18 June 2012 8:28:00.091 PM STROUD_PRB FORTRAN77 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 -0.113537E-17 -0.116376E-15 Y 0.326988E-16 -0.669871E-17 -0.887595E-16 Z 0.00000 0.953714E-17 -0.908299E-18 X*X 0.837758 0.837758 0.837758 X*Y 0.413276E-16 0.420088E-17 -0.496983E-16 X*Z 0.00000 -0.851530E-18 0.227075E-18 Y*Y 0.837758 0.837758 0.837758 Y*Z 0.00000 0.00000 -0.586315E-17 Z*Z 0.837758 0.837758 0.837758 X^3 -0.175415E-16 -0.316131E-17 -0.627578E-16 X*Y*Z 0.00000 0.126381E-16 0.332871E-18 Z*Z*Z 0.00000 -0.180241E-17 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 -0.158165E-18 -0.440002E-16 X^6 0.199466 0.199466 0.199466 R 3.15453 3.14291 3.14291 SIN(X) 0.490482E-16 -0.567687E-17 -0.875373E-16 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 0.00000 0.00000 1.00000 2.00000 9.42478 0.00000 0.00000 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 -0.217992E-16 X^4 0.392181 0.392699 X^5 -0.390120E-03 -0.163494E-16 X^6 0.245151 0.245437 R 2.09452 2.09637 SIN(X) -0.621017E-03 0.299739E-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 0.00000 -0.174393E-15 -0.141695E-15 -0.926465E-16 -0.544979E-16 X^2 0.785398 0.785398 0.785398 0.785398 0.785398 X^3 0.00000 -0.130795E-15 -0.108996E-15 -0.103546E-15 -0.190743E-16 X^4 0.196350 0.392699 0.392699 0.392699 0.392699 X^5 0.00000 -0.130795E-15 -0.762971E-16 -0.762971E-16 -0.544979E-17 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) 0.00000 -0.152594E-15 -0.871967E-16 -0.490482E-16 0.00000 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 0.566779E-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 0.959164E-15 0.117716E-14 X^4 9.42478 12.5664 12.5664 12.5664 12.5664 X^5 0.00000 -0.279029E-14 0.209272E-14 0.488302E-14 0.209272E-14 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 0.566779E-15 -0.125345E-15 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 -0.544979E-16 0.00000 -0.544979E-16 -0.272490E-17 X^2 0.785398 0.00000 0.785398 0.785398 X^3 -0.190743E-16 0.00000 -0.190743E-16 -0.217992E-16 X^4 0.392699 0.00000 0.392699 0.392699 X^5 -0.544979E-17 0.00000 -0.544979E-17 -0.163494E-16 X^6 0.245437 0.00000 0.245437 0.245437 R 2.09637 0.00000 2.09637 2.09637 SIN(X) 0.00000 0.00000 0.00000 0.299739E-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 -0.103546E-14 -0.414184E-15 -0.217992E-16 X^2 6.28319 6.28319 12.5664 12.5664 X^3 0.117716E-14 -0.222352E-14 -0.104636E-14 -0.697574E-15 X^4 12.5664 12.5664 25.1327 25.1327 X^5 0.209272E-14 -0.924285E-14 -0.715013E-14 -0.209272E-14 X^6 31.4159 31.4159 62.8319 62.8319 R 8.38550 8.38550 16.7710 16.7710 SIN(X) -0.125345E-15 -0.664875E-15 -0.790220E-15 0.102456E-14 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 -0.223224E-13 X^4 402.124 1206.37 1608.50 1608.50 X^5 1248.31 -1248.31 -0.161715E-02 -0.267868E-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 -0.170034E-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 -0.714315E-12 X^4 23793.4 79014.3 102808. 102944. X^5 151839. -152658. -818.194 -0.342871E-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 -0.470862E-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 ) (1-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 0.155431E-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 0.222045E-15 GW_02_XIU: 2 0.392699 0.166533E-15 CN_GEG_03_XIU: 2 0.392699 0.222045E-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 0.277556E-16 GW_02_XIU: 2 0.196350 0.138778E-15 CN_GEG_03_XIU: 2 0.196350 0.277556E-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 0.488498E-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 0.133227E-14 0.133227E-14 GW_02_XIU: 3 0.177636E-14 0.177636E-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 0.666134E-15 0.666134E-15 GW_02_XIU: 3 0.555112E-15 0.555112E-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 0.222045E-15 0.222045E-15 GW_02_XIU: 3 0.333067E-15 