02 March 2022 02:57:06 PM STROUD_TEST C 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: 1.000000 2.000000 Ball radius = 2.000000 Ball volume = 12.566371 Rule: F1 F3 F(X) 1 12.566371 12.566371 X 12.566371 12.566371 X^2 25.132741 25.132741 X^3 50.265482 50.265482 X^4 113.097336 113.097336 X^5 263.893783 263.893783 X^6 636.868863 628.318531 R 30.989157 31.108749 SIN(X) 6.102432 6.111611 EXP(X) 54.316802 54.281686 1/(1+R) 3.996849 3.925474 SQRT(R) 19.277156 19.378488 Spatial dimension N = 3 Ball center: 1.000000 2.000000 3.000000 Ball radius = 2.000000 Ball volume = 33.510322 Rule: F1 F3 F(X) 1 33.510322 33.510322 X 33.510322 33.510322 X^2 60.318579 60.318579 X^3 113.935094 113.935094 X^4 240.316878 240.316878 X^5 531.377957 531.377957 X^6 1224.697009 1203.772819 R 132.548551 132.561954 SIN(X) 18.418146 18.440781 EXP(X) 133.123885 133.038588 1/(1+R) 6.994492 6.990274 SQRT(R) 66.220129 66.227943 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 = 2.000000 Ball volume = 33.510322 Rule: MONOMIAL F1 F(X) 1 33.510322 33.510322 xyz 0.000000 0.000000 x^2 z^2 15.319004 15.319004 x^4y^2z^2 7.427396 7.610100 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 = 4.188790 Rule: #7 #14 #15 F(X) 1 4.188790 4.188790 4.188790 X 0.000000 0.000000 -0.000000 Y -0.000000 0.000000 -0.000000 Z 0.000000 0.000000 -0.000000 X*X 0.837758 0.837758 0.837758 X*Y 0.296192 -0.000000 -0.000000 X*Z 0.000000 -0.000000 -0.000000 Y*Y 0.837758 0.837758 0.837758 Y*Z 0.000000 0.000000 -0.000000 Z*Z 0.837758 0.837758 0.837758 X^3 0.000000 0.000000 -0.000000 X*Y*Z 0.000000 0.000000 -0.000000 Z*Z*Z 0.000000 0.000000 0.000000 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 -0.000000 -0.000000 -0.000000 X^6 0.199466 0.199466 0.199466 R 3.154534 3.142906 3.142906 SIN(X) -0.000000 -0.000000 -0.000000 EXP(X) 4.622909 4.622909 4.622909 1/(1+R) 3.347149 3.347932 3.347932 SQRT(R) 3.607050 3.592690 3.592690 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 = 3.141593 Rule: F1 F3 F(X) 1 3.141593 3.141593 X 0.000000 0.000000 X^2 0.785398 0.785398 X^3 0.000000 0.000000 X^4 0.392699 0.392699 X^5 0.000000 0.000000 X^6 0.229749 0.196350 R 2.074653 1.923825 SIN(X) 0.000000 0.000000 EXP(X) 3.550977 3.550929 1/(1+R) 1.942542 2.082507 SQRT(R) 2.494885 2.129062 Spatial dimension N = 3 Unit ball volume = 4.188790 Rule: F1 F3 F(X) 1 4.188790 4.188790 X 0.000000 0.000000 X^2 0.837758 0.837758 X^3 0.000000 0.000000 X^4 0.359039 0.359039 X^5 0.000000 0.000000 X^6 0.194741 0.153874 R 3.123589 2.973746 SIN(X) 0.000000 0.000000 EXP(X) 4.622902 4.622845 1/(1+R) 2.440326 2.577138 SQRT(R) 3.572543 3.234714 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 4.188790 BALL_UNIT_VOLUME_3D 3 4.188790 BALL_UNIT_VOLUME_ND TEST05 BALL_UNIT_VOLUME_ND computes the volume of the unit ball in ND. N Volume 2 3.141593 3 4.188790 4 4.934802 5 5.263789 6 5.167713 7 4.724766 8 4.058712 9 3.298509 10 2.550164 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 1.000000 4.188790 BALL_VOLUME_3D 3 1.000000 4.188790 BALL_VOLUME_ND 3 2.000000 33.510322 BALL_VOLUME_3D 3 2.000000 33.510322 BALL_VOLUME_ND 3 4.000000 268.082573 BALL_VOLUME_3D 3 4.000000 268.082573 BALL_VOLUME_ND TEST054 BALL_UNIT_VOLUME_ND computes the volume of the unit ball in N dimensions. N R Volume 2 0.500000 0.785398 2 1.000000 3.141593 2 2.000000 12.566371 3 0.500000 0.523599 3 1.000000 4.188790 3 2.000000 33.510322 4 0.500000 0.308425 4 1.000000 4.934802 4 2.000000 78.956835 5 0.500000 0.164493 5 1.000000 5.263789 5 2.000000 168.441248 6 0.500000 0.080746 6 1.000000 5.167713 6 2.000000 330.733618 7 0.500000 0.036912 7 1.000000 4.724766 7 2.000000 604.770044 8 0.500000 0.015854 8 1.000000 4.058712 8 2.000000 1039.030304 9 0.500000 0.006442 9 1.000000 3.298509 9 2.000000 1688.836558 10 0.500000 0.002490 10 1.000000 2.550164 10 2.000000 2611.367977 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 0.000000 0.000000 0.000000 1.000000 3.141593 1 0.000000 0.000000 0.000000 1.000000 3.141593 3.141593 X 0.000000 0.000000 0.000000 1.000000 -0.000000 -0.000000 X^2 0.000000 0.000000 0.000000 1.000000 0.785398 0.785398 X^3 0.000000 0.000000 0.000000 1.000000 -0.000000 -0.000000 X^4 0.000000 0.000000 0.000000 1.000000 0.392699 0.392699 X^5 0.000000 0.000000 0.000000 1.000000 -0.000000 -0.000000 X^6 0.000000 0.000000 0.000000 1.000000 0.245437 0.245437 R 0.000000 0.000000 0.000000 1.000000 2.096375 2.096375 SIN(X) 0.000000 0.000000 0.000000 1.000000 -0.000000 -0.000000 EXP(X) 0.000000 0.000000 0.000000 1.000000 3.550999 3.550999 1/(1+R) 0.000000 0.000000 0.000000 1.000000 1.926085 1.926085 SQRT(R) 0.000000 0.000000 0.000000 1.000000 2.517540 2.517540 Area 0.000000 0.000000 1.000000 2.000000 9.424778 1 0.000000 0.000000 1.000000 2.000000 9.424778 9.424778 X 0.000000 0.000000 1.000000 2.000000 0.000000 -0.000000 X^2 0.000000 0.000000 1.000000 2.000000 11.780972 11.780972 X^3 0.000000 0.000000 1.000000 2.000000 0.000000 -0.000000 X^4 0.000000 0.000000 1.000000 2.000000 24.740042 24.740042 X^5 0.000000 0.000000 1.000000 2.000000 -0.000000 -0.000000 X^6 0.000000 0.000000 1.000000 2.000000 62.586416 62.586416 R 0.000000 0.000000 1.000000 2.000000 14.660769 14.674623 SIN(X) 0.000000 0.000000 1.000000 2.000000 0.000000 0.000000 EXP(X) 0.000000 0.000000 1.000000 2.000000 16.437533 16.437533 1/(1+R) 0.000000 0.000000 1.000000 2.000000 3.735571 3.723083 SQRT(R) 0.000000 0.000000 1.000000 2.000000 11.703954 11.723818 Area 0.000000 0.000000 1.000000 3.000000 25.132741 1 0.000000 0.000000 1.000000 3.000000 25.132741 25.132741 X 0.000000 0.000000 1.000000 3.000000 -0.000000 -0.000000 X^2 0.000000 0.000000 1.000000 3.000000 62.831853 62.831853 X^3 0.000000 0.000000 1.000000 3.000000 -0.000000 -0.000000 X^4 0.000000 0.000000 1.000000 3.000000 285.884931 285.884931 X^5 0.000000 0.000000 1.000000 3.000000 -0.000000 -0.000000 X^6 0.000000 0.000000 1.000000 3.000000 1610.066235 1610.066235 R 0.000000 0.000000 1.000000 3.000000 54.454753 54.505742 SIN(X) 0.000000 0.000000 1.000000 3.000000 -0.000000 -0.000000 EXP(X) 0.000000 0.000000 1.000000 3.000000 70.968274 70.968274 1/(1+R) 0.000000 0.000000 1.000000 3.000000 8.210871 8.168636 SQRT(R) 0.000000 0.000000 1.000000 3.000000 36.665239 36.727029 TEST085 CIRCLE_ANNULUS_AREA_2D computes the area of a circular annulus. CENTER Radius1 Radius2 Area 0.000000 0.000000 0.000000 1.000000 3.141593 1.000000 0.000000 1.000000 2.000000 9.424778 3.000000 4.000000 1.000000 3.000000 25.132741 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 3.141593 3.141593 X -0.000728 -0.000000 X^2 0.784623 0.785398 X^3 -0.000661 -0.000000 X^4 0.392181 0.392699 X^5 -0.000390 -0.000000 X^6 0.245151 0.245437 R 2.094516 2.096375 SIN(X) -0.000621 -0.000000 EXP(X) 3.549748 3.550999 1/(1+R) 1.927896 1.926085 SQRT(R) 2.513561 2.517540 TEST10 CIRCLE_CUM approximates an integral over a circle. We use radius R = 3.000000 and center: CENTER = ( 0.000000, 0.000000). Order: 2 4 8 16 F(X) 1 28.274334 28.274334 28.274334 28.274334 X 0.000000 -0.000000 -0.000000 -0.000000 X^2 254.469005 127.234502 127.234502 127.234502 X^3 0.000000 -0.000000 -0.000000 -0.000000 X^4 2290.221044 1145.110522 858.832892 858.832892 X^5 0.000000 -0.000000 -0.000000 -0.000000 X^6 20611.989400 10305.994700 6441.246688 6441.246688 R 84.823002 84.823002 84.823002 84.823002 SIN(X) 0.000000 -0.000000 -0.000000 -0.000000 EXP(X) 284.656437 156.465385 138.047173 138.001159 1/(1+R) 7.068583 7.068583 7.068583 7.068583 SQRT(R) 48.972583 48.972583 48.972583 48.972583 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 1.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.523599 0.261799 0.250000 0.011799 1.000000 0.000000 1.047198 0.523599 0.433013 0.090586 1.000000 0.000000 1.570796 0.785398 0.500000 0.285398 1.000000 0.000000 2.094395 1.047198 0.433013 0.614185 1.000000 0.000000 2.617994 1.308997 0.250000 1.058997 1.000000 0.000000 3.141593 1.570796 0.000000 1.570796 1.000000 0.000000 3.665191 1.832596 -0.250000 2.082596 1.000000 0.000000 4.188790 2.094395 -0.433013 2.527408 1.000000 0.000000 4.712389 2.356194 -0.500000 2.856194 1.000000 0.000000 5.235988 2.617994 -0.433013 3.051007 1.000000 0.000000 5.759587 2.879793 -0.250000 3.129793 1.000000 0.000000 6.283185 3.141593 -0.000000 3.141593 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 = 50.000000 THETA1 THETA2 H W Area(THETA) Area(H) Area(W) 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.523599 1.703709 25.881905 29.498469 29.498469 29.498469 0.000000 1.047198 6.698730 50.000000 226.465184 226.465184 226.465184 0.000000 1.570796 14.644661 70.710678 713.495408 713.495408 713.495408 0.000000 2.094395 25.000000 86.602540 1535.462123 1535.462123 1535.462123 0.000000 2.617994 37.059048 96.592583 2647.492347 2647.492347 2647.492347 0.000000 3.141593 50.000000 100.000000 3926.990817 3926.990817 3926.990817 0.000000 3.665191 62.940952 96.592583 5206.489286 5206.489286 2647.492347 0.000000 4.188790 75.000000 86.602540 6318.519511 6318.519511 1535.462123 0.000000 4.712389 85.355339 70.710678 7140.486225 7140.486225 713.495408 0.000000 5.235988 93.301270 50.000000 7627.516450 7627.516450 226.465184 0.000000 5.759587 98.296291 25.881905 7824.483164 7824.483164 29.498469 0.000000 6.283185 100.000000 0.000000 7853.981634 7853.981634 0.000000 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 0.000000 0.000000 1.000000 0.000000 6.283185 3.141593 F 1 2 3 4 5 1 3.141593 3.141593 3.141593 3.141593 3.141593 X -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 X^2 0.785398 0.785398 0.785398 0.785398 0.785398 X^3 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 X^4 0.196350 0.392699 0.392699 0.392699 0.392699 X^5 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 X^6 0.049087 0.245437 0.245437 0.245437 0.245437 R 2.221441 2.117080 2.102290 2.098042 2.096375 SIN(X) -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 EXP(X) 3.542541 3.550999 3.550999 3.550999 3.550999 1/(1+R) 1.840302 1.908065 1.920618 1.924507 1.926085 SQRT(R) 2.641754 2.545301 2.526703 2.520359 2.517540 CENTER RADIUS THETA1 THETA2 Area 0.000000 0.000000 2.000000 0.000000 3.141593 6.283185 F 1 2 3 4 5 1 6.283185 6.283185 6.283185 6.283185 6.283185 X 0.000000 0.000000 0.000000 0.000000 0.000000 X^2 6.283185 6.283185 6.283185 6.283185 6.283185 X^3 0.000000 0.000000 0.000000 0.000000 0.000000 X^4 9.424778 12.566371 12.566371 12.566371 12.566371 X^5 0.000000 -0.000000 0.000000 0.000000 0.000000 X^6 15.707963 31.415927 31.415927 31.415927 31.415927 R 8.885766 8.468318 8.409159 8.392170 8.385499 SIN(X) 0.000000 0.000000 0.000000 0.000000 -0.000000 EXP(X) 9.839969 9.994199 9.994266 9.994266 9.994266 1/(1+R) 2.602581 2.768394 2.806112 2.819036 2.824584 SQRT(R) 7.472009 7.199199 7.146596 7.128652 7.120679 CENTER RADIUS THETA1 THETA2 Area 0.000000 0.000000 4.000000 0.000000 1.570796 12.566371 F 1 2 3 4 5 1 12.566371 12.566371 12.566371 12.566371 12.566371 X 22.773467 21.599074 21.429044 21.379069 21.358987 X^2 50.265482 50.265482 50.265482 50.265482 50.265482 X^3 120.646844 136.112618 136.468863 136.516259 136.527286 X^4 301.592895 402.123860 402.123860 402.123860 402.123860 X^5 772.372744 1251.560875 1248.478307 1248.329339 1248.310263 X^6 2010.619298 4021.238597 4021.238597 4021.238597 4021.238597 R 35.543064 33.873272 33.636637 33.568679 33.541995 SIN(X) 8.148528 7.093236 6.829501 6.770095 6.748000 EXP(X) 103.916121 119.054184 119.018924 118.975880 118.957489 1/(1+R) 3.282385 3.593545 3.680885 3.715112 3.731215 SQRT(R) 21.134032 20.362410 20.213626 20.162872 20.140322 CENTER RADIUS THETA1 THETA2 Area 0.000000 0.000000 8.000000 0.000000 0.785398 25.132741 F 1 2 3 4 5 1 25.132741 25.132741 25.132741 25.132741 25.132741 X 128.205848 122.035642 121.156077 120.901855 120.801384 X^2 659.776223 658.535556 658.306721 658.226697 658.189669 X^3 3423.258894 3852.696345 3861.175671 3861.964878 3862.023118 X^4 17895.848246 23808.195889 23798.432281 23795.017900 23793.438035 X^5 94198.389134 152324.381304 151890.157118 151851.355736 151839.463841 X^6 498915.108021 995946.804259 995596.467601 995473.742761 995416.920207 R 142.172254 135.493089 134.546549 134.274715 134.167980 SIN(X) -20.591291 -0.613778 -1.772041 -2.550657 -2.728531 EXP(X) 4580.191359 9276.306595 9550.421060 9552.767811 9552.254503 1/(1+R) 3.775468 4.237021 4.390051 4.458138 4.493559 SQRT(R) 59.776069 57.593593 57.172767 57.029215 56.965433 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 0.000000 0.000000 1.000000 0.000000 6.283185 3.141593 0.000000 3.141593 F Sector1 Sector2 Sum Circle 1 3.141593 0.000000 3.141593 3.141593 X -0.000000 0.000000 -0.000000 -0.000000 X^2 0.785398 0.000000 0.785398 0.785398 X^3 -0.000000 0.000000 -0.000000 -0.000000 X^4 0.392699 0.000000 0.392699 0.392699 X^5 -0.000000 0.000000 -0.000000 -0.000000 X^6 0.245437 0.000000 0.245437 0.245437 R 2.096375 0.000000 2.096375 2.096375 SIN(X) -0.000000 0.000000 -0.000000 -0.000000 EXP(X) 3.550999 0.000000 3.550999 3.550999 1/(1+R) 1.926085 0.000000 1.926085 1.926085 SQRT(R) 2.517540 0.000000 2.517540 2.517540 CENTER RADIUS THETA1 THETA2 Area1 Area2 Circle 0.000000 0.000000 2.000000 0.000000 3.141593 6.283185 6.283185 12.566371 F Sector1 Sector2 Sum Circle 1 6.283185 6.283185 12.566371 12.566371 X 0.000000 -0.000000 -0.000000 -0.000000 X^2 6.283185 6.283185 12.566371 12.566371 X^3 0.000000 -0.000000 -0.000000 -0.000000 X^4 12.566371 12.566371 25.132741 25.132741 X^5 0.000000 -0.000000 -0.000000 -0.000000 X^6 31.415927 31.415927 62.831853 62.831853 R 8.385499 8.385499 16.770998 16.770998 SIN(X) -0.000000 -0.000000 -0.000000 0.000000 EXP(X) 9.994266 9.994266 19.988532 19.988532 1/(1+R) 2.824584 2.824584 5.649168 5.649168 SQRT(R) 7.120679 7.120679 14.241358 14.241358 CENTER RADIUS THETA1 THETA2 Area1 Area2 Circle 0.000000 0.000000 4.000000 0.000000 1.570796 12.566371 37.699112 50.265482 F Sector1 Sector2 Sum Circle 1 12.566371 37.699112 50.265482 50.265482 X 21.358987 -21.402972 -0.043985 -0.000000 X^2 50.265482 150.796447 201.061930 201.061930 X^3 136.527286 -136.522672 0.004614 -0.000000 X^4 402.123860 1206.371579 1608.495439 1608.495439 X^5 1248.310263 -1248.311880 -0.001617 -0.000000 X^6 4021.238597 12063.715790 16084.954386 16084.954386 R 33.541995 100.625985 134.167980 134.167980 SIN(X) 6.748000 -6.792768 -0.044768 -0.000000 EXP(X) 118.957489 126.281394 245.238882 245.282112 1/(1+R) 3.731215 11.193646 14.924861 14.924861 SQRT(R) 20.140322 60.420966 80.561288 80.561288 CENTER RADIUS THETA1 THETA2 Area1 Area2 Circle 0.000000 0.000000 8.000000 0.000000 0.785398 25.132741 175.929189 201.061930 F Sector1 Sector2 Sum Circle 1 25.132741 175.929189 201.061930 201.061930 X 120.801384 -121.174781 -0.373397 -0.000000 X^2 658.189669 2555.614296 3213.803965 3216.990877 X^3 3862.023118 -3883.706321 -21.683203 -0.000000 X^4 23793.438035 79014.295099 102807.733134 102943.708073 X^5 151839.463841 -152657.657886 -818.194046 -0.000000 X^6 995416.920207 3117531.066718 4112947.986926 4117748.322913 R 134.167980 939.175862 1073.343842 1073.343842 SIN(X) -2.728531 2.770118 0.041587 -0.000000 EXP(X) 9552.254503 10508.429397 20060.683901 20099.816126 1/(1+R) 4.493559 31.454916 35.948475 35.948475 SQRT(R) 56.965433 398.758033 455.723467 455.723467 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 1.000000 and center: CENTER = ( 1.000000, 1.000000 ) Rule: 1 2 3 4 5 Function 1 3.141593 3.141593 3.141593 3.141593 3.141593 X 3.141593 3.141593 3.141593 3.141593 3.141593 X^2 3.141593 3.534292 3.926991 3.926991 3.926991 X^3 3.141593 4.319690 5.497787 5.497787 5.497787 X^4 3.141593 5.595962 8.639380 8.246681 8.377580 X^5 3.141593 7.559457 14.922565 12.959070 13.613568 X^6 3.141593 10.529244 27.488936 21.053034 23.169246 R 4.442883 4.587997 4.763043 4.726959 4.750687 SIN(X) 2.643559 2.481750 2.339750 2.326605 2.331020 EXP(X) 8.539734 9.084680 9.699175 9.652595 9.668010 1/(1+R) 1.301290 1.302073 1.286046 1.307523 1.293101 SQRT(R) 3.736004 3.769984 3.827845 3.797552 3.817872 Rule: 6 7 8 9 Function 1 3.141593 3.141593 3.141593 3.141593 X 3.141593 3.141593 3.141593 3.141593 X^2 3.926991 3.926991 3.926991 3.926991 X^3 5.497787 5.497787 5.497787 5.497787 X^4 8.377580 8.246681 8.246681 8.246681 X^5 13.613568 12.959070 12.959070 12.959070 X^6 23.169246 21.058488 21.058488 21.058488 R 4.750718 4.726999 4.726985 4.727000 SIN(X) 2.331020 2.326599 2.326599 2.326599 EXP(X) 9.668008 9.652617 9.652617 9.652617 1/(1+R) 1.293068 1.307487 1.307499 1.307486 SQRT(R) 3.817914 3.797554 3.797534 3.797555 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 1.000000 and center: CENTER = (1.000000, 1.000000). Rule: 1 2 3 4 5 'Function 1 3.141593 3.141593 3.141593 3.141593 3.141593 X 3.141593 3.141593 3.141593 3.141593 3.141593 X^2 3.141593 3.534292 3.926991 4.712389 3.926991 X^3 3.141593 4.319690 5.497787 7.853982 5.497787 X^4 3.141593 5.595962 8.050331 13.351769 8.639380 X^5 3.141593 7.559457 11.977322 22.776547 14.922565 X^6 3.141593 10.529244 17.916896 38.877209 27.488936 R 4.442883 4.587997 4.705089 4.942141 4.763043 SIN(X) 2.643559 2.481750 2.319941 2.009751 2.339750 EXP(X) 8.539734 9.084680 9.629626 10.765119 9.699175 1/(1+R) 1.301290 1.302073 1.320266 1.360350 1.286046 SQRT(R) 3.736004 3.769984 3.779523 3.793093 3.827845 Rule: 6 7 8 9 10 'Function 1 3.141593 3.141593 3.141593 3.141593 3.141593 X 3.141593 3.141593 3.141593 3.141593 3.141593 X^2 3.926991 3.926991 3.926991 3.926991 3.926991 X^3 5.497787 5.497787 5.497787 5.497787 5.497787 X^4 8.246681 8.246681 8.246681 8.246681 8.246681 X^5 12.959070 12.959070 12.959070 12.959070 12.959070 X^6 21.053034 21.107576 21.009401 21.058488 21.058488 R 4.726959 4.724407 4.726917 4.725828 4.727699 SIN(X) 2.326605 2.326544 2.326655 2.326599 2.326599 EXP(X) 9.652595 9.652809 9.652427 9.652617 9.652616 1/(1+R) 1.307523 1.308860 1.307845 1.308406 1.306936 SQRT(R) 3.797552 3.795630 3.797099 3.796264 3.798316 Rule: 11 12 13 'Function 1 3.141593 3.141593 3.141593 X 3.141593 3.141593 3.141593 X^2 3.926991 3.926991 3.926991 X^3 5.497787 5.497787 5.497787 X^4 8.246681 8.246681 8.246681 X^5 12.959070 12.959070 12.959070 X^6 21.058488 21.058488 21.058488 R 4.726999 4.727041 4.726985 SIN(X) 2.326599 2.326599 2.326599 EXP(X) 9.652617 9.652617 9.652617 1/(1+R) 1.307487 1.307445 1.307499 SQRT(R) 3.797554 3.797618 3.797534 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 3.141593 0.000000 CN_GEG_01_1: 1 3.141593 0.000000 CN_GEG_02_XIU: 2 3.141593 0.000000 GW_02_XIU: 2 3.141593 0.000000 CN_GEG_03_XIU: 2 3.141593 0.000000 EXACT 3.141593 N = 1 ALPHA = 0.000000 EXPON = 0 Degree = 0 CN_GEG_00_1: 1 2.000000 0.000000 CN_GEG_01_1: 1 2.000000 0.000000 CN_GEG_02_XIU: 2 2.000000 0.000000 GW_02_XIU: 2 2.000000 0.000000 CN_GEG_03_XIU: 2 2.000000 0.000000 EXACT 2.000000 N = 1 ALPHA = 0.500000 EXPON = 0 Degree = 0 CN_GEG_00_1: 1 1.570796 0.000000 CN_GEG_01_1: 1 1.570796 0.000000 CN_GEG_02_XIU: 2 1.570796 0.000000 GW_02_XIU: 2 1.570796 0.000000 CN_GEG_03_XIU: 2 1.570796 0.000000 EXACT 1.570796 N = 1 ALPHA = 1.000000 EXPON = 0 Degree = 0 CN_GEG_00_1: 1 1.333333 0.000000 CN_GEG_01_1: 1 1.333333 0.000000 CN_GEG_02_XIU: 2 1.333333 0.000000 GW_02_XIU: 2 1.333333 0.000000 CN_GEG_03_XIU: 2 1.333333 0.000000 EXACT 1.333333 N = 1 ALPHA = 1.500000 EXPON = 0 Degree = 0 CN_GEG_00_1: 1 1.178097 0.000000 CN_GEG_01_1: 1 1.178097 0.000000 CN_GEG_02_XIU: 2 1.178097 0.000000 GW_02_XIU: 2 1.178097 0.000000 CN_GEG_03_XIU: 2 1.178097 0.000000 EXACT 1.178097 N = 1 ALPHA = -0.500000 EXPON = 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 2 0.000000 0.000000 GW_02_XIU: 2 0.000000 0.000000 CN_GEG_03_XIU: 2 0.000000 0.000000 EXACT 0.000000 N = 1 ALPHA = 0.000000 EXPON = 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 2 0.000000 0.000000 GW_02_XIU: 2 0.000000 0.000000 CN_GEG_03_XIU: 2 0.000000 0.000000 EXACT 0.000000 N = 1 ALPHA = 0.500000 EXPON = 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 2 0.000000 0.000000 GW_02_XIU: 2 0.000000 0.000000 CN_GEG_03_XIU: 2 0.000000 0.000000 EXACT 0.000000 N = 1 ALPHA = 1.000000 EXPON = 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 2 0.000000 0.000000 GW_02_XIU: 2 0.000000 0.000000 CN_GEG_03_XIU: 2 0.000000 0.000000 EXACT 0.000000 N = 1 ALPHA = 1.500000 EXPON = 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 2 0.000000 0.000000 GW_02_XIU: 2 0.000000 0.000000 CN_GEG_03_XIU: 2 0.000000 0.000000 EXACT 0.000000 N = 1 ALPHA = -0.500000 EXPON = 2 Degree = 2 CN_GEG_02_XIU: 2 1.570796 0.000000 GW_02_XIU: 2 1.570796 0.000000 CN_GEG_03_XIU: 2 1.570796 0.000000 EXACT 1.570796 N = 1 ALPHA = 0.000000 EXPON = 2 Degree = 2 CN_GEG_02_XIU: 2 0.666667 0.000000 GW_02_XIU: 2 0.666667 0.000000 CN_GEG_03_XIU: 2 0.666667 0.000000 EXACT 0.666667 N = 1 ALPHA = 0.500000 EXPON = 2 Degree = 2 CN_GEG_02_XIU: 2 0.392699 0.000000 GW_02_XIU: 2 0.392699 0.000000 CN_GEG_03_XIU: 2 0.392699 0.000000 EXACT 0.392699 N = 1 ALPHA = 1.000000 EXPON = 2 Degree = 2 CN_GEG_02_XIU: 2 0.266667 0.000000 GW_02_XIU: 2 0.266667 0.000000 CN_GEG_03_XIU: 2 0.266667 0.000000 EXACT 0.266667 N = 1 ALPHA = 1.500000 EXPON = 2 Degree = 2 CN_GEG_02_XIU: 2 0.196350 0.000000 GW_02_XIU: 2 0.196350 0.000000 CN_GEG_03_XIU: 2 0.196350 0.000000 EXACT 0.196350 N = 2 ALPHA = -0.500000 EXPON = 0 0 Degree = 0 CN_GEG_00_1: 1 9.869604 0.000000 CN_GEG_01_1: 1 9.869604 0.000000 CN_GEG_02_XIU: 3 9.869604 0.000000 GW_02_XIU: 3 9.869604 0.000000 CN_GEG_03_XIU: 4 9.869604 0.000000 EXACT 9.869604 N = 2 ALPHA = 0.000000 EXPON = 0 0 Degree = 0 CN_GEG_00_1: 1 4.000000 0.000000 CN_GEG_01_1: 1 4.000000 0.000000 CN_GEG_02_XIU: 3 4.000000 0.000000 GW_02_XIU: 3 4.000000 0.000000 CN_GEG_03_XIU: 4 4.000000 0.000000 EXACT 4.000000 N = 2 ALPHA = 0.500000 EXPON = 0 0 Degree = 0 CN_GEG_00_1: 1 2.467401 0.000000 CN_GEG_01_1: 1 2.467401 0.000000 CN_GEG_02_XIU: 3 2.467401 0.000000 GW_02_XIU: 3 2.467401 0.000000 CN_GEG_03_XIU: 4 2.467401 0.000000 EXACT 2.467401 N = 2 ALPHA = 1.000000 EXPON = 0 0 Degree = 0 CN_GEG_00_1: 1 1.777778 0.000000 CN_GEG_01_1: 1 1.777778 0.000000 CN_GEG_02_XIU: 3 1.777778 0.000000 GW_02_XIU: 3 1.777778 0.000000 CN_GEG_03_XIU: 4 1.777778 0.000000 EXACT 1.777778 N = 2 ALPHA = 1.500000 EXPON = 0 0 Degree = 0 CN_GEG_00_1: 1 1.387913 0.000000 CN_GEG_01_1: 1 1.387913 0.000000 CN_GEG_02_XIU: 3 1.387913 0.000000 GW_02_XIU: 3 1.387913 0.000000 CN_GEG_03_XIU: 4 1.387913 0.000000 EXACT 1.387913 N = 2 ALPHA = -0.500000 EXPON = 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 3 0.000000 0.000000 GW_02_XIU: 3 0.000000 0.000000 CN_GEG_03_XIU: 4 0.000000 0.000000 EXACT 0.000000 N = 2 ALPHA = 0.000000 EXPON = 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 3 0.000000 0.000000 GW_02_XIU: 3 0.000000 0.000000 CN_GEG_03_XIU: 4 0.000000 0.000000 EXACT 0.000000 N = 2 ALPHA = 0.500000 EXPON = 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 3 0.000000 0.000000 GW_02_XIU: 3 0.000000 0.000000 CN_GEG_03_XIU: 4 0.000000 0.000000 EXACT 0.000000 N = 2 ALPHA = 1.000000 EXPON = 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 3 0.000000 0.000000 GW_02_XIU: 3 0.000000 0.000000 CN_GEG_03_XIU: 4 0.000000 0.000000 EXACT 0.000000 N = 2 ALPHA = 1.500000 EXPON = 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 3 0.000000 0.000000 GW_02_XIU: 3 0.000000 0.000000 CN_GEG_03_XIU: 4 0.000000 0.000000 EXACT 0.000000 N = 2 ALPHA = -0.500000 EXPON = 1 1 Degree = 2 CN_GEG_02_XIU: 3 0.000000 0.000000 GW_02_XIU: 3 0.000000 0.000000 CN_GEG_03_XIU: 4 0.000000 0.000000 EXACT 0.000000 N = 2 ALPHA = 0.000000 EXPON = 1 1 Degree = 2 CN_GEG_02_XIU: 3 0.000000 0.000000 GW_02_XIU: 3 0.000000 0.000000 CN_GEG_03_XIU: 4 0.000000 0.000000 EXACT 0.000000 N = 2 ALPHA = 0.500000 EXPON = 1 1 Degree = 2 CN_GEG_02_XIU: 3 0.000000 0.000000 GW_02_XIU: 3 0.000000 0.000000 CN_GEG_03_XIU: 4 0.000000 0.000000 EXACT 0.000000 N = 2 ALPHA = 1.000000 EXPON = 1 1 Degree = 2 CN_GEG_02_XIU: 3 0.000000 0.000000 GW_02_XIU: 3 0.000000 0.000000 CN_GEG_03_XIU: 4 0.000000 0.000000 EXACT 0.000000 N = 2 ALPHA = 1.500000 EXPON = 1 1 Degree = 2 CN_GEG_02_XIU: 3 0.000000 0.000000 GW_02_XIU: 3 0.000000 0.000000 CN_GEG_03_XIU: 4 0.000000 0.000000 EXACT 0.000000 N = 2 ALPHA = -0.500000 EXPON = 2 0 Degree = 2 CN_GEG_02_XIU: 3 4.934802 0.000000 GW_02_XIU: 3 4.934802 0.000000 CN_GEG_03_XIU: 4 4.934802 0.000000 EXACT 4.934802 N = 2 ALPHA = 0.000000 EXPON = 2 0 Degree = 2 CN_GEG_02_XIU: 3 1.333333 0.000000 GW_02_XIU: 3 1.333333 0.000000 CN_GEG_03_XIU: 4 1.333333 0.000000 EXACT 1.333333 N = 2 ALPHA = 0.500000 EXPON = 2 0 Degree = 2 CN_GEG_02_XIU: 3 0.616850 0.000000 GW_02_XIU: 3 0.616850 0.000000 CN_GEG_03_XIU: 4 0.616850 0.000000 EXACT 0.616850 N = 2 ALPHA = 1.000000 EXPON = 2 0 Degree = 2 CN_GEG_02_XIU: 3 0.355556 0.000000 GW_02_XIU: 3 0.355556 0.000000 CN_GEG_03_XIU: 4 0.355556 0.000000 EXACT 0.355556 N = 2 ALPHA = 1.500000 EXPON = 2 0 Degree = 2 CN_GEG_02_XIU: 3 0.231319 0.000000 GW_02_XIU: 3 0.231319 0.000000 CN_GEG_03_XIU: 4 0.231319 0.000000 EXACT 0.231319 N = 3 ALPHA = -0.500000 EXPON = 0 0 0 Degree = 0 CN_GEG_00_1: 1 31.006277 0.000000 CN_GEG_01_1: 1 31.006277 0.000000 CN_GEG_02_XIU: 4 31.006277 0.000000 GW_02_XIU: 4 31.006277 0.000000 CN_GEG_03_XIU: 6 31.006277 0.000000 EXACT 31.006277 N = 3 ALPHA = 0.000000 EXPON = 0 0 0 Degree = 0 CN_GEG_00_1: 1 8.000000 0.000000 CN_GEG_01_1: 1 8.000000 0.000000 CN_GEG_02_XIU: 4 8.000000 0.000000 GW_02_XIU: 4 8.000000 0.000000 CN_GEG_03_XIU: 6 8.000000 0.000000 EXACT 8.000000 N = 3 ALPHA = 0.500000 EXPON = 0 0 0 Degree = 0 CN_GEG_00_1: 1 3.875785 0.000000 CN_GEG_01_1: 1 3.875785 0.000000 CN_GEG_02_XIU: 4 3.875785 0.000000 GW_02_XIU: 4 3.875785 0.000000 CN_GEG_03_XIU: 6 3.875785 0.000000 EXACT 3.875785 N = 3 ALPHA = 1.000000 EXPON = 0 0 0 Degree = 0 CN_GEG_00_1: 1 2.370370 0.000000 CN_GEG_01_1: 1 2.370370 0.000000 CN_GEG_02_XIU: 4 2.370370 0.000000 GW_02_XIU: 4 2.370370 0.000000 CN_GEG_03_XIU: 6 2.370370 0.000000 EXACT 2.370370 N = 3 ALPHA = 1.500000 EXPON = 0 0 0 Degree = 0 CN_GEG_00_1: 1 1.635097 0.000000 CN_GEG_01_1: 1 1.635097 0.000000 CN_GEG_02_XIU: 4 1.635097 0.000000 GW_02_XIU: 4 1.635097 0.000000 CN_GEG_03_XIU: 6 1.635097 0.000000 EXACT 1.635097 N = 3 ALPHA = -0.500000 EXPON = 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 4 0.000000 0.000000 GW_02_XIU: 4 0.000000 0.000000 CN_GEG_03_XIU: 6 0.000000 0.000000 EXACT 0.000000 N = 3 ALPHA = 0.000000 EXPON = 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 4 0.000000 0.000000 GW_02_XIU: 4 0.000000 0.000000 CN_GEG_03_XIU: 6 