20 January 2017 08:58:13 PM SANDIA_RULES_PRB C version Test the SANDIA_RULES library. TEST01 CHEBYSHEV1_COMPUTE computes a Gauss-Chebyshev type 1 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) / sqrt ( 1 - x^2 ) dx. CHEBYSHEV1_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 3.141593 3.141593 0.000000e+00 1 1 0.000000 0.000000 0.000000e+00 1 2 0.000000 1.570796 1.570796e+00 1 3 0.000000 0.000000 0.000000e+00 1 4 0.000000 1.178097 1.178097e+00 2 0 3.141593 3.141593 0.000000e+00 2 1 0.000000 0.000000 0.000000e+00 2 2 1.570796 1.570796 0.000000e+00 2 3 0.000000 0.000000 2.220446e-16 2 4 0.785398 1.178097 3.926991e-01 2 5 0.000000 0.000000 2.220446e-16 2 6 0.392699 0.981748 5.890486e-01 3 0 3.141593 3.141593 0.000000e+00 3 1 0.000000 0.000000 0.000000e+00 3 2 1.570796 1.570796 2.220446e-16 3 3 0.000000 0.000000 0.000000e+00 3 4 1.178097 1.178097 2.220446e-16 3 5 0.000000 0.000000 0.000000e+00 3 6 0.883573 0.981748 9.817477e-02 3 7 0.000000 0.000000 0.000000e+00 3 8 0.662680 0.859029 1.963495e-01 4 0 3.141593 3.141593 0.000000e+00 4 1 -0.000000 0.000000 1.110223e-16 4 2 1.570796 1.570796 0.000000e+00 4 3 0.000000 0.000000 1.110223e-16 4 4 1.178097 1.178097 0.000000e+00 4 5 0.000000 0.000000 0.000000e+00 4 6 0.981748 0.981748 1.110223e-16 4 7 0.000000 0.000000 0.000000e+00 4 8 0.834486 0.859029 2.454369e-02 4 9 0.000000 0.000000 0.000000e+00 4 10 0.711767 0.773126 6.135923e-02 5 0 3.141593 3.141593 0.000000e+00 5 1 0.000000 0.000000 1.110223e-16 5 2 1.570796 1.570796 0.000000e+00 5 3 0.000000 0.000000 0.000000e+00 5 4 1.178097 1.178097 2.220446e-16 5 5 0.000000 0.000000 0.000000e+00 5 6 0.981748 0.981748 2.220446e-16 5 7 0.000000 0.000000 0.000000e+00 5 8 0.859029 0.859029 4.440892e-16 5 9 0.000000 0.000000 0.000000e+00 5 10 0.766990 0.773126 6.135923e-03 5 11 0.000000 0.000000 0.000000e+00 5 12 0.690291 0.708699 1.840777e-02 6 0 3.141593 3.141593 4.440892e-16 6 1 0.000000 0.000000 2.220446e-16 6 2 1.570796 1.570796 0.000000e+00 6 3 0.000000 0.000000 2.220446e-16 6 4 1.178097 1.178097 0.000000e+00 6 5 0.000000 0.000000 2.220446e-16 6 6 0.981748 0.981748 1.110223e-16 6 7 0.000000 0.000000 3.885781e-16 6 8 0.859029 0.859029 3.330669e-16 6 9 0.000000 0.000000 3.885781e-16 6 10 0.773126 0.773126 1.110223e-16 6 11 0.000000 0.000000 4.996004e-16 6 12 0.707165 0.708699 1.533981e-03 6 13 0.000000 0.000000 5.551115e-16 6 14 0.652709 0.658078 5.368933e-03 7 0 3.141593 3.141593 0.000000e+00 7 1 0.000000 0.000000 1.110223e-16 7 2 1.570796 1.570796 2.220446e-16 7 3 0.000000 0.000000 1.110223e-16 7 4 1.178097 1.178097 0.000000e+00 7 5 0.000000 0.000000 5.551115e-17 7 6 0.981748 0.981748 1.110223e-16 7 7 -0.000000 0.000000 5.551115e-17 7 8 0.859029 0.859029 1.110223e-16 7 9 -0.000000 0.000000 1.665335e-16 7 10 0.773126 0.773126 3.330669e-16 7 11 -0.000000 0.000000 2.775558e-16 7 12 0.708699 0.708699 6.661338e-16 7 13 -0.000000 0.000000 3.330669e-16 7 14 0.657694 0.658078 3.834952e-04 7 15 -0.000000 0.000000 4.440892e-16 7 16 0.615414 0.616948 1.533981e-03 8 0 3.141593 3.141593 0.000000e+00 8 1 0.000000 0.000000 2.220446e-16 8 2 1.570796 1.570796 0.000000e+00 8 3 0.000000 0.000000 5.551115e-17 8 4 1.178097 1.178097 0.000000e+00 8 5 0.000000 0.000000 0.000000e+00 8 6 0.981748 0.981748 1.110223e-16 8 7 0.000000 0.000000 0.000000e+00 8 8 0.859029 0.859029 3.330669e-16 8 9 -0.000000 0.000000 1.110223e-16 8 10 0.773126 0.773126 0.000000e+00 8 11 -0.000000 0.000000 5.551115e-17 8 12 0.708699 0.708699 0.000000e+00 8 13 -0.000000 0.000000 5.551115e-17 8 14 0.658078 0.658078 1.110223e-16 8 15 -0.000000 0.000000 5.551115e-17 8 16 0.616852 0.616948 9.587380e-05 8 17 -0.000000 0.000000 5.551115e-17 8 18 0.582242 0.582673 4.314321e-04 9 0 3.141593 3.141593 0.000000e+00 9 1 0.000000 0.000000 1.665335e-16 9 2 1.570796 1.570796 2.220446e-16 9 3 0.000000 0.000000 2.220446e-16 9 4 1.178097 1.178097 2.220446e-16 9 5 0.000000 0.000000 2.220446e-16 9 6 0.981748 0.981748 1.110223e-16 9 7 0.000000 0.000000 2.220446e-16 9 8 0.859029 0.859029 3.330669e-16 9 9 0.000000 0.000000 1.665335e-16 9 10 0.773126 0.773126 2.220446e-16 9 11 0.000000 0.000000 2.220446e-16 9 12 0.708699 0.708699 2.220446e-16 9 13 0.000000 0.000000 1.665335e-16 9 14 0.658078 0.658078 3.330669e-16 9 15 0.000000 0.000000 1.665335e-16 9 16 0.616948 0.616948 4.440892e-16 9 17 0.000000 0.000000 1.110223e-16 9 18 0.582649 0.582673 2.396845e-05 9 19 0.000000 0.000000 1.110223e-16 9 20 0.553420 0.553539 1.198422e-04 10 0 3.141593 3.141593 0.000000e+00 10 1 0.000000 0.000000 1.665335e-16 10 2 1.570796 1.570796 2.220446e-16 10 3 0.000000 0.000000 3.330669e-16 10 4 1.178097 1.178097 2.220446e-16 10 5 0.000000 0.000000 3.885781e-16 10 6 0.981748 0.981748 0.000000e+00 10 7 0.000000 0.000000 3.330669e-16 10 8 0.859029 0.859029 2.220446e-16 10 9 0.000000 0.000000 4.440892e-16 10 10 0.773126 0.773126 0.000000e+00 10 11 0.000000 0.000000 4.440892e-16 10 12 0.708699 0.708699 0.000000e+00 10 13 0.000000 0.000000 4.996004e-16 10 14 0.658078 0.658078 1.110223e-16 10 15 0.000000 0.000000 5.551115e-16 10 16 0.616948 0.616948 1.110223e-16 10 17 0.000000 0.000000 6.106227e-16 10 18 0.582673 0.582673 2.220446e-16 10 19 0.000000 0.000000 6.106227e-16 10 20 0.553533 0.553539 5.992112e-06 10 21 0.000000 0.000000 6.383782e-16 10 22 0.528346 0.528378 3.295662e-05 TEST02 CHEBYSHEV1_COMPUTE computes a Gauss-Chebyshev type 1 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) / sqrt(1-x^2) dx. Order = 1 0 0.0000000000000000 3.1415926535897931 Order = 2 0 -0.7071067811865475 1.5707963267948966 1 0.7071067811865476 1.5707963267948966 Order = 3 0 -0.8660254037844387 1.0471975511965976 1 0.0000000000000000 1.0471975511965976 2 0.8660254037844387 1.0471975511965976 Order = 4 0 -0.9238795325112867 0.7853981633974483 1 -0.3826834323650897 0.7853981633974483 2 0.3826834323650898 0.7853981633974483 3 0.9238795325112867 0.7853981633974483 Order = 5 0 -0.9510565162951535 0.6283185307179586 1 -0.5877852522924730 0.6283185307179586 2 0.0000000000000000 0.6283185307179586 3 0.5877852522924731 0.6283185307179586 4 0.9510565162951535 0.6283185307179586 Order = 6 0 -0.9659258262890682 0.5235987755982988 1 -0.7071067811865475 0.5235987755982988 2 -0.2588190451025206 0.5235987755982988 3 0.2588190451025207 0.5235987755982988 4 0.7071067811865476 0.5235987755982988 5 0.9659258262890683 0.5235987755982988 Order = 7 0 -0.9749279121818237 0.4487989505128276 1 -0.7818314824680295 0.4487989505128276 2 -0.4338837391175581 0.4487989505128276 3 0.0000000000000000 0.4487989505128276 4 0.4338837391175582 0.4487989505128276 5 0.7818314824680298 0.4487989505128276 6 0.9749279121818236 0.4487989505128276 Order = 8 0 -0.9807852804032304 0.3926990816987241 1 -0.8314696123025453 0.3926990816987241 2 -0.5555702330196020 0.3926990816987241 3 -0.1950903220161282 0.3926990816987241 4 0.1950903220161283 0.3926990816987241 5 0.5555702330196023 0.3926990816987241 6 0.8314696123025452 0.3926990816987241 7 0.9807852804032304 0.3926990816987241 Order = 9 0 -0.9848077530122080 0.3490658503988659 1 -0.8660254037844385 0.3490658503988659 2 -0.6427876096865394 0.3490658503988659 3 -0.3420201433256685 0.3490658503988659 4 0.0000000000000000 0.3490658503988659 5 0.3420201433256688 0.3490658503988659 6 0.6427876096865394 0.3490658503988659 7 0.8660254037844387 0.3490658503988659 8 0.9848077530122080 0.3490658503988659 Order = 10 0 -0.9876883405951377 0.3141592653589793 1 -0.8910065241883678 0.3141592653589793 2 -0.7071067811865475 0.3141592653589793 3 -0.4539904997395467 0.3141592653589793 4 -0.1564344650402306 0.3141592653589793 5 0.1564344650402309 0.3141592653589793 6 0.4539904997395468 0.3141592653589793 7 0.7071067811865476 0.3141592653589793 8 0.8910065241883679 0.3141592653589793 9 0.9876883405951378 0.3141592653589793 TEST03 CHEBYSHEV2_COMPUTE computes a Gauss-Chebyshev type 2 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) * sqrt ( 1 - x^2 ) dx. CHEBYSHEV2_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 1.570796 1.570796 0.000000e+00 1 1 0.000000 0.000000 0.000000e+00 1 2 0.000000 0.392699 3.926991e-01 1 3 0.000000 0.000000 0.000000e+00 1 4 0.000000 0.196350 1.963495e-01 2 0 1.570796 1.570796 0.000000e+00 2 1 0.000000 0.000000 1.110223e-16 2 2 0.392699 0.392699 1.110223e-16 2 3 0.000000 0.000000 1.526557e-16 2 4 0.098175 0.196350 9.817477e-02 2 5 0.000000 0.000000 7.285839e-17 2 6 0.024544 0.122718 9.817477e-02 3 0 1.570796 1.570796 0.000000e+00 3 1 -0.000000 0.000000 1.110223e-16 3 2 0.392699 0.392699 0.000000e+00 3 3 0.000000 0.000000 0.000000e+00 3 4 0.196350 0.196350 0.000000e+00 3 5 0.000000 0.000000 1.387779e-17 3 6 0.098175 0.122718 2.454369e-02 3 7 0.000000 0.000000 1.387779e-17 3 8 0.049087 0.085903 3.681554e-02 4 0 1.570796 1.570796 2.220446e-16 4 1 -0.000000 0.000000 5.551115e-17 4 2 0.392699 0.392699 0.000000e+00 4 3 0.000000 0.000000 0.000000e+00 4 4 0.196350 0.196350 2.775558e-17 4 5 0.000000 0.000000 1.387779e-17 4 6 0.122718 0.122718 1.387779e-17 4 7 0.000000 0.000000 2.775558e-17 4 8 0.079767 0.085903 6.135923e-03 4 9 0.000000 0.000000 2.775558e-17 4 10 0.052155 0.064427 1.227185e-02 5 0 1.570796 1.570796 0.000000e+00 5 1 0.000000 0.000000 5.551115e-17 5 2 0.392699 0.392699 5.551115e-17 5 3 0.000000 0.000000 9.714451e-17 5 4 0.196350 0.196350 2.775558e-17 5 5 0.000000 0.000000 4.163336e-17 5 6 0.122718 0.122718 1.387779e-17 5 7 0.000000 0.000000 1.387779e-17 5 8 0.085903 0.085903 1.387779e-17 5 9 0.000000 0.000000 6.938894e-18 5 10 0.062893 0.064427 1.533981e-03 5 11 0.000000 0.000000 0.000000e+00 5 12 0.046786 0.050621 3.834952e-03 6 0 1.570796 1.570796 0.000000e+00 6 1 -0.000000 0.000000 8.326673e-17 6 2 0.392699 0.392699 0.000000e+00 6 3 -0.000000 0.000000 1.387779e-17 6 4 0.196350 0.196350 2.775558e-17 6 5 0.000000 0.000000 1.387779e-17 6 6 0.122718 0.122718 1.387779e-17 6 7 0.000000 0.000000 1.387779e-17 6 8 0.085903 0.085903 1.387779e-17 6 9 0.000000 0.000000 2.081668e-17 6 10 0.064427 0.064427 1.387779e-17 6 11 0.000000 0.000000 2.081668e-17 6 12 0.050238 0.050621 3.834952e-04 6 13 0.000000 0.000000 2.775558e-17 6 14 0.039979 0.041130 1.150486e-03 7 0 1.570796 1.570796 0.000000e+00 7 1 -0.000000 0.000000 1.387779e-17 7 2 0.392699 0.392699 5.551115e-17 7 3 -0.000000 0.000000 2.081668e-17 7 4 0.196350 0.196350 0.000000e+00 7 5 -0.000000 0.000000 2.081668e-17 7 6 0.122718 0.122718 2.775558e-17 7 7 -0.000000 0.000000 1.387779e-17 7 8 0.085903 0.085903 0.000000e+00 7 9 -0.000000 0.000000 1.387779e-17 7 10 0.064427 0.064427 1.387779e-17 7 11 -0.000000 0.000000 1.387779e-17 7 12 0.050621 0.050621 0.000000e+00 7 13 -0.000000 0.000000 6.938894e-18 7 14 0.041034 0.041130 9.587380e-05 7 15 -0.000000 0.000000 6.938894e-18 7 16 0.033939 0.034275 3.355583e-04 8 0 1.570796 1.570796 2.220446e-16 8 1 0.000000 0.000000 4.857226e-17 8 2 0.392699 0.392699 0.000000e+00 8 3 0.000000 0.000000 4.163336e-17 8 4 0.196350 0.196350 0.000000e+00 8 5 0.000000 0.000000 1.387779e-17 8 6 0.122718 0.122718 1.387779e-17 8 7 0.000000 0.000000 1.734723e-17 8 8 0.085903 0.085903 1.387779e-17 8 9 0.000000 0.000000 2.081668e-17 8 10 0.064427 0.064427 0.000000e+00 8 11 0.000000 0.000000 1.387779e-17 8 12 0.050621 0.050621 6.938894e-18 8 13 0.000000 0.000000 1.387779e-17 8 14 0.041130 0.041130 6.938894e-18 8 15 0.000000 0.000000 1.734723e-17 8 16 0.034251 0.034275 2.396845e-05 8 17 0.000000 0.000000 1.561251e-17 8 18 0.029038 0.029134 9.587380e-05 9 0 1.570796 1.570796 2.220446e-16 9 1 -0.000000 0.000000 5.204170e-17 9 2 0.392699 0.392699 0.000000e+00 9 3 0.000000 0.000000 1.734723e-17 9 4 0.196350 0.196350 2.775558e-17 9 5 0.000000 0.000000 0.000000e+00 9 6 0.122718 0.122718 1.387779e-17 9 7 0.000000 0.000000 3.469447e-18 9 8 0.085903 0.085903 0.000000e+00 9 9 0.000000 0.000000 1.387779e-17 9 10 0.064427 0.064427 0.000000e+00 9 11 0.000000 0.000000 0.000000e+00 9 12 0.050621 0.050621 1.387779e-17 9 13 -0.000000 0.000000 1.734723e-18 9 14 0.041130 0.041130 2.081668e-17 9 15 -0.000000 0.000000 3.469447e-18 9 16 0.034275 0.034275 1.387779e-17 9 17 -0.000000 0.000000 3.469447e-18 9 18 0.029128 0.029134 5.992112e-06 9 19 -0.000000 0.000000 1.734723e-18 9 20 0.025134 0.025161 2.696451e-05 10 0 1.570796 1.570796 2.220446e-16 10 1 -0.000000 0.000000 1.040834e-17 10 2 0.392699 0.392699 5.551115e-17 10 3 0.000000 0.000000 1.387779e-17 10 4 0.196350 0.196350 0.000000e+00 10 5 0.000000 0.000000 1.040834e-17 10 6 0.122718 0.122718 1.387779e-17 10 7 0.000000 0.000000 1.387779e-17 10 8 0.085903 0.085903 1.387779e-17 10 9 0.000000 0.000000 1.387779e-17 10 10 0.064427 0.064427 0.000000e+00 10 11 0.000000 0.000000 1.387779e-17 10 12 0.050621 0.050621 6.938894e-18 10 13 0.000000 0.000000 1.040834e-17 10 14 0.041130 0.041130 1.387779e-17 