22 April 2014 04:16:32 PM SPHERE_TRIANGLE_QUAD_PRB C++ version Test the SPHERE_TRIANGLE_QUAD library. TEST01 Approximate the integral of a function 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 1000 points. QUAD_MC2 uses a Monte Carlo method with 10000 points. QUAD_MC3 uses a Monte Carlo method with 100000 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 1.49039 1.45444 1.45637 P(X,Y,Z) = Y -1.07421 -1.0461 -1.0444 P(X,Y,Z) = Z 1.26596 1.3098 1.32544 P(X,Y,Z) = X^2 1.2901 1.27525 1.27829 P(X,Y,Z) = Y^2 Z^2 0.19287 0.17703 0.182439 P(X,Y,Z) = X^2 Y^2 Z^2 0.0247888 0.0252832 0.0248802 P(X,Y,Z) = Y^2 Z^4 0.0837552 0.0861373 0.0851929 P(X,Y,Z) = Z^6 0.531964 0.479835 0.482496 P(X,Y,Z) = X Y^2 Z^4 0.00442363 0.00506373 0.00505596 P(X,Y,Z) = X^2 Y^4 Z^2 0.0066843 0.00717428 0.00725439 P(X,Y,Z) = X^6 Y^2 0.055871 0.0555553 0.0568885 P(X,Y,Z) = Z^8 0.37485 0.374319 0.377645 P(X,Y,Z) = X^6 Z^4 0.0158417 0.0150701 0.0150134 P(X,Y,Z) = X^4 Y^6 Z^2 0.000639512 0.000723393 0.000713386 P(X,Y,Z) = X^2 Y^4 Z^8 0.000443024 0.000452401 0.000453963 P(X,Y,Z) = X^16 0.379094 0.335969 0.339363 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 3.21916e-11 2 12 3.06905 3.21898e-11 4 48 3.06905 3.21907e-11 8 192 3.06905 3.21858e-11 16 768 3.06905 3.21654e-11 32 3072 3.06905 3.20708e-11 64 12288 3.06905 3.17062e-11 128 49152 3.06905 3.01297e-11 256 196608 3.06905 2.41891e-11 512 786432 3.06905 0 P(X,Y,Z) = X 1 3 1.51463 0.0494895 2 12 1.33913 0.126015 4 48 1.41618 0.0489648 8 192 1.44884 0.0163081 16 768 1.45778 0.00736701 32 3072 1.46323 0.00191226 64 12288 1.46467 0.000474421 128 49152 1.46503 0.000113076 256 196608 1.46512 2.26212e-05 512 786432 1.46514 0 P(X,Y,Z) = Y 1 3 -0.677827 0.37019 2 12 -0.694173 0.353843 4 48 -0.865286 0.18273 8 192 -1.00804 0.0399747 16 768 -1.03967 0.00834995 32 3072 -1.0459 0.00211192 64 12288 -1.04749 0.000523848 128 49152 -1.04789 0.00012485 256 196608 -1.04799 2.49764e-05 512 786432 -1.04802 0 P(X,Y,Z) = Z 1 3 0.705114 0.612961 2 12 0.814581 0.503494 4 48 1.03496 0.283116 8 192 1.2585 0.0595748 16 768 1.30671 0.0113608 32 3072 1.31522 0.00285993 64 12288 1.31737 0.00070935 128 49152 1.31791 0.000169059 256 196608 1.31804 3.38202e-05 512 786432 1.31808 0 P(X,Y,Z) = X^2 1 3 1.03742 0.244265 2 12 1.67384 0.392157 4 48 1.54651 0.264826 8 192 1.35022 0.0685309 16 768 1.2912 0.00951083 32 3072 1.28403 0.00234569 64 12288 1.28227 0.000582847 128 49152 1.28182 0.000138972 256 196608 1.28171 2.78044e-05 512 786432 1.28169 0 P(X,Y,Z) = Y^2 Z^2 1 3 0.0692846 0.112786 2 12 0.183284 0.0012132 4 48 0.145197 0.036874 8 192 0.172129 0.00994139 16 768 0.181049 0.001022 32 3072 0.181881 0.000189247 64 12288 