0.000000 0.000000 EXACT 0.000000 N = 3 ALPHA = 0.500000 EXPON = 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 4 0.000000 0.000000 GW_02_XIU: 4 0.000000 0.000000 CN_GEG_03_XIU: 6 0.000000 0.000000 EXACT 0.000000 N = 3 ALPHA = 1.000000 EXPON = 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 4 0.000000 0.000000 GW_02_XIU: 4 0.000000 0.000000 CN_GEG_03_XIU: 6 0.000000 0.000000 EXACT 0.000000 N = 3 ALPHA = 1.500000 EXPON = 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 4 0.000000 0.000000 GW_02_XIU: 4 0.000000 0.000000 CN_GEG_03_XIU: 6 0.000000 0.000000 EXACT 0.000000 N = 3 ALPHA = -0.500000 EXPON = 1 1 0 Degree = 2 CN_GEG_02_XIU: 4 0.000000 0.000000 GW_02_XIU: 4 0.000000 0.000000 CN_GEG_03_XIU: 6 0.000000 0.000000 EXACT 0.000000 N = 3 ALPHA = 0.000000 EXPON = 1 1 0 Degree = 2 CN_GEG_02_XIU: 4 0.000000 0.000000 GW_02_XIU: 4 0.000000 0.000000 CN_GEG_03_XIU: 6 0.000000 0.000000 EXACT 0.000000 N = 3 ALPHA = 0.500000 EXPON = 1 1 0 Degree = 2 CN_GEG_02_XIU: 4 0.000000 0.000000 GW_02_XIU: 4 0.000000 0.000000 CN_GEG_03_XIU: 6 0.000000 0.000000 EXACT 0.000000 N = 3 ALPHA = 1.000000 EXPON = 1 1 0 Degree = 2 CN_GEG_02_XIU: 4 0.000000 0.000000 GW_02_XIU: 4 0.000000 0.000000 CN_GEG_03_XIU: 6 0.000000 0.000000 EXACT 0.000000 N = 3 ALPHA = 1.500000 EXPON = 1 1 0 Degree = 2 CN_GEG_02_XIU: 4 0.000000 0.000000 GW_02_XIU: 4 0.000000 0.000000 CN_GEG_03_XIU: 6 0.000000 0.000000 EXACT 0.000000 N = 3 ALPHA = -0.500000 EXPON = 2 0 0 Degree = 2 CN_GEG_02_XIU: 4 15.503138 0.000000 GW_02_XIU: 4 15.503138 0.000000 CN_GEG_03_XIU: 6 15.503138 0.000000 EXACT 15.503138 N = 3 ALPHA = 0.000000 EXPON = 2 0 0 Degree = 2 CN_GEG_02_XIU: 4 2.666667 0.000000 GW_02_XIU: 4 2.666667 0.000000 CN_GEG_03_XIU: 6 2.666667 0.000000 EXACT 2.666667 N = 3 ALPHA = 0.500000 EXPON = 2 0 0 Degree = 2 CN_GEG_02_XIU: 4 0.968946 0.000000 GW_02_XIU: 4 0.968946 0.000000 CN_GEG_03_XIU: 6 0.968946 0.000000 EXACT 0.968946 N = 3 ALPHA = 1.000000 EXPON = 2 0 0 Degree = 2 CN_GEG_02_XIU: 4 0.474074 0.000000 GW_02_XIU: 4 0.474074 0.000000 CN_GEG_03_XIU: 6 0.474074 0.000000 EXACT 0.474074 N = 3 ALPHA = 1.500000 EXPON = 2 0 0 Degree = 2 CN_GEG_02_XIU: 4 0.272516 0.000000 GW_02_XIU: 4 0.272516 0.000000 CN_GEG_03_XIU: 6 0.272516 0.000000 EXACT 0.272516 N = 4 ALPHA = -0.500000 EXPON = 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 97.409091 0.000000 CN_GEG_01_1: 1 97.409091 0.000000 CN_GEG_02_XIU: 5 97.409091 0.000000 GW_02_XIU: 5 97.409091 0.000000 CN_GEG_03_XIU: 8 97.409091 0.000000 EXACT 97.409091 N = 4 ALPHA = 0.000000 EXPON = 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 16.000000 0.000000 CN_GEG_01_1: 1 16.000000 0.000000 CN_GEG_02_XIU: 5 16.000000 0.000000 GW_02_XIU: 5 16.000000 0.000000 CN_GEG_03_XIU: 8 16.000000 0.000000 EXACT 16.000000 N = 4 ALPHA = 0.500000 EXPON = 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 6.088068 0.000000 CN_GEG_01_1: 1 6.088068 0.000000 CN_GEG_02_XIU: 5 6.088068 0.000000 GW_02_XIU: 5 6.088068 0.000000 CN_GEG_03_XIU: 8 6.088068 0.000000 EXACT 6.088068 N = 4 ALPHA = 1.000000 EXPON = 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 3.160494 0.000000 CN_GEG_01_1: 1 3.160494 0.000000 CN_GEG_02_XIU: 5 3.160494 0.000000 GW_02_XIU: 5 3.160494 0.000000 CN_GEG_03_XIU: 8 3.160494 0.000000 EXACT 3.160494 N = 4 ALPHA = 1.500000 EXPON = 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 1.926303 0.000000 CN_GEG_01_1: 1 1.926303 0.000000 CN_GEG_02_XIU: 5 1.926303 0.000000 GW_02_XIU: 5 1.926303 0.000000 CN_GEG_03_XIU: 8 1.926303 0.000000 EXACT 1.926303 N = 4 ALPHA = -0.500000 EXPON = 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 5 0.000000 0.000000 GW_02_XIU: 5 0.000000 0.000000 CN_GEG_03_XIU: 8 -0.000000 0.000000 EXACT 0.000000 N = 4 ALPHA = 0.000000 EXPON = 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 5 0.000000 0.000000 GW_02_XIU: 5 0.000000 0.000000 CN_GEG_03_XIU: 8 -0.000000 0.000000 EXACT 0.000000 N = 4 ALPHA = 0.500000 EXPON = 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 5 0.000000 0.000000 GW_02_XIU: 5 0.000000 0.000000 CN_GEG_03_XIU: 8 -0.000000 0.000000 EXACT 0.000000 N = 4 ALPHA = 1.000000 EXPON = 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 5 0.000000 0.000000 GW_02_XIU: 5 0.000000 0.000000 CN_GEG_03_XIU: 8 -0.000000 0.000000 EXACT 0.000000 N = 4 ALPHA = 1.500000 EXPON = 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 5 0.000000 0.000000 GW_02_XIU: 5 0.000000 0.000000 CN_GEG_03_XIU: 8 -0.000000 0.000000 EXACT 0.000000 N = 4 ALPHA = -0.500000 EXPON = 1 1 0 0 Degree = 2 CN_GEG_02_XIU: 5 0.000000 0.000000 GW_02_XIU: 5 0.000000 0.000000 CN_GEG_03_XIU: 8 0.000000 0.000000 EXACT 0.000000 N = 4 ALPHA = 0.000000 EXPON = 1 1 0 0 Degree = 2 CN_GEG_02_XIU: 5 0.000000 0.000000 GW_02_XIU: 5 0.000000 0.000000 CN_GEG_03_XIU: 8 0.000000 0.000000 EXACT 0.000000 N = 4 ALPHA = 0.500000 EXPON = 1 1 0 0 Degree = 2 CN_GEG_02_XIU: 5 0.000000 0.000000 GW_02_XIU: 5 0.000000 0.000000 CN_GEG_03_XIU: 8 0.000000 0.000000 EXACT 0.000000 N = 4 ALPHA = 1.000000 EXPON = 1 1 0 0 Degree = 2 CN_GEG_02_XIU: 5 0.000000 0.000000 GW_02_XIU: 5 0.000000 0.000000 CN_GEG_03_XIU: 8 0.000000 0.000000 EXACT 0.000000 N = 4 ALPHA = 1.500000 EXPON = 1 1 0 0 Degree = 2 CN_GEG_02_XIU: 5 0.000000 0.000000 GW_02_XIU: 5 0.000000 0.000000 CN_GEG_03_XIU: 8 0.000000 0.000000 EXACT 0.000000 N = 4 ALPHA = -0.500000 EXPON = 2 0 0 0 Degree = 2 CN_GEG_02_XIU: 5 48.704546 0.000000 GW_02_XIU: 5 48.704546 0.000000 CN_GEG_03_XIU: 8 48.704546 0.000000 EXACT 48.704546 N = 4 ALPHA = 0.000000 EXPON = 2 0 0 0 Degree = 2 CN_GEG_02_XIU: 5 5.333333 0.000000 GW_02_XIU: 5 5.333333 0.000000 CN_GEG_03_XIU: 8 5.333333 0.000000 EXACT 5.333333 N = 4 ALPHA = 0.500000 EXPON = 2 0 0 0 Degree = 2 CN_GEG_02_XIU: 5 1.522017 0.000000 GW_02_XIU: 5 1.522017 0.000000 CN_GEG_03_XIU: 8 1.522017 0.000000 EXACT 1.522017 N = 4 ALPHA = 1.000000 EXPON = 2 0 0 0 Degree = 2 CN_GEG_02_XIU: 5 0.632099 0.000000 GW_02_XIU: 5 0.632099 0.000000 CN_GEG_03_XIU: 8 0.632099 0.000000 EXACT 0.632099 N = 4 ALPHA = 1.500000 EXPON = 2 0 0 0 Degree = 2 CN_GEG_02_XIU: 5 0.321050 0.000000 GW_02_XIU: 5 0.321050 0.000000 CN_GEG_03_XIU: 8 0.321050 0.000000 EXACT 0.321050 N = 5 ALPHA = -0.500000 EXPON = 0 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 306.019685 0.000000 CN_GEG_01_1: 1 306.019685 0.000000 CN_GEG_02_XIU: 6 306.019685 0.000000 GW_02_XIU: 6 306.019685 0.000000 CN_GEG_03_XIU: 10 306.019685 0.000000 EXACT 306.019685 N = 5 ALPHA = 0.000000 EXPON = 0 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 32.000000 0.000000 CN_GEG_01_1: 1 32.000000 0.000000 CN_GEG_02_XIU: 6 32.000000 0.000000 GW_02_XIU: 6 32.000000 0.000000 CN_GEG_03_XIU: 10 32.000000 0.000000 EXACT 32.000000 N = 5 ALPHA = 0.500000 EXPON = 0 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 9.563115 0.000000 CN_GEG_01_1: 1 9.563115 0.000000 CN_GEG_02_XIU: 6 9.563115 0.000000 GW_02_XIU: 6 9.563115 0.000000 CN_GEG_03_XIU: 10 9.563115 0.000000 EXACT 9.563115 N = 5 ALPHA = 1.000000 EXPON = 0 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 4.213992 0.000000 CN_GEG_01_1: 1 4.213992 0.000000 CN_GEG_02_XIU: 6 4.213992 0.000000 GW_02_XIU: 6 4.213992 0.000000 CN_GEG_03_XIU: 10 4.213992 0.000000 EXACT 4.213992 N = 5 ALPHA = 1.500000 EXPON = 0 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 2.269372 0.000000 CN_GEG_01_1: 1 2.269372 0.000000 CN_GEG_02_XIU: 6 2.269372 0.000000 GW_02_XIU: 6 2.269372 0.000000 CN_GEG_03_XIU: 10 2.269372 0.000000 EXACT 2.269372 N = 5 ALPHA = -0.500000 EXPON = 0 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 6 0.000000 0.000000 GW_02_XIU: 6 0.000000 0.000000 CN_GEG_03_XIU: 10 0.000000 0.000000 EXACT 0.000000 N = 5 ALPHA = 0.000000 EXPON = 0 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 6 0.000000 0.000000 GW_02_XIU: 6 0.000000 0.000000 CN_GEG_03_XIU: 10 0.000000 0.000000 EXACT 0.000000 N = 5 ALPHA = 0.500000 EXPON = 0 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 6 0.000000 0.000000 GW_02_XIU: 6 0.000000 0.000000 CN_GEG_03_XIU: 10 0.000000 0.000000 EXACT 0.000000 N = 5 ALPHA = 1.000000 EXPON = 0 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 6 0.000000 0.000000 GW_02_XIU: 6 0.000000 0.000000 CN_GEG_03_XIU: 10 0.000000 0.000000 EXACT 0.000000 N = 5 ALPHA = 1.500000 EXPON = 0 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 6 0.000000 0.000000 GW_02_XIU: 6 0.000000 0.000000 CN_GEG_03_XIU: 10 0.000000 0.000000 EXACT 0.000000 N = 5 ALPHA = -0.500000 EXPON = 1 1 0 0 0 Degree = 2 CN_GEG_02_XIU: 6 0.000000 0.000000 GW_02_XIU: 6 0.000000 0.000000 CN_GEG_03_XIU: 10 0.000000 0.000000 EXACT 0.000000 N = 5 ALPHA = 0.000000 EXPON = 1 1 0 0 0 Degree = 2 CN_GEG_02_XIU: 6 0.000000 0.000000 GW_02_XIU: 6 0.000000 0.000000 CN_GEG_03_XIU: 10 0.000000 0.000000 EXACT 0.000000 N = 5 ALPHA = 0.500000 EXPON = 1 1 0 0 0 Degree = 2 CN_GEG_02_XIU: 6 0.000000 0.000000 GW_02_XIU: 6 0.000000 0.000000 CN_GEG_03_XIU: 10 0.000000 0.000000 EXACT 0.000000 N = 5 ALPHA = 1.000000 EXPON = 1 1 0 0 0 Degree = 2 CN_GEG_02_XIU: 6 0.000000 0.000000 GW_02_XIU: 6 0.000000 0.000000 CN_GEG_03_XIU: 10 0.000000 0.000000 EXACT 0.000000 N = 5 ALPHA = 1.500000 EXPON = 1 1 0 0 0 Degree = 2 CN_GEG_02_XIU: 6 0.000000 0.000000 GW_02_XIU: 6 0.000000 0.000000 CN_GEG_03_XIU: 10 0.000000 0.000000 EXACT 0.000000 N = 5 ALPHA = -0.500000 EXPON = 2 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 6 153.009842 0.000000 GW_02_XIU: 6 153.009842 0.000000 CN_GEG_03_XIU: 10 153.009842 0.000000 EXACT 153.009842 N = 5 ALPHA = 0.000000 EXPON = 2 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 6 10.666667 0.000000 GW_02_XIU: 6 10.666667 0.000000 CN_GEG_03_XIU: 10 10.666667 0.000000 EXACT 10.666667 N = 5 ALPHA = 0.500000 EXPON = 2 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 6 2.390779 0.000000 GW_02_XIU: 6 2.390779 0.000000 CN_GEG_03_XIU: 10 2.390779 0.000000 EXACT 2.390779 N = 5 ALPHA = 1.000000 EXPON = 2 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 6 0.842798 0.000000 GW_02_XIU: 6 0.842798 0.000000 CN_GEG_03_XIU: 10 0.842798 0.000000 EXACT 0.842798 N = 5 ALPHA = 1.500000 EXPON = 2 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 6 0.378229 0.000000 GW_02_XIU: 6 0.378229 0.000000 CN_GEG_03_XIU: 10 0.378229 0.000000 EXACT 0.378229 N = 6 ALPHA = -0.500000 EXPON = 0 0 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 961.389194 0.000000 CN_GEG_01_1: 1 961.389194 0.000000 CN_GEG_02_XIU: 7 961.389194 0.000000 GW_02_XIU: 7 961.389194 0.000000 CN_GEG_03_XIU: 12 961.389194 0.000000 EXACT 961.389194 N = 6 ALPHA = 0.000000 EXPON = 0 0 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 64.000000 0.000000 CN_GEG_01_1: 1 64.000000 0.000000 CN_GEG_02_XIU: 7 64.000000 0.000000 GW_02_XIU: 7 64.000000 0.000000 CN_GEG_03_XIU: 12 64.000000 0.000000 EXACT 64.000000 N = 6 ALPHA = 0.500000 EXPON = 0 0 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 15.021706 0.000000 CN_GEG_01_1: 1 15.021706 0.000000 CN_GEG_02_XIU: 7 15.021706 0.000000 GW_02_XIU: 7 15.021706 0.000000 CN_GEG_03_XIU: 12 15.021706 0.000000 EXACT 15.021706 N = 6 ALPHA = 1.000000 EXPON = 0 0 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 5.618656 0.000000 CN_GEG_01_1: 1 5.618656 0.000000 CN_GEG_02_XIU: 7 5.618656 0.000000 GW_02_XIU: 7 5.618656 0.000000 CN_GEG_03_XIU: 12 5.618656 0.000000 EXACT 5.618656 N = 6 ALPHA = 1.500000 EXPON = 0 0 0 0 0 0 Degree = 0 CN_GEG_00_1: 1 2.673541 0.000000 CN_GEG_01_1: 1 2.673541 0.000000 CN_GEG_02_XIU: 7 2.673541 0.000000 GW_02_XIU: 7 2.673541 0.000000 CN_GEG_03_XIU: 12 2.673541 0.000000 EXACT 2.673541 N = 6 ALPHA = -0.500000 EXPON = 0 0 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 7 -0.000000 0.000000 GW_02_XIU: 7 -0.000000 0.000000 CN_GEG_03_XIU: 12 0.000000 0.000000 EXACT 0.000000 N = 6 ALPHA = 0.000000 EXPON = 0 0 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 7 -0.000000 0.000000 GW_02_XIU: 7 -0.000000 0.000000 CN_GEG_03_XIU: 12 0.000000 0.000000 EXACT 0.000000 N = 6 ALPHA = 0.500000 EXPON = 0 0 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 7 -0.000000 0.000000 GW_02_XIU: 7 -0.000000 0.000000 CN_GEG_03_XIU: 12 0.000000 0.000000 EXACT 0.000000 N = 6 ALPHA = 1.000000 EXPON = 0 0 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 7 -0.000000 0.000000 GW_02_XIU: 7 -0.000000 0.000000 CN_GEG_03_XIU: 12 0.000000 0.000000 EXACT 0.000000 N = 6 ALPHA = 1.500000 EXPON = 0 0 0 0 0 1 Degree = 1 CN_GEG_01_1: 1 0.000000 0.000000 CN_GEG_02_XIU: 7 -0.000000 0.000000 GW_02_XIU: 7 -0.000000 0.000000 CN_GEG_03_XIU: 12 0.000000 0.000000 EXACT 0.000000 N = 6 ALPHA = -0.500000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 0.000000 0.000000 GW_02_XIU: 7 0.000000 0.000000 CN_GEG_03_XIU: 12 0.000000 0.000000 EXACT 0.000000 N = 6 ALPHA = 0.000000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 0.000000 0.000000 GW_02_XIU: 7 0.000000 0.000000 CN_GEG_03_XIU: 12 0.000000 0.000000 EXACT 0.000000 N = 6 ALPHA = 0.500000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 0.000000 0.000000 GW_02_XIU: 7 0.000000 0.000000 CN_GEG_03_XIU: 12 0.000000 0.000000 EXACT 0.000000 N = 6 ALPHA = 1.000000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 0.000000 0.000000 GW_02_XIU: 7 0.000000 0.000000 CN_GEG_03_XIU: 12 -0.000000 0.000000 EXACT 0.000000 N = 6 ALPHA = 1.500000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 0.000000 0.000000 GW_02_XIU: 7 0.000000 0.000000 CN_GEG_03_XIU: 12 0.000000 0.000000 EXACT 0.000000 N = 6 ALPHA = -0.500000 EXPON = 2 0 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 480.694597 0.000000 GW_02_XIU: 7 480.694597 0.000000 CN_GEG_03_XIU: 12 480.694597 0.000000 EXACT 480.694597 N = 6 ALPHA = 0.000000 EXPON = 2 0 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 21.333333 0.000000 GW_02_XIU: 7 21.333333 0.000000 CN_GEG_03_XIU: 12 21.333333 0.000000 EXACT 21.333333 N = 6 ALPHA = 0.500000 EXPON = 2 0 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 3.755427 0.000000 GW_02_XIU: 7 3.755427 0.000000 CN_GEG_03_XIU: 12 3.755427 0.000000 EXACT 3.755427 N = 6 ALPHA = 1.000000 EXPON = 2 0 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 1.123731 0.000000 GW_02_XIU: 7 1.123731 0.000000 CN_GEG_03_XIU: 12 1.123731 0.000000 EXACT 1.123731 N = 6 ALPHA = 1.500000 EXPON = 2 0 0 0 0 0 Degree = 2 CN_GEG_02_XIU: 7 0.445590 0.000000 GW_02_XIU: 7 0.445590 0.000000 CN_GEG_03_XIU: 12 0.445590 0.000000 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 = 0.000000 BETA = 0.000000 EXPON = 0 Degree = 0 CN_JAC_00_1: 1 2.000000 0.000000 CN_JAC_01_1: 1 2.000000 0.000000 CN_JAC_02_XIU: 2 2.000000 0.000000 GW_02_XIU: 2 2.000000 0.000000 EXACT 2.000000 N = 1 ALPHA = 1.000000 BETA = 0.000000 EXPON = 0 Degree = 0 CN_JAC_00_1: 1 2.000000 0.000000 CN_JAC_01_1: 1 2.000000 0.000000 CN_JAC_02_XIU: 2 2.000000 0.000000 GW_02_XIU: 2 2.000000 0.000000 EXACT 2.000000 N = 1 ALPHA = 0.000000 BETA = 2.000000 EXPON = 0 Degree = 0 CN_JAC_00_1: 1 2.666667 0.000000 CN_JAC_01_1: 1 2.666667 0.000000 CN_JAC_02_XIU: 2 2.666667 0.000000 GW_02_XIU: 2 2.666667 0.000000 EXACT 2.666667 N = 1 ALPHA = 0.500000 BETA = 1.500000 EXPON = 0 Degree = 0 CN_JAC_00_1: 1 1.570796 0.000000 CN_JAC_01_1: 1 1.570796 0.000000 CN_JAC_02_XIU: 2 1.570796 0.000000 GW_02_XIU: 2 1.570796 0.000000 EXACT 1.570796 N = 1 ALPHA = 0.000000 BETA = 0.000000 EXPON = 1 Degree = 1 CN_JAC_01_1: 1 0.000000 0.000000 CN_JAC_02_XIU: 2 0.000000 0.000000 GW_02_XIU: 2 0.000000 0.000000 EXACT 0.000000 N = 1 ALPHA = 1.000000 BETA = 0.000000 EXPON = 1 Degree = 1 CN_JAC_01_1: 1 -0.666667 0.000000 CN_JAC_02_XIU: 2 -0.666667 0.000000 GW_02_XIU: 2 -0.666667 0.000000 EXACT -0.666667 N = 1 ALPHA = 0.000000 BETA = 2.000000 EXPON = 1 Degree = 1 CN_JAC_01_1: 1 1.333333 0.000000 CN_JAC_02_XIU: 2 1.333333 0.000000 GW_02_XIU: 2 1.333333 0.000000 EXACT 1.333333 N = 1 ALPHA = 0.500000 BETA = 1.500000 EXPON = 1 Degree = 1 CN_JAC_01_1: 1 0.392699 0.000000 CN_JAC_02_XIU: 2 0.392699 0.000000 GW_02_XIU: 2 0.392699 0.000000 EXACT 0.392699 N = 1 ALPHA = 0.000000 BETA = 0.000000 EXPON = 2 Degree = 2 CN_JAC_02_XIU: 2 0.666667 0.000000 GW_02_XIU: 2 0.666667 0.000000 EXACT 0.666667 N = 1 ALPHA = 1.000000 BETA = 0.000000 EXPON = 2 Degree = 2 CN_JAC_02_XIU: 2 0.666667 0.000000 GW_02_XIU: 2 0.666667 0.000000 EXACT 0.666667 N = 1 ALPHA = 0.000000 BETA = 2.000000 EXPON = 2 Degree = 2 CN_JAC_02_XIU: 2 1.066667 0.000000 GW_02_XIU: 2 1.066667 0.000000 EXACT 1.066667 N = 1 ALPHA = 0.500000 BETA = 1.500000 EXPON = 2 Degree = 2 CN_JAC_02_XIU: 2 0.392699 0.000000 GW_02_XIU: 2 0.392699 0.000000 EXACT 0.392699 N = 2 ALPHA = 0.000000 BETA = 0.000000 EXPON = 0 0 Degree = 0 CN_JAC_00_1: 1 4.000000 0.000000 CN_JAC_01_1: 1 4.000000 0.000000 CN_JAC_02_XIU: 3 4.000000 0.000000 GW_02_XIU: 3 4.000000 0.000000 EXACT 4.000000 N = 2 ALPHA = 1.000000 BETA = 0.000000 EXPON = 0 0 Degree = 0 CN_JAC_00_1: 1 4.000000 0.000000 CN_JAC_01_1: 1 4.000000 0.000000 CN_JAC_02_XIU: 3 4.000000 0.000000 GW_02_XIU: 3 4.000000 0.000000 EXACT 4.000000 N = 2 ALPHA = 0.000000 BETA = 2.000000 EXPON = 0 0 Degree = 0 CN_JAC_00_1: 1 7.111111 0.000000 CN_JAC_01_1: 1 7.111111 0.000000 CN_JAC_02_XIU: 3 7.111111 0.000000 GW_02_XIU: 3 7.111111 0.000000 EXACT 7.111111 N = 2 ALPHA = 0.500000 BETA = 1.500000 EXPON = 0 0 Degree = 0 CN_JAC_00_1: 1 2.467401 0.000000 CN_JAC_01_1: 1 2.467401 0.000000 CN_JAC_02_XIU: 3 2.467401 0.000000 GW_02_XIU: 3 2.467401 0.000000 EXACT 2.467401 N = 2 ALPHA = 0.000000 BETA = 0.000000 EXPON = 0 1 Degree = 1 CN_JAC_01_1: 1 0.000000 0.000000 CN_JAC_02_XIU: 3 0.000000 0.000000 GW_02_XIU: 3 0.000000 0.000000 EXACT 0.000000 N = 2 ALPHA = 1.000000 BETA = 0.000000 EXPON = 0 1 Degree = 1 CN_JAC_01_1: 1 -1.333333 0.000000 CN_JAC_02_XIU: 3 -1.333333 0.000000 GW_02_XIU: 3 -1.333333 0.000000 EXACT -1.333333 N = 2 ALPHA = 0.000000 BETA = 2.000000 EXPON = 0 1 Degree = 1 CN_JAC_01_1: 1 3.555556 0.000000 CN_JAC_02_XIU: 3 3.555556 0.000000 GW_02_XIU: 3 3.555556 0.000000 EXACT 3.555556 N = 2 ALPHA = 0.500000 BETA = 1.500000 EXPON = 0 1 Degree = 1 CN_JAC_01_1: 1 0.616850 0.000000 CN_JAC_02_XIU: 3 0.616850 0.000000 GW_02_XIU: 3 0.616850 0.000000 EXACT 0.616850 N = 2 ALPHA = 0.000000 BETA = 0.000000 EXPON = 1 1 Degree = 2 CN_JAC_02_XIU: 3 0.000000 0.000000 GW_02_XIU: 3 0.000000 0.000000 EXACT 0.000000 N = 2 ALPHA = 1.000000 BETA = 0.000000 EXPON = 1 1 Degree = 2 CN_JAC_02_XIU: 3 0.444444 0.000000 GW_02_XIU: 3 0.444444 0.000000 EXACT 0.444444 N = 2 ALPHA = 0.000000 BETA = 2.000000 EXPON = 1 1 Degree = 2 CN_JAC_02_XIU: 3 1.777778 0.000000 GW_02_XIU: 3 1.777778 0.000000 EXACT 1.777778 N = 2 ALPHA = 0.500000 BETA = 1.500000 EXPON = 1 1 Degree = 2 CN_JAC_02_XIU: 3 0.154213 0.000000 GW_02_XIU: 3 0.154213 0.000000 EXACT 0.154213 N = 2 ALPHA = 0.000000 BETA = 0.000000 EXPON = 2 0 Degree = 2 CN_JAC_02_XIU: 3 1.333333 0.000000 GW_02_XIU: 3 1.333333 0.000000 EXACT 1.333333 N = 2 ALPHA = 1.000000 BETA = 0.000000 EXPON = 2 0 Degree = 2 CN_JAC_02_XIU: 3 1.333333 0.000000 GW_02_XIU: 3 1.333333 0.000000 EXACT 1.333333 N = 2 ALPHA = 0.000000 BETA = 2.000000 EXPON = 2 0 Degree = 2 CN_JAC_02_XIU: 3 2.844444 0.000000 GW_02_XIU: 3 2.844444 0.000000 EXACT 2.844444 N = 2 ALPHA = 0.500000 BETA = 1.500000 EXPON = 2 0 Degree = 2 CN_JAC_02_XIU: 3 0.616850 0.000000 GW_02_XIU: 3 0.616850 0.000000 EXACT 0.616850 N = 3 ALPHA = 0.000000 BETA = 0.000000 EXPON = 0 0 0 Degree = 0 CN_JAC_00_1: 1 8.000000 0.000000 CN_JAC_01_1: 1 8.000000 0.000000 CN_JAC_02_XIU: 4 8.000000 0.000000 GW_02_XIU: 4 8.000000 0.000000 EXACT 8.000000 N = 3 ALPHA = 1.000000 BETA = 0.000000 EXPON = 0 0 0 Degree = 0 CN_JAC_00_1: 1 8.000000 0.000000 CN_JAC_01_1: 1 8.000000 0.000000 CN_JAC_02_XIU: 4 8.000000 0.000000 GW_02_XIU: 4 8.000000 0.000000 EXACT 8.000000 N = 3 ALPHA = 0.000000 BETA = 2.000000 EXPON = 0 0 0 Degree = 0 CN_JAC_00_1: 1 18.962963 0.000000 CN_JAC_01_1: 1 18.962963 0.000000 CN_JAC_02_XIU: 4 18.962963 0.000000 GW_02_XIU: 4 18.962963 0.000000 EXACT 18.962963 N = 3 ALPHA = 0.500000 BETA = 1.500000 EXPON = 0 0 0 Degree = 0 CN_JAC_00_1: 1 3.875785 0.000000 CN_JAC_01_1: 1 3.875785 0.000000 CN_JAC_02_XIU: 4 3.875785 0.000000 GW_02_XIU: 4 3.875785 0.000000 EXACT 3.875785 N = 3 ALPHA = 0.000000 BETA = 0.000000 EXPON = 0 0 1 Degree = 1 CN_JAC_01_1: 1 0.000000 0.000000 CN_JAC_02_XIU: 4 0.000000 0.000000 GW_02_XIU: 4 0.000000 0.000000 EXACT 0.000000 N = 3 ALPHA = 1.000000 BETA = 0.000000 EXPON = 0 0 1 Degree = 1 CN_JAC_01_1: 1 -2.666667 0.000000 CN_JAC_02_XIU: 4 -2.666667 0.000000 GW_02_XIU: 4 -2.666667 0.000000 EXACT -2.666667 N = 3 ALPHA = 0.000000 BETA = 2.000000 EXPON = 0 0 1 Degree = 1 CN_JAC_01_1: 1 9.481481 0.000000 CN_JAC_02_XIU: 4 9.481481 0.000000 GW_02_XIU: 4 9.481481 0.000000 EXACT 9.481481 N = 3 ALPHA = 0.500000 BETA = 1.500000 EXPON = 0 0 1 Degree = 1 CN_JAC_01_1: 1 0.968946 0.000000 CN_JAC_02_XIU: 4 0.968946 0.000000 GW_02_XIU: 4 0.968946 0.000000 EXACT 0.968946 N = 3 ALPHA = 0.000000 BETA = 0.000000 EXPON = 1 1 0 Degree = 2 CN_JAC_02_XIU: 4 0.000000 0.000000 GW_02_XIU: 4 0.000000 0.000000 EXACT 0.000000 N = 3 ALPHA = 1.000000 BETA = 0.000000 EXPON = 1 1 0 Degree = 2 CN_JAC_02_XIU: 4 0.888889 0.000000 GW_02_XIU: 4 0.888889 0.000000 EXACT 0.888889 N = 3 ALPHA = 0.000000 BETA = 2.000000 EXPON = 1 1 0 Degree = 2 CN_JAC_02_XIU: 4 4.740741 0.000000 GW_02_XIU: 4 4.740741 0.000000 EXACT 4.740741 N = 3 ALPHA = 0.500000 BETA = 1.500000 EXPON = 1 1 0 Degree = 2 CN_JAC_02_XIU: 4 0.242237 0.000000 GW_02_XIU: 4 0.242237 0.000000 EXACT 0.242237 N = 3 ALPHA = 0.000000 BETA = 0.000000 EXPON = 2 0 0 Degree = 2 CN_JAC_02_XIU: 4 2.666667 0.000000 GW_02_XIU: 4 2.666667 0.000000 EXACT 2.666667 N = 3 ALPHA = 1.000000 BETA = 0.000000 EXPON = 2 0 0 Degree = 2 CN_JAC_02_XIU: 4 2.666667 0.000000 GW_02_XIU: 4 2.666667 0.000000 EXACT 2.666667 N = 3 ALPHA = 0.000000 BETA = 2.000000 EXPON = 2 0 0 Degree = 2 CN_JAC_02_XIU: 4 7.585185 0.000000 GW_02_XIU: 4 7.585185 0.000000 EXACT 7.585185 N = 3 ALPHA = 0.500000 BETA = 1.500000 EXPON = 2 0 0 Degree = 2 CN_JAC_02_XIU: 4 0.968946 0.000000 GW_02_XIU: 4 0.968946 0.000000 EXACT 0.968946 N = 4 ALPHA = 0.000000 BETA = 0.000000 EXPON = 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 16.000000 0.000000 CN_JAC_01_1: 1 16.000000 0.000000 CN_JAC_02_XIU: 5 16.000000 0.000000 GW_02_XIU: 5 16.000000 0.000000 EXACT 16.000000 N = 4 ALPHA = 1.000000 BETA = 0.000000 EXPON = 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 16.000000 0.000000 CN_JAC_01_1: 1 16.000000 0.000000 CN_JAC_02_XIU: 5 16.000000 0.000000 GW_02_XIU: 5 16.000000 0.000000 EXACT 16.000000 N = 4 ALPHA = 0.000000 BETA = 2.000000 EXPON = 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 50.567901 0.000000 CN_JAC_01_1: 1 50.567901 0.000000 CN_JAC_02_XIU: 5 50.567901 0.000000 GW_02_XIU: 5 50.567901 0.000000 EXACT 50.567901 N = 4 ALPHA = 0.500000 BETA = 1.500000 EXPON = 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 6.088068 0.000000 CN_JAC_01_1: 1 6.088068 0.000000 CN_JAC_02_XIU: 5 6.088068 0.000000 GW_02_XIU: 5 6.088068 0.000000 EXACT 6.088068 N = 4 ALPHA = 0.000000 BETA = 0.000000 EXPON = 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 0.000000 0.000000 CN_JAC_02_XIU: 5 0.000000 0.000000 GW_02_XIU: 5 0.000000 0.000000 EXACT 0.000000 N = 4 ALPHA = 1.000000 BETA = 0.000000 EXPON = 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 -5.333333 0.000000 CN_JAC_02_XIU: 5 -5.333333 0.000000 GW_02_XIU: 5 -5.333333 0.000000 EXACT -5.333333 N = 4 ALPHA = 0.000000 BETA = 2.000000 EXPON = 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 25.283951 0.000000 CN_JAC_02_XIU: 5 25.283951 0.000000 GW_02_XIU: 5 25.283951 0.000000 EXACT 25.283951 N = 4 ALPHA = 0.500000 BETA = 1.500000 EXPON = 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 1.522017 0.000000 CN_JAC_02_XIU: 5 1.522017 0.000000 GW_02_XIU: 5 1.522017 0.000000 EXACT 1.522017 N = 4 ALPHA = 0.000000 BETA = 0.000000 EXPON = 1 1 0 0 Degree = 2 CN_JAC_02_XIU: 5 0.000000 0.000000 GW_02_XIU: 5 0.000000 0.000000 EXACT 0.000000 N = 4 ALPHA = 1.000000 BETA = 0.000000 EXPON = 1 1 0 0 Degree = 2 CN_JAC_02_XIU: 5 1.777778 0.000000 GW_02_XIU: 5 1.777778 0.000000 EXACT 1.777778 N = 4 ALPHA = 0.000000 BETA = 2.000000 EXPON = 1 1 0 0 Degree = 2 CN_JAC_02_XIU: 5 12.641975 0.000000 GW_02_XIU: 5 12.641975 0.000000 EXACT 12.641975 N = 4 ALPHA = 0.500000 BETA = 1.500000 EXPON = 1 1 0 0 Degree = 2 CN_JAC_02_XIU: 5 0.380504 0.000000 GW_02_XIU: 5 0.380504 0.000000 EXACT 0.380504 N = 4 ALPHA = 0.000000 BETA = 0.000000 EXPON = 2 0 0 0 Degree = 2 CN_JAC_02_XIU: 5 5.333333 0.000000 GW_02_XIU: 5 5.333333 0.000000 EXACT 5.333333 N = 4 ALPHA = 1.000000 BETA = 0.000000 EXPON = 2 0 0 0 Degree = 2 CN_JAC_02_XIU: 5 5.333333 0.000000 GW_02_XIU: 5 5.333333 0.000000 EXACT 5.333333 N = 4 ALPHA = 0.000000 BETA = 2.000000 EXPON = 2 0 0 0 Degree = 2 CN_JAC_02_XIU: 5 20.227160 0.000000 GW_02_XIU: 5 20.227160 0.000000 EXACT 20.227160 N = 4 ALPHA = 0.500000 BETA = 1.500000 EXPON = 2 0 0 0 Degree = 2 CN_JAC_02_XIU: 5 1.522017 0.000000 GW_02_XIU: 5 1.522017 0.000000 EXACT 1.522017 N = 5 ALPHA = 0.000000 BETA = 0.000000 EXPON = 0 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 32.000000 0.000000 CN_JAC_01_1: 1 32.000000 0.000000 CN_JAC_02_XIU: 6 32.000000 0.000000 GW_02_XIU: 6 32.000000 0.000000 EXACT 32.000000 N = 5 ALPHA = 1.000000 BETA = 0.000000 EXPON = 0 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 32.000000 0.000000 CN_JAC_01_1: 1 32.000000 0.000000 CN_JAC_02_XIU: 6 32.000000 0.000000 GW_02_XIU: 6 32.000000 0.000000 EXACT 32.000000 N = 5 ALPHA = 0.000000 BETA = 2.000000 EXPON = 0 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 134.847737 0.000000 CN_JAC_01_1: 1 134.847737 0.000000 CN_JAC_02_XIU: 6 134.847737 0.000000 GW_02_XIU: 6 134.847737 0.000000 EXACT 134.847737 N = 5 ALPHA = 0.500000 BETA = 1.500000 EXPON = 0 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 9.563115 0.000000 CN_JAC_01_1: 1 9.563115 0.000000 CN_JAC_02_XIU: 6 9.563115 0.000000 GW_02_XIU: 6 9.563115 0.000000 EXACT 9.563115 N = 5 ALPHA = 0.000000 BETA = 0.000000 EXPON = 0 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 0.000000 0.000000 CN_JAC_02_XIU: 6 0.000000 0.000000 GW_02_XIU: 6 0.000000 0.000000 EXACT 0.000000 N = 5 ALPHA = 1.000000 BETA = 0.000000 EXPON = 0 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 -10.666667 0.000000 CN_JAC_02_XIU: 6 -10.666667 0.000000 GW_02_XIU: 6 -10.666667 0.000000 EXACT -10.666667 N = 5 ALPHA = 0.000000 BETA = 2.000000 EXPON = 0 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 67.423868 