10 15 0.000000 0.000000 8.673617e-18 10 16 0.034275 0.034275 6.938894e-18 10 17 0.000000 0.000000 6.938894e-18 10 18 0.029134 0.029134 1.040834e-17 10 19 0.000000 0.000000 5.204170e-18 10 20 0.025159 0.025161 1.498028e-06 10 21 0.000000 0.000000 5.204170e-18 10 22 0.022008 0.022016 7.490141e-06 TEST04 CHEBYSHEV2_COMPUTE computes a Gauss-Chebyshev type 2 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) * sqrt(1-x^2) dx. Order = 1 0 0.0000000000000000 1.5707963267948966 Order = 2 0 -0.4999999999999998 0.7853981633974484 1 0.5000000000000001 0.7853981633974481 Order = 3 0 -0.7071067811865475 0.3926990816987243 1 0.0000000000000000 0.7853981633974483 2 0.7071067811865476 0.3926990816987240 Order = 4 0 -0.8090169943749473 0.2170787134227061 1 -0.3090169943749473 0.5683194499747424 2 0.3090169943749475 0.5683194499747423 3 0.8090169943749475 0.2170787134227060 Order = 5 0 -0.8660254037844387 0.1308996938995747 1 -0.4999999999999998 0.3926990816987242 2 0.0000000000000000 0.5235987755982988 3 0.5000000000000001 0.3926990816987240 4 0.8660254037844387 0.1308996938995747 Order = 6 0 -0.9009688679024190 0.0844886908915886 1 -0.6234898018587335 0.2743330560697779 2 -0.2225209339563143 0.4265764164360819 3 0.2225209339563144 0.4265764164360819 4 0.6234898018587336 0.2743330560697778 5 0.9009688679024191 0.0844886908915886 Order = 7 0 -0.9238795325112867 0.0575094490319132 1 -0.7071067811865475 0.1963495408493621 2 -0.3826834323650897 0.3351896326668110 3 0.0000000000000000 0.3926990816987241 4 0.3826834323650898 0.3351896326668110 5 0.7071067811865476 0.1963495408493620 6 0.9238795325112867 0.0575094490319131 Order = 8 0 -0.9396926207859083 0.0408329477091071 1 -0.7660444431189779 0.1442256007956728 2 -0.4999999999999998 0.2617993877991495 3 -0.1736481776669303 0.3385402270935190 4 0.1736481776669304 0.3385402270935190 5 0.5000000000000001 0.2617993877991494 6 0.7660444431189780 0.1442256007956727 7 0.9396926207859084 0.0408329477091071 Order = 9 0 -0.9510565162951535 0.0299995403716082 1 -0.8090169943749473 0.1085393567113530 2 -0.5877852522924730 0.2056199086476263 3 -0.3090169943749473 0.2841597249873712 4 0.0000000000000000 0.3141592653589793 5 0.3090169943749475 0.2841597249873711 6 0.5877852522924731 0.2056199086476263 7 0.8090169943749475 0.1085393567113530 8 0.9510565162951535 0.0299995403716082 Order = 10 0 -0.9594929736144974 0.0226689425018588 1 -0.8412535328311811 0.0834785409341891 2 -0.6548607339452850 0.1631221774548166 3 -0.4154150130018863 0.2363135602034873 4 -0.1423148382732850 0.2798149423030966 5 0.1423148382732851 0.2798149423030965 6 0.4154150130018864 0.2363135602034873 7 0.6548607339452851 0.1631221774548166 8 0.8412535328311812 0.0834785409341890 9 0.9594929736144974 0.0226689425018588 TEST05 CLENSHAW_CURTIS_COMPUTE computes a Clenshaw Curtis rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. LEGENDRE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = ORDER+1 if ORDER is odd, or N = ORDER if ORDER is even In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 2.000000 2.000000 0.000000e+00 1 1 0.000000 0.000000 0.000000e+00 1 2 0.000000 0.666667 6.666667e-01 1 3 0.000000 0.000000 0.000000e+00 1 4 0.000000 0.400000 4.000000e-01 2 0 2.000000 2.000000 0.000000e+00 2 1 0.000000 0.000000 0.000000e+00 2 2 2.000000 0.666667 1.333333e+00 2 3 0.000000 0.000000 0.000000e+00 2 4 2.000000 0.400000 1.600000e+00 3 0 2.000000 2.000000 0.000000e+00 3 1 0.000000 0.000000 0.000000e+00 3 2 0.666667 0.666667 1.110223e-16 3 3 0.000000 0.000000 0.000000e+00 3 4 0.666667 0.400000 2.666667e-01 3 5 0.000000 0.000000 0.000000e+00 3 6 0.666667 0.285714 3.809524e-01 4 0 2.000000 2.000000 4.440892e-16 4 1 0.000000 0.000000 4.579670e-16 4 2 0.666667 0.666667 1.110223e-16 4 3 0.000000 0.000000 2.636780e-16 4 4 0.333333 0.400000 6.666667e-02 4 5 0.000000 0.000000 9.714451e-17 4 6 0.250000 0.285714 3.571429e-02 5 0 2.000000 2.000000 0.000000e+00 5 1 0.000000 0.000000 1.387779e-16 5 2 0.666667 0.666667 1.110223e-16 5 3 0.000000 0.000000 1.110223e-16 5 4 0.400000 0.400000 0.000000e+00 5 5 0.000000 0.000000 8.326673e-17 5 6 0.266667 0.285714 1.904762e-02 5 7 0.000000 0.000000 5.551115e-17 5 8 0.200000 0.222222 2.222222e-02 6 0 2.000000 2.000000 0.000000e+00 6 1 0.000000 0.000000 1.387779e-16 6 2 0.666667 0.666667 1.110223e-16 6 3 0.000000 0.000000 1.665335e-16 6 4 0.400000 0.400000 0.000000e+00 6 5 0.000000 0.000000 1.110223e-16 6 6 0.283333 0.285714 2.380952e-03 6 7 0.000000 0.000000 1.110223e-16 6 8 0.212500 0.222222 9.722222e-03 7 0 2.000000 2.000000 0.000000e+00 7 1 0.000000 0.000000 2.740863e-16 7 2 0.666667 0.666667 0.000000e+00 7 3 0.000000 0.000000 1.353084e-16 7 4 0.400000 0.400000 0.000000e+00 7 5 0.000000 0.000000 5.204170e-17 7 6 0.285714 0.285714 0.000000e+00 7 7 0.000000 0.000000 1.040834e-17 7 8 0.221429 0.222222 7.936508e-04 7 9 -0.000000 0.000000 3.469447e-18 7 10 0.178571 0.181818 3.246753e-03 8 0 2.000000 2.000000 0.000000e+00 8 1 0.000000 0.000000 1.491862e-16 8 2 0.666667 0.666667 1.110223e-16 8 3 0.000000 0.000000 9.367507e-17 8 4 0.400000 0.400000 0.000000e+00 8 5 0.000000 0.000000 7.979728e-17 8 6 0.285714 0.285714 0.000000e+00 8 7 0.000000 0.000000 7.979728e-17 8 8 0.222024 0.222222 1.984127e-04 8 9 0.000000 0.000000 7.979728e-17 8 10 0.181101 0.181818 7.169913e-04 9 0 2.000000 2.000000 0.000000e+00 9 1 0.000000 0.000000 2.012279e-16 9 2 0.666667 0.666667 1.110223e-16 9 3 0.000000 0.000000 1.318390e-16 9 4 0.400000 0.400000 0.000000e+00 9 5 0.000000 0.000000 1.040834e-16 9 6 0.285714 0.285714 0.000000e+00 9 7 0.000000 0.000000 7.632783e-17 9 8 0.222222 0.222222 0.000000e+00 9 9 0.000000 0.000000 4.857226e-17 9 10 0.181746 0.181818 7.215007e-05 9 11 0.000000 0.000000 4.163336e-17 9 12 0.153571 0.153846 2.747253e-04 10 0 2.000000 2.000000 0.000000e+00 10 1 0.000000 0.000000 3.816392e-16 10 2 0.666667 0.666667 1.110223e-16 10 3 0.000000 0.000000 2.567391e-16 10 4 0.400000 0.400000 5.551115e-17 10 5 0.000000 0.000000 1.873501e-16 10 6 0.285714 0.285714 5.551115e-17 10 7 0.000000 0.000000 1.457168e-16 10 8 0.222222 0.222222 0.000000e+00 10 9 0.000000 0.000000 1.595946e-16 10 10 0.181796 0.181818 2.254690e-05 10 11 0.000000 0.000000 1.526557e-16 10 12 0.153757 0.153846 8.871337e-05 TEST06 CLENSHAW_CURTIS_COMPUTE computes a Clenshaw Curtis rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. Order = 1 0 0.0000000000000000 2.0000000000000000 Order = 2 0 -1.0000000000000000 1.0000000000000000 1 1.0000000000000000 1.0000000000000000 Order = 3 0 -1.0000000000000000 0.3333333333333334 1 0.0000000000000000 1.3333333333333333 2 1.0000000000000000 0.3333333333333334 Order = 4 0 -1.0000000000000000 0.1111111111111111 1 -0.4999999999999998 0.8888888888888888 2 0.5000000000000001 0.8888888888888892 3 1.0000000000000000 0.1111111111111111 Order = 5 0 -1.0000000000000000 0.0666666666666667 1 -0.7071067811865475 0.5333333333333333 2 0.0000000000000000 0.7999999999999999 3 0.7071067811865476 0.5333333333333334 4 1.0000000000000000 0.0666666666666667 Order = 6 0 -1.0000000000000000 0.0400000000000000 1 -0.8090169943749473 0.3607430412000112 2 -0.3090169943749473 0.5992569587999889 3 0.3090169943749475 0.5992569587999887 4 0.8090169943749475 0.3607430412000113 5 1.0000000000000000 0.0400000000000000 Order = 7 0 -1.0000000000000000 0.0285714285714286 1 -0.8660254037844387 0.2539682539682539 2 -0.4999999999999998 0.4571428571428571 3 0.0000000000000000 0.5206349206349206 4 0.5000000000000001 0.4571428571428573 5 0.8660254037844387 0.2539682539682539 6 1.0000000000000000 0.0285714285714286 Order = 8 0 -1.0000000000000000 0.0204081632653061 1 -0.9009688679024190 0.1901410072182084 2 -0.6234898018587335 0.3522424237181591 3 -0.2225209339563143 0.4372084057983264 4 0.2225209339563144 0.4372084057983264 5 0.6234898018587336 0.3522424237181591 6 0.9009688679024191 0.1901410072182084 7 1.0000000000000000 0.0204081632653061 Order = 9 0 -1.0000000000000000 0.0158730158730159 1 -0.9238795325112867 0.1462186492160181 2 -0.7071067811865475 0.2793650793650794 3 -0.3826834323650897 0.3617178587204897 4 0.0000000000000000 0.3936507936507936 5 0.3826834323650898 0.3617178587204898 6 0.7071067811865476 0.2793650793650794 7 0.9238795325112867 0.1462186492160182 8 1.0000000000000000 0.0158730158730159 Order = 10 0 -1.0000000000000000 0.0123456790123457 1 -0.9396926207859083 0.1165674565720371 2 -0.7660444431189779 0.2252843233381044 3 -0.4999999999999998 0.3019400352733685 4 -0.1736481776669303 0.3438625058041442 5 0.1736481776669304 0.3438625058041442 6 0.5000000000000001 0.3019400352733687 7 0.7660444431189780 0.2252843233381044 8 0.9396926207859084 0.1165674565720372 9 1.0000000000000000 0.0123456790123457 TEST07 FEJER2_COMPUTE computes a Fejer Type 2 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. LEGENDRE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = ORDER+1 if ORDER is odd, or N = ORDER if ORDER is even In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 2.000000 2.000000 0.000000e+00 1 1 0.000000 0.000000 0.000000e+00 1 2 0.000000 0.666667 6.666667e-01 1 3 0.000000 0.000000 0.000000e+00 1 4 0.000000 0.400000 4.000000e-01 2 0 2.000000 2.000000 0.000000e+00 2 1 0.000000 0.000000 3.330669e-16 2 2 0.500000 0.666667 1.666667e-01 2 3 0.000000 0.000000 2.498002e-16 2 4 0.125000 0.400000 2.750000e-01 3 0 2.000000 2.000000 0.000000e+00 3 1 0.000000 0.000000 0.000000e+00 3 2 0.666667 0.666667 1.110223e-16 3 3 0.000000 0.000000 8.326673e-17 3 4 0.333333 0.400000 6.666667e-02 3 5 0.000000 0.000000 6.938894e-17 3 6 0.166667 0.285714 1.190476e-01 4 0 2.000000 2.000000 0.000000e+00 4 1 0.000000 0.000000 1.110223e-16 4 2 0.666667 0.666667 1.110223e-16 4 3 0.000000 0.000000 5.551115e-17 4 4 0.375000 0.400000 2.500000e-02 4 5 0.000000 0.000000 5.551115e-17 4 6 0.239583 0.285714 4.613095e-02 5 0 2.000000 2.000000 0.000000e+00 5 1 0.000000 0.000000 1.110223e-16 5 2 0.666667 0.666667 0.000000e+00 5 3 0.000000 0.000000 1.110223e-16 5 4 0.400000 0.400000 0.000000e+00 5 5 0.000000 0.000000 2.775558e-17 5 6 0.275000 0.285714 1.071429e-02 5 7 0.000000 0.000000 1.387779e-17 5 8 0.200000 0.222222 2.222222e-02 6 0 2.000000 2.000000 2.220446e-16 6 1 0.000000 0.000000 5.551115e-17 6 2 0.666667 0.666667 0.000000e+00 6 3 0.000000 0.000000 1.110223e-16 6 4 0.400000 0.400000 1.110223e-16 6 5 0.000000 0.000000 8.326673e-17 6 6 0.281250 0.285714 4.464286e-03 6 7 0.000000 0.000000 1.110223e-16 6 8 0.211979 0.222222 1.024306e-02 7 0 2.000000 2.000000 0.000000e+00 7 1 -0.000000 0.000000 8.326673e-17 7 2 0.666667 0.666667 1.110223e-16 7 3 0.000000 0.000000 0.000000e+00 7 4 0.400000 0.400000 0.000000e+00 7 5 -0.000000 0.000000 2.775558e-17 7 6 0.285714 0.285714 0.000000e+00 7 7 0.000000 0.000000 0.000000e+00 7 8 0.220238 0.222222 1.984127e-03 7 9 -0.000000 0.000000 1.387779e-17 7 10 0.176786 0.181818 5.032468e-03 8 0 2.000000 2.000000 4.440892e-16 8 1 -0.000000 0.000000 2.775558e-17 8 2 0.666667 0.666667 0.000000e+00 8 3 0.000000 0.000000 1.110223e-16 8 4 0.400000 0.400000 5.551115e-17 8 5 0.000000 0.000000 8.326673e-17 8 6 0.285714 0.285714 0.000000e+00 8 7 0.000000 0.000000 8.326673e-17 8 8 0.221354 0.222222 8.680556e-04 8 9 0.000000 0.000000 9.714451e-17 8 10 0.179408 0.181818 2.409700e-03 9 0 2.000000 2.000000 0.000000e+00 9 1 -0.000000 0.000000 1.387779e-17 9 2 0.666667 0.666667 0.000000e+00 9 3 -0.000000 0.000000 1.387779e-17 9 4 0.400000 0.400000 5.551115e-17 9 5 -0.000000 0.000000 1.387779e-17 9 6 0.285714 0.285714 0.000000e+00 9 7 0.000000 0.000000 0.000000e+00 9 8 0.222222 0.222222 2.775558e-17 9 9 -0.000000 0.000000 2.775558e-17 9 10 0.181424 0.181818 3.945707e-04 9 11 -0.000000 0.000000 1.387779e-17 9 12 0.152654 0.153846 1.192384e-03 10 0 2.000000 2.000000 2.220446e-16 10 1 0.000000 0.000000 0.000000e+00 10 2 0.666667 0.666667 0.000000e+00 10 3 0.000000 0.000000 9.714451e-17 10 4 0.400000 0.400000 1.110223e-16 10 5 0.000000 0.000000 1.387779e-17 10 6 0.285714 0.285714 0.000000e+00 10 7 0.000000 0.000000 4.163336e-17 10 8 0.222222 0.222222 5.551115e-17 10 9 0.000000 0.000000 4.163336e-17 10 10 0.181641 0.181818 1.775568e-04 10 11 0.000000 0.000000 4.163336e-17 10 12 0.153266 0.153846 5.800948e-04 TEST08 FEJER2_COMPUTE computes a Fejer Type 2 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. Order = 1 0 0.0000000000000000 2.0000000000000000 Order = 2 0 -0.4999999999999998 1.0000000000000000 1 0.5000000000000001 1.0000000000000000 Order = 3 0 -0.7071067811865475 0.6666666666666667 1 0.0000000000000000 0.6666666666666666 2 0.7071067811865476 0.6666666666666666 Order = 4 0 -0.8090169943749473 0.4254644007500071 1 -0.3090169943749473 0.5745355992499930 2 0.3090169943749475 0.5745355992499930 3 