27 May 2021 03:28:23 PM SPHERE_TRIANGLE_QUAD_TEST C++ version Test the SPHERE_TRIANGLE_QUAD library. TEST01 Approximate an integral on a random spherical triangle. QUAD_01 uses centroids of spherical triangles. QUAD_02 uses vertices of spherical triangles. QUAD_03 uses midsides of spherical triangles. Vertices of random spherical triangle: V1: 0.795416 -0.223966 -0.563163 V2: -0.696321 -0.28431 0.659018 V3: 0.901719 0.397757 -0.169386 QUAD_01 QUAD_02 QUAD_03 P(X,Y,Z) = 1 3.06905 3.06905 3.06905 P(X,Y,Z) = X 3.04241 1.02385 1.51463 P(X,Y,Z) = Y -0.33597 -0.113063 -0.677827 P(X,Y,Z) = Z -0.223529 -0.0752233 0.705114 P(X,Y,Z) = X^2 3.016 1.97509 1.03742 P(X,Y,Z) = Y^2 Z^2 0.000195098 0.0568326 0.0692846 P(X,Y,Z) = X^2 Y^2 Z^2 0.000191725 0.031486 0.00748156 P(X,Y,Z) = Y^2 Z^4 1.03493e-06 0.0208925 0.0308005 P(X,Y,Z) = Z^6 4.58117e-07 0.116464 0.554106 P(X,Y,Z) = X Y^2 Z^4 1.02594e-06 -0.00663518 0.0115693 P(X,Y,Z) = X^2 Y^4 Z^2 2.29758e-06 0.00252144 0.00127823 P(X,Y,Z) = X^6 Y^2 0.0349039 0.109428 0.00540225 P(X,Y,Z) = Z^8 2.43015e-09 0.046748 0.44805 P(X,Y,Z) = X^6 Z^4 8.19592e-05 0.0485087 0.0164461 P(X,Y,Z) = X^4 Y^6 Z^2 2.70575e-08 0.000148405 3.58269e-05 P(X,Y,Z) = X^2 Y^4 Z^8 3.42959e-13 0.000131797 0.000122125 P(X,Y,Z) = X^16 2.6694 0.224836 0.243061 TEST02 Approximate the integral of a function on a random spherical triangle. SPHERE01_TRIANGLE_QUAD_00 uses the Monte Carlo method. QUAD_MC1 uses a Monte Carlo method with 100 points. QUAD_MC2 uses a Monte Carlo method with 1000 points. QUAD_MC3 uses a Monte Carlo method with 10000 points. Vertices of random spherical triangle: V1: 0.795416 -0.223966 -0.563163 V2: -0.696321 -0.28431 0.659018 V3: 0.901719 0.397757 -0.169386 QUAD_MC1 QUAD_MC2 QUAD_MC3 P(X,Y,Z) = 1 3.06905 3.06905 3.06905 P(X,Y,Z) = X 1.4315 1.52833 1.47665 P(X,Y,Z) = Y -0.94022 -1.03958 -1.06121 P(X,Y,Z) = Z 1.29126 1.29554 1.32762 P(X,Y,Z) = X^2 1.41225 1.22523 1.27077 P(X,Y,Z) = Y^2 Z^2 0.132389 0.18478 0.181499 P(X,Y,Z) = X^2 Y^2 Z^2 0.0251628 0.0253351 0.0255473 P(X,Y,Z) = Y^2 Z^4 0.0651289 0.0899393 0.0853924 P(X,Y,Z) = Z^6 0.545419 0.512305 0.472707 P(X,Y,Z) = X Y^2 Z^4 0.00127657 0.00642483 0.00545429 P(X,Y,Z) = X^2 Y^4 Z^2 0.00747278 0.0066768 0.00743173 P(X,Y,Z) = X^6 Y^2 0.0529985 0.0576074 0.0567267 P(X,Y,Z) = Z^8 0.295838 0.358571 0.389894 P(X,Y,Z) = X^6 Z^4 0.0173328 0.0166209 0.0155077 P(X,Y,Z) = X^4 Y^6 Z^2 0.000738518 0.00067729 0.000696674 P(X,Y,Z) = X^2 Y^4 Z^8 0.000479307 0.000481459 0.000472146 P(X,Y,Z) = X^16 0.291698 0.335782 0.337001 TEST03 Approximate the integral of a function on a random spherical triangle. SPHERE01_TRIANGLE_QUAD_ICOS1C approximates the integral of a function over a spherical triangle on the surface of the unit sphere using a centroid rule. We do not have an exact result, so we compare each estimate to the final one. Vertices of random spherical triangle: V1: 0.795416 -0.223966 -0.563163 V2: -0.696321 -0.28431 0.659018 V3: 0.901719 0.397757 -0.169386 FACTOR N RESULT P(X,Y,Z) = 1 1 1 3.06905 3.2335e-11 2 4 3.06905 3.23332e-11 4 16 3.06905 3.2331e-11 8 64 3.06905 3.23199e-11 16 256 3.06905 3.22986e-11 32 1024 3.06905 3.22005e-11 64 4096 3.06905 3.18474e-11 128 16384 