0.000000 CN_JAC_02_XIU: 6 67.423868 0.000000 GW_02_XIU: 6 67.423868 0.000000 EXACT 67.423868 N = 5 ALPHA = 0.500000 BETA = 1.500000 EXPON = 0 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 2.390779 0.000000 CN_JAC_02_XIU: 6 2.390779 0.000000 GW_02_XIU: 6 2.390779 0.000000 EXACT 2.390779 N = 5 ALPHA = 0.000000 BETA = 0.000000 EXPON = 1 1 0 0 0 Degree = 2 CN_JAC_02_XIU: 6 0.000000 0.000000 GW_02_XIU: 6 0.000000 0.000000 EXACT 0.000000 N = 5 ALPHA = 1.000000 BETA = 0.000000 EXPON = 1 1 0 0 0 Degree = 2 CN_JAC_02_XIU: 6 3.555556 0.000000 GW_02_XIU: 6 3.555556 0.000000 EXACT 3.555556 N = 5 ALPHA = 0.000000 BETA = 2.000000 EXPON = 1 1 0 0 0 Degree = 2 CN_JAC_02_XIU: 6 33.711934 0.000000 GW_02_XIU: 6 33.711934 0.000000 EXACT 33.711934 N = 5 ALPHA = 0.500000 BETA = 1.500000 EXPON = 1 1 0 0 0 Degree = 2 CN_JAC_02_XIU: 6 0.597695 0.000000 GW_02_XIU: 6 0.597695 0.000000 EXACT 0.597695 N = 5 ALPHA = 0.000000 BETA = 0.000000 EXPON = 2 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 6 10.666667 0.000000 GW_02_XIU: 6 10.666667 0.000000 EXACT 10.666667 N = 5 ALPHA = 1.000000 BETA = 0.000000 EXPON = 2 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 6 10.666667 0.000000 GW_02_XIU: 6 10.666667 0.000000 EXACT 10.666667 N = 5 ALPHA = 0.000000 BETA = 2.000000 EXPON = 2 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 6 53.939095 0.000000 GW_02_XIU: 6 53.939095 0.000000 EXACT 53.939095 N = 5 ALPHA = 0.500000 BETA = 1.500000 EXPON = 2 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 6 2.390779 0.000000 GW_02_XIU: 6 2.390779 0.000000 EXACT 2.390779 N = 6 ALPHA = 0.000000 BETA = 0.000000 EXPON = 0 0 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 64.000000 0.000000 CN_JAC_01_1: 1 64.000000 0.000000 CN_JAC_02_XIU: 7 64.000000 0.000000 GW_02_XIU: 7 64.000000 0.000000 EXACT 64.000000 N = 6 ALPHA = 1.000000 BETA = 0.000000 EXPON = 0 0 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 64.000000 0.000000 CN_JAC_01_1: 1 64.000000 0.000000 CN_JAC_02_XIU: 7 64.000000 0.000000 GW_02_XIU: 7 64.000000 0.000000 EXACT 64.000000 N = 6 ALPHA = 0.000000 BETA = 2.000000 EXPON = 0 0 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 359.593964 0.000000 CN_JAC_01_1: 1 359.593964 0.000000 CN_JAC_02_XIU: 7 359.593964 0.000000 GW_02_XIU: 7 359.593964 0.000000 EXACT 359.593964 N = 6 ALPHA = 0.500000 BETA = 1.500000 EXPON = 0 0 0 0 0 0 Degree = 0 CN_JAC_00_1: 1 15.021706 0.000000 CN_JAC_01_1: 1 15.021706 0.000000 CN_JAC_02_XIU: 7 15.021706 0.000000 GW_02_XIU: 7 15.021706 0.000000 EXACT 15.021706 N = 6 ALPHA = 0.000000 BETA = 0.000000 EXPON = 0 0 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 0.000000 0.000000 CN_JAC_02_XIU: 7 -0.000000 0.000000 GW_02_XIU: 7 -0.000000 0.000000 EXACT 0.000000 N = 6 ALPHA = 1.000000 BETA = 0.000000 EXPON = 0 0 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 -21.333333 0.000000 CN_JAC_02_XIU: 7 -21.333333 0.000000 GW_02_XIU: 7 -21.333333 0.000000 EXACT -21.333333 N = 6 ALPHA = 0.000000 BETA = 2.000000 EXPON = 0 0 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 179.796982 0.000000 CN_JAC_02_XIU: 7 179.796982 0.000000 GW_02_XIU: 7 179.796982 0.000000 EXACT 179.796982 N = 6 ALPHA = 0.500000 BETA = 1.500000 EXPON = 0 0 0 0 0 1 Degree = 1 CN_JAC_01_1: 1 3.755427 0.000000 CN_JAC_02_XIU: 7 3.755427 0.000000 GW_02_XIU: 7 3.755427 0.000000 EXACT 3.755427 N = 6 ALPHA = 0.000000 BETA = 0.000000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 7 0.000000 0.000000 GW_02_XIU: 7 0.000000 0.000000 EXACT 0.000000 N = 6 ALPHA = 1.000000 BETA = 0.000000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 7 7.111111 0.000000 GW_02_XIU: 7 7.111111 0.000000 EXACT 7.111111 N = 6 ALPHA = 0.000000 BETA = 2.000000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 7 89.898491 0.000000 GW_02_XIU: 7 89.898491 0.000000 EXACT 89.898491 N = 6 ALPHA = 0.500000 BETA = 1.500000 EXPON = 1 1 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 7 0.938857 0.000000 GW_02_XIU: 7 0.938857 0.000000 EXACT 0.938857 N = 6 ALPHA = 0.000000 BETA = 0.000000 EXPON = 2 0 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 7 21.333333 0.000000 GW_02_XIU: 7 21.333333 0.000000 EXACT 21.333333 N = 6 ALPHA = 1.000000 BETA = 0.000000 EXPON = 2 0 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 7 21.333333 0.000000 GW_02_XIU: 7 21.333333 0.000000 EXACT 21.333333 N = 6 ALPHA = 0.000000 BETA = 2.000000 EXPON = 2 0 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 7 143.837586 0.000000 GW_02_XIU: 7 143.837586 0.000000 EXACT 143.837586 N = 6 ALPHA = 0.500000 BETA = 1.500000 EXPON = 2 0 0 0 0 0 Degree = 2 CN_JAC_02_XIU: 7 3.755427 0.000000 GW_02_XIU: 7 3.755427 0.000000 EXACT 3.755427 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 2.000000 0.000000 CN_LEG_02_XIU: 2 2.000000 0.000000 GW_02_XIU: 2 2.000000 0.000000 CN_LEG_03_1: 2 2.000000 0.000000 CN_LEG_03_XIU: 2 2.000000 0.000000 EXACT 2.000000 N = 1 EXPON = 1 Degree = 1 CN_LEG_01_1: 1 0.000000 0.000000 CN_LEG_02_XIU: 2 0.000000 0.000000 GW_02_XIU: 2 0.000000 0.000000 CN_LEG_03_1: 2 0.000000 0.000000 CN_LEG_03_XIU: 2 0.000000 0.000000 EXACT 0.000000 N = 1 EXPON = 2 Degree = 2 CN_LEG_02_XIU: 2 0.666667 0.000000 GW_02_XIU: 2 0.666667 0.000000 CN_LEG_03_1: 2 0.666667 0.000000 CN_LEG_03_XIU: 2 0.666667 0.000000 EXACT 0.666667 N = 1 EXPON = 3 Degree = 3 CN_LEG_03_1: 2 0.000000 0.000000 CN_LEG_03_XIU: 2 0.000000 0.000000 EXACT 0.000000 N = 1 EXPON = 4 Degree = 4 EXACT 0.400000 N = 2 EXPON = 0 0 Degree = 0 CN_LEG_01_1: 1 4.000000 0.000000 CN_LEG_02_XIU: 3 4.000000 0.000000 GW_02_XIU: 3 4.000000 0.000000 CN_LEG_03_1: 4 4.000000 0.000000 CN_LEG_03_XIU: 4 4.000000 0.000000 CN_LEG_05_2: 9 4.000000 0.000000 EXACT 4.000000 N = 2 EXPON = 0 1 Degree = 1 CN_LEG_01_1: 1 0.000000 0.000000 CN_LEG_02_XIU: 3 0.000000 0.000000 GW_02_XIU: 3 0.000000 0.000000 CN_LEG_03_1: 4 -0.000000 0.000000 CN_LEG_03_XIU: 4 -0.000000 0.000000 CN_LEG_05_2: 9 0.000000 0.000000 EXACT 0.000000 N = 2 EXPON = 1 1 Degree = 2 CN_LEG_02_XIU: 3 0.000000 0.000000 GW_02_XIU: 3 0.000000 0.000000 CN_LEG_03_1: 4 -0.000000 0.000000 CN_LEG_03_XIU: 4 -0.000000 0.000000 CN_LEG_05_2: 9 0.000000 0.000000 EXACT 0.000000 N = 2 EXPON = 2 0 Degree = 2 CN_LEG_02_XIU: 3 1.333333 0.000000 GW_02_XIU: 3 1.333333 0.000000 CN_LEG_03_1: 4 1.333333 0.000000 CN_LEG_03_XIU: 4 1.333333 0.000000 CN_LEG_05_2: 9 1.333333 0.000000 EXACT 1.333333 N = 2 EXPON = 3 0 Degree = 3 CN_LEG_03_1: 4 0.000000 0.000000 CN_LEG_03_XIU: 4 0.000000 0.000000 CN_LEG_05_2: 9 0.000000 0.000000 EXACT 0.000000 N = 2 EXPON = 0 4 Degree = 4 CN_LEG_05_2: 9 0.800000 0.000000 EXACT 0.800000 N = 2 EXPON = 3 2 Degree = 5 CN_LEG_05_2: 9 0.000000 0.000000 EXACT 0.000000 N = 3 EXPON = 0 0 0 Degree = 0 CN_LEG_01_1: 1 8.000000 0.000000 CN_LEG_02_XIU: 4 8.000000 0.000000 GW_02_XIU: 4 8.000000 0.000000 CN_LEG_03_1: 6 8.000000 0.000000 CN_LEG_03_XIU: 6 8.000000 0.000000 CN_LEG_05_2: 19 8.000000 0.000000 EXACT 8.000000 N = 3 EXPON = 0 0 1 Degree = 1 CN_LEG_01_1: 1 0.000000 0.000000 CN_LEG_02_XIU: 4 0.000000 0.000000 GW_02_XIU: 4 0.000000 0.000000 CN_LEG_03_1: 6 0.000000 0.000000 CN_LEG_03_XIU: 6 0.000000 0.000000 CN_LEG_05_2: 19 0.000000 0.000000 EXACT 0.000000 N = 3 EXPON = 1 1 0 Degree = 2 CN_LEG_02_XIU: 4 0.000000 0.000000 GW_02_XIU: 4 0.000000 0.000000 CN_LEG_03_1: 6 0.000000 0.000000 CN_LEG_03_XIU: 6 0.000000 0.000000 CN_LEG_05_2: 19 0.000000 0.000000 EXACT 0.000000 N = 3 EXPON = 2 0 0 Degree = 2 CN_LEG_02_XIU: 4 2.666667 0.000000 GW_02_XIU: 4 2.666667 0.000000 CN_LEG_03_1: 6 2.666667 0.000000 CN_LEG_03_XIU: 6 2.666667 0.000000 CN_LEG_05_2: 19 2.666667 0.000000 EXACT 2.666667 N = 3 EXPON = 3 0 0 Degree = 3 CN_LEG_03_1: 6 0.000000 0.000000 CN_LEG_03_XIU: 6 0.000000 0.000000 CN_LEG_05_2: 19 0.000000 0.000000 EXACT 0.000000 N = 3 EXPON = 0 0 4 Degree = 4 CN_LEG_05_2: 19 1.600000 0.000000 EXACT 1.600000 N = 3 EXPON = 3 2 0 Degree = 5 CN_LEG_05_2: 19 0.000000 0.000000 EXACT 0.000000 N = 4 EXPON = 0 0 0 0 Degree = 0 CN_LEG_01_1: 1 16.000000 0.000000 CN_LEG_02_XIU: 5 16.000000 0.000000 GW_02_XIU: 5 16.000000 0.000000 CN_LEG_03_1: 8 16.000000 0.000000 CN_LEG_03_XIU: 8 16.000000 0.000000 CN_LEG_05_1(1): 22 16.000000 0.000000 CN_LEG_05_1(2): 22 16.000000 0.000000 CN_LEG_05_2: 33 16.000000 0.000000 EXACT 16.000000 N = 4 EXPON = 0 0 0 1 Degree = 1 CN_LEG_01_1: 1 0.000000 0.000000 CN_LEG_02_XIU: 5 0.000000 0.000000 GW_02_XIU: 5 0.000000 0.000000 CN_LEG_03_1: 8 -0.000000 0.000000 CN_LEG_03_XIU: 8 -0.000000 0.000000 CN_LEG_05_1(1): 22 0.000000 0.000000 CN_LEG_05_1(2): 22 0.000000 0.000000 CN_LEG_05_2: 33 0.000000 0.000000 EXACT 0.000000 N = 4 EXPON = 1 1 0 0 Degree = 2 CN_LEG_02_XIU: 5 0.000000 0.000000 GW_02_XIU: 5 0.000000 0.000000 CN_LEG_03_1: 8 0.000000 0.000000 CN_LEG_03_XIU: 8 0.000000 0.000000 CN_LEG_05_1(1): 22 -0.000000 0.000000 CN_LEG_05_1(2): 22 -0.000000 0.000000 CN_LEG_05_2: 33 0.000000 0.000000 EXACT 0.000000 N = 4 EXPON = 2 0 0 0 Degree = 2 CN_LEG_02_XIU: 5 5.333333 0.000000 GW_02_XIU: 5 5.333333 0.000000 CN_LEG_03_1: 8 5.333333 0.000000 CN_LEG_03_XIU: 8 5.333333 0.000000 CN_LEG_05_1(1): 22 5.333333 0.000000 CN_LEG_05_1(2): 22 5.333333 0.000000 CN_LEG_05_2: 33 5.333333 0.000000 EXACT 5.333333 N = 4 EXPON = 3 0 0 0 Degree = 3 CN_LEG_03_1: 8 -0.000000 0.000000 CN_LEG_03_XIU: 8 -0.000000 0.000000 CN_LEG_05_1(1): 22 0.000000 0.000000 CN_LEG_05_1(2): 22 0.000000 0.000000 CN_LEG_05_2: 33 0.000000 0.000000 EXACT 0.000000 N = 4 EXPON = 0 0 0 4 Degree = 4 CN_LEG_05_1(1): 22 3.200000 0.000000 CN_LEG_05_1(2): 22 3.200000 0.000000 CN_LEG_05_2: 33 3.200000 0.000000 EXACT 3.200000 N = 4 EXPON = 3 2 0 0 Degree = 5 CN_LEG_05_1(1): 22 -0.000000 0.000000 CN_LEG_05_1(2): 22 0.000000 0.000000 CN_LEG_05_2: 33 0.000000 0.000000 EXACT 0.000000 N = 5 EXPON = 0 0 0 0 0 Degree = 0 CN_LEG_01_1: 1 32.000000 0.000000 CN_LEG_02_XIU: 6 32.000000 0.000000 GW_02_XIU: 6 32.000000 0.000000 CN_LEG_03_1: 10 32.000000 0.000000 CN_LEG_03_XIU: 10 32.000000 0.000000 CN_LEG_05_1(1): 32 32.000000 0.000000 CN_LEG_05_1(2): 32 32.000000 0.000000 CN_LEG_05_2: 51 32.000000 0.000000 EXACT 32.000000 N = 5 EXPON = 0 0 0 0 1 Degree = 1 CN_LEG_01_1: 1 0.000000 0.000000 CN_LEG_02_XIU: 6 0.000000 0.000000 GW_02_XIU: 6 0.000000 0.000000 CN_LEG_03_1: 10 0.000000 0.000000 CN_LEG_03_XIU: 10 0.000000 0.000000 CN_LEG_05_1(1): 32 -0.000000 0.000000 CN_LEG_05_1(2): 32 0.000000 0.000000 CN_LEG_05_2: 51 0.000000 0.000000 EXACT 0.000000 N = 5 EXPON = 1 1 0 0 0 Degree = 2 CN_LEG_02_XIU: 6 0.000000 0.000000 GW_02_XIU: 6 0.000000 0.000000 CN_LEG_03_1: 10 0.000000 0.000000 CN_LEG_03_XIU: 10 0.000000 0.000000 CN_LEG_05_1(1): 32 0.000000 0.000000 CN_LEG_05_1(2): 32 -0.000000 0.000000 CN_LEG_05_2: 51 0.000000 0.000000 EXACT 0.000000 N = 5 EXPON = 2 0 0 0 0 Degree = 2 CN_LEG_02_XIU: 6 10.666667 0.000000 GW_02_XIU: 6 10.666667 0.000000 CN_LEG_03_1: 10 10.666667 0.000000 CN_LEG_03_XIU: 10 10.666667 0.000000 CN_LEG_05_1(1): 32 10.666667 0.000000 CN_LEG_05_1(2): 32 10.666667 0.000000 CN_LEG_05_2: 51 10.666667 0.000000 EXACT 10.666667 N = 5 EXPON = 3 0 0 0 0 Degree = 3 CN_LEG_03_1: 10 -0.000000 0.000000 CN_LEG_03_XIU: 10 -0.000000 0.000000 CN_LEG_05_1(1): 32 -0.000000 0.000000 CN_LEG_05_1(2): 32 0.000000 0.000000 CN_LEG_05_2: 51 0.000000 0.000000 EXACT 0.000000 N = 5 EXPON = 0 0 0 0 4 Degree = 4 CN_LEG_05_1(1): 32 6.400000 0.000000 CN_LEG_05_1(2): 32 6.400000 0.000000 CN_LEG_05_2: 51 6.400000 0.000000 EXACT 6.400000 N = 5 EXPON = 3 2 0 0 0 Degree = 5 CN_LEG_05_1(1): 32 0.000000 0.000000 CN_LEG_05_1(2): 32 -0.000000 0.000000 CN_LEG_05_2: 51 0.000000 0.000000 EXACT 0.000000 N = 6 EXPON = 0 0 0 0 0 0 Degree = 0 CN_LEG_01_1: 1 64.000000 0.000000 CN_LEG_02_XIU: 7 64.000000 0.000000 GW_02_XIU: 7 64.000000 0.000000 CN_LEG_03_1: 12 64.000000 0.000000 CN_LEG_03_XIU: 12 64.000000 0.000000 CN_LEG_05_1(1): 44 64.000000 0.000000 CN_LEG_05_2: 73 64.000000 0.000000 EXACT 64.000000 N = 6 EXPON = 0 0 0 0 0 1 Degree = 1 CN_LEG_01_1: 1 0.000000 0.000000 CN_LEG_02_XIU: 7 -0.000000 0.000000 GW_02_XIU: 7 -0.000000 0.000000 CN_LEG_03_1: 12 -0.000000 0.000000 CN_LEG_03_XIU: 12 -0.000000 0.000000 CN_LEG_05_1(1): 44 0.000000 0.000000 CN_LEG_05_2: 73 0.000000 0.000000 EXACT 0.000000 N = 6 EXPON = 1 1 0 0 0 0 Degree = 2 CN_LEG_02_XIU: 7 0.000000 0.000000 GW_02_XIU: 7 0.000000 0.000000 CN_LEG_03_1: 12 -0.000000 0.000000 CN_LEG_03_XIU: 12 -0.000000 0.000000 CN_LEG_05_1(1): 44 0.000000 0.000000 CN_LEG_05_2: 73 0.000000 0.000000 EXACT 0.000000 N = 6 EXPON = 2 0 0 0 0 0 Degree = 2 CN_LEG_02_XIU: 7 21.333333 0.000000 GW_02_XIU: 7 21.333333 0.000000 CN_LEG_03_1: 12 21.333333 0.000000 CN_LEG_03_XIU: 12 21.333333 0.000000 CN_LEG_05_1(1): 44 21.333333 0.000000 CN_LEG_05_2: 73 21.333333 0.000000 EXACT 21.333333 N = 6 EXPON = 3 0 0 0 0 0 Degree = 3 CN_LEG_03_1: 12 -0.000000 0.000000 CN_LEG_03_XIU: 12 -0.000000 0.000000 CN_LEG_05_1(1): 44 0.000000 0.000000 CN_LEG_05_2: 73 0.000000 0.000000 EXACT 0.000000 N = 6 EXPON = 0 0 0 0 0 4 Degree = 4 CN_LEG_05_1(1): 44 12.800000 0.000000 CN_LEG_05_2: 73 12.800000 0.000000 EXACT 12.800000 N = 6 EXPON = 3 2 0 0 0 0 Degree = 5 CN_LEG_05_1(1): 44 0.000000 0.000000 CN_LEG_05_2: 73 0.000000 0.000000 EXACT 0.000000 TEST17 CONE_UNIT_3D approximates integrals in a unit cone. Volume = 1.047198 F(X) CONE_3D 1 1.047198 X 0.000000 Y 0.000000 Z 0.261799 X*X 0.181917 X*Y 0.000000 X*Z 0.000000 Y*Y 0.181917 Y*Z 0.000000 Z*Z 0.104720 X^3 0.000000 X*Y*Z 0.000000 Z*Z*Z 0.052360 X^4 0.065312 X^2 Z^2 0.008663 Z^4 0.029920 X^5 0.000000 X^6 0.030394 R 0.680705 SIN(X) 0.000000 EXP(X) 1.140920 1/(1+R) 0.876763 SQRT(R) 0.837626 TEST18 CUBE_SHELL_ND approximates integrals in a cubical shell in ND. Inner radius = 0.000000 Outer radius = 1.000000 Spatial dimension N = 2 Volume = 4.000000 F(X) CUBE_SHELL_ND 1 4.000000 X 0.000000 X^2 1.333333 X^3 0.000000 X^4 0.555556 X^5 0.000000 X^6 0.259259 R 3.265986 SIN(X) 0.000000 EXP(X) 4.690178 1/(1+R) 2.202041 SQRT(R) 3.614408 Spatial dimension N = 3 Volume = 8.000000 F(X) CUBE_SHELL_ND 1 8.000000 X 0.000000 X^2 2.666667 X^3 0.000000 X^4 1.173333 X^5 0.000000 X^6 0.618667 R 8.000000 SIN(X) 0.000000 EXP(X) 9.383090 1/(1+R) 4.000000 SQRT(R) 8.000000 Spatial dimension N = 4 Volume = 16.000000 F(X) CUBE_SHELL_ND 1 16.000000 X 0.000000 X^2 5.333333 X^3 0.000000 X^4 2.370370 X^5 0.000000 X^6 1.316872 R 18.475209 SIN(X) 0.000000 EXP(X) 18.767282 1/(1+R) 7.425626 SQRT(R) 17.193119 Inner radius = 1.000000 Outer radius = 2.000000 Spatial dimension N = 2 Volume = 12.000000 F(X) CUBE_SHELL_ND 1 12.000000 X 0.000000 X^2 20.000000 X^3 0.000000 X^4 41.666667 X^5 0.000000 X^6 97.222222 R 21.908902 SIN(X) 0.000000 EXP(X) 23.877192 1/(1+R) 4.246672 SQRT(R) 16.214402 Spatial dimension N = 3 Volume = 56.000000 F(X) CUBE_SHELL_ND 1 56.000000 X 0.000000 X^2 82.666667 X^3 0.000000 X^4 161.081905 X^5 0.000000 X^6 376.136707 R 117.847359 SIN(X) 0.000000 EXP(X) 104.591841 1/(1+R) 18.038813 SQRT(R) 81.237012 Spatial dimension N = 4 Volume = 240.000000 F(X) CUBE_SHELL_ND 1 240.000000 X 0.000000 X^2 336.000000 X^3 0.000000 X^4 627.200000 X^5 0.000000 X^6 1463.466667 R 567.943659 SIN(X) 0.000000 EXP(X) 436.263723 1/(1+R) 71.292100 SQRT(R) 369.197072 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 8.000000 8.000000 8.000000 X 0.000000 -0.000000 0.000000 Y 0.000000 -0.000000 0.000000 Z 0.000000 -0.000000 0.000000 X*X 2.666667 2.666667 2.666667 X*Y 0.000000 0.000000 0.000000 X*Z 0.000000 0.000000 0.000000 Y*Y 2.666667 2.666667 2.666667 Y*Z 0.000000 -0.000000 0.000000 Z*Z 2.666667 2.666667 2.666667 X^3 0.000000 0.000000 0.000000 X*Y*Z 0.000000 -0.000000 0.000000 Z*Z*Z 0.000000 -0.000000 0.000000 X^4 0.888889 1.600000 0.888889 X^2 Z^2 0.888889 0.888889 0.888889 Z^4 0.888889 1.600000 0.888889 X^5 0.000000 -0.000000 0.000000 X^6 0.296296 1.142857 0.296296 R 8.000000 7.684820 8.000000 SIN(X) 0.000000 0.000000 -0.000000 EXP(X) 9.370784 9.401610 9.370784 1/(1+R) 5.656854 5.804472 5.656854 SQRT(R) 8.000000 7.745091 8.000000 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 4.000000 4.000000 4.000000 4.000000 4.000000 4.000000 4.000000 4.000000 4.000000 4.000000 4.000000 4.000000 4.000000 4.000000 4.000000 4.000000 4.000000 4.000000 4.000000 4.000000 X 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 X^2 0.000000 1.000000 1.185185 1.250000 1.280000 1.296296 1.306122 1.312500 1.316872 1.320000 0.000000 1.333333 1.333333 1.333333 1.333333 1.333333 1.333333 1.333333 1.333333 1.333333 X^3 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 X^4 0.000000 0.250000 0.526749 0.640625 0.696320 0.727366 0.746356 0.758789 0.767363 0.773520 0.000000 0.333333 0.800000 0.800000 0.800000 0.800000 0.800000 0.800000 0.800000 0.800000 X^5 0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 X^6 0.000000 0.062500 0.234111 0.356445 0.425984 0.467393 0.493633 0.511185 0.523457 0.532356 0.000000 0.083333 0.407407 0.571429 0.571429 0.571429 0.571429 0.571429 0.571429 0.571429 R 0.000000 2.828427 2.861290 2.995352 2.998983 3.030550 3.031424 3.043445 3.043749 3.049558 0.000000 3.771236 2.777124 3.293362 2.829515 3.334887 2.702383 3.567654 2.302236 4.248719 SIN(X) 0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 EXP(X) 4.000000 4.510504 4.614868 4.652193 4.669612 4.679112 4.684854 4.688585 4.691146 4.692979 4.000000 4.680672 4.700571 4.700803 4.700805 4.700805 4.700805 4.700805 4.700805 4.700805 1/(1+R) 4.000000 2.343146 2.426166 2.341041 2.357407 2.337447 2.343220 2.335661 2.338332 2.334695 4.000000 1.790861 2.580298 2.115550 2.553848 2.066628 2.683294 1.834029 3.081459 1.156910 SQRT(R) 0.000000 3.363586 3.177742 3.402767 3.349853 3.413020 3.389867 3.417118 3.404641 3.419163 0.000000 4.484781 2.847104 3.984154 2.790682 4.252750 2.245060 5.212636 0.574027 8.138653 Spatial dimension N = 3 Value of K = 5 F(X) CUBE_UNIT_ND 1 8.000000 8.000000 8.000000 8.000000 8.000000 8.000000 8.000000 8.000000 8.000000 8.000000 X 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 X^2 0.000000 2.000000 2.370370 2.500000 2.560000 0.000000 2.666667 2.666667 2.666667 2.666667 X^3 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 X^4 0.000000 0.500000 1.053498 1.281250 1.392640 0.000000 0.666667 1.600000 1.600000 1.600000 X^5 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 X^6 0.000000 0.125000 0.468221 0.712891 0.851968 0.000000 0.166667 0.814815 1.142857 1.142857 R 0.000000 6.928203 7.274463 7.488694 7.550318 0.000000 9.237604 7.340704 7.867975 7.543723 SIN(X) 0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 EXP(X) 8.000000 9.021008 9.229736 9.304386 9.339223 8.000000 9.361344 9.401141 9.401607 9.401610 1/(1+R) 8.000000 4.287187 4.293166 4.208491 4.204360 8.000000 3.049583 4.453996 4.005483 4.305373 SQRT(R) 0.000000 7.444839 7.451064 7.661227 7.665368 0.000000 9.926452 7.147243 8.152746 7.389495 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 25.000000 20.000000 0.600000 141.808339 1570.796327 0.218418 0.956318 0.973569 4.064610 0.656207 0.829509 0.561695 0.735852 4.411185 1.463767 0.415307 0.066119 0.987246 1.719008 0.086267 0.257578 0.109957 0.904305 1.201702 0.088978 (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^EXPON1 * X2^EXPON2 * ... XN^EXPONN 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 1.772454 0.000000 EN_R2_02_XIU: 2 1.772454 0.000000 GW_02_XIU: 2 1.772454 0.000000 EN_R2_03_1: 2 1.772454 0.000000 EN_R2_03_2: 2 1.772454 0.000000 EN_R2_03_XIU: 2 1.772454 0.000000 EN_R2_05_2: 3 1.772454 0.000000 EN_R2_05_4: 3 1.772454 0.000000 EN_R2_05_5: 3 1.772454 0.000000 EXACT 1.772454 N = 1 EXPON = 2 Degree = 2 EN_R2_02_XIU: 2 0.886227 0.000000 GW_02_XIU: 2 0.886227 0.000000 EN_R2_03_1: 2 0.886227 0.000000 EN_R2_03_2: 2 0.886227 0.000000 EN_R2_03_XIU: 2 0.886227 0.000000 EN_R2_05_2: 3 0.886227 0.000000 EN_R2_05_4: 3 0.886227 0.000000 EN_R2_05_5: 3 0.886227 0.000000 EXACT 0.886227 N = 1 EXPON = 0 Degree = 0 EN_R2_01_1: 1 1.772454 0.000000 EN_R2_02_XIU: 2 1.772454 0.000000 GW_02_XIU: 2 1.772454 0.000000 EN_R2_03_1: 2 1.772454 0.000000 EN_R2_03_2: 2 1.772454 0.000000 EN_R2_03_XIU: 2 1.772454 0.000000 EN_R2_05_2: 3 1.772454 0.000000 EN_R2_05_4: 3 1.772454 0.000000 EN_R2_05_5: 3 1.772454 0.000000 EXACT 1.772454 N = 1 EXPON = 6 Degree = 6 EXACT 3.323351 N = 1 EXPON = 2 Degree = 2 EN_R2_02_XIU: 2 0.886227 0.000000 GW_02_XIU: 2 0.886227 0.000000 EN_R2_03_1: 2 0.886227 0.000000 EN_R2_03_2: 2 0.886227 0.000000 EN_R2_03_XIU: 2 0.886227 0.000000 EN_R2_05_2: 3 0.886227 0.000000 EN_R2_05_4: 3 0.886227 0.000000 EN_R2_05_5: 3 0.886227 0.000000 EXACT 0.886227 N = 1 EXPON = 8 Degree = 8 EXACT 11.631728 N = 1 EXPON = 10 Degree = 10 EXACT 52.342778 N = 1 EXPON = 1 Degree = 1 EN_R2_01_1: 1 0.000000 0.000000 EN_R2_02_XIU: 2 0.000000 0.000000 GW_02_XIU: 2 0.000000 0.000000 EN_R2_03_1: 2 0.000000 0.000000 EN_R2_03_2: 2 0.000000 0.000000 EN_R2_03_XIU: 2 0.000000 0.000000 EN_R2_05_2: 3 0.000000 0.000000 EN_R2_05_4: 3 0.000000 0.000000 EN_R2_05_5: 3 0.000000 0.000000 EXACT 0.000000 N = 1 EXPON = 2 Degree = 2 EN_R2_02_XIU: 2 0.886227 0.000000 GW_02_XIU: 2 0.886227 0.000000 EN_R2_03_1: 2 0.886227 0.000000 EN_R2_03_2: 2 0.886227 0.000000 EN_R2_03_XIU: 2 0.886227 0.000000 EN_R2_05_2: 3 0.886227 0.000000 EN_R2_05_4: 3 0.886227 0.000000 EN_R2_05_5: 3 0.886227 0.000000 EXACT 0.886227 N = 2 EXPON = 0 0 Degree = 0 EN_R2_01_1: 1 3.141593 0.000000 EN_R2_02_XIU: 3 3.141593 0.000000 GW_02_XIU: 3 3.141593 0.000000 EN_R2_03_1: 4 3.141593 0.000000 EN_R2_03_2: 4 3.141593 0.000000 EN_R2_03_XIU: 4 3.141593 0.000000 EN_R2_05_1(1): 8 3.141593 0.000000 EN_R2_05_2: 9 3.141593 0.000000 EN_R2_05_4: 7 3.141593 0.000000 EN_R2_05_5: 9 3.141593 0.000000 EXACT 3.141593 N = 2 EXPON = 2 0 Degree = 2 EN_R2_02_XIU: 3 1.570796 0.000000 GW_02_XIU: 3 1.570796 0.000000 EN_R2_03_1: 4 1.570796 0.000000 EN_R2_03_2: 4 1.570796 0.000000 EN_R2_03_XIU: 4 1.570796 0.000000 EN_R2_05_1(1): 8 1.570796 0.000000 EN_R2_05_2: 9 1.570796 0.000000 EN_R2_05_4: 7 1.570796 0.000000 EN_R2_05_5: 9 1.570796 0.000000 EXACT 1.570796 N = 2 EXPON = 0 4 Degree = 4 EN_R2_05_1(1): 8 2.356194 0.000000 EN_R2_05_2: 9 2.356194 0.000000 EN_R2_05_4: 7 2.356194 0.000000 EN_R2_05_5: 9 2.356194 0.000000 EXACT 2.356194 N = 2 EXPON = 0 6 Degree = 6 EXACT 5.890486 N = 2 EXPON = 2 4 Degree = 6 EXACT 1.178097 N = 2 EXPON = 8 0 Degree = 8 EXACT 20.616702 N = 2 EXPON = 0 10 Degree = 10 EXACT 92.775158 N = 2 EXPON = 1 2 Degree = 3 EN_R2_03_1: 4 0.000000 0.000000 EN_R2_03_2: 4 0.000000 0.000000 EN_R2_03_XIU: 4 -0.000000 0.000000 EN_R2_05_1(1): 8 0.000000 0.000000 EN_R2_05_2: 9 0.000000 0.000000 EN_R2_05_4: 7 0.000000 0.000000 EN_R2_05_5: 9 0.000000 0.000000 EXACT 0.000000 N = 2 EXPON = 2 2 Degree = 4 EN_R2_05_1(1): 8 0.785398 0.000000 EN_R2_05_2: 9 0.785398 0.000000 EN_R2_05_4: 7 0.785398 0.000000 EN_R2_05_5: 9 0.785398 0.000000 EXACT 0.785398 N = 3 EXPON = 0 0 0 Degree = 0 EN_R2_01_1: 1 5.568328 0.000000 EN_R2_02_XIU: 4 5.568328 0.000000 GW_02_XIU: 4 5.568328 0.000000 EN_R2_03_1: 6 5.568328 0.000000 EN_R2_03_2: 8 5.568328 0.000000 EN_R2_03_XIU: 6 5.568328 0.000000 EN_R2_05_1(1): 14 5.568328 0.000000 EN_R2_05_1(2): 14 5.568328 0.000000 EN_R2_05_2: 19 5.568328 0.000000 EN_R2_05_3: 14 5.568328 0.000000 EN_R2_05_4: 15 5.568328 0.000000 EN_R2_05_5: 25 5.568328 0.000000 EN_R2_07_1(1): 27 5.568328 0.000000 EN_R2_07_1(2): 27 5.568328 0.000000 EN_R2_07_2: 52 5.568328 0.000000 EN_R2_07_3(1): 45 5.568328 0.000000 EN_R2_07_3(2): 45 5.568328 0.000000 EN_R2_09_1(1): 77 5.568328 0.000000 EN_R2_09_1(2): 77 5.568328 0.000000 EN_R2_11_1(1): 151 5.568328 0.000000 EN_R2_11_1(2): 151 5.568328 0.000000 EXACT 5.568328 N = 3 EXPON = 2 0 0 Degree = 2 EN_R2_02_XIU: 4 2.784164 0.000000 GW_02_XIU: 4 2.784164 0.000000 EN_R2_03_1: 6 2.784164 0.000000 EN_R2_03_2: 8 2.784164 0.000000 EN_R2_03_XIU: 6 2.784164 0.000000 EN_R2_05_1(1): 14 2.784164 0.000000 EN_R2_05_1(2): 14 2.784164 0.000000 EN_R2_05_2: 19 2.784164 0.000000 EN_R2_05_3: 14 2.784164 0.000000 EN_R2_05_4: 15 2.784164 0.000000 EN_R2_05_5: 25 2.784164 0.000000 EN_R2_07_1(1): 27 2.784164 0.000000 EN_R2_07_1(2): 27 2.784164 0.000000 EN_R2_07_2: 52 2.784164 0.000000 EN_R2_07_3(1): 45 2.784164 0.000000 EN_R2_07_3(2): 45 2.784164 0.000000 EN_R2_09_1(1): 77 2.784164 0.000000 EN_R2_09_1(2): 77 2.784164 0.000000 EN_R2_11_1(1): 151 2.784164 0.000000 EN_R2_11_1(2): 151 2.784164 0.000000 EXACT 2.784164 N = 3 EXPON = 0 4 0 Degree = 4 EN_R2_05_1(1): 14 4.176246 0.000000 EN_R2_05_1(2): 14 4.176246 0.000000 EN_R2_05_2: 19 4.176246 0.000000 EN_R2_05_3: 14 4.176246 0.000000 EN_R2_05_4: 15 4.176246 0.000000 EN_R2_05_5: 25 4.176246 0.000000 EN_R2_07_1(1): 27 4.176246 0.000000 EN_R2_07_1(2): 27 4.176246 0.000000 EN_R2_07_2: 52 4.176246 0.000000 EN_R2_07_3(1): 45 4.176246 0.000000 EN_R2_07_3(2): 45 4.176246 0.000000 EN_R2_09_1(1): 77 4.176246 0.000000 EN_R2_09_1(2): 77 4.176246 0.000000 EN_R2_11_1(1): 151 4.176246 0.000000 EN_R2_11_1(2): 151 4.176246 0.000000 EXACT 4.176246 N = 3 EXPON = 6 0 0 Degree = 6 EN_R2_07_1(1): 27 10.440615 0.000000 EN_R2_07_1(2): 27 10.440615 0.000000 EN_R2_07_2: 52 10.440615 0.000000 EN_R2_07_3(1): 45 10.440615 0.000000 EN_R2_07_3(2): 45 10.440615 0.000000 EN_R2_09_1(1): 77 10.440615 0.000000 EN_R2_09_1(2): 77 10.440615 0.000000 EN_R2_11_1(1): 151 10.440615 0.000000 EN_R2_11_1(2): 151 10.440615 0.000000 EXACT 10.440615 N = 3 EXPON = 2 4 0 Degree = 6 EN_R2_07_1(1): 27 2.088123 0.000000 EN_R2_07_1(2): 27 2.088123 0.000000 EN_R2_07_2: 52 2.088123 0.000000 EN_R2_07_3(1): 45 2.088123 0.000000 EN_R2_07_3(2): 45 2.088123 0.000000 EN_R2_09_1(1): 77 2.088123 0.000000 EN_R2_09_1(2): 77 2.088123 0.000000 EN_R2_11_1(1): 151 2.088123 0.000000 EN_R2_11_1(2): 151 2.088123 0.000000 EXACT 2.088123 N = 3 EXPON = 0 8 0 Degree = 8 EN_R2_09_1(1): 77 36.542152 0.000000 EN_R2_09_1(2): 77 36.542152 0.000000 EN_R2_11_1(1): 151 36.542152 0.000000 EN_R2_11_1(2): 151 36.542152 0.000000 EXACT 36.542152 N = 3 EXPON = 0 0 10 Degree = 10 EN_R2_11_1(1): 151 164.439686 0.000000 EN_R2_11_1(2): 151 164.439686 0.000000 EXACT 164.439686 N = 3 EXPON = 1 2 3 Degree = 6 EN_R2_07_1(1): 27 0.000000 0.000000 EN_R2_07_1(2): 27 0.000000 0.000000 EN_R2_07_2: 52 0.000000 0.000000 EN_R2_07_3(1): 45 0.000000 0.000000 EN_R2_07_3(2): 45 0.000000 0.000000 EN_R2_09_1(1): 77 0.000000 0.000000 EN_R2_09_1(2): 77 0.000000 0.000000 EN_R2_11_1(1): 151 0.000000 0.000000 EN_R2_11_1(2): 151 0.000000 0.000000 EXACT 0.000000 N = 3 EXPON = 2 2 2 Degree = 6 EN_R2_07_1(1): 27 0.696041 0.000000 EN_R2_07_1(2): 27 0.696041 0.000000 EN_R2_07_2: 52 0.696041 0.000000 EN_R2_07_3(1): 45 