0.8090169943749475 0.4254644007500070 Order = 5 0 -0.8660254037844387 0.3111111111111111 1 -0.4999999999999998 0.4000000000000001 2 0.0000000000000000 0.5777777777777777 3 0.5000000000000001 0.4000000000000000 4 0.8660254037844387 0.3111111111111111 Order = 6 0 -0.9009688679024190 0.2269152467244296 1 -0.6234898018587335 0.3267938603769863 2 -0.2225209339563143 0.4462908928985842 3 0.2225209339563144 0.4462908928985841 4 0.6234898018587336 0.3267938603769863 5 0.9009688679024191 0.2269152467244296 Order = 7 0 -0.9238795325112867 0.1779646809620499 1 -0.7071067811865475 0.2476190476190477 2 -0.3826834323650897 0.3934638904665215 3 0.0000000000000000 0.3619047619047619 4 0.3826834323650898 0.3934638904665215 5 0.7071067811865476 0.2476190476190476 6 0.9238795325112867 0.1779646809620499 Order = 8 0 -0.9396926207859083 0.1397697435050226 1 -0.7660444431189779 0.2063696457302284 2 -0.4999999999999998 0.3142857142857144 3 -0.1736481776669303 0.3395748964790348 4 0.1736481776669304 0.3395748964790348 5 0.5000000000000001 0.3142857142857143 6 0.7660444431189780 0.2063696457302284 7 0.9396926207859084 0.1397697435050225 Order = 9 0 -0.9510565162951535 0.1147810750857218 1 -0.8090169943749473 0.1654331942222276 2 -0.5877852522924730 0.2737903534857068 3 -0.3090169943749473 0.2790112502222169 4 0.0000000000000000 0.3339682539682539 5 0.3090169943749475 0.2790112502222170 6 0.5877852522924731 0.2737903534857068 7 0.8090169943749475 0.1654331942222276 8 0.9510565162951535 0.1147810750857217 Order = 10 0 -0.9594929736144974 0.0944195417398281 1 -0.8412535328311811 0.1411354380109716 2 -0.6548607339452850 0.2263866903636005 3 -0.4154150130018863 0.2530509772156453 4 -0.1423148382732850 0.2850073526699546 5 0.1423148382732851 0.2850073526699544 6 0.4154150130018864 0.2530509772156453 7 0.6548607339452851 0.2263866903636005 8 0.8412535328311812 0.1411354380109716 9 0.9594929736144974 0.0944195417398281 TEST09 GEGENBAUER_COMPUTE computes a generalized Gauss-Gegenbauer rule which is appropriate for integrands of the form Integral ( 0 <= x < +oo ) f(x) (1-x^2)^alpha dx. GEGENBAUER_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Alpha Estimate Exact Error 1 0 0.500000 1.570796 1.570796 1.110223e-15 1 1 0.500000 0.000000 0.000000 0.000000e+00 1 2 0.500000 0.000000 0.392699 3.926991e-01 1 3 0.500000 0.000000 0.000000 0.000000e+00 1 4 0.500000 0.000000 0.196350 1.963495e-01 2 0 0.500000 1.570796 1.570796 1.110223e-15 2 1 0.500000 0.000000 0.000000 0.000000e+00 2 2 0.500000 0.392699 0.392699 0.000000e+00 2 3 0.500000 0.000000 0.000000 0.000000e+00 2 4 0.500000 0.098175 0.196350 9.817477e-02 2 5 0.500000 0.000000 0.000000 0.000000e+00 2 6 0.500000 0.024544 0.122718 9.817477e-02 3 0 0.500000 1.570796 1.570796 1.110223e-15 3 1 0.500000 0.000000 0.000000 4.996004e-16 3 2 0.500000 0.392699 0.392699 0.000000e+00 3 3 0.500000 0.000000 0.000000 3.053113e-16 3 4 0.500000 0.196350 0.196350 5.551115e-17 3 5 0.500000 0.000000 0.000000 1.804112e-16 3 6 0.500000 0.098175 0.122718 2.454369e-02 3 7 0.500000 0.000000 0.000000 9.714451e-17 3 8 0.500000 0.049087 0.085903 3.681554e-02 4 0 0.500000 1.570796 1.570796 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2.500000 0.981748 0.981748 1.110223e-16 8 1 2.500000 0.000000 0.000000 2.602085e-17 8 2 2.500000 0.122718 0.122718 9.714451e-17 8 3 2.500000 0.000000 0.000000 4.336809e-19 8 4 2.500000 0.036816 0.036816 3.469447e-17 8 5 2.500000 0.000000 0.000000 2.168404e-18 8 6 2.500000 0.015340 0.015340 0.000000e+00 8 7 2.500000 0.000000 0.000000 2.168404e-19 8 8 2.500000 0.007670 0.007670 1.994932e-17 8 9 2.500000 0.000000 0.000000 0.000000e+00 8 10 2.500000 0.004314 0.004314 2.081668e-17 8 11 2.500000 0.000000 0.000000 0.000000e+00 8 12 2.500000 0.002637 0.002637 6.071532e-18 8 13 2.500000 0.000000 0.000000 0.000000e+00 8 14 2.500000 0.001714 0.001714 2.168404e-19 8 15 2.500000 0.000000 0.000000 0.000000e+00 8 16 2.500000 0.001166 0.001168 2.596582e-06 8 17 2.500000 0.000000 0.000000 0.000000e+00 8 18 2.500000 0.000819 0.000828 8.503806e-06 9 0 2.500000 0.981748 0.981748 1.110223e-16 9 1 2.500000 0.000000 0.000000 1.647987e-17 9 2 2.500000 0.122718 0.122718 2.775558e-17 9 3 2.500000 0.000000 0.000000 2.168404e-18 9 4 2.500000 0.036816 0.036816 0.000000e+00 9 5 2.500000 -0.000000 0.000000 6.505213e-19 9 6 2.500000 0.015340 0.015340 2.775558e-17 9 7 2.500000 0.000000 0.000000 4.336809e-19 9 8 2.500000 0.007670 0.007670 3.642919e-17 9 9 2.500000 0.000000 0.000000 0.000000e+00 9 10 2.500000 0.004314 0.004314 3.122502e-17 9 11 2.500000 -0.000000 0.000000 1.084202e-19 9 12 2.500000 0.002637 0.002637 1.214306e-17 9 13 2.500000 0.000000 0.000000 0.000000e+00 9 14 2.500000 0.001714 0.001714 4.119968e-18 9 15 2.500000 0.000000 0.000000 0.000000e+00 9 16 2.500000 0.001168 0.001168 2.168404e-19 9 17 2.500000 -0.000000 0.000000 5.421011e-20 9 18 2.500000 0.000827 0.000828 6.196389e-07 9 19 2.500000 0.000000 0.000000 0.000000e+00 9 20 2.500000 0.000603 0.000605 2.324729e-06 10 0 2.500000 0.981748 0.981748 4.440892e-16 10 1 2.500000 -0.000000 0.000000 1.327063e-16 10 2 2.500000 0.122718 0.122718 1.665335e-16 10 3 2.500000 -0.000000 0.000000 3.491131e-17 10 4 2.500000 0.036816 0.036816 6.938894e-17 10 5 2.500000 -0.000000 0.000000 1.214306e-17 10 6 2.500000 0.015340 0.015340 1.040834e-17 10 7 2.500000 -0.000000 0.000000 4.228388e-18 10 8 2.500000 0.007670 0.007670 1.474515e-17 10 9 2.500000 -0.000000 0.000000 1.409463e-18 10 10 2.500000 0.004314 0.004314 1.821460e-17 10 11 2.500000 -0.000000 0.000000 3.252607e-19 10 12 2.500000 0.002637 0.002637 6.071532e-18 10 13 2.500000 -0.000000 0.000000 1.626303e-19 10 14 2.500000 0.001714 0.001714 2.168404e-19 10 15 2.500000 -0.000000 0.000000 1.626303e-19 10 16 2.500000 0.001168 0.001168 3.035766e-18 10 17 2.500000 0.000000 0.000000 5.421011e-20 10 18 2.500000 0.000828 0.000828 4.011548e-18 10 19 2.500000 -0.000000 0.000000 2.710505e-20 10 20 2.500000 0.000605 0.000605 1.489517e-07 10 21 2.500000 0.000000 0.000000 2.710505e-20 10 22 2.500000 0.000453 0.000454 6.303847e-07 TEST10 GEGENBAUER_COMPUTE computes a generalized Gauss-Gegenbauer rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) (1-x^2)^alpha dx. Order = 1 ALPHA = 0.500000 0 0.0000000000000000 1.5707963267948968 Order = 2 ALPHA = 0.500000 0 -0.5000000000000000 0.7853981633974484 1 0.5000000000000000 0.7853981633974484 Order = 3 ALPHA = 0.500000 0 -0.7071067811865475 0.3926990816987239 1 0.0000000000000000 0.7853981633974484 2 0.7071067811865476 0.3926990816987245 Order = 4 ALPHA = 0.500000 0 -0.8090169943749475 0.2170787134227060 1 -0.3090169943749475 0.5683194499747424 2 0.3090169943749474 0.5683194499747424 3 0.8090169943749475 0.2170787134227060 Order = 5 ALPHA = 0.500000 0 -0.8660254037844387 0.1308996938995740 1 -0.5000000000000000 0.3926990816987242 2 0.0000000000000000 0.5235987755982989 3 0.5000000000000000 0.3926990816987242 4 0.8660254037844387 0.1308996938995745 Order = 6 ALPHA = 0.500000 0 -0.9009688679024191 0.0844886908915884 1 -0.6234898018587335 0.2743330560697777 2 -0.2225209339563144 0.4265764164360819 3 0.2225209339563144 0.4265764164360819 4 0.6234898018587335 0.2743330560697777 5 0.9009688679024191 0.0844886908915884 Order = 7 ALPHA = 0.500000 0 -0.9238795325112867 0.0575094490319133 1 -0.7071067811865475 0.1963495408493619 2 -0.3826834323650898 0.3351896326668111 3 0.0000000000000000 0.3926990816987242 4 0.3826834323650898 0.3351896326668108 5 0.7071067811865476 0.1963495408493624 6 0.9238795325112867 0.0575094490319133 Order = 8 ALPHA = 0.500000 0 -0.9396926207859084 0.0408329477091069 1 -0.7660444431189780 0.1442256007956728 2 -0.5000000000000000 0.2617993877991495 3 -0.1736481776669303 0.3385402270935191 4 0.1736481776669303 0.3385402270935191 5 0.5000000000000000 0.2617993877991495 6 0.7660444431189780 0.1442256007956728 7 0.9396926207859084 0.0408329477091075 Order = 9 ALPHA = 0.500000 0 -0.9510565162951536 0.0299995403716084 1 -0.8090169943749475 0.1085393567113530 2 -0.5877852522924731 0.2056199086476264 3 -0.3090169943749475 0.2841597249873712 4 0.0000000000000000 0.3141592653589794 5 0.3090169943749475 0.2841597249873712 6 0.5877852522924731 0.2056199086476264 7 0.8090169943749475 0.1085393567113530 8 0.9510565162951536 0.0299995403716084 Order = 10 ALPHA = 0.500000 0 -0.9594929736144974 0.0226689425018590 1 -0.8412535328311812 0.0834785409341889 2 -0.6548607339452851 0.1631221774548165 3 -0.4154150130018864 0.2363135602034873 4 -0.1423148382732851 0.2798149423030965 5 0.1423148382732851 0.2798149423030966 6 0.4154150130018864 0.2363135602034873 7 0.6548607339452851 0.1631221774548165 8 0.8412535328311812 0.0834785409341889 9 0.9594929736144974 0.0226689425018590 Order = 1 ALPHA = 1.000000 0 0.0000000000000000 1.3333333333333333 Order = 2 ALPHA = 1.000000 0 -0.4472135954999579 0.6666666666666665 1 0.4472135954999580 0.6666666666666667 Order = 3 ALPHA = 1.000000 0 -0.6546536707079772 0.3111111111111113 1 0.0000000000000000 0.7111111111111110 2 0.6546536707079772 0.3111111111111113 Order = 4 ALPHA = 1.000000 0 -0.7650553239294646 0.1569499125956942 1 -0.2852315164806451 0.5097167540709727 2 0.2852315164806451 0.5097167540709727 3 0.7650553239294647 0.1569499125956939 Order = 5 ALPHA = 1.000000 0 -0.8302238962785669 0.0860176821228073 1 -0.4688487934707142 0.3368394607343354 2 -0.0000000000000000 0.4876190476190476 3 0.4688487934707142 0.3368394607343354 4 0.8302238962785670 0.0860176821228076 Order = 6 ALPHA = 1.000000 0 -0.8717401485096066 0.0505835770160819 1 -0.5917001814331423 0.2216925320225178 2 -0.2092992179024789 0.3943905576280671 3 0.2092992179024789 0.3943905576280671 4 0.5917001814331423 0.2216925320225178 5 0.8717401485096067 0.0505835770160817 Order = 7 ALPHA = 1.000000 0 -0.8997579954114602 0.0315162001504988 1 -0.6771862795107377 0.1486404045834101 2 -0.3631174638261782 0.3007504247445493 3 0.0000000000000000 0.3715192743764172 4 0.3631174638261782 0.3007504247445493 5 0.6771862795107377 0.1486404045834101 6 0.8997579954114602 0.0315162001504988 Order = 8 ALPHA = 1.000000 0 -0.9195339081664589 0.0205900956491219 1 -0.7387738651055050 0.1021477023603581 2 -0.4779249498104445 0.2253365549698578 3 -0.1652789576663870 0.3185923136873282 4 0.1652789576663870 0.3185923136873282 5 0.4779249498104445 0.2253365549698580 6 0.7387738651055050 0.1021477023603581 7 0.9195339081664589 0.0205900956491219 Order = 9 ALPHA = 1.000000 0 -0.9340014304080592 0.0139910561244267 1 -0.7844834736631444 0.0719828561756607 2 -0.5652353269962050 0.1687989736144601 3 -0.2957581355869394 0.2617849830242735 4 -0.0000000000000000 0.3002175954556907 5 0.2957581355869394 0.2617849830242735 6 0.5652353269962050 0.1687989736144601 7 0.7844834736631444 0.0719828561756607 8 0.9340014304080592 0.0139910561244267 Order = 10 ALPHA = 1.000000 0 -0.9448992722228823 0.0098254048057683 1 -0.8192793216440066 0.0519391437742078 2 -0.6328761530318606 0.1273919483477328 3 -0.3995309409653490 0.2111657418760753 4 -0.1365529328549276 0.2663444278628824 5 0.1365529328549276 0.2663444278628824 6 0.3995309409653490 0.2111657418760753 7 0.6328761530318606 0.1273919483477326 8 0.8192793216440066 0.0519391437742078 9 0.9448992722228822 0.0098254048057681 Order = 1 ALPHA = 2.500000 0 0.0000000000000000 0.9817477042468103 Order = 2 ALPHA = 2.500000 0 -0.3535533905932737 0.4908738521234051 1 0.3535533905932737 0.4908738521234051 Order = 3 ALPHA = 2.500000 0 -0.5477225575051661 0.2045307717180856 1 0.0000000000000000 0.5726861608106394 2 0.5477225575051661 0.2045307717180856 Order = 4 ALPHA = 2.500000 0 -0.6660699417556469 0.0870080715314833 1 -0.2373833033084449 0.4038657805919217 2 0.2373833033084449 0.4038657805919217 3 0.6660699417556469 0.0870080715314835 Order = 5 ALPHA = 2.500000 0 -0.7434770045212192 0.0392893891317672 1 -0.4019050360892101 0.2454174450998077 2 -0.0000000000000000 0.4123340357836603 3 0.4019050360892101 0.2454174450998076 4 0.7434770045212192 0.0392893891317672 Order = 6 ALPHA = 2.500000 0 -0.7968373566406807 0.0189116433666931 1 -0.5196148456949133 0.1431407318203965 2 -0.1804179569647766 0.3288214769363154 3 0.1804179569647766 0.3288214769363154 4 0.5196148456949133 0.1431407318203969 5 0.7968373566406807 0.0189116433666931 Order = 7 ALPHA = 2.500000 0 -0.8351564541339112 0.0096585250210008 1 -0.6063141605450342 0.0834446705489847 2 -0.3186902924591673 0.2357822853526957 3 -0.0000000000000000 0.3239767424014473 4 0.3186902924591672 0.2357822853526958 5 0.6063141605450342 0.0834446705489847 6 0.8351564541339112 0.0096585250210010 Order = 8 ALPHA = 2.500000 0 -0.8635912403827584 0.0051997074221325 1 -0.6718232507247468 0.0494181881565223 2 -0.4260961495798479 