^0 * X4^1 * X5^0 0 1 5.555556e-01 1 7 5.551115e-16 2 28 1.332268e-15 3 84 7.771561e-16 F(X) = X1^0 * X2^2 * X3^0 * X4^1 * X5^0 0 1 2.222222e-01 1 7 1.110223e-16 2 28 2.220446e-16 3 84 0.000000e+00 F(X) = X1^1 * X2^0 * X3^1 * X4^1 * X5^0 0 1 5.555556e-01 1 7 0.000000e+00 2 28 1.110223e-15 3 84 7.771561e-16 F(X) = X1^0 * X2^1 * X3^1 * X4^1 * X5^0 0 1 5.555556e-01 1 7 0.000000e+00 2 28 1.110223e-15 3 84 9.992007e-16 F(X) = X1^0 * X2^0 * X3^2 * X4^1 * X5^0 0 1 2.222222e-01 1 7 1.110223e-16 2 28 4.440892e-16 3 84 0.000000e+00 F(X) = X1^1 * X2^0 * X3^0 * X4^2 * X5^0 0 1 2.222222e-01 1 7 1.110223e-16 2 28 2.220446e-16 3 84 4.440892e-16 F(X) = X1^0 * X2^1 * X3^0 * X4^2 * X5^0 0 1 2.222222e-01 1 7 1.110223e-16 2 28 2.220446e-16 3 84 0.000000e+00 F(X) = X1^0 * X2^0 * X3^1 * X4^2 * X5^0 0 1 2.222222e-01 1 7 1.110223e-16 2 28 4.440892e-16 3 84 2.220446e-16 F(X) = X1^0 * X2^0 * X3^0 * X4^3 * X5^0 0 1 7.407407e-01 1 7 0.000000e+00 2 28 2.220446e-16 3 84 8.881784e-16 F(X) = X1^2 * X2^0 * X3^0 * X4^0 * X5^1 0 1 2.222222e-01 1 7 0.000000e+00 2 28 2.220446e-16 3 84 2.220446e-16 F(X) = X1^1 * X2^1 * X3^0 * X4^0 * X5^1 0 1 5.555556e-01 1 7 2.220446e-16 2 28 8.881784e-16 3 84 2.220446e-16 F(X) = X1^0 * X2^2 * X3^0 * X4^0 * X5^1 0 1 2.222222e-01 1 7 0.000000e+00 2 28 2.220446e-16 3 84 2.220446e-16 F(X) = X1^1 * X2^0 * X3^1 * X4^0 * X5^1 0 1 5.555556e-01 1 7 2.220446e-16 2 28 1.110223e-15 3 84 6.661338e-16 F(X) = X1^0 * X2^1 * X3^1 * X4^0 * X5^1 0 1 5.555556e-01 1 7 2.220446e-16 2 28 1.110223e-15 3 84 7.771561e-16 F(X) = X1^0 * X2^0 * X3^2 * X4^0 * X5^1 0 1 2.222222e-01 1 7 0.000000e+00 2 28 4.440892e-16 3 84 2.220446e-16 F(X) = X1^1 * X2^0 * X3^0 * X4^1 * X5^1 0 1 5.555556e-01 1 7 2.220446e-16 2 28 1.110223e-15 3 84 5.551115e-16 F(X) = X1^0 * X2^1 * X3^0 * X4^1 * X5^1 0 1 5.555556e-01 1 7 2.220446e-16 2 28 0.000000e+00 3 84 7.771561e-16 F(X) = X1^0 * X2^0 * X3^1 * X4^1 * X5^1 0 1 5.555556e-01 1 7 2.220446e-16 2 28 8.881784e-16 3 84 4.440892e-16 F(X) = X1^0 * X2^0 * X3^0 * X4^2 * X5^1 0 1 2.222222e-01 1 7 0.000000e+00 2 28 4.440892e-16 3 84 2.220446e-16 F(X) = X1^1 * X2^0 * X3^0 * X4^0 * X5^2 0 1 2.222222e-01 1 7 2.220446e-16 2 28 0.000000e+00 3 84 2.220446e-16 