3.06905 3.02718e-11 256 65536 3.06905 2.42188e-11 512 262144 3.06905 0 P(X,Y,Z) = X 1 1 3.04241 1.57726 2 4 1.45806 0.00708841 4 16 1.49095 0.0257989 8 64 1.47159 0.00643866 16 256 1.4633 0.00184474 32 1024 1.46463 0.000518916 64 4096 1.46502 0.00012853 128 16384 1.46512 3.06221e-05 256 65536 1.46514 6.12542e-06 512 262144 1.46515 0 P(X,Y,Z) = Y 1 1 -0.33597 0.712051 2 4 -0.579449 0.468571 4 16 -0.877491 0.17053 8 64 -1.02115 0.0268672 16 256 -1.04377 0.00425179 32 1024 -1.04695 0.00106615 64 4096 -1.04776 0.000263938 128 16384 -1.04796 6.28743e-05 256 65536 -1.04801 1.25765e-05 512 262144 -1.04802 0 P(X,Y,Z) = Z 1 1 -0.223529 1.54161 2 4 0.592226 0.725855 4 16 1.03534 0.282738 8 64 1.27367 0.0444137 16 256 1.31187 0.00621313 32 1024 1.31654 0.00154251 64 4096 1.3177 0.000381847 128 16384 1.31799 9.09611e-05 256 65536 1.31806 1.81945e-05 512 262144 1.31808 0 P(X,Y,Z) = X^2 1 1 3.016 1.73431 2 4 2.11153 0.829843 4 16 1.62765 0.34597 8 64 1.34823 0.0665452 16 256 1.28772 0.00603714 32 1024 1.28311 0.00142382 64 4096 1.28203 0.000352758 128 16384 1.28177 8.40486e-05 256 65536 1.2817 1.68126e-05 512 262144 1.28168 0 P(X,Y,Z) = Y^2 Z^2 1 1 0.000195098 0.181876 2 4 0.111736 0.0703345 4 16 0.136024 0.0460475 8 64 0.170831 0.0112404 16 256 0.181307 0.00076404 32 1024 0.181964 0.00010682 64 4096 0.182045 2.63287e-05 128 16384 0.182065 6.26483e-06 256 65536 0.18207 1.25277e-06 512 262144 0.182071 0 P(X,Y,Z) = X^2 Y^2 Z^2 1 1 0.000191725 0.0247555 2 4 0.0245318 0.000415372 4 16 0.014438 0.0105092 8 64 0.0213259 0.00362133 16 256 0.024845 0.000102218 32 1024 0.0248874 5.97758e-05 64 4096 0.0249322 1.49509e-05 128 16384 0.0249436 3.56886e-06 256 65536 0.0249465 7.14229e-07 512 262144 0.0249472 0 P(X,Y,Z) = Y^2 Z^4 1 1 1.03493e-06 0.0852344 2 4 0.0694164 0.015819 4 16 0.0653743 0.0198612 8 64 0.0833725 0.00186292 16 256 0.0850634 0.000172057 32 1024 0.0852407 5.30859e-06 64 4096 0.085237 1.52499e-06 128 16384 0.0852358 3.74496e-07 256 65536 0.0852355 7.54704e-08 512 262144 0.0852354 0 P(X,Y,Z) = Z^6 1 1 4.58117e-07 0.481237 2 4 0.271422 0.209816 4 16 0.442027 0.0392103 8 64 0.473444 0.00779271 16 256 0.481089 0.000147876 32 1024 0.481376 0.00013849 64 4096 0.481272 3.48979e-05 128 16384 0.481245 8.3455e-06 256 65536 0.481239 1.67093e-06 512 262144 0.481237 0 P(X,Y,Z) = X Y^2 Z^4 1 1 1.02594e-06 0.00506752 2 4 -0.030979 0.0360475 4 16 -0.0163491 0.0214176 8 64 -0.00269072 0.00775926 16 256 0.00425011 0.000818428 32 1024 0.00494642 0.000122119 64 4096 0.00503826 3.02793e-05 128 16384 0.00506133 7.21577e-06 256 65536 0.0050671 1.44347e-06 512 262144 0.00506854 0 P(X,Y,Z) = X^2 Y^4 Z^2 1 1 2.29758e-06 0.00726244 2 4 0.00373937 0.00352537 4 16 0.00440256 0.00286218 8 64 0.00545547 0.00180927 16 256 0.00717495 8.97845e-05 32 1024 0.00724192 2.28228e-05 64 4096 0.00725901 5.7248e-06 128 16384 0.00726337 1.36605e-06 256 65536 0.00726447 2.73361e-07 512 262144 0.00726474 0 P(X,Y,Z) = X^6 Y^2 1 1 0.0349039 0.0221236 2 4 0.0363769 0.0206506 4 16 0.0863493 0.0293218 8 64 0.062461 0.0054335 16 256 0.0569218 0.000105664 32 1024 0.057113 8.55468e-05 64 4096 0.0570488 2.1256e-05 128 16384 0.0570326 5.07194e-06 256 65536 0.0570285 1.01494e-06 512 262144 