0.1615070338519726 3 -0.1459649294626305 0.2747489226927776 4 0.1459649294626305 0.2747489226927777 5 0.4260961495798479 0.1615070338519726 6 0.6718232507247468 0.0494181881565223 7 0.8635912403827584 0.0051997074221325 Order = 9 ALPHA = 2.500000 0 -0.8852662998295736 0.0029319009849924 1 -0.7224309555097356 0.0299209571936803 2 -0.5107538326640052 0.1088013928543096 3 -0.2643695362367075 0.2155149137501430 4 0.0000000000000000 0.2674093746805597 5 0.2643695362367075 0.2155149137501430 6 0.5107538326640052 0.1088013928543096 7 0.7224309555097356 0.0299209571936803 8 0.8852662998295736 0.0029319009849924 Order = 10 ALPHA = 2.500000 0 -0.9021636877146698 0.0017216382288932 1 -0.7622868838395971 0.0185564370059903 2 -0.5784393987065429 0.0731613283971410 3 -0.3610608688185057 0.1622660968500981 4 -0.1227285554678421 0.2351683516412825 5 0.1227285554678421 0.2351683516412824 6 0.3610608688185057 0.1622660968500981 7 0.5784393987065429 0.0731613283971409 8 0.7622868838395971 0.0185564370059903 9 0.9021636877146698 0.0017216382288932 TEST11 GEN_HERMITE_COMPUTE computes a generalized Gauss-Hermite rule which is appropriate for integrands of the form Integral ( -oo < x < +oo ) f(x) x^alpha exp(-x*x) dx. GEN_HERMITE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Alpha Estimate Exact Error 1 0 0.500000 1.225417e+00 1.225417e+00 0.000000e+00 1 1 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 1 2 0.500000 0.000000e+00 9.190625e-01 9.190625e-01 1 3 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 1 4 0.500000 0.000000e+00 1.608359e+00 1.608359e+00 2 0 0.500000 1.225417e+00 1.225417e+00 0.000000e+00 2 1 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 2 2 0.500000 9.190625e-01 9.190625e-01 1.110223e-16 2 3 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 2 4 0.500000 6.892969e-01 1.608359e+00 9.190625e-01 2 5 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 2 6 0.500000 5.169727e-01 4.422988e+00 3.906016e+00 3 0 0.500000 1.225417e+00 1.225417e+00 0.000000e+00 3 1 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 3 2 0.500000 9.190625e-01 9.190625e-01 1.110223e-16 3 3 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 3 4 0.500000 1.608359e+00 1.608359e+00 2.220446e-16 3 5 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 3 6 0.500000 2.814629e+00 4.422988e+00 1.608359e+00 3 7 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 3 8 0.500000 4.925601e+00 1.658621e+01 1.166061e+01 4 0 0.500000 1.225417e+00 1.225417e+00 2.220446e-16 4 1 0.500000 -2.775558e-17 0.000000e+00 2.775558e-17 4 2 0.500000 9.190625e-01 9.190625e-01 2.220446e-16 4 3 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 4 4 0.500000 1.608359e+00 1.608359e+00 4.440892e-16 4 5 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 4 6 0.500000 4.422988e+00 4.422988e+00 8.881784e-16 4 7 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 4 8 0.500000 1.336949e+01 1.658621e+01 3.216719e+00 4 9 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 4 10 0.500000 4.098803e+01 7.878448e+01 3.779645e+01 5 0 0.500000 1.225417e+00 1.225417e+00 0.000000e+00 5 1 0.500000 -1.387779e-17 0.000000e+00 1.387779e-17 5 2 0.500000 9.190625e-01 9.190625e-01 0.000000e+00 5 3 0.500000 -5.551115e-17 0.000000e+00 5.551115e-17 5 4 0.500000 1.608359e+00 1.608359e+00 0.000000e+00 5 5 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 5 6 0.500000 4.422988e+00 4.422988e+00 8.881784e-16 5 7 0.500000 -4.440892e-16 0.000000e+00 4.440892e-16 5 8 0.500000 1.658621e+01 1.658621e+01 3.552714e-15 5 9 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 5 10 0.500000 6.993850e+01 7.878448e+01 8.845977e+00 5 11 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 5 12 0.500000 3.048407e+02 4.530108e+02 1.481701e+02 6 0 0.500000 1.225417e+00 1.225417e+00 6.661338e-16 6 1 0.500000 3.295975e-17 0.000000e+00 3.295975e-17 6 2 0.500000 9.190625e-01 9.190625e-01 3.330669e-16 6 3 0.500000 1.387779e-17 0.000000e+00 1.387779e-17 6 4 0.500000 1.608359e+00 1.608359e+00 0.000000e+00 6 5 0.500000 5.551115e-17 0.000000e+00 5.551115e-17 6 6 0.500000 4.422988e+00 4.422988e+00 0.000000e+00 6 7 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 6 8 0.500000 1.658621e+01 1.658621e+01 0.000000e+00 6 9 0.500000 -1.776357e-15 0.000000e+00 1.776357e-15 6 10 0.500000 7.878448e+01 7.878448e+01 1.421085e-14 6 11 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 6 12 0.500000 4.264728e+02 4.530108e+02 2.653793e+01 6 13 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 6 14 0.500000 2.440816e+03 3.057823e+03 6.170069e+02 7 0 0.500000 1.225417e+00 1.225417e+00 0.000000e+00 7 1 0.500000 5.204170e-18 0.000000e+00 5.204170e-18 7 2 0.500000 9.190625e-01 9.190625e-01 4.440892e-16 7 3 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 7 4 0.500000 1.608359e+00 1.608359e+00 6.661338e-16 7 5 0.500000 5.551115e-17 0.000000e+00 5.551115e-17 7 6 0.500000 4.422988e+00 4.422988e+00 8.881784e-16 7 7 0.500000 2.220446e-16 0.000000e+00 2.220446e-16 7 8 0.500000 1.658621e+01 1.658621e+01 7.105427e-15 7 9 0.500000 1.776357e-15 0.000000e+00 1.776357e-15 7 10 0.500000 7.878448e+01 7.878448e+01 5.684342e-14 7 11 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 7 12 0.500000 4.530108e+02 4.530108e+02 4.547474e-13 7 13 0.500000 -5.684342e-14 0.000000e+00 5.684342e-14 7 14 0.500000 2.958305e+03 3.057823e+03 9.951724e+01 7 15 0.500000 0.000000e+00 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1.608359e+00 1.608359e+00 4.440892e-16 9 3 2.500000 -6.071532e-18 0.000000e+00 6.071532e-18 9 4 2.500000 4.422988e+00 4.422988e+00 0.000000e+00 9 5 2.500000 2.636780e-16 0.000000e+00 2.636780e-16 9 6 2.500000 1.658621e+01 1.658621e+01 3.552714e-15 9 7 2.500000 -2.553513e-15 0.000000e+00 2.553513e-15 9 8 2.500000 7.878448e+01 7.878448e+01 2.842171e-14 9 9 2.500000 7.105427e-15 0.000000e+00 7.105427e-15 9 10 2.500000 4.530108e+02 4.530108e+02 1.136868e-13 9 11 2.500000 -5.684342e-14 0.000000e+00 5.684342e-14 9 12 2.500000 3.057823e+03 3.057823e+03 9.094947e-13 9 13 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 9 14 2.500000 2.369813e+04 2.369813e+04 3.637979e-12 9 15 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 9 16 2.500000 2.073586e+05 2.073586e+05 0.000000e+00 9 17 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 9 18 2.500000 2.010874e+06 2.021746e+06 1.087226e+04 9 19 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 9 20 2.500000 2.111677e+07 2.173377e+07 6.170007e+05 10 0 2.500000 9.190625e-01 9.190625e-01 5.551115e-16 10 1 2.500000 1.299687e-17 0.000000e+00 1.299687e-17 10 2 2.500000 1.608359e+00 1.608359e+00 2.220446e-16 10 3 2.500000 1.426810e-16 0.000000e+00 1.426810e-16 10 4 2.500000 4.422988e+00 4.422988e+00 8.881784e-16 10 5 2.500000 -4.163336e-16 0.000000e+00 4.163336e-16 10 6 2.500000 1.658621e+01 1.658621e+01 7.105427e-15 10 7 2.500000 -4.440892e-16 0.000000e+00 4.440892e-16 10 8 2.500000 7.878448e+01 7.878448e+01 4.263256e-14 10 9 2.500000 5.329071e-15 0.000000e+00 5.329071e-15 10 10 2.500000 4.530108e+02 4.530108e+02 1.136868e-13 10 11 2.500000 7.105427e-14 0.000000e+00 7.105427e-14 10 12 2.500000 3.057823e+03 3.057823e+03 4.547474e-13 10 13 2.500000 4.547474e-13 0.000000e+00 4.547474e-13 10 14 2.500000 2.369813e+04 2.369813e+04 0.000000e+00 10 15 2.500000 1.818989e-12 0.000000e+00 1.818989e-12 10 16 2.500000 2.073586e+05 2.073586e+05 2.910383e-11 10 17 2.500000 -1.455192e-11 0.000000e+00 1.455192e-11 10 18 2.500000 2.021746e+06 2.021746e+06 0.000000e+00 10 19 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 10 20 2.500000 2.167941e+07 2.173377e+07 5.436129e+04 10 21 2.500000 3.725290e-09 0.000000e+00 3.725290e-09 10 22 2.500000 2.516073e+08 2.553718e+08 3.764519e+06 TEST12 GEN_HERMITE_COMPUTE computes a generalized Gauss-Hermite rule which is appropriate for integrands of the form Integral ( -oo < x < +oo ) f(x) x^alpha exp(-x*x) dx. Order = 1 ALPHA = 0.500000 0 0.0000000000000000 1.2254167024651776 Order = 2 ALPHA = 0.500000 0 -0.8660254037844386 0.6127083512325888 1 0.8660254037844386 0.6127083512325888 Order = 3 ALPHA = 0.500000 0 -1.3228756555322954 0.2625892933853952 1 0.0000000000000000 0.7002381156943872 2 1.3228756555322954 0.2625892933853952 Order = 4 ALPHA = 0.500000 0 -1.7529619663678659 0.0747721865343165 1 -0.6535475074298002 0.5379361646982722 2 0.6535475074298002 0.5379361646982722 3 1.7529619663678659 0.0747721865343165 Order = 5 ALPHA = 0.500000 0 -2.0995981508797583 0.0206908527402406 1 -1.0448385544294869 0.3373854564216621 2 0.0000000000000000 0.5092640841413724 3 1.0448385544294869 0.3373854564216621 4 2.0995981508797583 0.0206908527402406 Order = 6 ALPHA = 0.500000 0 -2.4311960068148712 0.0047584322858768 1 -1.4282643308502343 0.1432946705182554 2 -0.5471261076464519 0.4646552484284570 3 0.5471261076464519 0.4646552484284570 4 1.4282643308502343 0.1432946705182554 5 2.4311960068148712 0.0047584322858768 Order = 7 ALPHA = 0.500000 0 -2.7198800885562915 0.0011062894019685 1 -1.7473607788965217 0.0556473312506609 2 -0.8938582730216025 0.3522490969234108 3 0.0000000000000000 0.4074112673130975 4 0.8938582730216025 0.3522490969234108 5 1.7473607788965217 0.0556473312506609 6 2.7198800885562915 0.0011062894019685 Order = 8 ALPHA = 0.500000 0 -2.9990789683433161 0.0002288084584739 1 -2.0574394184774678 0.0178757746392672 2 -1.2417383409431890 0.1866121206001918 3 -0.4801606747408061 0.4079916475346560 4 0.4801606747408061 0.4079916475346560 5 1.2417383409431890 0.1866121206001918 6 2.0574394184774678 0.0178757746392672 7 2.9990789683433161 0.0002288084584739 Order = 9 ALPHA = 0.500000 0 -3.2511523261341324 0.0000482442834952 1 -2.3313221193007134 0.0055757541036437 2 -1.5374164086847444 0.0887579748998606 3 -0.7945417010067842 0.3467847917084955 4 0.0000000000000000 0.3430831724741876 5 0.7945417010067842 0.3467847917084955 6 1.5374164086847444 0.0887579748998606 7 2.3313221193007134 0.0055757541036437 8 3.2511523261341324 0.0000482442834952 Order = 10 ALPHA = 0.500000 0 -3.4966058807476776 0.0000093473340834 1 -2.5983971495446241 0.0015363564424026 2 -1.8279918123652743 0.0351763431437458 3 -1.1149053705666438 0.2117439807373521 4 -0.4330259998733385 0.3642423235750050 5 0.4330259998733385 0.3642423235750050 6 1.1149053705666438 0.2117439807373521 7 1.8279918123652743 0.0351763431437458 8 2.5983971495446241 0.0015363564424026 9 3.4966058807476776 0.0000093473340834 Order = 1 ALPHA = 1.000000 0 0.0000000000000000 1.0000000000000000 Order = 2 ALPHA = 1.000000 0 -1.0000000000000000 0.5000000000000000 1 1.0000000000000000 0.5000000000000000 Order = 3 ALPHA = 1.000000 0 -1.4142135623730951 0.2500000000000000 1 0.0000000000000000 0.5000000000000000 2 1.4142135623730951 0.2500000000000000 Order = 4 ALPHA = 1.000000 0 -1.8477590650225735 0.0732233047033631 1 -0.7653668647301796 0.4267766952966369 2 0.7653668647301796 0.4267766952966369 3 1.8477590650225735 0.0732233047033631 Order = 5 ALPHA = 1.000000 0 -2.1753277471610750 0.0223290993692602 1 -1.1260325006104943 0.3110042339640730 2 0.0000000000000000 0.3333333333333336 3 1.1260325006104943 0.3110042339640730 4 2.1753277471610750 0.0223290993692602 Order = 6 ALPHA = 1.000000 0 -2.5079762923395985 0.0051946282507931 1 -1.5146882056314566 0.1392588667846204 2 -0.6448058287449634 0.3555465049645868 3 0.6448058287449634 0.3555465049645868 4 1.5146882056314566 0.1392588667846204 5 2.5079762923395985 0.0051946282507931 Order = 7 ALPHA = 1.000000 0 -2.7854569612800755 0.0012954668385735 1 -1.8180779106881748 0.0591781927275502 2 -0.9673790505919010 0.3145263404338763 3 0.0000000000000000 0.2499999999999999 4 0.9673790505919010 0.3145263404338763 5 1.8180779106881748 0.0591781927275502 6 2.7854569612800755 0.0012954668385735 Order = 8 ALPHA = 1.000000 0 -3.0651379923750794 0.0002696473527807 1 -2.1299343409882683 0.0194439542575027 2 -1.3212725309936428 0.1787093462188998 3 -0.5679328213965030 0.3015770521708174 4 0.5679328213965030 0.3015770521708174 5 1.3212725309936428 0.1787093462188998 6 2.1299343409882683 0.0194439542575027 7 3.0651379923750794 0.0002696473527807 Order = 9 ALPHA = 1.000000 0 -3.3096667978337626 0.0000600630994212 1 -2.3939880433471465 0.0064714248102269 2 -1.6036318179826310 0.0928661670384225 3 -0.8621437977399313 0.3006023450519293 4 0.0000000000000000 0.2000000000000004 5 0.8621437977399313 0.3006023450519293 6 1.6036318179826310 0.0928661670384225 7 2.3939880433471465 0.0064714248102269 8 3.3096667978337626 0.0000600630994212 Order = 10 ALPHA = 1.000000 0 -3.5553903926679813 0.0000116849861929 1 -2.6619184821964095 0.0018058793399610 2 -1.8964244701650319 0.0379712248408538 3 -1.1888662915174761 0.1993334055415880 4 -0.5133812615572767 