F(X) = X1^0 * X2^1 * X3^0 * X4^0 * X5^2 0 1 2.222222e-01 1 7 2.220446e-16 2 28 0.000000e+00 3 84 8.881784e-16 F(X) = X1^0 * X2^0 * X3^1 * X4^0 * X5^2 0 1 2.222222e-01 1 7 2.220446e-16 2 28 2.220446e-16 3 84 2.220446e-16 F(X) = X1^0 * X2^0 * X3^0 * X4^1 * X5^2 0 1 2.222222e-01 1 7 2.220446e-16 2 28 2.220446e-16 3 84 2.220446e-16 F(X) = X1^0 * X2^0 * X3^0 * X4^0 * X5^3 0 1 7.407407e-01 1 7 0.000000e+00 2 28 4.440892e-16 3 84 2.220446e-16 F(X) = X1^4 * X2^0 * X3^0 * X4^0 * X5^0 0 1 9.027778e-01 1 7 1.171875e-01 2 28 1.332268e-15 3 84 8.881784e-16 F(X) = X1^3 * X2^1 * X3^0 * X4^0 * X5^0 0 1 6.111111e-01 1 7 9.375000e-02 2 28 0.000000e+00 3 84 6.661338e-16 F(X) = X1^2 * X2^2 * X3^0 * X4^0 * X5^0 0 1 4.166667e-01 1 7 2.031250e-01 2 28 6.661338e-16 3 84 6.661338e-16 F(X) = X1^1 * X2^3 * X3^0 * X4^0 * X5^0 0 1 6.111111e-01 1 7 9.375000e-02 2 28 4.440892e-16 3 84 4.440892e-16 F(X) = X1^0 * X2^4 * X3^0 * X4^0 * X5^0 0 1 9.027778e-01 1 7 1.171875e-01 2 28 1.332268e-15 3 84 6.661338e-16 F(X) = X1^3 * X2^0 * X3^1 * X4^0 * X5^0 0 1 6.111111e-01 1 7 9.375000e-02 2 28 8.881784e-16 3 84 8.881784e-16 F(X) = X1^2 * X2^1 * X3^1 * X4^0 * X5^0 0 1 1.666667e-01 1 7 3.125000e-02 2 28 1.776357e-15 3 84 7.771561e-16 F(X) = X1^1 * X2^2 * X3^1 * X4^0 * X5^0 0 1 1.666667e-01 1 7 3.125000e-02 2 28 1.776357e-15 3 84 6.661338e-16 F(X) = X1^0 * X2^3 * X3^1 * X4^0 * X5^0 0 1 6.111111e-01 1 7 9.375000e-02 2 28 0.000000e+00 3 84 4.440892e-16 F(X) = X1^2 * X2^0 * X3^2 * X4^0 * X5^0 0 1 4.166667e-01 1 7 2.031250e-01 2 28 6.661338e-16 3 84 8.881784e-16 F(X) = X1^1 * X2^1 * X3^2 * X4^0 * X5^0 0 1 1.666667e-01 1 7 3.125000e-02 2 28 6.661338e-16 3 84 6.661338e-16 F(X) = X1^0 * X2^2 * X3^2 * X4^0 * X5^0 0 1 4.166667e-01 1 7 2.031250e-01 2 28 6.661338e-16 3 84 6.661338e-16 F(X) = X1^1 * X2^0 * X3^3 * X4^0 * X5^0 0 1 6.111111e-01 1 7 9.375000e-02 2 28 1.110223e-15 3 84 4.440892e-16 F(X) = X1^0 * X2^1 * X3^3 * X4^0 * X5^0 0 1 6.111111e-01 1 7 9.375000e-02 2 28 6.661338e-16 3 84 8.881784e-16 F(X) = X1^0 * X2^0 * X3^4 * X4^0 * X5^0 0 1 9.027778e-01 1 7 1.171875e-01 2 28 8.881784e-16 3 84 1.332268e-15 F(X) = X1^3 * X2^0 * X3^0 * X4^1 * X5^0 0 1 6.111111e-01 1 7 9.375000e-02 2 28 4.440892e-16 