0.0570275 0 P(X,Y,Z) = Z^8 1 1 2.43015e-09 0.379869 2 4 0.170566 0.209303 4 16 0.381928 0.00205907 8 64 0.386589 0.00671972 16 256 0.380719 0.000850437 32 1024 0.380092 0.000222701 64 4096 0.379925 5.58843e-05 128 16384 0.379882 1.33488e-05 256 65536 0.379872 2.67192e-06 512 262144 0.379869 0 P(X,Y,Z) = X^6 Z^4 1 1 8.19592e-05 0.014852 2 4 0.00829619 0.00663782 4 16 0.0153499 0.00041587 8 64 0.0187221 0.00378806 16 256 0.0150526 0.000118569 32 1024 0.0149073 2.66853e-05 64 4096 0.0149276 6.38971e-06 128 16384 0.0149325 1.52315e-06 256 65536 0.0149337 3.0472e-07 512 262144 0.014934 0 P(X,Y,Z) = X^4 Y^6 Z^2 1 1 2.70575e-08 0.00070661 2 4 0.000125725 0.000580911 4 16 0.000256528 0.000450109 8 64 0.000705449 1.18736e-06 16 256 0.000762878 5.6241e-05 32 1024 0.000703781 2.8555e-06 64 4096 0.000705927 7.09464e-07 128 16384 0.000706467 1.69227e-07 256 65536 0.000706603 3.38609e-08 512 262144 0.000706637 0 P(X,Y,Z) = X^2 Y^4 Z^8 1 1 3.42959e-13 0.000456074 2 4 0.000920398 0.000464324 4 16 0.000373156 8.29179e-05 8 64 0.000241286 0.000214788 16 256 0.000414955 4.11189e-05 32 1024 0.000455436 6.38384e-07 64 4096 0.000455948 1.26461e-07 128 16384 0.000456044 3.02458e-08 256 65536 0.000456068 6.05599e-09 512 262144 0.000456074 0 P(X,Y,Z) = X^16 1 1 2.6694 2.3295 2 4 1.44954 1.10965 4 16 0.488517 0.148621 8 64 0.330088 0.00980781 16 256 0.346898 0.00700237 32 1024 0.341277 0.00138149 64 4096 0.340237 0.000341536 128 16384 0.339977 8.13609e-05 256 65536 0.339912 1.62743e-05 512 262144 0.339896 0 TEST04 Approximate the integral of a function on a random spherical triangle. SPHERE01_TRIANGLE_QUAD_ICOS1M approximates the integral of a function over a spherical triangle on the surface of the unit sphere using a midside rule. We do not have an exact result, so we compare each estimate to the final one. Vertices of random spherical triangle: V1: 0.795416 -0.223966 -0.563163 V2: -0.696321 -0.28431 0.659018 V3: 0.901719 0.397757 -0.169386 FACTOR N RESULT P(X,Y,Z) = 1 1 3 3.06905 3.219158e-11 2 12 3.06905 3.218981e-11 4 48 3.06905 3.219069e-11 8 192 3.06905 3.218581e-11 16 768 3.06905 3.216538e-11 32 3072 3.06905 3.207079e-11 64 12288 3.06905 3.170619e-11 128 49152 3.06905 3.012968e-11 256 196608 3.06905 2.418910e-11 512 786432 3.06905 0.000000e+00 P(X,Y,Z) = X 1 3 1.51463 4.948954e-02 2 12 1.33913 1.260150e-01 4 48 1.41618 4.896484e-02 8 192 1.44884 1.630814e-02 16 768 1.45778 7.367012e-03 32 3072 1.46323 1.912258e-03 64 12288 1.46467 4.744209e-04 128 49152 1.46503 1.130763e-04 256 196608 1.46512 2.262124e-05 512 786432 1.46514 0.000000e+00 P(X,Y,Z) = Y 1 3 -0.677827 3.701898e-01 2 12 -0.694173 3.538431e-01 4 48 -0.865286 1.827304e-01 8 192 -1.00804 3.997474e-02 16 768 -1.03967 8.349947e-03 32 3072 -1.0459 2.111923e-03 64 12288 -1.04749 5.238483e-04 128 49152 -1.04789 1.248505e-04 256 196608 -1.04799 2.497635e-05 512 786432 -1.04802 0.000000e+00 P(X,Y,Z) = Z 1 3 0.705114 6.129614e-01 2 12 0.814581 5.034938e-01 4 48 1.03496 2.831157e-01 8 192 1.2585 5.957482e-02 16 768 1.30671 1.136082e-02 32 3072 1.31522 2.859929e-03 64 12288 1.31737 7.093497e-04 128 49152 1.31791 1.690595e-04 256 196608 1.31804 3.382024e-05 512 786432 1.31808 0.000000e+00 