0.2608778052914040 5 0.5133812615572767 0.2608778052914040 6 1.1888662915174761 0.1993334055415880 7 1.8964244701650319 0.0379712248408538 8 2.6619184821964095 0.0018058793399610 9 3.5553903926679813 0.0000116849861929 Order = 1 ALPHA = 2.500000 0 0.0000000000000000 0.9190625268488832 Order = 2 ALPHA = 2.500000 0 -1.3228756555322954 0.4595312634244416 1 1.3228756555322954 0.4595312634244416 Order = 3 ALPHA = 2.500000 0 -1.6583123951776999 0.2924289858155538 1 0.0000000000000000 0.3342045552177757 2 1.6583123951776999 0.2924289858155538 Order = 4 ALPHA = 2.500000 0 -2.0995981508797583 0.0912117426015989 1 -1.0448385544294869 0.3683195208228427 2 1.0448385544294869 0.3683195208228427 3 2.0995981508797583 0.0912117426015989 Order = 5 ALPHA = 2.500000 0 -2.3846365914125589 0.0341953454149221 1 -1.3466656329231439 0.3362147032847793 2 0.0000000000000000 0.1782424294494804 3 1.3466656329231439 0.3362147032847793 4 2.3846365914125589 0.0341953454149221 Order = 6 ALPHA = 2.500000 0 -2.7198800885562915 0.0081840498746597 1 -1.7473607788965217 0.1699063099275073 2 -0.8938582730216025 0.2814409036222749 3 0.8938582730216025 0.2814409036222749 4 1.7473607788965217 0.1699063099275073 5 2.7198800885562915 0.0081840498746597 Order = 7 ALPHA = 2.500000 0 -2.9702203625084231 0.0024098131356743 1 -2.0150214027494173 0.0871896206077823 2 -1.1693928957378275 0.3136447466969384 3 0.0000000000000000 0.1125741659680933 4 1.1693928957378275 0.3136447466969384 5 2.0150214027494173 0.0871896206077823 6 2.9702203625084231 0.0024098131356743 Order = 8 ALPHA = 2.500000 0 -3.2511523261341324 0.0005099416639457 1 -2.3313221193007134 0.0303045738441493 2 -1.5374164086847444 0.2097927175810472 3 -0.7945417010067842 0.2189240303352995 4 0.7945417010067842 0.2189240303352995 5 1.5374164086847444 0.2097927175810472 6 2.3313221193007134 0.0303045738441493 7 3.2511523261341324 0.0005099416639457 Order = 9 ALPHA = 2.500000 0 -3.4765341365933926 0.0001303236536236 1 -2.5712913745972612 0.0117476928703869 2 -1.7891453520001153 0.1310178979856828 3 -1.0493474035985140 0.2774791172736727 4 0.0000000000000000 0.0783124632821511 5 1.0493474035985140 0.2774791172736727 6 1.7891453520001153 0.1310178979856828 7 2.5712913745972612 0.0117476928703869 8 3.4765341365933926 0.0001303236536236 Order = 10 ALPHA = 2.500000 0 -3.7229793194118348 0.0000256325065578 1 -2.8414549096381942 0.0033881904305600 2 -2.0875400348754183 0.0579439549110723 3 -1.3912394012659237 0.2236074413923168 4 -0.7226261238582256 0.1745660441839349 5 0.7226261238582256 0.1745660441839349 6 1.3912394012659237 0.2236074413923168 7 2.0875400348754183 0.0579439549110723 8 2.8414549096381942 0.0033881904305600 9 3.7229793194118348 0.0000256325065578 TEST13 GEN_LAGUERRE_COMPUTE computes a generalized Gauss-Laguerre rule which is appropriate for integrands of the form Integral ( 0 <= x < +oo ) f(x) x^alpha exp(-x) dx. GEN_LAGUERRE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Alpha Estimate Exact Error 1 0 0.500000 8.862269e-01 8.862269e-01 0.000000e+00 1 1 0.500000 1.329340e+00 1.329340e+00 0.000000e+00 1 2 0.500000 1.994011e+00 3.323351e+00 1.329340e+00 1 3 0.500000 2.991016e+00 1.163173e+01 8.640713e+00 1 4 0.500000 4.486524e+00 5.234278e+01 4.785625e+01 2 0 0.500000 8.862269e-01 8.862269e-01 1.110223e-16 2 1 0.500000 1.329340e+00 1.329340e+00 0.000000e+00 2 2 0.500000 3.323351e+00 3.323351e+00 0.000000e+00 2 3 0.500000 1.163173e+01 1.163173e+01 0.000000e+00 2 4 0.500000 4.569608e+01 5.234278e+01 6.646702e+00 2 5 0.500000 1.848614e+02 2.878853e+02 1.030239e+02 2 6 0.500000 7.529467e+02 1.871254e+03 1.118308e+03 3 0 0.500000 8.862269e-01 8.862269e-01 0.000000e+00 3 1 0.500000 1.329340e+00 1.329340e+00 2.220446e-16 3 2 0.500000 3.323351e+00 3.323351e+00 4.440892e-16 3 3 0.500000 1.163173e+01 1.163173e+01 3.552714e-15 3 4 0.500000 5.234278e+01 5.234278e+01 2.131628e-14 3 5 0.500000 2.878853e+02 2.878853e+02 1.136868e-13 3 6 0.500000 1.801464e+03 1.871254e+03 6.979037e+01 3 7 0.500000 1.204538e+04 1.403441e+04 1.989026e+03 3 8 0.500000 8.296657e+04 1.192925e+05 3.632589e+04 4 0 0.500000 8.862269e-01 8.862269e-01 0.000000e+00 4 1 0.500000 1.329340e+00 1.329340e+00 0.000000e+00 4 2 0.500000 3.323351e+00 3.323351e+00 4.440892e-16 4 3 0.500000 1.163173e+01 1.163173e+01 0.000000e+00 4 4 0.500000 5.234278e+01 5.234278e+01 7.105427e-15 4 5 0.500000 2.878853e+02 2.878853e+02 5.684342e-14 4 6 0.500000 1.871254e+03 1.871254e+03 0.000000e+00 4 7 0.500000 1.403441e+04 1.403441e+04 1.818989e-12 4 8 0.500000 1.180362e+05 1.192925e+05 1.256227e+03 4 9 0.500000 1.076120e+06 1.133278e+06 5.715831e+04 4 10 0.500000 1.031564e+07 1.189942e+07 1.583788e+06 5 0 0.500000 8.862269e-01 8.862269e-01 9.992007e-16 5 1 0.500000 1.329340e+00 1.329340e+00 4.440892e-16 5 2 0.500000 3.323351e+00 3.323351e+00 0.000000e+00 5 3 0.500000 1.163173e+01 1.163173e+01 0.000000e+00 5 4 0.500000 5.234278e+01 5.234278e+01 7.105427e-15 5 5 0.500000 2.878853e+02 2.878853e+02 5.684342e-14 5 6 0.500000 1.871254e+03 1.871254e+03 6.821210e-13 5 7 0.500000 1.403441e+04 1.403441e+04 7.275958e-12 5 8 0.500000 1.192925e+05 1.192925e+05 8.731149e-11 5 9 0.500000 1.133278e+06 1.133278e+06 1.164153e-09 5 10 0.500000 1.186488e+07 1.189942e+07 3.454623e+04 5 11 0.500000 1.345460e+08 1.368434e+08 2.297325e+06 5 12 0.500000 1.620549e+09 1.710542e+09 8.999294e+07 6 0 0.500000 8.862269e-01 8.862269e-01 4.440892e-16 6 1 0.500000 1.329340e+00 1.329340e+00 0.000000e+00 6 2 0.500000 3.323351e+00 3.323351e+00 0.000000e+00 6 3 0.500000 1.163173e+01 1.163173e+01 1.776357e-15 6 4 0.500000 5.234278e+01 5.234278e+01 0.000000e+00 6 5 0.500000 2.878853e+02 2.878853e+02 0.000000e+00 6 6 0.500000 1.871254e+03 1.871254e+03 2.273737e-13 6 7 0.500000 1.403441e+04 1.403441e+04 3.637979e-12 6 8 0.500000 1.192925e+05 1.192925e+05 5.820766e-11 6 9 0.500000 1.133278e+06 1.133278e+06 6.984919e-10 6 10 0.500000 1.189942e+07 1.189942e+07 9.313226e-09 6 11 0.500000 1.368434e+08 1.368434e+08 2.384186e-07 6 12 0.500000 1.709195e+09 1.710542e+09 1.347303e+06 6 13 0.500000 2.296904e+10 2.309232e+10 1.232782e+08 6 14 0.500000 3.283483e+11 3.348386e+11 6.490296e+09 7 0 0.500000 8.862269e-01 8.862269e-01 1.110223e-16 7 1 0.500000 1.329340e+00 1.329340e+00 8.881784e-16 7 2 0.500000 3.323351e+00 3.323351e+00 8.881784e-16 7 3 0.500000 1.163173e+01 1.163173e+01 0.000000e+00 7 4 0.500000 5.234278e+01 5.234278e+01 7.105427e-15 7 5 0.500000 2.878853e+02 2.878853e+02 0.000000e+00 7 6 0.500000 1.871254e+03 1.871254e+03 4.547474e-13 7 7 0.500000 1.403441e+04 1.403441e+04 1.818989e-12 7 8 0.500000 1.192925e+05 1.192925e+05 0.000000e+00 7 9 0.500000 1.133278e+06 1.133278e+06 0.000000e+00 7 10 0.500000 1.189942e+07 1.189942e+07 0.000000e+00 7 11 0.500000 1.368434e+08 1.368434e+08 2.980232e-07 7 12 0.500000 1.710542e+09 1.710542e+09 1.668930e-06 7 13 0.500000 2.309232e+10 2.309232e+10 9.918213e-05 7 14 0.500000 3.347679e+11 3.348386e+11 7.073341e+07 7 15 0.500000 5.181475e+12 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1.453966e+02 1 4 2.500000 4.987104e+02 1.871254e+03 1.372544e+03 2 0 2.500000 3.323351e+00 3.323351e+00 4.440892e-16 2 1 2.500000 1.163173e+01 1.163173e+01 0.000000e+00 2 2 2.500000 5.234278e+01 5.234278e+01 0.000000e+00 2 3 2.500000 2.878853e+02 2.878853e+02 5.684342e-14 2 4 2.500000 1.766569e+03 1.871254e+03 1.046856e+02 2 5 2.500000 1.136493e+04 1.403441e+04 2.669482e+03 2 6 2.500000 7.446087e+04 1.192925e+05 4.483159e+04 3 0 2.500000 3.323351e+00 3.323351e+00 2.220446e-15 3 1 2.500000 1.163173e+01 1.163173e+01 1.776357e-15 3 2 2.500000 5.234278e+01 5.234278e+01 7.105427e-15 3 3 2.500000 2.878853e+02 2.878853e+02 5.684342e-14 3 4 2.500000 1.871254e+03 1.871254e+03 4.547474e-13 3 5 2.500000 1.403441e+04 1.403441e+04 1.818989e-12 3 6 2.500000 1.175652e+05 1.192925e+05 1.727312e+03 3 7 2.500000 1.059868e+06 1.133278e+06 7.341075e+04 3 8 2.500000 9.974334e+06 1.189942e+07 1.925089e+06 4 0 2.500000 3.323351e+00 3.323351e+00 2.664535e-15 4 1 2.500000 1.163173e+01 1.163173e+01 5.329071e-15 4 2 2.500000 5.234278e+01 5.234278e+01 1.421085e-14 4 3 2.500000 2.878853e+02 2.878853e+02 5.684342e-14 4 4 2.500000 1.871254e+03 1.871254e+03 6.821210e-13 4 5 2.500000 1.403441e+04 1.403441e+04 5.456968e-12 4 6 2.500000 1.192925e+05 1.192925e+05 2.910383e-11 4 7 2.500000 1.133278e+06 1.133278e+06 2.328306e-10 4 8 2.500000 1.185451e+07 1.189942e+07 4.491010e+04 4 9 2.500000 1.339916e+08 1.368434e+08 2.851792e+06 4 10 2.500000 1.603083e+09 1.710542e+09 1.074586e+08 5 0 2.500000 3.323351e+00 3.323351e+00 1.332268e-15 5 1 2.500000 1.163173e+01 1.163173e+01 5.329071e-15 5 2 2.500000 5.234278e+01 5.234278e+01 2.842171e-14 5 3 2.500000 2.878853e+02 2.878853e+02 1.705303e-13 5 4 2.500000 1.871254e+03 1.871254e+03 9.094947e-13 5 5 2.500000 1.403441e+04 1.403441e+04 5.456968e-12 5 6 2.500000 1.192925e+05 1.192925e+05 4.365575e-11 5 7 2.500000 1.133278e+06 1.133278e+06 0.000000e+00 5 8 2.500000 1.189942e+07 1.189942e+07 0.000000e+00 5 9 2.500000 1.368434e+08 1.368434e+08 3.576279e-07 5 10 2.500000 1.708858e+09 1.710542e+09 1.684129e+06 5 11 2.500000 2.294327e+10 2.309232e+10 1.490454e+08 5 12 2.500000 3.272146e+11 3.348386e+11 7.624051e+09 6 0 2.500000 3.323351e+00 3.323351e+00 2.220446e-15 6 1 2.500000 1.163173e+01 1.163173e+01 7.105427e-15 6 2 2.500000 5.234278e+01 5.234278e+01 4.263256e-14 6 3 2.500000 2.878853e+02 2.878853e+02 2.273737e-13 6 4 2.500000 1.871254e+03 1.871254e+03 9.094947e-13 6 5 2.500000 1.403441e+04 1.403441e+04 5.456968e-12 6 6 2.500000 1.192925e+05 1.192925e+05 2.910383e-11 6 7 2.500000 1.133278e+06 1.133278e+06 0.000000e+00 6 8 2.500000 1.189942e+07 1.189942e+07 1.862645e-09 6 9 2.500000 1.368434e+08 1.368434e+08 2.980232e-07 6 10 2.500000 1.710542e+09 1.710542e+09 2.145767e-06 6 11 2.500000 2.309232e+10 2.309232e+10 9.918213e-05 6 12 2.500000 3.347527e+11 3.348386e+11 8.589057e+07 6 13 2.500000 5.179906e+12 5.189998e+12 1.009214e+10 6 14 2.500000 8.496252e+13 8.563497e+13 6.724588e+11 7 0 2.500000 3.323351e+00 3.323351e+00 1.776357e-15 7 1 2.500000 1.163173e+01 1.163173e+01 5.329071e-15 7 2 2.500000 5.234278e+01 5.234278e+01 1.421085e-14 7 3 2.500000 2.878853e+02 2.878853e+02 5.684342e-14 7 4 2.500000 1.871254e+03 1.871254e+03 0.000000e+00 7 5 2.500000 1.403441e+04 1.403441e+04 1.818989e-12 7 6 2.500000 1.192925e+05 1.192925e+05 0.000000e+00 7 7 2.500000 1.133278e+06 1.133278e+06 2.328306e-10 7 8 2.500000 1.189942e+07 1.189942e+07 1.862645e-09 7 9 2.500000 1.368434e+08 1.368434e+08 3.576279e-07 7 10 2.500000 1.710542e+09 1.710542e+09 1.430511e-06 7 11 2.500000 2.309232e+10 2.309232e+10 1.068115e-04 7 12 2.500000 3.348386e+11 3.348386e+11 1.342773e-03 7 13 2.500000 5.189998e+12 5.189998e+12 1.660156e-02 7 14 2.500000 8.562926e+13 8.563497e+13 5.711723e+09 7 15 2.500000 1.497752e+15 1.498612e+15 8.596143e+11 7 16 2.500000 2.765202e+16 2.772432e+16 7.230756e+13 8 0 2.500000 3.323351e+00 3.323351e+00 3.996803e-15 8 1 2.500000 1.163173e+01 1.163173e+01 1.065814e-14 8 2 2.500000 5.234278e+01 5.234278e+01 2.131628e-14 8 3 2.500000 2.878853e+02 2.878853e+02 0.000000e+00 8 4 2.500000 1.871254e+03 1.871254e+03 0.000000e+00 8 5 2.500000 1.403441e+04 1.403441e+04 1.818989e-12 8 6 2.500000 1.192925e+05 1.192925e+05 4.365575e-11 8 7 2.500000 1.133278e+06 1.133278e+06 6.984919e-10 8 8 2.500000 1.189942e+07 1.189942e+07 9.313226e-09 8 9 2.500000 1.368434e+08 1.368434e+08 5.066395e-07 8 10 2.500000 1.710542e+09 1.710542e+09 9.536743e-07 8 11 2.500000 2.309232e+10 2.309232e+10 1.449585e-04 8 12 2.500000 3.348386e+11 3.348386e+11 6.713867e-04 8 13 2.500000 5.189998e+12 5.189998e+12 3.906250e-03 8 14 2.500000 8.563497e+13 8.563497e+13 3.750000e-01 8 15 2.500000 1.498612e+15 1.498612e+15 1.000000e+00 8 16 2.500000 2.772384e+16 2.772432e+16 4.797847e+11 8 17 2.500000 5.405343e+17 5.406243e+17 8.995964e+13 8 18 2.500000 1.107348e+19 1.108280e+19 9.320178e+15 9 0 2.500000 3.323351e+00 3.323351e+00 2.220446e-15 9 1 2.500000 1.163173e+01 1.163173e+01 7.105427e-15 9 2 2.500000 5.234278e+01 5.234278e+01 2.842171e-14 9 3 2.500000 2.878853e+02 2.878853e+02 5.684342e-14 9 4 2.500000 1.871254e+03 1.871254e+03 2.273737e-13 9 5 2.500000 1.403441e+04 1.403441e+04 0.000000e+00 9 6 2.500000 1.192925e+05 1.192925e+05 0.000000e+00 9 7 2.500000 1.133278e+06 1.133278e+06 2.328306e-10 9 8 2.500000 1.189942e+07 1.189942e+07 1.862645e-09 9 9 2.500000 1.368434e+08 1.368434e+08 3.576279e-07 9 10 2.500000 1.710542e+09 1.710542e+09 9.536743e-07 9 11 2.500000 2.309232e+10 2.309232e+10 1.144409e-04 9 12 2.500000 3.348386e+11 3.348386e+11 1.220703e-03 9 13 2.500000 5.189998e+12 5.189998e+12 1.464844e-02 9 14 2.500000 8.563497e+13 8.563497e+13 1.562500e-01 9 15 2.500000 1.498612e+15 1.498612e+15 