0.696041 0.000000 EN_R2_07_3(2): 45 0.696041 0.000000 EN_R2_09_1(1): 77 0.696041 0.000000 EN_R2_09_1(2): 77 0.696041 0.000000 EN_R2_11_1(1): 151 0.696041 0.000000 EN_R2_11_1(2): 151 0.696041 0.000000 EXACT 0.696041 N = 4 EXPON = 0 0 0 0 Degree = 0 EN_R2_01_1: 1 9.869604 0.000000 EN_R2_02_XIU: 5 9.869604 0.000000 GW_02_XIU: 5 9.869604 0.000000 EN_R2_03_1: 8 9.869604 0.000000 EN_R2_03_2: 16 9.869604 0.000000 EN_R2_03_XIU: 8 9.869604 0.000000 EN_R2_05_1(1): 22 9.869604 0.000000 EN_R2_05_2: 33 9.869604 0.000000 EN_R2_05_3: 24 9.869604 0.000000 EN_R2_05_4: 31 9.869604 0.000000 EN_R2_05_5: 65 9.869604 0.000000 EN_R2_07_1(1): 49 9.869604 0.000000 EN_R2_07_1(2): 49 9.869604 0.000000 EN_R2_07_2: 96 9.869604 0.000000 EN_R2_07_3(1): 97 9.869604 0.000000 EN_R2_07_3(2): 97 9.869604 0.000000 EN_R2_09_1(1): 193 9.869604 0.000000 EN_R2_09_1(2): 193 9.869604 0.000000 EN_R2_11_1(1): 417 9.869604 0.000000 EN_R2_11_1(2): 417 9.869604 0.000000 EXACT 9.869604 N = 4 EXPON = 2 0 0 0 Degree = 2 EN_R2_02_XIU: 5 4.934802 0.000000 GW_02_XIU: 5 4.934802 0.000000 EN_R2_03_1: 8 4.934802 0.000000 EN_R2_03_2: 16 4.934802 0.000000 EN_R2_03_XIU: 8 4.934802 0.000000 EN_R2_05_1(1): 22 4.934802 0.000000 EN_R2_05_2: 33 4.934802 0.000000 EN_R2_05_3: 24 4.934802 0.000000 EN_R2_05_4: 31 4.934802 0.000000 EN_R2_05_5: 65 4.934802 0.000000 EN_R2_07_1(1): 49 4.934802 0.000000 EN_R2_07_1(2): 49 4.934802 0.000000 EN_R2_07_2: 96 4.934802 0.000000 EN_R2_07_3(1): 97 4.934802 0.000000 EN_R2_07_3(2): 97 4.934802 0.000000 EN_R2_09_1(1): 193 4.934802 0.000000 EN_R2_09_1(2): 193 4.934802 0.000000 EN_R2_11_1(1): 417 4.934802 0.000000 EN_R2_11_1(2): 417 4.934802 0.000000 EXACT 4.934802 N = 4 EXPON = 0 4 0 0 Degree = 4 EN_R2_05_1(1): 22 7.402203 0.000000 EN_R2_05_2: 33 7.402203 0.000000 EN_R2_05_3: 24 7.402203 0.000000 EN_R2_05_4: 31 7.402203 0.000000 EN_R2_05_5: 65 7.402203 0.000000 EN_R2_07_1(1): 49 7.402203 0.000000 EN_R2_07_1(2): 49 7.402203 0.000000 EN_R2_07_2: 96 7.402203 0.000000 EN_R2_07_3(1): 97 7.402203 0.000000 EN_R2_07_3(2): 97 7.402203 0.000000 EN_R2_09_1(1): 193 7.402203 0.000000 EN_R2_09_1(2): 193 7.402203 0.000000 EN_R2_11_1(1): 417 7.402203 0.000000 EN_R2_11_1(2): 417 7.402203 0.000000 EXACT 7.402203 N = 4 EXPON = 0 0 0 6 Degree = 6 EN_R2_07_1(1): 49 18.505508 0.000000 EN_R2_07_1(2): 49 18.505508 0.000000 EN_R2_07_2: 96 18.505508 0.000000 EN_R2_07_3(1): 97 18.505508 0.000000 EN_R2_07_3(2): 97 18.505508 0.000000 EN_R2_09_1(1): 193 18.505508 0.000000 EN_R2_09_1(2): 193 18.505508 0.000000 EN_R2_11_1(1): 417 18.505508 0.000000 EN_R2_11_1(2): 417 18.505508 0.000000 EXACT 18.505508 N = 4 EXPON = 2 4 0 0 Degree = 6 EN_R2_07_1(1): 49 3.701102 0.000000 EN_R2_07_1(2): 49 3.701102 0.000000 EN_R2_07_2: 96 3.701102 0.000000 EN_R2_07_3(1): 97 3.701102 0.000000 EN_R2_07_3(2): 97 3.701102 0.000000 EN_R2_09_1(1): 193 3.701102 0.000000 EN_R2_09_1(2): 193 3.701102 0.000000 EN_R2_11_1(1): 417 3.701102 0.000000 EN_R2_11_1(2): 417 3.701102 0.000000 EXACT 3.701102 N = 4 EXPON = 8 0 0 0 Degree = 8 EN_R2_09_1(1): 193 64.769279 0.000000 EN_R2_09_1(2): 193 64.769279 0.000000 EN_R2_11_1(1): 417 64.769279 0.000000 EN_R2_11_1(2): 417 64.769279 0.000000 EXACT 64.769279 N = 4 EXPON = 0 10 0 0 Degree = 10 EN_R2_11_1(1): 417 291.461755 0.000000 EN_R2_11_1(2): 417 291.461755 0.000000 EXACT 291.461755 N = 4 EXPON = 1 2 3 4 Degree = 10 EN_R2_11_1(1): 417 0.000000 0.000000 EN_R2_11_1(2): 417 0.000000 0.000000 EXACT 0.000000 N = 4 EXPON = 2 2 2 2 Degree = 8 EN_R2_09_1(1): 193 0.616850 0.000000 EN_R2_09_1(2): 193 0.616850 0.000000 EN_R2_11_1(1): 417 0.616850 0.000000 EN_R2_11_1(2): 417 0.616850 0.000000 EXACT 0.616850 N = 5 EXPON = 0 0 0 0 0 Degree = 0 EN_R2_01_1: 1 17.493418 0.000000 EN_R2_02_XIU: 6 17.493418 0.000000 GW_02_XIU: 6 17.493418 0.000000 EN_R2_03_1: 10 17.493418 0.000000 EN_R2_03_2: 32 17.493418 0.000000 EN_R2_03_XIU: 10 17.493418 0.000000 EN_R2_05_1(1): 32 17.493418 0.000000 EN_R2_05_1(2): 32 17.493418 0.000000 EN_R2_05_2: 51 17.493418 0.000000 EN_R2_05_3: 42 17.493418 0.000000 EN_R2_05_4: 63 17.493418 0.000000 EN_R2_05_5: 161 17.493418 0.000000 EN_R2_05_6: 192 17.493418 0.000000 EN_R2_07_2: 164 17.493418 0.000000 EN_R2_07_3(1): 181 17.493418 0.000000 EN_R2_09_1(1): 421 17.493418 0.000000 EN_R2_09_1(2): 421 17.493418 0.000000 EN_R2_11_1(1): 983 17.493418 0.000000 EN_R2_11_1(2): 983 17.493418 0.000000 EXACT 17.493418 N = 5 EXPON = 2 0 0 0 0 Degree = 2 EN_R2_02_XIU: 6 8.746709 0.000000 GW_02_XIU: 6 8.746709 0.000000 EN_R2_03_1: 10 8.746709 0.000000 EN_R2_03_2: 32 8.746709 0.000000 EN_R2_03_XIU: 10 8.746709 0.000000 EN_R2_05_1(1): 32 8.746709 0.000000 EN_R2_05_1(2): 32 8.746709 0.000000 EN_R2_05_2: 51 8.746709 0.000000 EN_R2_05_3: 42 8.746709 0.000000 EN_R2_05_4: 63 8.746709 0.000000 EN_R2_05_5: 161 8.746709 0.000000 EN_R2_05_6: 192 8.746709 0.000000 EN_R2_07_2: 164 8.746709 0.000000 EN_R2_07_3(1): 181 8.746709 0.000000 EN_R2_09_1(1): 421 8.746709 0.000000 EN_R2_09_1(2): 421 8.746709 0.000000 EN_R2_11_1(1): 983 8.746709 0.000000 EN_R2_11_1(2): 983 8.746709 0.000000 EXACT 8.746709 N = 5 EXPON = 0 4 0 0 0 Degree = 4 EN_R2_05_1(1): 32 13.120064 0.000000 EN_R2_05_1(2): 32 13.120064 0.000000 EN_R2_05_2: 51 13.120064 0.000000 EN_R2_05_3: 42 13.120064 0.000000 EN_R2_05_4: 63 13.120064 0.000000 EN_R2_05_5: 161 13.120064 0.000000 EN_R2_05_6: 192 13.120064 0.000000 EN_R2_07_2: 164 13.120064 0.000000 EN_R2_07_3(1): 181 13.120064 0.000000 EN_R2_09_1(1): 421 13.120064 0.000000 EN_R2_09_1(2): 421 13.120064 0.000000 EN_R2_11_1(1): 983 13.120064 0.000000 EN_R2_11_1(2): 983 13.120064 0.000000 EXACT 13.120064 N = 5 EXPON = 0 0 0 6 0 Degree = 6 EN_R2_07_2: 164 32.800159 0.000000 EN_R2_07_3(1): 181 32.800159 0.000000 EN_R2_09_1(1): 421 32.800159 0.000000 EN_R2_09_1(2): 421 32.800159 0.000000 EN_R2_11_1(1): 983 32.800159 0.000000 EN_R2_11_1(2): 983 32.800159 0.000000 EXACT 32.800159 N = 5 EXPON = 2 4 0 0 0 Degree = 6 EN_R2_07_2: 164 6.560032 0.000000 EN_R2_07_3(1): 181 6.560032 0.000000 EN_R2_09_1(1): 421 6.560032 0.000000 EN_R2_09_1(2): 421 6.560032 0.000000 EN_R2_11_1(1): 983 6.560032 0.000000 EN_R2_11_1(2): 983 6.560032 0.000000 EXACT 6.560032 N = 5 EXPON = 0 0 0 0 8 Degree = 8 EN_R2_09_1(1): 421 114.800558 0.000000 EN_R2_09_1(2): 421 114.800558 0.000000 EN_R2_11_1(1): 983 114.800558 0.000000 EN_R2_11_1(2): 983 114.800558 0.000000 EXACT 114.800558 N = 5 EXPON = 10 0 0 0 0 Degree = 10 EN_R2_11_1(1): 983 516.602510 0.000000 EN_R2_11_1(2): 983 516.602510 0.000000 EXACT 516.602510 N = 5 EXPON = 1 2 3 4 5 Degree = 15 EXACT 0.000000 N = 5 EXPON = 2 2 2 2 2 Degree = 10 EN_R2_11_1(1): 983 0.546669 0.000000 EN_R2_11_1(2): 983 0.546669 0.000000 EXACT 0.546669 N = 6 EXPON = 0 0 0 0 0 0 Degree = 0 EN_R2_01_1: 1 31.006277 0.000000 EN_R2_02_XIU: 7 31.006277 0.000000 GW_02_XIU: 7 31.006277 0.000000 EN_R2_03_1: 12 31.006277 0.000000 EN_R2_03_2: 64 31.006277 0.000000 EN_R2_03_XIU: 12 31.006277 0.000000 EN_R2_05_1(1): 44 31.006277 0.000000 EN_R2_05_1(2): 44 31.006277 0.000000 EN_R2_05_2: 73 31.006277 0.000000 EN_R2_05_3: 76 31.006277 0.000000 EN_R2_05_4: 127 31.006277 0.000000 EN_R2_05_5: 385 31.006277 0.000000 EN_R2_05_6: 448 31.006277 0.000000 EN_R2_07_1(1): 137 31.006277 0.000000 EN_R2_07_2: 272 31.006277 0.000000 EN_R2_07_3(1): 305 31.006277 0.000000 EN_R2_09_1(1): 825 31.006277 0.000000 EN_R2_09_1(2): 825 31.006277 0.000000 EXACT 31.006277 N = 6 EXPON = 2 0 0 0 0 0 Degree = 2 EN_R2_02_XIU: 7 15.503138 0.000000 GW_02_XIU: 7 15.503138 0.000000 EN_R2_03_1: 12 15.503138 0.000000 EN_R2_03_2: 64 15.503138 0.000000 EN_R2_03_XIU: 12 15.503138 0.000000 EN_R2_05_1(1): 44 15.503138 0.000000 EN_R2_05_1(2): 44 15.503138 0.000000 EN_R2_05_2: 73 15.503138 0.000000 EN_R2_05_3: 76 15.503138 0.000000 EN_R2_05_4: 127 15.503138 0.000000 EN_R2_05_5: 385 15.503138 0.000000 EN_R2_05_6: 448 15.503138 0.000000 EN_R2_07_1(1): 137 15.503138 0.000000 EN_R2_07_2: 272 15.503138 0.000000 EN_R2_07_3(1): 305 15.503138 0.000000 EN_R2_09_1(1): 825 15.503138 0.000000 EN_R2_09_1(2): 825 15.503138 0.000000 EXACT 15.503138 N = 6 EXPON = 0 4 0 0 0 0 Degree = 4 EN_R2_05_1(1): 44 23.254708 0.000000 EN_R2_05_1(2): 44 23.254708 0.000000 EN_R2_05_2: 73 23.254708 0.000000 EN_R2_05_3: 76 23.254708 0.000000 EN_R2_05_4: 127 23.254708 0.000000 EN_R2_05_5: 385 23.254708 0.000000 EN_R2_05_6: 448 23.254708 0.000000 EN_R2_07_1(1): 137 23.254708 0.000000 EN_R2_07_2: 272 23.254708 0.000000 EN_R2_07_3(1): 305 23.254708 0.000000 EN_R2_09_1(1): 825 23.254708 0.000000 EN_R2_09_1(2): 825 23.254708 0.000000 EXACT 23.254708 N = 6 EXPON = 0 0 0 6 0 0 Degree = 6 EN_R2_07_1(1): 137 58.136769 0.000000 EN_R2_07_2: 272 58.136769 0.000000 EN_R2_07_3(1): 305 58.136769 0.000000 EN_R2_09_1(1): 825 58.136769 0.000000 EN_R2_09_1(2): 825 58.136769 0.000000 EXACT 58.136769 N = 6 EXPON = 2 4 0 0 0 0 Degree = 6 EN_R2_07_1(1): 137 11.627354 0.000000 EN_R2_07_2: 272 11.627354 0.000000 EN_R2_07_3(1): 305 11.627354 0.000000 EN_R2_09_1(1): 825 11.627354 0.000000 EN_R2_09_1(2): 825 11.627354 0.000000 EXACT 11.627354 N = 6 EXPON = 0 0 0 0 8 0 Degree = 8 EN_R2_09_1(1): 825 203.478691 0.000000 EN_R2_09_1(2): 825 203.478691 0.000000 EXACT 203.478691 N = 6 EXPON = 0 0 0 0 0 10 Degree = 10 EXACT 915.654108 N = 6 EXPON = 1 2 3 4 5 6 Degree = 21 EXACT 0.000000 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 54.957195 0.000000 EN_R2_02_XIU: 8 54.957195 0.000000 GW_02_XIU: 8 54.957195 0.000000 EN_R2_03_1: 14 54.957195 0.000000 EN_R2_03_2: 128 54.957195 0.000000 EN_R2_03_XIU: 14 54.957195 0.000000 EN_R2_05_1(1): 58 54.957195 0.000000 EN_R2_05_2: 99 54.957195 0.000000 EN_R2_05_3: 142 54.957195 0.000000 EN_R2_05_4: 255 54.957195 0.000000 EN_R2_05_5: 897 54.957195 0.000000 EN_R2_05_6: 1024 54.957195 0.000000 EN_R2_07_1(1): 227 54.957195 0.000000 EN_R2_07_2: 452 54.957195 0.000000 EXACT 54.957195 N = 7 EXPON = 2 0 0 0 0 0 0 Degree = 2 EN_R2_02_XIU: 8 27.478597 0.000000 GW_02_XIU: 8 27.478597 0.000000 EN_R2_03_1: 14 27.478597 0.000000 EN_R2_03_2: 128 27.478597 0.000000 EN_R2_03_XIU: 14 27.478597 0.000000 EN_R2_05_1(1): 58 27.478597 0.000000 EN_R2_05_2: 99 27.478597 0.000000 EN_R2_05_3: 142 27.478597 0.000000 EN_R2_05_4: 255 27.478597 0.000000 EN_R2_05_5: 897 27.478597 0.000000 EN_R2_05_6: 1024 27.478597 0.000000 EN_R2_07_1(1): 227 27.478597 0.000000 EN_R2_07_2: 452 27.478597 0.000000 EXACT 27.478597 N = 7 EXPON = 0 4 0 0 0 0 0 Degree = 4 EN_R2_05_1(1): 58 41.217896 0.000000 EN_R2_05_2: 99 41.217896 0.000000 EN_R2_05_3: 142 41.217896 0.000000 EN_R2_05_4: 255 41.217896 0.000000 EN_R2_05_5: 897 41.217896 0.000000 EN_R2_05_6: 1024 41.217896 0.000000 EN_R2_07_1(1): 227 41.217896 0.000000 EN_R2_07_2: 452 41.217896 0.000000 EXACT 41.217896 N = 7 EXPON = 0 0 0 6 0 0 0 Degree = 6 EN_R2_07_1(1): 227 103.044740 0.000000 EN_R2_07_2: 452 103.044740 0.000000 EXACT 103.044740 N = 7 EXPON = 2 4 0 0 0 0 0 Degree = 6 EN_R2_07_1(1): 227 20.608948 0.000000 EN_R2_07_2: 452 20.608948 0.000000 EXACT 20.608948 N = 7 EXPON = 0 0 0 0 8 0 0 Degree = 8 EXACT 360.656589 N = 7 EXPON = 0 0 0 0 0 10 0 Degree = 10 EXACT 1622.954650 N = 7 EXPON = 1 2 3 4 5 6 7 Degree = 28 EXACT 0.000000 N = 7 EXPON = 2 2 2 2 2 2 2 Degree = 14 EXACT 0.429353 TEST2075 Demonstrate the use of quadrature rules for the region EPN_GLG, that is, the positive half space [0,+oo)^N, with the weight W(ALPHA;X) = product ( 1 <= I <= N ) X(I)^ALPHA exp ( -X(I) ) We use the formulas to integrate various monomials of the form X(1)^E(1) * X(2)^E(2) * ... X(N)^E(N) and compare to the exact integral. The precision of each formula is known, and we only use a formula if its precision indicates it should be able to produce an exact result. N = 1 ALPHA = -0.500000 EXPON = 0 Degree = 0 EPN_GLG_00_1: 1 1.772454 0.000000 EPN_GLG_01_1: 1 1.772454 0.000000 EPN_GLG_02_XIU: 2 1.772454 0.000000 GW_02_XIU: 2 1.772454 0.000000 EXACT 1.772454 N = 1 ALPHA = 0.000000 EXPON = 0 Degree = 0 EPN_GLG_00_1: 1 1.000000 0.000000 EPN_GLG_01_1: 1 1.000000 0.000000 EPN_GLG_02_XIU: 2 1.000000 0.000000 GW_02_XIU: 2 1.000000 0.000000 EXACT 1.000000 N = 1 ALPHA = 0.500000 EXPON = 0 Degree = 0 EPN_GLG_00_1: 1 0.886227 0.000000 EPN_GLG_01_1: 1 0.886227 0.000000 EPN_GLG_02_XIU: 2 0.886227 0.000000 GW_02_XIU: 2 0.886227 0.000000 EXACT 0.886227 N = 1 ALPHA = 1.000000 EXPON = 0 Degree = 0 EPN_GLG_00_1: 1 1.000000 0.000000 EPN_GLG_01_1: 1 1.000000 0.000000 EPN_GLG_02_XIU: 2 1.000000 0.000000 GW_02_XIU: 2 1.000000 0.000000 EXACT 1.000000 N = 1 ALPHA = 2.000000 EXPON = 0 Degree = 0 EPN_GLG_00_1: 1 2.000000 0.000000 EPN_GLG_01_1: 1 2.000000 0.000000 EPN_GLG_02_XIU: 2 2.000000 0.000000 GW_02_XIU: 2 2.000000 0.000000 EXACT 2.000000 N = 1 ALPHA = -0.500000 EXPON = 1 Degree = 1 EPN_GLG_01_1: 1 0.886227 0.000000 EPN_GLG_02_XIU: 2 0.886227 0.000000 GW_02_XIU: 2 0.886227 0.000000 EXACT 0.886227 N = 1 ALPHA = 0.000000 EXPON = 1 Degree = 1 EPN_GLG_01_1: 1 1.000000 0.000000 EPN_GLG_02_XIU: 2 1.000000 0.000000 GW_02_XIU: 2 1.000000 0.000000 EXACT 1.000000 N = 1 ALPHA = 0.500000 EXPON = 1 Degree = 1 EPN_GLG_01_1: 1 1.329340 0.000000 EPN_GLG_02_XIU: 2 1.329340 0.000000 GW_02_XIU: 2 1.329340 0.000000 EXACT 1.329340 N = 1 ALPHA = 1.000000 EXPON = 1 Degree = 1 EPN_GLG_01_1: 1 2.000000 0.000000 EPN_GLG_02_XIU: 2 2.000000 0.000000 GW_02_XIU: 2 2.000000 0.000000 EXACT 2.000000 N = 1 ALPHA = 2.000000 EXPON = 1 Degree = 1 EPN_GLG_01_1: 1 6.000000 0.000000 EPN_GLG_02_XIU: 2 6.000000 0.000000 GW_02_XIU: 2 6.000000 0.000000 EXACT 6.000000 N = 1 ALPHA = -0.500000 EXPON = 2 Degree = 2 EPN_GLG_02_XIU: 2 1.329340 0.000000 GW_02_XIU: 2 1.329340 0.000000 EXACT 1.329340 N = 1 ALPHA = 0.000000 EXPON = 2 Degree = 2 EPN_GLG_02_XIU: 2 2.000000 0.000000 GW_02_XIU: 2 2.000000 0.000000 EXACT 2.000000 N = 1 ALPHA = 0.500000 EXPON = 2 Degree = 2 EPN_GLG_02_XIU: 2 3.323351 0.000000 GW_02_XIU: 2 3.323351 0.000000 EXACT 3.323351 N = 1 ALPHA = 1.000000 EXPON = 2 Degree = 2 EPN_GLG_02_XIU: 2 6.000000 0.000000 GW_02_XIU: 2 6.000000 0.000000 EXACT 6.000000 N = 1 ALPHA = 2.000000 EXPON = 2 Degree = 2 EPN_GLG_02_XIU: 2 24.000000 0.000000 GW_02_XIU: 2 24.000000 0.000000 EXACT 24.000000 N = 2 ALPHA = -0.500000 EXPON = 0 0 Degree = 0 EPN_GLG_00_1: 1 3.141593 0.000000 EPN_GLG_01_1: 1 3.141593 0.000000 EPN_GLG_02_XIU: 3 3.141593 0.000000 GW_02_XIU: 3 3.141593 0.000000 EXACT 3.141593 N = 2 ALPHA = 0.000000 EXPON = 0 0 Degree = 0 EPN_GLG_00_1: 1 1.000000 0.000000 EPN_GLG_01_1: 1 1.000000 0.000000 EPN_GLG_02_XIU: 3 1.000000 0.000000 GW_02_XIU: 3 1.000000 0.000000 EXACT 1.000000 N = 2 ALPHA = 0.500000 EXPON = 0 0 Degree = 0 EPN_GLG_00_1: 1 0.785398 0.000000 EPN_GLG_01_1: 1 0.785398 0.000000 EPN_GLG_02_XIU: 3 0.785398 0.000000 GW_02_XIU: 3 0.785398 0.000000 EXACT 0.785398 N = 2 ALPHA = 1.000000 EXPON = 0 0 Degree = 0 EPN_GLG_00_1: 1 1.000000 0.000000 EPN_GLG_01_1: 1 1.000000 0.000000 EPN_GLG_02_XIU: 3 1.000000 0.000000 GW_02_XIU: 3 1.000000 0.000000 EXACT 1.000000 N = 2 ALPHA = 2.000000 EXPON = 0 0 Degree = 0 EPN_GLG_00_1: 1 4.000000 0.000000 EPN_GLG_01_1: 1 4.000000 0.000000 EPN_GLG_02_XIU: 3 4.000000 0.000000 GW_02_XIU: 3 4.000000 0.000000 EXACT 4.000000 N = 2 ALPHA = -0.500000 EXPON = 0 1 Degree = 1 EPN_GLG_01_1: 1 1.570796 0.000000 EPN_GLG_02_XIU: 3 1.570796 0.000000 GW_02_XIU: 3 1.570796 0.000000 EXACT 1.570796 N = 2 ALPHA = 0.000000 EXPON = 0 1 Degree = 1 EPN_GLG_01_1: 1 1.000000 0.000000 EPN_GLG_02_XIU: 3 1.000000 0.000000 GW_02_XIU: 3 1.000000 0.000000 EXACT 1.000000 N = 2 ALPHA = 0.500000 EXPON = 0 1 Degree = 1 EPN_GLG_01_1: 1 1.178097 0.000000 EPN_GLG_02_XIU: 3 1.178097 0.000000 GW_02_XIU: 3 1.178097 0.000000 EXACT 1.178097 N = 2 ALPHA = 1.000000 EXPON = 0 1 Degree = 1 EPN_GLG_01_1: 1 2.000000 0.000000 EPN_GLG_02_XIU: 3 2.000000 0.000000 GW_02_XIU: 3 2.000000 0.000000 EXACT 2.000000 N = 2 ALPHA = 2.000000 EXPON = 0 1 Degree = 1 EPN_GLG_01_1: 1 12.000000 0.000000 EPN_GLG_02_XIU: 3 12.000000 0.000000 GW_02_XIU: 3 12.000000 0.000000 EXACT 12.000000 N = 2 ALPHA = -0.500000 EXPON = 1 1 Degree = 2 EPN_GLG_02_XIU: 3 0.785398 0.000000 GW_02_XIU: 3 0.785398 0.000000 EXACT 0.785398 N = 2 ALPHA = 0.000000 EXPON = 1 1 Degree = 2 EPN_GLG_02_XIU: 3 1.000000 0.000000 GW_02_XIU: 3 1.000000 0.000000 EXACT 1.000000 N = 2 ALPHA = 0.500000 EXPON = 1 1 Degree = 2 EPN_GLG_02_XIU: 3 1.767146 0.000000 GW_02_XIU: 3 1.767146 0.000000 EXACT 1.767146 N = 2 ALPHA = 1.000000 EXPON = 1 1 Degree = 2 EPN_GLG_02_XIU: 3 4.000000 0.000000 GW_02_XIU: 3 4.000000 0.000000 EXACT 4.000000 N = 2 ALPHA = 2.000000 EXPON = 1 1 Degree = 2 EPN_GLG_02_XIU: 3 36.000000 0.000000 GW_02_XIU: 3 36.000000 0.000000 EXACT 36.000000 N = 2 ALPHA = -0.500000 EXPON = 2 0 Degree = 2 EPN_GLG_02_XIU: 3 2.356194 0.000000 GW_02_XIU: 3 2.356194 0.000000 EXACT 2.356194 N = 2 ALPHA = 0.000000 EXPON = 2 0 Degree = 2 EPN_GLG_02_XIU: 3 2.000000 0.000000 GW_02_XIU: 3 2.000000 0.000000 EXACT 2.000000 N = 2 ALPHA = 0.500000 EXPON = 2 0 Degree = 2 EPN_GLG_02_XIU: 3 2.945243 0.000000 GW_02_XIU: 3 2.945243 0.000000 EXACT 2.945243 N = 2 ALPHA = 1.000000 EXPON = 2 0 Degree = 2 EPN_GLG_02_XIU: 3 6.000000 0.000000 GW_02_XIU: 3 6.000000 0.000000 EXACT 6.000000 N = 2 ALPHA = 2.000000 EXPON = 2 0 Degree = 2 EPN_GLG_02_XIU: 3 48.000000 0.000000 GW_02_XIU: 3 48.000000 0.000000 EXACT 48.000000 N = 3 ALPHA = -0.500000 EXPON = 0 0 0 Degree = 0 EPN_GLG_00_1: 1 5.568328 0.000000 EPN_GLG_01_1: 1 5.568328 0.000000 EPN_GLG_02_XIU: 4 5.568328 0.000000 GW_02_XIU: 4 5.568328 0.000000 EXACT 5.568328 N = 3 ALPHA = 0.000000 EXPON = 0 0 0 Degree = 0 EPN_GLG_00_1: 1 1.000000 0.000000 EPN_GLG_01_1: 1 1.000000 0.000000 EPN_GLG_02_XIU: 4 1.000000 0.000000 GW_02_XIU: 4 1.000000 0.000000 EXACT 1.000000 N = 3 ALPHA = 0.500000 EXPON = 0 0 0 Degree = 0 EPN_GLG_00_1: 1 0.696041 0.000000 EPN_GLG_01_1: 1 0.696041 0.000000 EPN_GLG_02_XIU: 4 0.696041 0.000000 GW_02_XIU: 4 0.696041 0.000000 EXACT 0.696041 N = 3 ALPHA = 1.000000 EXPON = 0 0 0 Degree = 0 EPN_GLG_00_1: 1 1.000000 0.000000 EPN_GLG_01_1: 1 1.000000 0.000000 EPN_GLG_02_XIU: 4 1.000000 0.000000 GW_02_XIU: 4 1.000000 0.000000 EXACT 1.000000 N = 3 ALPHA = 2.000000 EXPON = 0 0 0 Degree = 0 EPN_GLG_00_1: 1 8.000000 0.000000 EPN_GLG_01_1: 1 8.000000 0.000000 EPN_GLG_02_XIU: 4 8.000000 0.000000 GW_02_XIU: 4 8.000000 0.000000 EXACT 8.000000 N = 3 ALPHA = -0.500000 EXPON = 0 0 1 Degree = 1 EPN_GLG_01_1: 1 2.784164 0.000000 EPN_GLG_02_XIU: 4 2.784164 0.000000 GW_02_XIU: 4 2.784164 0.000000 EXACT 2.784164 N = 3 ALPHA = 0.000000 EXPON = 0 0 1 Degree = 1 EPN_GLG_01_1: 1 1.000000 0.000000 EPN_GLG_02_XIU: 4 1.000000 0.000000 GW_02_XIU: 4 1.000000 0.000000 EXACT 1.000000 N = 3 ALPHA = 0.500000 EXPON = 0 0 1 Degree = 1 EPN_GLG_01_1: 1 1.044061 0.000000 EPN_GLG_02_XIU: 4 1.044061 0.000000 GW_02_XIU: 4 1.044061 0.000000 EXACT 1.044061 N = 3 ALPHA = 1.000000 EXPON = 0 0 1 Degree = 1 EPN_GLG_01_1: 1 2.000000 0.000000 EPN_GLG_02_XIU: 4 2.000000 0.000000 GW_02_XIU: 4 2.000000 0.000000 EXACT 2.000000 N = 3 ALPHA = 2.000000 EXPON = 0 0 1 Degree = 1 EPN_GLG_01_1: 1 24.000000 0.000000 EPN_GLG_02_XIU: 4 24.000000 0.000000 GW_02_XIU: 4 24.000000 0.000000 EXACT 24.000000 N = 3 ALPHA = -0.500000 EXPON = 1 1 0 Degree = 2 EPN_GLG_02_XIU: 4 1.392082 0.000000 GW_02_XIU: 4 1.392082 0.000000 EXACT 1.392082 N = 3 ALPHA = 0.000000 EXPON = 1 1 0 Degree = 2 EPN_GLG_02_XIU: 4 1.000000 0.000000 GW_02_XIU: 4 1.000000 0.000000 EXACT 1.000000 N = 3 ALPHA = 0.500000 EXPON = 1 1 0 Degree = 2 EPN_GLG_02_XIU: 4 1.566092 0.000000 GW_02_XIU: 4 1.566092 0.000000 EXACT 1.566092 N = 3 ALPHA = 1.000000 EXPON = 1 1 0 Degree = 2 EPN_GLG_02_XIU: 4 4.000000 0.000000 GW_02_XIU: 4 4.000000 0.000000 EXACT 4.000000 N = 3 ALPHA = 2.000000 EXPON = 1 1 0 Degree = 2 EPN_GLG_02_XIU: 4 72.000000 0.000000 GW_02_XIU: 4 72.000000 0.000000 EXACT 72.000000 N = 3 ALPHA = -0.500000 EXPON = 2 0 0 Degree = 2 EPN_GLG_02_XIU: 4 4.176246 0.000000 GW_02_XIU: 4 4.176246 0.000000 EXACT 4.176246 N = 3 ALPHA = 0.000000 EXPON = 2 0 0 Degree = 2 EPN_GLG_02_XIU: 4 2.000000 0.000000 GW_02_XIU: 4 2.000000 0.000000 EXACT 2.000000 N = 3 ALPHA = 0.500000 EXPON = 2 0 0 Degree = 2 EPN_GLG_02_XIU: 4 2.610154 0.000000 GW_02_XIU: 4 2.610154 0.000000 EXACT 2.610154 N = 3 ALPHA = 1.000000 EXPON = 2 0 0 Degree = 2 EPN_GLG_02_XIU: 4 6.000000 0.000000 GW_02_XIU: 4 6.000000 0.000000 EXACT 6.000000 N = 3 ALPHA = 2.000000 EXPON = 2 0 0 Degree = 2 EPN_GLG_02_XIU: 4 96.000000 0.000000 GW_02_XIU: 4 96.000000 0.000000 EXACT 96.000000 N = 4 ALPHA = -0.500000 EXPON = 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 9.869604 0.000000 EPN_GLG_01_1: 1 9.869604 0.000000 EPN_GLG_02_XIU: 5 9.869604 0.000000 GW_02_XIU: 5 9.869604 0.000000 EXACT 9.869604 N = 4 ALPHA = 0.000000 EXPON = 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 1.000000 0.000000 EPN_GLG_01_1: 1 1.000000 0.000000 EPN_GLG_02_XIU: 5 1.000000 0.000000 GW_02_XIU: 5 1.000000 0.000000 EXACT 1.000000 N = 4 ALPHA = 0.500000 EXPON = 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 0.616850 0.000000 EPN_GLG_01_1: 1 0.616850 0.000000 EPN_GLG_02_XIU: 5 0.616850 0.000000 GW_02_XIU: 5 0.616850 0.000000 EXACT 0.616850 N = 4 ALPHA = 1.000000 EXPON = 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 1.000000 0.000000 EPN_GLG_01_1: 1 1.000000 0.000000 EPN_GLG_02_XIU: 5 1.000000 0.000000 GW_02_XIU: 5 1.000000 0.000000 EXACT 1.000000 N = 4 ALPHA = 2.000000 EXPON = 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 16.000000 0.000000 EPN_GLG_01_1: 1 16.000000 0.000000 EPN_GLG_02_XIU: 5 16.000000 0.000000 GW_02_XIU: 5 16.000000 0.000000 EXACT 16.000000 N = 4 ALPHA = -0.500000 EXPON = 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 4.934802 0.000000 EPN_GLG_02_XIU: 5 4.934802 0.000000 GW_02_XIU: 5 4.934802 0.000000 EXACT 4.934802 N = 4 ALPHA = 0.000000 EXPON = 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 1.000000 0.000000 EPN_GLG_02_XIU: 5 1.000000 0.000000 GW_02_XIU: 5 1.000000 0.000000 EXACT 1.000000 N = 4 ALPHA = 0.500000 EXPON = 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 0.925275 0.000000 EPN_GLG_02_XIU: 5 0.925275 0.000000 GW_02_XIU: 5 0.925275 0.000000 EXACT 0.925275 N = 4 ALPHA = 1.000000 EXPON = 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 2.000000 0.000000 EPN_GLG_02_XIU: 5 2.000000 0.000000 GW_02_XIU: 5 2.000000 0.000000 EXACT 2.000000 N = 4 ALPHA = 2.000000 EXPON = 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 48.000000 0.000000 EPN_GLG_02_XIU: 5 48.000000 0.000000 GW_02_XIU: 5 48.000000 0.000000 EXACT 48.000000 N = 4 ALPHA = -0.500000 EXPON = 1 1 0 0 Degree = 2 EPN_GLG_02_XIU: 5 2.467401 0.000000 GW_02_XIU: 5 2.467401 0.000000 EXACT 2.467401 N = 4 ALPHA = 0.000000 EXPON = 1 1 0 0 Degree = 2 EPN_GLG_02_XIU: 5 1.000000 0.000000 GW_02_XIU: 5 1.000000 0.000000 EXACT 1.000000 N = 4 ALPHA = 0.500000 EXPON = 1 1 0 0 Degree = 2 EPN_GLG_02_XIU: 5 1.387913 0.000000 GW_02_XIU: 5 1.387913 0.000000 EXACT 1.387913 N = 4 ALPHA = 1.000000 EXPON = 1 1 0 0 Degree = 2 EPN_GLG_02_XIU: 5 4.000000 0.000000 GW_02_XIU: 5 4.000000 0.000000 EXACT 4.000000 N = 4 ALPHA = 2.000000 EXPON = 1 1 0 0 Degree = 2 EPN_GLG_02_XIU: 5 144.000000 0.000000 GW_02_XIU: 5 144.000000 0.000000 EXACT 144.000000 N = 4 ALPHA = -0.500000 EXPON = 2 0 0 0 Degree = 2 EPN_GLG_02_XIU: 5 7.402203 0.000000 GW_02_XIU: 5 7.402203 0.000000 EXACT 7.402203 N = 4 ALPHA = 0.000000 EXPON = 2 0 0 0 Degree = 2 EPN_GLG_02_XIU: 5 2.000000 0.000000 GW_02_XIU: 5 2.000000 0.000000 EXACT 2.000000 N = 4 ALPHA = 0.500000 EXPON = 2 0 0 0 Degree = 2 EPN_GLG_02_XIU: 5 2.313189 0.000000 GW_02_XIU: 5 2.313189 0.000000 EXACT 2.313189 N = 4 ALPHA = 1.000000 EXPON = 2 0 0 0 Degree = 2 EPN_GLG_02_XIU: 5 6.000000 0.000000 GW_02_XIU: 5 6.000000 0.000000 EXACT 6.000000 N = 4 ALPHA = 2.000000 EXPON = 2 0 0 0 Degree = 2 EPN_GLG_02_XIU: 5 192.000000 0.000000 GW_02_XIU: 5 192.000000 0.000000 EXACT 192.000000 N = 5 ALPHA = -0.500000 EXPON = 0 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 17.493418 0.000000 EPN_GLG_01_1: 1 17.493418 0.000000 EPN_GLG_02_XIU: 6 17.493418 0.000000 GW_02_XIU: 6 17.493418 0.000000 EXACT 17.493418 N = 5 ALPHA = 0.000000 EXPON = 0 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 1.000000 0.000000 EPN_GLG_01_1: 1 1.000000 0.000000 EPN_GLG_02_XIU: 6 1.000000 0.000000 GW_02_XIU: 6 1.000000 0.000000 EXACT 1.000000 N = 5 ALPHA = 0.500000 EXPON = 0 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 0.546669 0.000000 EPN_GLG_01_1: 1 0.546669 0.000000 EPN_GLG_02_XIU: 6 0.546669 0.000000 GW_02_XIU: 6 0.546669 0.000000 EXACT 0.546669 N = 5 ALPHA = 1.000000 EXPON = 0 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 1.000000 0.000000 EPN_GLG_01_1: 1 1.000000 0.000000 EPN_GLG_02_XIU: 6 1.000000 0.000000 GW_02_XIU: 6 1.000000 0.000000 EXACT 1.000000 N = 5 ALPHA = 2.000000 EXPON = 0 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 32.000000 0.000000 EPN_GLG_01_1: 1 32.000000 0.000000 EPN_GLG_02_XIU: 6 32.000000 0.000000 GW_02_XIU: 6 32.000000 0.000000 EXACT 32.000000 N = 5 ALPHA = -0.500000 EXPON = 0 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 8.746709 0.000000 EPN_GLG_02_XIU: 6 8.746709 0.000000 GW_02_XIU: 6 8.746709 0.000000 EXACT 8.746709 N = 5 ALPHA = 0.000000 EXPON = 0 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 1.000000 0.000000 EPN_GLG_02_XIU: 6 1.000000 0.000000 GW_02_XIU: 6 1.000000 0.000000 EXACT 1.000000 N = 5 ALPHA = 0.500000 EXPON = 0 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 0.820004 0.000000 EPN_GLG_02_XIU: 6 0.820004 0.000000 GW_02_XIU: 6 0.820004 0.000000 EXACT 0.820004 N = 5 ALPHA = 1.000000 EXPON = 0 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 2.000000 0.000000 EPN_GLG_02_XIU: 6 2.000000 0.000000 GW_02_XIU: 6 2.000000 0.000000 EXACT 2.000000 N = 5 ALPHA = 2.000000 EXPON = 0 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 96.000000 0.000000 EPN_GLG_02_XIU: 6 96.000000 0.000000 GW_02_XIU: 6 96.000000 0.000000 EXACT 96.000000 N = 5 ALPHA = -0.500000 EXPON = 1 1 0 0 0 Degree = 2 EPN_GLG_02_XIU: 6 4.373355 0.000000 GW_02_XIU: 6 4.373355 0.000000 EXACT 4.373355 N = 5 ALPHA = 0.000000 EXPON = 1 1 0 0 0 Degree = 2 EPN_GLG_02_XIU: 6 1.000000 0.000000 GW_02_XIU: 6 1.000000 0.000000 EXACT 1.000000 N = 5 ALPHA = 0.500000 EXPON = 1 1 0 0 0 Degree = 2 EPN_GLG_02_XIU: 6 1.230006 0.000000 GW_02_XIU: 6 1.230006 0.000000 EXACT 1.230006 N = 5 ALPHA = 1.000000 EXPON = 1 1 0 0 0 Degree = 2 EPN_GLG_02_XIU: 6 4.000000 0.000000 GW_02_XIU: 6 4.000000 0.000000 EXACT 4.000000 N = 5 ALPHA = 2.000000 EXPON = 1 1 0 0 0 Degree = 2 EPN_GLG_02_XIU: 6 288.000000 0.000000 GW_02_XIU: 6 288.000000 0.000000 EXACT 