15 September 2021 3:23:36.873 PM STROUD_TEST FORTRAN90 version Test the STROUD library. TEST01 For integrals in a ball in ND: BALL_F1_ND approximates the integral; BALL_F3_ND approximates the integral. Spatial dimension N = 2 Ball center: 1.0000 2.0000 Ball radius = 2.00000 Ball volume = 12.5664 Rule: F1 F3 F(X) 1 12.56637061 12.56637061 X 12.56637061 12.56637061 X^2 25.13274123 25.13274123 X^3 50.26548246 50.26548246 X^4 113.09733553 113.09733553 X^5 263.89378290 263.89378290 X^6 636.86886292 628.31853072 R 30.98915741 31.10874897 SIN(X) 6.10243156 6.11161086 EXP(X) 54.31680198 54.28168577 1/(1+R) 3.99684876 3.92547444 SQRT(R) 19.27715636 19.37848813 Spatial dimension N = 3 Ball center: 1.0000 2.0000 3.0000 Ball radius = 2.00000 Ball volume = 33.5103 Rule: F1 F3 F(X) 1 33.51032164 33.51032164 X 33.51032164 33.51032164 X^2 60.31857895 60.31857895 X^3 113.93509357 113.93509357 X^4 240.31687803 240.31687803 X^5 531.37795741 531.37795741 X^6 1224.69700910 1203.77281934 R 132.54855093 132.56195430 SIN(X) 18.41814625 18.44078084 EXP(X) 133.12388457 133.03858839 1/(1+R) 6.99449233 6.99027378 SQRT(R) 66.22012906 66.22794304 TEST02 For the integral of a monomial in a ball in ND: BALL_MONOMIAL_ND approximates the integral. BALL_F1_ND, which can handle general integrands, will be used for comparison. Spatial dimension N = 3 Ball radius = 2.00000 Ball volume = 33.5103 Rule: MONOMIAL F1 F(X) 1 33.51032164 33.51032164 xyz 0.00000000 0.00000000 x^2z^2 15.31900418 15.31900418 x^4y^2z^2 7.42739596 7.61010022 TEST03 For integrals in the unit ball in 3D: BALL_UNIT_07_3D uses a formula of degree 7; BALL_UNIT_14_3D uses a formula of degree 14; BALL_UNIT_15_3D uses a formula of degree 15. Unit ball volume = 4.18879 Rule: #7 #14 #15 F(X) 1 4.18879 4.18879 4.18879 X -0.145328E-16 0.170306E-16 -0.152708E-15 Y 0.326988E-16 0.454150E-18 -0.669603E-16 Z 0.00000 0.227075E-17 -0.908299E-18 X*X 0.837758 0.837758 0.837758 X*Y 0.413276E-16 -0.136245E-17 -0.401470E-16 X*Z 0.00000 -0.312228E-17 -0.158952E-17 Y*Y 0.837758 0.837758 0.837758 Y*Z 0.00000 0.193014E-17 -0.586315E-17 Z*Z 0.837758 0.837758 0.837758 X^3 -0.175415E-16 0.924975E-17 -0.102297E-15 X*Y*Z 0.00000 0.162713E-16 -0.928939E-19 Z*Z*Z 0.00000 0.00000 0.163494E-16 X^4 0.359039 0.359039 0.359039 X^2 Z^2 0.119680 0.119680 0.119680 Z^4 0.359039 0.359039 0.359039 X^5 0.485195E-17 -0.394005E-17 -0.688666E-16 X^6 0.199466 0.199466 0.199466 R 3.15453 3.14291 3.14291 SIN(X) 0.490482E-16 -0.131703E-16 -0.130000E-15 EXP(X) 4.62291 4.62291 4.62291 1/(1+R) 3.34715 3.34793 3.34793 SQRT(R) 3.60705 3.59269 3.59269 TEST04 For integrals inside the unit ball in ND: BALL_UNIT_F1_ND approximates the integral; BALL_UNIT_F3_ND approximates the integral. Spatial dimension N = 2 Unit ball volume = 3.14159 Rule: F1 F3 F(X) 1 3.14159265 3.14159265 X 0.00000000 0.00000000 X^2 0.78539816 0.78539816 X^3 0.00000000 0.00000000 X^4 0.39269908 0.39269908 X^5 0.00000000 0.00000000 X^6 0.22974928 0.19634954 R 2.07465287 1.92382475 SIN(X) 0.00000000 0.00000000 EXP(X) 3.55097673 3.55092935 1/(1+R) 1.94254245 2.08250740 SQRT(R) 2.49488479 2.12906206 Spatial dimension N = 3 Unit ball volume = 4.18879 Rule: F1 F3 F(X) 1 4.18879020 4.18879020 X 0.00000000 0.00000000 X^2 0.83775804 0.83775804 X^3 0.00000000 0.00000000 X^4 0.35903916 0.35903916 X^5 0.00000000 0.00000000 X^6 0.19474148 0.15387393 R 3.12358869 2.97374605 SIN(X) 0.00000000 0.00000000 EXP(X) 4.62290241 4.62284455 1/(1+R) 2.44032616 2.57713848 SQRT(R) 3.57254308 3.23471399 TEST045 In 3 dimensions: BALL_UNIT_VOLUME_3D gets the volume of the unit ball. BALL_UNIT_VOLUME_ND will be called for comparison. N Volume Method 3 4.18879020 BALL_UNIT_VOLUME_3D 3 4.18879020 BALL_UNIT_VOLUME_ND TEST05 BALL_UNIT_VOLUME_ND computes the volume of the unit ball in ND. N Volume 2 3.14159265 3 4.18879020 4 4.93480220 5 5.26378901 6 5.16771278 7 4.72476597 8 4.05871213 9 3.29850890 10 2.55016404 TEST052 In 3 dimensions: BALL_VOLUME_3D computes the volume of a unit ball. BALL_VOLUME_ND will be called for comparison. N R Volume Method 3 1.00000000 4.18879020 BALL_VOLUME_3D 3 1.00000000 4.18879020 BALL_VOLUME_ND 3 2.00000000 33.51032164 BALL_VOLUME_3D 3 2.00000000 33.51032164 BALL_VOLUME_ND 3 4.00000000 268.08257311 BALL_VOLUME_3D 3 4.00000000 268.08257311 BALL_VOLUME_ND TEST054 BALL_UNIT_VOLUME_ND computes the volume of the unit ball in N dimensions. N R Volume 2 0.50000000 0.78539816 2 1.00000000 3.14159265 2 2.00000000 12.56637061 3 0.50000000 0.52359878 3 1.00000000 4.18879020 3 2.00000000 33.51032164 4 0.50000000 0.30842514 4 1.00000000 4.93480220 4 2.00000000 78.95683521 5 0.50000000 0.16449341 5 1.00000000 5.26378901 5 2.00000000 168.44124845 6 0.50000000 0.08074551 6 1.00000000 5.16771278 6 2.00000000 330.73361792 7 0.50000000 0.03691223 7 1.00000000 4.72476597 7 2.00000000 604.77004420 8 0.50000000 0.01585434 8 1.00000000 4.05871213 8 2.00000000 1039.03030436 9 0.50000000 0.00644240 9 1.00000000 3.29850890 9 2.00000000 1688.83655820 10 0.50000000 0.00249039 10 1.00000000 2.55016404 10 2.00000000 2611.36797683 TEST07 CIRCLE_ANNULUS estimates integrals in a circular annulus. F CENTER Radius1 Radius2 NR Result Area 0.000000 0.000000 0.000000 1.000000 3.141593 1 0.000000 0.000000 0.000000 1.000000 1 3.141593 1 0.000000 0.000000 0.000000 1.000000 2 3.141593 1 0.000000 0.000000 0.000000 1.000000 3 3.141593 1 0.000000 0.000000 0.000000 1.000000 4 3.141593 X 0.000000 0.000000 0.000000 1.000000 1 -0.000000 X 0.000000 0.000000 0.000000 1.000000 2 -0.000000 X 0.000000 0.000000 0.000000 1.000000 3 -0.000000 X 0.000000 0.000000 0.000000 1.000000 4 -0.000000 X^2 0.000000 0.000000 0.000000 1.000000 1 0.785398 X^2 0.000000 0.000000 0.000000 1.000000 2 0.785398 X^2 0.000000 0.000000 0.000000 1.000000 3 0.785398 X^2 0.000000 0.000000 0.000000 1.000000 4 0.785398 X^3 0.000000 0.000000 0.000000 1.000000 1 -0.000000 X^3 0.000000 0.000000 0.000000 1.000000 2 -0.000000 X^3 0.000000 0.000000 0.000000 1.000000 3 -0.000000 X^3 0.000000 0.000000 0.000000 1.000000 4 -0.000000 X^4 0.000000 0.000000 0.000000 1.000000 1 0.392699 X^4 0.000000 0.000000 0.000000 1.000000 2 0.392699 X^4 0.000000 0.000000 0.000000 1.000000 3 0.392699 X^4 0.000000 0.000000 0.000000 1.000000 4 0.392699 X^5 0.000000 0.000000 0.000000 1.000000 1 -0.000000 X^5 0.000000 0.000000 0.000000 1.000000 2 -0.000000 X^5 0.000000 0.000000 0.000000 1.000000 3 -0.000000 X^5 0.000000 0.000000 0.000000 1.000000 4 -0.000000 X^6 0.000000 0.000000 0.000000 1.000000 1 0.196350 X^6 0.000000 0.000000 0.000000 1.000000 2 0.245437 X^6 0.000000 0.000000 0.000000 1.000000 3 0.245437 X^6 0.000000 0.000000 0.000000 1.000000 4 0.245437 R 0.000000 0.000000 0.000000 1.000000 1 2.221441 R 0.000000 0.000000 0.000000 1.000000 2 2.117080 R 0.000000 0.000000 0.000000 1.000000 3 2.102290 R 0.000000 0.000000 0.000000 1.000000 4 2.098042 SIN(X) 0.000000 0.000000 0.000000 1.000000 1 -0.000000 SIN(X) 0.000000 0.000000 0.000000 1.000000 2 -0.000000 SIN(X) 0.000000 0.000000 0.000000 1.000000 3 -0.000000 SIN(X) 0.000000 0.000000 0.000000 1.000000 4 -0.000000 EXP(X) 0.000000 0.000000 0.000000 1.000000 1 3.550929 EXP(X) 0.000000 0.000000 0.000000 1.000000 2 3.550999 EXP(X) 0.000000 0.000000 0.000000 1.000000 3 3.550999 EXP(X) 0.000000 0.000000 0.000000 1.000000 4 3.550999 1/(1+R) 0.000000 0.000000 0.000000 1.000000 1 1.840302 1/(1+R) 0.000000 0.000000 0.000000 1.000000 2 1.908065 1/(1+R) 0.000000 0.000000 0.000000 1.000000 3 1.920618 1/(1+R) 0.000000 0.000000 0.000000 1.000000 4 1.924507 SQRT(R) 0.000000 0.000000 0.000000 1.000000 1 2.641754 SQRT(R) 0.000000 0.000000 0.000000 1.000000 2 2.545301 SQRT(R) 0.000000 0.000000 0.000000 1.000000 3 2.526703 SQRT(R) 0.000000 0.000000 0.000000 1.000000 4 2.520359 Area 0.000000 0.000000 1.000000 2.000000 9.424778 1 0.000000 0.000000 1.000000 2.000000 1 9.424778 1 0.000000 0.000000 1.000000 2.000000 2 9.424778 1 0.000000 0.000000 1.000000 2.000000 3 9.424778 1 0.000000 0.000000 1.000000 2.000000 4 9.424778 X 0.000000 0.000000 1.000000 2.000000 1 -0.000000 X 0.000000 0.000000 1.000000 2.000000 2 -0.000000 X 0.000000 0.000000 1.000000 2.000000 3 -0.000000 X 0.000000 0.000000 1.000000 2.000000 4 -0.000000 X^2 0.000000 0.000000 1.000000 2.000000 1 11.780972 X^2 0.000000 0.000000 1.000000 2.000000 2 11.780972 X^2 0.000000 0.000000 1.000000 2.000000 3 11.780972 X^2 0.000000 0.000000 1.000000 2.000000 4 11.780972 X^3 0.000000 0.000000 1.000000 2.000000 1 -0.000000 X^3 0.000000 0.000000 1.000000 2.000000 2 -0.000000 X^3 0.000000 0.000000 1.000000 2.000000 3 -0.000000 X^3 0.000000 0.000000 1.000000 2.000000 4 -0.000000 X^4 0.000000 0.000000 1.000000 2.000000 1 29.452431 X^4 0.000000 0.000000 1.000000 2.000000 2 24.740042 X^4 0.000000 0.000000 1.000000 2.000000 3 24.740042 X^4 0.000000 0.000000 1.000000 2.000000 4 24.740042 X^5 0.000000 0.000000 1.000000 2.000000 1 -0.000000 X^5 0.000000 0.000000 1.000000 2.000000 2 -0.000000 X^5 0.000000 0.000000 1.000000 2.000000 3 -0.000000 X^5 0.000000 0.000000 1.000000 2.000000 4 -0.000000 X^6 0.000000 0.000000 1.000000 2.000000 1 73.631078 X^6 0.000000 0.000000 1.000000 2.000000 2 62.586416 X^6 0.000000 0.000000 1.000000 2.000000 3 62.586416 X^6 0.000000 0.000000 1.000000 2.000000 4 62.586416 R 0.000000 0.000000 1.000000 2.000000 1 14.901882 R 0.000000 0.000000 1.000000 2.000000 2 14.669400 R 0.000000 0.000000 1.000000 2.000000 3 14.661276 R 0.000000 0.000000 1.000000 2.000000 4 14.660802 SIN(X) 0.000000 0.000000 1.000000 2.000000 1 -0.000000 SIN(X) 0.000000 0.000000 1.000000 2.000000 2 -0.000000 SIN(X) 0.000000 0.000000 1.000000 2.000000 3 -0.000000 SIN(X) 0.000000 0.000000 1.000000 2.000000 4 -0.000000 EXP(X) 0.000000 0.000000 1.000000 2.000000 1 16.649409 EXP(X) 0.000000 0.000000 1.000000 2.000000 2 16.437636 EXP(X) 0.000000 0.000000 1.000000 2.000000 3 16.437533 EXP(X) 0.000000 0.000000 1.000000 2.000000 4 16.437533 1/(1+R) 0.000000 0.000000 1.000000 2.000000 1 3.651403 1/(1+R) 0.000000 0.000000 1.000000 2.000000 2 3.730817 1/(1+R) 0.000000 0.000000 1.000000 2.000000 3 3.735238 1/(1+R) 0.000000 0.000000 1.000000 2.000000 4 3.735547 SQRT(R) 0.000000 0.000000 1.000000 2.000000 1 11.851031 SQRT(R) 0.000000 0.000000 1.000000 2.000000 2 11.710792 SQRT(R) 0.000000 0.000000 1.000000 2.000000 3 11.704411 SQRT(R) 0.000000 0.000000 1.000000 2.000000 4 11.703987 TEST08 CIRCLE_ANNULUS estimates integrals in a circular annulus. CIRCLE_RT_SET sets up a rule for a circle; CIRCLE_RT_SUM applies the rule. RESULT1 = CIRCLE_ANNULUS result. RESULT2 = Difference of two CIRCLE_RT_SUM results. F CENTER Radius1 Radius2 Result1 Result2 Area 0.00000 0.00000 0.00000 1.00000 3.14159 1 0.00000 0.00000 0.00000 1.00000 3.14159 3.14159 X 0.00000 0.00000 0.00000 1.00000 -0.00000 -0.00000 X^2 0.00000 0.00000 0.00000 1.00000 0.78540 0.78540 X^3 0.00000 0.00000 0.00000 1.00000 -0.00000 -0.00000 X^4 0.00000 0.00000 0.00000 1.00000 0.39270 0.39270 X^5 0.00000 0.00000 0.00000 1.00000 -0.00000 -0.00000 X^6 0.00000 0.00000 0.00000 1.00000 0.24544 0.24544 R 0.00000 0.00000 0.00000 1.00000 2.09637 2.09637 SIN(X) 0.00000 0.00000 0.00000 1.00000 -0.00000 -0.00000 EXP(X) 0.00000 0.00000 0.00000 1.00000 3.55100 3.55100 1/(1+R) 0.00000 0.00000 0.00000 1.00000 1.92609 1.92609 SQRT(R) 0.00000 0.00000 0.00000 1.00000 2.51754 2.51754 Area 0.00000 0.00000 1.00000 2.00000 9.42478 1 0.00000 0.00000 1.00000 2.00000 9.42478 9.42478 X 0.00000 0.00000 1.00000 2.00000 -0.00000 -0.00000 X^2 0.00000 0.00000 1.00000 2.00000 11.78097 11.78097 X^3 0.00000 0.00000 1.00000 2.00000 -0.00000 -0.00000 X^4 0.00000 0.00000 1.00000 2.00000 24.74004 24.74004 X^5 0.00000 0.00000 1.00000 2.00000 -0.00000 -0.00000 X^6 0.00000 0.00000 1.00000 2.00000 62.58642 62.58642 R 0.00000 0.00000 1.00000 2.00000 14.66077 14.67462 SIN(X) 0.00000 0.00000 1.00000 2.00000 -0.00000 0.00000 EXP(X) 0.00000 0.00000 1.00000 2.00000 16.43753 16.43753 1/(1+R) 0.00000 0.00000 1.00000 2.00000 3.73557 3.72308 SQRT(R) 0.00000 0.00000 1.00000 2.00000 11.70395 11.72382 Area 0.00000 0.00000 1.00000 3.00000 25.13274 1 0.00000 0.00000 1.00000 3.00000 25.13274 25.13274 X 0.00000 0.00000 1.00000 3.00000 -0.00000 -0.00000 X^2 0.00000 0.00000 1.00000 3.00000 62.83185 62.83185 X^3 0.00000 0.00000 1.00000 3.00000 -0.00000 -0.00000 X^4 0.00000 0.00000 1.00000 3.00000 285.88493 285.88493 X^5 0.00000 0.00000 1.00000 3.00000 -0.00000 -0.00000 X^6 0.00000 0.00000 1.00000 3.00000 1610.06623 1610.06623 R 0.00000 0.00000 1.00000 3.00000 54.45475 54.50574 SIN(X) 0.00000 0.00000 1.00000 3.00000 0.00000 -0.00000 EXP(X) 0.00000 0.00000 1.00000 3.00000 70.96827 70.96827 1/(1+R) 0.00000 0.00000 1.00000 3.00000 8.21087 8.16864 SQRT(R) 0.00000 0.00000 1.00000 3.00000 36.66524 36.72703 TEST085 CIRCLE_ANNULUS_AREA_2D computes the area of a circular annulus. CENTER Radius1 Radius2 Area 0.00000 0.00000 0.00000 1.00000 3.14159 1.00000 0.00000 1.00000 2.00000 9.42478 3.00000 4.00000 1.00000 3.00000 25.13274 TEST09 CIRCLE_ANNULUS_SECTOR estimates an integral in a circular annulus sector. CIRCLE_RT_SET sets an integration rule in a circle. CIRCLE_RT_SUM uses an integration rule in a circle. To test CIRCLE_ANNULUS_SECTOR, we estimate an integral over 4 annular sectors that make up the unit circle, and add to get RESULT1. We will also estimate the integral over the unit circle using CIRCLE_RT_SET and CIRCLE_RT_SUM to get RESULT2. We will then compare RESULT1 and RESULT2. CIRCLE_ANNULUS_SECTOR computations will use NR = 5 CIRCLE_RT_SET/CIRCLE_RT_SUM will use rule 9 "RESULT1" is the sum of Annulus Sector calculations. "RESULT2" is for CIRCLE_RT_SET/CIRCLE_RT_SUM. F Result1 Result2 1 3.14159 3.14159 X -0.727989E-03 -0.272490E-17 X^2 0.784623 0.785398 X^3 -0.661095E-03 -0.435984E-16 X^4 0.392181 0.392699 X^5 -0.390120E-03 -0.245241E-16 X^6 0.245151 0.245437 R 2.09452 2.09637 SIN(X) -0.621017E-03 -0.136245E-16 EXP(X) 3.54975 3.55100 1/(1+R) 1.92790 1.92609 SQRT(R) 2.51356 2.51754 TEST10 CIRCLE_CUM approximates an integral over a circle. We use radius R = 3.00000 and center: CENTER = 0.00000 0.00000 Order: 2 4 8 16 F(X) 1 28.27433388 28.27433388 28.27433388 28.27433388 X 0.00000000 -0.00000000 -0.00000000 -0.00000000 X^2 254.46900494 127.23450247 127.23450247 127.23450247 X^3 0.00000000 0.00000000 0.00000000 -0.00000000 X^4 2290.22104447 1145.11052223 858.83289168 858.83289168 X^5 0.00000000 0.00000000 -0.00000000 -0.00000000 X^620611.9894002010305.99470010 6441.24668756 6441.24668756 R 84.82300165 84.82300165 84.82300165 84.82300165 SIN(X) 0.00000000 -0.00000000 -0.00000000 -0.00000000 EXP(X) 284.65643668 156.46538528 138.04717297 138.00115919 1/(1+R) 7.06858347 7.06858347 7.06858347 7.06858347 SQRT(R) 48.97258283 48.97258283 48.97258283 48.97258283 TEST11 LENS_HALF_AREA_2D computes the area of a circular half lens, defined by joining the endpoints of a circular arc. CIRCLE_SECTOR_AREA_2D computes the area of a circular sector, defined by joining the endpoints of a circular arc to the center. CIRCLE_TRIANGLE_AREA_2D computes the signed area of a triangle, defined by joining the endpoints of a circular arc and the center. R Theta1 Theta2 Sector Triangle Half Lens 1.000 0.000 0.00000000 0.00000000 0.00000000 0.00000000 1.000 0.000 0.52359878 0.26179939 0.25000000 0.01179939 1.000 0.000 1.04719755 0.52359878 0.43301270 0.09058607 1.000 0.000 1.57079633 0.78539816 0.50000000 0.28539816 1.000 0.000 2.09439510 1.04719755 0.43301270 0.61418485 1.000 0.000 2.61799388 1.30899694 0.25000000 1.05899694 1.000 0.000 3.14159265 1.57079633 0.00000000 1.57079633 1.000 0.000 3.66519143 1.83259571 -0.25000000 2.08259571 1.000 0.000 4.18879020 2.09439510 -0.43301270 2.52740780 1.000 0.000 4.71238898 2.35619449 -0.50000000 2.85619449 1.000 0.000 5.23598776 2.61799388 -0.43301270 3.05100658 1.000 0.000 5.75958653 2.87979327 -0.25000000 3.12979327 1.000 0.000 6.28318531 3.14159265 -0.00000000 3.14159265 TEST12 For the area of a circular half lens, LENS_HALF_AREA_2D uses two angles; LENS_HALF_H_AREA_2D works from the height; LENS_HALF_W_AREA_2D works from the width. The circle has radius R = 50.0000 THETA1 THETA2 H W Area(THETA) Area(H) Area(W) 0.00 0.00 0.00 0.00 0.0000 0.0000 0.0000 0.00 0.52 1.70 25.88 29.4985 29.4985 29.4985 0.00 1.05 6.70 50.00 226.4652 226.4652 226.4652 0.00 1.57 14.64 70.71 713.4954 713.4954 713.4954 0.00 2.09 25.00 86.60 1535.4621 1535.4621 1535.4621 0.00 2.62 37.06 96.59 2647.4923 2647.4923 2647.4923 0.00 3.14 50.00100.00 3926.9908 3926.9908 3926.9908 0.00 3.67 62.94 96.59 5206.4893 5206.4893 2647.4923 0.00 4.19 75.00 86.60 6318.5195 6318.5195 1535.4621 0.00 4.71 85.36 70.71 7140.4862 7140.4862 713.4954 0.00 5.24 93.30 50.00 7627.5164 7627.5164 226.4652 0.00 5.76 98.30 25.88 7824.4832 7824.4832 29.4985 0.00 6.28100.00 0.00 7853.9816 7853.9816 0.0000 TEST13 CIRCLE_SECTOR_AREA_2D computes the area of a circular sector. CIRCLE_SECTOR estimates an integral in a circular sector. The user can specify NR, the number of radial values used to approximated the integral. In this test, computations will use values of NR from 1 to 5 CENTER RADIUS THETA1 THETA2 Area 0.0000 0.0000 1.0000 0.0000 6.2832 3.1416 F 1 2 3 4 5 1 3.14159 3.14159 3.14159 3.14159 3.14159 X -0.871967E-16 -0.174393E-15 -0.185293E-15 -0.708473E-16 -0.599477E-16 X^2 0.785398 0.785398 0.785398 0.785398 0.785398 X^3 -0.653975E-16 -0.174393E-15 -0.130795E-15 -0.103546E-15 -0.299739E-16 X^4 0.196350 0.392699 0.392699 0.392699 0.392699 X^5 -0.272490E-16 -0.163494E-15 -0.708473E-16 -0.817469E-16 -0.108996E-16 X^6 0.490874E-01 0.245437 0.245437 0.245437 0.245437 R 2.22144 2.11708 2.10229 2.09804 2.09637 SIN(X) -0.871967E-16 -0.174393E-15 -0.152594E-15 -0.326988E-16 -0.326988E-16 EXP(X) 3.54254 3.55100 3.55100 3.55100 3.55100 1/(1+R) 1.84030 1.90807 1.92062 1.92451 1.92609 SQRT(R) 2.64175 2.54530 2.52670 2.52036 2.51754 CENTER RADIUS THETA1 THETA2 Area 0.0000 0.0000 2.0000 0.0000 3.1416 6.2832 F 1 2 3 4 5 1 6.28319 6.28319 6.28319 6.28319 6.28319 X 0.00000 0.523180E-15 0.435984E-15 0.479582E-15 0.621277E-15 X^2 6.28319 6.28319 6.28319 6.28319 6.28319 X^3 0.00000 0.348787E-15 0.156954E-14 0.261590E-15 0.479582E-15 X^4 9.42478 12.5664 12.5664 12.5664 12.5664 X^5 0.00000 -0.279029E-14 0.139515E-14 0.418544E-14 0.697574E-15 X^6 15.7080 31.4159 31.4159 31.4159 31.4159 R 8.88577 8.46832 8.40916 8.39217 8.38550 SIN(X) 0.00000 0.348787E-15 0.326988E-15 0.392385E-15 -0.817469E-16 EXP(X) 9.83997 9.99420 9.99427 9.99427 9.99427 1/(1+R) 2.60258 2.76839 2.80611 2.81904 2.82458 SQRT(R) 7.47201 7.19920 7.14660 7.12865 7.12068 CENTER RADIUS THETA1 THETA2 Area 0.0000 0.0000 4.0000 0.0000 1.5708 12.5664 F 1 2 3 4 5 1 12.5664 12.5664 12.5664 12.5664 12.5664 X 22.7735 21.5991 21.4290 21.3791 21.3590 X^2 50.2655 50.2655 50.2655 50.2655 50.2655 X^3 120.647 136.113 136.469 136.516 136.527 X^4 301.593 402.124 402.124 402.124 402.124 X^5 772.373 1251.56 1248.48 1248.33 1248.31 X^6 2010.62 4021.24 4021.24 4021.24 4021.24 R 35.5431 33.8733 33.6366 33.5687 33.5420 SIN(X) 8.14853 7.09324 6.82950 6.77009 6.74800 EXP(X) 103.916 119.054 119.019 118.976 118.957 1/(1+R) 3.28238 3.59355 3.68088 3.71511 3.73122 SQRT(R) 21.1340 20.3624 20.2136 20.1629 20.1403 CENTER RADIUS THETA1 THETA2 Area 0.0000 0.0000 8.0000 0.0000 0.7854 25.1327 F 1 2 3 4 5 1 25.1327 25.1327 25.1327 25.1327 25.1327 X 128.206 122.036 121.156 120.902 120.801 X^2 659.776 658.536 658.307 658.227 658.190 X^3 3423.26 3852.70 3861.18 3861.96 3862.02 X^4 17895.8 23808.2 23798.4 23795.0 23793.4 X^5 94198.4 152324. 151890. 151851. 151839. X^6 498915. 995947. 995596. 995474. 995417. R 142.172 135.493 134.547 134.275 134.168 SIN(X) -20.5913 -0.613778 -1.77204 -2.55066 -2.72853 EXP(X) 4580.19 9276.31 9550.42 9552.77 9552.25 1/(1+R) 3.77547 4.23702 4.39005 4.45814 4.49356 SQRT(R) 59.7761 57.5936 57.1728 57.0292 56.9654 TEST14 CIRCLE_SECTOR estimates integrals in a circular sector. CIRCLE_RT_SET sets an integration rule in a circle. CIRCLE_RT_SUM uses an integration rule in a circle. To test CIRCLE_SECTOR, we estimate an integral over a sector, and over its complement and add the results to get RESULT1. We also estimate the integral over the whole circle using CIRCLE_RT_SET and CIRCLE_RT_SUM to get RESULT2. We will then compare RESULT1 and RESULT2. CIRCLE_SECTOR computations will use NR = 5 CIRCLE_RT_SET/CIRCLE_RT_SUM will use rule 9 "Sector1" and "Sector2" are the CIRCLE_SECTOR computations for the sector and its complement. "Sum" is the sum of Sector1 and Sector2. "Circle" is the computation for CIRCLE_RT_SET + CIRCLE_RT_SUM. CENTER RADIUS THETA1 THETA2 Area1 Area2 Circle 0.0000 0.0000 1.0000 0.0000 6.2832 3.1416 0.0000 3.1416 F Sector1 Sector2 Sum Circle 1 3.14159 0.00000 3.14159 3.14159 X -0.599477E-16 0.00000 -0.599477E-16 -0.272490E-17 X^2 0.785398 0.00000 0.785398 0.785398 X^3 -0.299739E-16 0.00000 -0.299739E-16 -0.435984E-16 X^4 0.392699 0.00000 0.392699 0.392699 X^5 -0.108996E-16 0.00000 -0.108996E-16 -0.245241E-16 X^6 0.245437 0.00000 0.245437 0.245437 R 2.09637 0.00000 2.09637 2.09637 SIN(X) -0.326988E-16 0.00000 -0.326988E-16 -0.136245E-16 EXP(X) 3.55100 0.00000 3.55100 3.55100 1/(1+R) 1.92609 0.00000 1.92609 1.92609 SQRT(R) 2.51754 0.00000 2.51754 2.51754 CENTER RADIUS THETA1 THETA2 Area1 Area2 Circle 0.0000 0.0000 2.0000 0.0000 3.1416 6.2832 6.2832 12.5664 F Sector1 Sector2 Sum Circle 1 6.28319 6.28319 12.5664 12.5664 X 0.621277E-15 -0.120985E-14 -0.588578E-15 -0.217992E-16 X^2 6.28319 6.28319 12.5664 12.5664 X^3 0.479582E-15 -0.274670E-14 -0.226711E-14 -0.139515E-14 X^4 12.5664 12.5664 25.1327 25.1327 X^5 0.697574E-15 -0.101148E-13 -0.941724E-14 -0.313908E-14 X^6 31.4159 31.4159 62.8319 62.8319 R 8.38550 8.38550 16.7710 16.7710 SIN(X) -0.817469E-16 -0.664875E-15 -0.746622E-15 0.675775E-15 EXP(X) 9.99427 9.99427 19.9885 19.9885 1/(1+R) 2.82458 2.82458 5.64917 5.64917 SQRT(R) 7.12068 7.12068 14.2414 14.2414 CENTER RADIUS THETA1 THETA2 Area1 Area2 Circle 0.0000 0.0000 4.0000 0.0000 1.5708 12.5664 37.6991 50.2655 F Sector1 Sector2 Sum Circle 1 12.5664 37.6991 50.2655 50.2655 X 21.3590 -21.4030 -0.439855E-01 -0.174393E-15 X^2 50.2655 150.796 201.062 201.062 X^3 136.527 -136.523 0.461370E-02 -0.446447E-13 X^4 402.124 1206.37 1608.50 1608.50 X^5 1248.31 -1248.31 -0.161715E-02 -0.401802E-12 X^6 4021.24 12063.7 16085.0 16085.0 R 33.5420 100.626 134.168 134.168 SIN(X) 6.74800 -6.79277 -0.447682E-01 -0.159134E-14 EXP(X) 118.957 126.281 245.239 245.282 1/(1+R) 3.73122 11.1936 14.9249 14.9249 SQRT(R) 20.1403 60.4210 80.5613 80.5613 CENTER RADIUS THETA1 THETA2 Area1 Area2 Circle 0.0000 0.0000 8.0000 0.0000 0.7854 25.1327 175.9292 201.0619 F Sector1 Sector2 Sum Circle 1 25.1327 175.929 201.062 201.062 X 120.801 -121.175 -0.373397 -0.139515E-14 X^2 658.190 2555.61 3213.80 3216.99 X^3 3862.02 -3883.71 -21.6832 -0.142863E-11 X^4 23793.4 79014.3 102808. 102944. X^5 151839. -152658. -818.194 -0.514307E-10 X^6 995417. 0.311753E+07 0.411295E+07 0.411775E+07 R 134.168 939.176 1073.34 1073.34 SIN(X) -2.72853 2.77012 0.415870E-01 -0.174393E-14 EXP(X) 9552.25 10508.4 20060.7 20099.8 1/(1+R) 4.49356 31.4549 35.9485 35.9485 SQRT(R) 56.9654 398.758 455.723 455.723 TEST15 For R, Theta product rules on the unit circle, CIRCLE_RT_SET sets a rule. CIRCLE_RT_SUM uses the rule in an arbitrary circle. We use a radius 1.00000 and center: CENTER = 1.00000 1.00000 Rule: 1 2 3 4 5 Function 1 3.14159265 3.14159265 3.14159265 3.14159265 3.14159265 X 3.14159265 3.14159265 3.14159265 3.14159265 3.14159265 X^2 3.14159265 3.53429174 3.92699082 3.92699082 3.92699082 X^3 3.14159265 4.31968990 5.49778714 5.49778714 5.49778714 X^4 3.14159265 5.54687453 8.63937980 8.24668072 8.37758041 X^5 3.14159265 7.31402040 14.92256510 12.95906970 13.61356817 X^6 3.14159265 9.77452558 27.48893572 21.05303410 23.16924582 R 4.44288294 4.57763596 4.76304332 4.72695942 4.75068677 SIN(X) 2.64355906 2.48005053 2.33974956 2.32660549 2.33102004 EXP(X) 8.53973422 9.07905055 9.69917529 9.65259543 9.66800959 1/(1+R) 1.30129028 1.30812266 1.28604560 1.30752303 1.29310149 SQRT(R) 3.73600434 3.76141787 3.82784452 3.79755167 3.81787212 Rule: 6 7 8 9 Function 1 3.14159265 3.14159265 3.14159265 3.14159265 X 3.14159265 3.14159265 3.14159265 3.14159265 X^2 3.92699082 3.92699082 3.92699082 3.92699082 X^3 5.49778714 5.49778714 5.49778714 5.49778714 X^4 8.37758041 8.24668072 8.24668072 8.24668072 X^5 13.61356817 12.95906970 12.95906970 12.95906970 X^6 23.16924582 21.05848826 21.05848826 21.05848826 R 4.75071817 4.72699935 4.72698493 4.72700022 SIN(X) 2.33101961 2.32659943 2.32659943 2.32659943 EXP(X) 9.66800810 9.65261708 9.65261708 9.65261708 1/(1+R) 1.29306754 1.30748659 1.30749937 1.30748567 SQRT(R) 3.81791399 3.79755407 3.79753411 3.79755516 TEST16 CIRCLE_XY_SET sets a quadrature rule for the unit circle. CIRCLE_XY_SUM evaluates the quadrature rule in an arbitrary circle. We use a radius 1.00000 and center: CENTER = 1.00000 1.00000 Rule: 1 2 3 4 5 Function 1 3.14159265 3.14159265 3.14159265 3.14159265 3.14159265 X 3.14159265 3.14159265 3.14159265 3.14159265 3.14159265 X^2 3.14159265 3.53429174 3.92699082 4.71238898 3.92699082 X^3 3.14159265 4.31968990 5.49778714 7.85398163 5.49778714 X^4 3.14159265 5.59596191 8.05033117 13.35176878 8.63937980 X^5 3.14159265 7.55945732 11.97732199 22.77654674 14.92256510 X^6 3.14159265 10.52924413 17.91689560 38.87720909 27.48893572 R 4.44288294 4.58799703 4.70508854 4.94214052 4.76304332 SIN(X) 2.64355906 2.48175020 2.31994134 2.00975150 2.33974956 EXP(X) 8.53973422 9.08468013 9.62962605 10.76511925 9.69917529 1/(1+R) 1.30129028 1.30207255 1.32026645 1.36034952 1.28604560 SQRT(R) 3.73600434 3.76998355 3.77952335 3.79309320 3.82784452 Rule: 6 7 8 9 10 Function 1 3.14159265 3.14159265 3.14159265 3.14159265 3.14159265 X 3.14159265 3.14159265 3.14159265 3.14159265 3.14159265 X^2 3.92699082 3.92699082 3.92699082 3.92699082 3.92699082 X^3 5.49778714 5.49778714 5.49778714 5.49778714 5.49778714 X^4 8.24668072 8.24668072 8.24668072 8.24668072 8.24668072 X^5 12.95906970 12.95906970 12.95906970 12.95906970 12.95906970 X^6 21.05303410 21.10757564 21.00940087 21.05848826 21.05848826 R 4.72695942 4.72440680 4.72691668 4.72582845 4.72769882 SIN(X) 2.32660549 2.32654408 2.32665528 2.32659933 2.32659918 EXP(X) 9.65259543 9.65280913 9.65242673 9.65261676 9.65261624 1/(1+R) 1.30752303 1.30885950 1.30784480 1.30840608 1.30693595 SQRT(R) 3.79755167 3.79563041 3.79709944 3.79626392 3.79831648 Rule: 11 12 13 Function 1 3.14159265 3.14159265 3.14159265 X 3.14159265 3.14159265 3.14159265 X^2 3.92699082 3.92699082 3.92699082 X^3 5.49778714 5.49778714 5.49778714 X^4 8.24668072 8.24668072 8.24668072 X^5 12.95906970 12.95906970 12.95906970 X^6 21.05848826 21.05848826 21.05848826 R 4.72699935 4.72704100 4.72698493 SIN(X) 2.32659943 2.32659943 2.32659943 EXP(X) 9.65261708 9.65261708 9.65261708 1/(1+R) 1.30748659 1.30744489 1.30749937 SQRT(R) 3.79755407 3.79761812 3.79753411 TEST163 Demonstrate the use of quadrature rules for the region CN_GEG, that is, the hypercube [-1,+1]^N, with the weight W(ALPHA;X) = product ( 1 <= I <= N ) 7771-X(I)^2)^ALPHA We use the formulas to integrate various monomials of the form X(1)^E(1) * X(2)^E(2) * ... X(N)^E(N) and compare to the exact integral. The precision of each formula is known, and we only use a formula if its precision indicates it should be able to produce an exact result. N = 1 ALPHA = -0.500000 EXPON = 0 Degree = 0 CN_GEG_00_1: 1 3.14159 0.00000 CN_GEG_01_1: 1 3.14159 0.00000 CN_GEG_02_XIU: 2 3.14159 0.00000 GW_02_XIU: 2 3.14159 0.399680E-14 CN_GEG_03_XIU: 2 3.14159 0.00000 EXACT: 3.14159 N = 1 ALPHA = 0.00000 EXPON = 0 Degree = 0 CN_GEG_00_1: 1 2.00000 0.00000 CN_GEG_01_1: 1 2.00000 0.00000 CN_GEG_02_XIU: 2 2.00000 0.00000 GW_02_XIU: 2 2.00000 0.222045E-15 CN_GEG_03_XIU: 2 2.00000 0.00000 EXACT: 2.00000 N = 1 ALPHA = 0.500000 EXPON = 0 Degree = 0 CN_GEG_00_1: 1 1.57080 0.00000 CN_GEG_01_1: 1 1.57080 0.00000 CN_GEG_02_XIU: 2 1.57080 0.00000 GW_02_XIU: 2 1.57080 0.133227E-14 CN_GEG_03_XIU: 2 1.57080 0.00000 EXACT: 1.57080 N = 1 ALPHA = 1.00000 