3.500000e+00 9 16 2.500000 2.772432e+16 2.772432e+16 8.000000e+01 9 17 2.500000 5.406243e+17 5.406243e+17 2.560000e+02 9 18 2.500000 1.108275e+19 1.108280e+19 4.965772e+13 9 19 2.500000 2.382688e+20 2.382802e+20 1.134679e+16 9 20 2.500000 5.359884e+21 5.361304e+21 1.419416e+18 10 0 2.500000 3.323351e+00 3.323351e+00 4.440892e-15 10 1 2.500000 1.163173e+01 1.163173e+01 1.065814e-14 10 2 2.500000 5.234278e+01 5.234278e+01 3.552714e-14 10 3 2.500000 2.878853e+02 2.878853e+02 1.705303e-13 10 4 2.500000 1.871254e+03 1.871254e+03 6.821210e-13 10 5 2.500000 1.403441e+04 1.403441e+04 5.456968e-12 10 6 2.500000 1.192925e+05 1.192925e+05 0.000000e+00 10 7 2.500000 1.133278e+06 1.133278e+06 2.328306e-10 10 8 2.500000 1.189942e+07 1.189942e+07 3.725290e-09 10 9 2.500000 1.368434e+08 1.368434e+08 4.172325e-07 10 10 2.500000 1.710542e+09 1.710542e+09 2.384186e-07 10 11 2.500000 2.309232e+10 2.309232e+10 1.258850e-04 10 12 2.500000 3.348386e+11 3.348386e+11 1.037598e-03 10 13 2.500000 5.189998e+12 5.189998e+12 9.765625e-03 10 14 2.500000 8.563497e+13 8.563497e+13 2.656250e-01 10 15 2.500000 1.498612e+15 1.498612e+15 1.250000e+00 10 16 2.500000 2.772432e+16 2.772432e+16 4.000000e+01 10 17 2.500000 5.406243e+17 5.406243e+17 5.760000e+02 10 18 2.500000 1.108280e+19 1.108280e+19 8.192000e+04 10 19 2.500000 2.382802e+20 2.382802e+20 1.212416e+06 10 20 2.500000 5.361297e+21 5.361304e+21 6.207215e+15 10 21 2.500000 1.259889e+23 1.259906e+23 1.697673e+18 10 22 2.500000 3.086518e+24 3.086771e+24 2.522504e+20 TEST14 GEN_LAGUERRE_COMPUTE computes a generalized Gauss-Laguerre rule which is appropriate for integrands of the form Integral ( 0 <= x < +oo ) f(x) x^alpha exp(-x) dx. Order = 1 ALPHA = 0.500000 0 1.5000000000000000 0.8862269254527581 Order = 2 ALPHA = 0.500000 0 0.9188611699158102 0.7233630235462755 1 4.0811388300841900 0.1628639019064825 Order = 3 ALPHA = 0.500000 0 0.6663259077023709 0.5671862778403113 1 2.8007750541502565 0.3053717688445466 2 7.0328990381473728 0.0136688787679001 Order = 4 ALPHA = 0.500000 0 0.5235260767382691 0.4530087465586076 1 2.1566487632690943 0.3816169601717996 2 5.1373875461767113 0.0507946275722408 3 10.1824376138159245 0.0008065911501100 Order = 5 ALPHA = 0.500000 0 0.4313988071478514 0.3704505700074577 1 1.7597536984236963 0.4125843737694528 2 4.1044653628283152 0.0977798200531807 3 7.7467037795425568 0.0053734153411720 4 13.4576783520575809 0.0000387462814939 Order = 6 ALPHA = 0.500000 0 0.3669498773083708 0.3094240968362596 1 1.4885342923104523 0.4177521497070224 2 3.4340079684240714 0.1432858732209768 3 6.3490679256803793 0.0153324910226338 4 10.5404698584483434 0.0004306911960439 5 16.8209700778283811 0.0000016234698211 Order = 7 ALPHA = 0.500000 0 0.3193036339206300 0.2631245143958917 1 1.2907586229591526 0.4091418694141027 2 2.9583744586966501 0.1821177320927161 3 5.4090315972444332 0.0300533243012710 4 8.8040795780567755 0.0017608941175401 5 13.4685357432514810 0.0000285294712212 6 20.2499163658708774 0.0000000616600154 Order = 8 ALPHA = 0.500000 0 0.2826336481165992 0.2271393619524718 1 1.1398738015816137 0.3935945428036146 2 2.6015248434060294 0.2129089708672283 3 4.7241145375277904 0.0478774832031382 4 7.6052562992316135 0.0045425174747626 5 11.4171820765458296 0.0001624046001853 6 16.4994107976558162 0.0000016423774138 7 23.7300039959347089 0.0000000021739431 Order = 9 ALPHA = 0.500000 0 0.2535325549744191 0.1985712548680198 1 1.0208442777203899 0.3749207846631700 2 2.3230960770224658 0.2360748210008255 3 4.1993506006572927 0.0670961050032043 4 6.7139743166150287 0.0090085088966443 5 9.9720091595393487 0.0005426607386359 6 14.1540536712780458 0.0000127053668791 7 19.6119028191659481 0.0000000848430924 8 27.2512365230270603 0.0000000000722865 Order = 10 ALPHA = 0.500000 0 0.2298729805186562 0.1754708150466581 1 0.9244815469866572 0.3552233888020710 2 2.0994104627087982 0.2526835596756779 3 3.7828808737072901 0.0863561026953326 4 6.0199180277014612 0.0151097780348608 5 8.8803475979967086 0.0013282156283636 6 12.4748324048362047 0.0000541878002117 7 16.9908472935425543 0.0000008737475869 8 22.7910028949489458 0.0000000040196999 9 30.8064059170527216 0.0000000000022922 Order = 1 ALPHA = 1.000000 0 2.0000000000000000 1.0000000000000000 Order = 2 ALPHA = 1.000000 0 1.2679491924311228 0.7886751345948125 1 4.7320508075688776 0.2113248654051871 Order = 3 ALPHA = 1.000000 0 0.9358222275240878 0.5886814810396594 1 3.3054072893322788 0.3912160592223101 2 7.7587704831436328 0.0201024597380305 Order = 4 ALPHA = 1.000000 0 0.7432919279814316 0.4468705932187761 1 2.5716350076462784 0.4776357723638682 2 5.7311787516890993 0.0741777847310522 3 10.9538943126831914 0.0013158496863032 Order = 5 ALPHA = 1.000000 0 0.6170308532782703 0.3480145400233487 1 2.1129659585785241 0.5022806741324930 2 4.6108331510175322 0.1409159194944726 3 8.3990669712048422 0.0087198930261000 4 14.2601030659208305 0.0000689733235856 Order = 6 ALPHA = 1.000000 0 0.5276681217111292 0.2776501420298742 1 1.7962998096434089 0.4939105830503539 2 3.8766415204769125 0.2030042967437299 3 6.9188165667047219 0.0246688203691898 4 11.2346104290831157 0.0007630427674635 5 17.6459635523807137 0.0000031150393875 Order = 7 ALPHA = 1.000000 0 0.4610242198049946 0.2262105963831890 1 1.5635861896542627 0.4697087077412672 2 3.3520505025367342 0.2531225162324006 3 5.9162972490205394 0.0478031088842371 4 9.4206993830215904 0.0031004597560254 5 14.1941655480074633 0.0000544846688934 6 21.0921769079544177 0.0000001263339882 Order = 8 ALPHA = 1.000000 0 0.4093835732031851 0.1876325414057237 1 1.3849631848031396 0.4389853607311435 2 2.9562545561688625 0.2899960707813131 3 5.1819431010400709 0.0751413846166974 4 8.1617096881458178 0.0079326466487073 5 12.0700551268371541 0.0003086421368133 6 17.2497355261489886 0.0000033489582098 7 24.5859552436527800 0.0000000047213928 Order = 9 ALPHA = 1.000000 0 0.3681784529417413 0.1580309235007803 1 1.2433579621404816 0.4065825794555731 2 2.6460338413842073 0.3149824836650978 3 4.6168825146350603 0.1037755199516432 4 7.2217865393965708 0.0155776377242784 5 10.5673208077418668 0.0010248719698508 6 14.8359145152609297 0.0000258002113872 7 20.3821819854492503 0.0000001833559550 8 28.1183433810498933 0.0000000001654329 Order = 10 ALPHA = 1.000000 0 0.3345286763247525 0.1348565545313774 1 1.1282533558766366 0.3749243411054584 2 2.3958699247473065 0.3302471308042159 3 4.1668409879287669 0.1315288846413067 4 6.4873530313808114 0.0258420642117171 5 9.4283548133356057 0.0024896457958959 6 13.1017235803678016 0.0001094884764267 7 17.6964875668462263 0.0000018812754142 8 23.5777870883601537 0.0000000091526850 9 31.6828009748319381 0.0000000000055013 Order = 1 ALPHA = 2.500000 0 3.5000000000000000 3.3233509704478426 Order = 2 ALPHA = 2.500000 0 2.3786796564403576 2.4449968210461095 1 6.6213203435596428 0.8783541494017335 Order = 3 ALPHA = 2.500000 0 1.8248198030758733 1.6404916909909513 1 4.8137462264876980 1.5764206102831910 2 9.8614339704364280 0.1064386691737023 Order = 4 ALPHA = 2.500000 0 1.4862411426653499 1.1090225785851453 1 3.8376839879001006 1.8203241249078195 2 7.4820583668989809 0.3853099204909317 3 13.1940165025355700 0.0086943464639431 Order = 5 ALPHA = 2.500000 0 1.2558461409184438 0.7693461876473185 1 3.2070406266149099 1.7891436571378652 2 6.1239082233284119 0.7075497394015841 3 10.3161258663913475 0.0567556714589862 4 16.5970791427468853 0.0005557148020873 Order = 6 ALPHA = 2.500000 0 1.0882635378141619 0.5488342282458465 1 2.7608555713855094 1.6358288552513571 2 5.2130824709557162 0.9757018876753594 3 8.6088345005369984 0.1568919688876181 4 13.2736722693042051 0.0060640457339780 5 20.0552916500034115 0.0000299846536813 Order = 7 ALPHA = 2.500000 0 0.9606221295119329 0.4018644652334328 1 2.4266771014277406 1.4440235761293305 2 4.5506549675827701 1.1578194454504422 3 7.4324177726246079 0.2949071223080978 4 11.2446775658018687 0.0242175011439701 5 16.3271095162749198 0.0005174322318932 6 23.5578409467761567 0.0000014279506744 Order = 8 ALPHA = 2.500000 0 0.8600416237603892 0.3011867253184344 1 2.1662435289922479 1.2529637803437648 2 4.0437426773491278 1.2583457960796296 3 6.5589225384279040 0.4473442363684208 4 9.8172555052257042 0.0605825502790592 5 13.9994797507150555 0.0028904817699157 6 19.4574474318146251 0.0000373385286657 7 27.0968669437149430 0.0000000617599561 Order = 9 ALPHA = 2.500000 0 0.7786800215607390 0.2304206696494756 1 1.9571902235047764 1.0787334348681785 2 3.6417887950645977 1.2944983831865144 3 5.8793297220349769 0.5941550811279992 4 8.7400288949175451 0.1158251816851087 5 12.3346755858879824 0.0094316286919319 6 16.8507961274953821 0.0002842237871894 7 22.6510256983860536 0.0000023649776431 8 30.6664849311479486 0.0000000024738037 Order = 10 ALPHA = 2.500000 0 0.7114760698983055 0.1795029442066154 1 1.7854778061294867 0.9263170003438104 2 3.3144812176022094 1.2847675181016542 3 5.3330492524705839 0.7226374668786890 4 7.8907634025314399 0.1864748527585193 5 11.0616739762852863 0.0224379062050689 6 14.9610223243985114 0.0011891427230148 7 19.7822180787733188 0.0000240042379159 8 25.8976644597361130 0.0000001348994981 9 34.2621734121747465 0.0000000000930604 TEST15 HERMITE_COMPUTE computes a Gauss-Hermite rule which is appropriate for integrands of the form Integral ( -oo < x < +oo ) f(x) exp(-x*x) dx. HERMITE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 1.772454 1.772454 5.515810e-12 1 1 0.000000 0.000000 0.000000e+00 1 2 0.000000 0.886227 8.862269e-01 1 3 0.000000 0.000000 0.000000e+00 1 4 0.000000 1.329340 1.329340e+00 2 0 1.772454 1.772454 5.516254e-12 2 1 0.000000 0.000000 0.000000e+00 2 2 0.886227 0.886227 2.758349e-12 2 3 0.000000 0.000000 0.000000e+00 2 4 0.443113 1.329340 8.862269e-01 2 5 0.000000 0.000000 0.000000e+00 2 6 0.221557 3.323351 3.101794e+00 3 0 1.772454 1.772454 5.515366e-12 3 1 0.000000 0.000000 0.000000e+00 3 2 0.886227 0.886227 2.757683e-12 3 3 0.000000 0.000000 0.000000e+00 3 4 1.329340 1.329340 4.136913e-12 3 5 0.000000 0.000000 0.000000e+00 3 6 1.994011 3.323351 1.329340e+00 3 7 0.000000 0.000000 0.000000e+00 3 8 2.991016 11.631728 8.640713e+00 4 0 1.772454 1.772454 5.516032e-12 4 1 -0.000000 0.000000 2.775558e-17 4 2 0.886227 0.886227 2.758127e-12 4 3 -0.000000 0.000000 5.551115e-17 4 4 1.329340 1.329340 4.137357e-12 4 5 -0.000000 0.000000 1.110223e-16 4 6 3.323351 3.323351 1.034284e-11 4 7 0.000000 0.000000 0.000000e+00 4 8 8.973048 11.631728 2.658681e+00 4 9 0.000000 0.000000 0.000000e+00 4 10 24.426630 52.342778 2.791615e+01 5 0 1.772454 1.772454 5.515588e-12 5 1 0.000000 0.000000 0.000000e+00 5 2 0.886227 0.886227 2.757572e-12 5 3 0.000000 0.000000 0.000000e+00 5 4 1.329340 1.329340 4.136691e-12 5 5 0.000000 0.000000 0.000000e+00 5 6 3.323351 3.323351 1.034062e-11 5 7 0.000000 0.000000 0.000000e+00 5 8 11.631728 11.631728 3.619327e-11 5 9 0.000000 0.000000 0.000000e+00 5 10 45.696076 52.342778 6.646702e+00 5 11 0.000000 0.000000 0.000000e+00 5 12 184.861398 287.885278 1.030239e+02 6 0 1.772454 1.772454 5.516254e-12 6 1 -0.000000 0.000000 3.469447e-17 6 2 0.886227 0.886227 2.758238e-12 6 3 -0.000000 0.000000 2.775558e-17 6 4 1.329340 1.329340 4.137579e-12 6 5 -0.000000 0.000000 5.551115e-17 6 6 3.323351 3.323351 1.034328e-11 6 7 0.000000 0.000000 0.000000e+00 6 8 11.631728 11.631728 3.620215e-11 6 9 0.000000 0.000000 0.000000e+00 6 10 52.342778 52.342778 1.629061e-10 6 11 0.000000 0.000000 0.000000e+00 6 12 267.945172 287.885278 1.994011e+01 6 13 0.000000 0.000000 0.000000e+00 6 14 1442.542031 1871.254306 4.287123e+02 7 0 1.772454 1.772454 5.516698e-12 7 1 -0.000000 0.000000 8.239937e-18 7 2 0.886227 0.886227 2.758571e-12 7 3 0.000000 0.000000 3.469447e-18 7 4 1.329340 1.329340 4.138467e-12 7 5 -0.000000 0.000000 2.775558e-17 7 6 3.323351 3.323351 1.034550e-11 7 7 -0.000000 0.000000 2.220446e-16 7 8 11.631728 11.631728 3.620571e-11 7 9 -0.000000 0.000000 8.881784e-16 7 10 52.342778 52.342778 1.628848e-10 7 11 0.000000 0.000000 0.000000e+00 7 12 287.885278 287.885278 8.957386e-10 7 13 0.000000 0.000000 0.000000e+00 7 14 1801.463935 1871.254306 6.979037e+01 7 15 0.000000 0.000000 0.000000e+00 7 16 12045.381738 14034.407293 1.989026e+03 8 0 1.772454 1.772454 5.516032e-12 8 1 0.000000 0.000000 3.816392e-17 8 2 0.886227 0.886227 2.758682e-12 8 3 -0.000000 0.000000 1.387779e-17 8 4 1.329340 1.329340 4.138245e-12 8 5 0.000000 0.000000 2.289835e-16 8 6 3.323351 3.323351 1.034461e-11 8 7 -0.000000 0.000000 1.665335e-16 8 8 11.631728 11.631728 3.620393e-11 8 9 0.000000 0.000000 0.000000e+00 8 10 52.342778 52.342778 1.629132e-10 8 11 -0.000000 0.000000 3.552714e-15 8 12 287.885278 287.885278 8.959091e-10 8 13 0.000000 0.000000 