288.000000 N = 5 ALPHA = -0.500000 EXPON = 2 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 6 13.120064 0.000000 GW_02_XIU: 6 13.120064 0.000000 EXACT 13.120064 N = 5 ALPHA = 0.000000 EXPON = 2 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 6 2.000000 0.000000 GW_02_XIU: 6 2.000000 0.000000 EXACT 2.000000 N = 5 ALPHA = 0.500000 EXPON = 2 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 6 2.050010 0.000000 GW_02_XIU: 6 2.050010 0.000000 EXACT 2.050010 N = 5 ALPHA = 1.000000 EXPON = 2 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 6 6.000000 0.000000 GW_02_XIU: 6 6.000000 0.000000 EXACT 6.000000 N = 5 ALPHA = 2.000000 EXPON = 2 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 6 384.000000 0.000000 GW_02_XIU: 6 384.000000 0.000000 EXACT 384.000000 N = 6 ALPHA = -0.500000 EXPON = 0 0 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 31.006277 0.000000 EPN_GLG_01_1: 1 31.006277 0.000000 EPN_GLG_02_XIU: 7 31.006277 0.000000 GW_02_XIU: 7 31.006277 0.000000 EXACT 31.006277 N = 6 ALPHA = 0.000000 EXPON = 0 0 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 1.000000 0.000000 EPN_GLG_01_1: 1 1.000000 0.000000 EPN_GLG_02_XIU: 7 1.000000 0.000000 GW_02_XIU: 7 1.000000 0.000000 EXACT 1.000000 N = 6 ALPHA = 0.500000 EXPON = 0 0 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 0.484473 0.000000 EPN_GLG_01_1: 1 0.484473 0.000000 EPN_GLG_02_XIU: 7 0.484473 0.000000 GW_02_XIU: 7 0.484473 0.000000 EXACT 0.484473 N = 6 ALPHA = 1.000000 EXPON = 0 0 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 1.000000 0.000000 EPN_GLG_01_1: 1 1.000000 0.000000 EPN_GLG_02_XIU: 7 1.000000 0.000000 GW_02_XIU: 7 1.000000 0.000000 EXACT 1.000000 N = 6 ALPHA = 2.000000 EXPON = 0 0 0 0 0 0 Degree = 0 EPN_GLG_00_1: 1 64.000000 0.000000 EPN_GLG_01_1: 1 64.000000 0.000000 EPN_GLG_02_XIU: 7 64.000000 0.000000 GW_02_XIU: 7 64.000000 0.000000 EXACT 64.000000 N = 6 ALPHA = -0.500000 EXPON = 0 0 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 15.503138 0.000000 EPN_GLG_02_XIU: 7 15.503138 0.000000 GW_02_XIU: 7 15.503138 0.000000 EXACT 15.503138 N = 6 ALPHA = 0.000000 EXPON = 0 0 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 1.000000 0.000000 EPN_GLG_02_XIU: 7 1.000000 0.000000 GW_02_XIU: 7 1.000000 0.000000 EXACT 1.000000 N = 6 ALPHA = 0.500000 EXPON = 0 0 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 0.726710 0.000000 EPN_GLG_02_XIU: 7 0.726710 0.000000 GW_02_XIU: 7 0.726710 0.000000 EXACT 0.726710 N = 6 ALPHA = 1.000000 EXPON = 0 0 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 2.000000 0.000000 EPN_GLG_02_XIU: 7 2.000000 0.000000 GW_02_XIU: 7 2.000000 0.000000 EXACT 2.000000 N = 6 ALPHA = 2.000000 EXPON = 0 0 0 0 0 1 Degree = 1 EPN_GLG_01_1: 1 192.000000 0.000000 EPN_GLG_02_XIU: 7 192.000000 0.000000 GW_02_XIU: 7 192.000000 0.000000 EXACT 192.000000 N = 6 ALPHA = -0.500000 EXPON = 1 1 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 7 7.751569 0.000000 GW_02_XIU: 7 7.751569 0.000000 EXACT 7.751569 N = 6 ALPHA = 0.000000 EXPON = 1 1 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 7 1.000000 0.000000 GW_02_XIU: 7 1.000000 0.000000 EXACT 1.000000 N = 6 ALPHA = 0.500000 EXPON = 1 1 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 7 1.090064 0.000000 GW_02_XIU: 7 1.090064 0.000000 EXACT 1.090064 N = 6 ALPHA = 1.000000 EXPON = 1 1 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 7 4.000000 0.000000 GW_02_XIU: 7 4.000000 0.000000 EXACT 4.000000 N = 6 ALPHA = 2.000000 EXPON = 1 1 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 7 576.000000 0.000000 GW_02_XIU: 7 576.000000 0.000000 EXACT 576.000000 N = 6 ALPHA = -0.500000 EXPON = 2 0 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 7 23.254708 0.000000 GW_02_XIU: 7 23.254708 0.000000 EXACT 23.254708 N = 6 ALPHA = 0.000000 EXPON = 2 0 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 7 2.000000 0.000000 GW_02_XIU: 7 2.000000 0.000000 EXACT 2.000000 N = 6 ALPHA = 0.500000 EXPON = 2 0 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 7 1.816774 0.000000 GW_02_XIU: 7 1.816774 0.000000 EXACT 1.816774 N = 6 ALPHA = 1.000000 EXPON = 2 0 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 7 6.000000 0.000000 GW_02_XIU: 7 6.000000 0.000000 EXACT 6.000000 N = 6 ALPHA = 2.000000 EXPON = 2 0 0 0 0 0 Degree = 2 EPN_GLG_02_XIU: 7 768.000000 0.000000 GW_02_XIU: 7 768.000000 0.000000 EXACT 768.000000 TEST208 Demonstrate the use of quadrature rules for the region EPN_LAG, that is, the positive half space [0,+oo)^N, with the weight W(X) = product ( 1 <= I <= N ) exp ( -X(I) ) We use the formulas to integrate various monomials of the form X(1)^E(1) * X(2)^E(2) * ... X(N)^E(N) and compare to the exact integral. The precision of each formula is known, and we only use a formula if its precision indicates it should be able to produce an exact result. N = 1 EXPON = 0 Degree = 0 EPN_LAG_00_1: 1 1.000000 0.000000 EPN_LAG_01_1: 1 1.000000 0.000000 EPN_LAG_02_XIU: 2 1.000000 0.000000 GW_02_XIU: 2 1.000000 0.000000 EXACT 1.000000 N = 1 EXPON = 1 Degree = 1 EPN_LAG_01_1: 1 1.000000 0.000000 EPN_LAG_02_XIU: 2 1.000000 0.000000 GW_02_XIU: 2 1.000000 0.000000 EXACT 1.000000 N = 1 EXPON = 2 Degree = 2 EPN_LAG_02_XIU: 2 2.000000 0.000000 GW_02_XIU: 2 2.000000 0.000000 EXACT 2.000000 N = 2 EXPON = 0 0 Degree = 0 EPN_LAG_00_1: 1 1.000000 0.000000 EPN_LAG_01_1: 1 1.000000 0.000000 EPN_LAG_02_XIU: 3 1.000000 0.000000 GW_02_XIU: 3 1.000000 0.000000 EXACT 1.000000 N = 2 EXPON = 0 1 Degree = 1 EPN_LAG_01_1: 1 1.000000 0.000000 EPN_LAG_02_XIU: 3 1.000000 0.000000 GW_02_XIU: 3 1.000000 0.000000 EXACT 1.000000 N = 2 EXPON = 1 1 Degree = 2 EPN_LAG_02_XIU: 3 1.000000 0.000000 GW_02_XIU: 3 1.000000 0.000000 EXACT 1.000000 N = 2 EXPON = 2 0 Degree = 2 EPN_LAG_02_XIU: 3 2.000000 0.000000 GW_02_XIU: 3 2.000000 0.000000 EXACT 2.000000 N = 3 EXPON = 0 0 0 Degree = 0 EPN_LAG_00_1: 1 1.000000 0.000000 EPN_LAG_01_1: 1 1.000000 0.000000 EPN_LAG_02_XIU: 4 1.000000 0.000000 GW_02_XIU: 4 1.000000 0.000000 EXACT 1.000000 N = 3 EXPON = 0 0 1 Degree = 1 EPN_LAG_01_1: 1 1.000000 0.000000 EPN_LAG_02_XIU: 4 1.000000 0.000000 GW_02_XIU: 4 1.000000 0.000000 EXACT 1.000000 N = 3 EXPON = 1 1 0 Degree = 2 EPN_LAG_02_XIU: 4 1.000000 0.000000 GW_02_XIU: 4 1.000000 0.000000 EXACT 1.000000 N = 3 EXPON = 2 0 0 Degree = 2 EPN_LAG_02_XIU: 4 2.000000 0.000000 GW_02_XIU: 4 2.000000 0.000000 EXACT 2.000000 N = 4 EXPON = 0 0 0 0 Degree = 0 EPN_LAG_00_1: 1 1.000000 0.000000 EPN_LAG_01_1: 1 1.000000 0.000000 EPN_LAG_02_XIU: 5 1.000000 0.000000 GW_02_XIU: 5 1.000000 0.000000 EXACT 1.000000 N = 4 EXPON = 0 0 0 1 Degree = 1 EPN_LAG_01_1: 1 1.000000 0.000000 EPN_LAG_02_XIU: 5 1.000000 0.000000 GW_02_XIU: 5 1.000000 0.000000 EXACT 1.000000 N = 4 EXPON = 1 1 0 0 Degree = 2 EPN_LAG_02_XIU: 5 1.000000 0.000000 GW_02_XIU: 5 1.000000 0.000000 EXACT 1.000000 N = 4 EXPON = 2 0 0 0 Degree = 2 EPN_LAG_02_XIU: 5 2.000000 0.000000 GW_02_XIU: 5 2.000000 0.000000 EXACT 2.000000 N = 5 EXPON = 0 0 0 0 0 Degree = 0 EPN_LAG_00_1: 1 1.000000 0.000000 EPN_LAG_01_1: 1 1.000000 0.000000 EPN_LAG_02_XIU: 6 1.000000 0.000000 GW_02_XIU: 6 1.000000 0.000000 EXACT 1.000000 N = 5 EXPON = 0 0 0 0 1 Degree = 1 EPN_LAG_01_1: 1 1.000000 0.000000 EPN_LAG_02_XIU: 6 1.000000 0.000000 GW_02_XIU: 6 1.000000 0.000000 EXACT 1.000000 N = 5 EXPON = 1 1 0 0 0 Degree = 2 EPN_LAG_02_XIU: 6 1.000000 0.000000 GW_02_XIU: 6 1.000000 0.000000 EXACT 1.000000 N = 5 EXPON = 2 0 0 0 0 Degree = 2 EPN_LAG_02_XIU: 6 2.000000 0.000000 GW_02_XIU: 6 2.000000 0.000000 EXACT 2.000000 N = 6 EXPON = 0 0 0 0 0 0 Degree = 0 EPN_LAG_00_1: 1 1.000000 0.000000 EPN_LAG_01_1: 1 1.000000 0.000000 EPN_LAG_02_XIU: 7 1.000000 0.000000 GW_02_XIU: 7 1.000000 0.000000 EXACT 1.000000 N = 6 EXPON = 0 0 0 0 0 1 Degree = 1 EPN_LAG_01_1: 1 1.000000 0.000000 EPN_LAG_02_XIU: 7 1.000000 0.000000 GW_02_XIU: 7 1.000000 0.000000 EXACT 1.000000 N = 6 EXPON = 1 1 0 0 0 0 Degree = 2 EPN_LAG_02_XIU: 7 1.000000 0.000000 GW_02_XIU: 7 1.000000 0.000000 EXACT 1.000000 N = 6 EXPON = 2 0 0 0 0 0 Degree = 2 EPN_LAG_02_XIU: 7 2.000000 0.000000 GW_02_XIU: 7 2.000000 0.000000 EXACT 2.000000 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 2.000000 and center: CENTER = (0.000000, 0.000000) Rule: 1 2 3 4 Function 1 10.392305 10.392305 10.392305 10.392305 X 0.000000 0.000000 0.000000 0.000000 X^2 0.000000 8.660254 8.660254 8.660254 X^3 0.000000 0.000000 0.000000 0.000000 X^4 0.000000 14.433757 25.980762 14.549227 X^5 0.000000 0.000000 0.000000 0.000000 X^6 0.000000 24.056261 95.262794 29.874412 R 0.000000 13.416408 8.660254 11.572751 SIN(X) 0.000000 0.000000 0.000000 0.000000 EXP(X) 10.392305 15.358263 15.946930 15.371805 1/(1+R) 10.392305 4.536155 7.505553 5.757017 SQRT(R) 0.000000 11.807938 6.123724 9.459640 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 1.000000 0.000000 0.000000 0.000000 2 0.000000 1.000000 0.000000 0.000000 0.000000 4 0.000000 1.000000 0.000000 0.000000 0.000000 6 0.000000 1.000000 0.000000 0.000000 0.000000 8 0.000000 1.000000 0.000000 0.000000 0.000000 10 0.000000 1.000000 0.000000 0.000000 0.000000 12 0.000000 1.000000 0.000000 0.000000 0.000000 14 0.000000 1.000000 0.000000 0.000000 0.000000 16 0.000000 1.000000 0.000000 0.785398 0.039146 2 0.039576 1.000000 0.000000 0.785398 0.039146 4 0.039215 1.000000 0.000000 0.785398 0.039146 6 0.039168 1.000000 0.000000 0.785398 0.039146 8 0.039156 1.000000 0.000000 0.785398 0.039146 10 0.039151 1.000000 0.000000 0.785398 0.039146 12 0.039149 1.000000 0.000000 0.785398 0.039146 14 0.039148 1.000000 0.000000 0.785398 0.039146 16 0.039147 1.000000 0.000000 1.570796 0.285398 2 0.288675 1.000000 0.000000 1.570796 0.285398 4 0.285920 1.000000 0.000000 1.570796 0.285398 6 0.285569 1.000000 0.000000 1.570796 0.285398 8 0.285474 1.000000 0.000000 1.570796 0.285398 10 0.285438 1.000000 0.000000 1.570796 0.285398 12 0.285422 1.000000 0.000000 1.570796 0.285398 14 0.285413 1.000000 0.000000 1.570796 0.285398 16 0.285408 1.000000 0.000000 2.356194 0.824544 2 0.834710 1.000000 0.000000 2.356194 0.824544 4 0.826147 1.000000 0.000000 2.356194 0.824544 6 0.825066 1.000000 0.000000 2.356194 0.824544 8 0.824776 1.000000 0.000000 2.356194 0.824544 10 0.824667 1.000000 0.000000 2.356194 0.824544 12 0.824616 1.000000 0.000000 2.356194 0.824544 14 0.824590 1.000000 0.000000 2.356194 0.824544 16 0.824575 1.000000 0.000000 3.141593 1.570796 2 1.592226 1.000000 0.000000 3.141593 1.570796 4 1.574115 1.000000 0.000000 3.141593 1.570796 6 1.571876 1.000000 0.000000 3.141593 1.570796 8 1.571275 1.000000 0.000000 3.141593 1.570796 10 1.571050 1.000000 0.000000 3.141593 1.570796 12 1.570946 1.000000 0.000000 3.141593 1.570796 14 1.570892 1.000000 0.000000 3.141593 1.570796 16 1.570861 1.000000 0.000000 3.926991 2.317049 2 2.353750 1.000000 0.000000 3.926991 2.317049 4 2.322469 1.000000 0.000000 3.926991 2.317049 6 2.318807 1.000000 0.000000 3.926991 2.317049 8 2.317829 1.000000 0.000000 3.926991 2.317049 10 2.317461 1.000000 0.000000 3.926991 2.317049 12 2.317292 1.000000 0.000000 3.926991 2.317049 14 2.317205 1.000000 0.000000 3.926991 2.317049 16 2.317155 1.000000 0.000000 4.712389 2.856194 2 2.914214 1.000000 0.000000 4.712389 2.856194 4 2.863766 1.000000 0.000000 4.712389 2.856194 6 2.858613 1.000000 0.000000 4.712389 2.856194 8 2.857265 1.000000 0.000000 4.712389 2.856194 10 2.856760 1.000000 0.000000 4.712389 2.856194 12 2.856529 1.000000 0.000000 4.712389 2.856194 14 2.856408 1.000000 0.000000 4.712389 2.856194 16 2.856340 1.000000 0.000000 5.497787 3.102447 2 3.194939 1.000000 0.000000 5.497787 3.102447 4 3.113362 1.000000 0.000000 5.497787 3.102447 6 3.105545 1.000000 0.000000 5.497787 3.102447 8 3.103755 1.000000 0.000000 5.497787 3.102447 10 3.103128 1.000000 0.000000 5.497787 3.102447 12 3.102848 1.000000 0.000000 5.497787 3.102447 14 3.102703 1.000000 0.000000 5.497787 3.102447 16 3.102621 1.000000 0.000000 6.283185 3.141593 2 3.265986 1.000000 0.000000 6.283185 3.141593 4 3.160555 1.000000 0.000000 6.283185 3.141593 6 3.147728 1.000000 0.000000 6.283185 3.141593 8 3.144310 1.000000 0.000000 6.283185 3.141593 10 3.143028 1.000000 0.000000 6.283185 3.141593 12 3.142441 1.000000 0.000000 6.283185 3.141593 14 3.142135 1.000000 0.000000 6.283185 3.141593 16 3.141961 Integrate F(X,Y) = X R Theta1 Theta2 Area Order Integral 1.000000 0.000000 0.000000 0.000000 2 0.000000 1.000000 0.000000 0.000000 0.000000 4 0.000000 1.000000 0.000000 0.000000 0.000000 6 0.000000 1.000000 0.000000 0.000000 0.000000 8 0.000000 1.000000 0.000000 0.000000 0.000000 10 0.000000 1.000000 0.000000 0.000000 0.000000 12 0.000000 1.000000 0.000000 0.000000 0.000000 14 0.000000 1.000000 0.000000 0.000000 0.000000 16 0.000000 1.000000 0.000000 0.785398 0.039146 2 0.034921 1.000000 0.000000 0.785398 0.039146 4 0.034582 1.000000 0.000000 0.785398 0.039146 6 0.034539 1.000000 0.000000 0.785398 0.039146 8 0.034527 1.000000 0.000000 0.785398 0.039146 10 0.034523 1.000000 0.000000 0.785398 0.039146 12 0.034521 1.000000 0.000000 0.785398 0.039146 14 0.034520 1.000000 0.000000 0.785398 0.039146 16 0.034519 1.000000 0.000000 1.570796 0.285398 2 0.169102 1.000000 0.000000 1.570796 0.285398 4 0.167041 1.000000 0.000000 1.570796 0.285398 6 0.166788 1.000000 0.000000 1.570796 0.285398 8 0.166720 1.000000 0.000000 1.570796 0.285398 10 0.166695 1.000000 0.000000 1.570796 0.285398 12 0.166683 1.000000 0.000000 1.570796 0.285398 14 0.166677 1.000000 0.000000 1.570796 0.285398 16 0.166674 1.000000 0.000000 2.356194 0.824544 2 0.205492 1.000000 0.000000 2.356194 0.824544 4 0.201814 1.000000 0.000000 2.356194 0.824544 6 0.201387 1.000000 0.000000 2.356194 0.824544 8 0.201274 1.000000 0.000000 2.356194 0.824544 10 0.201232 1.000000 0.000000 2.356194 0.824544 12 0.201212 1.000000 0.000000 2.356194 0.824544 14 0.201202 1.000000 0.000000 2.356194 0.824544 16 0.201197 1.000000 0.000000 3.141593 1.570796 2 0.000000 1.000000 0.000000 3.141593 1.570796 4 0.000000 1.000000 0.000000 3.141593 1.570796 6 -0.000000 1.000000 0.000000 3.141593 1.570796 8 0.000000 1.000000 0.000000 3.141593 1.570796 10 -0.000000 1.000000 0.000000 3.141593 1.570796 12 0.000000 1.000000 0.000000 3.141593 1.570796 14 -0.000000 1.000000 0.000000 3.141593 1.570796 16 0.000000 1.000000 0.000000 3.926991 2.317049 2 -0.216881 1.000000 0.000000 3.926991 2.317049 4 -0.203381 1.000000 0.000000 3.926991 2.317049 6 -0.201876 1.000000 0.000000 3.926991 2.317049 8 -0.201488 1.000000 0.000000 3.926991 2.317049 10 -0.201344 1.000000 0.000000 3.926991 2.317049 12 -0.201278 1.000000 0.000000 3.926991 2.317049 14 -0.201244 1.000000 0.000000 3.926991 2.317049 16 -0.201225 1.000000 0.000000 4.712389 2.856194 2 -0.201184 1.000000 0.000000 4.712389 2.856194 4 -0.172233 1.000000 0.000000 4.712389 2.856194 6 -0.168431 1.000000 0.000000 4.712389 2.856194 8 -0.167438 1.000000 0.000000 4.712389 2.856194 10 -0.167072 1.000000 0.000000 4.712389 2.856194 12 -0.166905 1.000000 0.000000 4.712389 2.856194 14 -0.166819 1.000000 0.000000 4.712389 2.856194 16 -0.166770 1.000000 0.000000 5.497787 3.102447 2 -0.062085 1.000000 0.000000 5.497787 3.102447 4 -0.041397 1.000000 0.000000 5.497787 3.102447 6 -0.037089 1.000000 0.000000 5.497787 3.102447 8 -0.035702 1.000000 0.000000 5.497787 3.102447 10 -0.035148 1.000000 0.000000 5.497787 3.102447 12 -0.034891 1.000000 0.000000 5.497787 3.102447 14 -0.034756 1.000000 0.000000 5.497787 3.102447 16 -0.034679 1.000000 0.000000 6.283185 3.141593 2 -0.000000 1.000000 0.000000 6.283185 3.141593 4 0.000000 1.000000 0.000000 6.283185 3.141593 6 -0.000000 1.000000 0.000000 6.283185 3.141593 8 0.000000 1.000000 0.000000 6.283185 3.141593 10 -0.000000 1.000000 0.000000 6.283185 3.141593 12 0.000000 1.000000 0.000000 6.283185 3.141593 14 -0.000000 1.000000 0.000000 6.283185 3.141593 16 -0.000000 Integrate F(X,Y) = R R Theta1 Theta2 Area Order Integral 1.000000 0.000000 0.000000 0.000000 2 0.000000 1.000000 0.000000 0.000000 0.000000 4 0.000000 1.000000 0.000000 0.000000 0.000000 6 0.000000 1.000000 0.000000 0.000000 0.000000 8 0.000000 1.000000 0.000000 0.000000 0.000000 10 0.000000 1.000000 0.000000 0.000000 0.000000 12 0.000000 1.000000 0.000000 0.000000 0.000000 14 0.000000 1.000000 0.000000 0.000000 0.000000 16 0.000000 1.000000 0.000000 0.785398 0.039146 2 0.038402 1.000000 0.000000 0.785398 0.039146 4 0.038032 1.000000 0.000000 0.785398 0.039146 6 0.037986 1.000000 0.000000 0.785398 0.039146 8 0.037973 1.000000 0.000000 0.785398 0.039146 10 0.037968 1.000000 0.000000 0.785398 0.039146 12 0.037966 1.000000 0.000000 0.785398 0.039146 14 0.037965 1.000000 0.000000 0.785398 0.039146 16 0.037964 1.000000 0.000000 1.570796 0.285398 2 0.257008 1.000000 0.000000 1.570796 0.285398 4 0.253591 1.000000 0.000000 1.570796 0.285398 6 0.253232 1.000000 0.000000 1.570796 0.285398 8 0.253137 1.000000 0.000000 1.570796 0.285398 10 0.253101 1.000000 0.000000 1.570796 0.285398 12 0.253085 1.000000 0.000000 1.570796 0.285398 14 0.253076 1.000000 0.000000 1.570796 0.285398 16 0.253072 1.000000 0.000000 2.356194 0.824544 2 0.656948 1.000000 0.000000 2.356194 0.824544 4 0.639356 1.000000 0.000000 2.356194 0.824544 6 0.637932 1.000000 0.000000 2.356194 0.824544 8 0.637615 1.000000 0.000000 2.356194 0.824544 10 0.637503 1.000000 0.000000 2.356194 0.824544 12 0.637452 1.000000 0.000000 2.356194 0.824544 14 0.637426 1.000000 0.000000 2.356194 0.824544 16 0.637411 1.000000 0.000000 3.141593 1.570796 2 1.120706 1.000000 0.000000 3.141593 1.570796 4 1.058565 1.000000 0.000000 3.141593 1.570796 6 1.050890 1.000000 0.000000 3.141593 1.570796 8 1.048838 1.000000 0.000000 3.141593 1.570796 10 1.048065 1.000000 0.000000 3.141593 1.570796 12 1.047711 1.000000 0.000000 3.141593 1.570796 14 1.047526 1.000000 0.000000 3.141593 1.570796 16 1.047420 1.000000 0.000000 3.926991 2.317049 2 1.599399 1.000000 0.000000 3.926991 2.317049 4 1.477219 1.000000 0.000000 3.926991 2.317049 6 1.464100 1.000000 0.000000 3.926991 2.317049 8 1.460168 1.000000 0.000000 3.926991 2.317049 10 1.458613 1.000000 0.000000 3.926991 2.317049 12 1.457991 1.000000 0.000000 3.926991 2.317049 14 1.457652 1.000000 0.000000 3.926991 2.317049 16 1.457430 1.000000 0.000000 4.712389 2.856194 2 2.051285 1.000000 0.000000 4.712389 2.856194 4 1.866262 1.000000 0.000000 4.712389 2.856194 6 1.848476 1.000000 0.000000 4.712389 2.856194 8 1.844273 1.000000 0.000000 4.712389 2.856194 10 1.842892 1.000000 0.000000 4.712389 2.856194 12 1.842326 1.000000 0.000000 4.712389 2.856194 14 1.842033 1.000000 0.000000 4.712389 2.856194 16 1.841845 1.000000 0.000000 5.497787 3.102447 2 2.344954 1.000000 0.000000 5.497787 3.102447 4 2.090841 1.000000 0.000000 5.497787 3.102447 6 2.066759 1.000000 0.000000 5.497787 3.102447 8 2.060833 1.000000 0.000000 5.497787 3.102447 10 2.058689 1.000000 0.000000 5.497787 3.102447 12 2.057728 1.000000 0.000000 5.497787 3.102447 14 2.057235 1.000000 0.000000 5.497787 3.102447 16 2.056958 1.000000 0.000000 6.283185 3.141593 2 2.434322 1.000000 0.000000 6.283185 3.141593 4 2.139030 1.000000 0.000000 6.283185 3.141593 6 2.108515 1.000000 0.000000 6.283185 3.141593 8 2.100602 1.000000 0.000000 6.283185 3.141593 10 2.097662 1.000000 0.000000 6.283185 3.141593 12 2.096323 1.000000 0.000000 6.283185 3.141593 14 2.095627 1.000000 0.000000 6.283185 3.141593 16 2.095229 TEST22 OCTAHEDRON_UNIT_ND approximates integrals in a unit octahedron in N dimensions. F(X) N = 1 N = 2 N = 3 1 2.000000 2.000000 1.333333 X 0.000000 0.000000 0.000000 X^2 0.666667 0.333333 0.133333 X^3 0.000000 0.000000 0.000000 X^4 0.222222 0.111111 0.040000 X^5 0.000000 0.000000 0.000000 X^6 0.074074 0.037037 0.012000 R 1.154701 1.154701 0.730297 SIN(X) 0.000000 0.000000 0.000000 EXP(X) 2.342696 2.171348 1.401683 1/(1+R) 1.267949 1.267949 0.861481 SQRT(R) 1.519671 1.519671 0.986777 TEST23 PARALLELIPIPED_VOLUME_ND computes the volume of a parallelipiped in N dimensions. Spatial dimension N = 2 Parallelipiped vertices: 0.000000 1.000000 0.000000 0.000000 0.000000 1.000000 Volume is 1.000000 Spatial dimension N = 3 Parallelipiped vertices: 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 Volume is 1.000000 Spatial dimension N = 4 Parallelipiped vertices: 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000 Volume is 1.000000 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*X POLYGON_XY_2D integrates X*Y POLYGON_YY_2D integrates Y*Y F(X,Y) Integral 1 1.000000 X 0.500000 Y 0.500000 X*X 0.333333 X*Y 0.250000 Y*Y 0.333333 TEST25 For the unit pyramid, we approximate integrals with: PYRAMID_UNIT_O01_3D, a 1 point rule. PYRAMID_UNIT_O05_3D, a 5 point rule. PYRAMID_UNIT_O06_3D, a 6 point rule. PYRAMID_UNIT_O08_3D, an 8 point rule. PYRAMID_UNIT_O08b_3D, an 8 point rule. PYRAMID_UNIT_O09_3D, a 9 point rule. PYRAMID_UNIT_O13_3D, a 13 point rule. PYRAMID_UNIT_O18_3D, a 18 point rule. PYRAMID_UNIT_O27_3D, a 27 point rule. PYRAMID_UNIT_O48_3D, a 48 point rule. PYRAMID_UNIT_VOLUME_3D computes the volume of a unit pyramid. Volume = 1.333333 Order 1 X Y Z X*X 1 1.333333 0.000000 0.000000 0.333333 0.000000 5 1.333333 0.000000 0.000000 0.333333 0.266667 6 1.333333 0.000000 0.000000 0.333333 0.266667 8 1.333333 0.000000 0.000000 0.333333 0.266667 8 1.333333 0.000000 0.000000 0.333333 0.266667 9 1.333333 0.000000 0.000000 0.333333 0.266667 13 1.333333 0.000000 0.000000 0.333333 0.266667 18 1.333333 0.000000 -0.000000 0.333333 0.266667 27 1.333333 0.000000 -0.000000 0.333333 0.266667 48 1.333333 0.000000 0.000000 0.333333 0.266667 Order X*Y X*Z Y*Y Y*Z Z*Z 1 0.000000 0.000000 0.000000 0.000000 0.083333 5 0.000000 0.000000 0.266667 0.000000 0.133333 6 0.000000 0.000000 0.266667 0.000000 0.133333 8 0.000000 0.000000 0.266667 0.000000 0.133333 8 0.000000 0.000000 0.266667 0.000000 0.133333 9 0.000000 0.000000 0.266667 0.000000 0.133333 13 0.000000 0.000000 0.266667 0.000000 0.133333 18 0.000000 0.000000 0.266667 0.000000 0.133333 27 0.000000 0.000000 0.266667 0.000000 0.133333 48 0.000000 0.000000 0.266667 0.000000 0.133333 Order X^3 X*Y*Z Z*Z*Z X^4 X^2 Z^2 1 0.000000 0.000000 0.020833 0.000000 0.000000 5 0.000000 0.000000 0.076667 0.063210 0.007407 6 0.000000 0.000000 0.077315 0.063492 0.007407 8 0.000000 0.000000 0.066667 0.063210 0.011852 8 0.000000 0.000000 0.064762 0.063210 0.012698 9 0.000000 0.000000 0.066931 0.063492 0.012698 13 0.000000 0.000000 0.066650 0.066467 0.012698 18 0.000000 0.000000 0.066667 0.113778 0.011852 27 0.000000 0.000000 0.066667 0.114286 0.012698 48 0.000000 0.000000 0.066667 0.114286 0.012698 Order Z^4 X^5 X^6 R SIN(X) 1 0.005208 0.000000 0.000000 0.333333 0.000000 5 0.050889 0.000000 0.014983 0.942800 0.000000 6 0.052315 0.000000 0.015117 0.941763 0.000000 8 0.035556 0.000000 0.015861 0.941900 0.000000 8 0.033088 0.000000 0.016028 0.941565 0.000000 9 0.039204 0.000000 0.016440 0.939009 0.000000 13 0.037967 0.000000 0.017432 0.936256 0.000000 18 0.035556 0.000000 0.051390 0.882331 0.000000 27 0.038095 0.000000 0.053316 0.880646 0.000000 48 0.038095 0.000000 0.063492 0.906432 0.000000 Order EXP(X) 1/(1+R) SQRT(R) 1 1.333333 1.293523 0.666667 5 1.469321 1.088665 1.121187 6 1.469333 1.089038 1.120238 8 1.469323 1.089005 1.120379 8 1.469323 1.089132 1.120078 9 1.469335 1.090111 1.117794 13 1.469460 1.090942 1.115132 18 1.471479 1.103581 1.054689 27 1.471503 1.104285 1.051658 48 1.471518 1.102448 1.088539 TEST255 For the unit pyramid, PYRAMID_UNIT_MONOMIAL_3D returns the exact value of the integral of X^ALPHA Y^BETA Z^GAMMA Volume = 1.333333 ALPHA BETA GAMMA INTEGRAL 0 0 0 1.333333 0 0 1 0.333333 0 0 2 0.133333 0 0 3 0.066667 0 0 4 0.038095 0 1 0 0.000000 0 1 1 0.000000 0 1 2 0.000000 0 1 3 0.000000 0 2 0 0.266667 0 2 1 0.044444 0 2 2 0.012698 0 3 0 0.000000 0 3 1 0.000000 0 4 0 0.114286 1 0 0 0.000000 1 0 1 0.000000 1 0 2 0.000000 1 0 3 0.000000 1 1 0 0.000000 1 1 1 0.000000 1 1 2 0.000000 1 2 0 0.000000 1 2 1 0.000000 1 3 0 0.000000 2 0 0 0.266667 2 0 1 0.044444 2 0 2 0.012698 2 1 0 0.000000 2 1 1 0.000000 2 2 0 0.063492 3 0 0 0.000000 3 0 1 0.000000 3 1 0 0.000000 4 0 0 0.114286 TEST26 QMULT_1D approximates an integral on a one-dimensional interval. We use the interval: A = -1.000000 B = 1.000000 F(X) QMULT_1D 1 2.000000 X 0.000000 X^2 0.666667 X^3 0.400000 X^4 0.400000 X^5 0.000000 X^6 0.285714 R 1.003031 SIN(X) -0.000000 EXP(X) 2.350402 1/(1+R) 1.383282 SQRT(R) 1.343468 TEST27 SIMPLEX_ND approximates integrals inside an arbitrary simplex in ND. Spatial dimension N = 2 Simplex vertices: 0.0000001.0000000.000000 0.0000000.0000001.000000 F(X) SIMPLEX_ND 1 0.500000 X 0.166667 X^2 0.083333 X^3 0.050926 X^4 0.033179 X^5 0.021991 X^6 0.014639 R 0.268345 SIN(X) 0.158360 EXP(X) 0.718409 1/(1+R) 0.332444 SQRT(R) 0.357237 Spatial dimension N = 3 Simplex vertices: 0.0000001.0000000.0000000.000000 0.0000000.0000001.0000000.000000 0.0000000.0000000.0000001.000000 F(X) SIMPLEX_ND 1 0.166667 X 0.041667 X^2 0.016667 X^3 0.008689 X^4 0.004939 X^5 0.002871 X^6 0.001678 R 0.087120 SIN(X) 0.040242 EXP(X) 0.218347 1/(1+R) 0.110915 SQRT(R) 0.118586 Spatial dimension N = 4 Simplex vertices: 0.0000001.0000000.0000000.0000000.000000 0.0000000.0000001.0000000.0000000.000000 0.0000000.0000000.0000001.0000000.000000 0.0000000.0000000.0000000.0000001.000000 F(X) SIMPLEX_ND 1 0.041667 X 0.008333 X^2 0.002778 X^3 0.001272 X^4 0.000647 X^5 0.000338 X^6 0.000178 R 0.020809 SIN(X) 0.008124 EXP(X) 0.051631 1/(1+R) 0.028036 SQRT(R) 0.029112 TEST28 SIMPLEX_VOLUME_ND computes the volume of a simplex in N dimensions. Spatial dimension N = 2 Simplex vertices: 0.000000 1.000000 0.000000 0.000000 0.000000 1.000000 Volume is 0.500000 Spatial dimension N = 3 Simplex vertices: 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 Volume is 0.166667 Spatial dimension N = 4 Simplex vertices: 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000 Volume is 0.041667 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 0.500000 0.500000 0.500000 0.500000 0.500000 3 0.166667 0.166667 0.166667 0.166667 0.166667 4 0.041667 0.041667 0.041667 0.041667 0.041667 5 0.008333 0.008333 0.008333 0.008333 0.008333 6 0.001389 0.001389 0.001389 0.001389 0.001389 7 0.000198 0.000198 0.000198 0.000198 0.000198 8 0.000025 0.000025 0.000025 0.000025 0.000025 9 0.000003 0.000003 0.000003 0.000003 0.000003 10 0.000000 0.000000 0.000000 0.000000 0.000000 11 0.000000 0.000000 0.000000 0.000000 0.000000 12 0.000000 0.000000 0.000000 0.000000 0.000000 