EXPON = 0 Degree = 0 CN_GEG_00_1: 1 1.33333 0.00000 CN_GEG_01_1: 1 1.33333 0.00000 CN_GEG_02_XIU: 2 1.33333 0.00000 GW_02_XIU: 2 1.33333 0.00000 CN_GEG_03_XIU: 2 1.33333 0.00000 EXACT: 1.33333 N = 1 ALPHA = 1.50000 EXPON = 0 Degree = 0 CN_GEG_00_1: 1 1.17810 0.00000 CN_GEG_01_1: 1 1.17810 0.00000 CN_GEG_02_XIU: 2 1.17810 0.00000 GW_02_XIU: 2 1.17810 0.888178E-15 CN_GEG_03_XIU: 2 1.17810 0.00000 EXACT: 1.17810 N = 1 ALPHA = -0.500000 EXPON = 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 2 0.00000 0.00000 GW_02_XIU: 2 0.00000 0.00000 CN_GEG_03_XIU: 2 0.00000 0.00000 EXACT: 0.00000 N = 1 ALPHA = 0.00000 EXPON = 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 2 0.00000 0.00000 GW_02_XIU: 2 0.00000 0.00000 CN_GEG_03_XIU: 2 0.00000 0.00000 EXACT: 0.00000 N = 1 ALPHA = 0.500000 EXPON = 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 2 0.00000 0.00000 GW_02_XIU: 2 0.00000 0.00000 CN_GEG_03_XIU: 2 0.00000 0.00000 EXACT: 0.00000 N = 1 ALPHA = 1.00000 EXPON = 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 2 0.00000 0.00000 GW_02_XIU: 2 0.00000 0.00000 CN_GEG_03_XIU: 2 0.00000 0.00000 EXACT: 0.00000 N = 1 ALPHA = 1.50000 EXPON = 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 2 0.00000 0.00000 GW_02_XIU: 2 0.00000 0.00000 CN_GEG_03_XIU: 2 0.00000 0.00000 EXACT: 0.00000 N = 1 ALPHA = -0.500000 EXPON = 2 Degree = 2 CN_GEG_02_XIU: 2 1.57080 0.666134E-15 GW_02_XIU: 2 1.57080 0.133227E-14 CN_GEG_03_XIU: 2 1.57080 0.155431E-14 EXACT: 1.57080 N = 1 ALPHA = 0.00000 EXPON = 2 Degree = 2 CN_GEG_02_XIU: 2 0.666667 0.00000 GW_02_XIU: 2 0.666667 0.111022E-15 CN_GEG_03_XIU: 2 0.666667 0.222045E-15 EXACT: 0.666667 N = 1 ALPHA = 0.500000 EXPON = 2 Degree = 2 CN_GEG_02_XIU: 2 0.392699 0.277556E-15 GW_02_XIU: 2 0.392699 0.555112E-16 CN_GEG_03_XIU: 2 0.392699 0.277556E-15 EXACT: 0.392699 N = 1 ALPHA = 1.00000 EXPON = 2 Degree = 2 CN_GEG_02_XIU: 2 0.266667 0.555112E-16 GW_02_XIU: 2 0.266667 0.555112E-16 CN_GEG_03_XIU: 2 0.266667 0.555112E-16 EXACT: 0.266667 N = 1 ALPHA = 1.50000 EXPON = 2 Degree = 2 CN_GEG_02_XIU: 2 0.196350 0.555112E-16 GW_02_XIU: 2 0.196350 0.111022E-15 CN_GEG_03_XIU: 2 0.196350 0.555112E-16 EXACT: 0.196350 N = 2 ALPHA = -0.500000 EXPON = 0 0 Degree = 0 CN_GEG_00_1: 1 9.86960 0.00000 CN_GEG_01_1: 1 9.86960 0.00000 CN_GEG_02_XIU: 3 9.86960 0.00000 GW_02_XIU: 3 9.86960 0.248690E-13 CN_GEG_03_XIU: 4 9.86960 0.00000 EXACT: 9.86960 N = 2 ALPHA = 0.00000 EXPON = 0 0 Degree = 0 CN_GEG_00_1: 1 4.00000 0.00000 CN_GEG_01_1: 1 4.00000 0.00000 CN_GEG_02_XIU: 3 4.00000 0.00000 GW_02_XIU: 3 4.00000 0.888178E-15 CN_GEG_03_XIU: 4 4.00000 0.00000 EXACT: 4.00000 N = 2 ALPHA = 0.500000 EXPON = 0 0 Degree = 0 CN_GEG_00_1: 1 2.46740 0.00000 CN_GEG_01_1: 1 2.46740 0.00000 CN_GEG_02_XIU: 3 2.46740 0.00000 GW_02_XIU: 3 2.46740 0.399680E-14 CN_GEG_03_XIU: 4 2.46740 0.00000 EXACT: 2.46740 N = 2 ALPHA = 1.00000 EXPON = 0 0 Degree = 0 CN_GEG_00_1: 1 1.77778 0.00000 CN_GEG_01_1: 1 1.77778 0.00000 CN_GEG_02_XIU: 3 1.77778 0.00000 GW_02_XIU: 3 1.77778 0.00000 CN_GEG_03_XIU: 4 1.77778 0.00000 EXACT: 1.77778 N = 2 ALPHA = 1.50000 EXPON = 0 0 Degree = 0 CN_GEG_00_1: 1 1.38791 0.00000 CN_GEG_01_1: 1 1.38791 0.00000 CN_GEG_02_XIU: 3 1.38791 0.00000 GW_02_XIU: 3 1.38791 0.222045E-14 CN_GEG_03_XIU: 4 1.38791 0.00000 EXACT: 1.38791 N = 2 ALPHA = -0.500000 EXPON = 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 3 0.888178E-15 0.888178E-15 GW_02_XIU: 3 0.133227E-14 0.133227E-14 CN_GEG_03_XIU: 4 -0.160250E-15 0.160250E-15 EXACT: 0.00000 N = 2 ALPHA = 0.00000 EXPON = 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 3 0.444089E-15 0.444089E-15 GW_02_XIU: 3 0.444089E-15 0.444089E-15 CN_GEG_03_XIU: 4 -0.889617E-16 0.889617E-16 EXACT: 0.00000 N = 2 ALPHA = 0.500000 EXPON = 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 3 0.111022E-15 0.111022E-15 GW_02_XIU: 3 0.222045E-15 0.222045E-15 CN_GEG_03_XIU: 4 -0.513219E-16 0.513219E-16 EXACT: 0.00000 N = 2 ALPHA = 1.00000 EXPON = 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 3 0.166533E-15 0.166533E-15 GW_02_XIU: 3 0.166533E-15 0.166533E-15 CN_GEG_03_XIU: 4 -0.133364E-16 0.133364E-16 EXACT: 0.00000 N = 2 ALPHA = 1.50000 EXPON = 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 3 0.555112E-16 0.555112E-16 GW_02_XIU: 3 0.555112E-16 0.555112E-16 CN_GEG_03_XIU: 4 -0.213106E-16 0.213106E-16 EXACT: 0.00000 N = 2 ALPHA = -0.500000 EXPON = 1 1 Degree = 2 CN_GEG_02_XIU: 3 0.155431E-14 0.155431E-14 GW_02_XIU: 3 0.177636E-14 0.177636E-14 CN_GEG_03_XIU: 4 -0.302169E-15 0.302169E-15 EXACT: 0.00000 N = 2 ALPHA = 0.00000 EXPON = 1 1 Degree = 2 CN_GEG_02_XIU: 3 0.277556E-15 0.277556E-15 GW_02_XIU: 3 0.333067E-15 0.333067E-15 CN_GEG_03_XIU: 4 -0.816431E-16 0.816431E-16 EXACT: 0.00000 N = 2 ALPHA = 0.500000 EXPON = 1 1 Degree = 2 CN_GEG_02_XIU: 3 0.194289E-15 0.194289E-15 GW_02_XIU: 3 0.166533E-15 0.166533E-15 CN_GEG_03_XIU: 4 -0.377712E-16 0.377712E-16 EXACT: 0.00000 N = 2 ALPHA = 1.00000 EXPON = 1 1 Degree = 2 CN_GEG_02_XIU: 3 0.971445E-16 0.971445E-16 GW_02_XIU: 3 0.971445E-16 0.971445E-16 CN_GEG_03_XIU: 4 -0.217715E-16 0.217715E-16 EXACT: 0.00000 N = 2 ALPHA = 1.50000 EXPON = 1 1 Degree = 2 CN_GEG_02_XIU: 3 0.832667E-16 0.832667E-16 GW_02_XIU: 3 0.971445E-16 0.971445E-16 CN_GEG_03_XIU: 4 -0.141642E-16 0.141642E-16 EXACT: 0.00000 N = 2 ALPHA = -0.500000 EXPON = 2 0 Degree = 2 CN_GEG_02_XIU: 3 4.93480 0.888178E-15 GW_02_XIU: 3 4.93480 0.115463E-13 CN_GEG_03_XIU: 4 4.93480 0.355271E-14 EXACT: 4.93480 N = 2 ALPHA = 0.00000 EXPON = 2 0 Degree = 2 CN_GEG_02_XIU: 3 1.33333 0.222045E-15 GW_02_XIU: 3 1.33333 0.00000 CN_GEG_03_XIU: 4 1.33333 0.444089E-15 EXACT: 1.33333 N = 2 ALPHA = 0.500000 EXPON = 2 0 Degree = 2 CN_GEG_02_XIU: 3 0.616850 0.777156E-15 GW_02_XIU: 3 0.616850 0.333067E-15 CN_GEG_03_XIU: 4 0.616850 0.555112E-15 EXACT: 0.616850 N = 2 ALPHA = 1.00000 EXPON = 2 0 Degree = 2 CN_GEG_02_XIU: 3 0.355556 0.555112E-16 GW_02_XIU: 3 0.355556 0.555112E-16 CN_GEG_03_XIU: 4 0.355556 0.555112E-16 EXACT: 0.355556 N = 2 ALPHA = 1.50000 EXPON = 2 0 Degree = 2 CN_GEG_02_XIU: 3 0.231319 0.277556E-16 GW_02_XIU: 3 0.231319 0.388578E-15 CN_GEG_03_XIU: 4 0.231319 0.00000 EXACT: 0.231319 N = 3 ALPHA = -0.500000 EXPON = 0 0 0 Degree = 0 CN_GEG_00_1: 1 31.0063 0.00000 CN_GEG_01_1: 1 31.0063 0.00000 CN_GEG_02_XIU: 4 31.0063 0.00000 GW_02_XIU: 4 31.0063 0.117240E-12 CN_GEG_03_XIU: 6 31.0063 0.00000 EXACT: 31.0063 N = 3 ALPHA = 0.00000 EXPON = 0 0 0 Degree = 0 CN_GEG_00_1: 1 8.00000 0.00000 CN_GEG_01_1: 1 8.00000 0.00000 CN_GEG_02_XIU: 4 8.00000 0.00000 GW_02_XIU: 4 8.00000 0.266454E-14 CN_GEG_03_XIU: 6 8.00000 0.888178E-15 EXACT: 8.00000 N = 3 ALPHA = 0.500000 EXPON = 0 0 0 Degree = 0 CN_GEG_00_1: 1 3.87578 0.00000 CN_GEG_01_1: 1 3.87578 0.00000 CN_GEG_02_XIU: 4 3.87578 0.00000 GW_02_XIU: 4 3.87578 0.976996E-14 CN_GEG_03_XIU: 6 3.87578 0.00000 EXACT: 3.87578 N = 3 ALPHA = 1.00000 EXPON = 0 0 0 Degree = 0 CN_GEG_00_1: 1 2.37037 0.00000 CN_GEG_01_1: 1 2.37037 0.00000 CN_GEG_02_XIU: 4 2.37037 0.00000 GW_02_XIU: 4 2.37037 0.00000 CN_GEG_03_XIU: 6 2.37037 0.00000 EXACT: 2.37037 N = 3 ALPHA = 1.50000 EXPON = 0 0 0 Degree = 0 CN_GEG_00_1: 1 1.63510 0.00000 CN_GEG_01_1: 1 1.63510 0.00000 CN_GEG_02_XIU: 4 1.63510 0.00000 GW_02_XIU: 4 1.63510 0.377476E-14 CN_GEG_03_XIU: 6 1.63510 0.222045E-15 EXACT: 1.63510 N = 3 ALPHA = -0.500000 EXPON = 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 4 0.00000 0.00000 GW_02_XIU: 4 0.00000 0.00000 CN_GEG_03_XIU: 6 0.00000 0.00000 EXACT: 0.00000 N = 3 ALPHA = 0.00000 EXPON = 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 4 0.00000 0.00000 GW_02_XIU: 4 0.00000 0.00000 CN_GEG_03_XIU: 6 0.00000 0.00000 EXACT: 0.00000 N = 3 ALPHA = 0.500000 EXPON = 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 4 0.00000 0.00000 GW_02_XIU: 4 0.00000 0.00000 CN_GEG_03_XIU: 6 0.00000 0.00000 EXACT: 0.00000 N = 3 ALPHA = 1.00000 EXPON = 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 4 0.00000 0.00000 GW_02_XIU: 4 0.00000 0.00000 CN_GEG_03_XIU: 6 0.00000 0.00000 EXACT: 0.00000 N = 3 ALPHA = 1.50000 EXPON = 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 4 0.00000 0.00000 GW_02_XIU: 4 0.00000 0.00000 CN_GEG_03_XIU: 6 0.00000 0.00000 EXACT: 0.00000 N = 3 ALPHA = -0.500000 EXPON = 1 1 0 Degree = 2 CN_GEG_02_XIU: 4 0.949293E-15 0.949293E-15 GW_02_XIU: 4 0.949293E-15 0.949293E-15 CN_GEG_03_XIU: 6 0.665430E-16 0.665430E-16 EXACT: 0.00000 N = 3 ALPHA = 0.00000 EXPON = 1 1 0 Degree = 2 CN_GEG_02_XIU: 4 0.163286E-15 0.163286E-15 GW_02_XIU: 4 0.163286E-15 0.163286E-15 CN_GEG_03_XIU: 6 0.115352E-15 0.115352E-15 EXACT: 0.00000 N = 3 ALPHA = 0.500000 EXPON = 1 1 0 Degree = 2 CN_GEG_02_XIU: 4 0.593308E-16 0.593308E-16 GW_02_XIU: 4 0.593308E-16 0.593308E-16 CN_GEG_03_XIU: 6 0.319145E-16 0.319145E-16 EXACT: 0.00000 N = 3 ALPHA = 1.00000 EXPON = 1 1 0 Degree = 2 CN_GEG_02_XIU: 4 0.290287E-16 0.290287E-16 GW_02_XIU: 4 0.290287E-16 0.290287E-16 CN_GEG_03_XIU: 6 0.292848E-17 0.292848E-17 EXACT: 0.00000 N = 3 ALPHA = 1.50000 EXPON = 1 1 0 Degree = 2 CN_GEG_02_XIU: 4 0.166868E-16 0.166868E-16 GW_02_XIU: 4 0.166868E-16 0.166868E-16 CN_GEG_03_XIU: 6 -0.143238E-17 0.143238E-17 EXACT: 0.00000 N = 3 ALPHA = -0.500000 EXPON = 2 0 0 Degree = 2 CN_GEG_02_XIU: 4 15.5031 0.355271E-14 GW_02_XIU: 4 15.5031 0.550671E-13 CN_GEG_03_XIU: 6 15.5031 0.710543E-14 EXACT: 15.5031 N = 3 ALPHA = 0.00000 EXPON = 2 0 0 Degree = 2 CN_GEG_02_XIU: 4 2.66667 0.00000 GW_02_XIU: 4 2.66667 0.888178E-15 CN_GEG_03_XIU: 6 2.66667 0.133227E-14 EXACT: 2.66667 N = 3 ALPHA = 0.500000 EXPON = 2 0 0 Degree = 2 CN_GEG_02_XIU: 4 0.968946 0.999201E-15 GW_02_XIU: 4 0.968946 0.144329E-14 CN_GEG_03_XIU: 6 0.968946 0.122125E-14 EXACT: 0.968946 N = 3 ALPHA = 1.00000 EXPON = 2 0 0 Degree = 2 CN_GEG_02_XIU: 4 0.474074 0.111022E-15 GW_02_XIU: 4 0.474074 0.111022E-15 CN_GEG_03_XIU: 6 0.474074 0.111022E-15 EXACT: 0.474074 N = 3 ALPHA = 1.50000 EXPON = 2 0 0 Degree = 2 CN_GEG_02_XIU: 4 0.272516 0.00000 GW_02_XIU: 4 0.272516 0.666134E-15 CN_GEG_03_XIU: 6 0.272516 0.555112E-16 EXACT: 0.272516 N = 4 ALPHA = -0.500000 EXPON = 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 97.4091 0.00000 CN_GEG_01_1: 1 97.4091 0.00000 CN_GEG_02_XIU: 5 97.4091 0.00000 GW_02_XIU: 5 97.4091 0.497380E-12 CN_GEG_03_XIU: 8 97.4091 0.142109E-13 EXACT: 97.4091 N = 4 ALPHA = 0.00000 EXPON = 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 16.0000 0.00000 CN_GEG_01_1: 1 16.0000 0.00000 CN_GEG_02_XIU: 5 16.0000 0.00000 GW_02_XIU: 5 16.0000 0.710543E-14 CN_GEG_03_XIU: 8 16.0000 0.00000 EXACT: 16.0000 N = 4 ALPHA = 0.500000 EXPON = 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 6.08807 0.00000 CN_GEG_01_1: 1 6.08807 0.00000 CN_GEG_02_XIU: 5 6.08807 0.888178E-15 GW_02_XIU: 5 6.08807 0.195399E-13 CN_GEG_03_XIU: 8 6.08807 0.888178E-15 EXACT: 6.08807 N = 4 ALPHA = 1.00000 EXPON = 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 3.16049 0.00000 CN_GEG_01_1: 1 3.16049 0.00000 CN_GEG_02_XIU: 5 3.16049 0.00000 GW_02_XIU: 5 3.16049 0.00000 CN_GEG_03_XIU: 8 3.16049 0.00000 EXACT: 3.16049 N = 4 ALPHA = 1.50000 EXPON = 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 1.92630 0.222045E-15 CN_GEG_01_1: 1 1.92630 0.222045E-15 CN_GEG_02_XIU: 5 1.92630 0.222045E-15 GW_02_XIU: 5 1.92630 0.577316E-14 CN_GEG_03_XIU: 8 1.92630 0.444089E-15 EXACT: 1.92630 N = 4 ALPHA = -0.500000 EXPON = 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 5 0.710543E-14 0.710543E-14 GW_02_XIU: 5 0.710543E-14 0.710543E-14 CN_GEG_03_XIU: 8 -0.178287E-13 0.178287E-13 EXACT: 0.00000 N = 4 ALPHA = 0.00000 EXPON = 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 5 0.111022E-14 0.111022E-14 GW_02_XIU: 5 0.111022E-14 0.111022E-14 CN_GEG_03_XIU: 8 -0.253217E-14 0.253217E-14 EXACT: 0.00000 N = 4 ALPHA = 0.500000 EXPON = 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 5 0.333067E-15 0.333067E-15 GW_02_XIU: 5 0.333067E-15 0.333067E-15 CN_GEG_03_XIU: 8 -0.783978E-15 0.783978E-15 EXACT: 0.00000 N = 4 ALPHA = 1.00000 EXPON = 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 5 0.138778E-15 0.138778E-15 GW_02_XIU: 5 0.138778E-15 0.138778E-15 CN_GEG_03_XIU: 8 -0.377882E-15 0.377882E-15 EXACT: 0.00000 N = 4 ALPHA = 1.50000 EXPON = 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 5 0.832667E-16 0.832667E-16 GW_02_XIU: 5 0.832667E-16 0.832667E-16 CN_GEG_03_XIU: 8 -0.213172E-15 0.213172E-15 EXACT: 0.00000 N = 4 ALPHA = -0.500000 EXPON = 1 1 0 0 Degree = 2 CN_GEG_02_XIU: 5 0.444089E-14 0.444089E-14 GW_02_XIU: 5 0.444089E-14 0.444089E-14 CN_GEG_03_XIU: 8 0.570420E-15 0.570420E-15 EXACT: 0.00000 N = 4 ALPHA = 0.00000 EXPON = 1 1 0 0 Degree = 2 CN_GEG_02_XIU: 5 0.444089E-15 0.444089E-15 GW_02_XIU: 5 0.333067E-15 0.333067E-15 CN_GEG_03_XIU: 8 0.117517E-15 0.117517E-15 EXACT: 0.00000 N = 4 ALPHA = 0.500000 EXPON = 1 1 0 0 Degree = 2 CN_GEG_02_XIU: 5 0.194289E-15 0.194289E-15 GW_02_XIU: 5 0.194289E-15 0.194289E-15 CN_GEG_03_XIU: 8 0.178256E-16 0.178256E-16 EXACT: 0.00000 N = 4 ALPHA = 1.00000 EXPON = 1 1 0 0 Degree = 2 CN_GEG_02_XIU: 5 0.693889E-16 0.693889E-16 GW_02_XIU: 5 0.693889E-16 0.693889E-16 CN_GEG_03_XIU: 8 0.168063E-16 0.168063E-16 EXACT: 0.00000 N = 4 ALPHA = 1.50000 EXPON = 1 1 0 0 Degree = 2 CN_GEG_02_XIU: 5 0.485723E-16 0.485723E-16 GW_02_XIU: 5 0.485723E-16 0.485723E-16 CN_GEG_03_XIU: 8 0.809690E-17 0.809690E-17 EXACT: 0.00000 N = 4 ALPHA = -0.500000 EXPON = 2 0 0 0 Degree = 2 CN_GEG_02_XIU: 5 48.7045 0.142109E-13 GW_02_XIU: 5 48.7045 0.234479E-12 CN_GEG_03_XIU: 8 48.7045 0.284217E-13 EXACT: 48.7045 N = 4 ALPHA = 0.00000 EXPON = 2 0 0 0 Degree = 2 CN_GEG_02_XIU: 5 5.33333 0.00000 GW_02_XIU: 5 5.33333 0.266454E-14 CN_GEG_03_XIU: 8 5.33333 0.177636E-14 EXACT: 5.33333 N = 4 ALPHA = 0.500000 EXPON = 2 0 0 0 Degree = 2 CN_GEG_02_XIU: 5 1.52202 0.111022E-14 GW_02_XIU: 5 1.52202 0.355271E-14 CN_GEG_03_XIU: 8 1.52202 0.133227E-14 EXACT: 1.52202 N = 4 ALPHA = 1.00000 EXPON = 2 0 0 0 Degree = 2 CN_GEG_02_XIU: 5 0.632099 0.111022E-15 GW_02_XIU: 5 0.632099 0.111022E-15 CN_GEG_03_XIU: 8 0.632099 0.111022E-15 EXACT: 0.632099 N = 4 ALPHA = 1.50000 EXPON = 2 0 0 0 Degree = 2 CN_GEG_02_XIU: 5 0.321050 0.00000 GW_02_XIU: 5 0.321050 0.999201E-15 CN_GEG_03_XIU: 8 0.321050 0.00000 EXACT: 0.321050 N = 5 ALPHA = -0.500000 EXPON = 0 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 306.020 0.00000 CN_GEG_01_1: 1 306.020 0.00000 CN_GEG_02_XIU: 6 306.020 0.00000 GW_02_XIU: 6 306.020 0.193268E-11 CN_GEG_03_XIU: 10 306.020 0.00000 EXACT: 306.020 N = 5 ALPHA = 0.00000 EXPON = 0 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 32.0000 0.00000 CN_GEG_01_1: 1 32.0000 0.00000 CN_GEG_02_XIU: 6 32.0000 0.355271E-14 GW_02_XIU: 6 32.0000 0.213163E-13 CN_GEG_03_XIU: 10 32.0000 0.355271E-14 EXACT: 32.0000 N = 5 ALPHA = 0.500000 EXPON = 0 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 9.56312 0.00000 CN_GEG_01_1: 1 9.56312 0.00000 CN_GEG_02_XIU: 6 9.56312 0.00000 GW_02_XIU: 6 9.56312 0.390799E-13 CN_GEG_03_XIU: 10 9.56312 0.00000 EXACT: 9.56312 N = 5 ALPHA = 1.00000 EXPON = 0 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 4.21399 0.00000 CN_GEG_01_1: 1 4.21399 0.00000 CN_GEG_02_XIU: 6 4.21399 0.00000 GW_02_XIU: 6 4.21399 0.00000 CN_GEG_03_XIU: 10 4.21399 0.888178E-15 EXACT: 4.21399 N = 5 ALPHA = 1.50000 EXPON = 0 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 2.26937 0.444089E-15 CN_GEG_01_1: 1 2.26937 0.444089E-15 CN_GEG_02_XIU: 6 2.26937 0.444089E-15 GW_02_XIU: 6 2.26937 0.888178E-14 CN_GEG_03_XIU: 10 2.26937 0.00000 EXACT: 2.26937 N = 5 ALPHA = -0.500000 EXPON = 0 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 6 0.00000 0.00000 GW_02_XIU: 6 0.00000 0.00000 CN_GEG_03_XIU: 10 0.00000 0.00000 EXACT: 0.00000 N = 5 ALPHA = 0.00000 EXPON = 0 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 6 0.00000 0.00000 GW_02_XIU: 6 0.00000 0.00000 CN_GEG_03_XIU: 10 0.00000 0.00000 EXACT: 0.00000 N = 5 ALPHA = 0.500000 EXPON = 0 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 6 0.00000 0.00000 GW_02_XIU: 6 0.00000 0.00000 CN_GEG_03_XIU: 10 0.00000 0.00000 EXACT: 0.00000 N = 5 ALPHA = 1.00000 EXPON = 0 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 6 0.00000 0.00000 GW_02_XIU: 6 0.00000 0.00000 CN_GEG_03_XIU: 10 0.00000 0.00000 EXACT: 0.00000 N = 5 ALPHA = 1.50000 EXPON = 0 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 6 0.00000 0.00000 GW_02_XIU: 6 0.00000 0.00000 CN_GEG_03_XIU: 10 0.00000 0.00000 EXACT: 0.00000 N = 5 ALPHA = -0.500000 EXPON = 1 1 0 0 0 Degree = 2 CN_GEG_02_XIU: 6 0.177636E-13 0.177636E-13 GW_02_XIU: 6 0.177636E-13 0.177636E-13 CN_GEG_03_XIU: 10 -0.389893E-15 0.389893E-15 EXACT: 0.00000 N = 5 ALPHA = 0.00000 EXPON = 1 1 0 0 0 Degree = 2 CN_GEG_02_XIU: 6 0.666134E-15 0.666134E-15 GW_02_XIU: 6 0.666134E-15 0.666134E-15 CN_GEG_03_XIU: 10 0.143618E-15 0.143618E-15 EXACT: 0.00000 N = 5 ALPHA = 0.500000 EXPON = 1 1 0 0 0 Degree = 2 CN_GEG_02_XIU: 6 0.222045E-15 0.222045E-15 GW_02_XIU: 6 0.222045E-15 0.222045E-15 CN_GEG_03_XIU: 10 -0.609208E-17 0.609208E-17 EXACT: 0.00000 N = 5 ALPHA = 1.00000 EXPON = 1 1 0 0 0 Degree = 2 CN_GEG_02_XIU: 6 0.832667E-16 0.832667E-16 GW_02_XIU: 6 0.832667E-16 0.832667E-16 CN_GEG_03_XIU: 10 -0.135296E-16 0.135296E-16 EXACT: 0.00000 N = 5 ALPHA = 1.50000 EXPON = 1 1 0 0 0 Degree = 2 CN_GEG_02_XIU: 6 0.624500E-16 0.624500E-16 GW_02_XIU: 6 0.555112E-16 0.555112E-16 CN_GEG_03_XIU: 10 0.228882E-17 0.228882E-17 EXACT: 0.00000 N = 5 ALPHA = -0.500000 EXPON = 2 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 6 153.010 0.00000 GW_02_XIU: 6 153.010 0.966338E-12 CN_GEG_03_XIU: 10 153.010 0.852651E-13 EXACT: 153.010 N = 5 ALPHA = 0.00000 EXPON = 2 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 6 10.6667 0.355271E-14 GW_02_XIU: 6 10.6667 0.532907E-14 CN_GEG_03_XIU: 10 10.6667 0.532907E-14 EXACT: 10.6667 N = 5 ALPHA = 0.500000 EXPON = 2 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 6 2.39078 0.266454E-14 GW_02_XIU: 6 2.39078 0.754952E-14 CN_GEG_03_XIU: 10 2.39078 0.222045E-14 EXACT: 2.39078 N = 5 ALPHA = 1.00000 EXPON = 2 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 6 0.842798 0.222045E-15 GW_02_XIU: 6 0.842798 0.222045E-15 CN_GEG_03_XIU: 10 0.842798 0.222045E-15 EXACT: 0.842798 N = 5 ALPHA = 1.50000 EXPON = 2 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 6 0.378229 0.00000 GW_02_XIU: 6 0.378229 0.166533E-14 CN_GEG_03_XIU: 10 0.378229 0.00000 EXACT: 0.378229 N = 6 ALPHA = -0.500000 EXPON = 0 0 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 961.389 0.00000 CN_GEG_01_1: 1 961.389 0.00000 CN_GEG_02_XIU: 7 961.389 0.113687E-12 GW_02_XIU: 7 961.389 0.727596E-11 CN_GEG_03_XIU: 12 961.389 0.113687E-12 EXACT: 961.389 N = 6 ALPHA = 0.00000 EXPON = 0 0 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 64.0000 0.00000 CN_GEG_01_1: 1 64.0000 0.00000 CN_GEG_02_XIU: 7 64.0000 0.142109E-13 GW_02_XIU: 7 64.0000 0.355271E-13 CN_GEG_03_XIU: 12 64.0000 0.00000 EXACT: 64.0000 N = 6 ALPHA = 0.500000 EXPON = 0 0 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 15.0217 0.177636E-14 CN_GEG_01_1: 1 15.0217 0.177636E-14 CN_GEG_02_XIU: 7 15.0217 0.00000 GW_02_XIU: 7 15.0217 0.746070E-13 CN_GEG_03_XIU: 12 15.0217 0.177636E-14 EXACT: 15.0217 N = 6 ALPHA = 1.00000 EXPON = 0 0 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 5.61866 0.888178E-15 CN_GEG_01_1: 1 5.61866 0.888178E-15 CN_GEG_02_XIU: 7 5.61866 0.00000 GW_02_XIU: 7 5.61866 0.00000 CN_GEG_03_XIU: 12 5.61866 0.888178E-15 EXACT: 5.61866 N = 6 ALPHA = 1.50000 EXPON = 0 0 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 2.67354 0.444089E-15 CN_GEG_01_1: 1 2.67354 0.444089E-15 CN_GEG_02_XIU: 7 2.67354 0.00000 GW_02_XIU: 7 2.67354 0.128786E-13 CN_GEG_03_XIU: 12 2.67354 0.00000 EXACT: 2.67354 N = 6 ALPHA = -0.500000 EXPON = 0 0 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 7 -0.355271E-13 0.355271E-13 GW_02_XIU: 7 -0.639488E-13 0.639488E-13 CN_GEG_03_XIU: 12 -0.261950E-12 0.261950E-12 EXACT: 0.00000 N = 6 ALPHA = 0.00000 EXPON = 0 0 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 7 -0.355271E-14 0.355271E-14 GW_02_XIU: 7 -0.444089E-14 0.444089E-14 CN_GEG_03_XIU: 12 -0.141424E-13 0.141424E-13 EXACT: 0.00000 N = 6 ALPHA = 0.500000 EXPON = 0 0 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 7 -0.999201E-15 0.999201E-15 GW_02_XIU: 7 -0.666134E-15 0.666134E-15 CN_GEG_03_XIU: 12 -0.272994E-14 0.272994E-14 EXACT: 0.00000 N = 6 ALPHA = 1.00000 EXPON = 0 0 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 7 -0.249800E-15 0.249800E-15 GW_02_XIU: 7 -0.249800E-15 0.249800E-15 CN_GEG_03_XIU: 12 -0.942871E-15 0.942871E-15 EXACT: 0.00000 N = 6 ALPHA = 1.50000 EXPON = 0 0 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.00000 0.00000 CN_GEG_02_XIU: 7 -0.124900E-15 0.124900E-15 GW_02_XIU: 7 -0.111022E-15 0.111022E-15 CN_GEG_03_XIU: 12 -0.420227E-15 0.420227E-15 EXACT: 0.00000 N = 6 ALPHA = -0.500000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 0.426326E-13 0.426326E-13 GW_02_XIU: 7 0.426326E-13 0.426326E-13 CN_GEG_03_XIU: 12 -0.196227E-13 0.196227E-13 EXACT: 0.00000 N = 6 ALPHA = 0.00000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 0.222045E-14 0.222045E-14 GW_02_XIU: 7 0.222045E-14 0.222045E-14 CN_GEG_03_XIU: 12 -0.648815E-15 0.648815E-15 EXACT: 0.00000 N = 6 ALPHA = 0.500000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 0.222045E-15 0.222045E-15 GW_02_XIU: 7 0.333067E-15 0.333067E-15 CN_GEG_03_XIU: 12 -0.422801E-16 0.422801E-16 EXACT: 0.00000 N = 6 ALPHA = 1.00000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 0.555112E-16 0.555112E-16 GW_02_XIU: 7 0.555112E-16 0.555112E-16 CN_GEG_03_XIU: 12 -0.458725E-16 0.458725E-16 EXACT: 0.00000 N = 6 ALPHA = 1.50000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 0.277556E-16 0.277556E-16 GW_02_XIU: 7 0.208167E-16 0.208167E-16 CN_GEG_03_XIU: 12 -0.112508E-16 0.112508E-16 EXACT: 0.00000 N = 6 ALPHA = -0.500000 EXPON = 2 0 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 480.695 0.170530E-12 GW_02_XIU: 7 480.695 0.346745E-11 CN_GEG_03_XIU: 12 480.695 0.227374E-12 EXACT: 480.695 N = 6 ALPHA = 0.00000 EXPON = 2 0 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 21.3333 0.00000 GW_02_XIU: 7 21.3333 0.106581E-13 CN_GEG_03_XIU: 12 21.3333 0.710543E-14 EXACT: 21.3333 N = 6 ALPHA = 0.500000 EXPON = 2 0 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 3.75543 0.266454E-14 GW_02_XIU: 7 3.75543 0.159872E-13 CN_GEG_03_XIU: 12 3.75543 0.266454E-14 EXACT: 3.75543 N = 6 ALPHA = 1.00000 EXPON = 2 0 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 1.12373 0.222045E-15 GW_02_XIU: 7 1.12373 0.222045E-15 CN_GEG_03_XIU: 12 1.12373 0.444089E-15 EXACT: 1.12373 N = 6 ALPHA = 1.50000 EXPON = 2 0 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 0.445590 0.00000 GW_02_XIU: 7 0.445590 0.222045E-14 CN_GEG_03_XIU: 12 0.445590 0.111022E-15 EXACT: 0.445590 TEST165 Demonstrate the use of quadrature rules for the region CN_JAC, that is, the hypercube [-1,+1]^N, with the weight W(ALPHA,BETA;X) = product ( 1 <= I <= N ) (1-X(I))^ALPHA (1+X(I))^BETA We use the formulas to integrate various monomials of the form X(1)^E(1) * X(2)^E(2) * ... X(N)^E(N) and compare to the exact integral. The precision of each formula is known, and we only use a formula if its precision indicates it should be able to produce an exact result. N = 1 ALPHA = 0.00000 BETA = 0.00000 EXPON = 0 Degree = 0 CN_JAC_00_1: 1 2.00000 0.00000 CN_JAC_01_1: 1 2.00000 0.00000 CN_JAC_02_XIU: 2 2.00000 0.00000 GW_02_XIU: 2 2.00000 0.00000 EXACT: 2.00000 N = 1 ALPHA = 1.00000 BETA = 0.00000 EXPON = 0 Degree = 0 CN_JAC_00_1: 1 2.00000 0.00000 CN_JAC_01_1: 1 2.00000 0.00000 CN_JAC_02_XIU: 2 2.00000 0.00000 GW_02_XIU: 2 2.00000 0.00000 EXACT: 2.00000 N = 1 ALPHA = 0.00000 BETA = 2.00000 EXPON = 0 Degree = 0 CN_JAC_00_1: 1 2.66667 0.00000 CN_JAC_01_1: 1 2.66667 0.00000 CN_JAC_02_XIU: 2 2.66667 0.00000 GW_02_XIU: 2 2.66667 0.444089E-15 EXACT: 2.66667 N = 1 ALPHA = 0.500000 BETA = 1.50000 EXPON = 0 Degree = 0 CN_JAC_00_1: 1 1.57080 0.00000 CN_JAC_01_1: 1 1.57080 0.00000 CN_JAC_02_XIU: 2 1.57080 0.00000 GW_02_XIU: 2 1.57080 0.444089E-15 EXACT: 1.57080 N = 1 ALPHA = 0.00000 BETA = 0.00000 EXPON = 1 Degree = 1 CN_JAC_01_1: 1 0.00000 0.00000 CN_JAC_02_XIU: 2 0.00000 0.00000 GW_02_XIU: 2 0.00000 0.00000 EXACT: 0.00000 N = 1 ALPHA = 1.00000 BETA = 0.00000 EXPON = 1 Degree = 1 CN_JAC_01_1: 1 -0.666667 0.00000 CN_JAC_02_XIU: 2 -0.666667 0.111022E-15 GW_02_XIU: 2 -0.666667 0.111022E-15 EXACT: -0.666667 N = 1 ALPHA = 0.00000 BETA = 2.00000 EXPON = 1 Degree = 1 CN_JAC_01_1: 1 1.33333 0.00000 CN_JAC_02_XIU: 2 1.33333 0.222045E-15 GW_02_XIU: 2 1.33333 0.00000 EXACT: 1.33333 N = 1 ALPHA = 0.500000 BETA = 1.50000 EXPON = 1 Degree = 1 CN_JAC_01_1: 1 0.392699 0.555112E-16 CN_JAC_02_XIU: 2 0.392699 0.555112E-16 GW_02_XIU: 2 0.392699 0.00000 EXACT: 0.392699 N = 1 ALPHA = 0.00000 BETA = 0.00000 EXPON = 2 Degree = 2 CN_JAC_02_XIU: 2 0.666667 0.00000 GW_02_XIU: 2 0.666667 0.00000 EXACT: 0.666667 N = 1 ALPHA = 1.00000 BETA = 0.00000 EXPON = 2 Degree = 2 CN_JAC_02_XIU: 2 0.666667 0.222045E-15 GW_02_XIU: 2 0.666667 0.222045E-15 EXACT: 0.666667 N = 1 ALPHA = 0.00000 BETA = 2.00000 EXPON = 2 Degree = 2 CN_JAC_02_XIU: 2 1.06667 0.00000 GW_02_XIU: 2 1.06667 0.222045E-15 EXACT: 1.06667 N = 1 ALPHA = 0.500000 BETA = 1.50000 EXPON = 2 Degree = 2 CN_JAC_02_XIU: 2 0.392699 0.00000 GW_02_XIU: 2 0.392699 0.555112E-16 EXACT: 0.392699 N = 2 ALPHA = 0.00000 BETA = 0.00000 EXPON = 0 0 Degree = 0 CN_JAC_00_1: 1 4.00000 0.00000 CN_JAC_01_1: 1 4.00000 0.00000 CN_JAC_02_XIU: 3 4.00000 0.00000 GW_02_XIU: 3 4.00000 0.00000 EXACT: 4.00000 N = 2 ALPHA = 1.00000 BETA = 0.00000 EXPON = 0 0 Degree = 0 CN_JAC_00_1: 1 4.00000 0.00000 CN_JAC_01_1: 1 4.00000 0.00000 CN_JAC_02_XIU: 3 4.00000 0.00000 GW_02_XIU: 3 4.00000 0.00000 EXACT: 4.00000 N = 2 ALPHA = 0.00000 BETA = 2.00000 EXPON = 0 0 Degree = 0 CN_JAC_00_1: 1 7.11111 0.00000 CN_JAC_01_1: 1 7.11111 0.00000 CN_JAC_02_XIU: 3 7.11111 0.00000 GW_02_XIU: 3 7.11111 0.177636E-14 EXACT: 7.11111 N = 2 ALPHA = 0.500000 BETA = 1.50000 EXPON = 0 0 Degree = 0 CN_JAC_00_1: 1 2.46740 0.00000 CN_JAC_01_1: 1 2.46740 0.00000 CN_JAC_02_XIU: 3 2.46740 0.00000 GW_02_XIU: 3 2.46740 0.133227E-14 EXACT: 2.46740 N = 2 ALPHA = 0.00000 BETA = 0.00000 EXPON = 0 1 Degree = 1 CN_JAC_01_1: 1 0.00000 0.00000 CN_JAC_02_XIU: 3 0.444089E-15 0.444089E-15 GW_02_XIU: 3 0.444089E-15 0.444089E-15 EXACT: 0.00000 N = 2 ALPHA = 1.00000 BETA = 0.00000 EXPON = 0 1 Degree = 1 CN_JAC_01_1: 1 -1.33333 0.00000 CN_JAC_02_XIU: 3 -1.33333 0.666134E-15 GW_02_XIU: 3 -1.33333 0.666134E-15 EXACT: -1.33333 N = 2 ALPHA = 0.00000 BETA = 2.00000 EXPON = 0 1 Degree = 1 CN_JAC_01_1: 1 3.55556 0.00000 CN_JAC_02_XIU: 3 3.55556 0.444089E-15 GW_02_XIU: 3 3.55556 0.00000 EXACT: 3.55556 N = 2 ALPHA = 0.500000 BETA = 1.50000 EXPON = 0 1 Degree = 1 CN_JAC_01_1: 1 0.616850 0.00000 CN_JAC_02_XIU: 3 0.616850 0.111022E-15 GW_02_XIU: 3 0.616850 0.444089E-15 EXACT: 0.616850 N = 2 ALPHA = 0.00000 BETA = 0.00000 EXPON = 1 1 Degree = 2 CN_JAC_02_XIU: 3 0.277556E-15 0.277556E-15 GW_02_XIU: 3 0.277556E-15 0.277556E-15 EXACT: 0.00000 N = 2 ALPHA = 1.00000 BETA = 0.00000 EXPON = 1 1 Degree = 2 CN_JAC_02_XIU: 3 0.444444 0.555112E-16 GW_02_XIU: 3 0.444444 0.555112E-16 EXACT: 0.444444 N = 2 ALPHA = 0.00000 BETA = 2.00000 EXPON = 1 1 Degree = 2 CN_JAC_02_XIU: 3 1.77778 0.888178E-15 GW_02_XIU: 3 1.77778 0.222045E-15 EXACT: 1.77778 N = 2 ALPHA = 0.500000 BETA = 1.50000 EXPON = 1 1 Degree = 2 CN_JAC_02_XIU: 3 0.154213 0.138778E-15 GW_02_XIU: 3 0.154213 0.222045E-15 EXACT: 0.154213 N = 2 ALPHA = 0.00000 BETA = 0.00000 EXPON = 2 0 Degree = 2 CN_JAC_02_XIU: 3 1.33333 0.222045E-15 GW_02_XIU: 3 1.33333 0.222045E-15 EXACT: 1.33333 N = 2 ALPHA = 1.00000 BETA = 0.00000 EXPON = 2 0 Degree = 2 CN_JAC_02_XIU: 3 1.33333 0.222045E-15 GW_02_XIU: 3 1.33333 0.222045E-15 EXACT: 1.33333 N = 2 ALPHA = 0.00000 BETA = 2.00000 EXPON = 2 0 Degree = 2 CN_JAC_02_XIU: 3 2.84444 0.00000 GW_02_XIU: 3 2.84444 0.888178E-15 EXACT: 2.84444 N = 2 ALPHA = 0.500000 BETA = 1.50000 EXPON = 2 0 Degree = 2 