3 84 4.440892e-16 F(X) = X1^2 * X2^1 * X3^0 * X4^1 * X5^0 0 1 1.666667e-01 1 7 3.125000e-02 2 28 1.110223e-15 3 84 6.661338e-16 F(X) = X1^1 * X2^2 * X3^0 * X4^1 * X5^0 0 1 1.666667e-01 1 7 3.125000e-02 2 28 1.110223e-15 3 84 1.221245e-15 F(X) = X1^0 * X2^3 * X3^0 * X4^1 * X5^0 0 1 6.111111e-01 1 7 9.375000e-02 2 28 0.000000e+00 3 84 4.440892e-16 F(X) = X1^2 * X2^0 * X3^1 * X4^1 * X5^0 0 1 1.666667e-01 1 7 3.125000e-02 2 28 6.661338e-16 3 84 1.110223e-15 F(X) = X1^1 * X2^1 * X3^1 * X4^1 * X5^0 0 1 1.333333e+00 1 7 6.250000e-02 2 28 0.000000e+00 3 84 1.998401e-15 F(X) = X1^0 * X2^2 * X3^1 * X4^1 * X5^0 0 1 1.666667e-01 1 7 3.125000e-02 2 28 6.661338e-16 3 84 0.000000e+00 F(X) = X1^1 * X2^0 * X3^2 * X4^1 * X5^0 0 1 1.666667e-01 1 7 3.125000e-02 2 28 8.881784e-16 3 84 8.881784e-16 F(X) = X1^0 * X2^1 * X3^2 * X4^1 * X5^0 0 1 1.666667e-01 1 7 3.125000e-02 2 28 8.881784e-16 3 84 8.881784e-16 F(X) = X1^0 * X2^0 * X3^3 * X4^1 * X5^0 0 1 6.111111e-01 1 7 9.375000e-02 2 28 2.220446e-16 3 84 8.881784e-16 F(X) = X1^2 * X2^0 * X3^0 * X4^2 * X5^0 0 1 4.166667e-01 1 7 2.031250e-01 2 28 1.332268e-15 3 84 7.771561e-16 F(X) = X1^1 * X2^1 * X3^0 * X4^2 * X5^0 0 1 1.666667e-01 1 7 3.125000e-02 2 28 1.332268e-15 3 84 2.220446e-16 F(X) = X1^0 * X2^2 * X3^0 * X4^2 * X5^0 0 1 4.166667e-01 1 7 2.031250e-01 2 28 1.332268e-15 3 84 4.440892e-16 F(X) = X1^1 * X2^0 * X3^1 * X4^2 * X5^0 0 1 1.666667e-01 1 7 3.125000e-02 2 28 8.881784e-16 3 84 3.330669e-16 F(X) = X1^0 * X2^1 * X3^1 * X4^2 * X5^0 0 1 1.666667e-01 1 7 3.125000e-02 2 28 8.881784e-16 3 84 3.330669e-16 F(X) = X1^0 * X2^0 * X3^2 * X4^2 * X5^0 0 1 4.166667e-01 1 7 2.031250e-01 2 28 1.332268e-15 3 84 6.661338e-16 F(X) = X1^1 * X2^0 * X3^0 * X4^3 * X5^0 0 1 6.111111e-01 1 7 9.375000e-02 2 28 6.661338e-16 3 84 0.000000e+00 F(X) = X1^0 * X2^1 * X3^0 * X4^3 * X5^0 0 1 6.111111e-01 1 7 9.375000e-02 2 28 6.661338e-16 3 84 6.661338e-16 F(X) = X1^0 * X2^0 * X3^1 * X4^3 * X5^0 0 1 6.111111e-01 1 7 9.375000e-02 2 28 2.220446e-16 3 84 8.881784e-16 F(X) = X1^0 * X2^0 * X3^0 * X4^4 * X5^0 0 1 9.027778e-01 1 7 1.171875e-01 2 28 4.440892e-16 3 84 1.110223e-15 F(X) = X1^3 * X2^0 * X3^0 * X4^0 * X5^1 0 1 6.111111e-01 1 7 9.375000e-02 