P(X,Y,Z) = X^2 1 3 1.03742 2.442650e-01 2 12 1.67384 3.921575e-01 4 48 1.54651 2.648256e-01 8 192 1.35022 6.853094e-02 16 768 1.2912 9.510829e-03 32 3072 1.28403 2.345688e-03 64 12288 1.28227 5.828466e-04 128 49152 1.28182 1.389723e-04 256 196608 1.28171 2.780445e-05 512 786432 1.28169 0.000000e+00 P(X,Y,Z) = Y^2 Z^2 1 3 0.0692846 1.127861e-01 2 12 0.183284 1.213197e-03 4 48 0.145197 3.687397e-02 8 192 0.172129 9.941391e-03 16 768 0.181049 1.022002e-03 32 3072 0.181881 1.892473e-04 64 12288 0.182024 4.652055e-05 128 49152 0.18206 1.106178e-05 256 196608 0.182068 2.211625e-06 512 786432 0.182071 0.000000e+00 P(X,Y,Z) = X^2 Y^2 Z^2 1 3 0.00748156 1.746543e-02 2 12 0.0401385 1.519153e-02 4 48 0.0229795 1.967522e-03 8 192 0.0217482 3.198777e-03 16 768 0.0246428 3.041575e-04 32 3072 0.0248386 1.083999e-04 64 12288 0.0249196 2.743495e-05 128 49152 0.0249404 6.570594e-06 256 196608 0.0249457 1.316054e-06 512 786432 0.024947 0.000000e+00 P(X,Y,Z) = Y^2 Z^4 1 3 0.0308005 5.443498e-02 2 12 0.0987627 1.352715e-02 4 48 0.0578983 2.733723e-02 8 192 0.0835862 1.649282e-03 16 768 0.0851493 8.622280e-05 32 3072 0.0852587 2.322019e-05 64 12288 0.085242 6.462019e-06 128 49152 0.0852371 1.581686e-06 256 196608 0.0852358 3.185048e-07 512 786432 0.0852355 0.000000e+00 P(X,Y,Z) = Z^6 1 3 0.554106 7.286725e-02 2 12 0.302548 1.786901e-01 4 48 0.449237 3.200111e-02 8 192 0.477564 3.674416e-03 16 768 0.482445 1.206241e-03 32 3072 0.481715 4.759882e-04 64 12288 0.481359 1.207838e-04 128 49152 0.481267 2.894819e-05 256 196608 0.481244 5.799194e-06 512 786432 0.481239 0.000000e+00 P(X,Y,Z) = X Y^2 Z^4 1 3 0.0115693 6.501233e-03 2 12 0.0188167 1.374856e-02 4 48 -0.0116994 1.676752e-02 8 192 -0.00164606 6.714175e-03 16 768 0.00393746 1.130649e-03 32 3072 0.00483841 2.297064e-04 64 12288 0.00501091 5.720485e-05 128 49152 0.00505447 1.364736e-05 256 196608 0.00506538 2.730833e-06 512 786432 0.00506811 0.000000e+00 P(X,Y,Z) = X^2 Y^4 Z^2 1 3 0.00127823 5.986434e-03 2 12 0.00897193 1.707263e-03 4 48 0.00830424 1.039576e-03 8 192 0.00584431 1.420357e-03 16 768 0.00711873 1.459369e-04 32 3072 0.00722495 3.971578e-05 64 12288 0.00725461 1.005479e-05 128 49152 0.00726226 2.407240e-06 256 196608 0.00726419 4.821144e-07 512 786432 0.00726467 0.000000e+00 P(X,Y,Z) = X^6 Y^2 1 3 0.00540225 5.162545e-02 2 12 0.0416218 1.540594e-02 4 48 0.0703454 1.331768e-02 8 192 0.0625629 5.535213e-03 16 768 0.0572146 1.869190e-04 32 3072 0.0571604 1.327165e-04 64 12288 0.0570612 3.346656e-05 128 49152 0.0570357 8.012093e-06 256 196608 0.0570293 1.604624e-06 512 786432 0.0570277 0.000000e+00 P(X,Y,Z) = Z^8 1 3 0.44805 6.817929e-02 2 12 0.179639 2.002317e-01 4 48 0.357799 2.207122e-02 8 192 0.386048 6.177377e-03 16 768 0.382143 2.272817e-03 32 3072 0.380504 6.336408e-04 64 12288 0.380031 1.603001e-04 128 49152 0.379909 3.838817e-05 256 196608 0.379878 7.688763e-06 512 786432 0.379871 0.000000e+00 P(X,Y,Z) = X^6 Z^4 1 3 0.0164461 1.512224e-03 2 12 0.0151825 2.486669e-04 4 48 0.00718153 7.752349e-03 8 192 0.0165234 1.589541e-03 16 768 0.0148645 6.934039e-05 32 3072 0.0148742 5.970789e-05 64 12288 0.0149191 1.472977e-05 128 49152 0.0149304 3.515488e-06 256 196608 0.0149332 7.035176e-07 512 