0.182024 4.65206e-05 128 49152 0.18206 1.10618e-05 256 196608 0.182068 2.21162e-06 512 786432 0.182071 0 P(X,Y,Z) = X^2 Y^2 Z^2 1 3 0.00748156 0.0174654 2 12 0.0401385 0.0151915 4 48 0.0229795 0.00196752 8 192 0.0217482 0.00319878 16 768 0.0246428 0.000304158 32 3072 0.0248386 0.0001084 64 12288 0.0249196 2.74349e-05 128 49152 0.0249404 6.57059e-06 256 196608 0.0249457 1.31605e-06 512 786432 0.024947 0 P(X,Y,Z) = Y^2 Z^4 1 3 0.0308005 0.054435 2 12 0.0987627 0.0135271 4 48 0.0578983 0.0273372 8 192 0.0835862 0.00164928 16 768 0.0851493 8.62228e-05 32 3072 0.0852587 2.32202e-05 64 12288 0.085242 6.46202e-06 128 49152 0.0852371 1.58169e-06 256 196608 0.0852358 3.18505e-07 512 786432 0.0852355 0 P(X,Y,Z) = Z^6 1 3 0.554106 0.0728673 2 12 0.302548 0.17869 4 48 0.449237 0.0320011 8 192 0.477564 0.00367442 16 768 0.482445 0.00120624 32 3072 0.481715 0.000475988 64 12288 0.481359 0.000120784 128 49152 0.481267 2.89482e-05 256 196608 0.481244 5.79919e-06 512 786432 0.481239 0 P(X,Y,Z) = X Y^2 Z^4 1 3 0.0115693 0.00650123 2 12 0.0188167 0.0137486 4 48 -0.0116994 0.0167675 8 192 -0.00164606 0.00671418 16 768 0.00393746 0.00113065 32 3072 0.00483841 0.000229706 64 12288 0.00501091 5.72048e-05 128 49152 0.00505447 1.36474e-05 256 196608 0.00506538 2.73083e-06 512 786432 0.00506811 0 P(X,Y,Z) = X^2 Y^4 Z^2 1 3 0.00127823 0.00598643 2 12 0.00897193 0.00170726 4 48 0.00830424 0.00103958 8 192 0.00584431 0.00142036 16 768 0.00711873 0.000145937 32 3072 0.00722495 3.97158e-05 64 12288 0.00725461 1.00548e-05 128 49152 0.00726226 2.40724e-06 256 196608 0.00726419 4.82114e-07 512 786432 0.00726467 0 P(X,Y,Z) = X^6 Y^2 1 3 0.00540225 0.0516255 2 12 0.0416218 0.0154059 4 48 0.0703454 0.0133177 8 192 0.0625629 0.00553521 16 768 0.0572146 0.000186919 32 3072 0.0571604 0.000132717 64 12288 0.0570612 3.34666e-05 128 49152 0.0570357 8.01209e-06 256 196608 0.0570293 1.60462e-06 512 786432 0.0570277 0 P(X,Y,Z) = Z^8 1 3 0.44805 0.0681793 2 12 0.179639 0.200232 4 48 0.357799 0.0220712 8 192 0.386048 0.00617738 16 768 0.382143 0.00227282 32 3072 0.380504 0.000633641 64 12288 0.380031 0.0001603 128 49152 0.379909 3.83882e-05 256 196608 0.379878 7.68876e-06 512 786432 0.379871 0 P(X,Y,Z) = X^6 Z^4 1 3 0.0164461 0.00151222 2 12 0.0151825 0.000248667 4 48 0.00718153 0.00775235 8 192 0.0165234 0.00158954 16 768 0.0148645 6.93404e-05 32 3072 0.0148742 5.97079e-05 64 12288 0.0149191 1.47298e-05 128 49152 0.0149304 3.51549e-06 256 196608 0.0149332 7.03518e-07 512 786432 0.0149339 0 P(X,Y,Z) = X^4 Y^6 Z^2 1 3 3.58269e-05 0.000670801 2 12 0.000517082 0.000189546 4 48 0.00065595 5.06776e-05 8 192 0.000508199 0.000198429 16 768 0.000734295 2.76671e-05 32 3072 