CN_JAC_02_XIU: 3 0.616850 0.111022E-15 GW_02_XIU: 3 0.616850 0.444089E-15 EXACT: 0.616850 N = 3 ALPHA = 0.00000 BETA = 0.00000 EXPON = 0 0 0 Degree = 0 CN_JAC_00_1: 1 8.00000 0.00000 CN_JAC_01_1: 1 8.00000 0.00000 CN_JAC_02_XIU: 4 8.00000 0.00000 GW_02_XIU: 4 8.00000 0.00000 EXACT: 8.00000 N = 3 ALPHA = 1.00000 BETA = 0.00000 EXPON = 0 0 0 Degree = 0 CN_JAC_00_1: 1 8.00000 0.00000 CN_JAC_01_1: 1 8.00000 0.00000 CN_JAC_02_XIU: 4 8.00000 0.00000 GW_02_XIU: 4 8.00000 0.00000 EXACT: 8.00000 N = 3 ALPHA = 0.00000 BETA = 2.00000 EXPON = 0 0 0 Degree = 0 CN_JAC_00_1: 1 18.9630 0.00000 CN_JAC_01_1: 1 18.9630 0.00000 CN_JAC_02_XIU: 4 18.9630 0.00000 GW_02_XIU: 4 18.9630 0.710543E-14 EXACT: 18.9630 N = 3 ALPHA = 0.500000 BETA = 1.50000 EXPON = 0 0 0 Degree = 0 CN_JAC_00_1: 1 3.87578 0.00000 CN_JAC_01_1: 1 3.87578 0.00000 CN_JAC_02_XIU: 4 3.87578 0.00000 GW_02_XIU: 4 3.87578 0.310862E-14 EXACT: 3.87578 N = 3 ALPHA = 0.00000 BETA = 0.00000 EXPON = 0 0 1 Degree = 1 CN_JAC_01_1: 1 0.00000 0.00000 CN_JAC_02_XIU: 4 0.00000 0.00000 GW_02_XIU: 4 0.00000 0.00000 EXACT: 0.00000 N = 3 ALPHA = 1.00000 BETA = 0.00000 EXPON = 0 0 1 Degree = 1 CN_JAC_01_1: 1 -2.66667 0.00000 CN_JAC_02_XIU: 4 -2.66667 0.444089E-15 GW_02_XIU: 4 -2.66667 0.444089E-15 EXACT: -2.66667 N = 3 ALPHA = 0.00000 BETA = 2.00000 EXPON = 0 0 1 Degree = 1 CN_JAC_01_1: 1 9.48148 0.00000 CN_JAC_02_XIU: 4 9.48148 0.177636E-14 GW_02_XIU: 4 9.48148 0.177636E-14 EXACT: 9.48148 N = 3 ALPHA = 0.500000 BETA = 1.50000 EXPON = 0 0 1 Degree = 1 CN_JAC_01_1: 1 0.968946 0.111022E-15 CN_JAC_02_XIU: 4 0.968946 0.222045E-15 GW_02_XIU: 4 0.968946 0.777156E-15 EXACT: 0.968946 N = 3 ALPHA = 0.00000 BETA = 0.00000 EXPON = 1 1 0 Degree = 2 CN_JAC_02_XIU: 4 0.163286E-15 0.163286E-15 GW_02_XIU: 4 0.163286E-15 0.163286E-15 EXACT: 0.00000 N = 3 ALPHA = 1.00000 BETA = 0.00000 EXPON = 1 1 0 Degree = 2 CN_JAC_02_XIU: 4 0.888889 0.222045E-15 GW_02_XIU: 4 0.888889 0.222045E-15 EXACT: 0.888889 N = 3 ALPHA = 0.00000 BETA = 2.00000 EXPON = 1 1 0 Degree = 2 CN_JAC_02_XIU: 4 4.74074 0.177636E-14 GW_02_XIU: 4 4.74074 0.00000 EXACT: 4.74074 N = 3 ALPHA = 0.500000 BETA = 1.50000 EXPON = 1 1 0 Degree = 2 CN_JAC_02_XIU: 4 0.242237 0.555112E-16 GW_02_XIU: 4 0.242237 0.277556E-15 EXACT: 0.242237 N = 3 ALPHA = 0.00000 BETA = 0.00000 EXPON = 2 0 0 Degree = 2 CN_JAC_02_XIU: 4 2.66667 0.00000 GW_02_XIU: 4 2.66667 0.00000 EXACT: 2.66667 N = 3 ALPHA = 1.00000 BETA = 0.00000 EXPON = 2 0 0 Degree = 2 CN_JAC_02_XIU: 4 2.66667 0.888178E-15 GW_02_XIU: 4 2.66667 0.888178E-15 EXACT: 2.66667 N = 3 ALPHA = 0.00000 BETA = 2.00000 EXPON = 2 0 0 Degree = 2 CN_JAC_02_XIU: 4 7.58519 0.888178E-15 GW_02_XIU: 4 7.58519 0.355271E-14 EXACT: 7.58519 N = 3 ALPHA = 0.500000 BETA = 1.50000 EXPON = 2 0 0 Degree = 2 CN_JAC_02_XIU: 4 0.968946 0.111022E-15 GW_02_XIU: 4 0.968946 0.888178E-15 EXACT: 0.968946 N = 4 ALPHA = 0.00000 BETA = 0.00000 EXPON = 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 16.0000 0.00000 CN_JAC_01_1: 1 16.0000 0.00000 CN_JAC_02_XIU: 5 16.0000 0.00000 GW_02_XIU: 5 16.0000 0.00000 EXACT: 16.0000 N = 4 ALPHA = 1.00000 BETA = 0.00000 EXPON = 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 16.0000 0.00000 CN_JAC_01_1: 1 16.0000 0.00000 CN_JAC_02_XIU: 5 16.0000 0.00000 GW_02_XIU: 5 16.0000 0.00000 EXACT: 16.0000 N = 4 ALPHA = 0.00000 BETA = 2.00000 EXPON = 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 50.5679 0.00000 CN_JAC_01_1: 1 50.5679 0.00000 CN_JAC_02_XIU: 5 50.5679 0.00000 GW_02_XIU: 5 50.5679 0.213163E-13 EXACT: 50.5679 N = 4 ALPHA = 0.500000 BETA = 1.50000 EXPON = 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 6.08807 0.00000 CN_JAC_01_1: 1 6.08807 0.00000 CN_JAC_02_XIU: 5 6.08807 0.888178E-15 GW_02_XIU: 5 6.08807 0.710543E-14 EXACT: 6.08807 N = 4 ALPHA = 0.00000 BETA = 0.00000 EXPON = 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 0.00000 0.00000 CN_JAC_02_XIU: 5 0.111022E-14 0.111022E-14 GW_02_XIU: 5 0.111022E-14 0.111022E-14 EXACT: 0.00000 N = 4 ALPHA = 1.00000 BETA = 0.00000 EXPON = 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 -5.33333 0.00000 CN_JAC_02_XIU: 5 -5.33333 0.888178E-15 GW_02_XIU: 5 -5.33333 0.888178E-15 EXACT: -5.33333 N = 4 ALPHA = 0.00000 BETA = 2.00000 EXPON = 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 25.2840 0.00000 CN_JAC_02_XIU: 5 25.2840 0.710543E-14 GW_02_XIU: 5 25.2840 0.710543E-14 EXACT: 25.2840 N = 4 ALPHA = 0.500000 BETA = 1.50000 EXPON = 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 1.52202 0.222045E-15 CN_JAC_02_XIU: 5 1.52202 0.00000 GW_02_XIU: 5 1.52202 0.177636E-14 EXACT: 1.52202 N = 4 ALPHA = 0.00000 BETA = 0.00000 EXPON = 1 1 0 0 Degree = 2 CN_JAC_02_XIU: 5 0.444089E-15 0.444089E-15 GW_02_XIU: 5 0.444089E-15 0.444089E-15 EXACT: 0.00000 N = 4 ALPHA = 1.00000 BETA = 0.00000 EXPON = 1 1 0 0 Degree = 2 CN_JAC_02_XIU: 5 1.77778 0.00000 GW_02_XIU: 5 1.77778 0.00000 EXACT: 1.77778 N = 4 ALPHA = 0.00000 BETA = 2.00000 EXPON = 1 1 0 0 Degree = 2 CN_JAC_02_XIU: 5 12.6420 0.355271E-14 GW_02_XIU: 5 12.6420 0.00000 EXACT: 12.6420 N = 4 ALPHA = 0.500000 BETA = 1.50000 EXPON = 1 1 0 0 Degree = 2 CN_JAC_02_XIU: 5 0.380504 0.166533E-15 GW_02_XIU: 5 0.380504 0.610623E-15 EXACT: 0.380504 N = 4 ALPHA = 0.00000 BETA = 0.00000 EXPON = 2 0 0 0 Degree = 2 CN_JAC_02_XIU: 5 5.33333 0.00000 GW_02_XIU: 5 5.33333 0.00000 EXACT: 5.33333 N = 4 ALPHA = 1.00000 BETA = 0.00000 EXPON = 2 0 0 0 Degree = 2 CN_JAC_02_XIU: 5 5.33333 0.177636E-14 GW_02_XIU: 5 5.33333 0.177636E-14 EXACT: 5.33333 N = 4 ALPHA = 0.00000 BETA = 2.00000 EXPON = 2 0 0 0 Degree = 2 CN_JAC_02_XIU: 5 20.2272 0.355271E-14 GW_02_XIU: 5 20.2272 0.106581E-13 EXACT: 20.2272 N = 4 ALPHA = 0.500000 BETA = 1.50000 EXPON = 2 0 0 0 Degree = 2 CN_JAC_02_XIU: 5 1.52202 0.00000 GW_02_XIU: 5 1.52202 0.199840E-14 EXACT: 1.52202 N = 5 ALPHA = 0.00000 BETA = 0.00000 EXPON = 0 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 32.0000 0.00000 CN_JAC_01_1: 1 32.0000 0.00000 CN_JAC_02_XIU: 6 32.0000 0.355271E-14 GW_02_XIU: 6 32.0000 0.355271E-14 EXACT: 32.0000 N = 5 ALPHA = 1.00000 BETA = 0.00000 EXPON = 0 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 32.0000 0.00000 CN_JAC_01_1: 1 32.0000 0.00000 CN_JAC_02_XIU: 6 32.0000 0.355271E-14 GW_02_XIU: 6 32.0000 0.355271E-14 EXACT: 32.0000 N = 5 ALPHA = 0.00000 BETA = 2.00000 EXPON = 0 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 134.848 0.00000 CN_JAC_01_1: 1 134.848 0.00000 CN_JAC_02_XIU: 6 134.848 0.00000 GW_02_XIU: 6 134.848 0.852651E-13 EXACT: 134.848 N = 5 ALPHA = 0.500000 BETA = 1.50000 EXPON = 0 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 9.56312 0.00000 CN_JAC_01_1: 1 9.56312 0.00000 CN_JAC_02_XIU: 6 9.56312 0.00000 GW_02_XIU: 6 9.56312 0.142109E-13 EXACT: 9.56312 N = 5 ALPHA = 0.00000 BETA = 0.00000 EXPON = 0 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 0.00000 0.00000 CN_JAC_02_XIU: 6 0.00000 0.00000 GW_02_XIU: 6 0.00000 0.00000 EXACT: 0.00000 N = 5 ALPHA = 1.00000 BETA = 0.00000 EXPON = 0 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 -10.6667 0.00000 CN_JAC_02_XIU: 6 -10.6667 0.177636E-14 GW_02_XIU: 6 -10.6667 0.177636E-14 EXACT: -10.6667 N = 5 ALPHA = 0.00000 BETA = 2.00000 EXPON = 0 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 67.4239 0.00000 CN_JAC_02_XIU: 6 67.4239 0.142109E-13 GW_02_XIU: 6 67.4239 0.284217E-13 EXACT: 67.4239 N = 5 ALPHA = 0.500000 BETA = 1.50000 EXPON = 0 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 2.39078 0.00000 CN_JAC_02_XIU: 6 2.39078 0.00000 GW_02_XIU: 6 2.39078 0.266454E-14 EXACT: 2.39078 N = 5 ALPHA = 0.00000 BETA = 0.00000 EXPON = 1 1 0 0 0 Degree = 2 CN_JAC_02_XIU: 6 0.666134E-15 0.666134E-15 GW_02_XIU: 6 0.666134E-15 0.666134E-15 EXACT: 0.00000 N = 5 ALPHA = 1.00000 BETA = 0.00000 EXPON = 1 1 0 0 0 Degree = 2 CN_JAC_02_XIU: 6 3.55556 0.00000 GW_02_XIU: 6 3.55556 0.00000 EXACT: 3.55556 N = 5 ALPHA = 0.00000 BETA = 2.00000 EXPON = 1 1 0 0 0 Degree = 2 CN_JAC_02_XIU: 6 33.7119 0.213163E-13 GW_02_XIU: 6 33.7119 0.00000 EXACT: 33.7119 N = 5 ALPHA = 0.500000 BETA = 1.50000 EXPON = 1 1 0 0 0 Degree = 2 CN_JAC_02_XIU: 6 0.597695 0.111022E-15 GW_02_XIU: 6 0.597695 0.999201E-15 EXACT: 0.597695 N = 5 ALPHA = 0.00000 BETA = 0.00000 EXPON = 2 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 6 10.6667 0.355271E-14 GW_02_XIU: 6 10.6667 0.355271E-14 EXACT: 10.6667 N = 5 ALPHA = 1.00000 BETA = 0.00000 EXPON = 2 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 6 10.6667 0.177636E-14 GW_02_XIU: 6 10.6667 0.177636E-14 EXACT: 10.6667 N = 5 ALPHA = 0.00000 BETA = 2.00000 EXPON = 2 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 6 53.9391 0.00000 GW_02_XIU: 6 53.9391 0.355271E-13 EXACT: 53.9391 N = 5 ALPHA = 0.500000 BETA = 1.50000 EXPON = 2 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 6 2.39078 0.444089E-15 GW_02_XIU: 6 2.39078 0.399680E-14 EXACT: 2.39078 N = 6 ALPHA = 0.00000 BETA = 0.00000 EXPON = 0 0 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 64.0000 0.00000 CN_JAC_01_1: 1 64.0000 0.00000 CN_JAC_02_XIU: 7 64.0000 0.142109E-13 GW_02_XIU: 7 64.0000 0.142109E-13 EXACT: 64.0000 N = 6 ALPHA = 1.00000 BETA = 0.00000 EXPON = 0 0 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 64.0000 0.00000 CN_JAC_01_1: 1 64.0000 0.00000 CN_JAC_02_XIU: 7 64.0000 0.142109E-13 GW_02_XIU: 7 64.0000 0.142109E-13 EXACT: 64.0000 N = 6 ALPHA = 0.00000 BETA = 2.00000 EXPON = 0 0 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 359.594 0.00000 CN_JAC_01_1: 1 359.594 0.00000 CN_JAC_02_XIU: 7 359.594 0.00000 GW_02_XIU: 7 359.594 0.284217E-12 EXACT: 359.594 N = 6 ALPHA = 0.500000 BETA = 1.50000 EXPON = 0 0 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 15.0217 0.177636E-14 CN_JAC_01_1: 1 15.0217 0.177636E-14 CN_JAC_02_XIU: 7 15.0217 0.177636E-14 GW_02_XIU: 7 15.0217 0.266454E-13 EXACT: 15.0217 N = 6 ALPHA = 0.00000 BETA = 0.00000 EXPON = 0 0 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 0.00000 0.00000 CN_JAC_02_XIU: 7 -0.355271E-14 0.355271E-14 GW_02_XIU: 7 -0.355271E-14 0.355271E-14 EXACT: 0.00000 N = 6 ALPHA = 1.00000 BETA = 0.00000 EXPON = 0 0 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 -21.3333 0.00000 CN_JAC_02_XIU: 7 -21.3333 0.355271E-14 GW_02_XIU: 7 -21.3333 0.355271E-14 EXACT: -21.3333 N = 6 ALPHA = 0.00000 BETA = 2.00000 EXPON = 0 0 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 179.797 0.00000 CN_JAC_02_XIU: 7 179.797 0.284217E-13 GW_02_XIU: 7 179.797 0.113687E-12 EXACT: 179.797 N = 6 ALPHA = 0.500000 BETA = 1.50000 EXPON = 0 0 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 3.75543 0.888178E-15 CN_JAC_02_XIU: 7 3.75543 0.888178E-15 GW_02_XIU: 7 3.75543 0.488498E-14 EXACT: 3.75543 N = 6 ALPHA = 0.00000 BETA = 0.00000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 7 0.222045E-14 0.222045E-14 GW_02_XIU: 7 0.222045E-14 0.222045E-14 EXACT: 0.00000 N = 6 ALPHA = 1.00000 BETA = 0.00000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 7 7.11111 0.00000 GW_02_XIU: 7 7.11111 0.00000 EXACT: 7.11111 N = 6 ALPHA = 0.00000 BETA = 2.00000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 7 89.8985 0.284217E-13 GW_02_XIU: 7 89.8985 0.426326E-13 EXACT: 89.8985 N = 6 ALPHA = 0.500000 BETA = 1.50000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 7 0.938857 0.00000 GW_02_XIU: 7 0.938857 0.155431E-14 EXACT: 0.938857 N = 6 ALPHA = 0.00000 BETA = 0.00000 EXPON = 2 0 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 7 21.3333 0.00000 GW_02_XIU: 7 21.3333 0.00000 EXACT: 21.3333 N = 6 ALPHA = 1.00000 BETA = 0.00000 EXPON = 2 0 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 7 21.3333 0.355271E-14 GW_02_XIU: 7 21.3333 0.355271E-14 EXACT: 21.3333 N = 6 ALPHA = 0.00000 BETA = 2.00000 EXPON = 2 0 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 7 143.838 0.284217E-13 GW_02_XIU: 7 143.838 0.142109E-12 EXACT: 143.838 N = 6 ALPHA = 0.500000 BETA = 1.50000 EXPON = 2 0 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 7 3.75543 0.444089E-15 GW_02_XIU: 7 3.75543 0.577316E-14 EXACT: 3.75543 TEST167 Demonstrate the use of quadrature rules for the region CN_LEG, that is, the hypercube [-1,+1]^N, with the Legendre weight W(X) = 1 We use the formulas to integrate various monomials of the form X(1)^E(1) * X(2)^E(2) * ... X(N)^E(N) and compare to the exact integral. The precision of each formula is known, and we only use a formula if its precision indicates it should be able to produce an exact result. N = 1 EXPON = 0 Degree = 0 CN_LEG_01_1: 1 2.00000 0.00000 CN_LEG_02_XIU: 2 2.00000 0.00000 GW_02_XIU: 2 2.00000 0.00000 CN_LEG_03_1: 2 2.00000 0.00000 CN_LEG_03_XIU: 2 2.00000 0.00000 EXACT: 2.00000 N = 1 EXPON = 1 Degree = 1 CN_LEG_01_1: 1 0.00000 0.00000 CN_LEG_02_XIU: 2 0.00000 0.00000 GW_02_XIU: 2 0.00000 0.00000 CN_LEG_03_1: 2 0.00000 0.00000 CN_LEG_03_XIU: 2 0.00000 0.00000 EXACT: 0.00000 N = 1 EXPON = 2 Degree = 2 CN_LEG_02_XIU: 2 0.666667 0.00000 GW_02_XIU: 2 0.666667 0.00000 CN_LEG_03_1: 2 0.666667 0.222045E-15 CN_LEG_03_XIU: 2 0.666667 0.222045E-15 EXACT: 0.666667 N = 1 EXPON = 3 Degree = 3 CN_LEG_03_1: 2 0.00000 0.00000 CN_LEG_03_XIU: 2 0.00000 0.00000 EXACT: 0.00000 N = 1 EXPON = 4 Degree = 4 EXACT: 0.400000 N = 2 EXPON = 0 0 Degree = 0 CN_LEG_01_1: 1 4.00000 0.00000 CN_LEG_02_XIU: 3 4.00000 0.00000 GW_02_XIU: 3 4.00000 0.00000 CN_LEG_03_1: 4 4.00000 0.00000 CN_LEG_03_XIU: 4 4.00000 0.00000 CN_LEG_05_2: 9 4.00000 0.00000 EXACT: 4.00000 N = 2 EXPON = 0 1 Degree = 1 CN_LEG_01_1: 1 0.00000 0.00000 CN_LEG_02_XIU: 3 0.444089E-15 0.444089E-15 GW_02_XIU: 3 0.444089E-15 0.444089E-15 CN_LEG_03_1: 4 -0.889617E-16 0.889617E-16 CN_LEG_03_XIU: 4 -0.889617E-16 0.889617E-16 CN_LEG_05_2: 9 0.00000 0.00000 EXACT: 0.00000 N = 2 EXPON = 1 1 Degree = 2 CN_LEG_02_XIU: 3 0.277556E-15 0.277556E-15 GW_02_XIU: 3 0.277556E-15 0.277556E-15 CN_LEG_03_1: 4 -0.816431E-16 0.816431E-16 CN_LEG_03_XIU: 4 -0.816431E-16 0.816431E-16 CN_LEG_05_2: 9 0.00000 0.00000 EXACT: 0.00000 N = 2 EXPON = 2 0 Degree = 2 CN_LEG_02_XIU: 3 1.33333 0.222045E-15 GW_02_XIU: 3 1.33333 0.222045E-15 CN_LEG_03_1: 4 1.33333 0.444089E-15 CN_LEG_03_XIU: 4 1.33333 0.444089E-15 CN_LEG_05_2: 9 1.33333 0.222045E-15 EXACT: 1.33333 N = 2 EXPON = 3 0 Degree = 3 CN_LEG_03_1: 4 0.00000 0.00000 CN_LEG_03_XIU: 4 0.00000 0.00000 CN_LEG_05_2: 9 0.00000 0.00000 EXACT: 0.00000 N = 2 EXPON = 0 4 Degree = 4 CN_LEG_05_2: 9 0.800000 0.222045E-15 EXACT: 0.800000 N = 2 EXPON = 3 2 Degree = 5 CN_LEG_05_2: 9 0.00000 0.00000 EXACT: 0.00000 N = 3 EXPON = 0 0 0 Degree = 0 CN_LEG_01_1: 1 8.00000 0.00000 CN_LEG_02_XIU: 4 8.00000 0.00000 GW_02_XIU: 4 8.00000 0.00000 CN_LEG_03_1: 6 8.00000 0.888178E-15 CN_LEG_03_XIU: 6 8.00000 0.888178E-15 CN_LEG_05_2: 19 8.00000 0.00000 EXACT: 8.00000 N = 3 EXPON = 0 0 1 Degree = 1 CN_LEG_01_1: 1 0.00000 0.00000 CN_LEG_02_XIU: 4 0.00000 0.00000 GW_02_XIU: 4 0.00000 0.00000 CN_LEG_03_1: 6 0.00000 0.00000 CN_LEG_03_XIU: 6 0.00000 0.00000 CN_LEG_05_2: 19 0.00000 0.00000 EXACT: 0.00000 N = 3 EXPON = 1 1 0 Degree = 2 CN_LEG_02_XIU: 4 0.163286E-15 0.163286E-15 GW_02_XIU: 4 0.163286E-15 0.163286E-15 CN_LEG_03_1: 6 0.115352E-15 0.115352E-15 CN_LEG_03_XIU: 6 0.115352E-15 0.115352E-15 CN_LEG_05_2: 19 0.00000 0.00000 EXACT: 0.00000 N = 3 EXPON = 2 0 0 Degree = 2 CN_LEG_02_XIU: 4 2.66667 0.00000 GW_02_XIU: 4 2.66667 0.00000 CN_LEG_03_1: 6 2.66667 0.133227E-14 CN_LEG_03_XIU: 6 2.66667 0.133227E-14 CN_LEG_05_2: 19 2.66667 0.444089E-15 EXACT: 2.66667 N = 3 EXPON = 3 0 0 Degree = 3 CN_LEG_03_1: 6 0.00000 0.00000 CN_LEG_03_XIU: 6 0.00000 0.00000 CN_LEG_05_2: 19 0.00000 0.00000 EXACT: 0.00000 N = 3 EXPON = 0 0 4 Degree = 4 CN_LEG_05_2: 19 1.60000 0.444089E-15 EXACT: 1.60000 N = 3 EXPON = 3 2 0 Degree = 5 CN_LEG_05_2: 19 0.00000 0.00000 EXACT: 0.00000 N = 4 EXPON = 0 0 0 0 Degree = 0 CN_LEG_01_1: 1 16.0000 0.00000 CN_LEG_02_XIU: 5 16.0000 0.00000 GW_02_XIU: 5 16.0000 0.00000 CN_LEG_03_1: 8 16.0000 0.00000 CN_LEG_03_XIU: 8 16.0000 0.00000 CN_LEG_05_1(1): 22 16.0000 0.106581E-13 CN_LEG_05_1(2): 22 16.0000 0.177636E-13 CN_LEG_05_2: 33 16.0000 0.355271E-14 EXACT: 16.0000 N = 4 EXPON = 0 0 0 1 Degree = 1 CN_LEG_01_1: 1 0.00000 0.00000 CN_LEG_02_XIU: 5 0.111022E-14 0.111022E-14 GW_02_XIU: 5 0.111022E-14 0.111022E-14 CN_LEG_03_1: 8 -0.253217E-14 0.253217E-14 CN_LEG_03_XIU: 8 -0.253217E-14 0.253217E-14 CN_LEG_05_1(1): 22 0.00000 0.00000 CN_LEG_05_1(2): 22 0.832667E-16 0.832667E-16 CN_LEG_05_2: 33 0.00000 0.00000 EXACT: 0.00000 N = 4 EXPON = 1 1 0 0 Degree = 2 CN_LEG_02_XIU: 5 0.444089E-15 0.444089E-15 GW_02_XIU: 5 0.444089E-15 0.444089E-15 CN_LEG_03_1: 8 0.117517E-15 0.117517E-15 CN_LEG_03_XIU: 8 0.117517E-15 0.117517E-15 CN_LEG_05_1(1): 22 -0.115186E-14 0.115186E-14 CN_LEG_05_1(2): 22 -0.194289E-15 0.194289E-15 CN_LEG_05_2: 33 0.00000 0.00000 EXACT: 0.00000 N = 4 EXPON = 2 0 0 0 Degree = 2 CN_LEG_02_XIU: 5 5.33333 0.00000 GW_02_XIU: 5 5.33333 0.00000 CN_LEG_03_1: 8 5.33333 0.177636E-14 CN_LEG_03_XIU: 8 5.33333 0.177636E-14 CN_LEG_05_1(1): 22 5.33333 0.444089E-14 CN_LEG_05_1(2): 22 5.33333 0.444089E-14 CN_LEG_05_2: 33 5.33333 0.888178E-15 EXACT: 5.33333 N = 4 EXPON = 3 0 0 0 Degree = 3 CN_LEG_03_1: 8 -0.444089E-15 0.444089E-15 CN_LEG_03_XIU: 8 -0.444089E-15 0.444089E-15 CN_LEG_05_1(1): 22 0.107553E-15 0.107553E-15 CN_LEG_05_1(2): 22 0.111022E-15 0.111022E-15 CN_LEG_05_2: 33 0.00000 0.00000 EXACT: 0.00000 N = 4 EXPON = 0 0 0 4 Degree = 4 CN_LEG_05_1(1): 22 3.20000 0.310862E-14 CN_LEG_05_1(2): 22 3.20000 0.621725E-14 CN_LEG_05_2: 33 3.20000 0.888178E-15 EXACT: 3.20000 N = 4 EXPON = 3 2 0 0 Degree = 5 CN_LEG_05_1(1): 22 0.845678E-17 0.845678E-17 CN_LEG_05_1(2): 22 0.00000 0.00000 CN_LEG_05_2: 33 0.00000 0.00000 EXACT: 0.00000 N = 5 EXPON = 0 0 0 0 0 Degree = 0 CN_LEG_01_1: 1 32.0000 0.00000 CN_LEG_02_XIU: 6 32.0000 0.355271E-14 GW_02_XIU: 6 32.0000 0.355271E-14 CN_LEG_03_1: 10 32.0000 0.355271E-14 CN_LEG_03_XIU: 10 32.0000 0.355271E-14 CN_LEG_05_1(1): 32 32.0000 0.426326E-13 CN_LEG_05_1(2): 32 32.0000 0.568434E-13 CN_LEG_05_2: 51 32.0000 0.319744E-13 EXACT: 32.0000 N = 5 EXPON = 0 0 0 0 1 Degree = 1 CN_LEG_01_1: 1 0.00000 0.00000 CN_LEG_02_XIU: 6 0.00000 0.00000 GW_02_XIU: 6 0.00000 0.00000 CN_LEG_03_1: 10 0.00000 0.00000 CN_LEG_03_XIU: 10 0.00000 0.00000 CN_LEG_05_1(1): 32 -0.111022E-15 0.111022E-15 CN_LEG_05_1(2): 32 0.333067E-15 0.333067E-15 CN_LEG_05_2: 51 0.00000 0.00000 EXACT: 0.00000 N = 5 EXPON = 1 1 0 0 0 Degree = 2 CN_LEG_02_XIU: 6 0.666134E-15 0.666134E-15 GW_02_XIU: 6 0.666134E-15 0.666134E-15 CN_LEG_03_1: 10 0.143618E-15 0.143618E-15 CN_LEG_03_XIU: 10 0.143618E-15 0.143618E-15 CN_LEG_05_1(1): 32 0.999201E-15 0.999201E-15 CN_LEG_05_1(2): 32 -0.693889E-16 0.693889E-16 CN_LEG_05_2: 51 0.00000 0.00000 EXACT: 0.00000 N = 5 EXPON = 2 0 0 0 0 Degree = 2 CN_LEG_02_XIU: 6 10.6667 0.355271E-14 GW_02_XIU: 6 10.6667 0.355271E-14 CN_LEG_03_1: 10 10.6667 0.532907E-14 CN_LEG_03_XIU: 10 10.6667 0.532907E-14 CN_LEG_05_1(1): 32 10.6667 0.106581E-13 CN_LEG_05_1(2): 32 10.6667 0.00000 CN_LEG_05_2: 51 10.6667 0.177636E-14 EXACT: 10.6667 N = 5 EXPON = 3 0 0 0 0 Degree = 3 CN_LEG_03_1: 10 -0.222045E-15 0.222045E-15 CN_LEG_03_XIU: 10 -0.222045E-15 0.222045E-15 CN_LEG_05_1(1): 32 -0.444089E-15 0.444089E-15 CN_LEG_05_1(2): 32 0.166533E-15 0.166533E-15 CN_LEG_05_2: 51 0.00000 0.00000 EXACT: 0.00000 N = 5 EXPON = 0 0 0 0 4 Degree = 4 CN_LEG_05_1(1): 32 6.40000 0.799361E-14 CN_LEG_05_1(2): 32 6.40000 0.133227E-13 CN_LEG_05_2: 51 6.40000 0.177636E-14 EXACT: 6.40000 N = 5 EXPON = 3 2 0 0 0 Degree = 5 CN_LEG_05_1(1): 32 0.166533E-15 0.166533E-15 CN_LEG_05_1(2): 32 -0.109288E-15 0.109288E-15 CN_LEG_05_2: 51 0.00000 0.00000 EXACT: 0.00000 N = 6 EXPON = 0 0 0 0 0 0 Degree = 0 CN_LEG_01_1: 1 64.0000 0.00000 CN_LEG_02_XIU: 7 64.0000 0.142109E-13 GW_02_XIU: 7 64.0000 0.142109E-13 CN_LEG_03_1: 12 64.0000 0.00000 CN_LEG_03_XIU: 12 64.0000 0.00000 CN_LEG_05_1(1): 44 64.0000 0.284217E-13 CN_LEG_05_2: 73 64.0000 0.994760E-13 EXACT: 64.0000 N = 6 EXPON = 0 0 0 0 0 1 Degree = 1 CN_LEG_01_1: 1 0.00000 0.00000 CN_LEG_02_XIU: 7 -0.355271E-14 0.355271E-14 GW_02_XIU: 7 -0.355271E-14 0.355271E-14 CN_LEG_03_1: 12 -0.141424E-13 0.141424E-13 CN_LEG_03_XIU: 12 -0.141424E-13 0.141424E-13 CN_LEG_05_1(1): 44 0.666134E-15 0.666134E-15 CN_LEG_05_2: 73 0.00000 0.00000 EXACT: 0.00000 N = 6 EXPON = 1 1 0 0 0 0 Degree = 2 CN_LEG_02_XIU: 7 0.222045E-14 0.222045E-14 GW_02_XIU: 7 0.222045E-14 0.222045E-14 CN_LEG_03_1: 12 -0.648815E-15 0.648815E-15 CN_LEG_03_XIU: 12 -0.648815E-15 0.648815E-15 CN_LEG_05_1(1): 44 0.00000 0.00000 CN_LEG_05_2: 73 0.00000 0.00000 EXACT: 0.00000 N = 6 EXPON = 2 0 0 0 0 0 Degree = 2 CN_LEG_02_XIU: 7 21.3333 0.00000 GW_02_XIU: 7 21.3333 0.00000 CN_LEG_03_1: 12 21.3333 0.710543E-14 CN_LEG_03_XIU: 12 21.3333 0.710543E-14 CN_LEG_05_1(1): 44 21.3333 0.213163E-13 CN_LEG_05_2: 73 21.3333 0.710543E-14 EXACT: 21.3333 N = 6 EXPON = 3 0 0 0 0 0 Degree = 3 CN_LEG_03_1: 12 -0.355271E-14 0.355271E-14 CN_LEG_03_XIU: 12 -0.355271E-14 0.355271E-14 CN_LEG_05_1(1): 44 0.111022E-15 0.111022E-15 CN_LEG_05_2: 73 0.00000 0.00000 EXACT: 0.00000 N = 6 EXPON = 0 0 0 0 0 4 Degree = 4 CN_LEG_05_1(1): 44 12.8000 0.213163E-13 CN_LEG_05_2: 73 12.8000 0.710543E-14 EXACT: 12.8000 N = 6 EXPON = 3 2 0 0 0 0 Degree = 5 CN_LEG_05_1(1): 44 0.00000 0.00000 CN_LEG_05_2: 73 0.00000 0.00000 EXACT: 0.00000 TEST17 CONE_UNIT_3D approximates integrals in a unit cone. Volume = 1.04720 F(X) CONE_3D 1 1.04719754 X 0.00000000 Y 0.00000000 Z 0.26179939 X*X 0.18191701 X*Y 0.00000000 X*Z 0.00000000 Y*Y 0.18191701 Y*Z 0.00000000 Z*Z 0.10471975 X^3 0.00000000 X*Y*Z 0.00000000 Z*Z*Z 0.05235988 X^4 0.06531153 X^2 Z^2 0.00866271 Z^4 0.02991993 X^5 0.00000000 X^6 0.03039381 R 0.68070464 SIN(X) 0.00000000 EXP(X) 1.14091998 1/(1+R) 0.87676259 SQRT(R) 0.83762638 TEST18 CUBE_SHELL_ND approximates integrals in a cubical shell in ND. Inner radius = 0.00000 Outer radius = 1.00000 Spatial dimension N = 2 Volume = 4.00000 F(X) CUBE_SHELL_ND 1 4.00000000 X 0.00000000 X^2 1.33333333 X^3 0.00000000 X^4 0.55555556 X^5 0.00000000 X^6 0.25925926 R 3.26598632 SIN(X) 0.00000000 EXP(X) 4.69017805 1/(1+R) 2.20204103 SQRT(R) 3.61440801 Spatial dimension N = 3 Volume = 8.00000 F(X) CUBE_SHELL_ND 1 8.00000000 X 0.00000000 X^2 2.66666667 X^3 0.00000000 X^4 1.17333333 X^5 0.00000000 X^6 0.61866667 R 8.00000000 SIN(X) 0.00000000 EXP(X) 9.38309032 1/(1+R) 4.00000000 SQRT(R) 8.00000000 Spatial dimension N = 4 Volume = 16.0000 F(X) CUBE_SHELL_ND 1 16.00000000 X 0.00000000 X^2 5.33333333 X^3 0.00000000 X^4 2.37037037 X^5 0.00000000 X^6 1.31687243 R 18.47520861 SIN(X) 0.00000000 EXP(X) 18.76728156 1/(1+R) 7.42562584 SQRT(R) 17.19311891 Inner radius = 1.00000 Outer radius = 2.00000 Spatial dimension N = 2 Volume = 12.0000 F(X) CUBE_SHELL_ND 1 12.00000000 X 0.00000000 X^2 20.00000000 X^3 0.00000000 X^4 41.66666667 X^5 0.00000000 X^6 97.22222222 R 21.90890230 SIN(X) 0.00000000 EXP(X) 23.87719185 1/(1+R) 4.24667241 SQRT(R) 16.21440186 Spatial dimension N = 3 Volume = 56.0000 F(X) CUBE_SHELL_ND 1 56.00000000 X 0.00000000 X^2 82.66666667 X^3 0.00000000 X^4 161.08190476 X^5 0.00000000 X^6 376.13670748 R 117.84735890 SIN(X) 0.00000000 EXP(X) 104.59184063 1/(1+R) 18.03881301 SQRT(R) 81.23701188 Spatial dimension N = 4 Volume = 240.000 F(X) CUBE_SHELL_ND 1 240.00000000 X 0.00000000 X^2 336.00000000 X^3 0.00000000 X^4 627.20000000 X^5 0.00000000 X^6 1463.46666667 R 567.94365918 SIN(X) 0.00000000 EXP(X) 436.26372256 1/(1+R) 71.29209982 SQRT(R) 369.19707231 TEST19 CUBE_UNIT_3D approximates integrals in the unit cube in 3D. QMULT_3D approximates triple integrals. RECTANGLE_3D approximates integrals in a rectangular block. F(X) CUBE_UNIT_3D QMULT_3D RECTANGLE_3D 1 8.00000000 8.00000000 8.00000000 X 0.00000000 -0.00000000 0.00000000 Y 0.00000000 -0.00000000 0.00000000 Z 0.00000000 -0.00000000 0.00000000 X*X 2.66666667 2.66666667 2.66666667 X*Y 0.00000000 0.00000000 0.00000000 X*Z 0.00000000 0.00000000 0.00000000 Y*Y 2.66666667 2.66666667 2.66666667 Y*Z 0.00000000 -0.00000000 0.00000000 Z*Z 2.66666667 2.66666667 2.66666667 X^3 0.00000000 0.00000000 0.00000000 X*Y*Z 0.00000000 -0.00000000 0.00000000 Z*Z*Z 0.00000000 -0.00000000 0.00000000 X^4 0.88888889 1.60000000 0.88888889 X^2 Z^2 0.88888889 0.88888889 0.88888889 Z^4 0.88888889 1.60000000 0.88888889 X^5 0.00000000 -0.00000000 0.00000000 X^6 0.29629630 1.14285714 0.29629630 R 8.00000000 7.68481960 8.00000000 SIN(X) 0.00000000 0.00000000 -0.00000000 EXP(X) 9.37078435 9.40160955 9.37078435 1/(1+R) 5.65685425 5.80447213 5.65685425 SQRT(R) 8.00000000 7.74509074 8.00000000 TEST20 CUBE_UNIT_ND approximates integrals inside the unit cube in ND. Spatial dimension N = 2 Value of K = 10 F(X) CUBE_UNIT_ND 1 4.00000000 4.00000000 4.00000000 4.00000000 4.00000000 4.00000000 4.00000000 4.00000000 4.00000000 4.00000000 4.00000000 4.00000000 4.00000000 4.00000000 4.00000000 4.00000000 4.00000000 4.00000000 4.00000000 4.00000000 X 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 -0.00000000 -0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 -0.00000000 0.00000000 -0.00000000 0.00000000 -0.00000000 X^2 0.00000000 1.00000000 1.18518519 1.25000000 1.28000000 1.29629630 1.30612245 1.31250000 1.31687243 1.32000000 0.00000000 1.33333333 1.33333333 1.33333333 1.33333333 1.33333333 1.33333333 1.33333333 1.33333333 1.33333333 X^3 0.00000000 0.00000000 -0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 -0.00000000 0.00000000 0.00000000 -0.00000000 0.00000000 -0.00000000 0.00000000 -0.00000000 0.00000000 0.00000000 -0.00000000 X^4 0.00000000 0.25000000 0.52674897 0.64062500 0.69632000 0.72736626 0.74635569 0.75878906 0.76736270 0.77352000 0.00000000 0.33333333 0.80000000 0.80000000 0.80000000 0.80000000 0.80000000 0.80000000 0.80000000 0.80000000 X^5 0.00000000 0.00000000 0.00000000 0.00000000 -0.00000000 -0.00000000 0.00000000 0.00000000 -0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 -0.00000000 -0.00000000 -0.00000000 0.00000000 -0.00000000 0.00000000 0.00000000 X^6 0.00000000 0.06250000 0.23411065 0.35644531 0.42598400 0.46739255 0.49363300 0.51118469 0.52345729 0.53235600 0.00000000 0.08333333 0.40740741 0.57142857 0.57142857 0.57142857 0.57142857 0.57142857 0.57142857 0.57142857 R 0.00000000 2.82842712 2.86129015 2.99535239 2.99898282 3.03054964 3.03142393 3.04344523 3.04374925 3.04955799 0.00000000 3.77123617 2.77712362 3.29336224 2.82951536 3.33488705 2.70238318 3.56765353 2.30223564 4.24871948 SIN(X) 0.00000000 0.00000000 0.00000000 -0.00000000 0.00000000 0.00000000 -0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 -0.00000000 0.00000000 -0.00000000 -0.00000000 0.00000000 -0.00000000 0.00000000 EXP(X) 4.00000000 4.51050386 4.61486821 4.65219277 4.66961171 4.67911211 4.68485363 4.68858537 4.69114625 4.69297923 4.00000000 4.68067181 4.70057068 4.70080328 4.70080477 4.70080477 4.70080477 4.70080477 4.70080477 4.70080477 1/(1+R) 4.00000000 2.34314575 2.42616645 2.34104062 2.35740728 2.33744673 2.34321959 2.33566092 2.33833179 2.33469505 4.00000000 1.79086100 2.58029825 2.11554972 2.55384768 2.06662814 2.68329425 1.83402946 3.08145908 1.15690962 SQRT(R) 0.00000000 3.36358566 3.17774244 3.40276654 3.34985327 3.41302022 3.38986666 3.41711827 3.40464103 3.41916252 0.00000000 4.48478088 2.84710374 3.98415400 2.79068193 4.25275023 2.24505950 5.21263643 0.57402716 8.13865317 Spatial dimension N = 3 Value of K = 5 F(X) CUBE_UNIT_ND 1 8.00000000 8.00000000 8.00000000 8.00000000 8.00000000 8.00000000 8.00000000 8.00000000 8.00000000 8.00000000 X 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^2 0.00000000 2.00000000 2.37037037 2.50000000 2.56000000 0.00000000 2.66666667 2.66666667 2.66666667 2.66666667 X^3 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 -0.00000000 0.00000000 X^4 0.00000000 0.50000000 1.05349794 1.28125000 1.39264000 0.00000000 0.66666667 1.60000000 1.60000000 1.60000000 X^5 0.00000000 0.00000000 -0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 -0.00000000 0.00000000 0.00000000 X^6 0.00000000 0.12500000 0.46822131 0.71289062 0.85196800 0.00000000 0.16666667 0.81481481 1.14285714 1.14285714 R 