13 0.000000 0.000000 0.000000 0.000000 0.000000 14 0.000000 0.000000 0.000000 0.000000 0.000000 15 0.000000 0.000000 0.000000 0.000000 0.000000 16 0.000000 0.000000 0.000000 0.000000 0.000000 Check the integral of X N Volume #1 #3 #5 #5.2 2 0.500000 0.250000 0.166667 0.166667 0.166667 3 0.166667 0.055556 0.041667 0.041667 0.041667 4 0.041667 0.010417 0.008333 0.008333 0.008333 5 0.008333 0.001667 0.001389 0.001389 0.001389 6 0.001389 0.000231 0.000198 0.000198 0.000198 7 0.000198 0.000028 0.000025 0.000025 0.000025 8 0.000025 0.000003 0.000003 0.000003 0.000003 9 0.000003 0.000000 0.000000 0.000000 0.000000 10 0.000000 0.000000 0.000000 0.000000 0.000000 11 0.000000 0.000000 0.000000 0.000000 0.000000 12 0.000000 0.000000 0.000000 0.000000 0.000000 13 0.000000 0.000000 0.000000 0.000000 0.000000 14 0.000000 0.000000 0.000000 0.000000 0.000000 15 0.000000 0.000000 0.000000 0.000000 0.000000 16 0.000000 0.000000 0.000000 0.000000 0.000000 Check the integral of X^2 N Volume #1 #3 #5 #5.2 2 0.500000 0.125000 0.083333 0.083333 0.083333 3 0.166667 0.018519 0.016667 0.016667 0.016667 4 0.041667 0.002604 0.002778 0.002778 0.002778 5 0.008333 0.000333 0.000397 0.000397 0.000397 6 0.001389 0.000039 0.000050 0.000050 0.000050 7 0.000198 0.000004 0.000006 0.000006 0.000006 8 0.000025 0.000000 0.000001 0.000001 0.000001 9 0.000003 0.000000 0.000000 0.000000 0.000000 10 0.000000 0.000000 0.000000 0.000000 0.000000 11 0.000000 0.000000 0.000000 0.000000 0.000000 12 0.000000 0.000000 0.000000 0.000000 0.000000 13 0.000000 0.000000 0.000000 0.000000 0.000000 14 0.000000 0.000000 0.000000 0.000000 0.000000 15 0.000000 0.000000 0.000000 0.000000 0.000000 16 0.000000 0.000000 0.000000 0.000000 0.000000 Check the integral of X^3 N Volume #1 #3 #5 #5.2 2 0.500000 0.062500 0.050000 0.050000 0.050000 3 0.166667 0.006173 0.008333 0.008333 0.008333 4 0.041667 0.000651 0.001190 0.001190 0.001190 5 0.008333 0.000067 0.000149 0.000149 0.000149 6 0.001389 0.000006 0.000017 0.000017 0.000017 7 0.000198 0.000001 0.000002 0.000002 0.000002 8 0.000025 0.000000 0.000000 0.000000 0.000000 9 0.000003 0.000000 0.000000 0.000000 0.000000 10 0.000000 0.000000 0.000000 0.000000 0.000000 11 0.000000 0.000000 0.000000 0.000000 0.000000 12 0.000000 0.000000 0.000000 0.000000 0.000000 13 0.000000 0.000000 0.000000 0.000000 0.000000 14 0.000000 0.000000 0.000000 0.000000 0.000000 15 0.000000 0.000000 0.000000 0.000000 0.000000 16 0.000000 0.000000 0.000000 0.000000 0.000000 Check the integral of X^4 N Volume #1 #3 #5 #5.2 2 0.500000 0.031250 0.031111 0.033333 0.033333 3 0.166667 0.002058 0.004340 0.004762 0.004762 4 0.041667 0.000163 0.000533 0.000595 0.000595 5 0.008333 0.000013 0.000058 0.000066 0.000066 6 0.001389 0.000001 0.000006 0.000007 0.000007 7 0.000198 0.000000 0.000001 0.000001 0.000001 8 0.000025 0.000000 0.000000 0.000000 0.000000 9 0.000003 0.000000 0.000000 0.000000 0.000000 10 0.000000 0.000000 0.000000 0.000000 0.000000 11 0.000000 0.000000 0.000000 0.000000 0.000000 12 0.000000 0.000000 0.000000 0.000000 0.000000 13 0.000000 0.000000 0.000000 0.000000 0.000000 14 0.000000 0.000000 0.000000 0.000000 0.000000 15 0.000000 0.000000 0.000000 0.000000 0.000000 16 0.000000 0.000000 0.000000 0.000000 0.000000 Check the integral of X^5 N Volume #1 #3 #5 #5.2 2 0.500000 0.015625 0.019259 0.023810 0.023810 3 0.166667 0.000686 0.002242 0.002976 0.002976 4 0.041667 0.000041 0.000236 0.000331 0.000331 5 0.008333 0.000003 0.000023 0.000033 0.000033 6 0.001389 0.000000 0.000002 0.000003 0.000003 7 0.000198 0.000000 0.000000 0.000000 0.000000 8 0.000025 0.000000 0.000000 0.000000 0.000000 9 0.000003 0.000000 0.000000 0.000000 0.000000 10 0.000000 0.000000 0.000000 0.000000 0.000000 11 0.000000 0.000000 0.000000 0.000000 0.000000 12 0.000000 0.000000 0.000000 0.000000 0.000000 13 0.000000 0.000000 0.000000 0.000000 0.000000 14 0.000000 0.000000 0.000000 0.000000 0.000000 15 0.000000 0.000000 0.000000 0.000000 0.000000 16 0.000000 0.000000 0.000000 0.000000 0.000000 TEST30 For integrals on the unit sphere in 3D: SPHERE_UNIT_07_3D uses a formula of degree 7. SPHERE_UNIT_11_3D uses a formula of degree 11. SPHERE_UNIT_14_3D uses a formula of degree 14. SPHERE_UNIT_15_3D uses a formula of degree 15. Unit sphere area = 12.566371 F(X) S3S07 S3S11 S3S14 S3S15 1 12.566371 12.566371 12.566371 12.566371 X -0.000000 0.000000 0.000000 -0.000000 Y -0.000000 0.000000 0.000000 0.000000 Z 0.000000 -0.000000 0.000000 0.000000 X*X 4.188790 4.188790 4.188790 4.188790 X*Y 0.000000 -0.000000 -0.000000 -0.000000 X*Z -0.000000 0.000000 0.000000 0.000000 Y*Y 4.188790 4.188790 4.188790 4.188790 Y*Z -0.000000 -0.000000 -0.000000 -0.000000 Z*Z 4.188790 4.188790 4.188790 4.188790 X^3 -0.000000 0.000000 0.000000 -0.000000 X*Y*Z -0.000000 0.000000 -0.000000 0.000000 Z*Z*Z 0.000000 0.000000 -0.000000 -0.000000 X^4 2.513274 2.513274 2.513274 2.513274 X^2 Z^2 0.837758 0.837758 0.837758 0.837758 Z^4 2.513274 2.513274 2.513274 2.513274 X^5 -0.000000 0.000000 0.000000 -0.000000 X^6 1.795196 1.795196 1.795196 1.795196 R 12.566371 12.566371 12.566371 12.566371 SIN(X) 0.000000 -0.000000 0.000000 -0.000000 EXP(X) 14.768012 14.768014 14.768014 14.768014 1/(1+R) 8.885766 8.885766 8.885766 8.885766 SQRT(R) 12.566371 12.566371 12.566371 12.566371 TEST31 For integrals on the unit sphere in ND: SPHERE_UNIT_03_ND uses a formula of degree 3; SPHERE_UNIT_04_ND uses a formula of degree 4; SPHERE_UNIT_05_ND uses a formula of degree 5. SPHERE_UNIT_07_1_ND uses a formula of degree 7. SPHERE_UNIT_07_2_ND uses a formula of degree 7. SPHERE_UNIT_11_ND uses a formula of degree 11. Spatial dimension N = 3 Unit sphere area = 12.566371 Rule: #3 #4 #5 #7.1 #7.2 #11 Function 1 12.566371 12.566371 12.566371 12.566371 12.566371 12.566371 X 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 X^2 4.188790 4.188790 4.188790 4.188790 4.188790 4.188790 X^3 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 X^4 4.188790 2.513274 2.513274 2.513274 2.513274 2.513274 X^5 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 X^6 4.188790 1.675516 1.954769 1.795196 1.795196 1.795196 R 12.566371 12.566371 12.566371 12.566371 12.566371 12.566371 SIN(X) 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 EXP(X) 14.841221 14.767844 14.768245 14.768016 14.768011 14.768014 1/(1+R) 6.283185 6.283185 6.283185 6.283185 6.283185 6.283185 SQRT(R) 12.566371 12.566371 12.566371 12.566371 12.566371 12.566371 Spatial dimension N = 4 Unit sphere area = 19.739209 Rule: #3 #4 #5 #7.1 #7.2 #11 Function 1 19.739209 19.739209 19.739209 19.739209 19.739209 19.739209 X 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 X^2 4.934802 4.934802 4.934802 4.934802 4.934802 4.934802 X^3 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 X^4 4.934802 2.467401 2.467401 2.467401 2.467401 2.467401 X^5 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 X^6 4.934802 1.233701 1.850551 1.542126 1.542126 1.542126 R 19.739209 19.739209 19.739209 19.739209 19.739209 19.739209 SIN(X) 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 EXP(X) 22.419204 22.311147 22.312031 22.311589 22.311585 22.311587 1/(1+R) 9.869604 9.869604 9.869604 9.869604 9.869604 9.869604 SQRT(R) 19.739209 19.739209 19.739209 19.739209 19.739209 19.739209 Spatial dimension N = 5 Unit sphere area = 26.318945 Rule: #3 #4 #5 #7.1 #7.2 #11 Function 1 26.318945 26.318945 26.318945 26.318945 26.318945 26.318945 X 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 X^2 5.263789 5.263789 5.263789 5.263789 5.263789 5.263789 X^3 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 X^4 5.263789 2.255910 2.255910 2.255910 2.255910 2.255910 X^5 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 X^6 5.263789 0.751970 1.654334 1.253283 1.253283 1.253283 R 26.318945 26.318945 26.318945 26.318945 26.318945 26.318945 SIN(X) 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 EXP(X) 29.177607 29.045880 29.047172 29.046598 29.046594 29.046596 1/(1+R) 13.159473 13.159473 13.159473 13.159473 13.159473 13.159473 SQRT(R) 26.318945 26.318945 26.318945 26.318945 26.318945 26.318945 Spatial dimension N = 6 Unit sphere area = 31.006277 Rule: #3 #4 #5 #7.1 #7.2 #11 Function 1 31.006277 31.006277 31.006277 31.006277 31.006277 31.006277 X 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 X^2 5.167713 5.167713 5.167713 5.167713 5.167713 5.167713 X^3 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 X^4 5.167713 1.937892 1.937892 1.937892 1.937892 1.937892 X^5 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 X^6 5.167713 0.322982 1.399589 0.968946 0.968946 0.968946 R 31.006277 31.006277 31.006277 31.006277 31.006277 31.006277 SIN(X) 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 EXP(X) 33.812761 33.671315 33.672855 33.672239 33.672237 33.672238 1/(1+R) 15.503138 15.503138 15.503138 15.503138 15.503138 15.503138 SQRT(R) 31.006277 31.006277 31.006277 31.006277 31.006277 31.006277 Spatial dimension N = 7 Unit sphere area = 33.073362 Rule: #3 #4 #5 #7.1 #7.2 #11 Function 1 33.073362 33.073362 33.073362 33.073362 33.073362 33.073362 X 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 X^2 4.724766 4.724766 4.724766 4.724766 4.724766 4.724766 X^3 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 X^4 4.724766 1.574922 1.574922 1.574922 1.574922 1.574922 X^5 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 X^6 4.724766 -0.000000 1.124944 0.715874 0.715874 0.715874 R 33.073362 33.073362 33.073362 33.073362 33.073362 33.073362 SIN(X) 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 EXP(X) 35.639291 35.501347 35.502956 35.502371 35.502369 35.502370 1/(1+R) 16.536681 16.536681 16.536681 16.536681 16.536681 16.536681 SQRT(R) 33.073362 33.073362 33.073362 33.073362 33.073362 33.073362 Spatial dimension N = 8 Unit sphere area = 32.469697 Rule: #3 #4 #5 #7.1 #7.2 #11 Function 1 32.469697 32.469697 32.469697 32.469697 32.469697 32.469697 X 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 X^2 4.058712 4.058712 4.058712 4.058712 4.058712 4.058712 X^3 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 X^4 4.058712 1.217614 1.217614 1.217614 1.217614 1.217614 X^5 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 X^6 4.058712 -0.202936 0.862476 0.507339 0.507339 0.507339 R 32.469697 32.469697 32.469697 32.469697 32.469697 32.469697 SIN(X) 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 EXP(X) 34.673905 34.549482 34.551005 34.550498 34.550497 34.550498 1/(1+R) 16.234849 16.234849 16.234849 16.234849 16.234849 16.234849 SQRT(R) 32.469697 32.469697 32.469697 32.469697 32.469697 32.469697 Spatial dimension N = 9 Unit sphere area = 29.686580 Rule: #3 #4 #5 #7.1 #7.2 #11 Function 1 29.686580 29.686580 29.686580 29.686580 29.686580 29.686580 X 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 X^2 3.298509 3.298509 3.298509 3.298509 3.298509 3.298509 X^3 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 X^4 3.298509 0.899593 0.899593 0.899593 0.899593 0.899593 X^5 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 X^6 3.298509 -0.299864 0.633047 0.345997 0.345997 0.345997 R 29.686580 29.686580 29.686580 29.686580 29.686580 29.686580 SIN(X) 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 EXP(X) 31.477936 31.372879 31.374212 31.373802 31.373802 31.373802 1/(1+R) 14.843290 14.843290 14.843290 14.843290 14.843290 14.843290 SQRT(R) 29.686580 29.686580 29.686580 29.686580 29.686580 29.686580 Spatial dimension N = 10 Unit sphere area = 25.501640 Rule: #3 #4 #5 #7.1 #7.2 #11 Function 1 25.501640 25.501640 25.501640 25.501640 25.501640 25.501640 X 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 X^2 2.550164 2.550164 2.550164 2.550164 2.550164 2.550164 X^3 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 X^4 2.550164 0.637541 0.637541 0.637541 0.637541 0.637541 X^5 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 X^6 2.550164 -0.318771 0.446279 0.227693 0.227693 0.227693 R 25.501640 25.501640 25.501640 25.501640 25.501640 25.501640 SIN(X) 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 EXP(X) 26.886585 26.802824 26.803917 26.803605 26.803605 26.803605 1/(1+R) 12.750820 12.750820 12.750820 12.750820 12.750820 12.750820 SQRT(R) 25.501640 25.501640 25.501640 25.501640 25.501640 25.501640 TEST32 For integrals on a sphere in ND: SPHERE_05_ND uses a formula of degree 5. SPHERE_07_1_ND uses a formula of degree 7. Spatial dimension N = 2 Sphere center = 1.000000 1.000000 Sphere radius = 2.000000 Sphere area = 12.566371 Rule: #5 #7.1 Function 1 12.566371 12.566371 X 12.566371 12.566371 X^2 37.699112 37.699112 X^3 87.964594 87.964594 X^4 238.761042 238.761042 X^5 640.884901 640.884901 X^6 1771.858257 1771.858257 R 28.355955 28.355955 SIN(X) 2.367943 2.367943 EXP(X) 77.870103 77.870103 1/(1+R) 4.313206 4.313206 SQRT(R) 18.344181 18.344181 Spatial dimension N = 3 Sphere center = 1.000000 1.000000 1.000000 Sphere radius = 2.000000 Sphere area = 50.265482 Rule: #5 #7.1 Function 1 50.265482 50.265482 X 50.265482 50.265482 X^2 117.286126 117.286126 X^3 251.327412 251.327412 X^4 613.238886 613.238886 X^5 1524.719635 1524.719635 X^6 3968.739093 3927.888415 R 124.836332 125.495412 SIN(X) 19.189950 19.231787 EXP(X) 247.961123 247.785503 1/(1+R) 16.142768 15.707775 SQRT(R) 77.172607 77.792931 Spatial dimension N = 4 Sphere center = 1.000000 1.000000 1.000000 1.000000 Sphere radius = 2.000000 Sphere area = 157.913670 Rule: #5 #7.1 Function 1 157.913670 157.913670 X 157.913670 157.913670 X^2 315.827341 315.827341 X^3 631.654682 631.654682 X^4 1421.223034 1421.223034 X^5 3316.187079 3316.187079 X^6 8211.510862 8053.597191 R 425.599553 427.623680 SIN(X) 76.475316 76.638042 EXP(X) 683.473764 682.798772 1/(1+R) 47.544936 46.088021 SQRT(R) 251.965363 254.745231 TEST322 SPHERE_CAP_AREA_3D computes the volume of a 3D spherical cap, defined by a plane that cuts the sphere to a thickness of H units. SPHERE_CAP_AREA_ND computes the volume of an ND spherical cap, defined by a plane that cuts the sphere to a thickness of H units. Area of the total sphere in 3D = 12.566371 R H Cap Cap area_3d area_nd 1.000000 0.000000 0.000000 0.000000 1.000000 0.166667 1.047198 1.047198 1.000000 0.333333 2.094395 2.094395 1.000000 0.500000 3.141593 3.141593 1.000000 0.666667 4.188790 4.188790 1.000000 0.833333 5.235988 5.235988 1.000000 1.000000 6.283185 6.283185 1.000000 1.166667 7.330383 7.330383 1.000000 1.333333 8.377580 8.377580 1.000000 1.500000 9.424778 9.424778 1.000000 1.666667 10.471976 10.471976 1.000000 1.833333 11.519173 11.519173 1.000000 2.000000 12.566371 12.566371 1.000000 2.166667 12.566371 12.566371 TEST324 SPHERE_CAP_VOLUME_2D computes the volume (area) of a spherical cap, defined by a plane that cuts the sphere to a thickness of H units. SPHERE_CAP_VOLUME_ND does the same operation, but in N dimensions. Using a radius R = 1.000000 Volume of the total sphere in 2D = 3.141593 H Cap Cap vol_2d vol_nd 0.000000 0.000000 0.000000 0.166667 0.125043 0.125043 0.333333 0.344165 0.344165 0.500000 0.614185 0.614185 0.666667 0.916690 0.916690 0.833333 1.239013 1.239013 1.000000 1.570796 1.570796 1.166667 1.902580 1.902580 1.333333 2.224903 2.224903 1.500000 2.527408 2.527408 1.666667 2.797428 2.797428 1.833333 3.016549 3.016549 2.000000 3.141593 3.141593 2.166667 3.141593 3.141593 TEST326 SPHERE_CAP_VOLUME_3D computes the volume of a spherical cap, defined by a plane that cuts the sphere to a thickness of H units. SPHERE_CAP_VOLUME_ND does the same operation, but in N dimensions. Using a radius R = 1.000000 Volume of the total sphere in 3D = 4.188790 H Cap Cap volume_3d volume_nd 0.000000 0.000000 0.000000 0.166667 0.082418 0.082418 0.333333 0.310281 0.310281 0.500000 0.654498 0.654498 0.666667 1.085983 1.085983 0.833333 1.575644 1.575644 1.000000 2.094395 2.094395 1.166667 2.613146 2.613146 1.333333 3.102808 3.102808 1.500000 3.534292 3.534292 1.666667 3.878509 3.878509 1.833333 4.106372 4.106372 2.000000 4.188790 4.188790 2.166667 4.188790 4.188790 TEST33 For a sphere in ND: SPHERE_CAP_AREA_ND computes the area of a spherical cap. SPHERE_CAP_VOLUME_ND computes the volume of a spherical cap. Spatial dimension N = 2 Radius = 1.000000 Area = 6.283185 Volume = 3.141593 Sphere Sphere cap cap H area volume 0.000000 0.000000 0.000000 0.166667 1.171371 0.125043 0.333333 1.682137 0.344165 0.500000 2.094395 0.614185 0.666667 2.461919 0.916690 0.833333 2.806696 1.239013 1.000000 3.141593 1.570796 1.166667 3.476489 1.902580 1.333333 3.821266 2.224903 1.500000 4.188790 2.527408 1.666667 4.601048 2.797428 1.833333 5.111814 3.016549 2.000000 6.283185 3.141593 2.166667 6.283185 3.141593 Spatial dimension N = 3 Radius = 1.000000 Area = 12.566371 Volume = 4.188790 Sphere Sphere cap cap H area volume 0.000000 0.000000 0.000000 0.166667 1.047198 0.082418 0.333333 2.094395 0.310281 0.500000 3.141593 0.654498 0.666667 4.188790 1.085983 0.833333 5.235988 1.575644 1.000000 6.283185 2.094395 1.166667 7.330383 2.613146 1.333333 8.377580 3.102808 1.500000 9.424778 3.534292 1.666667 10.471976 3.878509 1.833333 11.519173 4.106372 2.000000 12.566371 4.188790 2.166667 12.566371 4.188790 Spatial dimension N = 4 Radius = 1.000000 Area = 19.739209 Volume = 4.934802 Sphere Sphere cap cap H area volume 0.000000 0.000000 0.000000 0.166667 0.785670 0.049022 0.333333 2.162450 0.251526 0.500000 3.859037 0.624672 0.666667 5.759731 1.147397 0.833333 7.784946 1.778925 1.000000 9.869604 2.467401 1.166667 11.954262 3.155877 1.333333 13.979477 3.787405 1.500000 15.880172 4.310130 1.666667 17.576758 4.683277 1.833333 18.953539 4.885780 2.000000 19.739209 4.934802 2.166667 19.739209 4.934802 Spatial dimension N = 5 Radius = 1.000000 Area = 26.318945 Volume = 5.263789 Sphere Sphere cap cap H area volume 0.000000 0.000000 0.000000 0.166667 0.517850 0.026781 0.333333 1.949551 0.186832 0.500000 4.112335 0.544884 0.666667 6.823430 1.104746 0.833333 9.900066 1.824531 1.000000 13.159473 2.631895 1.166667 16.418879 3.439258 1.333333 19.495515 4.159043 1.500000 22.206610 4.718905 1.666667 24.369394 5.076957 1.833333 25.801095 5.237008 2.000000 26.318945 5.263789 2.166667 26.318945 5.263789 TEST335 For integrals inside a spherical shell in ND: SPHERE_SHELL_03_ND approximates the integral. We compare these results with those computed by from the difference of two ball integrals: BALL_F1_ND approximates the integral; BALL_F3_ND approximates the integral Spatial dimension N = 2 Sphere center: 0.000000 0.000000 Inner sphere radius = 0.000000 Outer sphere radius = 1.000000 Spherical shell volume = 3.141593 Rule: #3 F1(R2)-F1(R1) F3(R2)-F3(R1) F(X) 1 3.141593 3.141593 3.141593 X 0.000000 0.000000 0.000000 X^2 0.785398 0.785398 0.785398 X^3 0.000000 0.000000 0.000000 X^4 0.392699 0.392699 0.392699 X^5 0.000000 0.000000 0.000000 X^6 0.196350 0.229749 0.196350 R 2.221441 2.074653 1.923825 SIN(X) 0.000000 0.000000 0.000000 EXP(X) 3.550929 3.550977 3.550929 1/(1+R) 1.840302 1.942542 2.082507 SQRT(R) 2.641754 2.494885 2.129062 Spatial dimension N = 3 Sphere center: 0.000000 0.000000 0.000000 Inner sphere radius = 0.000000 Outer sphere radius = 1.000000 Spherical shell volume = 4.188790 Rule: #3 F1(R2)-F1(R1) F3(R2)-F3(R1) F(X) 1 4.188790 4.188790 4.188790 X 0.000000 0.000000 0.000000 X^2 0.837758 0.837758 0.837758 X^3 0.000000 0.000000 0.000000 X^4 0.502655 0.359039 0.359039 X^5 0.000000 0.000000 0.000000 X^6 0.301593 0.194741 0.153874 R 3.244623 3.123589 2.973746 SIN(X) 0.000000 0.000000 0.000000 EXP(X) 4.629037 4.622902 4.622845 1/(1+R) 2.360418 2.440326 2.577138 SQRT(R) 3.686603 3.572543 3.234714 Spatial dimension N = 2 Sphere center: 1.000000 -1.000000 Inner sphere radius = 2.000000 Outer sphere radius = 3.000000 Spherical shell volume = 15.707963 Rule: #3 F1(R2)-F1(R1) F3(R2)-F3(R1) F(X) 1 15.707963 15.707963 15.707963 X 15.707963 15.707963 15.707963 X^2 66.758844 66.758844 66.758844 X^3 168.860605 168.860605 168.860605 X^4 653.843971 583.158136 583.158136 X^5 2185.370390 1831.941216 1831.941216 X^6 7915.831739 6147.213698 5936.628368 R 43.447124 42.937671 43.454796 SIN(X) 1.124966 -0.640965 -0.438804 EXP(X) 158.827813 147.703482 146.739733 1/(1+R) 4.436877 4.628686 4.314252 SQRT(R) 25.748011 25.420336 25.866913 Spatial dimension N = 3 Sphere center: 1.000000 -1.000000 2.000000 Inner sphere radius = 2.000000 Outer sphere radius = 3.000000 Spherical shell volume = 79.587014 Rule: #3 F1(R2)-F1(R1) F3(R2)-F3(R1) F(X) 1 79.587014 79.587014 79.587014 X 79.587014 79.587014 79.587014 X^2 256.353961 256.353961 256.353961 X^3 609.887854 609.887854 609.887854 X^4 2318.014770 1879.450325 1879.450325 X^5 7736.386861 5543.564637 5543.564637 X^6 28246.523471 17553.404673 16769.932716 R 268.542325 266.437150 267.899537 SIN(X) 25.736531 14.534141 15.300794 EXP(X) 623.423319 555.792149 552.271338 1/(1+R) 19.709955 20.670278 19.839008 SQRT(R) 143.860695 142.331488 143.557556 TEST34: SPHERE_UNIT_AREA_ND evaluates the area of the unit sphere in N dimensions. SPHERE_UNIT_AREA_VALUES returns some test values. dim_num Exact Computed Area Area 1 2.000000 2.000000 2 6.283185 6.283185 3 12.566371 12.566371 4 19.739209 19.739209 5 26.318945 26.318945 6 31.006277 31.006277 7 33.073362 33.073362 8 32.469697 32.469697 9 29.686580 29.686580 10 25.501640 25.501640 11 20.725143 20.725143 12 16.023153 16.023153 13 11.838174 11.838174 14 8.389703 8.389703 15 5.721649 5.721649 16 3.765290 3.765290 17 2.396679 2.396679 18 1.478626 1.478626 19 0.885810 0.885810 20 0.516138 0.516138 TEST345: SPHERE_UNIT_VOLUME_ND evaluates the area of the unit sphere in N dimensions. SPHERE_UNIT_VOLUME_VALUES returns some test values. dim_num Exact Computed Volume Volume 1 2.000000 2.000000 2 3.141593 3.141593 3 4.188790 4.188790 4 4.934802 4.934802 5 5.263789 5.263789 6 5.167713 5.167713 7 4.724766 4.724766 8 4.058712 4.058712 9 3.298509 3.298509 10 2.550164 2.550164 11 1.884104 1.884104 12 1.335263 1.335263 13 0.910629 0.910629 14 0.599265 0.599265 15 0.381443 0.381443 16 0.235331 0.235331 17 0.140981 0.140981 18 0.082146 0.082146 19 0.046622 0.046622 20 0.025807 0.025807 TEST35 SQUARE_UNIT_SET sets up a quadrature rule on a unit square. RECTANGLE_SUB_2D applies it to subrectangles of an arbitrary rectangle. The corners of the rectangle are: 1.000000 2.000000 3.000000 3.000000 Using unit square integration rule number 2 Order of rule is 4 Function Subdivisions Integral 1 1 2 2.000000 1 2 4 2.000000 1 3 6 2.000000 1 4 8 2.000000 1 5 10 2.000000 Function Subdivisions Integral X 1 2 4.000000 X 2 4 4.000000 X 3 6 4.000000 X 4 8 4.000000 X 5 10 4.000000 Function Subdivisions Integral X^2 1 2 8.666667 X^2 2 4 8.666667 X^2 3 6 8.666667 X^2 4 8 8.666667 X^2 5 10 8.666667 Function Subdivisions Integral X^3 1 2 20.000000 X^3 2 4 20.000000 X^3 3 6 20.000000 X^3 4 8 20.000000 X^3 5 10 20.000000 Function Subdivisions Integral X^4 1 2 48.222222 X^4 2 4 48.388889 X^4 3 6 48.397805 X^4 4 8 48.399306 X^4 5 10 48.399716 Function Subdivisions Integral X^5 1 2 119.555556 X^5 2 4 121.222222 X^5 3 6 121.311385 X^5 4 8 121.326389 X^5 5 10 121.330489 Function Subdivisions Integral X^6 1 2 301.407407 X^6 2 4 311.574074 X^6 3 6 312.143982 X^6 4 8 312.240741 X^6 5 10 312.267269 Function Subdivisions Integral R 1 2 6.476680 R 2 4 6.477047 R 3 6 6.477063 R 4 8 6.477066 R 5 10 6.477067 Function Subdivisions Integral SIN(X) 1 2 1.523822 SIN(X) 2 4 1.529929 SIN(X) 3 6 1.530224 SIN(X) 4 8 1.530272 SIN(X) 5 10 1.530286 Function Subdivisions Integral EXP(X) 1 2 17.310313 EXP(X) 2 4 17.363359 EXP(X) 3 6 17.366472 EXP(X) 4 8 17.367006 EXP(X) 5 10 17.367153 Function Subdivisions Integral 1/(1+R) 1 2 0.476772 1/(1+R) 2 4 0.476678 1/(1+R) 3 6 0.476673 1/(1+R) 4 8 0.476672 1/(1+R) 5 10 0.476672 Function Subdivisions Integral SQRT(R) 1 2 3.591205 SQRT(R) 2 4 3.591407 SQRT(R) 3 6 3.591416 SQRT(R) 4 8 3.591418 SQRT(R) 5 10 3.591418 TEST36 SQUARE_UNIT_SET sets up quadrature on the unit square; SQUARE_SUM carries it out on an arbitrary square. Square center: CENTER = ( 2.000000, 2.000000 ) Square radius is 3.000000 Rule: 1 2 3 4 5 Function 1 36.000000 36.000000 36.000000 36.000000 36.000000 X 72.000000 72.000000 72.000000 72.000000 72.000000 X^2 144.000000 252.000000 252.000000 252.000000 252.000000 X^3 288.000000 936.000000 936.000000 936.000000 936.000000 X^4 576.000000 3492.000000 3751.200000 3751.200000 3751.200000 X^5 1152.000000 13032.000000 15624.000000 15624.000000 15624.000000 X^6 2304.000000 48636.000000 66365.280000 66965.142857 66965.142857 R 101.823376 118.261521 122.814408 122.506823 123.004363 SIN(X) 32.734707 -5.255771 2.112732 1.525863 1.549446 EXP(X) 266.006020 775.295144 880.200264 888.093331 888.379741 1/(1+R) 9.403339 11.756502 9.673977 9.691780 9.476938 SQRT(R) 60.544542 61.034611 64.329899 64.228695 64.572032 Rule: 6 Function 1 36.000000 X 72.000000 X^2 252.000000 X^3 936.000000 X^4 3751.200000 X^5 15624.000000 X^6 66965.142857 R 122.495515 SIN(X) 1.539841 EXP(X) 888.271678 1/(1+R) 9.716640 SQRT(R) 64.202132 TEST37 SQUARE_UNIT_SET sets up quadrature on the unit square; SQUARE_UNIT_SUM carries it out on the unit square. Rule: 1 2 3 4 5 Function 1 1.000000 1.000000 1.000000 1.000000 1.000000 X 0.000000 0.000000 0.000000 0.000000 0.000000 X^2 0.000000 0.333333 0.333333 0.333333 0.333333 X^3 0.000000 0.000000 0.000000 0.000000 0.000000 X^4 0.000000 0.111111 0.200000 0.200000 0.200000 X^5 0.000000 0.000000 0.000000 0.000000 0.000000 X^6 0.000000 0.037037 0.120000 0.142857 0.142857 R 0.000000 0.816497 0.720617 0.774832 0.773460 SIN(X) 0.000000 0.000000 0.000000 0.000000 0.000000 EXP(X) 1.000000 1.171348 1.175168 1.175201 1.175201 1/(1+R) 1.000000 0.550510 0.623098 0.575052 0.576522 SQRT(R) 0.000000 0.903602 0.757659 0.868229 0.863466 Rule: 6 Function 1 1.000000 X -0.000000 X^2 0.333333 X^3 0.000000 X^4 0.200000 X^5 0.000000 X^6 0.142857 R 0.766081 SIN(X) 0.000000 EXP(X) 1.175201 1/(1+R) 0.582304 SQRT(R) 0.857542 TEST38 For integrals inside an arbitrary tetrahedron: TETRA_07 uses a formula of degree 7; TETRA_TPRODUCT uses a triangular product formula of varying degree. Tetrahedron vertices: 1.000000 2.000000 6.000000 4.000000 2.000000 6.000000 1.000000 3.000000 6.000000 1.000000 2.000000 8.000000 Tetrahedron unit volume = 0.166667 Tetrahedron Volume = 1.000000 F(X) TETRA_07 TETRA_TPRODUCT(1:4) TETRA_TPRODUCT(5:8) TETRA_TPRODUCT(9) 1 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 X 1.750000 1.750000 1.750000 