0.333067E-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 0.222045E-15 0.222045E-15 GW_02_XIU: 3 0.222045E-15 0.222045E-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 0.111022E-15 0.111022E-15 GW_02_XIU: 3 0.111022E-15 0.111022E-15 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 0.133227E-14 0.133227E-14 GW_02_XIU: 3 0.155431E-14 0.155431E-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 0.138778E-15 0.138778E-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 0.832667E-16 0.832667E-16 GW_02_XIU: 3 0.832667E-16 0.832667E-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 0.693889E-16 0.693889E-16 GW_02_XIU: 3 0.832667E-16 0.832667E-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 0.555112E-15 GW_02_XIU: 3 0.616850 0.666134E-15 CN_GEG_03_XIU: 4 0.616850 0.444089E-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 0.555112E-16 GW_02_XIU: 3 0.231319 0.416334E-15 CN_GEG_03_XIU: 4 0.231319 0.277556E-16 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 0.115463E-13 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 0.596701E-16 0.596701E-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 0.168063E-16 0.168063E-16 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 0.777156E-15 GW_02_XIU: 4 0.968946 0.210942E-14 CN_GEG_03_XIU: 6 0.968946 0.666134E-15 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 0.00000 GW_02_XIU: 5 6.08807 0.239808E-13 CN_GEG_03_XIU: 8 6.08807 0.00000 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 0.355271E-14 0.355271E-14 GW_02_XIU: 5 0.355271E-14 0.355271E-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 0.666134E-15 0.666134E-15 GW_02_XIU: 5 0.666134E-15 0.666134E-15 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 0.222045E-15 0.222045E-15 GW_02_XIU: 5 0.222045E-15 0.222045E-15 CN_GEG_03_XIU: 8 -0.839489E-15 0.839489E-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 0.832667E-16 0.832667E-16 GW_02_XIU: 5 0.832667E-16 0.832667E-16 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 0.277556E-16 0.277556E-16 GW_02_XIU: 5 0.277556E-16 0.277556E-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 0.266454E-14 0.266454E-14 GW_02_XIU: 5 0.266454E-14 0.266454E-14 CN_GEG_03_XIU: 8 -0.120594E-14 0.120594E-14 EXACT: 0.00000 N = 4 ALPHA = 0.00000 EXPON = 1 1 0 0 Degree = 2 CN_GEG_02_XIU: 5 0.222045E-15 0.222045E-15 GW_02_XIU: 5 0.111022E-15 0.111022E-15 CN_GEG_03_XIU: 8 -0.104528E-15 0.104528E-15 EXACT: 0.00000 N = 4 ALPHA = 0.500000 EXPON = 1 1 0 0 Degree = 2 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 -0.376855E-16 0.376855E-16 EXACT: 0.00000 N = 4 ALPHA = 1.00000 EXPON = 1 1 0 0 Degree = 2 CN_GEG_02_XIU: 5 0.277556E-16 0.277556E-16 GW_02_XIU: 5 0.277556E-16 0.277556E-16 CN_GEG_03_XIU: 8 -0.109493E-16 0.109493E-16 EXACT: 0.00000 N = 4 ALPHA = 1.50000 EXPON = 1 1 0 0 Degree = 2 CN_GEG_02_XIU: 5 0.277556E-16 0.277556E-16 GW_02_XIU: 5 0.277556E-16 0.277556E-16 CN_GEG_03_XIU: 8 -0.578088E-17 0.578088E-17 EXACT: 0.00000 N = 4 ALPHA = -0.500000 EXPON = 2 0 0 0 Degree = 2 CN_GEG_02_XIU: 5 48.7045 0.213163E-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 0.355271E-14 CN_GEG_03_XIU: 8 5.33333 0.266454E-14 EXACT: 5.33333 N = 4 ALPHA = 0.500000 EXPON = 2 0 0 0 Degree = 2 CN_GEG_02_XIU: 5 1.52202 0.888178E-15 GW_02_XIU: 5 1.52202 0.510703E-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 0.222045E-15 EXACT: 0.632099 N = 4 ALPHA = 1.50000 EXPON = 2 0 0 0 Degree = 2 CN_GEG_02_XIU: 5 0.321050 0.555112E-16 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 0.479616E-13 CN_GEG_03_XIU: 10 9.56312 0.177636E-14 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 -0.338477E-16 0.338477E-16 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 0.142259E-16 0.142259E-16 EXACT: 0.00000 N = 5 ALPHA = 1.50000 EXPON = 1 1 0 0 0 Degree = 2 CN_GEG_02_XIU: 6 0.485723E-16 0.485723E-16 GW_02_XIU: 6 0.485723E-16 0.485723E-16 