0.00000000 6.92820323 7.27446305 7.48869361 7.55031790 0.00000000 9.23760431 7.34070424 7.86797462 7.54372317 SIN(X) 0.00000000 0.00000000 0.00000000 -0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 -0.00000000 0.00000000 EXP(X) 8.00000000 9.02100772 9.22973643 9.30438554 9.33922342 8.00000000 9.36134363 9.40114136 9.40160657 9.40160954 1/(1+R) 8.00000000 4.28718708 4.29316635 4.20849132 4.20436029 8.00000000 3.04958277 4.45399564 4.00548256 4.30537348 SQRT(R) 0.00000000 7.44483887 7.45106414 7.66122741 7.66536795 0.00000000 9.92645183 7.14724341 8.15274554 7.38949520 TEST205 ELLIPSE_AREA_2D returns the area of an ellipse. ELLIPSE_ECCENTRICITY_2D returns the eccentricity of an ellipse. ELLIPSE_CIRCUMFERENCE_2D returns the circumference of an ellipse. R1 R2 E Circum Area 25.0000 20.0000 0.6000 141.8083 1570.7963 0.5053 0.7742 0.7576 4.0640 1.2289 0.7591 0.0445 0.9983 3.0557 0.1062 0.6496 0.0885 0.9907 2.6683 0.1805 0.2307 0.1091 0.8811 1.1019 0.0791 (For the first example, the eccentricity should be 0.6, the circumference should be about 141.8). TEST207 Demonstrate the use of Stroud rules for the region EN_R2, that is, all of N-dimensional space, with the weight function W(X) = exp ( - X1^2 - X2^2 ... -XN^2 ) We use the formulas to integrate various monomials of the form X1^ALPHA1 * X2^ALPHA2 * ... XN^ALPHAN and compare to the exact integral. The precision of each formula is known, and we only use a formula if its precision indicates it should be able to produce an exact result. N = 1 EXPON = 0 Degree = 0 EN_R2_01_1: 1 1.77245 0.222045E-15 EN_R2_02_XIU: 2 1.77245 0.222045E-15 GW_02_XIU: 2 1.77245 0.222045E-15 EN_R2_03_1: 2 1.77245 0.222045E-15 EN_R2_03_2: 2 1.77245 0.222045E-15 EN_R2_03_XIU: 2 1.77245 0.222045E-15 EN_R2_05_2: 3 1.77245 0.222045E-15 EN_R2_05_4: 3 1.77245 0.222045E-15 EN_R2_05_5: 3 1.77245 0.444089E-15 EXACT: 1.77245 N = 1 EXPON = 2 Degree = 2 EN_R2_02_XIU: 2 0.886227 0.111022E-15 GW_02_XIU: 2 0.886227 0.111022E-15 EN_R2_03_1: 2 0.886227 0.111022E-15 EN_R2_03_2: 2 0.886227 0.111022E-15 EN_R2_03_XIU: 2 0.886227 0.333067E-15 EN_R2_05_2: 3 0.886227 0.111022E-15 EN_R2_05_4: 3 0.443113 0.443113 EN_R2_05_5: 3 0.443113 0.443113 EXACT: 0.886227 N = 1 EXPON = 0 Degree = 0 EN_R2_01_1: 1 1.77245 0.222045E-15 EN_R2_02_XIU: 2 1.77245 0.222045E-15 GW_02_XIU: 2 1.77245 0.222045E-15 EN_R2_03_1: 2 1.77245 0.222045E-15 EN_R2_03_2: 2 1.77245 0.222045E-15 EN_R2_03_XIU: 2 1.77245 0.222045E-15 EN_R2_05_2: 3 1.77245 0.222045E-15 EN_R2_05_4: 3 1.77245 0.222045E-15 EN_R2_05_5: 3 1.77245 0.444089E-15 EXACT: 1.77245 N = 1 EXPON = 6 Degree = 6 EXACT: 3.32335 N = 1 EXPON = 2 Degree = 2 EN_R2_02_XIU: 2 0.886227 0.111022E-15 GW_02_XIU: 2 0.886227 0.111022E-15 EN_R2_03_1: 2 0.886227 0.111022E-15 EN_R2_03_2: 2 0.886227 0.111022E-15 EN_R2_03_XIU: 2 0.886227 0.333067E-15 EN_R2_05_2: 3 0.886227 0.111022E-15 EN_R2_05_4: 3 0.443113 0.443113 EN_R2_05_5: 3 0.443113 0.443113 EXACT: 0.886227 N = 1 EXPON = 8 Degree = 8 EXACT: 11.6317 N = 1 EXPON = 10 Degree = 10 EXACT: 52.3428 N = 1 EXPON = 1 Degree = 1 EN_R2_01_1: 1 0.00000 0.00000 EN_R2_02_XIU: 2 0.00000 0.00000 GW_02_XIU: 2 0.00000 0.00000 EN_R2_03_1: 2 0.00000 0.00000 EN_R2_03_2: 2 0.00000 0.00000 EN_R2_03_XIU: 2 0.00000 0.00000 EN_R2_05_2: 3 0.00000 0.00000 EN_R2_05_4: 3 -0.361801 0.361801 EN_R2_05_5: 3 -0.361801 0.361801 EXACT: 0.00000 N = 1 EXPON = 2 Degree = 2 EN_R2_02_XIU: 2 0.886227 0.111022E-15 GW_02_XIU: 2 0.886227 0.111022E-15 EN_R2_03_1: 2 0.886227 0.111022E-15 EN_R2_03_2: 2 0.886227 0.111022E-15 EN_R2_03_XIU: 2 0.886227 0.333067E-15 EN_R2_05_2: 3 0.886227 0.111022E-15 EN_R2_05_4: 3 0.443113 0.443113 EN_R2_05_5: 3 0.443113 0.443113 EXACT: 0.886227 N = 2 EXPON = 0 0 Degree = 0 EN_R2_01_1: 1 3.14159 0.444089E-15 EN_R2_02_XIU: 3 3.14159 0.444089E-15 GW_02_XIU: 3 3.14159 0.444089E-15 EN_R2_03_1: 4 3.14159 0.444089E-15 EN_R2_03_2: 4 3.14159 0.444089E-15 EN_R2_03_XIU: 4 3.14159 0.444089E-15 EN_R2_05_1(1): 8 3.14159 0.133227E-14 EN_R2_05_2: 9 3.14159 0.444089E-15 EN_R2_05_4: 7 NaN NaN EN_R2_05_5: 9 3.14159 0.444089E-15 EXACT: 3.14159 N = 2 EXPON = 2 0 Degree = 2 EN_R2_02_XIU: 3 1.57080 0.666134E-15 GW_02_XIU: 3 1.57080 0.666134E-15 EN_R2_03_1: 4 1.57080 0.222045E-15 EN_R2_03_2: 4 1.57080 0.222045E-15 EN_R2_03_XIU: 4 1.57080 0.222045E-15 EN_R2_05_1(1): 8 1.57080 0.199840E-14 EN_R2_05_2: 9 1.57080 0.00000 EN_R2_05_4: 7 NaN NaN EN_R2_05_5: 9 0.785398 0.785398 EXACT: 1.57080 N = 2 EXPON = 0 4 Degree = 4 EN_R2_05_1(1): 8 2.35619 0.710543E-14 EN_R2_05_2: 9 2.35619 0.888178E-15 EN_R2_05_4: 7 NaN NaN EN_R2_05_5: 9 0.589049 1.76715 EXACT: 2.35619 N = 2 EXPON = 6 0 Degree = 6 EXACT: 5.89049 N = 2 EXPON = 2 4 Degree = 6 EXACT: 1.17810 N = 2 EXPON = 0 8 Degree = 8 EXACT: 20.6167 N = 2 EXPON = 10 0 Degree = 10 EXACT: 92.7752 N = 2 EXPON = 1 2 Degree = 3 EN_R2_03_1: 4 0.00000 0.00000 EN_R2_03_2: 4 0.00000 0.00000 EN_R2_03_XIU: 4 -0.961835E-16 0.961835E-16 EN_R2_05_1(1): 8 0.00000 0.00000 EN_R2_05_2: 9 0.00000 0.00000 EN_R2_05_4: 7 NaN NaN EN_R2_05_5: 9 -0.256543 0.256543 EXACT: 0.00000 N = 2 EXPON = 2 2 Degree = 4 EN_R2_05_1(1): 8 0.785398 0.133227E-14 EN_R2_05_2: 9 0.785398 0.111022E-15 EN_R2_05_4: 7 NaN NaN EN_R2_05_5: 9 0.196350 0.589049 EXACT: 0.785398 N = 3 EXPON = 0 0 0 Degree = 0 EN_R2_01_1: 1 5.56833 0.888178E-15 EN_R2_02_XIU: 4 5.56833 0.266454E-14 GW_02_XIU: 4 5.56833 0.266454E-14 EN_R2_03_1: 6 5.56833 0.177636E-14 EN_R2_03_2: 8 5.56833 0.177636E-14 EN_R2_03_XIU: 6 5.56833 0.177636E-14 EN_R2_05_1(1): 14 5.56833 0.00000 EN_R2_05_1(2): 14 5.56833 0.00000 EN_R2_05_2: 19 5.56833 0.177636E-14 EN_R2_05_3: 14 5.56833 0.888178E-15 EN_R2_05_4: 15 4.51963 1.04870 EN_R2_05_5: 25 5.48068 0.876474E-01 EN_R2_07_1(1): 27 4.15655 1.41178 EN_R2_07_1(2): 27 5.56774 0.592876E-03 EN_R2_07_2: 52 4.85240 0.715928 EN_R2_07_3(1): 45 5.56833 0.156319E-12 EN_R2_07_3(2): 45 5.56833 0.337508E-13 EN_R2_09_1(1): 77 5.56833 0.532907E-14 EN_R2_09_1(2): 77 5.56833 0.532907E-14 EN_R2_11_1(1): 151 5.56833 0.710543E-14 EN_R2_11_1(2): 151 5.56833 0.230926E-13 EXACT: 5.56833 N = 3 EXPON = 2 0 0 Degree = 2 EN_R2_02_XIU: 4 2.78416 0.00000 GW_02_XIU: 4 2.78416 0.00000 EN_R2_03_1: 6 2.78416 0.888178E-15 EN_R2_03_2: 8 2.78416 0.00000 EN_R2_03_XIU: 6 2.78416 0.00000 EN_R2_05_1(1): 14 2.78416 0.444089E-15 EN_R2_05_1(2): 14 2.78416 0.488498E-14 EN_R2_05_2: 19 2.78416 0.444089E-15 EN_R2_05_3: 14 2.78416 0.444089E-15 EN_R2_05_4: 15 1.04406 1.74010 EN_R2_05_5: 25 1.04406 1.74010 EN_R2_07_1(1): 27 1.67765 1.10651 EN_R2_07_1(2): 27 2.77796 0.620518E-02 EN_R2_07_2: 52 2.33671 0.447455 EN_R2_07_3(1): 45 2.78416 0.142109E-13 EN_R2_07_3(2): 45 2.78416 0.159872E-13 EN_R2_09_1(1): 77 2.78416 0.266454E-14 EN_R2_09_1(2): 77 2.78416 0.266454E-14 EN_R2_11_1(1): 151 2.78416 0.444089E-15 EN_R2_11_1(2): 151 2.78416 0.124345E-13 EXACT: 2.78416 N = 3 EXPON = 0 4 0 Degree = 4 EN_R2_05_1(1): 14 4.17625 0.177636E-14 EN_R2_05_1(2): 14 4.17625 0.213163E-13 EN_R2_05_2: 19 4.17625 0.266454E-14 EN_R2_05_3: 14 4.17625 0.177636E-14 EN_R2_05_4: 15 1.97212 2.20413 EN_R2_05_5: 25 1.04406 3.13218 EN_R2_07_1(1): 27 3.34368 0.832563 EN_R2_07_1(2): 27 4.11390 0.623470E-01 EN_R2_07_2: 52 3.72879 0.447455 EN_R2_07_3(1): 45 4.17625 0.461853E-13 EN_R2_07_3(2): 45 4.17625 0.417444E-13 EN_R2_09_1(1): 77 4.17625 0.248690E-13 EN_R2_09_1(2): 77 4.17625 0.248690E-13 EN_R2_11_1(1): 151 4.17625 0.301981E-13 EN_R2_11_1(2): 151 4.17625 0.310862E-13 EXACT: 4.17625 N = 3 EXPON = 0 0 6 Degree = 6 EN_R2_07_1(1): 27 9.83158 0.609036 EN_R2_07_1(2): 27 9.83158 0.609036 EN_R2_07_2: 52 9.83158 0.609036 EN_R2_07_3(1): 45 10.4406 0.197176E-12 EN_R2_07_3(2): 45 10.4406 0.168754E-12 EN_R2_09_1(1): 77 10.4406 0.122569E-12 EN_R2_09_1(2): 77 10.4406 0.122569E-12 EN_R2_11_1(1): 151 10.4406 0.213163E-13 EN_R2_11_1(2): 151 10.4406 0.621725E-13 EXACT: 10.4406 N = 3 EXPON = 2 4 0 Degree = 6 EN_R2_07_1(1): 27 1.56609 0.522031 EN_R2_07_1(2): 27 1.56609 0.522031 EN_R2_07_2: 52 1.56609 0.522031 EN_R2_07_3(1): 45 2.08812 0.377476E-13 EN_R2_07_3(2): 45 2.08812 0.328626E-13 EN_R2_09_1(1): 77 2.08812 0.128786E-13 EN_R2_09_1(2): 77 2.08812 0.128786E-13 EN_R2_11_1(1): 151 2.08812 0.230926E-13 EN_R2_11_1(2): 151 2.08812 0.235367E-13 EXACT: 2.08812 N = 3 EXPON = 8 0 0 Degree = 8 EN_R2_09_1(1): 77 36.5422 0.653699E-12 EN_R2_09_1(2): 77 36.5422 0.653699E-12 EN_R2_11_1(1): 151 36.5422 0.163425E-12 EN_R2_11_1(2): 151 36.5422 0.355271E-13 EXACT: 36.5422 N = 3 EXPON = 0 10 0 Degree = 10 EN_R2_11_1(1): 151 164.440 0.454747E-12 EN_R2_11_1(2): 151 164.440 0.108002E-11 EXACT: 164.440 N = 3 EXPON = 1 2 3 Degree = 6 EN_R2_07_1(1): 27 0.870051E-01 0.870051E-01 EN_R2_07_1(2): 27 0.870051E-01 0.870051E-01 EN_R2_07_2: 52 0.870051E-01 0.870051E-01 EN_R2_07_3(1): 45 0.00000 0.00000 EN_R2_07_3(2): 45 0.00000 0.00000 EN_R2_09_1(1): 77 0.00000 0.00000 EN_R2_09_1(2): 77 0.00000 0.00000 EN_R2_11_1(1): 151 0.00000 0.00000 EN_R2_11_1(2): 151 0.00000 0.00000 EXACT: 0.00000 N = 3 EXPON = 2 2 2 Degree = 6 EN_R2_07_1(1): 27 0.870051E-01 0.609036 EN_R2_07_1(2): 27 0.870051E-01 0.609036 EN_R2_07_2: 52 0.870051E-01 0.609036 EN_R2_07_3(1): 45 0.696041 0.555112E-15 EN_R2_07_3(2): 45 0.696041 0.117684E-13 EN_R2_09_1(1): 77 0.696041 0.155431E-14 EN_R2_09_1(2): 77 0.696041 0.155431E-14 EN_R2_11_1(1): 151 0.696041 0.666134E-14 EN_R2_11_1(2): 151 0.696041 0.732747E-14 EXACT: 0.696041 N = 4 EXPON = 0 0 0 0 Degree = 0 EN_R2_01_1: 1 9.86960 0.355271E-14 EN_R2_02_XIU: 5 9.86960 0.710543E-14 GW_02_XIU: 5 9.86960 0.710543E-14 EN_R2_03_1: 8 9.86960 0.177636E-14 EN_R2_03_2: 16 9.86960 0.177636E-14 EN_R2_03_XIU: 8 9.86960 0.177636E-14 EN_R2_05_1(1): 22 9.86960 0.355271E-14 EN_R2_05_2: 33 9.86960 0.532907E-14 EN_R2_05_3: 24 9.86960 0.177636E-14 EN_R2_05_4: 31 3.73055 6.13905 EN_R2_05_5: 65 13.1242 3.25461 EN_R2_07_1(1): 49 6.54096 3.32864 EN_R2_07_1(2): 49 9.80557 0.640360E-01 EN_R2_07_2: 96 19.8553 9.98566 EN_R2_07_3(1): 97 9.86960 0.131806E-11 EN_R2_07_3(2): 97 9.86960 0.213163E-13 EN_R2_09_1(1): 193 9.86960 0.159872E-13 EN_R2_09_1(2): 193 9.86960 0.337508E-13 EN_R2_11_1(1): 417 9.86960 0.600409E-11 EN_R2_11_1(2): 417 9.86960 0.131450E-12 EXACT: 9.86960 N = 4 EXPON = 2 0 0 0 Degree = 2 EN_R2_02_XIU: 5 4.93480 0.888178E-15 GW_02_XIU: 5 4.93480 0.888178E-15 EN_R2_03_1: 8 4.93480 0.888178E-15 EN_R2_03_2: 16 4.93480 0.888178E-15 EN_R2_03_XIU: 8 4.93480 0.177636E-14 EN_R2_05_1(1): 22 4.93480 0.532907E-14 EN_R2_05_2: 33 4.93480 0.266454E-14 EN_R2_05_3: 24 4.93480 0.355271E-14 EN_R2_05_4: 31 1.23370 3.70110 EN_R2_05_5: 65 1.23370 3.70110 EN_R2_07_1(1): 49 2.63269 2.30212 EN_R2_07_1(2): 49 4.76952 0.165285 EN_R2_07_2: 96 3.70110 1.23370 EN_R2_07_3(1): 97 4.93480 0.104805E-12 EN_R2_07_3(2): 97 4.93480 0.310862E-13 EN_R2_09_1(1): 193 4.93480 0.888178E-14 EN_R2_09_1(2): 193 4.93480 0.355271E-14 EN_R2_11_1(1): 417 4.93480 0.380140E-12 EN_R2_11_1(2): 417 4.93480 0.117240E-12 EXACT: 4.93480 N = 4 EXPON = 0 4 0 0 Degree = 4 EN_R2_05_1(1): 22 7.40220 0.710543E-14 EN_R2_05_2: 33 7.40220 0.00000 EN_R2_05_3: 24 7.40220 0.00000 EN_R2_05_4: 31 2.62161 4.78059 EN_R2_05_5: 65 1.38791 6.01429 EN_R2_07_1(1): 49 5.82110 1.58111 EN_R2_07_1(2): 49 6.97855 0.423656 EN_R2_07_2: 96 6.39982 1.00238 EN_R2_07_3(1): 97 7.40220 0.116351E-12 EN_R2_07_3(2): 97 7.40220 0.968114E-13 EN_R2_09_1(1): 193 7.40220 0.541789E-13 EN_R2_09_1(2): 193 7.40220 0.337508E-13 EN_R2_11_1(1): 417 7.40220 0.116351E-12 EN_R2_11_1(2): 417 7.40220 0.621725E-13 EXACT: 7.40220 N = 4 EXPON = 0 0 6 0 Degree = 6 EN_R2_07_1(1): 49 17.4260 1.07949 EN_R2_07_1(2): 49 17.4260 1.07949 EN_R2_07_2: 96 17.4260 1.07949 EN_R2_07_3(1): 97 18.5055 0.312639E-12 EN_R2_07_3(2): 97 18.5055 0.323297E-12 EN_R2_09_1(1): 193 18.5055 0.216716E-12 EN_R2_09_1(2): 193 18.5055 0.262901E-12 EN_R2_11_1(1): 417 18.5055 0.184741E-12 EN_R2_11_1(2): 417 18.5055 0.213163E-13 EXACT: 18.5055 N = 4 EXPON = 2 4 0 0 Degree = 6 EN_R2_07_1(1): 49 2.69872 1.00238 EN_R2_07_1(2): 49 2.69872 1.00238 EN_R2_07_2: 96 2.69872 1.00238 EN_R2_07_3(1): 97 3.70110 0.599520E-13 EN_R2_07_3(2): 97 3.70110 0.635048E-13 EN_R2_09_1(1): 193 3.70110 0.235367E-13 EN_R2_09_1(2): 193 3.70110 0.177636E-13 EN_R2_11_1(1): 417 3.70110 0.208722E-13 EN_R2_11_1(2): 417 3.70110 0.386358E-13 EXACT: 3.70110 N = 4 EXPON = 0 0 0 8 Degree = 8 EN_R2_09_1(1): 193 64.7693 0.112266E-11 EN_R2_09_1(2): 193 64.7693 0.108002E-11 EN_R2_11_1(1): 417 64.7693 0.966338E-12 EN_R2_11_1(2): 417 64.7693 0.190425E-11 EXACT: 64.7693 N = 4 EXPON = 10 0 0 0 Degree = 10 EN_R2_11_1(1): 417 291.462 0.426326E-11 EN_R2_11_1(2): 417 291.462 0.164846E-11 EXACT: 291.462 N = 4 EXPON = 1 2 3 4 Degree = 10 EN_R2_11_1(1): 417 0.00000 0.00000 EN_R2_11_1(2): 417 0.00000 0.00000 EXACT: 0.00000 N = 4 EXPON = 2 2 2 2 Degree = 8 EN_R2_09_1(1): 193 0.616850 0.123235E-13 EN_R2_09_1(2): 193 0.616850 0.555112E-15 EN_R2_11_1(1): 417 0.616850 0.666134E-15 EN_R2_11_1(2): 417 0.616850 0.145439E-13 EXACT: 0.616850 N = 5 EXPON = 0 0 0 0 0 Degree = 0 EN_R2_01_1: 1 17.4934 0.710543E-14 EN_R2_02_XIU: 6 17.4934 0.142109E-13 GW_02_XIU: 6 17.4934 0.142109E-13 EN_R2_03_1: 10 17.4934 0.710543E-14 EN_R2_03_2: 32 17.4934 0.106581E-13 EN_R2_03_XIU: 10 17.4934 0.710543E-14 EN_R2_05_1(1): 32 17.4934 0.106581E-13 EN_R2_05_1(2): 32 17.4934 0.355271E-14 EN_R2_05_2: 51 17.4934 0.284217E-13 EN_R2_05_3: 42 17.4934 0.142109E-13 EN_R2_05_4: 63 9.58494 7.90848 EN_R2_05_5: 161 -15.4653 32.9587 EN_R2_05_6: 192 -15.9907 33.4841 EN_R2_07_2: 164 11.8532 5.64024 EN_R2_07_3(1): 181 17.4934 0.742517E-12 EN_R2_07_3(2): 181 17.4934 0.166622E-11 EN_R2_09_1(1): 421 17.4934 0.213163E-13 EN_R2_09_1(2): 421 17.4934 0.461853E-13 EN_R2_11_1(1): 983 17.4935 0.795467E-04 EN_R2_11_1(2): 983 17.4934 0.117844E-10 EXACT: 17.4934 N = 5 EXPON = 2 0 0 0 0 Degree = 2 EN_R2_02_XIU: 6 8.74671 0.177636E-14 GW_02_XIU: 6 8.74671 0.177636E-14 EN_R2_03_1: 10 8.74671 0.177636E-14 EN_R2_03_2: 32 8.74671 0.532907E-14 EN_R2_03_XIU: 10 8.74671 0.355271E-14 EN_R2_05_1(1): 32 8.74671 0.337508E-12 EN_R2_05_1(2): 32 8.74671 0.337508E-13 EN_R2_05_2: 51 8.74671 0.177636E-14 EN_R2_05_3: 42 8.74671 0.532907E-14 EN_R2_05_4: 63 1.36667 7.38004 EN_R2_05_5: 161 1.36667 7.38004 EN_R2_05_6: 192 1.36667 7.38004 EN_R2_07_2: 164 5.81812 2.92859 EN_R2_07_3(1): 181 8.74671 0.467182E-12 EN_R2_07_3(2): 181 8.74671 0.444089E-13 EN_R2_09_1(1): 421 8.74671 0.479616E-13 EN_R2_09_1(2): 421 8.74671 0.266454E-13 EN_R2_11_1(1): 983 8.74671 0.400819E-06 EN_R2_11_1(2): 983 8.74671 0.129674E-12 EXACT: 8.74671 N = 5 EXPON = 0 4 0 0 0 Degree = 4 EN_R2_05_1(1): 32 13.1201 0.452971E-12 EN_R2_05_1(2): 32 13.1201 0.799361E-13 EN_R2_05_2: 51 13.1201 0.710543E-14 EN_R2_05_3: 42 13.1201 0.142109E-13 EN_R2_05_4: 63 3.09779 10.0223 EN_R2_05_5: 161 1.64001 11.4801 EN_R2_05_6: 192 1.64001 11.4801 EN_R2_07_2: 164 10.9941 2.12594 EN_R2_07_3(1): 181 13.1201 0.298428E-12 EN_R2_07_3(2): 181 13.1201 0.159872E-12 EN_R2_09_1(1): 421 13.1201 0.888178E-14 EN_R2_09_1(2): 421 13.1201 0.710543E-13 EN_R2_11_1(1): 983 13.1201 0.762219E-07 EN_R2_11_1(2): 983 13.1201 0.817124E-13 EXACT: 13.1201 N = 5 EXPON = 0 0 6 0 0 Degree = 6 EN_R2_07_2: 164 30.8185 1.98168 EN_R2_07_3(1): 181 32.8002 0.632383E-12 EN_R2_07_3(2): 181 32.8002 0.575540E-12 EN_R2_09_1(1): 421 32.8002 0.369482E-12 EN_R2_09_1(2): 421 32.8002 0.348166E-12 EN_R2_11_1(1): 983 32.8002 0.144940E-07 EN_R2_11_1(2): 983 32.8002 0.426326E-12 EXACT: 32.8002 N = 5 EXPON = 2 4 0 0 0 Degree = 6 EN_R2_07_2: 164 4.64669 1.91334 EN_R2_07_3(1): 181 6.56003 0.117240E-12 EN_R2_07_3(2): 181 6.56003 0.122569E-12 EN_R2_09_1(1): 421 6.56003 0.479616E-13 EN_R2_09_1(2): 421 6.56003 0.417444E-13 EN_R2_11_1(1): 983 6.56003 0.162537E-12 EN_R2_11_1(2): 983 6.56003 0.808242E-13 EXACT: 6.56003 N = 5 EXPON = 0 0 0 8 0 Degree = 8 EN_R2_09_1(1): 421 114.801 0.150635E-11 EN_R2_09_1(2): 421 114.801 0.153477E-11 EN_R2_11_1(1): 983 114.801 0.275533E-08 EN_R2_11_1(2): 983 114.801 0.190425E-11 EXACT: 114.801 N = 5 EXPON = 0 0 0 0 10 Degree = 10 EN_R2_11_1(1): 983 516.603 0.521936E-09 EN_R2_11_1(2): 983 516.603 0.227374E-12 EXACT: 516.603 N = 5 EXPON = 1 2 3 4 5 Degree = 15 EXACT: 0.00000 N = 5 EXPON = 2 2 2 2 2 Degree = 10 EN_R2_11_1(1): 983 0.546669 0.355271E-14 EN_R2_11_1(2): 983 0.546669 0.355271E-14 EXACT: 0.546669 N = 6 EXPON = 0 0 0 0 0 0 Degree = 0 EN_R2_01_1: 1 31.0063 0.177636E-13 EN_R2_02_XIU: 7 31.0063 0.284217E-13 GW_02_XIU: 7 31.0063 0.284217E-13 EN_R2_03_1: 12 31.0063 0.248690E-13 EN_R2_03_2: 64 31.0063 0.355271E-14 EN_R2_03_XIU: 12 31.0063 0.248690E-13 EN_R2_05_1(1): 44 31.0063 0.142109E-13 EN_R2_05_1(2): 44 31.0063 0.142109E-13 EN_R2_05_2: 73 31.0063 0.710543E-14 EN_R2_05_3: 76 31.0063 0.124345E-12 EN_R2_05_4: 127 13.9301 17.0762 EN_R2_05_5: 385 8.98682 22.0195 EN_R2_05_6: 448 347.712 316.706 EN_R2_07_1(1): 137 18.7254 12.2809 EN_R2_07_2: 272 18.5795 12.4267 EN_R2_07_3(1): 305 31.0062 0.112129E-03 EN_R2_07_3(2): 305 31.0063 0.246914E-11 EN_R2_09_1(1): 825 31.0063 0.891731E-12 EN_R2_09_1(2): 825 31.0063 0.579092E-12 EXACT: 31.0063 N = 6 EXPON = 2 0 0 0 0 0 Degree = 2 EN_R2_02_XIU: 7 15.5031 0.106581E-13 GW_02_XIU: 7 15.5031 0.106581E-13 EN_R2_03_1: 12 15.5031 0.106581E-13 EN_R2_03_2: 64 15.5031 0.177636E-14 EN_R2_03_XIU: 12 15.5031 0.710543E-14 EN_R2_05_1(1): 44 15.5031 0.603961E-13 EN_R2_05_1(2): 44 15.5031 0.337508E-13 EN_R2_05_2: 73 15.5031 0.177636E-14 EN_R2_05_3: 76 15.5031 0.355271E-14 EN_R2_05_4: 127 1.45342 14.0497 EN_R2_05_5: 385 1.45342 14.0497 EN_R2_05_6: 448 1.45342 14.0497 EN_R2_07_1(1): 137 7.31741 8.18573 EN_R2_07_2: 272 9.18076 6.32237 EN_R2_07_3(1): 305 15.5031 0.307665E-11 EN_R2_07_3(2): 305 15.5031 0.692779E-13 EN_R2_09_1(1): 825 15.5031 0.181188E-12 EN_R2_09_1(2): 825 15.5031 0.248690E-13 EXACT: 15.5031 N = 6 EXPON = 0 4 0 0 0 0 Degree = 4 EN_R2_05_1(1): 44 23.2547 0.223821E-12 EN_R2_05_1(2): 44 23.2547 0.142109E-12 EN_R2_05_2: 73 23.2547 0.142109E-13 EN_R2_05_3: 76 23.2547 0.994760E-13 EN_R2_05_4: 127 3.43168 19.8230 EN_R2_05_5: 385 1.81677 21.4379 EN_R2_05_6: 448 1.81677 21.4379 EN_R2_07_1(1): 137 17.8002 5.45452 EN_R2_07_2: 272 18.9671 4.28759 EN_R2_07_3(1): 305 23.2547 0.120437E-11 EN_R2_07_3(2): 305 23.2547 0.291323E-12 EN_R2_09_1(1): 825 23.2547 0.230926E-12 EN_R2_09_1(2): 825 23.2547 0.461853E-13 EXACT: 23.2547 N = 6 EXPON = 0 0 6 0 0 0 Degree = 6 EN_R2_07_1(1): 137 54.5032 3.63355 EN_R2_07_2: 272 54.5032 3.63355 EN_R2_07_3(1): 305 58.1368 0.112266E-11 EN_R2_07_3(2): 305 58.1368 0.930811E-12 EN_R2_09_1(1): 825 58.1368 0.220268E-12 EN_R2_09_1(2): 825 58.1368 0.412115E-12 EXACT: 58.1368 N = 6 EXPON = 2 4 0 0 0 0 Degree = 6 EN_R2_07_1(1): 137 8.05436 3.57299 EN_R2_07_2: 272 8.05436 3.57299 EN_R2_07_3(1): 305 11.6274 0.216716E-12 EN_R2_07_3(2): 305 11.6274 0.220268E-12 EN_R2_09_1(1): 825 11.6274 0.122569E-12 EN_R2_09_1(2): 825 11.6274 0.781597E-13 EXACT: 11.6274 N = 6 EXPON = 0 0 0 8 0 0 Degree = 8 EN_R2_09_1(1): 825 203.479 0.503064E-11 EN_R2_09_1(2): 825 203.479 0.443379E-11 EXACT: 203.479 N = 6 EXPON = 0 0 0 0 10 0 Degree = 10 EXACT: 915.654 N = 6 EXPON = 1 2 3 4 5 6 Degree = 21 EXACT: 0.00000 N = 6 EXPON = 2 2 2 2 2 2 Degree = 12 EXACT: 0.484473 N = 7 EXPON = 0 0 0 0 0 0 0 Degree = 0 EN_R2_01_1: 1 54.9572 0.284217E-13 EN_R2_02_XIU: 8 54.9572 0.639488E-13 GW_02_XIU: 8 54.9572 0.639488E-13 EN_R2_03_1: 14 54.9572 0.213163E-13 EN_R2_03_2: 128 54.9572 0.149214E-12 EN_R2_03_XIU: 14 54.9572 0.213163E-13 EN_R2_05_1(1): 58 54.9572 0.568434E-13 EN_R2_05_2: 99 54.9572 0.781597E-13 EN_R2_05_3: 142 54.9572 0.248690E-12 EN_R2_05_4: 255 141.569 86.6117 EN_R2_05_5: 897 14.8842 40.0730 EN_R2_05_6: 1024 844.435 789.478 EN_R2_07_1(1): 227 117.284 62.3267 EN_R2_07_2: 452 29.4562 25.5010 EXACT: 54.9572 N = 7 EXPON = 2 0 0 0 0 0 0 Degree = 2 EN_R2_02_XIU: 8 27.4786 0.213163E-13 GW_02_XIU: 8 27.4786 0.213163E-13 EN_R2_03_1: 14 27.4786 0.142109E-13 EN_R2_03_2: 128 27.4786 0.746070E-13 EN_R2_03_XIU: 14 27.4786 0.106581E-13 EN_R2_05_1(1): 58 27.4786 0.142109E-12 EN_R2_05_2: 99 27.4786 0.710543E-14 EN_R2_05_3: 142 27.4786 0.117240E-12 EN_R2_05_4: 255 1.50274 25.9759 EN_R2_05_5: 897 1.50274 25.9759 EN_R2_05_6: 1024 1.50274 25.9759 EN_R2_07_1(1): 227 13.2028 14.2758 EN_R2_07_2: 452 14.6219 12.8567 EXACT: 27.4786 N = 7 EXPON = 0 4 0 0 0 0 0 Degree = 4 EN_R2_05_1(1): 58 41.2179 0.461853E-12 EN_R2_05_2: 99 41.2179 0.142109E-13 EN_R2_05_3: 142 41.2179 0.198952E-12 EN_R2_05_4: 255 3.64950 37.5684 EN_R2_05_5: 897 1.93209 39.2858 EN_R2_05_6: 1024 1.93209 39.2858 EN_R2_07_1(1): 227 31.5099 9.70800 EN_R2_07_2: 452 32.8845 8.33335 EXACT: 41.2179 N = 7 EXPON = 0 0 6 0 0 0 0 Degree = 6 EN_R2_07_1(1): 227 96.4434 6.60130 EN_R2_07_2: 452 96.4434 6.60130 EXACT: 103.045 N = 7 EXPON = 2 4 0 0 0 0 0 Degree = 6 EN_R2_07_1(1): 227 14.0613 6.54763 EN_R2_07_2: 452 14.0613 6.54763 EXACT: 20.6089 N = 7 EXPON = 0 0 0 8 0 0 0 Degree = 8 EXACT: 360.657 N = 7 EXPON = 0 0 0 0 10 0 0 Degree = 10 EXACT: 1622.95 N = 7 EXPON = 1 2 3 4 5 6 7 Degree = 28 EXACT: 0.00000 N = 7 EXPON = 2 2 2 2 2 2 2 Degree = 14 EXACT: 0.429353 TEST2075 Demonstrate the use of quadrature rules for the region EPN_GLG, that is, the positive half space [0,+oo)^N, with the weight W(ALPHA;X) = product ( 1 <= I <= N ) X(I)^ALPHA exp ( -X(I) ) We use the formulas to integrate various monomials of the form X(1)^E(1) * X(2)^E(2) * ... X(N)^E(N) and compare to the exact integral. The precision of each formula is known, and we only use a formula if its precision indicates it should be able to produce an exact result. N = 1 ALPHA = -0.500000 EXPON = 0 Degree = 0 EPN_GLG_00_1: 1 1.77245 0.00000 EPN_GLG_01_1: 1 1.77245 0.00000 EPN_GLG_02_XIU: 2 1.77245 0.00000 GW_02_XIU: 2 1.77245 0.00000 EXACT: 1.77245 N = 1 ALPHA = 0.00000 EXPON = 0 Degree = 0 EPN_GLG_00_1: 1 1.00000 0.00000 EPN_GLG_01_1: 1 1.00000 0.00000 EPN_GLG_02_XIU: 2 1.00000 0.00000 GW_02_XIU: 2 1.00000 0.00000 EXACT: 1.00000 N = 1 ALPHA = 0.500000 EXPON = 0 Degree = 0 EPN_GLG_00_1: 1 0.886227 0.00000 EPN_GLG_01_1: 1 0.886227 0.00000 EPN_GLG_02_XIU: 2 0.886227 0.00000 GW_02_XIU: 2 0.886227 0.00000 EXACT: 0.886227 N = 1 ALPHA = 1.00000 EXPON = 0 Degree = 0 EPN_GLG_00_1: 1 1.00000 0.00000 EPN_GLG_01_1: 1 1.00000 0.00000 EPN_GLG_02_XIU: 2 1.00000 0.00000 GW_02_XIU: 2 1.00000 0.00000 EXACT: 1.00000 N = 1 ALPHA = 2.00000 EXPON = 0 Degree = 0 EPN_GLG_00_1: 1 2.00000 0.00000 EPN_GLG_01_1: 1 2.00000 0.00000 EPN_GLG_02_XIU: 2 2.00000 0.00000 GW_02_XIU: 2 2.00000 0.00000 EXACT: 2.00000 N = 1 ALPHA = -0.500000 EXPON = 1 Degree = 1 EPN_GLG_01_1: 1 0.886227 0.00000 EPN_GLG_02_XIU: 2 0.886227 0.222045E-15 GW_02_XIU: 2 0.886227 0.222045E-15 EXACT: 0.886227 N = 1 ALPHA = 0.00000 EXPON = 1 Degree = 1 EPN_GLG_01_1: 1 1.00000 0.00000 EPN_GLG_02_XIU: 2 1.00000 0.00000 GW_02_XIU: 2 1.00000 0.00000 EXACT: 1.00000 N = 1 ALPHA = 0.500000 EXPON = 1 Degree = 1 EPN_GLG_01_1: 1 1.32934 0.00000 EPN_GLG_02_XIU: 2 1.32934 0.00000 GW_02_XIU: 2 1.32934 0.00000 EXACT: 1.32934 N = 1 ALPHA = 1.00000 EXPON = 1 Degree = 1 EPN_GLG_01_1: 1 2.00000 0.00000 EPN_GLG_02_XIU: 2 2.00000 0.00000 GW_02_XIU: 2 2.00000 0.00000 EXACT: 2.00000 N = 1 ALPHA = 2.00000 EXPON = 1 Degree = 1 EPN_GLG_01_1: 1 6.00000 0.00000 EPN_GLG_02_XIU: 2 6.00000 0.00000 GW_02_XIU: 2 6.00000 0.00000 EXACT: 6.00000 N = 1 ALPHA = -0.500000 EXPON = 2 Degree = 2 EPN_GLG_02_XIU: 2 1.32934 0.00000 GW_02_XIU: 2 1.32934 0.00000 EXACT: 1.32934 N = 1 ALPHA = 0.00000 EXPON = 2 Degree = 2 EPN_GLG_02_XIU: 2 2.00000 0.00000 GW_02_XIU: 2 2.00000 0.00000 EXACT: 2.00000 N = 1 ALPHA = 0.500000 EXPON = 2 Degree = 2 EPN_GLG_02_XIU: 2 3.32335 0.00000 GW_02_XIU: 2 3.32335 0.00000 EXACT: 3.32335 N = 1 ALPHA = 1.00000 EXPON = 2 Degree = 2 EPN_GLG_02_XIU: 2 6.00000 0.00000 GW_02_XIU: 2 6.00000 0.00000 EXACT: 6.00000 N = 1 ALPHA = 2.00000 EXPON = 2 Degree = 2 EPN_GLG_02_XIU: 2 24.0000 0.710543E-14 GW_02_XIU: 2 24.0000 0.710543E-14 EXACT: 24.0000 N = 2 ALPHA = -0.500000 EXPON = 0 0 Degree = 0 EPN_GLG_00_1: 1 3.14159 0.00000 EPN_GLG_01_1: 1 3.14159 0.00000 EPN_GLG_02_XIU: 3 3.14159 0.00000 GW_02_XIU: 3 3.14159 0.00000 EXACT: 3.14159 N = 2 ALPHA = 0.00000 EXPON = 0 0 Degree = 0 EPN_GLG_00_1: 1 1.00000 0.00000 EPN_GLG_01_1: 1 1.00000 0.00000 EPN_GLG_02_XIU: 3 1.00000 0.00000 GW_02_XIU: 3 1.00000 0.00000 EXACT: 1.00000 N = 2 ALPHA = 0.500000 EXPON = 0 0 Degree = 0 EPN_GLG_00_1: 1 0.785398 0.00000 EPN_GLG_01_1: 1 0.785398 0.00000 EPN_GLG_02_XIU: 3 0.785398 0.00000 GW_02_XIU: 3 0.785398 0.00000 EXACT: 0.785398 N = 2 ALPHA = 1.00000 EXPON = 0 0 Degree = 0 EPN_GLG_00_1: 1 1.00000 0.00000 EPN_GLG_01_1: 1 1.00000 0.00000 EPN_GLG_02_XIU: 3 1.00000 0.00000 GW_02_XIU: 3 1.00000 0.00000 EXACT: 1.00000 N = 2 ALPHA = 2.00000 EXPON = 0 0 Degree = 0 EPN_GLG_00_1: 1 4.00000 0.00000 EPN_GLG_01_1: 1 4.00000 0.00000 EPN_GLG_02_XIU: 3 4.00000 0.00000 GW_02_XIU: 3 4.00000 0.00000 EXACT: 4.00000 N = 2 ALPHA = -0.500000 EXPON = 0 1 Degree = 1 EPN_GLG_01_1: 1 1.57080 0.00000 EPN_GLG_02_XIU: 3 1.57080 0.444089E-15 GW_02_XIU: 3 1.57080 0.444089E-15 EXACT: 1.57080 N = 2 ALPHA = 0.00000 EXPON = 0 1 Degree = 1 EPN_GLG_01_1: 1 1.00000 0.00000 EPN_GLG_02_XIU: 3 1.00000 0.222045E-15 GW_02_XIU: 3 1.00000 0.222045E-15 EXACT: 1.00000 N = 2 ALPHA = 0.500000 EXPON = 0 1 Degree = 1 EPN_GLG_01_1: 1 1.17810 0.222045E-15 EPN_GLG_02_XIU: 3 1.17810 0.00000 GW_02_XIU: 3 1.17810 0.00000 EXACT: 1.17810 N = 2 ALPHA = 1.00000 EXPON = 0 1 Degree = 1 EPN_GLG_01_1: 1 2.00000 0.00000 EPN_GLG_02_XIU: 3 2.00000 0.444089E-15 GW_02_XIU: 3 2.00000 0.444089E-15 EXACT: 2.00000 N = 2 ALPHA = 2.00000 EXPON = 0 1 Degree = 1 EPN_GLG_01_1: 1 12.0000 0.00000 EPN_GLG_02_XIU: 3 12.0000 0.177636E-14 GW_02_XIU: 3 12.0000 0.177636E-14 EXACT: 12.0000 N = 2 ALPHA = -0.500000 EXPON = 1 1 Degree = 2 EPN_GLG_02_XIU: 3 0.785398 0.111022E-15 GW_02_XIU: 3 0.785398 0.111022E-15 EXACT: 0.785398 N = 2 ALPHA = 0.00000 EXPON = 1 1 Degree = 2 EPN_GLG_02_XIU: 3 1.00000 0.222045E-15 GW_02_XIU: 3 1.00000 0.222045E-15 EXACT: 1.00000 N = 2 ALPHA = 0.500000 EXPON = 1 1 Degree = 2 EPN_GLG_02_XIU: 3 1.76715 0.444089E-15 GW_02_XIU: 3 1.76715 0.444089E-15 EXACT: 1.76715 N = 2 ALPHA = 1.00000 EXPON = 1 1 Degree = 2 EPN_GLG_02_XIU: 3 4.00000 0.444089E-15 GW_02_XIU: 3 4.00000 0.444089E-15 EXACT: 4.00000 N = 2 ALPHA = 2.00000 EXPON = 1 1 Degree = 2 EPN_GLG_02_XIU: 3 36.0000 0.00000 GW_02_XIU: 3 36.0000 0.00000 EXACT: 36.0000 N = 2 ALPHA = -0.500000 EXPON = 2 0 Degree = 2 EPN_GLG_02_XIU: 3 2.35619 0.888178E-15 GW_02_XIU: 3 2.35619 0.888178E-15 EXACT: 2.35619 N = 2 ALPHA = 0.00000 EXPON = 2 0 Degree = 2 EPN_GLG_02_XIU: 3 2.00000 0.888178E-15 GW_02_XIU: 3 2.00000 0.888178E-15 EXACT: 2.00000 N = 2 ALPHA = 0.500000 EXPON = 2 0 Degree = 2 EPN_GLG_02_XIU: 3 2.94524 0.444089E-15 GW_02_XIU: 3 2.94524 0.444089E-15 EXACT: 2.94524 N = 2 ALPHA = 1.00000 EXPON = 2 0 Degree = 2 EPN_GLG_02_XIU: 3 6.00000 0.888178E-15 GW_02_XIU: 3 6.00000 0.888178E-15 EXACT: 6.00000 N = 2 ALPHA = 2.00000 EXPON = 2 0 Degree = 2 EPN_GLG_02_XIU: 3 48.0000 0.00000 GW_02_XIU: 3 48.0000 0.00000 EXACT: 48.0000 N = 3 ALPHA = -0.500000 EXPON = 0 0 0 Degree = 0 EPN_GLG_00_1: 1 5.56833 0.00000 EPN_GLG_01_1: 1 5.56833 0.00000 EPN_GLG_02_XIU: 4 5.56833 0.00000 GW_02_XIU: 4 5.56833 0.00000 EXACT: 5.56833 N = 3 ALPHA = 0.00000 EXPON = 0 0 0 Degree = 0 EPN_GLG_00_1: 1 1.00000 0.00000 EPN_GLG_01_1: 1 1.00000 0.00000 EPN_GLG_02_XIU: 4 1.00000 0.00000 GW_02_XIU: 4 1.00000 0.00000 EXACT: 1.00000 N = 3 ALPHA = 0.500000 EXPON = 0 0 0 Degree = 0 EPN_GLG_00_1: 1 0.696041 0.00000 EPN_GLG_01_1: 1 0.696041 0.00000 EPN_GLG_02_XIU: 4 0.696041 0.00000 GW_02_XIU: 4 0.696041 0.00000 EXACT: 0.696041 N = 3 ALPHA = 1.00000 EXPON = 0 0 0 Degree = 0 EPN_GLG_00_1: 1 1.00000 0.00000 EPN_GLG_01_1: 1 1.00000 0.00000 EPN_GLG_02_XIU: 4 1.00000 0.00000 GW_02_XIU: 4 1.00000 0.00000 EXACT: 1.00000 N = 3 ALPHA = 2.00000 EXPON = 0 0 0 Degree = 0 EPN_GLG_00_1: 1 8.00000 0.00000 EPN_GLG_01_1: 1 8.00000 0.00000 EPN_GLG_02_XIU: 4 8.00000 0.00000 GW_02_XIU: 4 8.00000 0.00000 EXACT: 8.00000 N = 3 ALPHA = -0.500000 EXPON = 0 0 1 Degree = 1 EPN_GLG_01_1: 1 2.78416 0.00000 EPN_GLG_02_XIU: 4 2.78416 0.00000 GW_02_XIU: 4 2.78416 0.00000 EXACT: 2.78416 N = 3 ALPHA = 0.00000 EXPON = 0 0 1 Degree = 1 EPN_GLG_01_1: 1 1.00000 0.00000 EPN_GLG_02_XIU: 4 1.00000 0.00000 GW_02_XIU: 