1.750000 1.750000 1.750000 1.750000 1.750000 1.750000 1.750000 Y 2.250000 2.250000 2.250000 2.250000 2.250000 2.250000 2.250000 2.250000 2.250000 2.250000 Z 6.500000 6.500000 6.500000 6.500000 6.500000 6.500000 6.500000 6.500000 6.500000 6.500000 X*X 3.400000 3.062500 3.400000 3.400000 3.400000 3.400000 3.400000 3.400000 3.400000 3.400000 X*Y 3.900000 3.937500 3.900000 3.900000 3.900000 3.900000 3.900000 3.900000 3.900000 3.900000 X*Z 11.300000 11.375000 11.300000 11.300000 11.300000 11.300000 11.300000 11.300000 11.300000 11.300000 Y*Y 5.100000 5.062500 5.100000 5.100000 5.100000 5.100000 5.100000 5.100000 5.100000 5.100000 Y*Z 14.600000 14.625000 14.600000 14.600000 14.600000 14.600000 14.600000 14.600000 14.600000 14.600000 Z*Z 42.400000 42.250000 42.400000 42.400000 42.400000 42.400000 42.400000 42.400000 42.400000 42.400000 X^3 7.300000 5.359375 7.300000 7.300000 7.300000 7.300000 7.300000 7.300000 7.300000 7.300000 X*Y*Z 25.150000 25.593750 25.150000 25.150000 25.150000 25.150000 25.150000 25.150000 25.150000 25.150000 Z*Z*Z 277.600000 274.625000 277.600000 277.600000 277.600000 277.600000 277.600000 277.600000 277.600000 277.600000 X^4 17.114286 9.378906 16.988800 17.114286 17.114286 17.114286 17.114286 17.114286 17.114286 17.114286 X^2 Z^2 140.171429 129.390625 140.153600 140.171429 140.171429 140.171429 140.171429 140.171429 140.171429 140.171429 Z^4 1824.457142 1785.062500 1824.438044 1824.457143 1824.457143 1824.457143 1824.457143 1824.457143 1824.457143 1824.457143 X^5 43.160714 16.413086 41.839360 43.160714 43.160714 43.160714 43.160714 43.160714 43.160714 43.160714 X^6 115.428571 28.722900 106.838733 115.376228 115.428571 115.428571 115.428571 115.428571 115.428571 115.428571 R 7.127473 7.097535 7.127513 7.127473 7.127473 7.127473 7.127473 7.127473 7.127473 7.127473 SIN(X) 0.835961 0.983986 0.831740 0.836018 0.835961 0.835962 0.835962 0.835962 0.835962 0.835962 EXP(X) 6.998384 5.754603 6.952523 6.997700 6.998384 6.998390 6.998390 6.998390 6.998390 6.998390 1/(1+R) 0.139196 0.139516 0.139194 0.139196 0.139196 0.139196 0.139196 0.139196 0.139196 0.139196 SQRT(R) 2.669087 2.664120 2.669098 2.669087 2.669087 2.669087 2.669087 2.669087 2.669087 2.669087 TEST39 TETRA_UNIT_SET sets quadrature rules for the unit tetrahedron; TETRA_UNIT_SUM applies them to the unit tetrahedron. Rule: 1 2 3 4 5 Function 1 0.166667 0.166667 0.166667 0.166667 0.166667 X 0.041667 0.041667 0.041667 0.041667 0.041667 Y 0.041667 0.041667 0.041667 0.041667 0.041667 Z 0.041667 0.041667 0.041667 0.041667 0.041667 X*X 0.010417 0.041667 0.016667 0.016667 0.016667 X*Y 0.010417 0.000000 0.008333 0.008333 0.008333 X*Z 0.010417 0.000000 0.008333 0.008333 0.008333 Y*Y 0.010417 0.041667 0.016667 0.016667 0.016667 Y*Z 0.010417 0.000000 0.008333 0.008333 0.008333 Z*Z 0.010417 0.041667 0.016667 0.016667 0.016667 X^3 0.002604 0.041667 0.008689 0.004167 0.008333 X*Y*Z 0.002604 0.000000 0.001508 0.000000 0.001389 Z*Z*Z 0.002604 0.041667 0.008689 0.004167 0.008333 X^4 0.000651 0.041667 0.004939 -0.002083 0.004340 X^2 Z^2 0.000651 0.000000 0.000576 0.002083 0.000637 Z^4 0.000651 0.041667 0.004939 -0.002083 0.004340 X^5 0.000163 0.041667 0.002871 -0.005208 0.002242 X^6 0.000041 0.041667 0.001678 -0.006771 0.001144 R 0.072169 0.125000 0.087120 0.095711 0.088289 SIN(X) 0.041234 0.035061 0.040242 0.040930 0.040296 EXP(X) 0.214004 0.238262 0.218347 0.217220 0.218257 1/(1+R) 0.152944 0.130055 0.146894 0.145081 0.146620 SQRT(R) 0.109673 0.125000 0.118586 0.129800 0.119842 Rule: 6 7 8 Function 1 0.166667 0.166667 0.027778 X 0.041667 0.041667 0.006944 Y 0.041667 0.041667 0.006944 Z 0.041667 0.041667 0.006944 X*X 0.016667 0.016667 0.002778 X*Y 0.008333 0.008333 0.001389 X*Z 0.008333 0.008333 0.001389 Y*Y 0.016667 0.016667 0.002778 Y*Z 0.008333 0.008333 0.001389 Z*Z 0.016667 0.016667 0.002778 X^3 0.008333 0.008333 0.001389 X*Y*Z 0.001389 0.001389 0.000231 Z*Z*Z 0.008333 0.008333 0.001389 X^4 0.005556 0.004762 0.000794 X^2 Z^2 0.000926 0.000794 0.000132 Z^4 0.005556 0.004762 0.000794 X^5 0.004630 0.002790 0.000505 X^6 0.004321 0.001507 0.000349 R 0.087184 0.087657 0.014643 SIN(X) 0.040316 0.040301 0.006717 EXP(X) 0.218333 0.218279 0.036380 1/(1+R) 0.147242 0.147032 0.024487 SQRT(R) 0.118235 0.119963 0.019925 TEST40 TETRA_UNIT_SET sets quadrature rules for the unit tetrahedron; TETRA_SUM applies them to an arbitrary tetrahedron. Tetrahedron vertices: 1.000000 2.000000 6.000000 4.000000 2.000000 6.000000 1.000000 3.000000 6.000000 1.000000 2.000000 8.000000 Rule: 1 2 3 4 5 Function 1 1.000000 1.000000 1.000000 1.000000 1.000000 X 1.750000 1.750000 1.750000 1.750000 1.750000 Y 2.250000 2.250000 2.250000 2.250000 2.250000 Z 6.500000 6.500000 6.500000 6.500000 6.500000 X*X 3.062500 4.750000 3.400000 3.400000 3.400000 X*Y 3.937500 3.750000 3.900000 3.900000 3.900000 X*Z 11.375000 11.000000 11.300000 11.300000 11.300000 Y*Y 5.062500 5.250000 5.100000 5.100000 5.100000 Y*Z 14.625000 14.500000 14.600000 14.600000 14.600000 Z*Z 42.250000 43.000000 42.400000 42.400000 42.400000 X^3 5.359375 16.750000 7.357652 6.625000 7.300000 X*Y*Z 25.593750 23.500000 25.154271 25.100000 25.150000 Z*Z*Z 274.625000 290.000000 277.617082 277.400000 277.600000 X^4 9.378906 64.750000 17.431063 11.087500 16.909375 X^2 Z^2 129.390625 178.000000 139.953557 142.450000 140.137500 Z^4 1785.062500 1996.000000 1824.884133 1819.000000 1824.416667 X^5 16.413086 256.750000 44.014822 7.843750 41.066406 X^6 28.722900 1024.750000 115.615324 -57.865625 102.262305 R 7.097535 7.243848 7.127426 7.128685 7.127528 SIN(X) 0.983986 0.441903 0.834842 0.775585 0.828742 EXP(X) 5.754603 15.688249 7.021198 5.802816 6.928720 1/(1+R) 0.139516 0.138036 0.139197 0.139148 0.139194 SQRT(R) 2.664120 2.688104 2.669075 2.669399 2.669101 Rule: 6 7 8 Function 1 1.000000 1.000000 0.166667 X 1.750000 1.750000 0.291667 Y 2.250000 2.250000 0.375000 Z 6.500000 6.500000 1.083333 X*X 3.400000 3.400000 0.566667 X*Y 3.900000 3.900000 0.650000 X*Z 11.300000 11.300000 1.883333 Y*Y 5.100000 5.100000 0.850000 Y*Z 14.600000 14.600000 2.433333 Z*Z 42.400000 42.400000 7.066667 X^3 7.300000 7.300000 1.216667 X*Y*Z 25.150000 25.150000 4.191667 Z*Z*Z 277.600000 277.600000 46.266667 X^4 17.500000 17.114286 2.852381 X^2 Z^2 140.200000 140.171429 23.361905 Z^4 1824.533333 1824.457143 304.076190 X^5 47.500000 42.889509 7.206367 X^6 145.900000 111.713058 19.395203 R 7.127357 7.127469 1.187912 SIN(X) 0.847297 0.837317 0.139281 EXP(X) 7.168051 6.974256 1.167247 1/(1+R) 0.139201 0.139196 0.023199 SQRT(R) 2.669056 2.669086 0.444848 TEST41 TRIANGLE_UNIT_SET sets up a quadrature rule on a triangle. TRIANGLE_SUB applies it to subtriangles of an arbitrary triangle. Triangle vertices: 0.000000 0.000000 0.000000 1.000000 1.000000 0.000000 Using unit triangle quadrature rule 3 Rule order = 3 Function Nsub Result 1 1 0.500000 1 2 0.500000 1 3 0.500000 1 4 0.500000 1 5 0.500000 X 1 0.166667 X 2 0.166667 X 3 0.166667 X 4 0.166667 X 5 0.166667 X^2 1 0.083333 X^2 2 0.083333 X^2 3 0.083333 X^2 4 0.083333 X^2 5 0.083333 X^3 1 0.050926 X^3 2 0.050058 X^3 3 0.050011 X^3 4 0.050004 X^3 5 0.050001 X^4 1 0.033179 X^4 2 0.033324 X^4 3 0.033331 X^4 4 0.033333 X^4 5 0.033333 X^5 1 0.021991 X^5 2 0.023769 X^5 3 0.023804 X^5 4 0.023808 X^5 5 0.023809 X^6 1 0.014639 X^6 2 0.017789 X^6 3 0.017848 X^6 4 0.017855 X^6 5 0.017856 R 1 0.268345 R 2 0.270376 R 3 0.270492 R 4 0.270519 R 5 0.270528 SIN(X) 1 0.158360 SIN(X) 2 0.158519 SIN(X) 3 0.158527 SIN(X) 4 0.158528 SIN(X) 5 0.158529 EXP(X) 1 0.718409 EXP(X) 2 0.718291 EXP(X) 3 0.718284 EXP(X) 4 0.718282 EXP(X) 5 0.718282 1/(1+R) 1 0.332444 1/(1+R) 2 0.330667 1/(1+R) 3 0.330536 1/(1+R) 4 0.330506 1/(1+R) 5 0.330495 SQRT(R) 1 0.357237 SQRT(R) 2 0.359497 SQRT(R) 3 0.359708 SQRT(R) 4 0.359769 SQRT(R) 5 0.359794 TEST42 TRIANGLE_UNIT_SET sets up a quadrature in the unit triangle, TRIANGLE_UNIT_SUM applies it. Rule: 1 2 3 4 5 Function 1 0.500000 0.500000 0.500000 0.500000 0.500000 X 0.166667 0.166667 0.166667 0.166667 0.166667 X^2 0.055556 0.166667 0.083333 0.083333 0.083333 X^3 0.018519 0.166667 0.050926 0.041667 0.050000 X^4 0.006173 0.166667 0.033179 0.020833 0.031111 X^5 0.002058 0.166667 0.021991 0.010417 0.019259 X^6 0.000686 0.166667 0.014639 0.005208 0.011798 R 0.235702 0.333333 0.268345 0.284518 0.270478 SIN(X) 0.163597 0.140245 0.158360 0.159809 0.158492 EXP(X) 0.697806 0.786380 0.718409 0.716240 0.718141 1/(1+R) 0.339811 0.333333 0.332444 0.319853 0.330905 SQRT(R) 0.343295 0.333333 0.357237 0.375852 0.359598 Rule: 6 7 8 9 10 Function 1 0.500000 0.500000 0.500000 0.500000 0.500000 X 0.166667 0.169444 0.166667 0.166667 0.166667 X^2 0.083333 0.086574 0.083333 0.083333 0.083333 X^3 0.050000 0.052855 0.050000 0.050000 0.050000 X^4 0.031944 0.034195 0.033333 0.033333 0.033333 X^5 0.020833 0.022507 0.023935 0.023457 0.023810 X^6 0.013681 0.014899 0.018088 0.016930 0.017775 R 0.270487 0.272066 0.270438 0.270744 0.270044 SIN(X) 0.158505 0.160821 0.158530 0.158526 0.158529 EXP(X) 0.718192 0.723176 0.718283 0.718277 0.718282 1/(1+R) 0.330704 0.330513 0.330634 0.330160 0.330940 SQRT(R) 0.359901 0.360298 0.359485 0.360778 0.359106 Rule: 11 12 13 14 15 Function 1 0.500000 0.500000 0.500000 0.500000 0.500000 X 0.166667 0.166667 0.166667 0.166667 0.166667 X^2 0.083333 0.083333 0.083333 0.083333 0.083333 X^3 0.050000 0.050000 0.050000 0.050000 0.050000 X^4 0.033333 0.033333 0.033333 0.036111 0.033333 X^5 0.023810 0.023810 0.023810 0.030093 0.023810 X^6 0.017784 0.017857 0.017857 0.027392 0.017857 R 0.270906 0.270472 0.270469 0.269873 0.276233 SIN(X) 0.158529 0.158529 0.158529 0.158579 0.158529 EXP(X) 0.718282 0.718282 0.718282 0.718466 0.718282 1/(1+R) 0.330147 0.330552 0.330556 0.330856 0.322383 SQRT(R) 0.360518 0.359673 0.359669 0.354823 0.372565 Rule: 16 17 18 19 20 Function 1 0.500000 0.500000 0.500000 0.500000 0.500000 X 0.166667 0.166667 0.166667 0.166667 0.166667 X^2 0.083333 0.083333 0.083333 0.083333 0.083333 X^3 0.050000 0.050000 0.050000 0.050000 0.050000 X^4 0.033333 0.033333 0.033333 0.033333 0.033333 X^5 0.023810 0.023810 0.023810 0.023810 0.023810 X^6 0.017857 0.017857 0.017857 0.017857 0.017857 R 0.270536 0.270485 0.268132 0.270517 0.270534 SIN(X) 0.158529 0.158529 0.158529 0.158529 0.158529 EXP(X) 0.718282 0.718282 0.718282 0.718282 0.718282 1/(1+R) 0.330486 0.330536 0.334869 0.330506 0.330489 SQRT(R) 0.359820 0.359697 0.353414 0.359774 0.359813 TEST425 TRIANGLE_UNIT_SET sets up a quadrature in the unit triangle, Estimate integral of X^A * Y^B. A = 0 B = 0 Rule QUAD ERROR 1 1.000000 0.000000e+00 2 1.000000 0.000000e+00 3 1.000000 0.000000e+00 4 1.000000 0.000000e+00 5 1.000000 0.000000e+00 6 1.000000 1.110223e-16 7 1.000000 0.000000e+00 8 1.000000 9.992007e-16 9 1.000000 1.998401e-15 10 1.000000 0.000000e+00 11 1.000000 6.661338e-16 12 1.000000 1.998401e-15 13 1.000000 1.110223e-15 14 1.000000 0.000000e+00 15 1.000000 1.999998e-10 16 1.000000 2.220446e-16 17 1.000000 6.661338e-16 18 1.000000 5.551115e-16 19 1.000000 2.220446e-16 20 1.000000 2.220446e-16 A = 0 B = 1 Rule QUAD ERROR 1 1.000000 0.000000e+00 2 1.000000 0.000000e+00 3 1.000000 1.110223e-16 4 1.000000 0.000000e+00 5 1.000000 0.000000e+00 6 1.000000 0.000000e+00 7 1.000000 0.000000e+00 8 1.000000 6.661338e-16 9 1.000000 4.497402e-12 10 1.000000 1.110223e-16 11 1.000000 8.881784e-16 12 1.000000 1.554312e-15 13 1.000000 4.440892e-16 14 1.000000 0.000000e+00 15 1.326073 3.260726e-01 16 1.000000 4.360734e-12 17 1.000000 2.220446e-16 18 1.056574 5.657395e-02 19 1.000000 2.220446e-16 20 1.000000 0.000000e+00 A = 0 B = 2 Rule QUAD ERROR 1 0.666667 3.333333e-01 2 2.000000 1.000000e+00 3 1.000000 0.000000e+00 4 1.000000 0.000000e+00 5 1.000000 2.220446e-16 6 1.000000 4.440892e-16 7 1.000000 0.000000e+00 8 1.000000 0.000000e+00 9 1.000000 4.528600e-13 10 1.000000 2.220446e-16 11 1.000000 1.665335e-15 12 1.000000 3.108624e-15 13 1.000000 1.110223e-15 14 1.000000 0.000000e+00 15 1.483663 4.836633e-01 16 1.000000 5.130341e-12 17 1.000000 2.220446e-16 18 1.126251 1.262514e-01 19 1.000000 3.330669e-16 20 1.000000 1.110223e-16 A = 0 B = 3 Rule QUAD ERROR 1 0.370370 6.296296e-01 2 3.333333 2.333333e+00 3 1.018519 1.851852e-02 4 0.833333 1.666667e-01 5 1.000000 0.000000e+00 6 1.000000 7.771561e-16 7 1.000000 2.220446e-16 8 1.000000 1.110223e-15 9 1.000000 2.486900e-14 10 1.000000 2.220446e-16 11 1.000000 2.331468e-15 12 1.000000 4.440892e-15 13 1.000000 1.998401e-15 14 1.000000 1.110223e-16 15 1.641254 6.412540e-01 16 1.000000 3.687051e-12 17 1.000000 4.440892e-16 18 1.170979 1.709789e-01 19 1.000000 4.440892e-16 20 1.000000 2.220446e-16 A = 0 B = 4 Rule QUAD ERROR 1 0.185185 8.148148e-01 2 5.000000 4.000000e+00 3 0.995370 4.629630e-03 4 0.625000 3.750000e-01 5 0.933333 6.666667e-02 6 0.958333 4.166667e-02 7 0.958333 4.166667e-02 8 1.000000 2.220446e-15 9 1.000000 3.330669e-15 10 1.000000 4.440892e-16 11 1.000000 2.886580e-15 12 1.000000 5.107026e-15 13 1.000000 2.442491e-15 14 1.083333 8.333333e-02 15 2.130152 1.130152e+00 16 1.000000 9.678924e-13 17 1.000000 6.661338e-16 18 1.194457 1.944571e-01 19 1.000000 2.220446e-16 20 1.000000 0.000000e+00 A = 0 B = 5 Rule QUAD ERROR 1 0.086420 9.135802e-01 2 7.000000 6.000000e+00 3 0.923611 7.638889e-02 4 0.437500 5.625000e-01 5 0.808889 1.911111e-01 6 0.875000 1.250000e-01 7 0.875000 1.250000e-01 8 1.005274 5.273935e-03 9 0.985185 1.481481e-02 10 1.000000 7.771561e-16 11 1.000000 3.552714e-15 12 1.000000 5.995204e-15 13 1.000000 2.886580e-15 14 1.263889 2.638889e-01 15 3.762087 2.762087e+00 16 1.000000 2.267075e-12 17 1.000000 1.110223e-15 18 1.202942 2.029416e-01 19 1.000000 0.000000e+00 20 1.000000 1.110223e-16 A = 0 B = 6 Rule QUAD ERROR 1 0.038409 9.615912e-01 2 9.333333 8.333333e+00 3 0.819787 1.802126e-01 4 0.291667 7.083333e-01 5 0.660662 3.393383e-01 6 0.766111 2.338889e-01 7 0.766975 2.330247e-01 8 1.012949 1.294949e-02 9 0.948082 5.191770e-02 10 0.995414 4.585538e-03 11 0.995927 4.072611e-03 12 1.000000 6.661338e-15 13 1.000000 2.886580e-15 14 1.533951 5.339506e-01 15 9.514781 8.514781e+00 16 1.000000 5.616840e-12 17 1.000000 1.110223e-15 18 1.200851 2.008511e-01 19 1.000000 0.000000e+00 20 1.000000 0.000000e+00 A = 0 B = 7 Rule QUAD ERROR 1 0.016461 9.835391e-01 2 12.000000 1.100000e+01 3 0.702418 2.975823e-01 4 0.187500 8.125000e-01 5 0.516101 4.838993e-01 6 0.648333 3.516667e-01 7 0.650926 3.490741e-01 8 1.016902 1.690162e-02 9 0.889877 1.101235e-01 10 0.981708 1.829176e-02 11 0.983203 1.679708e-02 12 1.000198 1.980392e-04 13 1.000000 3.108624e-15 14 1.882407 8.824074e-01 15 33.730634 3.273063e+01 16 1.000000 8.920309e-12 17 1.000000 1.776357e-15 18 1.191475 1.914754e-01 19 1.000000 2.220446e-16 20 1.000000 2.220446e-16 A = 0 B = 8 Rule QUAD ERROR 1 0.006859 9.931413e-01 2 15.000000 1.400000e+01 3 0.585294 4.147055e-01 4 0.117188 8.828125e-01 5 0.389922 6.100780e-01 6 0.533854 4.661458e-01 7 0.538484 4.615162e-01 8 1.012161 1.216139e-02 9 0.816165 1.838354e-01 10 0.956602 4.339756e-02 11 0.959183 4.081739e-02 12 1.000439 4.393770e-04 13 0.999572 4.279584e-04 14 2.299961 1.299961e+00 15 161.221957 1.602220e+02 16 1.000000 1.214051e-11 17 1.000000 1.998401e-15 18 1.177418 1.774182e-01 19 1.000000 1.110223e-16 20 1.000000 4.440892e-16 A = 0 B = 9 Rule QUAD ERROR 1 0.002794 9.972057e-01 2 18.333333 1.733333e+01 3 0.476896 5.231043e-01 4 0.071615 9.283854e-01 5 0.287142 7.128585e-01 6 0.429942 5.700579e-01 7 0.436368 5.636324e-01 8 0.996255 3.744597e-03 9 0.733569 2.664306e-01 10 0.920084 7.991597e-02 11 0.923540 7.645961e-02 12 1.000086 8.581607e-05 13 0.997886 2.113584e-03 14 2.779903 1.779903e+00 15 1002.829779 1.001830e+03 16 1.000000 1.529477e-11 17 1.000091 9.148595e-05 18 1.160729 1.607286e-01 19 1.000000 4.440892e-16 20 1.000000 0.000000e+00 A = 0 B = 10 Rule QUAD ERROR 1 0.001118 9.988823e-01 2 22.000000 2.100000e+01 3 0.381514 6.184856e-01 4 0.042969 9.570312e-01 5 0.207231 7.927692e-01 6 0.339994 6.600058e-01 7 0.347660 6.523402e-01 8 0.968918 3.108196e-02 9 0.647992 3.520083e-01 10 0.873671 1.263288e-01 11 0.877655 1.223451e-01 12 0.998219 1.780634e-03 13 0.993975 6.025082e-03 14 3.317690 2.317690e+00 15 7878.783989 7.877784e+03 16 1.000000 1.841771e-11 17 1.000489 4.886059e-04 18 1.142929 1.429288e-01 19 1.000000 7.771561e-16 20 1.000000 6.661338e-16 A = 1 B = 0 Rule QUAD ERROR 1 1.000000 0.000000e+00 2 1.000000 0.000000e+00 3 1.000000 1.110223e-16 4 1.000000 0.000000e+00 5 1.000000 0.000000e+00 6 1.000000 1.110223e-16 7 1.016667 1.666667e-02 8 1.000000 6.661338e-16 9 1.000000 4.497513e-12 10 1.000000 1.110223e-16 11 1.000000 8.881784e-16 12 1.000000 1.554312e-15 13 1.000000 6.661338e-16 14 1.000000 0.000000e+00 15 1.000000 3.407273e-10 16 1.000000 8.721912e-12 17 1.000000 2.220446e-16 18 1.000000 2.220446e-16 19 1.000000 0.000000e+00 20 1.000000 0.000000e+00 A = 1 B = 1 Rule QUAD ERROR 1 1.333333 3.333333e-01 2 0.000000 1.000000e+00 3 1.000000 0.000000e+00 4 1.000000 0.000000e+00 5 1.000000 2.220446e-16 6 1.000000 2.220446e-16 7 1.033333 3.333333e-02 8 1.000000 8.881784e-16 9 1.000000 1.754019e-11 10 1.000000 2.220446e-16 11 1.000000 4.440892e-16 12 1.000000 3.330669e-16 13 1.000000 1.443290e-15 14 1.000000 0.000000e+00 15 1.326073 3.260726e-01 16 1.000000 2.051026e-12 17 1.000000 2.220446e-16 18 1.475334 4.753338e-01 19 1.000000 1.110223e-16 20 1.000000 4.440892e-16 A = 1 B = 2 Rule QUAD ERROR 1 1.111111 1.111111e-01 2 0.000000 1.000000e+00 3 0.972222 2.777778e-02 4 1.250000 2.500000e-01 5 1.000000 4.440892e-16 6 1.000000 2.220446e-16 7 1.041667 4.166667e-02 8 1.000000 4.440892e-16 9 1.000000 1.459333e-11 10 1.000000 2.220446e-16 11 1.000000 4.440892e-16 12 1.000000 8.881784e-16 13 1.000000 9.992007e-16 14 1.000000 0.000000e+00 15 1.483663 4.836633e-01 16 1.000000 1.090572e-11 17 1.000000 0.000000e+00 18 2.061874 1.061874e+00 19 1.000000 0.000000e+00 20 1.000000 2.220446e-16 A = 1 B = 3 Rule QUAD ERROR 1 0.740741 2.592593e-01 2 0.000000 1.000000e+00 3 1.064815 6.481481e-02 4 1.250000 2.500000e-01 5 1.133333 1.333333e-01 6 1.083333 8.333333e-02 7 1.125000 1.250000e-01 8 1.000000 8.881784e-16 9 1.000000 1.867850e-11 10 1.000000 4.440892e-16 11 1.000000 7.771561e-16 12 1.000000 1.998401e-15 13 1.000000 0.000000e+00 14 0.833333 1.666667e-01 15 1.341599 3.415994e-01 16 1.000000 1.728107e-11 17 1.000000 2.220446e-16 18 2.780757 1.780757e+00 19 1.000000 0.000000e+00 20 1.000000 1.110223e-16 A = 1 B = 4 Rule QUAD ERROR 1 0.432099 5.679012e-01 2 0.000000 1.000000e+00 3 1.174769 1.747685e-01 4 1.093750 9.375000e-02 5 1.244444 2.444444e-01 6 1.166667 1.666667e-01 7 1.203125 2.031250e-01 8 0.986815 1.318484e-02 9 1.037037 3.703704e-02 10 1.000000 4.440892e-16 11 1.000000 1.776357e-15 12 1.000000 2.886580e-15 13 1.000000 1.110223e-15 14 0.631944 3.680556e-01 15 0.933348 6.665154e-02 16 1.000000 2.037992e-11 17 1.000000 2.220446e-16 18 3.613047 2.613047e+00 19 1.000000 3.330669e-16 20 1.000000 0.000000e+00 A = 1 B = 5 Rule QUAD ERROR 1 0.230453 7.695473e-01 2 0.000000 1.000000e+00 3 1.235082 2.350823e-01 4 0.875000 1.250000e-01 5 1.253570 2.535704e-01 6 1.201667 2.016667e-01 7 1.228241 2.282407e-01 8 0.982247 1.775274e-02 9 1.096494 9.649383e-02 10 1.013757 1.375661e-02 11 1.012218 1.221783e-02 12 1.000000 3.774758e-15 13 1.000000 1.776357e-15 14 0.453704 5.462963e-01 15 0.523274 4.767259e-01 16 1.000000 2.118083e-11 17 1.000000 3.330669e-16 18 4.533686 3.533686e+00 19 1.000000 1.110223e-16 20 1.000000 0.000000e+00 A = 1 B = 6 Rule QUAD ERROR 1 0.115226 8.847737e-01 2 0.000000 1.000000e+00 3 1.230581 2.305813e-01 4 0.656250 3.437500e-01 5 1.166625 1.666252e-01 6 1.178333 1.783333e-01 7 1.195023 1.950231e-01 8 0.999117 8.829594e-04 9 1.151802 1.518025e-01 10 1.043386 4.338624e-02 11 1.040463 4.046303e-02 12 0.999307 6.931371e-04 13 1.000000 2.220446e-15 14 0.314352 6.856481e-01 15 0.257577 7.424229e-01 16 1.000000 2.081046e-11 17 1.000000 4.440892e-16 18 5.520729 4.520729e+00 19 1.000000 1.110223e-16 20 1.000000 0.000000e+00 A = 1 B = 7 Rule QUAD ERROR 1 0.054870 9.451303e-01 2 0.000000 1.000000e+00 3 1.170910 1.709105e-01 4 0.468750 5.312500e-01 5 1.020816 2.081580e-02 6 1.106250 1.062500e-01 7 1.116319 1.163194e-01 8 1.035863 3.586257e-02 9 1.184724 1.847243e-01 10 1.082131 8.213142e-02 11 1.079284 7.928415e-02 12 0.999233 7.673120e-04 13 1.001712 1.711833e-03 14 0.212191 7.878086e-01 15 0.120566 8.794335e-01 16 1.000000 2.006262e-11 17 1.000000 5.551115e-16 18 6.556325 5.556325e+00 19 1.000000 2.220446e-16 20 1.000000 1.110223e-16 A = 1 B = 8 Rule QUAD ERROR 1 0.025149 9.748514e-01 2 0.000000 1.000000e+00 3 1.073089 7.308903e-02 4 0.322266 6.777344e-01 5 0.852434 1.475661e-01 6 1.001458 1.458333e-03 7 1.008749 8.748875e-03 8 1.083738 8.373832e-02 9 1.187843 1.878431e-01 10 1.120935 1.209353e-01 11 1.119573 1.195726e-01 12 1.002030 2.030401e-03 13 1.007157 7.157358e-03 14 0.140223 8.597769e-01 15 0.056599 9.434009e-01 16 1.000000 1.940226e-11 17 0.999588 4.116868e-04 18 7.626030 6.626030e+00 19 1.000000 8.881784e-16 20 1.000000 2.220446e-16 A = 1 B = 9 Rule QUAD ERROR 1 0.011177 9.888228e-01 2 0.000000 1.000000e+00 3 0.953802 4.619766e-02 4 0.214844 7.851562e-01 5 0.686695 3.133048e-01 6 0.879682 1.203183e-01 7 0.887068 1.129321e-01 8 1.132942 1.329422e-01 9 1.161458 1.614578e-01 10 1.152148 1.521482e-01 11 1.152968 1.529678e-01 12 1.009418 9.418068e-03 13 1.017444 1.744391e-02 14 0.090967 9.090328e-01 15 0.027312 9.726882e-01 16 1.000000 1.905964e-11 17 0.998106 1.894114e-03 18 8.717905 7.717905e+00 19 1.000000 1.554312e-15 20 1.000000 1.110223e-16 A = 2 B = 0 Rule QUAD ERROR 1 0.666667 3.333333e-01 2 2.000000 1.000000e+00 3 1.000000 0.000000e+00 4 1.000000 0.000000e+00 5 1.000000 2.220446e-16 6 1.000000 4.440892e-16 7 1.038889 3.888889e-02 8 1.000000 0.000000e+00 9 1.000000 4.529710e-13 10 1.000000 2.220446e-16 11 1.000000 1.665335e-15 12 1.000000 3.108624e-15 13 1.000000 1.110223e-15 14 1.000000 0.000000e+00 15 1.000000 4.579244e-10 16 1.000000 1.949474e-11 17 1.000000 4.440892e-16 18 1.000000 0.000000e+00 19 1.000000 3.330669e-16 20 1.000000 0.000000e+00 A = 2 B = 1 Rule QUAD ERROR 1 1.111111 1.111111e-01 2 0.000000 1.000000e+00 3 0.972222 2.777778e-02 4 1.250000 2.500000e-01 5 1.000000 2.220446e-16 6 1.000000 2.220446e-16 7 1.097222 9.722222e-02 8 1.000000 4.440892e-16 9 1.000000 1.459355e-11 10 1.000000 2.220446e-16 11 1.000000 5.551115e-16 12 1.000000 8.881784e-16 13 1.000000 8.881784e-16 14 1.000000 0.000000e+00 15 1.326073 3.260726e-01 16 1.000000 1.123146e-11 17 1.000000 0.000000e+00 18 2.039797 1.039797e+00 19 1.000000 1.110223e-16 20 1.000000 2.220446e-16 A = 2 B = 2 Rule QUAD ERROR 1 1.111111 1.111111e-01 2 0.000000 1.000000e+00 3 0.763889 2.361111e-01 4 1.875000 8.750000e-01 5 0.800000 2.000000e-01 6 0.875000 1.250000e-01 7 1.020833 2.083333e-02 8 1.000000 4.440892e-16 9 1.000000 4.098388e-12 10 1.000000 2.220446e-16 11 1.000000 0.000000e+00 12 1.000000 2.664535e-15 13 1.000000 9.992007e-16 14 1.250000 2.500000e-01 15 1.743502 7.435024e-01 16 1.000000 1.843525e-12 17 1.000000 2.220446e-16 18 3.739045 2.739045e+00 19 1.000000 2.220446e-16 20 1.000000 2.220446e-16 A = 2 B = 3 Rule QUAD ERROR 1 0.864198 1.358025e-01 2 0.000000 1.000000e+00 3 0.729167 2.708333e-01 4 2.187500 1.187500e+00 5 0.808889 1.911111e-01 6 0.875000 1.250000e-01 7 1.045139 4.513889e-02 8 1.005274 5.273935e-03 9 0.985185 1.481481e-02 10 1.000000 5.551115e-16 11 1.000000 2.220446e-16 12 1.000000 3.996803e-15 13 1.000000 0.000000e+00 14 1.263889 2.638889e-01 15 2.228652 1.228652e+00 16 1.000000 9.533485e-12 17 1.000000 0.000000e+00 18 6.382553 5.382553e+00 19 1.000000 2.220446e-16 20 1.000000 2.220446e-16 A = 2 B = 4 Rule QUAD ERROR 1 0.576132 4.238683e-01 2 0.000000 1.000000e+00 3 0.819187 1.808128e-01 4 2.187500 1.187500e+00 5 0.949926 5.007407e-02 6 0.962500 3.750000e-02 7 1.126157 1.261574e-01 8 0.958434 4.156566e-02 9 1.030123 3.012346e-02 10 0.962963 3.703704e-02 11 0.983741 1.625887e-02 12 1.000000 5.551115e-15 13 1.000000 8.881784e-16 14 1.134259 1.342593e-01 15 2.324836 1.324836e+00 16 1.000000 1.797784e-11 17 1.000000 1.110223e-16 18 10.120140 9.120140e+00 19 1.000000 3.330669e-16 20 1.000000 0.000000e+00 A = 2 B = 5 Rule QUAD ERROR 1 0.345679 6.543210e-01 2 0.000000 1.000000e+00 3 0.937114 6.288580e-02 4 1.968750 9.687500e-01 5 1.080676 8.067556e-02 6 1.067500 6.750000e-02 7 1.193403 1.934028e-01 8 0.881502 1.184980e-01 9 1.140741 1.407407e-01 10 0.909751 9.024943e-02 11 0.967270 3.272971e-02 12 1.000891 8.911763e-04 13 1.000000 1.332268e-15 14 0.943056 5.694444e-02 15 1.819692 8.196915e-01 16 1.000000 2.247758e-11 17 1.000000 1.110223e-16 18 15.030887 1.403089e+01 19 1.000000 4.440892e-16 20 1.000000 1.110223e-16 A = 2 B = 6 Rule QUAD ERROR 1 0.192044 8.079561e-01 2 0.000000 1.000000e+00 3 1.028485 2.848508e-02 4 1.640625 6.406250e-01 5 1.133495 1.334953e-01 6 1.144792 1.447917e-01 7 1.217303 2.173032e-01 8 0.811392 1.886079e-01 9 1.286387 2.863868e-01 10 0.866501 1.334989e-01 11 0.970273 2.972680e-02 12 0.996224 3.775557e-03 13 0.994117 5.883009e-03 14 0.742670 2.573302e-01 15 1.070181 7.018063e-02 16 1.000000 2.380052e-11 17 1.000000 4.440892e-16 18 21.158426 2.015843e+01 19 1.000000 8.881784e-16 20 1.000000 0.000000e+00 A = 2 B = 7 Rule QUAD ERROR 1 0.100594 8.994056e-01 2 0.000000 1.000000e+00 3 1.074121 7.412051e-02 4 1.289062 2.890625e-01 5 1.103432 1.034316e-01 6 1.175625 1.756250e-01 7 1.195875 1.958751e-01 8 0.769683 2.303172e-01 9 1.433446 4.334464e-01 10 0.846090 1.539098e-01 11 1.001166 1.166369e-03 12 0.981678 1.832186e-02 13 0.979223 2.077746e-02 14 0.560892 4.391075e-01 15 0.497553 5.024474e-01 16 1.000000 2.303535e-11 17 1.000729 7.290941e-04 18 28.521870 2.752187e+01 19 1.000000 1.332268e-15 20 1.000000 2.220446e-16 A = 2 B = 8 Rule QUAD ERROR 1 0.050297 9.497028e-01 2 0.000000 1.000000e+00 3 1.073286 7.328550e-02 4 0.966797 3.320312e-02 5 1.012554 1.255422e-02 6 1.158724 1.587240e-01 7 1.137832 1.378320e-01 8 0.761318 2.386820e-01 9 1.557612 5.576121e-01 10 0.849523 1.504772e-01 11 1.057002 5.700238e-02 12 0.957945 4.205474e-02 13 0.956754 4.324599e-02 14 0.409352 5.906475e-01 15 0.197751 8.022492e-01 16 1.000000 2.111444e-11 17 1.002323 2.322910e-03 18 37.119245 3.611925e+01 19 1.000000 1.332268e-15 20 1.000000 2.220446e-16 A = 3 B = 0 Rule QUAD ERROR 1 0.370370 6.296296e-01 2 3.333333 2.333333e+00 3 1.018519 1.851852e-02 4 0.833333 1.666667e-01 5 1.000000 0.000000e+00 6 1.000000 6.661338e-16 7 1.057099 