786432 0.0149339 0.000000e+00 P(X,Y,Z) = X^4 Y^6 Z^2 1 3 3.58269e-05 6.708010e-04 2 12 0.000517082 1.895464e-04 4 48 0.00065595 5.067755e-05 8 192 0.000508199 1.984292e-04 16 768 0.000734295 2.766714e-05 32 3072 0.000701638 4.990253e-06 64 12288 0.000705373 1.254905e-06 128 49152 0.000706328 3.001804e-07 256 196608 0.000706568 6.010621e-08 512 786432 0.000706628 0.000000e+00 P(X,Y,Z) = X^2 Y^4 Z^8 1 3 0.000122125 3.339477e-04 2 12 0.00130941 8.533406e-04 4 48 0.000476343 2.027045e-05 8 192 0.000376033 8.004015e-05 16 768 0.000426762 2.931059e-05 32 3072 0.000455141 9.318745e-07 64 12288 0.000455856 2.168647e-07 128 49152 0.000456021 5.221537e-08 256 196608 0.000456062 1.047229e-08 512 786432 0.000456073 0.000000e+00 P(X,Y,Z) = X^16 1 3 0.243061 9.683965e-02 2 12 0.658949 3.190482e-01 4 48 0.60303 2.631289e-01 8 192 0.355423 1.552225e-02 16 768 0.351409 1.150791e-02 32 3072 0.342565 2.664184e-03 64 12288 0.340562 6.607952e-04 128 49152 0.340059 1.575037e-04 256 196608 0.339933 3.150928e-05 512 786432 0.339901 0.000000e+00 TEST05 Approximate an integral on a random spherical triangle. SPHERE01_TRIANGLE_QUAD_ICOS1V approximates the integral of a function over a spherical triangle on the surface of the unit sphere using a vertex rule. We do not have an exact result, so we compare each estimate to the final one. Vertices of random spherical triangle: V1: 0.795416 -0.223966 -0.563163 V2: -0.696321 -0.28431 0.659018 V3: 0.901719 0.397757 -0.169386 FACTOR N RESULT P(X,Y,Z) = 1 1 1 3.06905 3.219158e-11 2 4 3.06905 3.218981e-11 4 16 3.06905 3.219069e-11 8 64 3.06905 3.218581e-11 16 256 3.06905 3.216538e-11 32 1024 3.06905 3.207079e-11 64 4096 3.06905 3.170619e-11 128 16384 3.06905 3.012968e-11 256 65536 3.06905 2.418910e-11 512 262144 3.06905 0.000000e+00 P(X,Y,Z) = X 1 1 1.02385 4.412758e-01 2 4 1.03168 4.334481e-01 4 16 1.24163 2.235016e-01 8 64 1.37792 8.720784e-02 16 256 1.44148 2.365057e-02 32 1024 1.45906 6.063496e-03 64 4096 1.46362 1.510267e-03 128 16384 1.46477 3.603249e-04 256 65536 1.46506 7.210199e-05 512 262144 1.46513 0.000000e+00 P(X,Y,Z) = Y 1 1 -0.113063 9.349412e-01 2 4 -0.740073 3.079309e-01 4 16 -0.89914 1.488638e-01 8 64 -0.980003 6.800125e-02 16 256 -1.02766 2.033962e-02 32 1024 -1.04277 5.229150e-03 64 4096 -1.0467 1.302302e-03 128 16384 -1.04769 3.106991e-04 256 65536 -1.04794 6.217128e-05 512 262144 -1.048 0.000000e+00 P(X,Y,Z) = Z 1 1 -0.0752233 1.393283e+00 2 4 0.932754 3.853057e-01 4 16 1.13812 1.799406e-01 8 64 1.23184 8.621917e-02 16 256 1.29167 2.639289e-02 32 1024 1.31127 6.787871e-03 64 4096 1.31637 1.690311e-03 128 16384 1.31766 4.032579e-04 256 65536 1.31798 8.069187e-05 512 262144 1.31806 0.000000e+00 P(X,Y,Z) = X^2 1 1 1.97509 6.933902e-01 2 4 1.01402 2.676726e-01 4 16 1.21785 6.384827e-02 8 64 1.31306 3.136629e-02 16 256 1.3009 1.920540e-02 32 1024 1.28678 5.085250e-03 64 4096 1.28297 1.271448e-03 128 16384 1.282 3.036393e-04 256 65536 1.28176 6.077372e-05 512 262144 1.2817 0.000000e+00 P(X,Y,Z) = Y^2 Z^2 1 1 0.0568326 1.252372e-01 2 4 0.0776178 1.044519e-01 4 16 0.187192 5.122158e-03 8 64 0.182395 3.255797e-04 16 256 0.180598 1.471675e-03 32 1024 0.18163 4.395023e-04 64 4096 0.181962 1.072905e-04 