4 1.00000 0.00000 EXACT: 1.00000 N = 3 ALPHA = 0.500000 EXPON = 0 0 1 Degree = 1 EPN_GLG_01_1: 1 1.04406 0.00000 EPN_GLG_02_XIU: 4 1.04406 0.00000 GW_02_XIU: 4 1.04406 0.00000 EXACT: 1.04406 N = 3 ALPHA = 1.00000 EXPON = 0 0 1 Degree = 1 EPN_GLG_01_1: 1 2.00000 0.00000 EPN_GLG_02_XIU: 4 2.00000 0.00000 GW_02_XIU: 4 2.00000 0.00000 EXACT: 2.00000 N = 3 ALPHA = 2.00000 EXPON = 0 0 1 Degree = 1 EPN_GLG_01_1: 1 24.0000 0.00000 EPN_GLG_02_XIU: 4 24.0000 0.00000 GW_02_XIU: 4 24.0000 0.00000 EXACT: 24.0000 N = 3 ALPHA = -0.500000 EXPON = 1 1 0 Degree = 2 EPN_GLG_02_XIU: 4 1.39208 0.222045E-15 GW_02_XIU: 4 1.39208 0.222045E-15 EXACT: 1.39208 N = 3 ALPHA = 0.00000 EXPON = 1 1 0 Degree = 2 EPN_GLG_02_XIU: 4 1.00000 0.111022E-15 GW_02_XIU: 4 1.00000 0.111022E-15 EXACT: 1.00000 N = 3 ALPHA = 0.500000 EXPON = 1 1 0 Degree = 2 EPN_GLG_02_XIU: 4 1.56609 0.444089E-15 GW_02_XIU: 4 1.56609 0.444089E-15 EXACT: 1.56609 N = 3 ALPHA = 1.00000 EXPON = 1 1 0 Degree = 2 EPN_GLG_02_XIU: 4 4.00000 0.00000 GW_02_XIU: 4 4.00000 0.00000 EXACT: 4.00000 N = 3 ALPHA = 2.00000 EXPON = 1 1 0 Degree = 2 EPN_GLG_02_XIU: 4 72.0000 0.00000 GW_02_XIU: 4 72.0000 0.00000 EXACT: 72.0000 N = 3 ALPHA = -0.500000 EXPON = 2 0 0 Degree = 2 EPN_GLG_02_XIU: 4 4.17625 0.266454E-14 GW_02_XIU: 4 4.17625 0.266454E-14 EXACT: 4.17625 N = 3 ALPHA = 0.00000 EXPON = 2 0 0 Degree = 2 EPN_GLG_02_XIU: 4 2.00000 0.00000 GW_02_XIU: 4 2.00000 0.00000 EXACT: 2.00000 N = 3 ALPHA = 0.500000 EXPON = 2 0 0 Degree = 2 EPN_GLG_02_XIU: 4 2.61015 0.444089E-15 GW_02_XIU: 4 2.61015 0.444089E-15 EXACT: 2.61015 N = 3 ALPHA = 1.00000 EXPON = 2 0 0 Degree = 2 EPN_GLG_02_XIU: 4 6.00000 0.00000 GW_02_XIU: 4 6.00000 0.00000 EXACT: 6.00000 N = 3 ALPHA = 2.00000 EXPON = 2 0 0 Degree = 2 EPN_GLG_02_XIU: 4 96.0000 0.284217E-13 GW_02_XIU: 4 96.0000 0.284217E-13 EXACT: 96.0000 N = 4 ALPHA = -0.500000 EXPON = 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 9.86960 0.177636E-14 EPN_GLG_01_1: 1 9.86960 0.177636E-14 EPN_GLG_02_XIU: 5 9.86960 0.177636E-14 GW_02_XIU: 5 9.86960 0.177636E-14 EXACT: 9.86960 N = 4 ALPHA = 0.00000 EXPON = 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 1.00000 0.00000 EPN_GLG_01_1: 1 1.00000 0.00000 EPN_GLG_02_XIU: 5 1.00000 0.00000 GW_02_XIU: 5 1.00000 0.00000 EXACT: 1.00000 N = 4 ALPHA = 0.500000 EXPON = 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 0.616850 0.111022E-15 EPN_GLG_01_1: 1 0.616850 0.111022E-15 EPN_GLG_02_XIU: 5 0.616850 0.111022E-15 GW_02_XIU: 5 0.616850 0.111022E-15 EXACT: 0.616850 N = 4 ALPHA = 1.00000 EXPON = 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 1.00000 0.00000 EPN_GLG_01_1: 1 1.00000 0.00000 EPN_GLG_02_XIU: 5 1.00000 0.00000 GW_02_XIU: 5 1.00000 0.00000 EXACT: 1.00000 N = 4 ALPHA = 2.00000 EXPON = 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 16.0000 0.00000 EPN_GLG_01_1: 1 16.0000 0.00000 EPN_GLG_02_XIU: 5 16.0000 0.00000 GW_02_XIU: 5 16.0000 0.00000 EXACT: 16.0000 N = 4 ALPHA = -0.500000 EXPON = 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 4.93480 0.888178E-15 EPN_GLG_02_XIU: 5 4.93480 0.177636E-14 GW_02_XIU: 5 4.93480 0.177636E-14 EXACT: 4.93480 N = 4 ALPHA = 0.00000 EXPON = 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 1.00000 0.00000 EPN_GLG_02_XIU: 5 1.00000 0.00000 GW_02_XIU: 5 1.00000 0.00000 EXACT: 1.00000 N = 4 ALPHA = 0.500000 EXPON = 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 0.925275 0.111022E-15 EPN_GLG_02_XIU: 5 0.925275 0.111022E-15 GW_02_XIU: 5 0.925275 0.111022E-15 EXACT: 0.925275 N = 4 ALPHA = 1.00000 EXPON = 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 2.00000 0.00000 EPN_GLG_02_XIU: 5 2.00000 0.00000 GW_02_XIU: 5 2.00000 0.00000 EXACT: 2.00000 N = 4 ALPHA = 2.00000 EXPON = 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 48.0000 0.00000 EPN_GLG_02_XIU: 5 48.0000 0.00000 GW_02_XIU: 5 48.0000 0.00000 EXACT: 48.0000 N = 4 ALPHA = -0.500000 EXPON = 1 1 0 0 Degree = 2 EPN_GLG_02_XIU: 5 2.46740 0.444089E-15 GW_02_XIU: 5 2.46740 0.444089E-15 EXACT: 2.46740 N = 4 ALPHA = 0.00000 EXPON = 1 1 0 0 Degree = 2 EPN_GLG_02_XIU: 5 1.00000 0.222045E-15 GW_02_XIU: 5 1.00000 0.222045E-15 EXACT: 1.00000 N = 4 ALPHA = 0.500000 EXPON = 1 1 0 0 Degree = 2 EPN_GLG_02_XIU: 5 1.38791 0.444089E-15 GW_02_XIU: 5 1.38791 0.444089E-15 EXACT: 1.38791 N = 4 ALPHA = 1.00000 EXPON = 1 1 0 0 Degree = 2 EPN_GLG_02_XIU: 5 4.00000 0.00000 GW_02_XIU: 5 4.00000 0.00000 EXACT: 4.00000 N = 4 ALPHA = 2.00000 EXPON = 1 1 0 0 Degree = 2 EPN_GLG_02_XIU: 5 144.000 0.00000 GW_02_XIU: 5 144.000 0.00000 EXACT: 144.000 N = 4 ALPHA = -0.500000 EXPON = 2 0 0 0 Degree = 2 EPN_GLG_02_XIU: 5 7.40220 0.266454E-14 GW_02_XIU: 5 7.40220 0.266454E-14 EXACT: 7.40220 N = 4 ALPHA = 0.00000 EXPON = 2 0 0 0 Degree = 2 EPN_GLG_02_XIU: 5 2.00000 0.444089E-15 GW_02_XIU: 5 2.00000 0.444089E-15 EXACT: 2.00000 N = 4 ALPHA = 0.500000 EXPON = 2 0 0 0 Degree = 2 EPN_GLG_02_XIU: 5 2.31319 0.444089E-15 GW_02_XIU: 5 2.31319 0.444089E-15 EXACT: 2.31319 N = 4 ALPHA = 1.00000 EXPON = 2 0 0 0 Degree = 2 EPN_GLG_02_XIU: 5 6.00000 0.888178E-15 GW_02_XIU: 5 6.00000 0.888178E-15 EXACT: 6.00000 N = 4 ALPHA = 2.00000 EXPON = 2 0 0 0 Degree = 2 EPN_GLG_02_XIU: 5 192.000 0.284217E-13 GW_02_XIU: 5 192.000 0.284217E-13 EXACT: 192.000 N = 5 ALPHA = -0.500000 EXPON = 0 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 17.4934 0.355271E-14 EPN_GLG_01_1: 1 17.4934 0.355271E-14 EPN_GLG_02_XIU: 6 17.4934 0.355271E-14 GW_02_XIU: 6 17.4934 0.355271E-14 EXACT: 17.4934 N = 5 ALPHA = 0.00000 EXPON = 0 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 1.00000 0.00000 EPN_GLG_01_1: 1 1.00000 0.00000 EPN_GLG_02_XIU: 6 1.00000 0.111022E-15 GW_02_XIU: 6 1.00000 0.111022E-15 EXACT: 1.00000 N = 5 ALPHA = 0.500000 EXPON = 0 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 0.546669 0.111022E-15 EPN_GLG_01_1: 1 0.546669 0.111022E-15 EPN_GLG_02_XIU: 6 0.546669 0.111022E-15 GW_02_XIU: 6 0.546669 0.111022E-15 EXACT: 0.546669 N = 5 ALPHA = 1.00000 EXPON = 0 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 1.00000 0.00000 EPN_GLG_01_1: 1 1.00000 0.00000 EPN_GLG_02_XIU: 6 1.00000 0.111022E-15 GW_02_XIU: 6 1.00000 0.111022E-15 EXACT: 1.00000 N = 5 ALPHA = 2.00000 EXPON = 0 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 32.0000 0.00000 EPN_GLG_01_1: 1 32.0000 0.00000 EPN_GLG_02_XIU: 6 32.0000 0.355271E-14 GW_02_XIU: 6 32.0000 0.355271E-14 EXACT: 32.0000 N = 5 ALPHA = -0.500000 EXPON = 0 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 8.74671 0.177636E-14 EPN_GLG_02_XIU: 6 8.74671 0.355271E-14 GW_02_XIU: 6 8.74671 0.355271E-14 EXACT: 8.74671 N = 5 ALPHA = 0.00000 EXPON = 0 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 1.00000 0.00000 EPN_GLG_02_XIU: 6 1.00000 0.00000 GW_02_XIU: 6 1.00000 0.00000 EXACT: 1.00000 N = 5 ALPHA = 0.500000 EXPON = 0 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 0.820004 0.111022E-15 EPN_GLG_02_XIU: 6 0.820004 0.111022E-15 GW_02_XIU: 6 0.820004 0.111022E-15 EXACT: 0.820004 N = 5 ALPHA = 1.00000 EXPON = 0 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 2.00000 0.00000 EPN_GLG_02_XIU: 6 2.00000 0.00000 GW_02_XIU: 6 2.00000 0.00000 EXACT: 2.00000 N = 5 ALPHA = 2.00000 EXPON = 0 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 96.0000 0.00000 EPN_GLG_02_XIU: 6 96.0000 0.142109E-13 GW_02_XIU: 6 96.0000 0.142109E-13 EXACT: 96.0000 N = 5 ALPHA = -0.500000 EXPON = 1 1 0 0 0 Degree = 2 EPN_GLG_02_XIU: 6 4.37335 0.00000 GW_02_XIU: 6 4.37335 0.00000 EXACT: 4.37335 N = 5 ALPHA = 0.00000 EXPON = 1 1 0 0 0 Degree = 2 EPN_GLG_02_XIU: 6 1.00000 0.00000 GW_02_XIU: 6 1.00000 0.00000 EXACT: 1.00000 N = 5 ALPHA = 0.500000 EXPON = 1 1 0 0 0 Degree = 2 EPN_GLG_02_XIU: 6 1.23001 0.222045E-15 GW_02_XIU: 6 1.23001 0.222045E-15 EXACT: 1.23001 N = 5 ALPHA = 1.00000 EXPON = 1 1 0 0 0 Degree = 2 EPN_GLG_02_XIU: 6 4.00000 0.888178E-15 GW_02_XIU: 6 4.00000 0.888178E-15 EXACT: 4.00000 N = 5 ALPHA = 2.00000 EXPON = 1 1 0 0 0 Degree = 2 EPN_GLG_02_XIU: 6 288.000 0.00000 GW_02_XIU: 6 288.000 0.00000 EXACT: 288.000 N = 5 ALPHA = -0.500000 EXPON = 2 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 6 13.1201 0.532907E-14 GW_02_XIU: 6 13.1201 0.532907E-14 EXACT: 13.1201 N = 5 ALPHA = 0.00000 EXPON = 2 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 6 2.00000 0.00000 GW_02_XIU: 6 2.00000 0.00000 EXACT: 2.00000 N = 5 ALPHA = 0.500000 EXPON = 2 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 6 2.05001 0.00000 GW_02_XIU: 6 2.05001 0.00000 EXACT: 2.05001 N = 5 ALPHA = 1.00000 EXPON = 2 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 6 6.00000 0.00000 GW_02_XIU: 6 6.00000 0.00000 EXACT: 6.00000 N = 5 ALPHA = 2.00000 EXPON = 2 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 6 384.000 0.00000 GW_02_XIU: 6 384.000 0.00000 EXACT: 384.000 N = 6 ALPHA = -0.500000 EXPON = 0 0 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 31.0063 0.710543E-14 EPN_GLG_01_1: 1 31.0063 0.710543E-14 EPN_GLG_02_XIU: 7 31.0063 0.355271E-14 GW_02_XIU: 7 31.0063 0.355271E-14 EXACT: 31.0063 N = 6 ALPHA = 0.00000 EXPON = 0 0 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 1.00000 0.00000 EPN_GLG_01_1: 1 1.00000 0.00000 EPN_GLG_02_XIU: 7 1.00000 0.222045E-15 GW_02_XIU: 7 1.00000 0.222045E-15 EXACT: 1.00000 N = 6 ALPHA = 0.500000 EXPON = 0 0 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 0.484473 0.111022E-15 EPN_GLG_01_1: 1 0.484473 0.111022E-15 EPN_GLG_02_XIU: 7 0.484473 0.555112E-16 GW_02_XIU: 7 0.484473 0.555112E-16 EXACT: 0.484473 N = 6 ALPHA = 1.00000 EXPON = 0 0 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 1.00000 0.00000 EPN_GLG_01_1: 1 1.00000 0.00000 EPN_GLG_02_XIU: 7 1.00000 0.222045E-15 GW_02_XIU: 7 1.00000 0.222045E-15 EXACT: 1.00000 N = 6 ALPHA = 2.00000 EXPON = 0 0 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 64.0000 0.00000 EPN_GLG_01_1: 1 64.0000 0.00000 EPN_GLG_02_XIU: 7 64.0000 0.142109E-13 GW_02_XIU: 7 64.0000 0.142109E-13 EXACT: 64.0000 N = 6 ALPHA = -0.500000 EXPON = 0 0 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 15.5031 0.355271E-14 EPN_GLG_02_XIU: 7 15.5031 0.355271E-14 GW_02_XIU: 7 15.5031 0.355271E-14 EXACT: 15.5031 N = 6 ALPHA = 0.00000 EXPON = 0 0 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 1.00000 0.00000 EPN_GLG_02_XIU: 7 1.00000 0.00000 GW_02_XIU: 7 1.00000 0.00000 EXACT: 1.00000 N = 6 ALPHA = 0.500000 EXPON = 0 0 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 0.726710 0.222045E-15 EPN_GLG_02_XIU: 7 0.726710 0.222045E-15 GW_02_XIU: 7 0.726710 0.222045E-15 EXACT: 0.726710 N = 6 ALPHA = 1.00000 EXPON = 0 0 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 2.00000 0.00000 EPN_GLG_02_XIU: 7 2.00000 0.00000 GW_02_XIU: 7 2.00000 0.00000 EXACT: 2.00000 N = 6 ALPHA = 2.00000 EXPON = 0 0 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 192.000 0.00000 EPN_GLG_02_XIU: 7 192.000 0.00000 GW_02_XIU: 7 192.000 0.00000 EXACT: 192.000 N = 6 ALPHA = -0.500000 EXPON = 1 1 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 7 7.75157 0.888178E-15 GW_02_XIU: 7 7.75157 0.888178E-15 EXACT: 7.75157 N = 6 ALPHA = 0.00000 EXPON = 1 1 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 7 1.00000 0.00000 GW_02_XIU: 7 1.00000 0.00000 EXACT: 1.00000 N = 6 ALPHA = 0.500000 EXPON = 1 1 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 7 1.09006 0.222045E-15 GW_02_XIU: 7 1.09006 0.222045E-15 EXACT: 1.09006 N = 6 ALPHA = 1.00000 EXPON = 1 1 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 7 4.00000 0.00000 GW_02_XIU: 7 4.00000 0.00000 EXACT: 4.00000 N = 6 ALPHA = 2.00000 EXPON = 1 1 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 7 576.000 0.00000 GW_02_XIU: 7 576.000 0.00000 EXACT: 576.000 N = 6 ALPHA = -0.500000 EXPON = 2 0 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 7 23.2547 0.710543E-14 GW_02_XIU: 7 23.2547 0.710543E-14 EXACT: 23.2547 N = 6 ALPHA = 0.00000 EXPON = 2 0 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 7 2.00000 0.00000 GW_02_XIU: 7 2.00000 0.00000 EXACT: 2.00000 N = 6 ALPHA = 0.500000 EXPON = 2 0 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 7 1.81677 0.222045E-15 GW_02_XIU: 7 1.81677 0.222045E-15 EXACT: 1.81677 N = 6 ALPHA = 1.00000 EXPON = 2 0 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 7 6.00000 0.888178E-15 GW_02_XIU: 7 6.00000 0.888178E-15 EXACT: 6.00000 N = 6 ALPHA = 2.00000 EXPON = 2 0 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 7 768.000 0.00000 GW_02_XIU: 7 768.000 0.00000 EXACT: 768.000 TEST208 Demonstrate the use of quadrature rules for the region EPN_LAG, that is, the positive half space [0,+oo)^N, with the weight W(X) = product ( 1 <= I <= N ) exp ( -X(I) ) We use the formulas to integrate various monomials of the form X(1)^E(1) * X(2)^E(2) * ... X(N)^E(N) and compare to the exact integral. The precision of each formula is known, and we only use a formula if its precision indicates it should be able to produce an exact result. N = 1 EXPON = 0 Degree = 0 EPN_LAG_00_1: 1 1.00000 0.00000 EPN_LAG_01_1: 1 1.00000 0.00000 EPN_LAG_02_XIU: 2 1.00000 0.00000 GW_02_XIU: 2 1.00000 0.00000 EXACT: 1.00000 N = 1 EXPON = 1 Degree = 1 EPN_LAG_01_1: 1 1.00000 0.00000 EPN_LAG_02_XIU: 2 1.00000 0.00000 GW_02_XIU: 2 1.00000 0.00000 EXACT: 1.00000 N = 1 EXPON = 2 Degree = 2 EPN_LAG_02_XIU: 2 2.00000 0.00000 GW_02_XIU: 2 2.00000 0.00000 EXACT: 2.00000 N = 2 EXPON = 0 0 Degree = 0 EPN_LAG_00_1: 1 1.00000 0.00000 EPN_LAG_01_1: 1 1.00000 0.00000 EPN_LAG_02_XIU: 3 1.00000 0.00000 GW_02_XIU: 3 1.00000 0.00000 EXACT: 1.00000 N = 2 EXPON = 0 1 Degree = 1 EPN_LAG_01_1: 1 1.00000 0.00000 EPN_LAG_02_XIU: 3 1.00000 0.222045E-15 GW_02_XIU: 3 1.00000 0.222045E-15 EXACT: 1.00000 N = 2 EXPON = 1 1 Degree = 2 EPN_LAG_02_XIU: 3 1.00000 0.222045E-15 GW_02_XIU: 3 1.00000 0.222045E-15 EXACT: 1.00000 N = 2 EXPON = 2 0 Degree = 2 EPN_LAG_02_XIU: 3 2.00000 0.888178E-15 GW_02_XIU: 3 2.00000 0.888178E-15 EXACT: 2.00000 N = 3 EXPON = 0 0 0 Degree = 0 EPN_LAG_00_1: 1 1.00000 0.00000 EPN_LAG_01_1: 1 1.00000 0.00000 EPN_LAG_02_XIU: 4 1.00000 0.00000 GW_02_XIU: 4 1.00000 0.00000 EXACT: 1.00000 N = 3 EXPON = 0 0 1 Degree = 1 EPN_LAG_01_1: 1 1.00000 0.00000 EPN_LAG_02_XIU: 4 1.00000 0.00000 GW_02_XIU: 4 1.00000 0.00000 EXACT: 1.00000 N = 3 EXPON = 1 1 0 Degree = 2 EPN_LAG_02_XIU: 4 1.00000 0.111022E-15 GW_02_XIU: 4 1.00000 0.111022E-15 EXACT: 1.00000 N = 3 EXPON = 2 0 0 Degree = 2 EPN_LAG_02_XIU: 4 2.00000 0.00000 GW_02_XIU: 4 2.00000 0.00000 EXACT: 2.00000 N = 4 EXPON = 0 0 0 0 Degree = 0 EPN_LAG_00_1: 1 1.00000 0.00000 EPN_LAG_01_1: 1 1.00000 0.00000 EPN_LAG_02_XIU: 5 1.00000 0.00000 GW_02_XIU: 5 1.00000 0.00000 EXACT: 1.00000 N = 4 EXPON = 0 0 0 1 Degree = 1 EPN_LAG_01_1: 1 1.00000 0.00000 EPN_LAG_02_XIU: 5 1.00000 0.00000 GW_02_XIU: 5 1.00000 0.00000 EXACT: 1.00000 N = 4 EXPON = 1 1 0 0 Degree = 2 EPN_LAG_02_XIU: 5 1.00000 0.222045E-15 GW_02_XIU: 5 1.00000 0.222045E-15 EXACT: 1.00000 N = 4 EXPON = 2 0 0 0 Degree = 2 EPN_LAG_02_XIU: 5 2.00000 0.444089E-15 GW_02_XIU: 5 2.00000 0.444089E-15 EXACT: 2.00000 N = 5 EXPON = 0 0 0 0 0 Degree = 0 EPN_LAG_00_1: 1 1.00000 0.00000 EPN_LAG_01_1: 1 1.00000 0.00000 EPN_LAG_02_XIU: 6 1.00000 0.111022E-15 GW_02_XIU: 6 1.00000 0.111022E-15 EXACT: 1.00000 N = 5 EXPON = 0 0 0 0 1 Degree = 1 EPN_LAG_01_1: 1 1.00000 0.00000 EPN_LAG_02_XIU: 6 1.00000 0.00000 GW_02_XIU: 6 1.00000 0.00000 EXACT: 1.00000 N = 5 EXPON = 1 1 0 0 0 Degree = 2 EPN_LAG_02_XIU: 6 1.00000 0.00000 GW_02_XIU: 6 1.00000 0.00000 EXACT: 1.00000 N = 5 EXPON = 2 0 0 0 0 Degree = 2 EPN_LAG_02_XIU: 6 2.00000 0.00000 GW_02_XIU: 6 2.00000 0.00000 EXACT: 2.00000 N = 6 EXPON = 0 0 0 0 0 0 Degree = 0 EPN_LAG_00_1: 1 1.00000 0.00000 EPN_LAG_01_1: 1 1.00000 0.00000 EPN_LAG_02_XIU: 7 1.00000 0.222045E-15 GW_02_XIU: 7 1.00000 0.222045E-15 EXACT: 1.00000 N = 6 EXPON = 0 0 0 0 0 1 Degree = 1 EPN_LAG_01_1: 1 1.00000 0.00000 EPN_LAG_02_XIU: 7 1.00000 0.00000 GW_02_XIU: 7 1.00000 0.00000 EXACT: 1.00000 N = 6 EXPON = 1 1 0 0 0 0 Degree = 2 EPN_LAG_02_XIU: 7 1.00000 0.00000 GW_02_XIU: 7 1.00000 0.00000 EXACT: 1.00000 N = 6 EXPON = 2 0 0 0 0 0 Degree = 2 EPN_LAG_02_XIU: 7 2.00000 0.00000 GW_02_XIU: 7 2.00000 0.00000 EXACT: 2.00000 TEST21 HEXAGON_UNIT_SET sets a quadrature rule for the unit hexagon. HEXAGON_SUM evaluates the quadrature rule in an arbitrary hexagon. We use a radius 2.00000 and center: CENTER = 0.00000 0.00000 Rule: 1 2 3 4 Function 1 10.39230485 10.39230485 10.39230485 10.39230485 X 0.00000000 0.00000000 0.00000000 0.00000000 X^2 0.00000000 8.66025404 8.66025404 8.66025404 X^3 0.00000000 0.00000000 0.00000000 0.00000000 X^4 0.00000000 14.43375673 25.98076211 14.54922678 X^5 0.00000000 0.00000000 0.00000000 0.00000000 X^6 0.00000000 24.05626122 95.26279442 29.87441233 R 0.00000000 13.41640786 8.66025404 11.57275125 SIN(X) 0.00000000 0.00000000 0.00000000 0.00000000 EXP(X) 10.39230485 15.35826291 15.94692966 15.37180468 1/(1+R) 10.39230485 4.53615453 7.50555350 5.75701708 SQRT(R) 0.00000000 11.80793803 6.12372436 9.45964010 TEST215 LENS_HALF_2D approximates an integral within a circular half lens, defined by joining the endpoints of a circular arc. Integrate F(X,Y) = 1 R Theta1 Theta2 Area Order Integral 1.00000 0.00000 0.00000 0.00000 2 0.00000 1.00000 0.00000 0.00000 0.00000 4 0.00000 1.00000 0.00000 0.00000 0.00000 6 0.00000 1.00000 0.00000 0.00000 0.00000 8 0.00000 1.00000 0.00000 0.00000 0.00000 10 0.00000 1.00000 0.00000 0.00000 0.00000 12 0.00000 1.00000 0.00000 0.00000 0.00000 14 0.00000 1.00000 0.00000 0.00000 0.00000 16 0.00000 1.00000 0.00000 0.785398 0.391457E-01 2 0.395760E-01 1.00000 0.00000 0.785398 0.391457E-01 4 0.392147E-01 1.00000 0.00000 0.785398 0.391457E-01 6 0.391683E-01 1.00000 0.00000 0.785398 0.391457E-01 8 0.391557E-01 1.00000 0.00000 0.785398 0.391457E-01 10 0.391510E-01 1.00000 0.00000 0.785398 0.391457E-01 12 0.391488E-01 1.00000 0.00000 0.785398 0.391457E-01 14 0.391477E-01 1.00000 0.00000 0.785398 0.391457E-01 16 0.391471E-01 1.00000 0.00000 1.57080 0.285398 2 0.288675 1.00000 0.00000 1.57080 0.285398 4 0.285920 1.00000 0.00000 1.57080 0.285398 6 0.285569 1.00000 0.00000 1.57080 0.285398 8 0.285474 1.00000 0.00000 1.57080 0.285398 10 0.285438 1.00000 0.00000 1.57080 0.285398 12 0.285422 1.00000 0.00000 1.57080 0.285398 14 0.285413 1.00000 0.00000 1.57080 0.285398 16 0.285408 1.00000 0.00000 2.35619 0.824544 2 0.834710 1.00000 0.00000 2.35619 0.824544 4 0.826147 1.00000 0.00000 2.35619 0.824544 6 0.825066 1.00000 0.00000 2.35619 0.824544 8 0.824776 1.00000 0.00000 2.35619 0.824544 10 0.824667 1.00000 0.00000 2.35619 0.824544 12 0.824616 1.00000 0.00000 2.35619 0.824544 14 0.824590 1.00000 0.00000 2.35619 0.824544 16 0.824575 1.00000 0.00000 3.14159 1.57080 2 1.59223 1.00000 0.00000 3.14159 1.57080 4 1.57411 1.00000 0.00000 3.14159 1.57080 6 1.57188 1.00000 0.00000 3.14159 1.57080 8 1.57128 1.00000 0.00000 3.14159 1.57080 10 1.57105 1.00000 0.00000 3.14159 1.57080 12 1.57095 1.00000 0.00000 3.14159 1.57080 14 1.57089 1.00000 0.00000 3.14159 1.57080 16 1.57086 1.00000 0.00000 3.92699 2.31705 2 2.35375 1.00000 0.00000 3.92699 2.31705 4 2.32247 1.00000 0.00000 3.92699 2.31705 6 2.31881 1.00000 0.00000 3.92699 2.31705 8 2.31783 1.00000 0.00000 3.92699 2.31705 10 2.31746 1.00000 0.00000 3.92699 2.31705 12 2.31729 1.00000 0.00000 3.92699 2.31705 14 2.31720 1.00000 0.00000 3.92699 2.31705 16 2.31715 1.00000 0.00000 4.71239 2.85619 2 2.91421 1.00000 0.00000 4.71239 2.85619 4 2.86377 1.00000 0.00000 4.71239 2.85619 6 2.85861 1.00000 0.00000 4.71239 2.85619 8 2.85727 1.00000 0.00000 4.71239 2.85619 10 2.85676 1.00000 0.00000 4.71239 2.85619 12 2.85653 1.00000 0.00000 4.71239 2.85619 14 2.85641 1.00000 0.00000 4.71239 2.85619 16 2.85634 1.00000 0.00000 5.49779 3.10245 2 3.19494 1.00000 0.00000 5.49779 3.10245 4 3.11336 1.00000 0.00000 5.49779 3.10245 6 3.10555 1.00000 0.00000 5.49779 3.10245 8 3.10376 1.00000 0.00000 5.49779 3.10245 10 3.10313 1.00000 0.00000 5.49779 3.10245 12 3.10285 1.00000 0.00000 5.49779 3.10245 14 3.10270 1.00000 0.00000 5.49779 3.10245 16 3.10262 1.00000 0.00000 6.28319 3.14159 2 3.26599 1.00000 0.00000 6.28319 3.14159 4 3.16056 1.00000 0.00000 6.28319 3.14159 6 3.14773 1.00000 0.00000 6.28319 3.14159 8 3.14431 1.00000 0.00000 6.28319 3.14159 10 3.14303 1.00000 0.00000 6.28319 3.14159 12 3.14244 1.00000 0.00000 6.28319 3.14159 14 3.14214 1.00000 0.00000 6.28319 3.14159 16 3.14196 Integrate F(X,Y) = X R Theta1 Theta2 Area Order Integral 1.00000 0.00000 0.00000 0.00000 2 0.00000 1.00000 0.00000 0.00000 0.00000 4 0.00000 1.00000 0.00000 0.00000 0.00000 6 0.00000 1.00000 0.00000 0.00000 0.00000 8 0.00000 1.00000 0.00000 0.00000 0.00000 10 0.00000 1.00000 0.00000 0.00000 0.00000 12 0.00000 1.00000 0.00000 0.00000 0.00000 14 0.00000 1.00000 0.00000 0.00000 0.00000 16 0.00000 1.00000 0.00000 0.785398 0.391457E-01 2 0.349205E-01 1.00000 0.00000 0.785398 0.391457E-01 4 0.345818E-01 1.00000 0.00000 0.785398 0.391457E-01 6 0.345387E-01 1.00000 0.00000 0.785398 0.391457E-01 8 0.345271E-01 1.00000 0.00000 0.785398 0.391457E-01 10 0.345227E-01 1.00000 0.00000 0.785398 0.391457E-01 12 0.345207E-01 1.00000 0.00000 0.785398 0.391457E-01 14 0.345197E-01 1.00000 0.00000 0.785398 0.391457E-01 16 0.345191E-01 1.00000 0.00000 1.57080 0.285398 2 0.169102 1.00000 0.00000 1.57080 0.285398 4 0.167041 1.00000 0.00000 1.57080 0.285398 6 0.166788 1.00000 0.00000 1.57080 0.285398 8 0.166720 1.00000 0.00000 1.57080 0.285398 10 0.166695 1.00000 0.00000 1.57080 0.285398 12 0.166683 1.00000 0.00000 1.57080 0.285398 14 0.166677 1.00000 0.00000 1.57080 0.285398 16 0.166674 1.00000 0.00000 2.35619 0.824544 2 0.205492 1.00000 0.00000 2.35619 0.824544 4 0.201814 1.00000 0.00000 2.35619 0.824544 6 0.201387 1.00000 0.00000 2.35619 0.824544 8 0.201274 1.00000 0.00000 2.35619 0.824544 10 0.201232 1.00000 0.00000 2.35619 0.824544 12 0.201212 1.00000 0.00000 2.35619 0.824544 14 0.201202 1.00000 0.00000 2.35619 0.824544 16 0.201197 1.00000 0.00000 3.14159 1.57080 2 0.00000 1.00000 0.00000 3.14159 1.57080 4 0.520417E-17 1.00000 0.00000 3.14159 1.57080 6 -0.108420E-17 1.00000 0.00000 3.14159 1.57080 8 0.409286E-17 1.00000 0.00000 3.14159 1.57080 10 -0.405221E-17 1.00000 0.00000 3.14159 1.57080 12 0.689146E-17 1.00000 0.00000 3.14159 1.57080 14 -0.187533E-17 1.00000 0.00000 3.14159 1.57080 16 0.612913E-17 1.00000 0.00000 3.92699 2.31705 2 -0.216881 1.00000 0.00000 3.92699 2.31705 4 -0.203381 1.00000 0.00000 3.92699 2.31705 6 -0.201876 1.00000 0.00000 3.92699 2.31705 8 -0.201488 1.00000 0.00000 3.92699 2.31705 10 -0.201344 1.00000 0.00000 3.92699 2.31705 12 -0.201278 1.00000 0.00000 3.92699 2.31705 14 -0.201244 1.00000 0.00000 3.92699 2.31705 16 -0.201225 1.00000 0.00000 4.71239 2.85619 2 -0.201184 1.00000 0.00000 4.71239 2.85619 4 -0.172233 1.00000 0.00000 4.71239 2.85619 6 -0.168431 1.00000 0.00000 4.71239 2.85619 8 -0.167438 1.00000 0.00000 4.71239 2.85619 10 -0.167072 1.00000 0.00000 4.71239 2.85619 12 -0.166905 1.00000 0.00000 4.71239 2.85619 14 -0.166819 1.00000 0.00000 4.71239 2.85619 16 -0.166770 1.00000 0.00000 5.49779 3.10245 2 -0.620848E-01 1.00000 0.00000 5.49779 3.10245 4 -0.413975E-01 1.00000 0.00000 5.49779 3.10245 6 -0.370893E-01 1.00000 0.00000 5.49779 3.10245 8 -0.357024E-01 1.00000 0.00000 5.49779 3.10245 10 -0.351485E-01 1.00000 0.00000 5.49779 3.10245 12 -0.348907E-01 1.00000 0.00000 5.49779 3.10245 14 -0.347560E-01 1.00000 0.00000 5.49779 3.10245 16 -0.346791E-01 1.00000 0.00000 6.28319 3.14159 2 -0.555112E-16 1.00000 0.00000 6.28319 3.14159 4 0.194289E-15 1.00000 0.00000 6.28319 3.14159 6 -0.693889E-17 1.00000 0.00000 6.28319 3.14159 8 0.260209E-17 1.00000 0.00000 6.28319 3.14159 10 -0.524754E-16 1.00000 0.00000 6.28319 3.14159 12 0.783878E-16 1.00000 0.00000 6.28319 3.14159 14 -0.473254E-16 1.00000 0.00000 6.28319 3.14159 16 -0.149715E-15 Integrate F(X,Y) = R R Theta1 Theta2 Area Order Integral 1.00000 0.00000 0.00000 0.00000 2 0.00000 1.00000 0.00000 0.00000 0.00000 4 0.00000 1.00000 0.00000 0.00000 0.00000 6 0.00000 1.00000 0.00000 0.00000 0.00000 8 0.00000 1.00000 0.00000 0.00000 0.00000 10 0.00000 1.00000 0.00000 0.00000 0.00000 12 0.00000 1.00000 0.00000 0.00000 0.00000 14 0.00000 1.00000 0.00000 0.00000 0.00000 16 0.00000 1.00000 0.00000 0.785398 0.391457E-01 2 0.384017E-01 1.00000 0.00000 0.785398 0.391457E-01 4 0.380323E-01 1.00000 0.00000 0.785398 0.391457E-01 6 0.379857E-01 1.00000 0.00000 0.785398 0.391457E-01 8 0.379732E-01 1.00000 0.00000 0.785398 0.391457E-01 10 0.379684E-01 1.00000 0.00000 0.785398 0.391457E-01 12 0.379663E-01 1.00000 0.00000 0.785398 0.391457E-01 14 0.379651E-01 1.00000 0.00000 0.785398 0.391457E-01 16 0.379645E-01 1.00000 0.00000 1.57080 0.285398 2 0.257008 1.00000 0.00000 1.57080 0.285398 4 0.253591 1.00000 0.00000 1.57080 0.285398 6 0.253232 1.00000 0.00000 1.57080 0.285398 8 0.253137 1.00000 0.00000 1.57080 0.285398 10 0.253101 1.00000 0.00000 1.57080 0.285398 12 0.253085 1.00000 0.00000 1.57080 0.285398 14 0.253076 1.00000 0.00000 1.57080 0.285398 16 0.253072 1.00000 0.00000 2.35619 0.824544 2 0.656948 1.00000 0.00000 2.35619 0.824544 4 0.639356 1.00000 0.00000 2.35619 0.824544 6 0.637932 1.00000 0.00000 2.35619 0.824544 8 0.637615 1.00000 0.00000 2.35619 0.824544 10 0.637503 1.00000 0.00000 2.35619 0.824544 12 0.637452 1.00000 0.00000 2.35619 0.824544 14 0.637426 1.00000 0.00000 2.35619 0.824544 16 0.637411 1.00000 0.00000 3.14159 1.57080 2 1.12071 1.00000 0.00000 3.14159 1.57080 4 1.05856 1.00000 0.00000 3.14159 1.57080 6 1.05089 1.00000 0.00000 3.14159 1.57080 8 1.04884 1.00000 0.00000 3.14159 1.57080 10 1.04807 1.00000 0.00000 3.14159 1.57080 12 1.04771 1.00000 0.00000 3.14159 1.57080 14 1.04753 1.00000 0.00000 3.14159 1.57080 16 1.04742 1.00000 0.00000 3.92699 2.31705 2 1.59940 1.00000 0.00000 3.92699 2.31705 4 1.47722 1.00000 0.00000 3.92699 2.31705 6 1.46410 1.00000 0.00000 3.92699 2.31705 8 1.46017 1.00000 0.00000 3.92699 2.31705 10 1.45861 1.00000 0.00000 3.92699 2.31705 12 1.45799 1.00000 0.00000 3.92699 2.31705 14 1.45765 1.00000 0.00000 3.92699 2.31705 16 1.45743 1.00000 0.00000 4.71239 2.85619 2 2.05128 1.00000 0.00000 4.71239 2.85619 4 1.86626 1.00000 0.00000 4.71239 2.85619 6 1.84848 1.00000 0.00000 4.71239 2.85619 8 1.84427 1.00000 0.00000 4.71239 2.85619 10 1.84289 1.00000 0.00000 4.71239 2.85619 12 1.84233 1.00000 0.00000 4.71239 2.85619 14 1.84203 1.00000 0.00000 4.71239 2.85619 16 1.84184 1.00000 0.00000 5.49779 3.10245 2 2.34495 1.00000 0.00000 5.49779 3.10245 4 2.09084 1.00000 0.00000 5.49779 3.10245 6 2.06676 1.00000 0.00000 5.49779 3.10245 8 2.06083 1.00000 0.00000 5.49779 3.10245 10 2.05869 1.00000 0.00000 5.49779 3.10245 12 2.05773 1.00000 0.00000 5.49779 3.10245 14 2.05723 1.00000 0.00000 5.49779 3.10245 16 2.05696 1.00000 0.00000 6.28319 3.14159 2 2.43432 1.00000 0.00000 6.28319 3.14159 4 2.13903 1.00000 0.00000 6.28319 3.14159 6 2.10851 1.00000 0.00000 6.28319 3.14159 8 2.10060 1.00000 0.00000 6.28319 3.14159 10 2.09766 1.00000 0.00000 6.28319 3.14159 12 2.09632 1.00000 0.00000 6.28319 3.14159 14 2.09563 1.00000 0.00000 6.28319 3.14159 16 2.09523 TEST22 OCTAHEDRON_UNIT_ND approximates integrals in a unit octahedron in N dimensions. F(X) N = 1 N = 2 N = 3 1 2.00000000 2.00000000 1.33333333 X 0.00000000 0.00000000 0.00000000 X^2 0.66666667 0.33333333 0.13333333 X^3 0.00000000 0.00000000 0.00000000 X^4 0.22222222 0.11111111 0.04000000 X^5 0.00000000 0.00000000 0.00000000 X^6 0.07407407 0.03703704 0.01200000 R 1.15470054 1.15470054 0.73029674 SIN(X) 0.00000000 0.00000000 0.00000000 EXP(X) 2.34269609 2.17134804 1.40168342 1/(1+R) 1.26794919 1.26794919 0.86148084 SQRT(R) 1.51967137 1.51967137 0.98677707 TEST23 PARALLELIPIPED_VOLUME_ND computes the volume of a parallelipiped in N dimensions. Spatial dimension N = 2 Parallelipiped vertices: Col 1 2 3 Row 1: 0. 1. 0. 2: 0. 0. 1. Volume is 1.00000 Spatial dimension N = 3 Parallelipiped vertices: Col 1 2 3 4 Row 1: 0. 1. 0. 0. 2: 0. 0. 1. 0. 3: 0. 0. 0. 1. Volume is 1.00000 Spatial dimension N = 4 Parallelipiped vertices: Col 1 2 3 4 5 Row 1: 0. 1. 0. 0. 0. 