CN_GEG_03_XIU: 10 0.922771E-17 0.922771E-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 0.976996E-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 0.00000 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 0.177636E-14 GW_02_XIU: 7 15.0217 0.905942E-13 CN_GEG_03_XIU: 12 15.0217 0.355271E-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 -0.247739E-12 0.247739E-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 -0.777156E-15 0.777156E-15 GW_02_XIU: 7 -0.666134E-15 0.666134E-15 CN_GEG_03_XIU: 12 -0.261892E-14 0.261892E-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 -0.406349E-15 0.406349E-15 EXACT: 0.00000 N = 6 ALPHA = -0.500000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 0.284217E-13 0.284217E-13 GW_02_XIU: 7 0.142109E-13 0.142109E-13 CN_GEG_03_XIU: 12 -0.125173E-13 0.125173E-13 EXACT: 0.00000 N = 6 ALPHA = 0.00000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 0.133227E-14 0.133227E-14 GW_02_XIU: 7 0.133227E-14 0.133227E-14 CN_GEG_03_XIU: 12 -0.426771E-15 0.426771E-15 EXACT: 0.00000 N = 6 ALPHA = 0.500000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 0.111022E-15 0.111022E-15 GW_02_XIU: 7 0.222045E-15 0.222045E-15 CN_GEG_03_XIU: 12 -0.977912E-16 0.977912E-16 EXACT: 0.00000 N = 6 ALPHA = 1.00000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 0.277556E-16 0.277556E-16 GW_02_XIU: 7 0.277556E-16 0.277556E-16 CN_GEG_03_XIU: 12 -0.319947E-16 0.319947E-16 EXACT: 0.00000 N = 6 ALPHA = 1.50000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 0.00000 0.00000 GW_02_XIU: 7 0.693889E-17 0.693889E-17 CN_GEG_03_XIU: 12 -0.431190E-17 0.431190E-17 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 0.222045E-14 GW_02_XIU: 7 3.75543 0.199840E-13 CN_GEG_03_XIU: 12 3.75543 0.310862E-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 0.666134E-15 0.666134E-15 GW_02_XIU: 3 0.666134E-15 0.666134E-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 0.00000 GW_02_XIU: 3 0.444444 0.00000 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 0.666134E-15 0.666134E-15 GW_02_XIU: 5 0.666134E-15 0.666134E-15 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 0.222045E-15 0.222045E-15 GW_02_XIU: 5 0.222045E-15 0.222045E-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 0.888178E-15 GW_02_XIU: 5 5.33333 0.888178E-15 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 0.177636E-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 0.444089E-15 GW_02_XIU: 6 3.55556 0.444089E-15 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 0.133227E-14 0.133227E-14 GW_02_XIU: 7 0.133227E-14 0.133227E-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 0.177636E-14 GW_02_XIU: 7 7.11111 0.177636E-14 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 0.142109E-13 GW_02_XIU: 7 89.8985 0.568434E-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 0.222045E-15 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 0.666134E-15 0.666134E-15 GW_02_XIU: 3 0.666134E-15 0.666134E-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 -0.111022E-15 0.111022E-15 CN_LEG_03_XIU: 6 -0.111022E-15 0.111022E-15 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 0.666134E-15 0.666134E-15 GW_02_XIU: 5 0.666134E-15 0.666134E-15 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 0.222045E-15 0.222045E-15 GW_02_XIU: 5 0.222045E-15 0.222045E-15 CN_LEG_03_1: 8 -0.104528E-15 0.104528E-15 CN_LEG_03_XIU: 8 -0.104528E-15 0.104528E-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 0.266454E-14 CN_LEG_03_XIU: 8 5.33333 0.266454E-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 0.00000 0.00000 CN_LEG_03_XIU: 8 0.00000 0.00000 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 0.133227E-14 0.133227E-14 GW_02_XIU: 7 0.133227E-14 0.133227E-14 CN_LEG_03_1: 12 -0.426771E-15 0.426771E-15 CN_LEG_03_XIU: 12 -0.426771E-15 0.426771E-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 