128 16384 0.182044 2.546492e-05 256 65536 0.182065 5.088916e-06 512 262144 0.18207 0.000000e+00 P(X,Y,Z) = X^2 Y^2 Z^2 1 1 0.031486 6.539653e-03 2 4 0.0137496 1.119676e-02 4 16 0.0352328 1.028640e-02 8 64 0.0281084 3.162043e-03 16 256 0.0239672 9.792018e-04 32 1024 0.0246982 2.481339e-04 64 4096 0.0248819 6.447466e-05 128 16384 0.0249308 1.554982e-05 256 65536 0.0249433 3.119995e-06 512 262144 0.0249464 0.000000e+00 P(X,Y,Z) = Y^2 Z^4 1 1 0.0208925 6.434330e-02 2 4 0.0341358 5.109997e-02 4 16 0.099839 1.460327e-02 8 64 0.0826407 2.595089e-03 16 256 0.0854851 2.493434e-04 32 1024 0.0853055 6.971717e-05 64 4096 0.0852566 2.079269e-05 128 16384 0.0852409 5.173011e-06 256 65536 0.0852368 1.045824e-06 512 262144 0.0852358 0.000000e+00 P(X,Y,Z) = Z^6 1 1 0.116464 3.647782e-01 2 4 0.540575 5.933233e-02 4 16 0.451272 2.997064e-02 8 64 0.488918 7.675749e-03 16 256 0.486678 5.435005e-03 32 1024 0.482693 1.450286e-03 64 4096 0.481619 3.759449e-04 128 16384 0.481333 9.059945e-05 256 65536 0.481261 1.817475e-05 512 262144 0.481243 0.000000e+00 P(X,Y,Z) = X Y^2 Z^4 1 1 -0.00663518 1.170200e-02 2 4 0.0071491 2.082276e-03 4 16 0.0268923 2.182544e-02 8 64 0.00663022 1.563396e-03 16 256 0.00350267 1.564158e-03 32 1024 0.00451886 5.479674e-04 64 4096 0.00492913 1.376950e-04 128 16384 0.0050339 3.292421e-05 256 65536 0.00506023 6.591853e-06 512 262144 0.00506683 0.000000e+00 P(X,Y,Z) = X^2 Y^4 Z^2 1 1 0.00252144 4.743023e-03 2 4 0.00175347 5.510988e-03 4 16 0.00808782 8.233558e-04 8 64 0.00918832 1.923862e-03 16 256 0.00699039 2.740703e-04 32 1024 0.00717608 8.837553e-05 64 4096 0.00724156 2.289898e-05 128 16384 0.00725894 5.521616e-06 256 65536 0.00726335 1.107834e-06 512 262144 0.00726446 0.000000e+00 P(X,Y,Z) = X^6 Y^2 1 1 0.109428 5.239939e-02 2 4 0.0146093 4.241896e-02 4 16 0.0298538 2.717453e-02 8 64 0.0544624 2.565858e-03 16 256 0.0586195 1.591170e-03 32 1024 0.0572951 2.668008e-04 64 4096 0.0570979 6.963750e-05 128 16384 0.0570451 1.680352e-05 256 65536 0.0570317 3.371963e-06 512 262144 0.0570283 0.000000e+00 P(X,Y,Z) = Z^8 1 1 0.046748 3.331276e-01 2 4 0.425583 4.570770e-02 4 16 0.312031 6.784417e-02 8 64 0.371732 8.143230e-03 16 256 0.385578 5.701991e-03 32 1024 0.381699 1.823217e-03 64 4096 0.380346 4.706989e-04 128 16384 0.379989 1.133272e-04 256 65536 0.379898 2.272873e-05 512 262144 0.379876 0.000000e+00 P(X,Y,Z) = X^6 Z^4 1 1 0.0485087 3.357522e-02 2 4 0.0210268 6.093286e-03 4 16 0.010823 4.110431e-03 8 64 0.00673142 8.202061e-03 16 256 0.0140475 8.859677e-04 32 1024 0.0147777 1.557647e-04 64 4096 0.0148938 3.964622e-05 128 16384 0.014924 9.486027e-06 256 65536 0.0149316 1.899530e-06 512 262144 0.0149335 0.000000e+00 P(X,Y,Z) = X^4 Y^6 Z^2 1 1 0.000148405 5.581966e-04 2 4 5.94834e-05 6.471183e-04 4 16 0.000426022 2.805793e-04 8 64 0.000621124 8.547743e-05 16 256 0.000549768 1.568335e-04 32 1024 0.000695456 1.114579e-05 64 4096 0.000703726 2.875356e-06 128 16384 0.00070591 6.920393e-07 256 65536 0.000706463 1.387832e-07 512 262144 0.000706602 0.000000e+00 P(X,Y,Z) = X^2 Y^4 Z^8 1 1 0.000131797 3.242716e-04 2 4 0.000153809 3.022593e-04 4 16 0.00123251 7.764395e-04 8 64 0.00084853 3.924619e-04 16 256 0.000518639 6.257057e-05 32 1024 