2 28 2.220446e-16 3 84 8.881784e-16 F(X) = X1^2 * X2^1 * X3^0 * X4^0 * X5^1 0 1 1.666667e-01 1 7 3.125000e-02 2 28 1.776357e-15 3 84 8.881784e-16 F(X) = X1^1 * X2^2 * X3^0 * X4^0 * X5^1 0 1 1.666667e-01 1 7 3.125000e-02 2 28 4.440892e-16 3 84 6.661338e-16 F(X) = X1^0 * X2^3 * X3^0 * X4^0 * X5^1 0 1 6.111111e-01 1 7 9.375000e-02 2 28 2.220446e-16 3 84 1.332268e-15 F(X) = X1^2 * X2^0 * X3^1 * X4^0 * X5^1 0 1 1.666667e-01 1 7 3.125000e-02 2 28 1.776357e-15 3 84 7.771561e-16 F(X) = X1^1 * X2^1 * X3^1 * X4^0 * X5^1 0 1 1.333333e+00 1 7 6.250000e-02 2 28 1.332268e-15 3 84 2.109424e-15 F(X) = X1^0 * X2^2 * X3^1 * X4^0 * X5^1 0 1 1.666667e-01 1 7 3.125000e-02 2 28 1.332268e-15 3 84 6.661338e-16 F(X) = X1^1 * X2^0 * X3^2 * X4^0 * X5^1 0 1 1.666667e-01 1 7 3.125000e-02 2 28 1.110223e-15 3 84 1.110223e-15 F(X) = X1^0 * X2^1 * X3^2 * X4^0 * X5^1 0 1 1.666667e-01 1 7 3.125000e-02 2 28 1.110223e-15 3 84 8.881784e-16 F(X) = X1^0 * X2^0 * X3^3 * X4^0 * X5^1 0 1 6.111111e-01 1 7 9.375000e-02 2 28 2.220446e-16 3 84 4.440892e-16 F(X) = X1^2 * X2^0 * X3^0 * X4^1 * X5^1 0 1 1.666667e-01 1 7 3.125000e-02 2 28 6.661338e-16 3 84 3.330669e-16 F(X) = X1^1 * X2^1 * X3^0 * X4^1 * X5^1 0 1 1.333333e+00 1 7 6.250000e-02 2 28 2.442491e-15 3 84 1.887379e-15 F(X) = X1^0 * X2^2 * X3^0 * X4^1 * X5^1 0 1 1.666667e-01 1 7 3.125000e-02 2 28 1.332268e-15 3 84 6.661338e-16 F(X) = X1^1 * X2^0 * X3^1 * X4^1 * X5^1 0 1 1.333333e+00 1 7 6.250000e-02 2 28 8.881784e-16 3 84 1.221245e-15 F(X) = X1^0 * X2^1 * X3^1 * X4^1 * X5^1 0 1 1.333333e+00 1 7 6.250000e-02 2 28 2.220446e-15 3 84 1.332268e-15 F(X) = X1^0 * X2^0 * X3^2 * X4^1 * X5^1 0 1 1.666667e-01 1 7 3.125000e-02 2 28 6.661338e-16 3 84 8.881784e-16 F(X) = X1^1 * X2^0 * X3^0 * X4^2 * X5^1 0 1 1.666667e-01 1 7 3.125000e-02 2 28 1.110223e-15 3 84 6.661338e-16 F(X) = X1^0 * X2^1 * X3^0 * X4^2 * X5^1 0 1 1.666667e-01 1 7 3.125000e-02 2 28 1.110223e-15 3 84 1.554312e-15 F(X) = X1^0 * X2^0 * X3^1 * X4^2 * X5^1 0 1 1.666667e-01 1 7 3.125000e-02 2 28 6.661338e-16 3 84 0.000000e+00 F(X) = X1^0 * X2^0 * X3^0 * X4^3 * X5^1 0 1 6.111111e-01 1 7 9.375000e-02 2 28 2.220446e-16 3 84 0.000000e+00 F(X) = X1^2 * X2^0 * X3^0 * X4^0 * X5^2 0 1 4.166667e-01 