1.00000000 1.18518519 1.25000000 1.28000000 1.29629630 1.30612245 1.31250000 1.31687243 1.32000000 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 0.25000000 0.52674897 0.64062500 0.69632000 0.72736626 0.74635569 0.75878906 0.76736270 0.77352000 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 0.06250000 0.23411065 0.35644531 0.42598400 0.46739255 0.49363300 0.51118469 0.52345729 0.53235600 R 0.00000000 2.82842712 2.86129015 2.99535239 2.99898282 3.03054964 3.03142393 3.04344523 3.04374925 3.04955799 0.00000000 2.82842712 2.86129015 2.99535239 2.99898282 3.03054964 3.03142393 3.04344523 3.04374925 3.04955799 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 4.51050386 4.61486821 4.65219277 4.66961171 4.67911211 4.68485363 4.68858537 4.69114625 4.69297923 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 2.34314575 2.42616645 2.34104062 2.35740728 2.33744673 2.34321959 2.33566092 2.33833179 2.33469505 SQRT(R) 0.00000000 3.36358566 3.17774244 3.40276654 3.34985327 3.41302022 3.38986666 3.41711827 3.40464103 3.41916252 0.00000000 3.36358566 3.17774244 3.40276654 3.34985327 3.41302022 3.38986666 3.41711827 3.40464103 3.41916252 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 2.00000000 2.37037037 2.50000000 2.56000000 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 0.50000000 1.05349794 1.28125000 1.39264000 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 0.71289063 0.85196800 0.00000000 0.12500000 0.46822131 0.71289063 0.85196800 R 0.00000000 6.92820323 7.27446305 7.48869361 7.55031790 0.00000000 6.92820323 7.27446305 7.48869361 7.55031790 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 9.02100772 9.22973643 9.30438554 9.33922342 1/(1+R) 8.00000000 4.28718708 4.29316635 4.20849132 4.20436029 8.00000000 4.28718708 4.29316635 4.20849132 4.20436029 SQRT(R) 0.00000000 7.44483887 7.45106414 7.66122741 7.66536795 0.00000000 7.44483887 7.45106414 7.66122741 7.66536795 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 0.2184 0.9563 0.9736 4.0646 0.6562 0.8295 0.5617 0.7359 4.4112 1.4638 0.4153 0.0661 0.9872 1.7190 0.0863 0.2576 0.1100 0.9043 1.2017 0.0890 (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 3.07614 0.654498E-01 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 0.785398 0.785398 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 1.11265 1.24355 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 -0.160319 0.160319 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 0.196350 0.589049 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 4.74236 0.825969 EN_R2_05_5: 25 5.03934 0.528991 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 0.444089E-15 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 0.426326E-12 EN_R2_11_1(2): 151 164.440 0.105160E-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 9.07455 0.795051 EN_R2_05_5: 65 -58.7503 68.6199 EN_R2_07_1(1): 49 -26.8335 36.7031 EN_R2_07_1(2): 49 -23.5689 33.4385 EN_R2_07_2: 96 8.02751 1.84209 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 0.177636E-14 GW_02_XIU: 5 4.93480 0.177636E-14 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 0.420641E-11 EN_R2_11_1(2): 417 291.462 0.159162E-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 16.0008 1.49262 EN_R2_05_5: 161 -559.246 576.740 EN_R2_05_6: 192 -562.420 579.913 EN_R2_07_2: 164 12.5654 4.92799 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 0.522050E-09 EN_R2_11_1(2): 983 516.603 0.341061E-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 -1047.32 1078.33 EN_R2_05_5: 385 -2656.94 2687.94 EN_R2_05_6: 448 -2664.78 2695.79 EN_R2_07_1(1): 137 -1719.85 1750.86 EN_R2_07_2: 272 21.2118 9.79445 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 43.3958 11.5614 EN_R2_05_5: 897 -4.94966 59.9069 EN_R2_05_6: 1024 -16.9350 71.8922 EN_R2_07_1(1): 227 44.9405 10.0167 EN_R2_07_2: 452 -47.9552 102.912 