0.000000e+00 8 14 1871.254306 1871.254306 5.823495e-09 8 15 0.000000 0.000000 0.000000e+00 8 16 13755.245812 14034.407293 2.791615e+02 8 17 0.000000 0.000000 0.000000e+00 8 18 109103.067919 119292.461995 1.018939e+04 9 0 1.772454 1.772454 5.515588e-12 9 1 -0.000000 0.000000 2.049142e-17 9 2 0.886227 0.886227 2.757905e-12 9 3 -0.000000 0.000000 6.743738e-17 9 4 1.329340 1.329340 4.137135e-12 9 5 -0.000000 0.000000 5.204170e-18 9 6 3.323351 3.323351 1.034373e-11 9 7 0.000000 0.000000 2.775558e-16 9 8 11.631728 11.631728 3.620748e-11 9 9 0.000000 0.000000 6.661338e-16 9 10 52.342778 52.342778 1.629630e-10 9 11 0.000000 0.000000 0.000000e+00 9 12 287.885278 287.885278 8.964207e-10 9 13 0.000000 0.000000 0.000000e+00 9 14 1871.254306 1871.254306 5.828042e-09 9 15 -0.000000 0.000000 2.273737e-13 9 16 14034.407293 14034.407293 4.372123e-08 9 17 0.000000 0.000000 0.000000e+00 9 18 118036.235327 119292.461995 1.256227e+03 9 19 0.000000 0.000000 0.000000e+00 9 20 1076120.075605 1133278.388949 5.715831e+04 10 0 1.772454 1.772454 5.515810e-12 10 1 0.000000 0.000000 2.482823e-17 10 2 0.886227 0.886227 2.757794e-12 10 3 -0.000000 0.000000 2.921925e-17 10 4 1.329340 1.329340 4.136913e-12 10 5 0.000000 0.000000 6.938894e-18 10 6 3.323351 3.323351 1.034151e-11 10 7 -0.000000 0.000000 1.526557e-16 10 8 11.631728 11.631728 3.619149e-11 10 9 0.000000 0.000000 5.551115e-16 10 10 52.342778 52.342778 1.628422e-10 10 11 0.000000 0.000000 0.000000e+00 10 12 287.885278 287.885278 8.955112e-10 10 13 -0.000000 0.000000 5.684342e-14 10 14 1871.254306 1871.254306 5.820084e-09 10 15 -0.000000 0.000000 1.136868e-13 10 16 14034.407293 14034.407293 4.364483e-08 10 17 0.000000 0.000000 1.818989e-12 10 18 119292.461994 119292.461995 3.709138e-07 10 19 0.000000 0.000000 0.000000e+00 10 20 1126997.255611 1133278.388949 6.281133e+03 10 21 0.000000 0.000000 0.000000e+00 10 22 11550820.183881 11899423.083962 3.486029e+05 TEST16 HERMITE_COMPUTE computes a Gauss-Hermite rule which is appropriate for integrands of the form Integral ( -oo < x < +oo ) f(x) exp(-x*x) dx. Order = 1 0 0.0000000000000000 1.7724538509000001 Order = 2 0 -0.7071067811865475 0.8862269254499998 1 0.7071067811865475 0.8862269254499998 Order = 3 0 -1.2247448713915889 0.2954089751500001 1 0.0000000000000000 1.1816359006000001 2 1.2247448713915889 0.2954089751500001 Order = 4 0 -1.6506801238857847 0.0813128354469921 1 -0.5246476232752904 0.8049140900030078 2 0.5246476232752904 0.8049140900030078 3 1.6506801238857847 0.0813128354469921 Order = 5 0 -2.0201828704560856 0.0199532420589838 1 -0.9585724646138185 0.3936193231510163 2 0.0000000000000000 0.9453087204800000 3 0.9585724646138185 0.3936193231510163 4 2.0201828704560856 0.0199532420589838 Order = 6 0 -2.3506049736744923 0.0045300099054947 1 -1.3358490740136970 0.1570673203223678 2 -0.4360774119276165 0.7246295952221373 3 0.4360774119276165 0.7246295952221373 4 1.3358490740136970 0.1570673203223678 5 2.3506049736744923 0.0045300099054947 Order = 7 0 -2.6519613568352334 0.0009717812450965 1 -1.6735516287674714 0.0545155828189573 2 -0.8162878828589647 0.4256072526088032 3 0.0000000000000000 0.8102646175542857 4 0.8162878828589647 0.4256072526088032 5 1.6735516287674714 0.0545155828189573 6 2.6519613568352334 0.0009717812450965 Order = 8 0 -2.9306374202572441 0.0001996040722107 1 -1.9816567566958430 0.0170779830073603 2 -1.1571937124467802 0.2078023258142450 3 -0.3811869902073221 0.6611470125561838 4 0.3811869902073221 0.6611470125561838 5 1.1571937124467802 0.2078023258142450 6 1.9816567566958430 0.0170779830073603 7 2.9306374202572441 0.0001996040722107 Order = 9 0 -3.1909932017815272 0.0000396069772631 1 -2.2665805845318432 0.0049436242755216 2 -1.4685532892166679 0.0884745273941013 3 -0.7235510187528376 0.4326515590012093 4 0.0000000000000000 0.7202352156038097 5 0.7235510187528376 0.4326515590012093 6 1.4685532892166679 0.0884745273941013 7 2.2665805845318432 0.0049436242755216 8 3.1909932017815272 0.0000396069772631 Order = 10 0 -3.4361591188377378 0.0000076404328552 1 -2.5327316742327897 0.0013436457467771 2 -1.7566836492998819 0.0338743944553756 3 -1.0366108297895136 0.2401386110815674 4 -0.3429013272237046 0.6108626337334246 5 0.3429013272237046 0.6108626337334246 6 1.0366108297895136 0.2401386110815674 7 1.7566836492998819 0.0338743944553756 8 2.5327316742327897 0.0013436457467771 9 3.4361591188377378 0.0000076404328552 TEST17 JACOBI_COMPUTE computes a Gauss-Jacobi rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) (1-x)^alpha (1+x)^beta dx. JACOBI_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Alpha Beta Estimate Exact Error 1 0 0.500000 0.500000 1.570796 1.570796 1.110223e-15 1 1 0.500000 0.500000 0.000000 0.000000 0.000000e+00 1 2 0.500000 0.500000 0.000000 0.392699 3.926991e-01 1 3 0.500000 0.500000 0.000000 0.000000 0.000000e+00 1 4 0.500000 0.500000 0.000000 0.196350 1.963495e-01 2 0 0.500000 0.500000 1.570796 1.570796 1.110223e-15 2 1 0.500000 0.500000 0.000000 0.000000 0.000000e+00 2 2 0.500000 0.500000 0.392699 0.392699 0.000000e+00 2 3 0.500000 0.500000 0.000000 0.000000 0.000000e+00 2 4 0.500000 0.500000 0.098175 0.196350 9.817477e-02 2 5 0.500000 0.500000 0.000000 0.000000 0.000000e+00 2 6 0.500000 0.500000 0.024544 0.122718 9.817477e-02 3 0 0.500000 0.500000 1.570796 1.570796 1.110223e-15 3 1 0.500000 0.500000 0.000000 0.000000 4.996004e-16 3 2 0.500000 0.500000 0.392699 0.392699 0.000000e+00 3 3 0.500000 0.500000 0.000000 0.000000 3.053113e-16 3 4 0.500000 0.500000 0.196350 0.196350 8.326673e-17 3 5 0.500000 0.500000 0.000000 0.000000 1.804112e-16 3 6 0.500000 0.500000 0.098175 0.122718 2.454369e-02 3 7 0.500000 0.500000 0.000000 0.000000 9.714451e-17 3 8 0.500000 0.500000 0.049087 0.085903 3.681554e-02 4 0 0.500000 0.500000 1.570796 1.570796 8.881784e-16 4 1 0.500000 0.500000 -0.000000 0.000000 2.775558e-17 4 2 0.500000 0.500000 0.392699 0.392699 0.000000e+00 4 3 0.500000 0.500000 -0.000000 0.000000 1.387779e-17 4 4 0.500000 0.500000 0.196350 0.196350 5.551115e-17 4 5 0.500000 0.500000 0.000000 0.000000 0.000000e+00 4 6 0.500000 0.500000 0.122718 0.122718 1.387779e-17 4 7 0.500000 0.500000 0.000000 0.000000 0.000000e+00 4 8 0.500000 0.500000 0.079767 0.085903 6.135923e-03 4 9 0.500000 0.500000 0.000000 0.000000 0.000000e+00 4 10 0.500000 0.500000 0.052155 0.064427 1.227185e-02 5 0 0.500000 0.500000 1.570796 1.570796 1.998401e-15 5 1 0.500000 0.500000 0.000000 0.000000 4.440892e-16 5 2 0.500000 0.500000 0.392699 0.392699 6.106227e-16 5 3 0.500000 0.500000 0.000000 0.000000 2.914335e-16 5 4 0.500000 0.500000 0.196350 0.196350 5.551115e-16 5 5 0.500000 0.500000 0.000000 0.000000 2.359224e-16 5 6 0.500000 0.500000 0.122718 0.122718 3.608225e-16 5 7 0.500000 0.500000 0.000000 0.000000 1.734723e-16 5 8 0.500000 0.500000 0.085903 0.085903 2.498002e-16 5 9 0.500000 0.500000 0.000000 0.000000 1.318390e-16 5 10 0.500000 0.500000 0.062893 0.064427 1.533981e-03 5 11 0.500000 0.500000 0.000000 0.000000 9.714451e-17 5 12 0.500000 0.500000 0.046786 0.050621 3.834952e-03 6 0 0.500000 0.500000 1.570796 1.570796 1.776357e-15 6 1 0.500000 0.500000 -0.000000 0.000000 1.387779e-17 6 2 0.500000 0.500000 0.392699 0.392699 4.996004e-16 6 3 0.500000 0.500000 0.000000 0.000000 1.387779e-17 6 4 0.500000 0.500000 0.196350 0.196350 3.885781e-16 6 5 0.500000 0.500000 -0.000000 0.000000 6.938894e-18 6 6 0.500000 0.500000 0.122718 0.122718 2.359224e-16 6 7 0.500000 0.500000 0.000000 0.000000 0.000000e+00 6 8 0.500000 0.500000 0.085903 0.085903 1.804112e-16 6 9 0.500000 0.500000 0.000000 0.000000 0.000000e+00 6 10 0.500000 0.500000 0.064427 0.064427 4.163336e-17 6 11 0.500000 0.500000 0.000000 0.000000 0.000000e+00 6 12 0.500000 0.500000 0.050238 0.050621 3.834952e-04 6 13 0.500000 0.500000 0.000000 0.000000 0.000000e+00 6 14 0.500000 0.500000 0.039979 0.041130 1.150486e-03 7 0 0.500000 0.500000 1.570796 1.570796 8.881784e-16 7 1 0.500000 0.500000 0.000000 0.000000 3.191891e-16 7 2 0.500000 0.500000 0.392699 0.392699 3.330669e-16 7 3 0.500000 0.500000 0.000000 0.000000 2.081668e-16 7 4 0.500000 0.500000 0.196350 0.196350 1.665335e-16 7 5 0.500000 0.500000 0.000000 0.000000 1.179612e-16 7 6 0.500000 0.500000 0.122718 0.122718 2.081668e-16 7 7 0.500000 0.500000 0.000000 0.000000 6.245005e-17 7 8 0.500000 0.500000 0.085903 0.085903 1.804112e-16 7 9 0.500000 0.500000 0.000000 0.000000 3.469447e-17 7 10 0.500000 0.500000 0.064427 0.064427 3.191891e-16 7 11 0.500000 0.500000 0.000000 0.000000 2.081668e-17 7 12 0.500000 0.500000 0.050621 0.050621 5.967449e-16 7 13 0.500000 0.500000 0.000000 0.000000 1.040834e-17 7 14 0.500000 0.500000 0.041034 0.041130 9.587380e-05 7 15 0.500000 0.500000 0.000000 0.000000 3.469447e-18 7 16 0.500000 0.500000 0.033939 0.034275 3.355583e-04 8 0 0.500000 0.500000 1.570796 1.570796 4.440892e-16 8 1 0.500000 0.500000 0.000000 0.000000 5.620504e-16 8 2 0.500000 0.500000 0.392699 0.392699 1.665335e-16 8 3 0.500000 0.500000 0.000000 0.000000 5.065393e-16 8 4 0.500000 0.500000 0.196350 0.196350 1.110223e-16 8 5 0.500000 0.500000 0.000000 0.000000 4.406198e-16 8 6 0.500000 0.500000 0.122718 0.122718 1.665335e-16 8 7 0.500000 0.500000 0.000000 0.000000 3.851086e-16 8 8 0.500000 0.500000 0.085903 0.085903 1.665335e-16 8 9 0.500000 0.500000 0.000000 0.000000 3.434752e-16 8 10 0.500000 0.500000 0.064427 0.064427 3.330669e-16 8 11 0.500000 0.500000 0.000000 0.000000 3.053113e-16 8 12 0.500000 0.500000 0.050621 0.050621 6.314393e-16 8 13 0.500000 0.500000 0.000000 0.000000 2.671474e-16 8 14 0.500000 0.500000 0.041130 0.041130 8.326673e-17 8 15 0.500000 0.500000 0.000000 0.000000 2.324529e-16 8 16 0.500000 0.500000 0.034251 0.034275 2.396845e-05 8 17 0.500000 0.500000 0.000000 0.000000 2.099015e-16 8 18 0.500000 0.500000 0.029038 0.029134 9.587380e-05 9 0 0.500000 0.500000 1.570796 1.570796 2.220446e-16 9 1 0.500000 0.500000 -0.000000 0.000000 6.938894e-18 9 2 0.500000 0.500000 0.392699 0.392699 5.551115e-16 9 3 0.500000 0.500000 -0.000000 0.000000 3.469447e-18 9 4 0.500000 0.500000 0.196350 0.196350 3.330669e-16 9 5 0.500000 0.500000 -0.000000 0.000000 6.938894e-18 9 6 0.500000 0.500000 0.122718 0.122718 3.747003e-16 9 7 0.500000 0.500000 -0.000000 0.000000 3.469447e-18 9 8 0.500000 0.500000 0.085903 0.085903 3.608225e-16 9 9 0.500000 0.500000 0.000000 0.000000 0.000000e+00 9 10 0.500000 0.500000 0.064427 0.064427 4.996004e-16 9 11 0.500000 0.500000 0.000000 0.000000 0.000000e+00 9 12 0.500000 0.500000 0.050621 0.050621 7.771561e-16 9 13 0.500000 0.500000 -0.000000 0.000000 1.734723e-18 9 14 0.500000 0.500000 0.041130 0.041130 2.151057e-16 9 15 0.500000 0.500000 0.000000 0.000000 0.000000e+00 9 16 0.500000 0.500000 0.034275 0.034275 7.632783e-17 9 17 0.500000 0.500000 0.000000 0.000000 0.000000e+00 9 18 0.500000 0.500000 0.029128 0.029134 5.992112e-06 9 19 0.500000 0.500000 0.000000 0.000000 0.000000e+00 9 20 0.500000 0.500000 0.025134 0.025161 2.696451e-05 10 0 0.500000 0.500000 1.570796 1.570796 1.332268e-15 10 1 0.500000 0.500000 -0.000000 0.000000 2.428613e-17 10 2 0.500000 0.500000 0.392699 0.392699 5.551115e-17 10 3 0.500000 0.500000 0.000000 0.000000 1.734723e-17 10 4 0.500000 0.500000 0.196350 0.196350 8.326673e-17 10 5 0.500000 0.500000 0.000000 0.000000 3.469447e-18 10 6 0.500000 0.500000 0.122718 0.122718 1.526557e-16 10 7 0.500000 0.500000 0.000000 0.000000 0.000000e+00 10 8 0.500000 0.500000 0.085903 0.085903 1.804112e-16 10 9 0.500000 0.500000 0.000000 0.000000 0.000000e+00 10 10 0.500000 0.500000 0.064427 0.064427 3.469447e-16 10 11 0.500000 0.500000 0.000000 0.000000 0.000000e+00 10 12 0.500000 0.500000 0.050621 0.050621 6.591949e-16 10 13 0.500000 0.500000 -0.000000 0.000000 1.734723e-18 10 14 0.500000 0.500000 0.041130 0.041130 1.110223e-16 10 15 0.500000 0.500000 0.000000 0.000000 0.000000e+00 10 16 0.500000 0.500000 0.034275 0.034275 2.775558e-17 10 17 0.500000 0.500000 0.000000 0.000000 0.000000e+00 10 18 0.500000 0.500000 0.029134 0.029134 1.457168e-16 10 19 0.500000 0.500000 0.000000 0.000000 1.734723e-18 10 20 0.500000 0.500000 0.025159 0.025161 1.498028e-06 10 21 0.500000 0.500000 0.000000 0.000000 0.000000e+00 10 22 0.500000 0.500000 0.022008 0.022016 7.490141e-06 1 0 0.500000 1.000000 1.508494 1.508494 2.220446e-16 1 1 0.500000 1.000000 0.215499 0.215499 2.775558e-17 1 2 0.500000 1.000000 0.030786 0.359165 3.283797e-01 1 3 0.500000 1.000000 0.004398 0.111015 1.066168e-01 1 4 0.500000 1.000000 0.000628 0.174308 1.736799e-01 2 0 0.500000 1.000000 1.508494 1.508494 6.661338e-16 2 1 0.500000 1.000000 0.215499 0.215499 2.498002e-16 2 2 0.500000 1.000000 0.359165 0.359165 1.665335e-16 2 3 0.500000 1.000000 0.111015 0.111015 2.775558e-17 2 4 0.500000 1.000000 0.096371 