1 7 2.031250e-01 2 28 1.332268e-15 3 84 8.881784e-16 F(X) = X1^1 * X2^1 * X3^0 * X4^0 * X5^2 0 1 1.666667e-01 1 7 3.125000e-02 2 28 1.554312e-15 3 84 0.000000e+00 F(X) = X1^0 * X2^2 * X3^0 * X4^0 * X5^2 0 1 4.166667e-01 1 7 2.031250e-01 2 28 6.661338e-16 3 84 6.661338e-16 F(X) = X1^1 * X2^0 * X3^1 * X4^0 * X5^2 0 1 1.666667e-01 1 7 3.125000e-02 2 28 0.000000e+00 3 84 1.110223e-16 F(X) = X1^0 * X2^1 * X3^1 * X4^0 * X5^2 0 1 1.666667e-01 1 7 3.125000e-02 2 28 0.000000e+00 3 84 6.661338e-16 F(X) = X1^0 * X2^0 * X3^2 * X4^0 * X5^2 0 1 4.166667e-01 1 7 2.031250e-01 2 28 6.661338e-16 3 84 6.661338e-16 F(X) = X1^1 * X2^0 * X3^0 * X4^1 * X5^2 0 1 1.666667e-01 1 7 3.125000e-02 2 28 0.000000e+00 3 84 6.661338e-16 F(X) = X1^0 * X2^1 * X3^0 * X4^1 * X5^2 0 1 1.666667e-01 1 7 3.125000e-02 2 28 0.000000e+00 3 84 6.661338e-16 F(X) = X1^0 * X2^0 * X3^1 * X4^1 * X5^2 0 1 1.666667e-01 1 7 3.125000e-02 2 28 6.661338e-16 3 84 7.771561e-16 F(X) = X1^0 * X2^0 * X3^0 * X4^2 * X5^2 0 1 4.166667e-01 1 7 2.031250e-01 2 28 6.661338e-16 3 84 8.881784e-16 F(X) = X1^1 * X2^0 * X3^0 * X4^0 * X5^3 0 1 6.111111e-01 1 7 9.375000e-02 2 28 4.440892e-16 3 84 4.440892e-16 F(X) = X1^0 * X2^1 * X3^0 * X4^0 * X5^3 0 1 6.111111e-01 1 7 9.375000e-02 2 28 4.440892e-16 3 84 8.881784e-16 F(X) = X1^0 * X2^0 * X3^1 * X4^0 * X5^3 0 1 6.111111e-01 1 7 9.375000e-02 2 28 8.881784e-16 3 84 4.440892e-16 F(X) = X1^0 * X2^0 * X3^0 * X4^1 * X5^3 0 1 6.111111e-01 1 7 9.375000e-02 2 28 8.881784e-16 3 84 2.220446e-16 F(X) = X1^0 * X2^0 * X3^0 * X4^0 * X5^4 0 1 9.027778e-01 1 7 1.171875e-01 2 28 0.000000e+00 3 84 4.440892e-16 SIMPLEX_GM_RULE_TEST08 GM_GENERAL_RULE_SET determines the weights and abscissas of a Grundmann-Moeller quadrature rule for the M dimensional general simplex, using a rule of index RULE, which will have degree of exactness 2*RULE+1. Here we use M = 3 RULE = 2 DEGREE = 5 Simplex vertices: 1 0 0 2 0 0 1 2 0 1 0 3 POINT W X Y Z 1 0.304762 1.125000 0.250000 0.375000 2 0.304762 1.375000 0.250000 0.375000 3 0.304762 1.625000 0.250000 0.375000 4 0.304762 1.125000 0.750000 0.375000 5 0.304762 1.375000 0.750000 0.375000 6 0.304762 1.125000 1.250000 0.375000 7 0.304762 1.125000 0.250000 