0.000701638 4.99025e-06 64 12288 0.000705373 1.25491e-06 128 49152 0.000706328 3.0018e-07 256 196608 0.000706568 6.01062e-08 512 786432 0.000706628 0 P(X,Y,Z) = X^2 Y^4 Z^8 1 3 0.000122125 0.000333948 2 12 0.00130941 0.000853341 4 48 0.000476343 2.02704e-05 8 192 0.000376033 8.00402e-05 16 768 0.000426762 2.93106e-05 32 3072 0.000455141 9.31875e-07 64 12288 0.000455856 2.16865e-07 128 49152 0.000456021 5.22154e-08 256 196608 0.000456062 1.04723e-08 512 786432 0.000456073 0 P(X,Y,Z) = X^16 1 3 0.243061 0.0968396 2 12 0.658949 0.319048 4 48 0.60303 0.263129 8 192 0.355423 0.0155222 16 768 0.351409 0.0115079 32 3072 0.342565 0.00266418 64 12288 0.340562 0.000660795 128 49152 0.340059 0.000157504 256 196608 0.339933 3.15093e-05 512 786432 0.339901 0 TEST05 Approximate the integral of a function 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 3.21916e-11 2 4 3.06905 3.21898e-11 4 16 3.06905 3.21907e-11 8 64 3.06905 3.21858e-11 16 256 3.06905 3.21654e-11 32 1024 3.06905 3.20708e-11 64 4096 3.06905 3.17062e-11 128 16384 3.06905 3.01297e-11 256 65536 3.06905 2.41891e-11 512 262144 3.06905 0 P(X,Y,Z) = X 1 1 1.02385 0.441276 2 4 1.03168 0.433448 4 16 1.24163 0.223502 8 64 1.37792 0.0872078 16 256 1.44148 0.0236506 32 1024 1.45906 0.0060635 64 4096 1.46362 0.00151027 128 16384 1.46477 0.000360325 256 65536 1.46506 7.2102e-05 512 262144 1.46513 0 P(X,Y,Z) = Y 1 1 -0.113063 0.934941 2 4 -0.740073 0.307931 4 16 -0.89914 0.148864 8 64 -0.980003 0.0680013 16 256 -1.02766 0.0203396 32 1024 -1.04277 0.00522915 64 4096 -1.0467 0.0013023 128 16384 -1.04769 0.000310699 256 65536 -1.04794 6.21713e-05 512 262144 -1.048 0 P(X,Y,Z) = Z 1 1 -0.0752233 1.39328 2 4 0.932754 0.385306 4 16 1.13812 0.179941 8 64 1.23184 0.0862192 16 256 1.29167 0.0263929 32 1024 1.31127 0.00678787 64 4096 1.31637 0.00169031 128 16384 1.31766 0.000403258 256 65536 1.31798 8.06919e-05 512 262144 1.31806 0 P(X,Y,Z) = X^2 1 1 1.97509 0.69339 2 4 1.01402 0.267673 4 16 1.21785 0.0638483 8 64 1.31306 0.0313663 16 256 1.3009 0.0192054 32 1024 1.28678 0.00508525 64 4096 1.28297 0.00127145 128 16384 1.282 0.000303639 256 65536 1.28176 6.07737e-05 512 262144 1.2817 0 P(X,Y,Z) = Y^2 Z^2 1 1 0.0568326 0.125237 2 4 0.0776178 0.104452 4 16 0.187192 0.00512216 8 64 0.182395 0.00032558 16 256 0.180598 0.00147167 32 1024 0.18163 0.000439502 64 4096 0.181962 0.00010729 128 16384 0.182044 2.54649e-05 256 65536 0.182065 5.08892e-06 512 262144 0.18207 0 P(X,Y,Z) = X^2 Y^2 Z^2 1 1 0.031486 0.00653965 2 4 0.0137496 0.0111968 4 16 0.0352328 0.0102864 8 64 0.0281084 0.00316204 16 256 0.0239672 0.000979202 32 1024 0.0246982 0.000248134 64 4096 0.0248819 6.44747e-05 128 16384 0.0249308 