0.00045455 1.518924e-06 64 4096 0.000455586 4.823801e-07 128 16384 0.000455951 1.177650e-07 256 65536 0.000456045 2.370001e-08 512 262144 0.000456068 0.000000e+00 P(X,Y,Z) = X^16 1 1 0.224836 1.150803e-01 2 4 0.176743 1.631728e-01 4 16 0.432764 9.284726e-02 8 64 0.48864 1.487233e-01 16 256 0.362191 2.227504e-02 32 1024 0.346395 6.478664e-03 64 4096 0.341533 1.616714e-03 128 16384 0.340302 3.858167e-04 256 65536 0.339993 7.720742e-05 512 262144 0.339916 0.000000e+00 TEST06 Approximate the integral of a function on a random spherical triangle. SPHERE01_TRIANGLE_QUAD_ICOS2V approximates the integral of a function over a spherical triangle on the surface of the unit sphere using a vertex rule. We do not have an exact result, so we compare each estimate to the final one. Vertices of random spherical triangle: V1: 0.795416 -0.223966 -0.563163 V2: -0.696321 -0.28431 0.659018 V3: 0.901719 0.397757 -0.169386 FACTOR N RESULT P(X,Y,Z) = 1 1 1 3.06905 2.017941e-11 2 4 3.06905 2.018075e-11 4 16 3.06905 2.018075e-11 8 64 3.06905 2.017764e-11 16 256 3.06905 2.017408e-11 32 1024 3.06905 2.010259e-11 64 4096 3.06905 1.997424e-11 128 16384 3.06905 1.918155e-11 256 65536 3.06905 1.581624e-11 512 262144 3.06905 0.000000e+00 P(X,Y,Z) = X 1 1 1.02385 4.412912e-01 2 4 1.03168 4.334635e-01 4 16 1.32669 1.384495e-01 8 64 1.42968 3.546067e-02 16 256 1.45628 8.859718e-03 32 1024 1.46293 2.212647e-03 64 4096 1.4646 5.469410e-04 128 16384 1.46501 1.302415e-04 256 65536 1.46512 2.604919e-05 512 262144 1.46514 0.000000e+00 P(X,Y,Z) = Y 1 1 -0.113063 9.349564e-01 2 4 -0.740073 3.079461e-01 4 16 -0.950421 9.759772e-02 8 64 -1.02545 2.256487e-02 16 256 -1.04243 5.593517e-03 32 1024 -1.04662 1.396310e-03 64 4096 -1.04767 3.451115e-04 128 16384 -1.04794 8.217805e-05 256 65536 -1.048 1.643604e-05 512 262144 -1.04802 0.000000e+00 P(X,Y,Z) = Z 1 1 -0.0752233 1.393298e+00 2 4 0.932754 3.853204e-01 4 16 1.15162 1.664562e-01 8 64 1.26901 4.906858e-02 16 256 1.30563 1.244588e-02 32 1024 1.31496 3.118342e-03 64 4096 1.3173 7.714556e-04 128 16384 1.31789 1.837426e-04 256 65536 1.31804 3.675168e-05 512 262144 1.31807 0.000000e+00 P(X,Y,Z) = X^2 1 1 1.97509 6.934157e-01 2 4 1.01402 2.676471e-01 4 16 1.20058 8.109491e-02 8 64 1.26091 2.075673e-02 16 256 1.27636 5.314255e-03 32 1024 1.28033 1.338868e-03 64 4096 1.28134 3.316677e-04 128 16384 1.28159 7.902167e-05 256 65536 1.28165 1.580700e-05 512 262144 1.28167 0.000000e+00 P(X,Y,Z) = Y^2 Z^2 1 1 0.0568326 1.252416e-01 2 4 0.0776178 1.044564e-01 4 16 0.146194 3.588061e-02 8 64 0.193741 1.166680e-02 16 256 0.184932 2.858311e-03 32 1024 0.182775 7.005118e-04 64 4096 0.182247 1.727032e-04 128 16384 0.182115 4.110173e-05 256 65536 0.182082 8.219481e-06 512 262144 0.182074 0.000000e+00 P(X,Y,Z) = X^2 Y^2 Z^2 1 1 0.031486 6.537995e-03 2 4 0.0137496 1.119842e-02 4 16 0.0274828 2.534756e-03 8 64 0.0287035 3.755470e-03 16 256 0.0255744 6.263993e-04 32 1024 0.0251049 1.568860e-04 64 4096 0.0249868 3.880144e-05 128 16384 0.0249573 9.241183e-06 256 65536 0.0249499 1.848378e-06 512 262144 0.024948 0.000000e+00 P(X,Y,Z) = Y^2 Z^4 1 1 0.0208925 6.434383e-02 2 4 0.0341358 5.110049e-02 4 16 0.0406425 4.459375e-02 8 64 0.0876674 2.431069e-03 