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 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 0.666134E-15 GW_02_XIU: 3 1.57080 0.666134E-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 0.333067E-15 GW_02_XIU: 3 1.00000 0.333067E-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 0.00000 GW_02_XIU: 3 0.785398 0.00000 EXACT: 0.785398 N = 2 ALPHA = 0.00000 EXPON = 1 1 Degree = 2 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 = 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 0.00000 GW_02_XIU: 5 1.00000 0.00000 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 0.177636E-14 GW_02_XIU: 5 7.40220 0.177636E-14 EXACT: 7.40220 N = 4 ALPHA = 0.00000 EXPON = 2 0 0 0 Degree = 2 EPN_GLG_02_XIU: 5 2.00000 0.00000 GW_02_XIU: 5 2.00000 0.00000 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 0.00000 GW_02_XIU: 5 6.00000 0.00000 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 0.888178E-15 GW_02_XIU: 6 4.37335 0.888178E-15 EXACT: 4.37335 N = 5 ALPHA = 0.00000 EXPON = 1 1 0 0 0 Degree = 2 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 = 1 1 0 0 0 Degree = 2 EPN_GLG_02_XIU: 6 1.23001 0.00000 GW_02_XIU: 6 1.23001 0.00000 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 0.266454E-14 GW_02_XIU: 7 7.75157 0.266454E-14 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 0.444089E-15 GW_02_XIU: 7 4.00000 0.444089E-15 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 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 0.333067E-15 GW_02_XIU: 3 1.00000 0.333067E-15 EXACT: 1.00000 N = 2 EXPON = 1 1 Degree = 2 EPN_LAG_02_XIU: 3 1.00000 0.00000 GW_02_XIU: 3 1.00000 0.00000 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 0.00000 GW_02_XIU: 5 1.00000 0.00000 EXACT: 1.00000 N = 4 EXPON = 2 0 0 0 Degree = 2 EPN_LAG_02_XIU: 5 2.00000 0.00000 GW_02_XIU: 5 2.00000 0.00000 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 0.111022E-15 GW_02_XIU: 6 1.00000 0.111022E-15 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: 0.00000 1.00000 0.00000 2: 0.00000 0.00000 1.00000 Volume is 1.00000 Spatial dimension N = 3 Parallelipiped vertices Col 1 2 3 4 Row 1: 0.00000 1.00000 0.00000 0.00000 2: 0.00000 0.00000 1.00000 0.00000 3: 0.00000 0.00000 0.00000 1.00000 Volume is 1.00000 Spatial dimension N = 4 Parallelipiped vertices Col 1 2 3 4 5 Row 1: 0.00000 1.00000 0.00000 0.00000 0.00000 2: 0.00000 0.00000 1.00000 0.00000 0.00000 3: 0.00000 0.00000 0.00000 1.00000 0.00000 4: 0.00000 0.00000 0.00000 0.00000 1.00000 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 1.00000 Y 1.00000 X^2 1.00000 XY 1.00000 Y^2 1.00000 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 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: 0.00000 1.00000 0.00000 2: 0.00000 0.00000 1.00000 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: 0.00000 1.00000 0.00000 0.00000 2: 0.00000 0.00000 1.00000 0.00000 3: 0.00000 0.00000 0.00000 1.00000 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: 0.00000 1.00000 0.00000 0.00000 0.00000 2: 0.00000 0.00000 1.00000 0.00000 0.00000 3: 0.00000 0.00000 0.00000 1.00000 0.00000 4: 0.00000 0.00000 0.00000 0.00000 1.00000 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: 0.00000 1.00000 0.00000 2: 0.00000 0.00000 1.00000 Volume is 0.500000 Spatial dimension N = 3 Simplex vertices: Col 1 2 3 4 Row 1: 0.00000 1.00000 0.00000 0.00000 2: 0.00000 0.00000 1.00000 0.00000 3: 0.00000 0.00000 0.00000 1.00000 Volume is 0.166667 Spatial dimension N = 4 Simplex vertices: Col 1 2 3 4 5 Row 1: 0.00000 1.00000 0.00000 0.00000 0.00000 2: 0.00000 0.00000 1.00000 0.00000 0.00000 3: 0.00000 0.00000 0.00000 1.00000 0.00000 4: 0.00000 0.00000 0.00000 0.00000 1.00000 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 0.51758E-09 0.51758E-09 0.51758E-09 0.51758E-09 0.51758E-09 I4_FACTORIAL - Fatal error! I4_FACTORIAL(N) cannot be computed as an integer for 13 < N. Input value N = 14