1.55498e-05 256 65536 0.0249433 3.12e-06 512 262144 0.0249464 0 P(X,Y,Z) = Y^2 Z^4 1 1 0.0208925 0.0643433 2 4 0.0341358 0.0511 4 16 0.099839 0.0146033 8 64 0.0826407 0.00259509 16 256 0.0854851 0.000249343 32 1024 0.0853055 6.97172e-05 64 4096 0.0852566 2.07927e-05 128 16384 0.0852409 5.17301e-06 256 65536 0.0852368 1.04582e-06 512 262144 0.0852358 0 P(X,Y,Z) = Z^6 1 1 0.116464 0.364778 2 4 0.540575 0.0593323 4 16 0.451272 0.0299706 8 64 0.488918 0.00767575 16 256 0.486678 0.005435 32 1024 0.482693 0.00145029 64 4096 0.481619 0.000375945 128 16384 0.481333 9.05994e-05 256 65536 0.481261 1.81747e-05 512 262144 0.481243 0 P(X,Y,Z) = X Y^2 Z^4 1 1 -0.00663518 0.011702 2 4 0.0071491 0.00208228 4 16 0.0268923 0.0218254 8 64 0.00663022 0.0015634 16 256 0.00350267 0.00156416 32 1024 0.00451886 0.000547967 64 4096 0.00492913 0.000137695 128 16384 0.0050339 3.29242e-05 256 65536 0.00506023 6.59185e-06 512 262144 0.00506683 0 P(X,Y,Z) = X^2 Y^4 Z^2 1 1 0.00252144 0.00474302 2 4 0.00175347 0.00551099 4 16 0.00808782 0.000823356 8 64 0.00918832 0.00192386 16 256 0.00699039 0.00027407 32 1024 0.00717608 8.83755e-05 64 4096 0.00724156 2.2899e-05 128 16384 0.00725894 5.52162e-06 256 65536 0.00726335 1.10783e-06 512 262144 0.00726446 0 P(X,Y,Z) = X^6 Y^2 1 1 0.109428 0.0523994 2 4 0.0146093 0.042419 4 16 0.0298538 0.0271745 8 64 0.0544624 0.00256586 16 256 0.0586195 0.00159117 32 1024 0.0572951 0.000266801 64 4096 0.0570979 6.96375e-05 128 16384 0.0570451 1.68035e-05 256 65536 0.0570317 3.37196e-06 512 262144 0.0570283 0 P(X,Y,Z) = Z^8 1 1 0.046748 0.333128 2 4 0.425583 0.0457077 4 16 0.312031 0.0678442 8 64 0.371732 0.00814323 16 256 0.385578 0.00570199 32 1024 0.381699 0.00182322 64 4096 0.380346 0.000470699 128 16384 0.379989 0.000113327 256 65536 0.379898 2.27287e-05 512 262144 0.379876 0 P(X,Y,Z) = X^6 Z^4 1 1 0.0485087 0.0335752 2 4 0.0210268 0.00609329 4 16 0.010823 0.00411043 8 64 0.00673142 0.00820206 16 256 0.0140475 0.000885968 32 1024 0.0147777 0.000155765 64 4096 0.0148938 3.96462e-05 128 16384 0.014924 9.48603e-06 256 65536 0.0149316 1.89953e-06 512 262144 0.0149335 0 P(X,Y,Z) = X^4 Y^6 Z^2 1 1 0.000148405 0.000558197 2 4 5.94834e-05 0.000647118 4 16 0.000426022 0.000280579 8 64 0.000621124 8.54774e-05 16 256 0.000549768 0.000156833 32 1024 0.000695456 1.11458e-05 64 4096 0.000703726 2.87536e-06 128 16384 0.00070591 6.92039e-07 256 65536 0.000706463 1.38783e-07 512 262144 0.000706602 0 P(X,Y,Z) = X^2 Y^4 Z^8 1 1 0.000131797 0.000324272 2 4 0.000153809 0.000302259 4 16 0.00123251 0.00077644 8 64 0.00084853 0.000392462 16 256 0.000518639 6.25706e-05 32 1024 0.00045455 1.51892e-06 64 4096 0.000455586 4.8238e-07 128 16384 