2: 0. 0. 1. 0. 0. 3: 0. 0. 0. 1. 0. 4: 0. 0. 0. 0. 1. Volume is 1.00000 TEST24 For a polygon in 2D: POLYGON_1_2D integrates 1 POLYGON_X_2D integrates X POLYGON_Y_2D integrates Y POLYGON_XX_2D integrates X^2 POLYGON_XY_2D integrates X*Y POLYGON_YY_2D integrates Y^2 F(X,Y) Integral 1 1.00000 X 0.500000 Y 0.500000 X^2 0.333333 XY 0.250000 Y^2 0.333333 TEST25 For the unit pyramid, we approximate integrals with: PYRAMID_UNIT_O01_3D, a 1 point rule. PYRAMID_UNIT_O05_3D, a 5 point rule. PYRAMID_UNIT_O06_3D, a 6 point rule. PYRAMID_UNIT_O08_3D, an 8 point rule. PYRAMID_UNIT_O08b_3D, an 8 point rule. PYRAMID_UNIT_O09_3D, a 9 point rule. PYRAMID_UNIT_O13_3D, a 13 point rule. PYRAMID_UNIT_O18_3D, a 18 point rule. PYRAMID_UNIT_O27_3D, a 27 point rule. PYRAMID_UNIT_O48_3D, a 48 point rule. PYRAMID_UNIT_VOLUME_3D computes the volume of a unit pyramid. Volume = 1.33333 Order 1 X Y Z X*X 1 1.33333 0.00000 0.00000 0.333333 0.00000 5 1.33333 0.00000 0.00000 0.333333 0.266667 6 1.33333 0.00000 0.00000 0.333333 0.266667 8 1.33333 0.00000 0.00000 0.333333 0.266667 8 1.33333 0.00000 0.00000 0.333333 0.266667 9 1.33333 0.00000 0.00000 0.333333 0.266667 13 1.33333 0.00000 0.00000 0.333333 0.266667 18 1.33333 0.00000 -0.925186E-17 0.333333 0.266667 27 1.33333 0.00000 -0.289121E-18 0.333333 0.266667 48 1.33333 0.00000 0.00000 0.333333 0.266667 Order X*Y X*Z Y*Y Y*Z Z*Z 1 0.00000 0.00000 0.00000 0.00000 0.833333E-01 5 0.00000 0.00000 0.266667 0.00000 0.133333 6 0.00000 0.00000 0.266667 0.00000 0.133333 8 0.00000 0.00000 0.266667 0.00000 0.133333 8 0.00000 0.00000 0.266667 0.00000 0.133333 9 0.00000 0.00000 0.266667 0.00000 0.133333 13 0.00000 0.00000 0.266667 0.00000 0.133333 18 0.00000 0.00000 0.266667 0.115648E-17 0.133333 27 0.00000 0.00000 0.266667 0.115648E-17 0.133333 48 0.00000 0.00000 0.266667 0.00000 0.133333 Order X^3 X*Y*Z Z*Z*Z X^4 X^2 Z^2 1 0.00000 0.00000 0.208333E-01 0.00000 0.00000 5 0.00000 0.00000 0.766667E-01 0.632099E-01 0.740741E-02 6 0.00000 0.00000 0.773148E-01 0.634921E-01 0.740741E-02 8 0.00000 0.00000 0.666667E-01 0.632099E-01 0.118519E-01 8 0.00000 0.00000 0.647619E-01 0.632099E-01 0.126984E-01 9 0.00000 0.00000 0.669312E-01 0.634921E-01 0.126984E-01 13 0.00000 0.00000 0.666504E-01 0.664669E-01 0.126984E-01 18 0.00000 0.00000 0.666667E-01 0.113778 0.118519E-01 27 0.00000 0.00000 0.666667E-01 0.114286 0.126984E-01 48 0.00000 0.00000 0.666667E-01 0.114286 0.126984E-01 Order Z^4 X^5 X^6 R SIN(X) 1 0.520833E-02 0.00000 0.00000 0.333333 0.00000 5 0.508889E-01 0.00000 0.149831E-01 0.942800 0.00000 6 0.523148E-01 0.00000 0.151172E-01 0.941763 0.00000 8 0.355556E-01 0.00000 0.158610E-01 0.941900 0.00000 8 0.330884E-01 0.00000 0.160282E-01 0.941565 0.00000 9 0.392038E-01 0.00000 0.164399E-01 0.939009 0.00000 13 0.379673E-01 0.00000 0.174323E-01 0.936256 0.00000 18 0.355556E-01 0.00000 0.513896E-01 0.882331 0.00000 27 0.380952E-01 0.00000 0.533163E-01 0.880646 0.00000 48 0.380952E-01 0.00000 0.634921E-01 0.906432 0.00000 Order EXP(X) 1/(1+R) SQRT(R) 1 1.33333 1.29352 0.666667 5 1.46932 1.08867 1.12119 6 1.46933 1.08904 1.12024 8 1.46932 1.08901 1.12038 8 1.46932 1.08913 1.12008 9 1.46934 1.09011 1.11779 13 1.46946 1.09094 1.11513 18 1.47148 1.10358 1.05469 27 1.47150 1.10428 1.05166 48 1.47152 1.10245 1.08854 TEST255 For the unit pyramid, PYRAMID_UNIT_MONOMIAL_3D returns the exact value of the integral of X^ALPHA Y^BETA Z^GAMMA Volume = 1.33333 ALPHA BETA GAMMA INTEGRAL 0 0 0 1.33333 0 0 1 0.333333 0 0 2 0.133333 0 0 3 0.666667E-01 0 0 4 0.380952E-01 0 1 0 0.00000 0 1 1 0.00000 0 1 2 0.00000 0 1 3 0.00000 0 2 0 0.266667 0 2 1 0.444444E-01 0 2 2 0.126984E-01 0 3 0 0.00000 0 3 1 0.00000 0 4 0 0.114286 1 0 0 0.00000 1 0 1 0.00000 1 0 2 0.00000 1 0 3 0.00000 1 1 0 0.00000 1 1 1 0.00000 1 1 2 0.00000 1 2 0 0.00000 1 2 1 0.00000 1 3 0 0.00000 2 0 0 0.266667 2 0 1 0.444444E-01 2 0 2 0.126984E-01 2 1 0 0.00000 2 1 1 0.00000 2 2 0 0.634921E-01 3 0 0 0.00000 3 0 1 0.00000 3 1 0 0.00000 4 0 0 0.114286 TEST26 QMULT_1D approximates an integral on a one-dimensional interval. We use the interval: A = -1.00000 B = 1.00000 F(X) QMULT_1D 1 2.00000000 X 0.00000000 X^2 0.66666667 X^3 0.00000000 X^4 0.40000000 X^5 0.00000000 X^6 0.28571429 R 1.00303105 SIN(X) -0.00000000 EXP(X) 2.35040239 1/(1+R) 1.38328245 SQRT(R) 1.34346834 TEST27 SIMPLEX_ND approximates integrals inside an arbitrary simplex in ND. Spatial dimension N = 2 Simplex vertices: Col 1 2 3 Row 1: 0. 1. 0. 2: 0. 0. 1. F(X) SIMPLEX_ND 1 0.50000000 X 0.16666667 X^2 0.08333333 X^3 0.05092593 X^4 0.03317901 X^5 0.02199074 X^6 0.01463906 R 0.26834513 SIN(X) 0.15836034 EXP(X) 0.71840914 1/(1+R) 0.33244390 SQRT(R) 0.35723722 Spatial dimension N = 3 Simplex vertices: Col 1 2 3 4 Row 1: 0. 1. 0. 0. 2: 0. 0. 1. 0. 3: 0. 0. 0. 1. F(X) SIMPLEX_ND 1 0.16666667 X 0.04166667 X^2 0.01666667 X^3 0.00868921 X^4 0.00493921 X^5 0.00287107 X^6 0.00167794 R 0.08712002 SIN(X) 0.04024220 EXP(X) 0.21834713 1/(1+R) 0.11091482 SQRT(R) 0.11858573 Spatial dimension N = 4 Simplex vertices: Col 1 2 3 4 5 Row 1: 0. 1. 0. 0. 0. 2: 0. 0. 1. 0. 0. 3: 0. 0. 0. 1. 0. 4: 0. 0. 0. 0. 1. F(X) SIMPLEX_ND 1 0.04166667 X 0.00833333 X^2 0.00277778 X^3 0.00127217 X^4 0.00064736 X^5 0.00033823 X^6 0.00017780 R 0.02080883 SIN(X) 0.00812411 EXP(X) 0.05163098 1/(1+R) 0.02803639 SQRT(R) 0.02911181 TEST28 SIMPLEX_VOLUME_ND computes the volume of a simplex in N dimensions. Spatial dimension N = 2 Simplex vertices: Col 1 2 3 Row 1: 0. 1. 0. 2: 0. 0. 1. Volume is 0.500000 Spatial dimension N = 3 Simplex vertices: Col 1 2 3 4 Row 1: 0. 1. 0. 0. 2: 0. 0. 1. 0. 3: 0. 0. 0. 1. Volume is 0.166667 Spatial dimension N = 4 Simplex vertices: Col 1 2 3 4 5 Row 1: 0. 1. 0. 0. 0. 2: 0. 0. 1. 0. 0. 3: 0. 0. 0. 1. 0. 4: 0. 0. 0. 0. 1. Volume is 0.416667E-01 TEST29 For integrals in the unit simplex in ND, SIMPLEX_UNIT_01_ND uses a formula of degree 1. SIMPLEX_UNIT_03_ND uses a formula of degree 3. SIMPLEX_UNIT_05_ND uses a formula of degree 5. SIMPLEX_UNIT_05_2_ND uses a formula of degree 5. Check the integral of 1 N Volume #1 #3 #5 #5.2 2 0.50000 0.50000 0.50000 0.50000 0.50000 3 0.16667 0.16667 0.16667 0.16667 0.16667 4 0.41667E-01 0.41667E-01 0.41667E-01 0.41667E-01 0.41667E-01 5 0.83333E-02 0.83333E-02 0.83333E-02 0.83333E-02 0.83333E-02 6 0.13889E-02 0.13889E-02 0.13889E-02 0.13889E-02 0.13889E-02 7 0.19841E-03 0.19841E-03 0.19841E-03 0.19841E-03 0.19841E-03 8 0.24802E-04 0.24802E-04 0.24802E-04 0.24802E-04 0.24802E-04 9 0.27557E-05 0.27557E-05 0.27557E-05 0.27557E-05 0.27557E-05 10 0.27557E-06 0.27557E-06 0.27557E-06 0.27557E-06 0.27557E-06 11 0.25052E-07 0.25052E-07 0.25052E-07 0.25052E-07 0.25052E-07 12 0.20877E-08 0.20877E-08 0.20877E-08 0.20877E-08 0.20877E-08 13 0.16059E-09 0.16059E-09 0.16059E-09 0.16059E-09 0.16059E-09 14 0.11471E-10 0.11471E-10 0.11471E-10 0.11471E-10 0.11471E-10 15 0.76472E-12 0.76472E-12 0.76472E-12 0.76472E-12 0.76472E-12 16 0.47795E-13 0.47795E-13 0.47795E-13 0.47795E-13 0.47795E-13 Check the integral of X N Volume #1 #3 #5 #5.2 2 0.50000 0.25000 0.16667 0.16667 0.16667 3 0.16667 0.55556E-01 0.41667E-01 0.41667E-01 0.41667E-01 4 0.41667E-01 0.10417E-01 0.83333E-02 0.83333E-02 0.83333E-02 5 0.83333E-02 0.16667E-02 0.13889E-02 0.13889E-02 0.13889E-02 6 0.13889E-02 0.23148E-03 0.19841E-03 0.19841E-03 0.19841E-03 7 0.19841E-03 0.28345E-04 0.24802E-04 0.24802E-04 0.24802E-04 8 0.24802E-04 0.31002E-05 0.27557E-05 0.27557E-05 0.27557E-05 9 0.27557E-05 0.30619E-06 0.27557E-06 0.27557E-06 0.27557E-06 10 0.27557E-06 0.27557E-07 0.25052E-07 0.25052E-07 0.25052E-07 11 0.25052E-07 0.22775E-08 0.20877E-08 0.20877E-08 0.20877E-08 12 0.20877E-08 0.17397E-09 0.16059E-09 0.16059E-09 0.16059E-09 13 0.16059E-09 0.12353E-10 0.11471E-10 0.11471E-10 0.11471E-10 14 0.11471E-10 0.81934E-12 0.76472E-12 0.76472E-12 0.76472E-12 15 0.76472E-12 0.50981E-13 0.47795E-13 0.47795E-13 0.47795E-13 16 0.47795E-13 0.29872E-14 0.28115E-14 0.28115E-14 0.28115E-14 Check the integral of X^2 N Volume #1 #3 #5 #5.2 2 0.50000 0.12500 0.83333E-01 0.83333E-01 0.83333E-01 3 0.16667 0.18519E-01 0.16667E-01 0.16667E-01 0.16667E-01 4 0.41667E-01 0.26042E-02 0.27778E-02 0.27778E-02 0.27778E-02 5 0.83333E-02 0.33333E-03 0.39683E-03 0.39683E-03 0.39683E-03 6 0.13889E-02 0.38580E-04 0.49603E-04 0.49603E-04 0.49603E-04 7 0.19841E-03 0.40492E-05 0.55115E-05 0.55115E-05 0.55115E-05 8 0.24802E-04 0.38752E-06 0.55115E-06 0.55115E-06 0.55115E-06 9 0.27557E-05 0.34021E-07 0.50104E-07 0.50104E-07 0.50104E-07 10 0.27557E-06 0.27557E-08 0.41754E-08 0.41754E-08 0.41754E-08 11 0.25052E-07 0.20704E-09 0.32118E-09 0.32118E-09 0.32118E-09 12 0.20877E-08 0.14498E-10 0.22941E-10 0.22941E-10 0.22941E-10 13 0.16059E-09 0.95024E-12 0.15294E-11 0.15294E-11 0.15294E-11 14 0.11471E-10 0.58524E-13 0.95590E-13 0.95590E-13 0.95590E-13 15 0.76472E-12 0.33987E-14 0.56229E-14 0.56229E-14 0.56229E-14 16 0.47795E-13 0.18670E-15 0.31238E-15 0.31238E-15 0.31238E-15 Check the integral of X^3 N Volume #1 #3 #5 #5.2 2 0.50000 0.62500E-01 0.50000E-01 0.50000E-01 0.50000E-01 3 0.16667 0.61728E-02 0.83333E-02 0.83333E-02 0.83333E-02 4 0.41667E-01 0.65104E-03 0.11905E-02 0.11905E-02 0.11905E-02 5 0.83333E-02 0.66667E-04 0.14881E-03 0.14881E-03 0.14881E-03 6 0.13889E-02 0.64300E-05 0.16534E-04 0.16534E-04 0.16534E-04 7 0.19841E-03 0.57846E-06 0.16534E-05 0.16534E-05 0.16534E-05 8 0.24802E-04 0.48441E-07 0.15031E-06 0.15031E-06 0.15031E-06 9 0.27557E-05 0.37802E-08 0.12526E-07 0.12526E-07 0.12526E-07 10 0.27557E-06 0.27557E-09 0.96354E-09 0.96354E-09 0.96354E-09 11 0.25052E-07 0.18822E-10 0.68824E-10 0.68824E-10 0.68824E-10 12 0.20877E-08 0.12081E-11 0.45883E-11 0.45883E-11 0.45883E-11 13 0.16059E-09 0.73095E-13 0.28677E-12 0.28677E-12 0.28677E-12 14 0.11471E-10 0.41803E-14 0.16869E-13 0.16869E-13 0.16869E-13 15 0.76472E-12 0.22658E-15 0.93715E-15 0.93715E-15 0.93715E-15 16 0.47795E-13 0.11669E-16 0.49324E-16 0.49324E-16 0.49324E-16 Check the integral of X^4 N Volume #1 #3 #5 #5.2 2 0.50000 0.31250E-01 0.31111E-01 0.33333E-01 0.33333E-01 3 0.16667 0.20576E-02 0.43403E-02 0.47619E-02 0.47619E-02 4 0.41667E-01 0.16276E-03 0.53288E-03 0.59524E-03 0.59524E-03 5 0.83333E-02 0.13333E-04 0.58387E-04 0.66138E-04 0.66138E-04 6 0.13889E-02 0.10717E-05 0.57739E-05 0.66138E-05 0.66138E-05 7 0.19841E-03 0.82638E-07 0.52014E-06 0.60125E-06 0.60125E-06 8 0.24802E-04 0.60551E-08 0.43019E-07 0.50104E-07 0.50104E-07 9 0.27557E-05 0.42002E-09 0.32881E-08 0.38542E-08 0.38542E-08 10 0.27557E-06 0.27557E-10 0.23359E-09 0.27530E-09 0.27530E-09 11 0.25052E-07 0.17111E-11 0.15499E-10 0.18353E-10 0.18353E-10 12 0.20877E-08 0.10068E-12 0.96472E-12 0.11471E-11 0.11471E-11 13 0.16059E-09 0.56227E-14 0.56543E-13 0.67475E-13 0.67475E-13 14 0.11471E-10 0.29859E-15 0.31312E-14 0.37486E-14 0.37486E-14 15 0.76472E-12 0.15106E-16 0.16433E-15 0.19730E-15 0.19730E-15 16 0.47795E-13 0.72929E-18 0.81952E-17 0.98648E-17 0.98648E-17 Check the integral of X^5 N Volume #1 #3 #5 #5.2 2 0.50000 0.15625E-01 0.19259E-01 0.23810E-01 0.23810E-01 3 0.16667 0.68587E-03 0.22425E-02 0.29762E-02 0.29762E-02 4 0.41667E-01 0.40690E-04 0.23615E-03 0.33069E-03 0.33069E-03 5 0.83333E-02 0.26667E-05 0.22649E-04 0.33069E-04 0.33069E-04 6 0.13889E-02 0.17861E-06 0.19913E-05 0.30063E-05 0.30063E-05 7 0.19841E-03 0.11805E-07 0.16147E-06 0.25052E-06 0.25052E-06 8 0.24802E-04 0.75688E-09 0.12141E-07 0.19271E-07 0.19271E-07 9 0.27557E-05 0.46669E-10 0.85073E-09 0.13765E-08 0.13765E-08 10 0.27557E-06 0.27557E-11 0.55789E-10 0.91766E-10 0.91766E-10 11 0.25052E-07 0.15555E-12 0.34375E-11 0.57354E-11 0.57354E-11 12 0.20877E-08 0.83899E-14 0.19970E-12 0.33737E-12 0.33737E-12 13 0.16059E-09 0.43252E-15 0.10973E-13 0.18743E-13 0.18743E-13 14 0.11471E-10 0.21328E-16 0.57193E-15 0.98648E-15 0.98648E-15 15 0.76472E-12 0.10070E-17 0.28348E-16 0.49324E-16 0.49324E-16 16 0.47795E-13 0.45581E-19 0.13394E-17 0.23488E-17 0.23488E-17 TEST30 For integrals on the unit sphere in 3D: SPHERE_UNIT_07_3D uses a formula of degree 7. SPHERE_UNIT_11_3D uses a formula of degree 11. SPHERE_UNIT_14_3D uses a formula of degree 14. SPHERE_UNIT_15_3D uses a formula of degree 15. Unit sphere area = 12.5664 F(X) S3S07 S3S11 S3S14 S3S15 1 12.56637061 12.56637061 12.56637061 12.56637061 X -0.00000000 0.00000000 0.00000000 -0.00000000 Y -0.00000000 -0.00000000 -0.00000000 -0.00000000 Z 0.00000000 -0.00000000 -0.00000000 0.00000000 X*X 4.18879020 4.18879020 4.18879020 4.18879020 X*Y 0.00000000 -0.00000000 -0.00000000 -0.00000000 X*Z -0.00000000 -0.00000000 -0.00000000 -0.00000000 Y*Y 4.18879020 4.18879020 4.18879020 4.18879020 Y*Z -0.00000000 0.00000000 -0.00000000 0.00000000 Z*Z 4.18879020 4.18879020 4.18879020 4.18879020 X^3 -0.00000000 0.00000000 0.00000000 -0.00000000 X*Y*Z -0.00000000 0.00000000 -0.00000000 0.00000000 Z*Z*Z 0.00000000 0.00000000 0.00000000 -0.00000000 X^4 2.51327412 2.51327412 2.51327412 2.51327412 X^2 Z^2 0.83775804 0.83775804 0.83775804 0.83775804 Z^4 2.51327412 2.51327412 2.51327412 2.51327412 X^5 -0.00000000 0.00000000 0.00000000 -0.00000000 X^6 1.79519580 1.79519580 1.79519580 1.79519580 R 12.56637061 12.56637061 12.56637061 12.56637061 SIN(X) 0.00000000 -0.00000000 0.00000000 -0.00000000 EXP(X) 14.76801224 14.76801375 14.76801375 14.76801375 1/(1+R) 8.88576588 8.88576588 8.88576588 8.88576588 SQRT(R) 12.56637061 12.56637061 12.56637061 12.56637061 TEST31 For integrals on the unit sphere in ND: SPHERE_UNIT_03_ND uses a formula of degree 3; SPHERE_UNIT_04_ND uses a formula of degree 4; SPHERE_UNIT_05_ND uses a formula of degree 5. SPHERE_UNIT_07_1_ND uses a formula of degree 7. SPHERE_UNIT_07_2_ND uses a formula of degree 7. SPHERE_UNIT_11_ND uses a formula of degree 11. Spatial dimension N = 3 Unit sphere area = 12.5664 Rule: #3 #4 #5 #7.1 #7.2 #11 Function 1 12.56637061 12.56637061 12.56637061 12.56637061 12.56637061 12.56637061 X 0.00000000 -0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^2 4.18879020 4.18879020 4.18879020 4.18879020 4.18879020 4.18879020 X^3 0.00000000 -0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^4 4.18879020 2.51327412 2.51327412 2.51327412 2.51327412 2.51327412 X^5 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^6 4.18879020 1.67551608 1.95476876 1.79519580 1.79519580 1.79519580 R 12.56637061 12.56637061 12.56637061 12.56637061 12.56637061 12.56637061 SIN(X) 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 EXP(X) 14.84122146 14.76784403 14.76824477 14.76801578 14.76801143 14.76801375 1/(1+R) 6.28318531 6.28318531 6.28318531 6.28318531 6.28318531 6.28318531 SQRT(R) 12.56637061 12.56637061 12.56637061 12.56637061 12.56637061 12.56637061 Spatial dimension N = 4 Unit sphere area = 19.7392 Rule: #3 #4 #5 #7.1 #7.2 #11 Function 1 19.73920880 19.73920880 19.73920880 19.73920880 19.73920880 19.73920880 X 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^2 4.93480220 4.93480220 4.93480220 4.93480220 4.93480220 4.93480220 X^3 0.00000000 -0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^4 4.93480220 2.46740110 2.46740110 2.46740110 2.46740110 2.46740110 X^5 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^6 4.93480220 1.23370055 1.85055083 1.54212569 1.54212569 1.54212569 R 19.73920880 19.73920880 19.73920880 19.73920880 19.73920880 19.73920880 SIN(X) 0.00000000 -0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 EXP(X) 22.41920431 22.31114714 22.31203102 22.31158908 22.31158468 22.31158712 1/(1+R) 9.86960440 9.86960440 9.86960440 9.86960440 9.86960440 9.86960440 SQRT(R) 19.73920880 19.73920880 19.73920880 19.73920880 19.73920880 19.73920880 Spatial dimension N = 5 Unit sphere area = 26.3189 Rule: #3 #4 #5 #7.1 #7.2 #11 Function 1 26.31894507 26.31894507 26.31894507 26.31894507 26.31894507 26.31894507 X 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^2 5.26378901 5.26378901 5.26378901 5.26378901 5.26378901 5.26378901 X^3 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^4 5.26378901 2.25590958 2.25590958 2.25590958 2.25590958 2.25590958 X^5 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^6 5.26378901 0.75196986 1.65433369 1.25328310 1.25328310 1.25328310 R 26.31894507 26.31894507 26.31894507 26.31894507 26.31894507 26.31894507 SIN(X) 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 EXP(X) 29.17760695 29.04588011 29.04717196 29.04659780 29.04659423 29.04659641 1/(1+R) 13.15947253 13.15947253 13.15947253 13.15947253 13.15947253 13.15947253 SQRT(R) 26.31894507 26.31894507 26.31894507 26.31894507 26.31894507 26.31894507 Spatial dimension N = 6 Unit sphere area = 31.0063 Rule: #3 #4 #5 #7.1 #7.2 #11 Function 1 31.00627668 31.00627668 31.00627668 31.00627668 31.00627668 31.00627668 X 0.00000000 -0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^2 5.16771278 5.16771278 5.16771278 5.16771278 5.16771278 5.16771278 X^3 0.00000000 -0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^4 5.16771278 1.93789229 1.93789229 1.93789229 1.93789229 1.93789229 X^5 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^6 5.16771278 0.32298205 1.39958888 0.96894615 0.96894615 0.96894615 R 31.00627668 31.00627668 31.00627668 31.00627668 31.00627668 31.00627668 SIN(X) 0.00000000 -0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 EXP(X) 33.81276142 33.67131491 33.67285530 33.67223914 33.67223669 33.67223846 1/(1+R) 15.50313834 15.50313834 15.50313834 15.50313834 15.50313834 15.50313834 SQRT(R) 31.00627668 31.00627668 31.00627668 31.00627668 31.00627668 31.00627668 Spatial dimension N = 7 Unit sphere area = 33.0734 Rule: #3 #4 #5 #7.1 #7.2 #11 Function 1 33.07336179 33.07336179 33.07336179 33.07336179 33.07336179 33.07336179 X 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^2 4.72476597 4.72476597 4.72476597 4.72476597 4.72476597 4.72476597 X^3 0.00000000 -0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^4 4.72476597 1.57492199 1.57492199 1.57492199 1.57492199 1.57492199 X^5 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^6 4.72476597 -0.00000000 1.12494428 0.71587363 0.71587363 0.71587363 R 33.07336179 33.07336179 33.07336179 33.07336179 33.07336179 33.07336179 SIN(X) 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 EXP(X) 35.63929069 35.50134667 35.50295555 35.50237050 35.50236909 35.50237042 1/(1+R) 16.53668090 16.53668090 16.53668090 16.53668090 16.53668090 16.53668090 SQRT(R) 33.07336179 33.07336179 33.07336179 33.07336179 33.07336179 33.07336179 Spatial dimension N = 8 Unit sphere area = 32.4697 Rule: #3 #4 #5 #7.1 #7.2 #11 Function 1 32.46969701 32.46969701 32.46969701 32.46969701 32.46969701 32.46969701 X 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^2 4.05871213 4.05871213 4.05871213 4.05871213 4.05871213 4.05871213 X^3 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^4 4.05871213 1.21761364 1.21761364 1.21761364 1.21761364 1.21761364 X^5 0.00000000 -0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^6 4.05871213 -0.20293561 0.86247633 0.50733902 0.50733902 0.50733902 R 32.46969701 32.46969701 32.46969701 32.46969701 32.46969701 32.46969701 SIN(X) 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 EXP(X) 34.67390497 34.54948212 34.55100538 34.55049762 34.55049700 34.55049794 1/(1+R) 16.23484851 16.23484851 16.23484851 16.23484851 16.23484851 16.23484851 SQRT(R) 32.46969701 32.46969701 32.46969701 32.46969701 32.46969701 32.46969701 Spatial dimension N = 9 Unit sphere area = 29.6866 Rule: #3 #4 #5 #7.1 #7.2 #11 Function 1 29.68658012 29.68658012 29.68658012 29.68658012 29.68658012 29.68658013 X 0.00000000 -0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^2 3.29850890 3.29850890 3.29850890 3.29850890 3.29850890 3.29850890 X^3 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^4 3.29850890 0.89959334 0.89959334 0.89959334 0.89959334 0.89959334 X^5 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^6 3.29850890 -0.29986445 0.63304716 0.34599744 0.34599744 0.34599744 R 29.68658012 29.68658012 29.68658012 29.68658012 29.68658012 29.68658013 SIN(X) 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 EXP(X) 31.47793643 31.37287851 31.37421200 31.37380169 31.37380158 31.37380221 1/(1+R) 14.84329006 14.84329006 14.84329006 14.84329006 14.84329006 14.84329006 SQRT(R) 29.68658012 29.68658012 29.68658012 29.68658012 29.68658012 29.68658013 Spatial dimension N = 10 Unit sphere area = 25.5016 Rule: #3 #4 #5 #7.1 #7.2 #11 Function 1 25.50164040 25.50164040 25.50164040 25.50164040 25.50164040 25.50164040 X 0.00000000 -0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^2 2.55016404 2.55016404 2.55016404 2.55016404 2.55016404 2.55016404 X^3 0.00000000 -0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^4 2.55016404 0.63754101 0.63754101 0.63754101 0.63754101 0.63754101 X^5 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 X^6 2.55016404 -0.31877050 0.44627871 0.22769322 0.22769322 0.22769322 R 25.50164040 25.50164040 25.50164040 25.50164040 25.50164040 25.50164040 SIN(X) 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 EXP(X) 26.88658510 26.80282384 26.80391717 26.80360479 26.80360494 26.80360535 1/(1+R) 12.75082020 12.75082020 12.75082020 12.75082020 12.75082020 12.75082020 SQRT(R) 25.50164040 25.50164040 25.50164040 25.50164040 25.50164040 25.50164040 TEST32 For integrals on a sphere in ND: SPHERE_05_ND uses a formula of degree 5. SPHERE_07_1_ND uses a formula of degree 7. Spatial dimension N = 2 Sphere center = 1.00000 1.00000 Sphere radius = 2.00000 Sphere area = 12.5664 Rule: #5 #7.1 Function 1 12.56637061 12.56637061 X 12.56637061 12.56637061 X^2 37.69911184 37.69911184 X^3 87.96459430 87.96459430 X^4 238.76104167 238.76104167 X^5 640.88490133 640.88490133 X^6 1771.85825662 1771.85825662 R 28.35595549 28.35595549 SIN(X) 2.36794306 2.36794306 EXP(X) 77.87010284 77.87010284 1/(1+R) 4.31320569 4.31320569 SQRT(R) 18.34418070 18.34418070 Spatial dimension N = 3 Sphere center = 1.00000 1.00000 1.00000 Sphere radius = 2.00000 Sphere area = 50.2655 Rule: #5 #7.1 Function 1 50.26548246 50.26548246 X 50.26548246 50.26548246 X^2 117.28612573 117.28612573 X^3 251.32741229 251.32741229 X^4 613.23888598 613.23888598 X^5 1524.71963454 1524.71963454 X^6 3968.73909269 3927.88841489 R 124.83633168 125.49541167 SIN(X) 19.18995050 19.23178699 EXP(X) 247.96112298 247.78550279 1/(1+R) 16.14276841 15.70777500 SQRT(R) 77.17260732 77.79293076 Spatial dimension N = 4 Sphere center = 1.00000 1.00000 1.00000 1.00000 Sphere radius = 2.00000 Sphere area = 157.914 Rule: #5 #7.1 Function 1 157.91367042 157.91367042 X 157.91367042 157.91367042 X^2 315.82734083 315.82734083 X^3 631.65468167 631.65468167 X^4 1421.22303376 1421.22303376 X^5 3316.18707877 3316.18707877 X^6 8211.51086171 8053.59719129 R 425.59955279 427.62368030 SIN(X) 76.47531628 76.63804172 EXP(X) 683.47376394 682.79877244 1/(1+R) 47.54493620 46.08802094 SQRT(R) 251.96536302 254.74523082 TEST322 SPHERE_CAP_AREA_3D computes the volume of a 3D spherical cap, defined by a plane that cuts the sphere to a thickness of H units. SPHERE_CAP_AREA_ND computes the volume of an ND spherical cap, defined by a plane that cuts the sphere to a thickness of H units. Area of the total sphere in 3D = 12.5664 R H Cap Cap area_3d area_nd 1.000000 0.000000 0.000000 0.000000 1.000000 0.166667 1.047198 1.047198 1.000000 0.333333 2.094395 2.094395 1.000000 0.500000 3.141593 3.141593 1.000000 0.666667 4.188790 4.188790 1.000000 0.833333 5.235988 5.235988 1.000000 1.000000 6.283185 6.283185 1.000000 1.166667 7.330383 7.330383 1.000000 1.333333 8.377580 8.377580 1.000000 1.500000 9.424778 9.424778 1.000000 1.666667 10.471976 10.471976 1.000000 1.833333 11.519173 11.519173 1.000000 2.000000 12.566371 12.566371 1.000000 2.166667 12.566371 12.566371 TEST324 SPHERE_CAP_VOLUME_2D computes the volume (area) of a spherical cap, defined by a plane that cuts the sphere to a thickness of H units. SPHERE_CAP_VOLUME_ND does the same operation, but in N dimensions. Using a radius R = 1.00000 Volume of the total sphere in 2D = 3.14159 H Cap Cap vol_2d vol_nd 0.000000 0.000000 0.000000 0.166667 0.125043 0.125043 0.333333 0.344165 0.344165 0.500000 0.614185 0.614185 0.666667 0.916690 0.916690 0.833333 1.239013 1.239013 1.000000 1.570796 1.570796 1.166667 1.902580 1.902580 1.333333 2.224903 2.224903 1.500000 2.527408 2.527408 1.666667 2.797428 2.797428 1.833333 3.016549 3.016549 2.000000 3.141593 3.141593 2.166667 3.141593 3.141593 TEST326 SPHERE_CAP_VOLUME_3D computes the volume of a spherical cap, defined by a plane that cuts the sphere to a thickness of H units. SPHERE_CAP_VOLUME_ND does the same operation, but in N dimensions. Using a radius R = 1.00000 Volume of the total sphere in 3D = 4.18879 H Cap Cap volume_3d volume_nd 0.000000 0.000000 0.000000 0.166667 0.082418 0.082418 0.333333 0.310281 0.310281 0.500000 0.654498 0.654498 0.666667 1.085983 1.085983 0.833333 1.575644 1.575644 1.000000 2.094395 2.094395 1.166667 2.613146 2.613146 1.333333 3.102808 3.102808 1.500000 3.534292 3.534292 1.666667 3.878509 3.878509 1.833333 4.106372 4.106372 2.000000 4.188790 4.188790 2.166667 4.188790 4.188790 TEST33 For a sphere in ND: SPHERE_CAP_AREA_ND computes the area of a spherical cap. SPHERE_CAP_VOLUME_ND computes the volume of a spherical cap. Spatial dimension N = 2 Radius = 1.00000 Area = 6.28319 Volume = 3.14159 Sphere Sphere cap cap H area volume 0.0000 0.00000 0.00000 0.1667 1.17137 0.125043 0.3333 1.68214 0.344165 0.5000 2.09440 0.614185 0.6667 2.46192 0.916690 0.8333 2.80670 1.23901 1.0000 3.14159 1.57080 1.1667 3.47649 1.90258 1.3333 3.82127 2.22490 1.5000 4.18879 2.52741 1.6667 4.60105 2.79743 1.8333 5.11181 3.01655 2.0000 6.28319 3.14159 2.1667 6.28319 3.14159 Spatial dimension N = 3 Radius = 1.00000 Area = 12.5664 Volume = 4.18879 Sphere Sphere cap cap H area volume 0.0000 0.00000 0.00000 0.1667 1.04720 0.824183E-01 0.3333 2.09440 0.310281 0.5000 3.14159 0.654498 0.6667 4.18879 1.08598 0.8333 5.23599 1.57564 1.0000 6.28319 2.09440 1.1667 7.33038 2.61315 1.3333 8.37758 3.10281 1.5000 9.42478 3.53429 1.6667 10.4720 3.87851 1.8333 11.5192 4.10637 2.0000 12.5664 4.18879 2.1667 12.5664 4.18879 Spatial dimension N = 4 Radius = 1.00000 Area = 19.7392 Volume = 4.93480 Sphere Sphere cap cap H area volume 0.0000 0.00000 0.00000 0.1667 0.785670 0.490225E-01 0.3333 2.16245 0.251526 0.5000 3.85904 0.624672 0.6667 5.75973 1.14740 0.8333 7.78495 1.77893 1.0000 9.86960 2.46740 1.1667 11.9543 3.15588 1.3333 13.9795 3.78740 1.5000 15.8802 4.31013 1.6667 17.5768 4.68328 1.8333 18.9535 4.88578 2.0000 19.7392 4.93480 2.1667 19.7392 4.93480 Spatial dimension N = 5 Radius = 1.00000 Area = 26.3189 Volume = 5.26379 Sphere Sphere cap cap H area volume 0.0000 0.00000 0.00000 0.1667 0.517850 0.267809E-01 0.3333 1.94955 0.186832 0.5000 4.11234 0.544884 0.6667 6.82343 1.10475 0.8333 9.90007 1.82453 1.0000 13.1595 2.63189 1.1667 16.4189 3.43926 1.3333 19.4955 4.15904 1.5000 22.2066 4.71890 1.6667 24.3694 5.07696 1.8333 25.8011 5.23701 2.0000 26.3189 5.26379 2.1667 26.3189 5.26379 TEST335 For integrals inside a spherical shell in ND: SPHERE_SHELL_03_ND approximates the integral. We compare these results with those computed by from the difference of two ball integrals: BALL_F1_ND approximates the integral; BALL_F3_ND approximates the integral. Spatial dimension N = 2 Sphere center: 0.0000 0.0000 Inner sphere radius = 0.00000 Outer sphere radius = 1.00000 Spherical shell volume = 3.14159 Rule: #3 F1(R2)-F1(R1) F3(R2)-F3(R1) F(X) 1 3.14159265 3.14159265 3.14159265 X 0.00000000 0.00000000 0.00000000 X^2 0.78539816 0.78539816 0.78539816 X^3 0.00000000 0.00000000 0.00000000 X^4 0.39269908 0.39269908 0.39269908 X^5 0.00000000 0.00000000 0.00000000 X^6 0.19634954 0.22974928 0.19634954 R 2.22144147 2.07465287 1.92382475 SIN(X) 0.00000000 0.00000000 0.00000000 EXP(X) 3.55092935 3.55097673 3.55092935 1/(1+R) 1.84030237 1.94254245 2.08250740 SQRT(R) 2.64175400 2.49488479 2.12906206 Spatial dimension N = 3 Sphere center: 0.0000 0.0000 0.0000 Inner sphere radius = 0.00000 Outer sphere radius = 1.00000 Spherical shell volume = 4.18879 Rule: #3 