16 256 0.0862801 1.043851e-03 32 1024 0.0854657 2.293890e-04 64 4096 0.0852917 5.543181e-05 128 16384 0.0852494 1.312917e-05 256 65536 0.0852389 2.622445e-06 512 262144 0.0852363 0.000000e+00 P(X,Y,Z) = Z^6 1 1 0.116464 3.647652e-01 2 4 0.540575 5.934534e-02 4 16 0.546179 6.494888e-02 8 64 0.456209 2.502101e-02 16 256 0.473917 7.312173e-03 32 1024 0.479449 1.780336e-03 64 4096 0.480791 4.383568e-04 128 16384 0.481125 1.042909e-04 256 65536 0.481209 2.085431e-05 512 262144 0.48123 0.000000e+00 P(X,Y,Z) = X Y^2 Z^4 1 1 -0.00663518 1.170515e-02 2 4 0.0071491 2.079128e-03 4 16 0.00651056 1.440587e-03 8 64 0.0107935 5.723544e-03 16 256 0.00610262 1.032650e-03 32 1024 0.00531467 2.446961e-04 64 4096 0.00512993 5.995711e-05 128 16384 0.00508422 1.424822e-05 256 65536 0.00507282 2.848308e-06 512 262144 0.00506997 0.000000e+00 P(X,Y,Z) = X^2 Y^4 Z^2 1 1 0.00252144 4.743639e-03 2 4 0.00175347 5.511604e-03 4 16 0.0122044 4.939307e-03 8 64 0.00801522 7.501488e-04 16 256 0.00751583 2.507524e-04 32 1024 0.00732775 6.267025e-05 64 4096 0.0072806 1.552237e-05 128 16384 0.00726877 3.698321e-06 256 65536 0.00726582 7.397921e-07 512 262144 0.00726508 0.000000e+00 P(X,Y,Z) = X^6 Y^2 1 1 0.109428 5.240003e-02 2 4 0.0146093 4.241832e-02 4 16 0.0802003 2.317264e-02 8 64 0.0590635 2.035808e-03 16 256 0.0575134 4.857154e-04 32 1024 0.0571508 1.231815e-04 64 4096 0.0570582 3.051320e-05 128 16384 0.0570349 7.269388e-06 256 65536 0.0570291 1.454090e-06 512 262144 0.0570276 0.000000e+00 P(X,Y,Z) = Z^8 1 1 0.046748 3.331141e-01 2 4 0.425583 4.572123e-02 4 16 0.478457 9.859475e-02 8 64 0.363942 1.592017e-02 16 256 0.373394 6.468115e-03 32 1024 0.378331 1.530873e-03 64 4096 0.379487 3.752503e-04 128 16384 0.379773 8.918333e-05 256 65536 0.379844 1.782873e-05 512 262144 0.379862 0.000000e+00 P(X,Y,Z) = X^6 Z^4 1 1 0.0485087 3.357501e-02 2 4 0.0210268 6.093079e-03 4 16 0.00896571 5.967973e-03 8 64 0.0127033 2.230433e-03 16 256 0.0145117 4.219679e-04 32 1024 0.0148257 1.080093e-04 64 4096 0.0149069 2.681292e-05 128 16384 0.0149273 6.391549e-06 256 65536 0.0149324 1.278685e-06 512 262144 0.0149337 0.000000e+00 P(X,Y,Z) = X^4 Y^6 Z^2 1 1 0.000148405 5.582704e-04 2 4 5.94834e-05 6.471921e-04 4 16 0.00126435 5.576747e-04 8 64 0.000764571 5.789534e-05 16 256 0.000733678 2.700224e-05 32 1024 0.000713657 6.981733e-06 64 4096 0.000708408 1.732169e-06 128 16384 0.000707088 4.128351e-07 256 65536 0.000706758 8.258753e-08 512 262144 0.000706675 0.000000e+00 P(X,Y,Z) = X^2 Y^4 Z^8 1 1 0.000131797 3.242877e-04 2 4 0.000153809 3.022753e-04 4 16 0.000173367 2.827174e-04 8 64 0.00053741 8.132511e-05 16 256 0.000462217 6.131950e-06 32 1024 0.000458233 2.148827e-06 64 4096 0.000456602 5.177553e-07 128 16384 0.000456207 1.226313e-07 256 65536 0.000456109 2.449532e-08 512 262144 0.000456085 0.000000e+00 P(X,Y,Z) = X^16 1 1 0.224836 1.150521e-01 2 4 0.176743 1.631445e-01 4 16 0.209105 1.307834e-01 8 64 0.331754 8.133866e-03 16 256 0.337299 2.589413e-03 32 1024 0.339246 6.418876e-04 64 4096 0.33973 1.583192e-04 128 16384 0.33985 3.767790e-05 256 65536 0.33988 7.534708e-06 512 262144 0.339888 0.000000e+00 SPHERE_TRIANGLE_QUAD_TEST Normal end of execution. 27 May 2021 03:29:36 PM