0.000455951 1.17765e-07 256 65536 0.000456045 2.37e-08 512 262144 0.000456068 0 P(X,Y,Z) = X^16 1 1 0.224836 0.11508 2 4 0.176743 0.163173 4 16 0.432764 0.0928473 8 64 0.48864 0.148723 16 256 0.362191 0.022275 32 1024 0.346395 0.00647866 64 4096 0.341533 0.00161671 128 16384 0.340302 0.000385817 256 65536 0.339993 7.72074e-05 512 262144 0.339916 0 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 2.01794e-11 2 4 3.06905 2.01807e-11 4 16 3.06905 2.01807e-11 8 64 3.06905 2.01776e-11 16 256 3.06905 2.01741e-11 32 1024 3.06905 2.01026e-11 64 4096 3.06905 1.99742e-11 128 16384 3.06905 1.91815e-11 256 65536 3.06905 1.58162e-11 512 262144 3.06905 0 P(X,Y,Z) = X 1 1 1.02385 0.441291 2 4 1.03168 0.433463 4 16 1.32669 0.13845 8 64 1.42968 0.0354607 16 256 1.45628 0.00885972 32 1024 1.46293 0.00221265 64 4096 1.4646 0.000546941 128 16384 1.46501 0.000130242 256 65536 1.46512 2.60492e-05 512 262144 1.46514 0 P(X,Y,Z) = Y 1 1 -0.113063 0.934956 2 4 -0.740073 0.307946 4 16 -0.950421 0.0975977 8 64 -1.02545 0.0225649 16 256 -1.04243 0.00559352 32 1024 -1.04662 0.00139631 64 4096 -1.04767 0.000345111 128 16384 -1.04794 8.2178e-05 256 65536 -1.048 1.6436e-05 512 262144 -1.04802 0 P(X,Y,Z) = Z 1 1 -0.0752233 1.3933 2 4 0.932754 0.38532 4 16 1.15162 0.166456 8 64 1.26901 0.0490686 16 256 1.30563 0.0124459 32 1024 1.31496 0.00311834 64 4096 1.3173 0.000771456 128 16384 1.31789 0.000183743 256 65536 1.31804 3.67517e-05 512 262144 1.31807 0 P(X,Y,Z) = X^2 1 1 1.97509 0.693416 2 4 1.01402 0.267647 4 16 1.20058 0.0810949 8 64 1.26091 0.0207567 16 256 1.27636 0.00531425 32 1024 1.28033 0.00133887 64 4096 1.28134 0.000331668 128 16384 1.28159 7.90217e-05 256 65536 1.28165 1.5807e-05 512 262144 1.28167 0 P(X,Y,Z) = Y^2 Z^2 1 1 0.0568326 0.125242 2 4 0.0776178 0.104456 4 16 0.146194 0.0358806 8 64 0.193741 0.0116668 16 256 0.184932 0.00285831 32 1024 0.182775 0.000700512 64 4096 0.182247 0.000172703 128 16384 0.182115 4.11017e-05 256 65536 0.182082 8.21948e-06 512 262144 0.182074 0 P(X,Y,Z) = X^2 Y^2 Z^2 1 1 0.031486 0.006538 2 4 0.0137496 0.0111984 4 16 0.0274828 0.00253476 8 64 0.0287035 0.00375547 16 256 0.0255744 0.000626399 32 1024 0.0251049 0.000156886 64 4096 0.0249868 3.88014e-05 128 16384 0.0249573 9.24118e-06 256 65536 0.0249499 1.84838e-06 512 262144 0.024948 0 P(X,Y,Z) = Y^2 Z^4 1 1 0.0208925 0.0643438 2 4 0.0341358 0.0511005 4 16 0.0406425 0.0445937 8 64 0.0876674 0.00243107 16 256 0.0862801 0.00104385 32 1024 0.0854657 0.000229389 64 4096 0.0852917 5.54318e-05 128 16384 0.0852494 1.31292e-05 256 65536 0.0852389 2.62245e-06 512 262144 0.0852363 0 P(X,Y,Z) = Z^6 1 1 0.116464 0.364765 2 4 0.540575 0.0593453 4 16 0.546179 0.0649489 