F1(R2)-F1(R1) F3(R2)-F3(R1) F(X) 1 4.18879020 4.18879020 4.18879020 X 0.00000000 0.00000000 0.00000000 X^2 0.83775804 0.83775804 0.83775804 X^3 0.00000000 0.00000000 0.00000000 X^4 0.50265482 0.35903916 0.35903916 X^5 0.00000000 0.00000000 0.00000000 X^6 0.30159289 0.19474148 0.15387393 R 3.24462294 3.12358869 2.97374605 SIN(X) 0.00000000 0.00000000 0.00000000 EXP(X) 4.62903657 4.62290241 4.62284455 1/(1+R) 2.36041816 2.44032616 2.57713848 SQRT(R) 3.68660342 3.57254308 3.23471399 Spatial dimension N = 2 Sphere center: 1.0000 -1.0000 Inner sphere radius = 2.00000 Outer sphere radius = 3.00000 Spherical shell volume = 15.7080 Rule: #3 F1(R2)-F1(R1) F3(R2)-F3(R1) F(X) 1 15.70796327 15.70796327 15.70796327 X 15.70796327 15.70796327 15.70796327 X^2 66.75884389 66.75884389 66.75884389 X^3 168.86060513 168.86060513 168.86060513 X^4 653.84397103 583.15813632 583.15813632 X^5 2185.37038965 1831.94121612 1831.94121612 X^6 7915.83173934 6147.21369777 5936.62836758 R 43.44712361 42.93767114 43.45479571 SIN(X) 1.12496633 -0.64096453 -0.43880414 EXP(X) 158.82781270 147.70348235 146.73973318 1/(1+R) 4.43687726 4.62868555 4.31425177 SQRT(R) 25.74801054 25.42033572 25.86691267 Spatial dimension N = 3 Sphere center: 1.0000 -1.0000 2.0000 Inner sphere radius = 2.00000 Outer sphere radius = 3.00000 Spherical shell volume = 79.5870 Rule: #3 F1(R2)-F1(R1) F3(R2)-F3(R1) F(X) 1 79.58701389 79.58701389 79.58701389 X 79.58701389 79.58701389 79.58701389 X^2 256.35396053 256.35396053 256.35396053 X^3 609.88785382 609.88785382 609.88785382 X^4 2318.01476979 1879.45032503 1879.45032503 X^5 7736.38686054 5543.56463673 5543.56463673 X^628246.5234714517553.4046726616769.93271603 R 268.54232501 266.43714983 267.89953673 SIN(X) 25.73653131 14.53414127 15.30079367 EXP(X) 623.42331946 555.79214858 552.27133790 1/(1+R) 19.70995483 20.67027756 19.83900758 SQRT(R) 143.86069457 142.33148757 143.55755569 TEST34: SPHERE_UNIT_AREA_ND evaluates the area of the unit sphere in N dimensions. SPHERE_UNIT_AREA_VALUES returns some test values. dim_num Exact Computed Area Area 1 2.000000 2.000000 2 6.283185 6.283185 3 12.566371 12.566371 4 19.739209 19.739209 5 26.318945 26.318945 6 31.006277 31.006277 7 33.073362 33.073362 8 32.469697 32.469697 9 29.686580 29.686580 10 25.501640 25.501640 11 20.725143 20.725143 12 16.023153 16.023153 13 11.838174 11.838174 14 8.389703 8.389703 15 5.721649 5.721649 16 3.765290 3.765290 17 2.396679 2.396679 18 1.478626 1.478626 19 0.885810 0.885810 20 0.516138 0.516138 TEST345: SPHERE_UNIT_VOLUME_ND evaluates the area of the unit sphere in N dimensions. SPHERE_UNIT_VOLUME_VALUES returns some test values. dim_num Exact Computed Volume Volume 1 2.000000 2.000000 2 3.141593 3.141593 3 4.188790 4.188790 4 4.934802 4.934802 5 5.263789 5.263789 6 5.167713 5.167713 7 4.724766 4.724766 8 4.058712 4.058712 9 3.298509 3.298509 10 2.550164 2.550164 11 1.884104 1.884104 12 1.335263 1.335263 13 0.910629 0.910629 14 0.599265 0.599265 15 0.381443 0.381443 16 0.235331 0.235331 17 0.140981 0.140981 18 0.082146 0.082146 19 0.046622 0.046622 20 0.025807 0.025807 TEST35 SQUARE_UNIT_SET sets up a quadrature rule on a unit square. RECTANGLE_SUB_2D applies it to subrectangles of an arbitrary rectangle. The corners of the rectangle are: 1.00000 2.00000 3.00000 3.00000 Using unit square integration rule number 2 Order of rule is 4 Function Subdivisions Integral 1 1 2 2.00000000 1 2 4 2.00000000 1 3 6 2.00000000 1 4 8 2.00000000 1 5 10 2.00000000 Function Subdivisions Integral X 1 2 4.00000000 X 2 4 4.00000000 X 3 6 4.00000000 X 4 8 4.00000000 X 5 10 4.00000000 Function Subdivisions Integral X^2 1 2 8.66666667 X^2 2 4 8.66666667 X^2 3 6 8.66666667 X^2 4 8 8.66666667 X^2 5 10 8.66666667 Function Subdivisions Integral X^3 1 2 20.00000000 X^3 2 4 20.00000000 X^3 3 6 20.00000000 X^3 4 8 20.00000000 X^3 5 10 20.00000000 Function Subdivisions Integral X^4 1 2 48.22222222 X^4 2 4 48.38888889 X^4 3 6 48.39780521 X^4 4 8 48.39930556 X^4 5 10 48.39971556 Function Subdivisions Integral X^5 1 2 119.55555556 X^5 2 4 121.22222222 X^5 3 6 121.31138546 X^5 4 8 121.32638889 X^5 5 10 121.33048889 Function Subdivisions Integral X^6 1 2 301.40740741 X^6 2 4 311.57407407 X^6 3 6 312.14398212 X^6 4 8 312.24074074 X^6 5 10 312.26726874 Function Subdivisions Integral R 1 2 6.47667970 R 2 4 6.47704736 R 3 6 6.47706336 R 4 8 6.47706591 R 5 10 6.47706659 Function Subdivisions Integral SIN(X) 1 2 1.52382214 SIN(X) 2 4 1.52992902 SIN(X) 3 6 1.53022383 SIN(X) 4 8 1.53027249 SIN(X) 5 10 1.53028569 Function Subdivisions Integral EXP(X) 1 2 17.31031282 EXP(X) 2 4 17.36335922 EXP(X) 3 6 17.36647206 EXP(X) 4 8 17.36700581 EXP(X) 5 10 17.36715270 Function Subdivisions Integral 1/(1+R) 1 2 0.47677239 1/(1+R) 2 4 0.47667751 1/(1+R) 3 6 0.47667300 1/(1+R) 4 8 0.47667227 1/(1+R) 5 10 0.47667207 Function Subdivisions Integral SQRT(R) 1 2 3.59120476 SQRT(R) 2 4 3.59140707 SQRT(R) 3 6 3.59141625 SQRT(R) 4 8 3.59141772 SQRT(R) 5 10 3.59141812 TEST36 SQUARE_UNIT_SET sets up quadrature on the unit square; SQUARE_SUM carries it out on an arbitrary square. Square center: CENTER = 2.00000 2.00000 Square radius is 3.00000 Rule: 1 2 3 4 5 Function 1 36.0000000 36.0000000 36.0000000 36.0000000 36.0000000 X 72.0000000 72.0000000 72.0000000 72.0000000 72.0000000 X^2 144.0000000 252.0000000 252.0000000 252.0000000 252.0000000 X^3 288.0000000 936.0000000 936.0000000 936.0000000 936.0000000 X^4 576.0000000 3492.0000000 3751.2000000 3751.2000000 3751.2000000 X^5 1152.000000013032.000000015624.000000015624.000000015624.0000000 X^6 2304.000000048636.000000066365.280000066965.142857166965.1428571 R 101.8233765 118.2615212 122.8144082 122.5068234 123.0043634 SIN(X) 32.7347074 -5.2557713 2.1127317 1.5258634 1.5494465 EXP(X) 266.0060196 775.2951436 880.2002636 888.0933305 888.3797409 1/(1+R) 9.4033395 11.7565024 9.6739772 9.6917800 9.4769384 SQRT(R) 60.5445419 61.0346112 64.3298985 64.2286949 64.5720321 Rule: 6 Function 1 36.0000000 X 72.0000000 X^2 252.0000000 X^3 936.0000000 X^4 3751.2000000 X^515624.0000000 X^666965.1428571 R 122.4955145 SIN(X) 1.5398407 EXP(X) 888.2716780 1/(1+R) 9.7166397 SQRT(R) 64.2021316 TEST37 SQUARE_UNIT_SET sets up quadrature on the unit square; SQUARE_UNIT_SUM carries it out on the unit square. Rule: 1 2 3 4 5 Function 1 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 X 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 X^2 0.0000000 0.3333333 0.3333333 0.3333333 0.3333333 X^3 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 X^4 0.0000000 0.1111111 0.2000000 0.2000000 0.2000000 X^5 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 X^6 0.0000000 0.0370370 0.1200000 0.1428571 0.1428571 R 0.0000000 0.8164966 0.7206172 0.7748323 0.7734600 SIN(X) 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 EXP(X) 1.0000000 1.1713480 1.1751685 1.1752011 1.1752012 1/(1+R) 1.0000000 0.5505103 0.6230984 0.5750525 0.5765221 SQRT(R) 0.0000000 0.9036020 0.7576586 0.8682289 0.8634664 Rule: 6 Function 1 1.0000000 X -0.0000000 X^2 0.3333333 X^3 0.0000000 X^4 0.2000000 X^5 0.0000000 X^6 0.1428571 R 0.7660810 SIN(X) 0.0000000 EXP(X) 1.1752012 1/(1+R) 0.5823041 SQRT(R) 0.8575418 TEST38 For integrals inside an arbitrary tetrahedron: TETRA_07 uses a formula of degree 7; TETRA_TPRODUCT uses a triangular product formula of varying degree. Tetrahedron vertices: 1. 2. 6. 4. 2. 6. 1. 3. 6. 1. 2. 8. Tetrahedron unit volume = 0.166667 Tetrahedron Volume = 1.00000 F(X) TETRA_07 TETRA_TPRODUCT(1:4) TETRA_TPRODUCT(5:8) TETRA_TPRODUCT(9) 1 0.9999999997 1.0000000000 1.0000000000 1.0000000000 1.0000000000 1.0000000000 1.0000000000 1.0000000000 1.0000000000 1.0000000000 X 1.7499999994 1.7500000000 1.7500000000 1.7500000000 1.7500000000 1.7500000000 1.7500000000 1.7500000000 1.7500000000 1.7500000000 Y 2.2499999993 2.2500000000 2.2500000000 2.2500000000 2.2500000000 2.2500000000 2.2500000000 2.2500000000 2.2500000000 2.2500000000 Z 6.4999999981 6.5000000000 6.5000000000 6.5000000000 6.5000000000 6.5000000000 6.5000000000 6.5000000000 6.5000000000 6.5000000000 X*X 3.3999999985 3.0625000000 3.4000000000 3.4000000000 3.4000000000 3.4000000000 3.4000000000 3.4000000000 3.4000000000 3.4000000000 X*Y 3.8999999987 3.9375000000 3.9000000000 3.9000000000 3.9000000000 3.9000000000 3.9000000000 3.9000000000 3.9000000000 3.9000000000 X*Z 11.2999999961 11.3750000000 11.3000000000 11.3000000000 11.3000000000 11.3000000000 11.3000000000 11.3000000000 11.3000000000 11.3000000000 Y*Y 5.0999999986 5.0625000000 5.1000000000 5.1000000000 5.1000000000 5.1000000000 5.1000000000 5.1000000000 5.1000000000 5.1000000000 Y*Z 14.5999999959 14.6250000000 14.6000000000 14.6000000000 14.6000000000 14.6000000000 14.6000000000 14.6000000000 14.6000000000 14.6000000000 Z*Z 42.3999999879 42.2500000000 42.4000000000 42.4000000000 42.4000000000 42.4000000000 42.4000000000 42.4000000000 42.4000000000 42.4000000000 X^3 7.2999999961 5.3593750000 7.3000000000 7.3000000000 7.3000000000 7.3000000000 7.3000000000 7.3000000000 7.3000000000 7.3000000000 X*Y*Z 25.1499999915 25.5937500000 25.1500000000 25.1500000000 25.1500000000 25.1500000000 25.1500000000 25.1500000000 25.1500000000 25.1500000000 Z*Z*Z 277.5999999221 274.6250000000 277.6000000000 277.6000000000 277.6000000000 277.6000000000 277.6000000000 277.6000000000 277.6000000000 277.6000000000 X^4 17.1142857031 9.3789062500 16.9888000000 17.1142857143 17.1142857143 17.1142857143 17.1142857143 17.1142857143 17.1142857143 17.1142857143 X^2 Z^2 140.1714285128 129.3906250000 140.1536000000 140.1714285714 140.1714285714 140.1714285714 140.1714285714 140.1714285714 140.1714285714 140.1714285714 Z^4 1824.4571423551 1785.0625000000 1824.4380444444 1824.4571428571 1824.4571428571 1824.4571428571 1824.4571428571 1824.4571428571 1824.4571428571 1824.4571428571 X^5 43.1607142522 16.4130859375 41.8393600000 43.1607142857 43.1607142857 43.1607142857 43.1607142857 43.1607142857 43.1607142857 43.1607142857 X^6 115.4285713245 28.7229003906 106.8387328000 115.3762280040 115.4285714286 115.4285714286 115.4285714286 115.4285714286 115.4285714286 115.4285714286 R 7.1274728916 7.0975347833 7.1275134646 7.1274727322 7.1274728937 7.1274728930 7.1274728930 7.1274728930 7.1274728930 7.1274728930 SIN(X) 0.8359612123 0.9839859469 0.8317403439 0.8360175131 0.8359612125 0.8359616476 0.8359616453 0.8359616454 0.8359616454 0.8359616454 EXP(X) 6.9983841175 5.7546026760 6.9525225440 6.9977000010 6.9983841208 6.9983898561 6.9983898868 6.9983898869 6.9983898869 6.9983898869 1/(1+R) 0.1391955081 0.1395160213 0.1391937051 0.1391955206 0.1391955081 0.1391955082 0.1391955082 0.1391955082 0.1391955082 0.1391955082 SQRT(R) 2.6690873652 2.6641198891 2.6690979682 2.6690873156 2.6690873660 2.6690873658 2.6690873658 2.6690873658 2.6690873658 2.6690873658 TEST39 TETRA_UNIT_SET sets quadrature rules for the unit tetrahedron; TETRA_UNIT_SUM applies them to the unit tetrahedron. Rule: 1 2 3 4 5 Function 1 0.166667 0.166667 0.166667 0.166667 0.166667 X 0.041667 0.041667 0.041667 0.041667 0.041667 Y 0.041667 0.041667 0.041667 0.041667 0.041667 Z 0.041667 0.041667 0.041667 0.041667 0.041667 X*X 0.010417 0.041667 0.016667 0.016667 0.016667 X*Y 0.010417 0.000000 0.008333 0.008333 0.008333 X*Z 0.010417 0.000000 0.008333 0.008333 0.008333 Y*Y 0.010417 0.041667 0.016667 0.016667 0.016667 Y*Z 0.010417 0.000000 0.008333 0.008333 0.008333 Z*Z 0.010417 0.041667 0.016667 0.016667 0.016667 X^3 0.002604 0.041667 0.008689 0.004167 0.008333 X*Y*Z 0.002604 0.000000 0.001508 0.000000 0.001389 Z*Z*Z 0.002604 0.041667 0.008689 0.004167 0.008333 X^4 0.000651 0.041667 0.004939 -0.002083 0.004340 X^2 Z^2 0.000651 0.000000 0.000576 0.002083 0.000637 Z^4 0.000651 0.041667 0.004939 -0.002083 0.004340 X^5 0.000163 0.041667 0.002871 -0.005208 0.002242 X^6 0.000041 0.041667 0.001678 -0.006771 0.001144 R 0.072169 0.125000 0.087120 0.095711 0.088289 SIN(X) 0.041234 0.035061 0.040242 0.040930 0.040296 EXP(X) 0.214004 0.238262 0.218347 0.217220 0.218257 1/(1+R) 0.152944 0.130055 0.146894 0.145081 0.146620 SQRT(R) 0.109673 0.125000 0.118586 0.129800 0.119842 Rule: 6 7 8 Function 1 0.166667 0.166667 0.027778 X 0.041667 0.041667 0.006944 Y 0.041667 0.041667 0.006944 Z 0.041667 0.041667 0.006944 X*X 0.016667 0.016667 0.002778 X*Y 0.008333 0.008333 0.001389 X*Z 0.008333 0.008333 0.001389 Y*Y 0.016667 0.016667 0.002778 Y*Z 0.008333 0.008333 0.001389 Z*Z 0.016667 0.016667 0.002778 X^3 0.008333 0.008333 0.001389 X*Y*Z 0.001389 0.001389 0.000231 Z*Z*Z 0.008333 0.008333 0.001389 X^4 0.005556 0.004762 0.000794 X^2 Z^2 0.000926 0.000794 0.000132 Z^4 0.005556 0.004762 0.000794 X^5 0.004630 0.002790 0.000505 X^6 0.004321 0.001507 0.000349 R 0.087184 0.087657 0.014643 SIN(X) 0.040316 0.040301 0.006717 EXP(X) 0.218333 0.218279 0.036380 1/(1+R) 0.147242 0.147032 0.024487 SQRT(R) 0.118235 0.119963 0.019925 TEST40 TETRA_UNIT_SET sets quadrature rules for the unit tetrahedron; TETRA_SUM applies them to an arbitrary tetrahedron. Tetrahedron vertices: 1.00 2.00 6.00 4.00 2.00 6.00 1.00 3.00 6.00 1.00 2.00 8.00 Rule: 1 2 3 4 5 Function 1 1.000000 1.000000 1.000000 1.000000 1.000000 X 1.750000 1.750000 1.750000 1.750000 1.750000 Y 2.250000 2.250000 2.250000 2.250000 2.250000 Z 6.500000 6.500000 6.500000 6.500000 6.500000 X*X 3.062500 4.750000 3.400000 3.400000 3.400000 X*Y 3.937500 3.750000 3.900000 3.900000 3.900000 X*Z 11.37500 11.00000 11.30000 11.30000 11.30000 Y*Y 5.062500 5.250000 5.100000 5.100000 5.100000 Y*Z 14.62500 14.50000 14.60000 14.60000 14.60000 Z*Z 42.25000 43.00000 42.40000 42.40000 42.40000 X^3 5.359375 16.75000 7.357652 6.625000 7.300000 X*Y*Z 25.59375 23.50000 25.15427 25.10000 25.15000 Z*Z*Z 274.6250 290.0000 277.6171 277.4000 277.6000 X^4 9.378906 64.75000 17.43106 11.08750 16.90937 X^2 Z^2 129.3906 178.0000 139.9536 142.4500 140.1375 Z^4 1785.062 1996.000 1824.884 1819.000 1824.417 X^5 16.41309 256.7500 44.01482 7.843750 41.06641 X^6 28.72290 1024.750 115.6153 -57.86563 102.2623 R 7.097535 7.243848 7.127426 7.128685 7.127528 SIN(X) 0.9839859 0.4419026 0.8348421 0.7755854 0.8287419 EXP(X) 5.754603 15.68825 7.021198 5.802816 6.928720 1/(1+R) 0.1395160 0.1380361 0.1391971 0.1391481 0.1391936 SQRT(R) 2.664120 2.688104 2.669075 2.669399 2.669101 Rule: 6 7 8 Function 1 1.000000 1.000000 0.1666667 X 1.750000 1.750000 0.2916667 Y 2.250000 2.250000 0.3750000 Z 6.500000 6.500000 1.083333 X*X 3.400000 3.400000 0.5666667 X*Y 3.900000 3.900000 0.6500000 X*Z 11.30000 11.30000 1.883333 Y*Y 5.100000 5.100000 0.8500000 Y*Z 14.60000 14.60000 2.433333 Z*Z 42.40000 42.40000 7.066667 X^3 7.300000 7.300000 1.216667 X*Y*Z 25.15000 25.15000 4.191667 Z*Z*Z 277.6000 277.6000 46.26667 X^4 17.50000 17.11429 2.852381 X^2 Z^2 140.2000 140.1714 23.36190 Z^4 1824.533 1824.457 304.0762 X^5 47.50000 42.88951 7.206367 X^6 145.9000 111.7131 19.39520 R 7.127357 7.127469 1.187912 SIN(X) 0.8472970 0.8373174 0.1392812 EXP(X) 7.168051 6.974256 1.167247 1/(1+R) 0.1392013 0.1391958 0.2319924E-01 SQRT(R) 2.669056 2.669086 0.4448479 TEST41 TRIANGLE_UNIT_SET sets up a quadrature rule on a triangle. TRIANGLE_SUB applies it to subtriangles of an arbitrary triangle. Triangle vertices: 0.00000 0.00000 0.00000 1.00000 1.00000 0.00000 Using unit triangle quadrature rule 3 Rule order = 3 Function Nsub Result 1 1 0.50000000 1 2 0.50000000 1 3 0.50000000 1 4 0.50000000 1 5 0.50000000 X 1 0.16666667 X 2 0.16666667 X 3 0.16666667 X 4 0.16666667 X 5 0.16666667 X^2 1 0.08333333 X^2 2 0.08333333 X^2 3 0.08333333 X^2 4 0.08333333 X^2 5 0.08333333 X^3 1 0.05092593 X^3 2 0.05005787 X^3 3 0.05001143 X^3 4 0.05000362 X^3 5 0.05000148 X^4 1 0.03317901 X^4 2 0.03332369 X^4 3 0.03333143 X^4 4 0.03333273 X^4 5 0.03333309 X^5 1 0.02199074 X^5 2 0.02376905 X^5 3 0.02380421 X^5 4 0.02380814 X^5 5 0.02380901 X^6 1 0.01463906 X^6 2 0.01778878 X^6 3 0.01784849 X^6 4 0.01785494 X^6 5 0.01785634 R 1 0.26834513 R 2 0.27037578 R 3 0.27049152 R 4 0.27051881 R 5 0.27052815 SIN(X) 1 0.15836034 SIN(X) 2 0.15851906 SIN(X) 3 0.15852707 SIN(X) 4 0.15852840 SIN(X) 5 0.15852876 EXP(X) 1 0.71840914 EXP(X) 2 0.71829061 EXP(X) 3 0.71828359 EXP(X) 4 0.71828239 EXP(X) 5 0.71828206 1/(1+R) 1 0.33244390 1/(1+R) 2 0.33066673 1/(1+R) 3 0.33053631 1/(1+R) 4 0.33050558 1/(1+R) 5 0.33049514 SQRT(R) 1 0.35723722 SQRT(R) 2 0.35949749 SQRT(R) 3 0.35970810 SQRT(R) 4 0.35976936 SQRT(R) 5 0.35979390 TEST42 TRIANGLE_UNIT_SET sets up a quadrature in the unit triangle, TRIANGLE_UNIT_SUM applies it. Rule: 1 2 3 4 5 Function 1 0.50000000 0.50000000 0.50000000 0.50000000 0.50000000 X 0.16666667 0.16666667 0.16666667 0.16666667 0.16666667 X^2 0.05555556 0.16666667 0.08333333 0.08333333 0.08333333 X^3 0.01851852 0.16666667 0.05092593 0.04166667 0.05000000 X^4 0.00617284 0.16666667 0.03317901 0.02083333 0.03111111 X^5 0.00205761 0.16666667 0.02199074 0.01041667 0.01925926 X^6 0.00068587 0.16666667 0.01463906 0.00520833 0.01179753 R 0.23570226 0.33333333 0.26834513 0.28451780 0.27047836 SIN(X) 0.16359735 0.14024516 0.15836034 0.15980851 0.15849241 EXP(X) 0.69780621 0.78638030 0.71840914 0.71624042 0.71814138 1/(1+R) 0.33981138 0.33333333 0.33244390 0.31985330 0.33090480 SQRT(R) 0.34329452 0.33333333 0.35723722 0.37585166 0.35959762 Rule: 6 7 8 9 10 Function 1 0.50000000 0.50000000 0.50000000 0.50000000 0.50000000 X 0.16666667 0.16666667 0.16666667 0.16666667 0.16666667 X^2 0.08333333 0.08333333 0.08333333 0.08333333 0.08333333 X^3 0.05000000 0.05000000 0.05000000 0.05000000 0.05000000 X^4 0.03194444 0.03194444 0.03333333 0.03333333 0.03333333 X^5 0.02083333 0.02083333 0.02393509 0.02345679 0.02380952 X^6 0.01368056 0.01369599 0.01808838 0.01693004 0.01777526 R 0.27048707 0.26996240 0.27043774 0.27074353 0.27004434 SIN(X) 0.15850517 0.15850516 0.15853001 0.15852637 0.15852906 EXP(X) 0.71819224 0.71819227 0.71828325 0.71827724 0.71828165 1/(1+R) 0.33070418 0.33118484 0.33063446 0.33016016 0.33094037 SQRT(R) 0.35990071 0.35909867 0.35948536 0.36077832 0.35910626 Rule: 11 12 13 14 15 Function 1 0.50000000 0.50000000 0.50000000 0.50000000 0.50000000 X 0.16666667 0.16666667 0.16666667 0.16666667 0.16666667 X^2 0.08333333 0.08333333 0.08333333 0.08333333 0.08333333 X^3 0.05000000 0.05000000 0.05000000 0.05000000 0.05000000 X^4 0.03333333 0.03333333 0.03333333 0.03611111 0.03333333 X^5 0.02380952 0.02380952 0.02380952 0.03009259 0.02380952 X^6 0.01778442 0.01785714 0.01785714 0.02739198 0.01785714 R 0.27090574 0.27047231 0.27046881 0.26987314 0.27048405 SIN(X) 0.15852906 0.15852901 0.15852902 0.15857899 0.15852902 EXP(X) 0.71828167 0.71828183 0.71828183 0.71846601 0.71828183 1/(1+R) 0.33014739 0.33055163 0.33055595 0.33085644 0.33053935 SQRT(R) 0.36051785 0.35967327 0.35966938 0.35482320 0.35970261 Rule: 16 17 18 19 20 Function 1 0.50000000 0.50000000 0.50000000 0.50000000 0.50000000 X 0.16666667 0.16666667 0.16666667 0.16666667 0.16666667 X^2 0.08333333 0.08333333 0.08333333 0.08333333 0.08333333 X^3 0.05000000 0.05000000 0.05000000 0.05000000 0.05000000 X^4 0.03333333 0.03333333 0.03333333 0.03333333 0.03333333 X^5 0.02380952 0.02380952 0.02380952 0.02380952 0.02380952 X^6 0.01785714 0.01785714 0.01785714 0.01785714 0.01785714 R 0.27053617 0.27048536 0.27051176 0.27051652 0.27053379 SIN(X) 0.15852902 0.15852902 0.15852902 0.15852902 0.15852902 EXP(X) 0.71828183 0.71828183 0.71828183 0.71828183 0.71828183 1/(1+R) 0.33048626 0.33053558 0.33051042 0.33050553 0.33048864 SQRT(R) 0.35982024 0.35969714 0.35976034 0.35977423 0.35981252 TEST425 TRIANGLE_UNIT_SET sets up a quadrature in the unit triangle, Estimate integral of X^A * Y^B. A = 0 B = 0 Rule QUAD ERROR 1 1.00000 0.00000000 2 1.00000 0.00000000 3 1.00000 0.00000000 4 1.00000 0.00000000 5 1.00000 0.00000000 6 1.00000 0.00000000 7 1.00000 0.00000000 8 1.00000 0.00000000 9 1.00000 0.00000000 10 1.00000 0.00000000 11 1.00000 0.00000000 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.00000 0.00000000 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 0 B = 1 Rule QUAD ERROR 1 1.00000 0.00000000 2 1.00000 0.00000000 3 1.00000 0.00000000 4 1.00000 0.00000000 5 1.00000 0.00000000 6 1.00000 0.00000000 7 1.00000 0.00000000 8 1.00000 0.00000000 9 1.00000 0.00000000 10 1.00000 0.00000000 11 1.00000 0.00000000 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.00000 0.00000000 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 0 B = 2 Rule QUAD ERROR 1 0.666667 0.33333333 2 2.00000 1.00000000 3 1.00000 0.00000000 4 1.00000 0.00000000 5 1.00000 0.00000000 6 1.00000 0.00000000 7 1.00000 0.00000000 8 1.00000 0.00000000 9 1.00000 0.00000000 10 1.00000 0.00000000 11 1.00000 0.00000000 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.00000 0.00000000 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 0 B = 3 Rule QUAD ERROR 1 0.370370 0.62962963 2 3.33333 2.33333333 3 1.01852 0.01851852 4 0.833333 0.16666667 5 1.00000 0.00000000 6 1.00000 0.00000000 7 1.00000 0.00000000 8 1.00000 0.00000000 9 1.00000 0.00000000 10 1.00000 0.00000000 11 1.00000 0.00000000 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.00000 0.00000000 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 0 B = 4 Rule QUAD ERROR 1 0.185185 0.81481481 2 5.00000 4.00000000 3 0.995370 0.00462963 4 0.625000 0.37500000 5 0.933333 0.06666667 6 0.958333 0.04166667 7 0.958333 0.04166667 8 1.00000 0.00000000 9 1.00000 0.00000000 10 1.00000 0.00000000 11 1.00000 0.00000000 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.08333 0.08333333 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 0 B = 5 Rule QUAD ERROR 1 0.864198E-01 0.91358025 2 7.00000 6.00000000 3 0.923611 0.07638889 4 0.437500 0.56250000 5 0.808889 0.19111111 6 0.875000 0.12500000 7 0.875000 0.12500000 8 1.00527 0.00527393 9 0.985185 0.01481481 10 1.00000 0.00000000 11 1.00000 0.00000000 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.26389 0.26388889 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 0 B = 6 Rule QUAD ERROR 1 0.384088E-01 0.96159122 2 9.33333 8.33333333 3 0.819787 0.18021262 4 0.291667 0.70833333 5 0.660662 0.33933827 6 0.766111 0.23388889 7 0.766975 0.23302469 8 1.01295 0.01294949 9 0.948082 0.05191770 10 0.995414 0.00458554 11 0.995927 0.00407261 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.53395 0.53395062 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 0 B = 7 Rule QUAD ERROR 1 0.164609E-01 0.98353909 2 12.0000 11.00000000 3 0.702418 0.29758230 4 0.187500 0.81250000 5 0.516101 0.48389926 6 0.648333 0.35166667 7 0.650926 0.34907407 8 1.01690 0.01690162 9 0.889877 0.11012346 10 0.981708 0.01829176 11 0.983203 0.01679708 12 1.00020 0.00019804 13 1.00000 0.00000000 14 1.88241 0.88240741 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 0 B = 8 Rule QUAD ERROR 1 0.685871E-02 0.99314129 2 15.0000 14.00000000 3 0.585294 0.41470550 4 0.117188 0.88281250 5 0.389922 0.61007802 6 0.533854 0.46614583 7 0.538484 0.46151620 8 1.01216 0.01216139 9 0.816165 0.18383539 10 0.956602 0.04339756 11 0.959183 0.04081739 12 1.00044 0.00043938 13 0.999572 0.00042796 14 2.29996 1.29996142 15 0.999733 0.00026706 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 0 B = 9 Rule QUAD ERROR 1 0.279429E-02 0.99720571 2 18.3333 17.33333333 3 0.476896 0.52310429 4 0.716146E-01 0.92838542 5 0.287142 0.71285845 6 0.429942 0.57005787 7 0.436368 0.56363240 8 0.996255 0.00374460 9 0.733569 0.26643061 10 0.920084 0.07991597 11 0.923540 0.07645961 12 1.00009 0.00008582 13 0.997886 0.00211358 14 2.77990 1.77990326 15 0.998637 0.00136299 16 1.00000 0.00000000 17 1.00009 0.00009149 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 0 B = 10 Rule QUAD ERROR 1 0.111772E-02 0.99888228 2 22.0000 21.00000000 3 0.381514 0.61848561 4 0.429688E-01 0.95703125 5 0.207231 0.79276920 6 0.339994 0.66000579 7 0.347660 0.65234018 8 0.968918 0.03108196 9 0.647992 0.35200829 10 0.873671 0.12632881 11 0.877655 0.12234509 12 0.998219 0.00178063 13 0.993975 0.00602508 14 3.31769 2.31769047 15 0.995992 0.00400760 16 1.00000 0.00000000 17 1.00049 0.00048861 18 0.999970 0.00003026 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 1 B = 0 Rule QUAD ERROR 1 1.00000 0.00000000 2 1.00000 0.00000000 3 1.00000 0.00000000 4 1.00000 0.00000000 5 1.00000 0.00000000 6 1.00000 0.00000000 7 1.00000 0.00000000 8 1.00000 0.00000000 9 1.00000 0.00000000 10 1.00000 0.00000000 11 1.00000 0.00000000 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.00000 0.00000000 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 1 B = 1 Rule QUAD ERROR 1 1.33333 0.33333333 2 0.00000 1.00000000 3 1.00000 0.00000000 4 1.00000 0.00000000 5 1.00000 0.00000000 6 1.00000 0.00000000 7 1.00000 0.00000000 8 1.00000 0.00000000 9 1.00000 0.00000000 10 1.00000 0.00000000 11 1.00000 0.00000000 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.00000 0.00000000 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 1 B = 2 Rule QUAD ERROR 1 1.11111 0.11111111 2 0.00000 1.00000000 3 0.972222 0.02777778 4 1.25000 0.25000000 5 1.00000 0.00000000 6 1.00000 0.00000000 7 1.00000 0.00000000 8 1.00000 0.00000000 9 1.00000 0.00000000 10 1.00000 0.00000000 11 1.00000 0.00000000 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.00000 0.00000000 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 1 B = 3 Rule QUAD ERROR 1 0.740741 0.25925926 2 0.00000 1.00000000 3 1.06481 0.06481481 4 1.25000 0.25000000 5 1.13333 0.13333333 6 1.08333 0.08333333 7 1.08333 0.08333333 8 1.00000 0.00000000 9 1.00000 0.00000000 10 1.00000 0.00000000 11 1.00000 0.00000000 12 1.00000 0.00000000 13 1.00000 0.00000000 14 0.833333 0.16666667 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 1 B = 4 Rule QUAD ERROR 1 0.432099 0.56790123 2 0.00000 1.00000000 3 1.17477 0.17476852 4 1.09375 0.09375000 5 1.24444 0.24444444 6 1.16667 0.16666667 7 1.16667 0.16666667 8 0.986815 0.01318484 9 1.03704 0.03703704 10 1.00000 0.00000000 11 1.00000 0.00000000 12 1.00000 0.00000000 13 1.00000 0.00000000 14 0.631944 0.36805556 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 1 B = 5 Rule QUAD ERROR 1 0.230453 0.76954733 2 0.00000 1.00000000 3 1.23508 0.23508230 4 0.875000 0.12500000 5 1.25357 0.25357037 6 1.20167 0.20166667 7 1.19907 0.19907407 8 0.982247 0.01775274 9 1.09649 0.09649383 10 1.01376 0.01375661 11 1.01222 0.01221783 12 1.00000 0.00000000 13 1.00000 0.00000000 14 0.453704 0.54629630 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 1 B = 6 Rule QUAD ERROR 1 0.115226 0.88477366 2 0.00000 1.00000000 3 1.23058 0.23058128 4 0.656250 0.34375000 5 1.16663 0.16662519 6 1.17833 0.17833333 7 1.17315 0.17314815 8 0.999117 0.00088296 9 1.15180 0.15180247 10 1.04339 0.04338624 11 1.04046 0.04046303 12 0.999307 0.00069314 13 1.00000 0.00000000 14 0.314352 0.68564815 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 1 B = 7 Rule QUAD ERROR 1 0.548697E-01 0.94513032 2 0.00000 1.00000000 3 1.17091 0.17091049 4 0.468750 0.53125000 5 1.02082 0.02081580 6 1.10625 0.10625000 7 1.10069 0.10069444 8 1.03586 0.03586257 9 1.18472 0.18472428 10 1.08213 0.08213142 11 1.07928 0.07928415 12 0.999233 0.00076731 13 1.00171 0.00171183 14 0.212191 0.78780864 15 1.00057 0.00056689 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 1 B = 8 Rule QUAD ERROR 1 0.251486E-01 0.97485139 2 0.00000 1.00000000 3 1.07309 0.07308903 4 0.322266 0.67773438 5 0.852434 0.14756609 6 1.00146 0.00145833 7 0.998007 0.00199331 8 1.08374 0.08373832 9 1.18784 0.18784307 10 1.12094 0.12093530 11 1.11957 0.11957262 12 1.00203 0.00203040 13 1.00716 0.00715736 14 0.140223 0.85977688 15 1.00253 0.00253185 16 1.00000 0.00000000 17 0.999588 0.00041169 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 1 B = 9 Rule QUAD ERROR 1 0.111772E-01 0.98882284 2 0.00000 1.00000000 3 0.953802 0.04619766 4 0.214844 0.78515625 5 0.686695 0.31330475 6 0.879682 0.12031829 7 0.879906 0.12009352 8 1.13294 0.13294222 9 1.16146 0.16145783 10 1.15215 0.15214823 11 1.15297 0.15296778 12 1.00942 0.00941807 13 1.01744 0.01744391 14 0.909672E-01 0.90903278 15 1.00658 0.00658406 16 1.00000 0.00000000 17 0.998106 