1.125000 8 0.304762 1.375000 0.250000 1.125000 9 0.304762 1.125000 0.750000 1.125000 10 0.304762 1.125000 0.250000 1.875000 11 -0.578571 1.166667 0.333333 0.500000 12 -0.578571 1.500000 0.333333 0.500000 13 -0.578571 1.166667 1.000000 0.500000 14 -0.578571 1.166667 0.333333 1.500000 15 0.266667 1.250000 0.500000 0.750000 SIMPLEX_GM_RULE_TEST09 GM_UNIT_RULE_SET determines the weights and abscissas of a Grundmann-Moeller quadrature rule for the M dimensional unit simplex, using a rule of index RULE, which will have degree of exactness 2*RULE+1. In this test, look at all the monomials up to some maximum degree, choose a few low order rules and determine the quadrature error for each. Simplex volume = 0.166667 N 1 X Y Z X^2 XY XZ Y^2 YZ Z^2 1 0.166667 0.0416667 0.0416667 0.0416667 0.0104167 0.0104167 0.0104167 0.0104167 0.0104167 0.0104167 5 0.166667 0.0416667 0.0416667 0.0416667 0.0166667 0.00833333 0.00833333 0.0166667 0.00833333 0.0166667 15 0.166667 0.0416667 0.0416667 0.0416667 0.0166667 0.00833333 0.00833333 0.0166667 0.00833333 0.0166667 35 0.166667 0.0416667 0.0416667 0.0416667 0.0166667 0.00833333 0.00833333 0.0166667 0.00833333 0.0166667 70 0.166667 0.0416667 0.0416667 0.0416667 0.0166667 0.00833333 0.00833333 0.0166667 0.00833333 0.0166667 126 0.166667 0.0416667 0.0416667 0.0416667 0.0166667 0.00833333 0.00833333 0.0166667 0.00833333 0.0166667 SIMPLEX_GM_RULE_TEST10 GM_GENERAL_RULE_SET determines the weights and abscissas of a Grundmann-Moeller quadrature rule for the M dimensional general simplex, using a rule of index RULE, which will have degree of exactness 2*RULE+1. In this test, look at all the monomials up to some maximum degree, choose a few low order rules and determine the quadrature error for each. Simplex vertices: 1 0 0 2 0 0 1 2 0 1 0 3 Simplex volume = 1 N 1 X Y Z X^2 XY XZ Y^2 YZ Z^2 1 1 1.25 0.5 0.75 1.5625 0.625 0.9375 0.25 0.375 0.5625 5 1 1.25 0.5 0.75 1.6 0.6 0.9 0.4 0.3 0.9 15 1 1.25 0.5 0.75 1.6 0.6 0.9 0.4 0.3 0.9 35 1 1.25 0.5 0.75 1.6 0.6 0.9 0.4 0.3 0.9 70 1 1.25 0.5 0.75 1.6 0.6 0.9 0.4 0.3 0.9 126 1 1.25 0.5 0.75 1.6 0.6 0.9 0.4 0.3 0.9 simplex_gm_rule_test(): Normal end of execution. 08-Jan-2022 09:34:46