8 64 0.456209 0.025021 16 256 0.473917 0.00731217 32 1024 0.479449 0.00178034 64 4096 0.480791 0.000438357 128 16384 0.481125 0.000104291 256 65536 0.481209 2.08543e-05 512 262144 0.48123 0 P(X,Y,Z) = X Y^2 Z^4 1 1 -0.00663518 0.0117051 2 4 0.0071491 0.00207913 4 16 0.00651056 0.00144059 8 64 0.0107935 0.00572354 16 256 0.00610262 0.00103265 32 1024 0.00531467 0.000244696 64 4096 0.00512993 5.99571e-05 128 16384 0.00508422 1.42482e-05 256 65536 0.00507282 2.84831e-06 512 262144 0.00506997 0 P(X,Y,Z) = X^2 Y^4 Z^2 1 1 0.00252144 0.00474364 2 4 0.00175347 0.0055116 4 16 0.0122044 0.00493931 8 64 0.00801522 0.000750149 16 256 0.00751583 0.000250752 32 1024 0.00732775 6.26702e-05 64 4096 0.0072806 1.55224e-05 128 16384 0.00726877 3.69832e-06 256 65536 0.00726582 7.39792e-07 512 262144 0.00726508 0 P(X,Y,Z) = X^6 Y^2 1 1 0.109428 0.0524 2 4 0.0146093 0.0424183 4 16 0.0802003 0.0231726 8 64 0.0590635 0.00203581 16 256 0.0575134 0.000485715 32 1024 0.0571508 0.000123181 64 4096 0.0570582 3.05132e-05 128 16384 0.0570349 7.26939e-06 256 65536 0.0570291 1.45409e-06 512 262144 0.0570276 0 P(X,Y,Z) = Z^8 1 1 0.046748 0.333114 2 4 0.425583 0.0457212 4 16 0.478457 0.0985948 8 64 0.363942 0.0159202 16 256 0.373394 0.00646812 32 1024 0.378331 0.00153087 64 4096 0.379487 0.00037525 128 16384 0.379773 8.91833e-05 256 65536 0.379844 1.78287e-05 512 262144 0.379862 0 P(X,Y,Z) = X^6 Z^4 1 1 0.0485087 0.033575 2 4 0.0210268 0.00609308 4 16 0.00896571 0.00596797 8 64 0.0127033 0.00223043 16 256 0.0145117 0.000421968 32 1024 0.0148257 0.000108009 64 4096 0.0149069 2.68129e-05 128 16384 0.0149273 6.39155e-06 256 65536 0.0149324 1.27868e-06 512 262144 0.0149337 0 P(X,Y,Z) = X^4 Y^6 Z^2 1 1 0.000148405 0.00055827 2 4 5.94834e-05 0.000647192 4 16 0.00126435 0.000557675 8 64 0.000764571 5.78953e-05 16 256 0.000733678 2.70022e-05 32 1024 0.000713657 6.98173e-06 64 4096 0.000708408 1.73217e-06 128 16384 0.000707088 4.12835e-07 256 65536 0.000706758 8.25875e-08 512 262144 0.000706675 0 P(X,Y,Z) = X^2 Y^4 Z^8 1 1 0.000131797 0.000324288 2 4 0.000153809 0.000302275 4 16 0.000173367 0.000282717 8 64 0.00053741 8.13251e-05 16 256 0.000462217 6.13195e-06 32 1024 0.000458233 2.14883e-06 64 4096 0.000456602 5.17755e-07 128 16384 0.000456207 1.22631e-07 256 65536 0.000456109 2.44953e-08 512 262144 0.000456085 0 P(X,Y,Z) = X^16 1 1 0.224836 0.115052 2 4 0.176743 0.163145 4 16 0.209105 0.130783 8 64 0.331754 0.00813387 16 256 0.337299 0.00258941 32 1024 0.339246 0.000641888 64 4096 0.33973 0.000158319 128 16384 0.33985 3.76779e-05 256 65536 0.33988 7.53471e-06 512 262144 0.339888 0 SPHERE_TRIANGLE_QUAD_PRB Normal end of execution. 22 April 2014 04:20:10 PM