0.00189411 18 1.00015 0.00015132 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 2 B = 0 Rule QUAD ERROR 1 0.666667 0.33333333 2 2.00000 1.00000000 3 1.00000 0.00000000 4 1.00000 0.00000000 5 1.00000 0.00000000 6 1.00000 0.00000000 7 1.00000 0.00000000 8 1.00000 0.00000000 9 1.00000 0.00000000 10 1.00000 0.00000000 11 1.00000 0.00000000 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.00000 0.00000000 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 2 B = 1 Rule QUAD ERROR 1 1.11111 0.11111111 2 0.00000 1.00000000 3 0.972222 0.02777778 4 1.25000 0.25000000 5 1.00000 0.00000000 6 1.00000 0.00000000 7 1.00000 0.00000000 8 1.00000 0.00000000 9 1.00000 0.00000000 10 1.00000 0.00000000 11 1.00000 0.00000000 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.00000 0.00000000 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 2 B = 2 Rule QUAD ERROR 1 1.11111 0.11111111 2 0.00000 1.00000000 3 0.763889 0.23611111 4 1.87500 0.87500000 5 0.800000 0.20000000 6 0.875000 0.12500000 7 0.875000 0.12500000 8 1.00000 0.00000000 9 1.00000 0.00000000 10 1.00000 0.00000000 11 1.00000 0.00000000 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.25000 0.25000000 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 2 B = 3 Rule QUAD ERROR 1 0.864198 0.13580247 2 0.00000 1.00000000 3 0.729167 0.27083333 4 2.18750 1.18750000 5 0.808889 0.19111111 6 0.875000 0.12500000 7 0.875000 0.12500000 8 1.00527 0.00527393 9 0.985185 0.01481481 10 1.00000 0.00000000 11 1.00000 0.00000000 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.26389 0.26388889 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 2 B = 4 Rule QUAD ERROR 1 0.576132 0.42386831 2 0.00000 1.00000000 3 0.819187 0.18081276 4 2.18750 1.18750000 5 0.949926 0.05007407 6 0.962500 0.03750000 7 0.956019 0.04398148 8 0.958434 0.04156566 9 1.03012 0.03012346 10 0.962963 0.03703704 11 0.983741 0.01625887 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.13426 0.13425926 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 2 B = 5 Rule QUAD ERROR 1 0.345679 0.65432099 2 0.00000 1.00000000 3 0.937114 0.06288580 4 1.96875 0.96875000 5 1.08068 0.08067556 6 1.06750 0.06750000 7 1.04028 0.04027778 8 0.881502 0.11849802 9 1.14074 0.14074074 10 0.909751 0.09024943 11 0.967270 0.03272971 12 1.00089 0.00089118 13 1.00000 0.00000000 14 0.943056 0.05694444 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 2 B = 6 Rule QUAD ERROR 1 0.192044 0.80795610 2 0.00000 1.00000000 3 1.02849 0.02848508 4 1.64062 0.64062500 5 1.13350 0.13349531 6 1.14479 0.14479167 7 1.08970 0.08969907 8 0.811392 0.18860789 9 1.28639 0.28638683 10 0.866501 0.13349890 11 0.970273 0.02972680 12 0.996224 0.00377556 13 0.994117 0.00588301 14 0.742670 0.25733025 15 0.997732 0.00226757 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 2 B = 7 Rule QUAD ERROR 1 0.100594 0.89940558 2 0.00000 1.00000000 3 1.07412 0.07412051 4 1.28906 0.28906250 5 1.10343 0.10343164 6 1.17562 0.17562500 7 1.09561 0.09561471 8 0.769683 0.23031716 9 1.43345 0.43344636 10 0.846090 0.15390976 11 1.00117 0.00116637 12 0.981678 0.01832186 13 0.979223 0.02077746 14 0.560892 0.43910751 15 0.990804 0.00919627 16 1.00000 0.00000000 17 1.00073 0.00072909 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 2 B = 8 Rule QUAD ERROR 1 0.502972E-01 0.94970279 2 0.00000 1.00000000 3 1.07329 0.07328550 4 0.966797 0.03320312 5 1.01255 0.01255422 6 1.15872 0.15872396 7 1.06264 0.06263664 8 0.761318 0.23868198 9 1.55761 0.55761207 10 0.849523 0.15047725 11 1.05700 0.05700238 12 0.957945 0.04205474 13 0.956754 0.04324599 14 0.409352 0.59064751 15 0.978158 0.02184193 16 1.00000 0.00000000 17 1.00232 0.00232291 18 0.999303 0.00069732 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 3 B = 0 Rule QUAD ERROR 1 0.370370 0.62962963 2 3.33333 2.33333333 3 1.01852 0.01851852 4 0.833333 0.16666667 5 1.00000 0.00000000 6 1.00000 0.00000000 7 1.00000 0.00000000 8 1.00000 0.00000000 9 1.00000 0.00000000 10 1.00000 0.00000000 11 1.00000 0.00000000 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.00000 0.00000000 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 3 B = 1 Rule QUAD ERROR 1 0.740741 0.25925926 2 0.00000 1.00000000 3 1.06481 0.06481481 4 1.25000 0.25000000 5 1.13333 0.13333333 6 1.08333 0.08333333 7 1.08333 0.08333333 8 1.00000 0.00000000 9 1.00000 0.00000000 10 1.00000 0.00000000 11 1.00000 0.00000000 12 1.00000 0.00000000 13 1.00000 0.00000000 14 0.833333 0.16666667 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 3 B = 2 Rule QUAD ERROR 1 0.864198 0.13580247 2 0.00000 1.00000000 3 0.729167 0.27083333 4 2.18750 1.18750000 5 0.808889 0.19111111 6 0.875000 0.12500000 7 0.875000 0.12500000 8 1.00527 0.00527393 9 0.985185 0.01481481 10 1.00000 0.00000000 11 1.00000 0.00000000 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.26389 0.26388889 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 3 B = 3 Rule QUAD ERROR 1 0.768176 0.23182442 2 0.00000 1.00000000 3 0.516118 0.48388203 4 2.91667 1.91666667 5 0.594568 0.40543210 6 0.738889 0.26111111 7 0.756173 0.24382716 8 1.06350 0.06349583 9 0.917202 0.08279835 10 1.05115 0.05114638 11 1.01215 0.01215469 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.51235 0.51234568 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 3 B = 4 Rule QUAD ERROR 1 0.576132 0.42386831 2 0.00000 1.00000000 3 0.481610 0.51838992 4 3.28125 2.28125000 5 0.591526 0.40847407 6 0.729167 0.27083333 7 0.761574 0.23842593 8 1.05737 0.05737044 9 0.905679 0.09432099 10 1.05367 0.05366591 11 0.999136 0.00086444 12 0.999505 0.00049510 13 1.00000 0.00000000 14 1.57176 0.57175926 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 3 B = 5 Rule QUAD ERROR 1 0.384088 0.61591221 2 0.00000 1.00000000 3 0.544624 0.45537551 4 3.28125 2.28125000 5 0.694511 0.30548938 6 0.802083 0.19791667 7 0.818287 0.18171296 8 0.972131 0.02786944 9 0.992329 0.00767078 10 0.992477 0.00752259 11 0.961034 0.03896621 12 1.01154 0.01153677 13 1.01166 0.01165761 14 1.48534 0.48533951 15 1.00529 0.00529101 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 3 B = 6 Rule QUAD ERROR 1 0.234720 0.76527968 2 0.00000 1.00000000 3 0.635854 0.36414633 4 3.00781 2.00781250 5 0.800594 0.19940618 6 0.903681 0.09631944 7 0.873050 0.12695045 8 0.846318 0.15368207 9 1.16057 0.16056991 10 0.896388 0.10361195 11 0.920984 0.07901635 12 1.03039 0.03039232 13 1.02893 0.02893207 14 1.30875 0.30874914 15 1.01621 0.01620895 16 1.00000 0.00000000 17 0.999369 0.00063062 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 3 B = 7 Rule QUAD ERROR 1 0.134126 0.86587410 2 0.00000 1.00000000 3 0.718176 0.28182394 4 2.57812 1.57812500 5 0.860170 0.13982982 6 0.995347 0.00465278 7 0.904171 0.09582905 8 0.720497 0.27950348 9 1.37501 0.37500503 10 0.797080 0.20291962 11 0.902218 0.09778175 12 1.04271 0.04270623 13 1.03934 0.03934337 14 1.09161 0.09160665 15 1.02814 0.02814171 16 1.00000 0.00000000 17 0.999861 0.00013938 18 1.00188 0.00188136 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 4 B = 0 Rule QUAD ERROR 1 0.185185 0.81481481 2 5.00000 4.00000000 3 0.995370 0.00462963 4 0.625000 0.37500000 5 0.933333 0.06666667 6 0.958333 0.04166667 7 0.958333 0.04166667 8 1.00000 0.00000000 9 1.00000 0.00000000 10 1.00000 0.00000000 11 1.00000 0.00000000 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.08333 0.08333333 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 4 B = 1 Rule QUAD ERROR 1 0.432099 0.56790123 2 0.00000 1.00000000 3 1.17477 0.17476852 4 1.09375 0.09375000 5 1.24444 0.24444444 6 1.16667 0.16666667 7 1.16667 0.16666667 8 0.986815 0.01318484 9 1.03704 0.03703704 10 1.00000 0.00000000 11 1.00000 0.00000000 12 1.00000 0.00000000 13 1.00000 0.00000000 14 0.631944 0.36805556 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 4 B = 2 Rule QUAD ERROR 1 0.576132 0.42386831 2 0.00000 1.00000000 3 0.819187 0.18081276 4 2.18750 1.18750000 5 0.949926 0.05007407 6 0.962500 0.03750000 7 0.956019 0.04398148 8 0.958434 0.04156566 9 1.03012 0.03012346 10 0.962963 0.03703704 11 0.983741 0.01625887 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.13426 0.13425926 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 4 B = 3 Rule QUAD ERROR 1 0.576132 0.42386831 2 0.00000 1.00000000 3 0.481610 0.51838992 4 3.28125 2.28125000 5 0.591526 0.40847407 6 0.729167 0.27083333 7 0.761574 0.23842593 8 1.05737 0.05737044 9 0.905679 0.09432099 10 1.05367 0.05366591 11 0.999136 0.00086444 12 0.999505 0.00049510 13 1.00000 0.00000000 14 1.57176 0.57175926 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 4 B = 4 Rule QUAD ERROR 1 0.480110 0.51989026 2 0.00000 1.00000000 3 0.320698 0.67930170 4 4.10156 3.10156250 5 0.414538 0.58546173 6 0.601563 0.39843750 7 0.698785 0.30121528 8 1.12991 0.12990697 9 0.799671 0.20032922 10 1.13775 0.13774610 11 1.00434 0.00433838 12 0.983592 0.01640758 13 0.985455 0.01454491 14 1.85667 0.85667438 15 0.992063 0.00793651 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 4 B = 5 Rule QUAD ERROR 1 0.352080 0.64791952 2 0.00000 1.00000000 3 0.293630 0.70637038 4 4.51172 3.51171875 5 0.402555 0.59744527 6 0.585521 0.41447917 7 0.715439 0.28456147 8 1.10610 0.10610483 9 0.796322 0.20367846 10 1.14249 0.14249035 11 0.974400 0.02559962 12 0.982741 0.01725909 13 0.985505 0.01449536 14 1.96312 0.96312371 15 0.986185 0.01381540 16 1.00000 0.00000000 17 1.00018 0.00017599 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 4 B = 6 Rule QUAD ERROR 1 0.234720 0.76527968 2 0.00000 1.00000000 3 0.332597 0.66740314 4 4.51172 3.51171875 5 0.467461 0.53253898 6 0.643003 0.35699653 7 0.750509 0.24949096 8 0.998828 0.00117160 9 0.905044 0.09495601 10 1.06917 0.06917320 11 0.916307 0.08369330 12 1.00786 0.00785815 13 1.01024 0.01024312 14 1.91031 0.91031164 15 0.992343 0.00765656 16 1.00000 0.00000000 17 0.997408 0.00259195 18 0.996691 0.00330876 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 5 B = 0 Rule QUAD ERROR 1 0.864198E-01 0.91358025 2 7.00000 6.00000000 3 0.923611 0.07638889 4 0.437500 0.56250000 5 0.808889 0.19111111 6 0.875000 0.12500000 7 0.875000 0.12500000 8 1.00527 0.00527393 9 0.985185 0.01481481 10 1.00000 0.00000000 11 1.00000 0.00000000 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.26389 0.26388889 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 5 B = 1 Rule QUAD ERROR 1 0.230453 0.76954733 2 0.00000 1.00000000 3 1.23508 0.23508230 4 0.875000 0.12500000 5 1.25357 0.25357037 6 1.20167 0.20166667 7 1.19907 0.19907407 8 0.982247 0.01775274 9 1.09649 0.09649383 10 1.01376 0.01375661 11 1.01222 0.01221783 12 1.00000 0.00000000 13 1.00000 0.00000000 14 0.453704 0.54629630 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 5 B = 2 Rule QUAD ERROR 1 0.345679 0.65432099 2 0.00000 1.00000000 3 0.937114 0.06288580 4 1.96875 0.96875000 5 1.08068 0.08067556 6 1.06750 0.06750000 7 1.04028 0.04027778 8 0.881502 0.11849802 9 1.14074 0.14074074 10 0.909751 0.09024943 11 0.967270 0.03272971 12 1.00089 0.00089118 13 1.00000 0.00000000 14 0.943056 0.05694444 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 5 B = 3 Rule QUAD ERROR 1 0.384088 0.61591221 2 0.00000 1.00000000 3 0.544624 0.45537551 4 3.28125 2.28125000 5 0.694511 0.30548938 6 0.802083 0.19791667 7 0.818287 0.18171296 8 0.972131 0.02786944 9 0.992329 0.00767078 10 0.992477 0.00752259 11 0.961034 0.03896621 12 1.01154 0.01153677 13 1.01166 0.01165761 14 1.48534 0.48533951 15 1.00794 0.00793651 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 5 B = 4 Rule QUAD ERROR 1 0.352080 0.64791952 2 0.00000 1.00000000 3 0.293630 0.70637038 4 4.51172 3.51171875 5 0.402555 0.59744527 6 0.585521 0.41447917 7 0.715439 0.28456147 8 1.10610 0.10610483 9 0.796322 0.20367846 10 1.14249 0.14249035 11 0.974400 0.02559962 12 0.982741 0.01725909 13 0.985505 0.01449536 14 1.96312 0.96312371 15 0.999118 0.00088183 16 1.00000 0.00000000 17 1.00018 0.00017599 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 5 B = 5 Rule QUAD ERROR 1 0.281664 0.71833562 2 0.00000 1.00000000 3 0.187868 0.81213206 4 5.41406 4.41406250 5 0.273552 0.72644773 6 0.473229 0.52677083 7 0.710487 0.28951260 8 1.16618 0.16618451 9 0.679036 0.32096351 10 1.23958 0.23957884 11 0.968337 0.03166255 12 0.950326 0.04967432 13 0.955574 0.04442596 14 2.29237 1.29237397 15 0.977562 0.02243778 16 1.00000 0.00000000 17 1.00302 0.00301864 18 1.00398 0.00397707 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 6 B = 0 Rule QUAD ERROR 1 0.384088E-01 0.96159122 2 9.33333 8.33333333 3 0.819787 0.18021262 4 0.291667 0.70833333 5 0.660662 0.33933827 6 0.766111 0.23388889 7 0.766975 0.23302469 8 1.01295 0.01294949 9 0.948082 0.05191770 10 0.995414 0.00458554 11 0.995927 0.00407261 12 1.00000 0.00000000 13 1.00000 0.00000000 14 1.53395 0.53395062 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 6 B = 1 Rule QUAD ERROR 1 0.115226 0.88477366 2 0.00000 1.00000000 3 1.23058 0.23058128 4 0.656250 0.34375000 5 1.16663 0.16662519 6 1.17833 0.17833333 7 1.17315 0.17314815 8 0.999117 0.00088296 9 1.15180 0.15180247 10 1.04339 0.04338624 11 1.04046 0.04046303 12 0.999307 0.00069314 13 1.00000 0.00000000 14 0.314352 0.68564815 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 6 B = 2 Rule QUAD ERROR 1 0.192044 0.80795610 2 0.00000 1.00000000 3 1.02849 0.02848508 4 1.64062 0.64062500 5 1.13350 0.13349531 6 1.14479 0.14479167 7 1.08970 0.08969907 8 0.811392 0.18860789 9 1.28639 0.28638683 10 0.866501 0.13349890 11 0.970273 0.02972680 12 0.996224 0.00377556 13 0.994117 0.00588301 14 0.742670 0.25733025 15 0.994709 0.00529101 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 6 B = 3 Rule QUAD ERROR 1 0.234720 0.76527968 2 0.00000 1.00000000 3 0.635854 0.36414633 4 3.00781 2.00781250 5 0.800594 0.19940618 6 0.903681 0.09631944 7 0.873050 0.12695045 8 0.846318 0.15368207 9 1.16057 0.16056991 10 0.896388 0.10361195 11 0.920984 0.07901635 12 1.03039 0.03039232 13 1.02893 0.02893207 14 1.30875 0.30874914 15 1.01529 0.01528513 16 1.00000 0.00000000 17 0.999369 0.00063062 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 6 B = 4 Rule QUAD ERROR 1 0.234720 0.76527968 2 0.00000 1.00000000 3 0.332597 0.66740314 4 4.51172 3.51171875 5 0.467461 0.53253898 6 0.643003 0.35699653 7 0.750509 0.24949096 8 0.998828 0.00117160 9 0.905044 0.09495601 10 1.06917 0.06917320 11 0.916307 0.08369330 12 1.00786 0.00785815 13 1.01024 0.01024312 14 1.91031 0.91031164 15 1.02067 0.02067411 16 1.00000 0.00000000 17 0.997408 0.00259195 18 0.996691 0.00330876 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 7 B = 0 Rule QUAD ERROR 1 0.164609E-01 0.98353909 2 12.0000 11.00000000 3 0.702418 0.29758230 4 0.187500 0.81250000 5 0.516101 0.48389926 6 0.648333 0.35166667 7 0.650926 0.34907407 8 1.01690 0.01690162 9 0.889877 0.11012346 10 0.981708 0.01829176 11 0.983203 0.01679708 12 1.00020 0.00019804 13 1.00000 0.00000000 14 1.88241 0.88240741 15 1.00000 0.00000000 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 7 B = 1 Rule QUAD ERROR 1 0.548697E-01 0.94513032 2 0.00000 1.00000000 3 1.17091 0.17091049 4 0.468750 0.53125000 5 1.02082 0.02081580 6 1.10625 0.10625000 7 1.10069 0.10069444 8 1.03586 0.03586257 9 1.18472 0.18472428 10 1.08213 0.08213142 11 1.07928 0.07928415 12 0.999233 0.00076731 13 1.00171 0.00171183 14 0.212191 0.78780864 15 1.00227 0.00226757 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 7 B = 2 Rule QUAD ERROR 1 0.100594 0.89940558 2 0.00000 1.00000000 3 1.07412 0.07412051 4 1.28906 0.28906250 5 1.10343 0.10343164 6 1.17562 0.17562500 7 1.09561 0.09561471 8 0.769683 0.23031716 9 1.43345 0.43344636 10 0.846090 0.15390976 11 1.00117 0.00116637 12 0.981678 0.01832186 13 0.979223 0.02077746 14 0.560892 0.43910751 15 0.982951 0.01704880 16 1.00000 0.00000000 17 1.00073 0.00072909 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 7 B = 3 Rule QUAD ERROR 1 0.134126 0.86587410 2 0.00000 1.00000000 3 0.718176 0.28182394 4 2.57812 1.57812500 5 0.860170 0.13982982 6 0.995347 0.00465278 7 0.904171 0.09582905 8 0.720497 0.27950348 9 1.37501 0.37500503 10 0.797080 0.20291962 11 0.902218 0.09778175 12 1.04271 0.04270623 13 1.03934 0.03934337 14 1.09161 0.09160665 15 1.01144 0.01143585 16 1.00000 0.00000000 17 0.999861 0.00013938 18 1.00188 0.00188136 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 8 B = 0 Rule QUAD ERROR 1 0.685871E-02 0.99314129 2 15.0000 14.00000000 3 0.585294 0.41470550 4 0.117188 0.88281250 5 0.389922 0.61007802 6 0.533854 0.46614583 7 0.538484 0.46151620 8 1.01216 0.01216139 9 0.816165 0.18383539 10 0.956602 0.04339756 11 0.959183 0.04081739 12 1.00044 0.00043938 13 0.999572 0.00042796 14 2.29996 1.29996142 15 0.999433 0.00056689 16 1.00000 0.00000000 17 1.00000 0.00000000 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 8 B = 1 Rule QUAD ERROR 1 0.251486E-01 0.97485139 2 0.00000 1.00000000 3 1.07309 0.07308903 4 0.322266 0.67773438 5 0.852434 0.14756609 6 1.00146 0.00145833 7 0.998007 0.00199331 8 1.08374 0.08373832 9 1.18784 0.18784307 10 1.12094 0.12093530 11 1.11957 0.11957262 12 1.00203 0.00203040 13 1.00716 0.00715736 14 0.140223 0.85977688 15 1.00951 0.00951121 16 1.00000 0.00000000 17 0.999588 0.00041169 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 8 B = 2 Rule QUAD ERROR 1 0.502972E-01 0.94970279 2 0.00000 1.00000000 3 1.07329 0.07328550 4 0.966797 0.03320312 5 1.01255 0.01255422 6 1.15872 0.15872396 7 1.06264 0.06263664 8 0.761318 0.23868198 9 1.55761 0.55761207 10 0.849523 0.15047725 11 1.05700 0.05700238 12 0.957945 0.04205474 13 0.956754 0.04324599 14 0.409352 0.59064751 15 0.968709 0.03129112 16 1.00000 0.00000000 17 1.00232 0.00232291 18 0.999303 0.00069732 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 9 B = 0 Rule QUAD ERROR 1 0.279429E-02 0.99720571 2 18.3333 17.33333333 3 0.476896 0.52310429 4 0.716146E-01 0.92838542 5 0.287142 0.71285845 6 0.429942 0.57005787 7 0.436368 0.56363240 8 0.996255 0.00374460 9 0.733569 0.26643061 10 0.920084 0.07991597 11 0.923540 0.07645961 12 1.00009 0.00008582 13 0.997886 0.00211358 14 2.77990 1.77990326 15 0.997194 0.00280647 16 1.00000 0.00000000 17 1.00009 0.00009149 18 1.00000 0.00000000 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 9 B = 1 Rule QUAD ERROR 1 0.111772E-01 0.98882284 2 0.00000 1.00000000 3 0.953802 0.04619766 4 0.214844 0.78515625 5 0.686695 0.31330475 6 0.879682 0.12031829 7 0.879906 0.12009352 8 1.13294 0.13294222 9 1.16146 0.16145783 10 1.15215 0.15214823 11 1.15297 0.15296778 12 1.00942 0.00941807 13 1.01744 0.01744391 14 0.909672E-01 0.90903278 15 1.02311 0.02311199 16 1.00000 0.00000000 17 0.998106 0.00189411 18 1.00015 0.00015132 19 1.00000 0.00000000 20 1.00000 0.00000000 A = 10 B = 0 Rule QUAD ERROR 1 0.111772E-02 0.99888228 2 22.0000 21.00000000 3 0.381514 0.61848561 4 0.429688E-01 0.95703125 5 0.207231 0.79276920 6 0.339994 0.66000579 7 0.347660 0.65234018 8 0.968918 0.03108196 9 0.647992 0.35200829 10 0.873671 0.12632881 11 0.877655 0.12234509 12 0.998219 0.00178063 13 0.993975 0.00602508 14 3.31769 2.31769047 15 0.992010 0.00799017 16 1.00000 0.00000000 17 1.00049 0.00048861 18 0.999970 0.00003026 19 1.00000 0.00000000 20 1.00000 0.00000000 TEST43 TRIANGLE_UNIT_PRODUCT_SET sets up a product quadrature rule in the unit triangle, TRIANGLE_UNIT_SUM applies it. Rule Order: 1 2 3 4 5 Function 1 0.50000000 0.50000000 0.50000000 0.50000000 0.50000000 X 0.16666667 0.16666667 0.16666667 0.16666667 0.16666667 X^2 0.05555556 0.08333333 0.08333333 0.08333333 0.08333333 X^3 0.01851852 0.05000000 0.05000000 0.05000000 0.05000000 X^4 0.00617284 0.03166667 0.03333333 0.03333333 0.03333333 X^5 0.00205761 0.02033333 0.02380952 0.02380952 0.02380952 X^6 0.00068587 0.01310000 0.01775510 0.01785714 0.01785714 R 0.23570226 0.26944774 0.27028994 0.27048405 0.27052000 SIN(X) 0.16359735 0.15850111 0.15852907 0.15852902 0.15852902 EXP(X) 0.69780621 0.71817557 0.71828161 0.71828183 0.71828183 1/(1+R) 0.33981138 0.33160874 0.33072840 0.33053935 0.33050247 SQRT(R) 0.34329452 0.35846440 0.35941775 0.35970261 0.35977705 Rule Order: 6 7 8 Function 1 0.50000000 0.50000000 0.50000000 X 0.16666667 0.16666667 0.16666667 X^2 0.08333333 0.08333333 0.08333333 X^3 0.05000000 0.05000000 0.05000000 X^4 0.03333333 0.03333333 0.03333333 X^5 0.02380952 0.02380952 0.02380952 X^6 0.01785714 0.01785714 0.01785714 R 0.27053097 0.27053468 0.27053617 SIN(X) 0.15852902 0.15852902 0.15852902 EXP(X) 0.71828183 0.71828183 0.71828183 1/(1+R) 0.33049147 0.33048775 0.33048626 SQRT(R) 0.35980415 0.35981516 0.35982024 TEST44 TRIANGLE_UNIT_SET sets up quadrature in the unit triangle, TRIANGLE_SUM applies it to an arbitrary triangle. Rule: 1 2 3 4 5 Function 1 3.00000000 3.00000000 3.00000000 3.00000000 3.00000000 X 5.00000000 5.00000000 5.00000000 5.00000000 5.00000000 X^2 8.33333333 11.00000000 9.00000000 9.00000000 9.00000000 X^3 13.88888889 29.00000000 17.44444444 17.00000000 17.40000000 X^4 23.14814815 83.00000000 35.96296296 33.00000000 35.58666667 X^5 38.58024691 245.00000000 77.59259259 65.00000000 75.63111111 X^6 64.30041152 731.00000000 172.62139918 129.00000000 164.51691852 R 7.81024968 8.69959685 8.06377119 8.13021250 8.07051336 SIN(X) 2.98622387 1.82406198 2.66696168 2.66006584 2.66305511 EXP(X) 15.88347015 25.52210058 17.89959429 17.49639403 17.83963153 1/(1+R) 0.83254321 0.84966074 0.83102747 0.81783689 0.82980745 SQRT(R) 4.84053190 4.99802971 4.89405319 4.92554875 4.89711156 Rule: 6 7 8 9 10 Function 1 3.00000000 3.00000000 3.00000000 3.00000000 3.00000000 X 5.00000000 5.00000000 5.00000000 5.00000000 5.00000000 X^2 9.00000000 9.00000000 9.00000000 9.00000000 9.00000000 X^3 17.40000000 17.40000000 17.40000000 17.40000000 17.40000000 X^4 35.66666667 35.66666667 35.80000000 35.80000000 35.80000000 X^5 76.33333333 76.33333333 77.59553799 77.50370370 77.57142857 X^6 168.25333333 168.25925926 175.51916732 174.52335802 175.25427060 R 8.07058628 8.07041532 8.07109893 8.07075431 8.07086192 SIN(X) 2.66628016 2.66627210 2.67142339 2.67159802 2.67147376 EXP(X) 17.85922999 17.85927426 17.89702742 17.89224309 17.89574097 1/(1+R) 0.82965093 0.82970842 0.82948069 0.82936411 0.82950094 SQRT(R) 4.89730546 4.89720275 4.89766891 4.89766442 4.89757853 Rule: 11 12 13 14 15 Function 1 3.00000000 3.00000000 3.00000000 3.00000000 3.00000000 X 5.00000000 5.00000000 5.00000000 5.00000000 5.00000000 X^2 9.00000000 9.00000000 9.00000000 9.00000000 9.00000000 X^3 17.40000000 17.40000000 17.40000000 17.40000000 17.40000000 X^4 35.80000000 35.80000000 35.80000000 36.06666667 35.80000000 X^5 77.57142857 77.57142857 77.57142857 80.11111111 77.57142856 X^6 175.25778781 175.28571429 175.28571429 190.18518519 175.28571427 R 8.07109403 8.07096332 8.07096095 8.07163688 8.07096044 SIN(X) 2.67146896 2.67143334 2.67143322 2.68143637 2.67143328 EXP(X) 17.89576670 17.89603871 17.89603592 17.97443427 17.89603636 1/(1+R) 0.82941974 0.82945918 0.82946016 0.82922831 0.82945976 SQRT(R) 4.89772042 4.89764526 4.89764368 4.89816331 4.89764387 Rule: 16 17 18 19 20 Function 1 3.00000000 3.00000000 3.00000000 3.00000000 3.00000000 X 5.00000000 5.00000000 5.00000000 5.00000000 5.00000000 X^2 9.00000000 9.00000000 9.00000000 9.00000000 9.00000000 X^3 17.40000000 17.40000000 17.40000000 17.40000000 17.40000000 X^4 35.80000000 35.80000000 35.80000000 35.80000000 35.80000000 X^5 77.57142857 77.57142857 77.57142857 77.57142857 77.57142857 X^6 175.28571429 175.28571429 175.28571429 175.28571429 175.28571429 R 8.07096333 8.07096259 8.07096363 8.07096332 8.07096333 SIN(X) 2.67143338 2.67143338 2.67143338 2.67143338 2.67143338 EXP(X) 17.89603716 17.89603720 17.89603715 17.89603716 17.89603716 1/(1+R) 0.82945845 0.82945869 0.82945835 0.82945846 0.82945845 SQRT(R) 4.89764586 4.89764542 4.89764604 4.89764586 4.89764586 TEST45 TORUS_1 approximates integrals on a torus. The degree N will be varied. Inner radius = 0.500000 Outer radius = 1.00000 Area = 19.7392 F(X) 1 4 16 64 256 1 19.73920880 19.73920880 19.73920880 19.73920880 19.73920880 X 0.00000000 -0.00000000 0.00000000 -0.00000000 0.00000000 Y -0.00000000 -0.00000000 0.00000000 -0.00000000 0.00000000 Z 0.00000000 0.00000000 0.00000000 -0.00000000 -0.00000000 X*X 44.41321980 17.27180770 17.27180770 17.27180770 17.27180770 X*Y -0.00000000 -0.00000000 -0.00000000 0.00000000 -0.00000000 X*Z 0.00000000 -0.00000000 0.00000000 -0.00000000 0.00000000 Y*Y 0.00000000 17.27180770 17.27180770 17.27180770 17.27180770 Y*Z 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 Z*Z -0.00000000 9.86960440 9.86960440 9.86960440 9.86960440 X^3 0.00000000 -0.00000000 -0.00000000 0.00000000 -0.00000000 X*Y*Z 0.00000000 -0.00000000 0.00000000 0.00000000 0.00000000 Z*Z*Z 0.00000000 0.00000000 -0.00000000 -0.00000000 -0.00000000 X^4 99.92974456 23.59452302 23.59452302 23.59452302 23.59452302 X^2 Z^2 0.00000000 4.93480220 4.93480220 4.93480220 4.93480220 Z^4 -0.00000000 7.40220330 7.40220330 7.40220330 7.40220330 X^5 0.00000000 5.51309933 -0.00000000 -0.00000000 -0.00000000 X^6 224.84192526 39.22782218 37.87846220 37.87846220 37.87846220 R 29.60881320 30.56781839 30.54421892 30.54421810 30.54421810 SIN(X) 0.00000000 0.04214551 0.00000000 -0.00000000 0.00000000 EXP(X) 46.43470458 29.46458972 29.41250219 29.41250219 29.41250219 1/(1+R) 10.94934300 10.46758450 10.47160632 10.47160632 10.47160632 SQRT(R) 24.17549475 24.89700189 24.87161513 24.87161386 24.87161386 TEST46 For the interior of a torus, TORUS_5S2, TORUS_6S2, and TORUS_5S2 approximate integrals. Inner radius = 0.500000 Outer radius = 1.00000 Volume = 9.86960 Rule: #5S2 #6S2 #14S F(X) 1 9.86960440 9.86960440 9.86960440 X 0.00000000 -0.00000000 -0.00000000 Y -0.00000000 -0.00000000 0.00000000 Z 0.00000000 0.00000000 0.00000000 X*X 4.93480220 4.93480219 4.93480220 X*Y 0.00000000 0.00000000 -0.00000000 X*Z 0.00000000 0.00000000 -0.00000000 Y*Y 4.93480220 4.93480219 4.93480220 Y*Z 0.00000000 0.00000000 -0.00000000 Z*Z 2.46740110 2.46740110 2.46740110 X^3 0.00000000 -0.00000000 -0.00000000 X*Y*Z 0.00000000 0.00000000 -0.00000000 Z*Z*Z 0.00000000 0.00000000 -0.00000000 X^4 4.85769592 4.85769591 4.85769591 X^2 Z^2 0.92527541 0.92527541 0.92527541 Z^4 1.23370055 1.23370055 1.23370055 X^5 0.00000000 0.00000000 0.00000000 X^6 7.20943759 6.12031132 6.12031131 R 10.78710055 10.77891886 10.78668274 SIN(X) -0.00000000 -0.00007317 -0.00000000 EXP(X) 12.54974574 12.54821237 12.54813208 1/(1+R) 6.71114069 6.71072788 6.71137682 SQRT(R) 10.25427136 10.24378739 10.25520737 TEST47 For integrals inside a torus with square cross-section: TORUS_SQUARE_5C2 approximates the integral; TORUS_SQUARE_14C approximates the integral. Inner radius = 1.00000 Outer radius = 0.125000 Volume = 0.392699 F(X) 5C2 14C 1 0.39269908 0.39269908 X 0.00000000 0.00000000 Y 0.00000000 0.00000000 Z 0.00000000 0.00000000 X*X 0.19941750 0.19941750 X*Y 0.00000000 0.00000000 X*Z 0.00000000 -0.00000000 Y*Y 0.19941750 0.19941750 Y*Z 0.00000000 -0.00000000 Z*Z 0.00204531 0.00204531 X^3 -0.00000000 -0.00000000 X*Y*Z 0.00000000 -0.00000000 Z*Z*Z 0.00000000 0.00000000 X^4 0.15496801 0.15496801 X^2 Z^2 0.00103863 0.00103863 Z^4 0.00001917 0.00001917 X^5 0.00000000 0.00000000 X^6 0.14998609 0.13635099 R 0.39575389 0.39576462 SIN(X) 0.00000000 -0.00000000 EXP(X) 0.49907710 0.49905742 1/(1+R) 0.27678210 0.27677774 SQRT(R) 0.39396739 0.39397703 TEST48 For evenly spaced angles between 0 and 2*PI: TVEC_EVEN TVEC_EVEN2 TVEC_EVEN3 TVEC_EVEN: 1 0.0000000 2 1.5707963 3 3.1415927 4 4.7123890 TVEC_EVEN2: 1 0.78539816 2 2.3561945 3 3.9269908 4 5.4977871 TVEC_EVEN3: 1 0.0000000 2 2.0943951 3 4.1887902 4 6.2831853 TEST49 For evenly spaced angles between THETA1 and THETA2: TVEC_EVEN_BRACKET TVEC_EVEN_BRACKET2. TVEC_EVEN_BRACKET3. THETA1 = 30.0000 THETA2 = 90.0000 TVEC_EVEN_BRACKET 1 30.000000 2 50.000000 3 70.000000 4 90.000000 TVEC_EVEN_BRACKET2 1 40.000000 2 50.000000 3 60.000000 4 70.000000 5 80.000000 TVEC_EVEN_BRACKET3 1 40.000000 2 60.000000 3 80.000000 STROUD_TEST Normal end of execution. 15 September 2021 3:23:37.088 PM