5.709877e-02 8 1.000000 1.110223e-15 9 1.000000 2.486900e-14 10 1.000000 3.330669e-16 11 1.000000 2.331468e-15 12 1.000000 4.440892e-15 13 1.000000 1.776357e-15 14 1.000000 1.110223e-16 15 1.000000 5.668598e-10 16 1.000000 2.500289e-11 17 1.000000 4.440892e-16 18 1.000000 2.220446e-16 19 1.000000 3.330669e-16 20 1.000000 2.220446e-16 A = 3 B = 1 Rule QUAD ERROR 1 0.740741 2.592593e-01 2 0.000000 1.000000e+00 3 1.064815 6.481481e-02 4 1.250000 2.500000e-01 5 1.133333 1.333333e-01 6 1.083333 8.333333e-02 7 1.254630 2.546296e-01 8 1.000000 8.881784e-16 9 1.000000 1.867850e-11 10 1.000000 4.440892e-16 11 1.000000 8.881784e-16 12 1.000000 1.776357e-15 13 1.000000 2.220446e-16 14 0.833333 1.666667e-01 15 1.271727 2.717271e-01 16 1.000000 2.061917e-11 17 1.000000 2.220446e-16 18 2.743242 1.743242e+00 19 1.000000 1.110223e-16 20 1.000000 1.110223e-16 A = 3 B = 2 Rule QUAD ERROR 1 0.864198 1.358025e-01 2 0.000000 1.000000e+00 3 0.729167 2.708333e-01 4 2.187500 1.187500e+00 5 0.808889 1.911111e-01 6 0.875000 1.250000e-01 7 1.174769 1.747685e-01 8 1.005274 5.273935e-03 9 0.985185 1.481481e-02 10 1.000000 6.661338e-16 11 1.000000 0.000000e+00 12 1.000000 4.218847e-15 13 1.000000 2.220446e-16 14 1.263889 2.638889e-01 15 1.758834 7.588342e-01 16 1.000000 1.701361e-11 17 1.000000 0.000000e+00 18 6.377950 5.377950e+00 19 1.000000 2.220446e-16 20 1.000000 0.000000e+00 A = 3 B = 3 Rule QUAD ERROR 1 0.768176 2.318244e-01 2 0.000000 1.000000e+00 3 0.516118 4.838820e-01 4 2.916667 1.916667e+00 5 0.594568 4.054321e-01 6 0.738889 2.611111e-01 7 1.155864 1.558642e-01 8 1.063496 6.349583e-02 9 0.917202 8.279835e-02 10 1.051146 5.114638e-02 11 1.012155 1.215469e-02 12 1.000000 6.217249e-15 13 1.000000 1.110223e-15 14 1.512346 5.123457e-01 15 2.922332 1.922332e+00 16 1.000000 4.539813e-12 17 1.000000 2.220446e-16 18 13.177549 1.217755e+01 19 1.000000 1.110223e-16 20 1.000000 1.110223e-16 A = 3 B = 4 Rule QUAD ERROR 1 0.576132 4.238683e-01 2 0.000000 1.000000e+00 3 0.481610 5.183899e-01 4 3.281250 2.281250e+00 5 0.591526 4.084741e-01 6 0.729167 2.708333e-01 7 1.211227 2.112269e-01 8 1.057370 5.737044e-02 9 0.905679 9.432099e-02 10 1.053666 5.366591e-02 11 0.999136 8.644443e-04 12 0.999505 4.950979e-04 13 1.000000 2.220446e-15 14 1.571759 5.717593e-01 15 4.668043 3.668043e+00 16 1.000000 8.978596e-12 17 1.000000 2.220446e-16 18 24.423056 2.342306e+01 19 1.000000 2.220446e-16 20 1.000000 2.220446e-16 A = 3 B = 5 Rule QUAD ERROR 1 0.384088 6.159122e-01 2 0.000000 1.000000e+00 3 0.544624 4.553755e-01 4 3.281250 2.281250e+00 5 0.694511 3.054894e-01 6 0.802083 1.979167e-01 7 1.267940 2.679398e-01 8 0.972131 2.786944e-02 9 0.992329 7.670782e-03 10 0.992477 7.522586e-03 11 0.961034 3.896621e-02 12 1.011537 1.153677e-02 13 1.011658 1.165761e-02 14 1.485340 4.853395e-01 15 5.955250 4.955250e+00 16 1.000000 1.939005e-11 17 1.000000 2.220446e-16 18 41.386725 4.038672e+01 19 1.000000 2.220446e-16 20 1.000000 0.000000e+00 A = 3 B = 6 Rule QUAD ERROR 1 0.234720 7.652797e-01 2 0.000000 1.000000e+00 3 0.635854 3.641463e-01 4 3.007812 2.007812e+00 5 0.800594 1.994062e-01 6 0.903681 9.631944e-02 7 1.285231 2.852313e-01 8 0.846318 1.536821e-01 9 1.160570 1.605699e-01 10 0.896388 1.036119e-01 11 0.920984 7.901635e-02 12 1.030392 3.039232e-02 13 1.028932 2.893207e-02 14 1.308749 3.087491e-01 15 5.563686 4.563686e+00 16 1.000000 2.584200e-11 17 0.999369 6.306245e-04 18 65.334352 6.433435e+01 19 1.000000 4.440892e-16 20 1.000000 0.000000e+00 A = 3 B = 7 Rule QUAD ERROR 1 0.134126 8.658741e-01 2 0.000000 1.000000e+00 3 0.718176 2.818239e-01 4 2.578125 1.578125e+00 5 0.860170 1.398298e-01 6 0.995347 4.652778e-03 7 1.257470 2.574696e-01 8 0.720497 2.795035e-01 9 1.375005 3.750050e-01 10 0.797080 2.029196e-01 11 0.902218 9.778175e-02 12 1.042706 4.270623e-02 13 1.039343 3.934337e-02 14 1.091607 9.160665e-02 15 3.712810 2.712810e+00 16 1.000000 2.879763e-11 17 0.999861 1.393829e-04 18 97.503822 9.650382e+01 19 1.000000 5.551115e-16 20 1.000000 0.000000e+00 A = 4 B = 0 Rule QUAD ERROR 1 0.185185 8.148148e-01 2 5.000000 4.000000e+00 3 0.995370 4.629630e-03 4 0.625000 3.750000e-01 5 0.933333 6.666667e-02 6 0.958333 4.166667e-02 7 1.025849 2.584877e-02 8 1.000000 2.220446e-15 9 1.000000 3.552714e-15 10 1.000000 4.440892e-16 11 1.000000 2.886580e-15 12 1.000000 5.107026e-15 13 1.000000 2.442491e-15 14 1.083333 8.333333e-02 15 1.000000 6.712765e-10 16 1.000000 2.719480e-11 17 1.000000 6.661338e-16 18 1.000000 2.220446e-16 19 1.000000 2.220446e-16 20 1.000000 0.000000e+00 A = 4 B = 1 Rule QUAD ERROR 1 0.432099 5.679012e-01 2 0.000000 1.000000e+00 3 1.174769 1.747685e-01 4 1.093750 9.375000e-02 5 1.244444 2.444444e-01 6 1.166667 1.666667e-01 7 1.402971 4.029707e-01 8 0.986815 1.318484e-02 9 1.037037 3.703704e-02 10 1.000000 5.551115e-16 11 1.000000 1.665335e-15 12 1.000000 2.664535e-15 13 1.000000 8.881784e-16 14 0.631944 3.680556e-01 15 1.206060 2.060598e-01 16 1.000000 2.332357e-11 17 1.000000 1.110223e-16 18 3.562986 2.562986e+00 19 1.000000 1.110223e-16 20 1.000000 1.110223e-16 A = 4 B = 2 Rule QUAD ERROR 1 0.576132 4.238683e-01 2 0.000000 1.000000e+00 3 0.819187 1.808128e-01 4 2.187500 1.187500e+00 5 0.949926 5.007407e-02 6 0.962500 3.750000e-02 7 1.428627 4.286265e-01 8 0.958434 4.156566e-02 9 1.030123 3.012346e-02 10 0.962963 3.703704e-02 11 0.983741 1.625887e-02 12 1.000000 5.329071e-15 13 1.000000 8.881784e-16 14 1.134259 1.342593e-01 15 1.619406 6.194063e-01 16 1.000000 2.948708e-11 17 1.000000 0.000000e+00 18 10.112792 9.112792e+00 19 1.000000 2.220446e-16 20 1.000000 0.000000e+00 A = 4 B = 3 Rule QUAD ERROR 1 0.576132 4.238683e-01 2 0.000000 1.000000e+00 3 0.481610 5.183899e-01 4 3.281250 2.281250e+00 5 0.591526 4.084741e-01 6 0.729167 2.708333e-01 7 1.470486 4.704861e-01 8 1.057370 5.737044e-02 9 0.905679 9.432099e-02 10 1.053666 5.366591e-02 11 0.999136 8.644443e-04 12 0.999505 4.950979e-04 13 1.000000 2.220446e-15 14 1.571759 5.717593e-01 15 3.110126 2.110126e+00 16 1.000000 2.143774e-11 17 1.000000 2.220446e-16 18 24.422282 2.342228e+01 19 1.000000 1.110223e-16 20 1.000000 2.220446e-16 A = 4 B = 4 Rule QUAD ERROR 1 0.480110 5.198903e-01 2 0.000000 1.000000e+00 3 0.320698 6.793017e-01 4 4.101562 3.101562e+00 5 0.414538 5.854617e-01 6 0.601563 3.984375e-01 7 1.584925 5.849248e-01 8 1.129907 1.299070e-01 9 0.799671 2.003292e-01 10 1.137746 1.377461e-01 11 1.004338 4.338376e-03 12 0.983592 1.640758e-02 13 0.985455 1.454491e-02 14 1.856674 8.566744e-01 15 6.940971 5.940971e+00 16 1.000000 6.638468e-12 17 1.000000 2.220446e-16 18 51.487415 5.048742e+01 19 1.000000 0.000000e+00 20 1.000000 0.000000e+00 A = 4 B = 5 Rule QUAD ERROR 1 0.352080 6.479195e-01 2 0.000000 1.000000e+00 3 0.293630 7.063704e-01 4 4.511719 3.511719e+00 5 0.402555 5.974453e-01 6 0.585521 4.144792e-01 7 1.690193 6.901926e-01 8 1.106105 1.061048e-01 9 0.796322 2.036785e-01 10 1.142490 1.424904e-01 11 0.974400 2.559962e-02 12 0.982741 1.725909e-02 13 0.985505 1.449536e-02 14 1.963124 9.631237e-01 15 13.897430 1.289743e+01 16 1.000000 8.099521e-12 17 1.000176 1.759882e-04 18 97.510098 9.651010e+01 19 1.000000 4.440892e-16 20 1.000000 2.220446e-16 A = 4 B = 6 Rule QUAD ERROR 1 0.234720 7.652797e-01 2 0.000000 1.000000e+00 3 0.332597 6.674031e-01 4 4.511719 3.511719e+00 5 0.467461 5.325390e-01 6 0.643003 3.569965e-01 7 1.725263 7.252631e-01 8 0.998828 1.171600e-03 9 0.905044 9.495601e-02 10 1.069173 6.917320e-02 11 0.916307 8.369330e-02 12 1.007858 7.858145e-03 13 1.010243 1.024312e-02 14 1.910312 9.103116e-01 15 21.577744 2.057774e+01 16 1.000000 1.993050e-11 17 0.997408 2.591954e-03 18 169.890739 1.688907e+02 19 1.000000 0.000000e+00 20 1.000000 2.220446e-16 A = 5 B = 0 Rule QUAD ERROR 1 0.086420 9.135802e-01 2 7.000000 6.000000e+00 3 0.923611 7.638889e-02 4 0.437500 5.625000e-01 5 0.808889 1.911111e-01 6 0.875000 1.250000e-01 7 0.945306 5.469393e-02 8 1.005274 5.273935e-03 9 0.985185 1.481481e-02 10 1.000000 7.771561e-16 11 1.000000 3.441691e-15 12 1.000000 5.995204e-15 13 1.000000 2.886580e-15 14 1.263889 2.638889e-01 15 1.000000 7.678156e-10 16 1.000000 2.874323e-11 17 1.000000 8.881784e-16 18 1.000000 4.440892e-16 19 1.000000 1.110223e-16 20 1.000000 1.110223e-16 A = 5 B = 1 Rule QUAD ERROR 1 0.230453 7.695473e-01 2 0.000000 1.000000e+00 3 1.235082 2.350823e-01 4 0.875000 1.250000e-01 5 1.253570 2.535704e-01 6 1.201667 2.016667e-01 7 1.480298 4.802984e-01 8 0.982247 1.775274e-02 9 1.096494 9.649383e-02 10 1.013757 1.375661e-02 11 1.012218 1.221783e-02 12 1.000000 3.774758e-15 13 1.000000 1.776357e-15 14 0.453704 5.462963e-01 15 1.147940 1.479403e-01 16 1.000000 1.962541e-11 17 1.000000 3.330669e-16 18 4.473902 3.473902e+00 19 1.000000 2.220446e-16 20 1.000000 1.110223e-16 A = 5 B = 2 Rule QUAD ERROR 1 0.345679 6.543210e-01 2 0.000000 1.000000e+00 3 0.937114 6.288580e-02 4 1.968750 9.687500e-01 5 1.080676 8.067556e-02 6 1.067500 6.750000e-02 7 1.673032 6.730324e-01 8 0.881502 1.184980e-01 9 1.140741 1.407407e-01 10 0.909751 9.024943e-02 11 0.967270 3.272971e-02 12 1.000891 8.911763e-04 13 1.000000 1.554312e-15 14 0.943056 5.694444e-02 15 1.428752 4.287522e-01 16 1.000000 3.592704e-11 17 1.000000 1.110223e-16 18 15.020982 1.402098e+01 19 1.000000 3.330669e-16 20 1.000000 0.000000e+00 A = 5 B = 3 Rule QUAD ERROR 1 0.384088 6.159122e-01 2 0.000000 1.000000e+00 3 0.544624 4.553755e-01 4 3.281250 2.281250e+00 5 0.694511 3.054894e-01 6 0.802083 1.979167e-01 7 1.872878 8.728781e-01 8 0.972131 2.786944e-02 9 0.992329 7.670782e-03 10 0.992477 7.522586e-03 11 0.961034 3.896621e-02 12 1.011537 1.153677e-02 13 1.011658 1.165761e-02 14 1.485340 4.853395e-01 15 2.862390 1.862390e+00 16 1.000000 3.623701e-11 17 1.000000 2.220446e-16 18 41.385512 4.038551e+01 19 1.000000 0.000000e+00 20 1.000000 0.000000e+00 A = 5 B = 4 Rule QUAD ERROR 1 0.352080 6.479195e-01 2 0.000000 1.000000e+00 3 0.293630 7.063704e-01 4 4.511719 3.511719e+00 5 0.402555 5.974453e-01 6 0.585521 4.144792e-01 7 2.165501 1.165501e+00 8 1.106105 1.061048e-01 9 0.796322 2.036785e-01 10 1.142490 1.424904e-01 11 0.974400 2.559962e-02 12 0.982741 1.725909e-02 13 0.985505 1.449536e-02 14 1.963124 9.631237e-01 15 8.058801 7.058801e+00 16 1.000000 2.432399e-11 17 1.000176 1.759882e-04 18 97.509981 9.650998e+01 19 1.000000 2.220446e-16 20 1.000000 2.220446e-16 A = 5 B = 5 Rule QUAD ERROR 1 0.281664 7.183356e-01 2 0.000000 1.000000e+00 3 0.187868 8.121321e-01 4 5.414062 4.414062e+00 5 0.273552 7.264477e-01 6 0.473229 5.267708e-01 7 2.450563 1.450563e+00 8 1.166185 1.661845e-01 9 0.679036 3.209635e-01 10 1.239579 2.395788e-01 11 0.968337 3.166255e-02 12 0.950326 4.967432e-02 13 0.955574 4.442596e-02 14 2.292374 1.292374e+00 15 23.244480 2.224448e+01 16 1.000000 8.464118e-12 17 1.003019 3.018643e-03 18 203.677701 2.026777e+02 19 1.000000 4.440892e-16 20 1.000000 4.440892e-16 A = 6 B = 0 Rule QUAD ERROR 1 0.038409 9.615912e-01 2 9.333333 8.333333e+00 3 0.819787 1.802126e-01 4 0.291667 7.083333e-01 5 0.660662 3.393383e-01 6 0.766111 2.338889e-01 7 0.834331 1.656693e-01 8 1.012949 1.294949e-02 9 0.948082 5.191770e-02 10 0.995414 4.585538e-03 11 0.995927 4.072611e-03 12 1.000000 6.439294e-15 13 1.000000 3.108624e-15 14 1.533951 5.339506e-01 15 1.000000 8.497045e-10 16 1.000000 3.178280e-11 17 1.000000 1.332268e-15 18 1.000000 2.220446e-16 19 1.000000 2.220446e-16 20 1.000000 1.110223e-16 A = 6 B = 1 Rule QUAD ERROR 1 0.115226 8.847737e-01 2 0.000000 1.000000e+00 3 1.230581 2.305813e-01 4 0.656250 3.437500e-01 5 1.166625 1.666252e-01 6 1.178333 1.783333e-01 7 1.476247 4.762474e-01 8 0.999117 8.829594e-04 9 1.151802 1.518025e-01 10 1.043386 4.338624e-02 11 1.040463 4.046303e-02 12 0.999307 6.931371e-04 13 1.000000 2.220446e-15 14 0.314352 6.856481e-01 15 1.102502 1.025015e-01 16 1.000000 1.147393e-11 17 1.000000 4.440892e-16 18 5.454145 4.454145e+00 19 1.000000 1.110223e-16 20 1.000000 0.000000e+00 A = 6 B = 2 Rule QUAD ERROR 1 0.192044 8.079561e-01 2 0.000000 1.000000e+00 3 1.028485 2.848508e-02 4 1.640625 6.406250e-01 5 1.133495 1.334953e-01 6 1.144792 1.447917e-01 7 1.847447 8.474473e-01 8 0.811392 1.886079e-01 9 1.286387 2.863868e-01 10 0.866501 1.334989e-01 11 0.970273 2.972680e-02 12 0.996224 3.775557e-03 13 0.994117 5.883009e-03 14 0.742670 2.573302e-01 15 1.241529 2.415285e-01 16 1.000000 3.572032e-11 17 1.000000 3.330669e-16 18 21.146164 2.014616e+01 19 1.000000 7.771561e-16 20 1.000000 2.220446e-16 A = 6 B = 3 Rule QUAD ERROR 1 0.234720 7.652797e-01 2 0.000000 1.000000e+00 3 0.635854 3.641463e-01 4 3.007812 2.007812e+00 5 0.800594 1.994062e-01 6 0.903681 9.631944e-02 7 2.262255 1.262255e+00 8 0.846318 1.536821e-01 9 1.160570 1.605699e-01 10 0.896388 1.036119e-01 11 0.920984 7.901635e-02 12 1.030392 3.039232e-02 13 1.028932 2.893207e-02 14 1.308749 3.087491e-01 15 2.379168 1.379168e+00 16 1.000000 4.616441e-11 17 0.999369 6.306245e-04 18 65.332696 6.433270e+01 19 1.000000 4.440892e-16 20 1.000000 2.220446e-16 A = 6 B = 4 Rule QUAD ERROR 1 0.234720 7.652797e-01 2 0.000000 1.000000e+00 3 0.332597 6.674031e-01 4 4.511719 3.511719e+00 5 0.467461 5.325390e-01 6 0.643003 3.569965e-01 7 2.834317 1.834317e+00 8 0.998828 1.171600e-03 9 0.905044 9.495601e-02 10 1.069173 6.917320e-02 11 0.916307 8.369330e-02 12 1.007858 7.858145e-03 13 1.010243 1.024312e-02 14 1.910312 9.103116e-01 15 7.676168 6.676168e+00 16 1.000000 4.018341e-11 17 0.997408 2.591954e-03 18 169.890556 1.688906e+02 19 1.000000 2.220446e-16 20 1.000000 2.220446e-16 A = 7 B = 0 Rule QUAD ERROR 1 0.016461 9.835391e-01 2 12.000000 1.100000e+01 3 0.702418 2.975823e-01 4 0.187500 8.125000e-01 5 0.516101 4.838993e-01 6 0.648333 3.516667e-01 7 0.711784 2.882159e-01 8 1.016902 1.690162e-02 9 0.889877 1.101235e-01 10 0.981708 1.829176e-02 11 0.983203 1.679708e-02 12 1.000198 1.980392e-04 13 1.000000 3.108624e-15 14 1.882407 8.824074e-01 15 1.000000 9.115336e-10 16 1.000000 3.758549e-11 17 1.000000 1.554312e-15 18 1.000000 4.440892e-16 19 1.000000 0.000000e+00 20 1.000000 0.000000e+00 A = 7 B = 1 Rule QUAD ERROR 1 0.054870 9.451303e-01 2 0.000000 1.000000e+00 3 1.170910 1.709105e-01 4 0.468750 5.312500e-01 5 1.020816 2.081580e-02 6 1.106250 1.062500e-01 7 1.404985 4.049854e-01 8 1.035863 3.586257e-02 9 1.184724 1.847243e-01 10 1.082131 8.213142e-02 11 1.079284 7.928415e-02 12 0.999233 7.673120e-04 13 1.001712 1.711833e-03 14 0.212191 7.878086e-01 15 1.071614 7.161425e-02 16 1.000000 1.082801e-12 17 1.000000 5.551115e-16 18 6.485790 5.485790e+00 19 1.000000 2.220446e-16 20 1.000000 0.000000e+00 A = 7 B = 2 Rule QUAD ERROR 1 0.100594 8.994056e-01 2 0.000000 1.000000e+00 3 1.074121 7.412051e-02 4 1.289062 2.890625e-01 5 1.103432 1.034316e-01 6 1.175625 1.756250e-01 7 1.932415 9.324149e-01 8 0.769683 2.303172e-01 9 1.433446 4.334464e-01 10 0.846090 1.539098e-01 11 1.001166 1.166369e-03 12 0.981678 1.832186e-02 13 0.979223 2.077746e-02 14 0.560892 4.391075e-01 15 1.082499 8.249901e-02 16 1.000000 2.975220e-11 17 1.000729 7.290941e-04 18 28.507579 2.750758e+01 19 1.000000 1.332268e-15 20 1.000000 2.220446e-16 A = 7 B = 3 Rule QUAD ERROR 1 0.134126 8.658741e-01 2 0.000000 1.000000e+00 3 0.718176 2.818239e-01 4 2.578125 1.578125e+00 5 0.860170 1.398298e-01 6 0.995347 4.652778e-03 7 2.577771 1.577771e+00 8 0.720497 2.795035e-01 9 1.375005 3.750050e-01 10 0.797080 2.029196e-01 11 0.902218 9.778175e-02 12 1.042706 4.270623e-02 13 1.039343 3.934337e-02 14 1.091607 9.160665e-02 15 1.849725 8.497249e-01 16 1.000000 5.043688e-11 17 0.999861 1.393829e-04 18 97.501716 9.650172e+01 19 1.000000 6.661338e-16 20 1.000000 0.000000e+00 A = 8 B = 0 Rule QUAD ERROR 1 0.006859 9.931413e-01 2 15.000000 1.400000e+01 3 0.585294 4.147055e-01 4 0.117188 8.828125e-01 5 0.389922 6.100780e-01 6 0.533854 4.661458e-01 7 0.591152 4.088479e-01 8 1.012161 1.216139e-02 9 0.816165 1.838354e-01 10 0.956602 4.339756e-02 11 0.959183 4.081739e-02 12 1.000439 4.393770e-04 13 0.999572 4.279584e-04 14 2.299961 1.299961e+00 15 0.999433 5.668944e-04 16 1.000000 4.671130e-11 17 1.000000 1.998401e-15 18 1.000000 9.992007e-16 19 1.000000 2.220446e-16 20 1.000000 4.440892e-16 A = 8 B = 1 Rule QUAD ERROR 1 0.025149 9.748514e-01 2 0.000000 1.000000e+00 3 1.073089 7.308903e-02 4 0.322266 6.777344e-01 5 0.852434 1.475661e-01 6 1.001458 1.458333e-03 7 1.287682 2.876823e-01 8 1.083738 8.373832e-02 9 1.187843 1.878431e-01 10 1.120935 1.209353e-01 11 1.119573 1.195726e-01 12 1.002030 2.030401e-03 13 1.007157 7.157358e-03 14 0.140223 8.597769e-01 15 1.055669 5.566901e-02 16 1.000000 9.668488e-12 17 0.999588 4.116868e-04 18 7.554137 6.554137e+00 19 1.000000 8.881784e-16 20 1.000000 0.000000e+00 A = 8 B = 2 Rule QUAD ERROR 1 0.050297 9.497028e-01 2 0.000000 1.000000e+00 3 1.073286 7.328550e-02 4 0.966797 3.320312e-02 5 1.012554 1.255422e-02 6 1.158724 1.587240e-01 7 1.931663 9.316634e-01 8 0.761318 2.386820e-01 9 1.557612 5.576121e-01 10 0.849523 1.504772e-01 11 1.057002 5.700238e-02 12 0.957945 4.205474e-02 13 0.956754 4.324599e-02 14 0.409352 5.906475e-01 15 0.960182 3.981790e-02 16 1.000000 1.940981e-11 17 1.002323 2.322910e-03 18 37.103355 3.610335e+01 19 1.000000 9.992007e-16 20 1.000000 2.220446e-16 A = 9 B = 0 Rule QUAD ERROR 1 0.002794 9.972057e-01 2 18.333333 1.733333e+01 3 0.476896 5.231043e-01 4 0.071615 9.283854e-01 5 0.287142 7.128585e-01 6 0.429942 5.700579e-01 7 0.480476 5.195239e-01 8 0.996255 3.744597e-03 9 0.733569 2.664306e-01 10 0.920084 7.991597e-02 11 0.923540 7.645961e-02 12 1.000086 8.581607e-05 13 0.997886 2.113584e-03 14 2.779903 1.779903e+00 15 0.997194 2.806473e-03 16 1.000000 5.923961e-11 17 1.000091 9.148595e-05 18 1.000000 1.332268e-15 19 1.000000 4.440892e-16 20 1.000000 2.220446e-16 A = 9 B = 1 Rule QUAD ERROR 1 0.011177 9.888228e-01 2 0.000000 1.000000e+00 3 0.953802 4.619766e-02 4 0.214844 7.851562e-01 5 0.686695 3.133048e-01 6 0.879682 1.203183e-01 7 1.144557 1.445573e-01 8 1.132942 1.329422e-01 9 1.161458 1.614578e-01 10 1.152148 1.521482e-01 11 1.152968 1.529678e-01 12 1.009418 9.418068e-03 13 1.017444 1.744391e-02 14 0.090967 9.090328e-01 15 1.053459 5.345945e-02 16 1.000000 1.936118e-11 17 0.998106 1.894114e-03 18 8.646853 7.646853e+00 19 1.000000 1.554312e-15 20 1.000000 1.110223e-16 A = 10 B = 0 Rule QUAD ERROR 1 0.001118 9.988823e-01 2 22.000000 2.100000e+01 3 0.381514 6.184856e-01 4 0.042969 9.570312e-01 5 0.207231 7.927692e-01 6 0.339994 6.600058e-01 7 0.383663 6.163372e-01 8 0.968918 3.108196e-02 9 0.647992 3.520083e-01 10 0.873671 1.263288e-01 11 0.877655 1.223451e-01 12 0.998219 1.780634e-03 13 0.993975 6.025082e-03 14 3.317690 2.317690e+00 15 0.992010 7.990165e-03 16 1.000000 7.496015e-11 17 1.000489 4.886059e-04 18 0.999970 3.026403e-05 19 1.000000 7.771561e-16 20 1.000000 4.440892e-16 TEST43 TRIANGLE_UNIT_PRODUCT_SET sets up a product quadrature rule in the unit triangle, TRIANGLE_UNIT_SUM applies it. Rule Order: 1 2 3 4 5 Function 1 0.500000 0.500000 0.500000 0.500000 0.500000 X 0.166667 0.166667 0.166667 0.166667 0.166667 X^2 0.055556 0.083333 0.083333 0.083333 0.083333 X^3 0.018519 0.050000 0.050000 0.050000 0.050000 X^4 0.006173 0.031667 0.033333 0.033333 0.033333 X^5 0.002058 0.020333 0.023810 0.023810 0.023810 X^6 0.000686 0.013100 0.017755 0.017857 0.017857 R 0.235702 0.269448 0.270290 0.270484 0.270520 SIN(X) 0.163597 0.158501 0.158529 0.158529 0.158529 EXP(X) 0.697806 0.718176 0.718282 0.718282 0.718282 1/(1+R) 0.339811 0.331609 0.330728 0.330539 0.330502 SQRT(R) 0.343295 0.358464 0.359418 0.359703 0.359777 Rule Order: 6 7 8 Function 1 0.500000 0.500000 0.500000 X 0.166667 0.166667 0.166667 X^2 0.083333 0.083333 0.083333 X^3 0.050000 0.050000 0.050000 X^4 0.033333 0.033333 0.033333 X^5 0.023810 0.023810 0.023810 X^6 0.017857 0.017857 0.017857 R 0.270531 0.270535 0.270536 SIN(X) 0.158529 0.158529 0.158529 EXP(X) 0.718282 0.718282 0.718282 1/(1+R) 0.330491 0.330488 0.330486 SQRT(R) 0.359804 0.359815 0.359820 TEST44 TRIANGLE_UNIT_SET sets up quadrature in the unit triangle, TRIANGLE_SUM applies it to an arbitrary triangle. Rule: 1 2 3 4 5 Function 1 3.000000 3.000000 3.000000 3.000000 3.000000 X 5.000000 5.000000 5.000000 5.000000 5.000000 X^2 8.333333 11.000000 9.000000 9.000000 9.000000 X^3 13.888889 29.000000 17.444444 17.000000 17.400000 X^4 23.148148 83.000000 35.962963 33.000000 35.586667 X^5 38.580247 245.000000 77.592593 65.000000 75.631111 X^6 64.300412 731.000000 172.621399 129.000000 164.516919 R 7.810250 8.699597 8.063771 8.130212 8.070513 SIN(X) 2.986224 1.824062 2.666962 2.660066 2.663055 EXP(X) 15.883470 25.522101 17.899594 17.496394 17.839632 1/(1+R) 0.832543 0.849661 0.831027 0.817837 0.829807 SQRT(R) 4.840532 4.998030 4.894053 4.925549 4.897112 Rule: 6 7 8 9 10 Function 1 3.000000 3.000000 3.000000 3.000000 3.000000 X 5.000000 5.000000 5.000000 5.000000 5.000000 X^2 9.000000 9.000000 9.000000 9.000000 9.000000 X^3 17.400000 17.400000 17.400000 17.400000 17.400000 X^4 35.666667 35.666667 35.800000 35.800000 35.800000 X^5 76.333333 76.333333 77.595538 77.503704 77.571429 X^6 168.253333 168.259259 175.519167 174.523358 175.254271 R 8.070586 8.052964 8.071099 8.070754 8.070862 SIN(X) 2.666280 2.666272 2.671423 2.671598 2.671474 EXP(X) 17.859230 17.859274 17.897027 17.892243 17.895741 1/(1+R) 0.829651 0.831470 0.829481 0.829364 0.829501 SQRT(R) 4.897305 4.891249 4.897669 4.897664 4.897579 Rule: 11 12 13 14 15 Function 1 3.000000 3.000000 3.000000 3.000000 3.000000 X 5.000000 5.000000 5.000000 5.000000 8.260726 X^2 9.000000 9.000000 9.000000 9.000000 68.660391 X^3 17.400000 17.400000 17.400000 17.400000 2981.185967 X^4 35.800000 35.800000 35.800000 36.066667 699384.698867 X^5 77.571429 77.571429 77.571429 80.1111111053914594.148228 X^6 175.257788 175.285714 175.285714 190.18518512377587786033.957031 R 8.071094 8.070963 8.070961 8.071637 14.391527 SIN(X) 2.671469 2.671433 2.671433 2.681436 1.707667 EXP(X) 17.895767 17.896039 17.896036 17.97443443414282.682041 1/(1+R) 0.829420 0.829459 0.829460 0.829228 0.851903 SQRT(R) 4.897720 4.897645 4.897644 4.898163 5.562446 Rule: 16 17 18 19 20 Function 1 3.000000 3.000000 3.000000 3.000000 3.000000 X 5.000000 5.000000 5.113148 5.000000 5.000000 X^2 9.000000 9.000000 9.478799 9.000000 9.000000 X^3 17.400000 17.400000 18.907301 17.400000 17.400000 X^4 35.800000 35.800000 40.031268 35.800000 35.800000 X^5 77.571429 77.571429 88.804736 77.571429 77.571429 X^6 175.285714 175.285714 204.236733 175.285714 175.285714 R 8.070963 8.070963 8.400637 8.070963 8.070963 SIN(X) 2.671433 2.671433 2.613613 2.671433 2.671433 EXP(X) 17.896037 17.896037 18.830738 17.896037 17.896037 1/(1+R) 0.829458 0.829459 0.804234 0.829458 0.829458 SQRT(R) 4.897646 4.897645 4.998644 4.897646 4.897646 TEST45 TORUS_1 approximates integrals on a torus. The order N will be varied. Inner radius = 0.500000 Outer radius = 1.000000 Area = 19.739209 F(X) 1 4 16 64 256 1 19.739209 19.739209 19.739209 19.739209 19.739209 X 0.000000 -0.000000 0.000000 -0.000000 0.000000 Y -0.000000 -0.000000 0.000000 -0.000000 0.000000 Z 0.000000 0.000000 0.000000 -0.000000 -0.000000 X*X 44.413220 17.271808 17.271808 17.271808 17.271808 X*Y -0.000000 -0.000000 -0.000000 0.000000 -0.000000 X*Z 0.000000 -0.000000 0.000000 -0.000000 0.000000 Y*Y 0.000000 17.271808 17.271808 17.271808 17.271808 Y*Z 0.000000 0.000000 0.000000 0.000000 0.000000 Z*Z -0.000000 9.869604 9.869604 9.869604 9.869604 X^3 0.000000 -0.000000 -0.000000 0.000000 -0.000000 X*Y*Z 0.000000 -0.000000 0.000000 0.000000 0.000000 Z*Z*Z 0.000000 0.000000 -0.000000 -0.000000 -0.000000 X^4 99.929745 23.594523 23.594523 23.594523 23.594523 X^2 Z^2 0.000000 4.934802 4.934802 4.934802 4.934802 Z^4 -0.000000 7.402203 7.402203 7.402203 7.402203 X^5 0.000000 5.513099 -0.000000 -0.000000 -0.000000 X^6 224.841925 39.227822 37.878462 37.878462 37.878462 R 29.608813 30.567818 30.544219 30.544218 30.544218 SIN(X) 0.000000 0.042146 0.000000 -0.000000 0.000000 EXP(X) 46.434705 29.464590 29.412502 29.412502 29.412502 1/(1+R) 10.949343 10.467585 10.471606 10.471606 10.471606 SQRT(R) 24.175495 24.897002 24.871615 24.871614 24.871614 TEST46 For the interior of a torus, TORUS_5S2, TORUS_6S2, and TORUS_5S2 approximate integrals. Inner radius = 0.500000 Outer radius = 1.000000 Volume = 9.869604 Rule: #5S2 #6S2 #14S F(X) 1 9.869604 9.869604 9.869604 X 0.000000 -0.000000 -0.000000 Y -0.000000 -0.000000 0.000000 Z 0.000000 0.000000 0.000000 X*X 4.934802 4.934802 4.934802 X*Y 0.000000 0.000000 -0.000000 X*Z 0.000000 0.000000 -0.000000 Y*Y 4.934802 4.934802 4.934802 Y*Z 0.000000 0.000000 -0.000000 Z*Z 2.467401 2.467401 2.467401 X^3 0.000000 -0.000000 -0.000000 X*Y*Z 0.000000 0.000000 -0.000000 Z*Z*Z 0.000000 0.000000 -0.000000 X^4 4.857696 4.857696 4.857696 X^2 Z^2 0.925275 0.925275 0.925275 Z^4 1.233701 1.233701 1.233701 X^5 0.000000 0.000000 0.000000 X^6 7.209438 6.120311 6.120311 R 10.787101 10.778919 10.786683 SIN(X) -0.000000 -0.000073 -0.000000 EXP(X) 12.549746 12.548212 12.548132 1/(1+R) 6.711141 6.710728 6.711377 SQRT(R) 10.254271 10.243787 10.255207 TEST47 For integrals inside a torus with square cross-section: TORUS_SQUARE_5C2 approximates the integral; TORUS_SQUARE_14C approximates the integral. Inner radius = 1.000000 Outer radius = 0.125000 Volume = 0.392699 F(X) 5C2 14C 1 0.392699 0.392699 X 0.000000 0.000000 Y 0.000000 0.000000 Z 0.000000 0.000000 X*X 0.199418 0.199418 X*Y 0.000000 0.000000 X*Z 0.000000 -0.000000 Y*Y 0.199418 0.199418 Y*Z 0.000000 -0.000000 Z*Z 0.002045 0.002045 X^3 -0.000000 -0.000000 X*Y*Z 0.000000 -0.000000 Z*Z*Z 0.000000 0.000000 X^4 0.154968 0.154968 X^2 Z^2 0.001039 0.001039 Z^4 0.000019 0.000019 X^5 0.000000 0.000000 X^6 0.149986 0.136351 R 0.395754 0.395765 SIN(X) 0.000000 -0.000000 EXP(X) 0.499077 0.499057 1/(1+R) 0.276782 0.276778 SQRT(R) 0.393967 0.393977 TEST48 For evenly spaced angles between 0 and 2*PI: TVEC_EVEN TVEC_EVEN2 TVEC_EVEN3 TVEC_EVEN: 0: 0.000000 1: 1.570796 2: 3.141593 3: 4.712389 TVEC_EVEN2: 0: 0.785398 1: 2.356194 2: 3.926991 3: 5.497787 TVEC_EVEN3: 0: 0.000000 1: 2.094395 2: 4.188790 3: 6.283185 TEST49 For evenly spaced angles between THETA1 and THETA2: TVEC_EVEN_BRACKET TVEC_EVEN_BRACKET2. TVEC_EVEN_BRACKET3. THETA1 = 30.000000 THETA2 = 90.000000 TVEC_EVEN_BRACKET 0: 30.000000 1: 50.000000 2: 70.000000 3: 90.000000 TVEC_EVEN_BRACKET2 0: 40.000000 1: 50.000000 2: 60.000000 3: 70.000000 4: 80.000000 TVEC_EVEN_BRACKET3 0: 40.000000 1: 60.000000 2: 80.000000 STROUD_TEST Normal end of execution. 02 March 2022 02:57:06 PM