0.174308 7.793713e-02 2 5 0.500000 1.000000 0.041071 0.070828 2.975781e-02 2 6 0.500000 1.000000 0.027910 0.106701 7.879089e-02 3 0 0.500000 1.000000 1.508494 1.508494 2.220446e-16 3 1 0.500000 1.000000 0.215499 0.215499 2.498002e-16 3 2 0.500000 1.000000 0.359165 0.359165 5.551115e-17 3 3 0.500000 1.000000 0.111015 0.111015 1.110223e-16 3 4 0.500000 1.000000 0.174308 0.174308 2.775558e-17 3 5 0.500000 1.000000 0.070828 0.070828 6.938894e-17 3 6 0.500000 1.000000 0.087742 0.106701 1.895882e-02 3 7 0.500000 1.000000 0.042566 0.050350 7.783096e-03 3 8 0.500000 1.000000 0.044985 0.073531 2.854635e-02 4 0 0.500000 1.000000 1.508494 1.508494 0.000000e+00 4 1 0.500000 1.000000 0.215499 0.215499 2.220446e-16 4 2 0.500000 1.000000 0.359165 0.359165 5.551115e-17 4 3 0.500000 1.000000 0.111015 0.111015 1.942890e-16 4 4 0.500000 1.000000 0.174308 0.174308 0.000000e+00 4 5 0.500000 1.000000 0.070828 0.070828 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2.500000 2.500000 0.000000 0.000000 2.168404e-18 8 6 2.500000 2.500000 0.015340 0.015340 3.469447e-18 8 7 2.500000 2.500000 0.000000 0.000000 2.168404e-19 8 8 2.500000 2.500000 0.007670 0.007670 1.994932e-17 8 9 2.500000 2.500000 0.000000 0.000000 0.000000e+00 8 10 2.500000 2.500000 0.004314 0.004314 2.081668e-17 8 11 2.500000 2.500000 0.000000 0.000000 0.000000e+00 8 12 2.500000 2.500000 0.002637 0.002637 5.637851e-18 8 13 2.500000 2.500000 0.000000 0.000000 0.000000e+00 8 14 2.500000 2.500000 0.001714 0.001714 2.168404e-19 8 15 2.500000 2.500000 0.000000 0.000000 0.000000e+00 8 16 2.500000 2.500000 0.001166 0.001168 2.596582e-06 8 17 2.500000 2.500000 0.000000 0.000000 0.000000e+00 8 18 2.500000 2.500000 0.000819 0.000828 8.503806e-06 9 0 2.500000 2.500000 0.981748 0.981748 1.110223e-16 9 1 2.500000 2.500000 0.000000 0.000000 1.647987e-17 9 2 2.500000 2.500000 0.122718 0.122718 2.775558e-17 9 3 2.500000 2.500000 0.000000 0.000000 2.168404e-18 9 4 2.500000 2.500000 0.036816 0.036816 6.938894e-18 9 5 2.500000 2.500000 -0.000000 0.000000 6.505213e-19 9 6 2.500000 2.500000 0.015340 0.015340 3.122502e-17 9 7 2.500000 2.500000 0.000000 0.000000 4.336809e-19 9 8 2.500000 2.500000 0.007670 0.007670 3.642919e-17 9 9 2.500000 2.500000 0.000000 0.000000 0.000000e+00 9 10 2.500000 2.500000 0.004314 0.004314 3.122502e-17 9 11 2.500000 2.500000 -0.000000 0.000000 1.084202e-19 9 12 2.500000 2.500000 0.002637 0.002637 1.170938e-17 9 13 2.500000 2.500000 0.000000 0.000000 0.000000e+00 9 14 2.500000 2.500000 0.001714 0.001714 4.119968e-18 9 15 2.500000 2.500000 0.000000 0.000000 0.000000e+00 9 16 2.500000 2.500000 0.001168 0.001168 2.168404e-19 9 17 2.500000 2.500000 -0.000000 0.000000 5.421011e-20 9 18 2.500000 2.500000 0.000827 0.000828 6.196389e-07 9 19 2.500000 2.500000 0.000000 0.000000 0.000000e+00 9 20 2.500000 2.500000 0.000603 0.000605 2.324729e-06 10 0 2.500000 2.500000 0.981748 0.981748 4.440892e-16 10 1 2.500000 2.500000 -0.000000 0.000000 1.327063e-16 10 2 2.500000 2.500000 0.122718 0.122718 1.665335e-16 10 3 2.500000 2.500000 -0.000000 0.000000 3.491131e-17 10 4 2.500000 2.500000 0.036816 0.036816 6.245005e-17 10 5 2.500000 2.500000 -0.000000 0.000000 1.214306e-17 10 6 2.500000 2.500000 0.015340 0.015340 6.938894e-18 10 7 2.500000 2.500000 -0.000000 0.000000 4.228388e-18 10 8 2.500000 2.500000 0.007670 0.007670 1.474515e-17 10 9 2.500000 2.500000 -0.000000 0.000000 1.409463e-18 10 10 2.500000 2.500000 0.004314 0.004314 1.821460e-17 10 11 2.500000 2.500000 -0.000000 0.000000 3.252607e-19 10 12 2.500000 2.500000 0.002637 0.002637 5.637851e-18 10 13 2.500000 2.500000 -0.000000 0.000000 1.626303e-19 10 14 2.500000 2.500000 0.001714 0.001714 2.168404e-19 10 15 2.500000 2.500000 -0.000000 0.000000 1.626303e-19 10 16 2.500000 2.500000 0.001168 0.001168 3.035766e-18 10 17 2.500000 2.500000 0.000000 0.000000 5.421011e-20 10 18 2.500000 2.500000 0.000828 0.000828 4.119968e-18 10 19 2.500000 2.500000 -0.000000 0.000000 2.710505e-20 10 20 2.500000 2.500000 0.000605 0.000605 1.489517e-07 10 21 2.500000 2.500000 0.000000 0.000000 2.710505e-20 10 22 2.500000 2.500000 0.000453 0.000454 6.303847e-07 TEST18 JACOBI_COMPUTE computes a Gauss-Jacobi rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) (1-x)^alpha (1+x)^beta dx. Order = 1 ALPHA = 0.500000 BETA = 0.500000 0 0.0000000000000000 1.5707963267948968 Order = 2 ALPHA = 0.500000 BETA = 0.500000 0 -0.5000000000000000 0.7853981633974484 1 0.5000000000000000 0.7853981633974484 Order = 3 ALPHA = 0.500000 BETA = 0.500000 0 -0.7071067811865475 0.3926990816987239 1 0.0000000000000000 0.7853981633974484 2 0.7071067811865476 0.3926990816987245 Order = 4 ALPHA = 0.500000 BETA = 0.500000 0 -0.8090169943749475 0.2170787134227060 1 -0.3090169943749475 0.5683194499747424 2 0.3090169943749474 0.5683194499747424 3 0.8090169943749475 0.2170787134227060 Order = 5 ALPHA = 0.500000 BETA = 0.500000 0 -0.8660254037844387 0.1308996938995740 1 -0.5000000000000000 0.3926990816987242 2 0.0000000000000000 0.5235987755982989 3 0.5000000000000000 0.3926990816987242 4 0.8660254037844387 0.1308996938995745 Order = 6 ALPHA = 0.500000 BETA = 0.500000 0 -0.9009688679024191 0.0844886908915884 1 -0.6234898018587335 0.2743330560697777 2 -0.2225209339563144 0.4265764164360819 3 0.2225209339563144 0.4265764164360819 4 0.6234898018587335 0.2743330560697777 5 0.9009688679024191 0.0844886908915884 Order = 7 ALPHA = 0.500000 BETA = 0.500000 0 -0.9238795325112867 0.0575094490319133 1 -0.7071067811865475 0.1963495408493619 2 -0.3826834323650898 0.3351896326668111 3 0.0000000000000000 0.3926990816987242 4 0.3826834323650898 0.3351896326668108 5 0.7071067811865476 0.1963495408493624 6 0.9238795325112867 0.0575094490319133 Order = 8 ALPHA = 0.500000 BETA = 0.500000 0 -0.9396926207859084 0.0408329477091069 1 -0.7660444431189780 0.1442256007956728 2 -0.5000000000000000 0.2617993877991495 3 -0.1736481776669303 0.3385402270935191 4 0.1736481776669303 0.3385402270935191 5 0.5000000000000000 0.2617993877991495 6 0.7660444431189780 0.1442256007956728 7 0.9396926207859084 0.0408329477091075 Order = 9 ALPHA = 0.500000 BETA = 0.500000 0 -0.9510565162951536 0.0299995403716084 1 -0.8090169943749475 0.1085393567113530 2 -0.5877852522924731 0.2056199086476264 3 -0.3090169943749475 0.2841597249873712 4 0.0000000000000000 0.3141592653589794 5 0.3090169943749475 0.2841597249873712 6 0.5877852522924731 0.2056199086476264 7 0.8090169943749475 0.1085393567113530 8 0.9510565162951536 0.0299995403716084 Order = 10 ALPHA = 0.500000 BETA = 0.500000 0 -0.9594929736144974 0.0226689425018590 1 -0.8412535328311812 0.0834785409341889 2 -0.6548607339452851 0.1631221774548165 3 -0.4154150130018864 0.2363135602034873 4 -0.1423148382732851 0.2798149423030965 5 0.1423148382732851 0.2798149423030966 6 0.4154150130018864 0.2363135602034873 7 0.6548607339452851 0.1631221774548165 8 0.8412535328311812 0.0834785409341889 9 0.9594929736144974 0.0226689425018590 Order = 1 ALPHA = 0.500000 BETA = 1.000000 0 0.1428571428571428 1.5084944665313014 Order = 2 ALPHA = 0.500000 BETA = 1.000000 0 -0.3785434359039293 0.6707847483124826 1 0.5603616177221111 0.8377097182188185 Order = 3 ALPHA = 0.500000 BETA = 1.000000 0 -0.6191289006136984 0.2886624303931601 1 0.0828374856848099 0.7711842727220675 2 0.7362914149288884 0.4486477634160737 Order = 4 ALPHA = 0.500000 BETA = 1.000000 0 -0.7447173440086010 0.1380862915234588 1 -0.2308286409011868 0.5011891464890379 2 0.3610180502309864 0.6101454555318188 3 0.8250542504682752 0.2590735729869860 Order = 5 ALPHA = 0.500000 BETA = 1.000000 0 -0.8175993274098121 0.0729555935075893 1 -0.4326084003843104 0.3103351805927452 2 0.0582907697984681 0.5176668424406611 3 0.5335962093870673 0.4466360854739414 4 0.8757120529564132 0.1609007645163642 Order = 6 ALPHA = 0.500000 BETA = 1.000000 0 -0.8633993972645075 0.0417787392098552 1 -0.5667404500118451 0.1951823756778569 2 -0.1659246338882320 0.3864855437976816 3 0.2649141017150741 0.4544246037858499 4 0.6461272538533950 0.3245589588523248 5 0.9072453478183379 0.1060642452077330 Order = 7 ALPHA = 0.500000 BETA = 1.000000 0 -0.8939724663908039 0.0255150146724793 1 -0.6594047339183108 0.1266052232513150 2 -0.3307207485144988 0.2785906683424280 3 0.0449578958653259 0.3894994287201925 4 0.4138036987843730 0.3759820326953335 5 0.7229723820004327 0.2389626799471577 6 0.9281704237863851 0.0733394189023952 Order = 8 ALPHA = 0.500000 BETA = 1.000000 0 -0.9153619779765982 0.0164120900902137 1 -0.7257160980084200 0.0848554733563242 2 -0.4533780733947834 0.2002825508076700 3 -0.1294870729403755 0.3123493200327135 4 0.2089491097631811 0.3577748934099902 5 0.5232643292074005 0.3048404655795938 6 0.7775490329551161 0.1792709060668855 7 0.9427521789659081 0.0527087671879100 Order = 9 ALPHA = 0.500000 BETA = 1.000000 0 -0.9308966361094718 0.0110143919419965 1 -0.7746386880679894 0.0586443046921291 2 -0.5463185334791457 0.1453933047463929 3 -0.2672381326388559 0.2442199669465212 4 0.0365863464439503 0.3121710958691359 5 0.3368351070820349 0.3145887170547920 6 0.6055223420849569 0.2462380941383862 7 0.8176042531316312 0.1371247634867023 8 0.9533131723221203 0.0390998276552453 Order = 10 ALPHA = 0.500000 BETA = 1.000000 0 -0.9425275499488474 0.0076569987337496 1 -0.8116864125172604 0.0416630238085871 2 -0.6180513952640334 0.1070695285744137 3 -0.3766418843493626 0.1896495013540224 4 -0.1061646285278261 0.2614695484772918 5 0.1724237674317270 0.2937514863341334 6 0.4375388507874068 0.2709981593343723 7 0.6686402341966643 0.1996303237647666 8 0.8478228797503713 0.1068280415109269 9 0.9612042779760442 0.0297778546390378 Order = 1 ALPHA = 0.500000 BETA = 2.500000 0 0.4000000000000000 1.9634954084936209 Order = 2 ALPHA = 0.500000 BETA = 2.500000 0 -0.1055161125036901 0.6949608437695367 1 0.6769446839322616 1.2685345647240831 Order = 3 ALPHA = 0.500000 BETA = 2.500000 0 -0.3949726057630553 0.2155740450619749 1 0.2647662774451411 0.9816213720815016 2 0.7968729949845809 0.7662999913501441 Order = 4 ALPHA = 0.500000 BETA = 2.500000 0 -0.5673928432278891 0.0735270499348779 1 -0.0438129709574188 0.4997101315556028 2 0.4782610448380588 0.9082548004475427 3 0.8602174966199764 0.4820034265555977 Order = 5 ALPHA = 0.500000 BETA = 2.500000 0 -0.6766083904771950 0.0281042710539255 1 -0.2615276527389886 0.2379108911216858 2 0.1979083288674991 0.6351035815608072 3 0.6116098169968442 0.7438254266897738 4 0.8978486665826095 0.3185512380674287 Order = 6 ALPHA = 0.500000 BETA = 2.500000 0 -0.7496281758956908 0.0118933444385382 1 -0.4162703993964171 0.1152628351592500 2 -0.0240992331493380 0.3830143850747026 3 0.3678575869061302 0.6464461334770021 4 0.7000843065535390 0.5868443910765002 5 0.9220559149817765 0.2200343192676289 Order = 7 ALPHA = 0.500000 BETA = 2.500000 0 -0.8006672147724484 0.0054858163722489 1 -0.5286792390323163 0.0581588649983062 2 -0.1959098000069028 0.2221608406878788 3 0.1580287890648371 0.4638578133115204 4 0.4905660544916727 0.5963414735975124 5 0.7616344330284673 0.4597291058720926 6 0.9385563889913962 0.1577614936540615 Order = 8 ALPHA = 0.500000 BETA = 2.500000 0 -0.8376646856786829 0.0027198968451134 1 -0.6123079224732051 0.0307056736312335 2 -0.3290736457267179 0.1289926195321417 3 -0.0152678951698322 0.3092909316152765 4 0.2984865079144489 0.4881539875384043 5 0.5815096611955630 0.5250491653932977 6 0.8061111752639282 0.3618821131546534 7 0.9503120678323927 0.1167010207835005 Order = 9 ALPHA = 0.500000 BETA = 2.500000 0 -0.8653022034444986 0.0014331423151996 1 -0.6759411199023827 0.0169372413862485 2 -0.4333019473215542 0.0761680132420717 3 -0.1566993909441265 0.2010801278076634 4 0.1315346053479575 0.3633349438080311 5 0.4080752758535235 0.4761564190243817 6 0.6505301580495421 0.4521871712839320 7 0.8392624591522134 0.2875734496942861 8 0.9589850203521829 0.0886248999318051 Order = 10 ALPHA = 0.500000 BETA = 2.500000 0 -0.8864732152715590 0.0007951842411665 1 -0.7253515462755796 0.0097270000118212 2 -0.5159038845260623 0.0460145772880254 3 -0.2721183637403178 0.1303012691755842 4 -0.0105450517531137 0.2590099865557985 5 0.2510118405448966 0.3871376587183347 6 0.4947368451601082 0.4448595186308595 7 0.7040250190649781 0.3857617289382685 8 0.8646161230141249 0.2310686902877482 9 0.9655674511738290 0.0688197946460142 Order = 1 ALPHA = 1.000000 BETA = 0.500000 0 -0.1428571428571428 1.5084944665313014 Order = 2 ALPHA = 1.000000 BETA = 0.500000 0 -0.5603616177221111 0.8377097182188190 1 0.3785434359039293 0.6707847483124826 Order = 3 ALPHA = 1.000000 BETA = 0.500000 0 -0.7362914149288884 0.4486477634160737 1 -0.0828374856848099 0.7711842727220676 2 0.6191289006136984 0.2886624303931601 Order = 4 ALPHA = 1.000000 BETA = 0.500000 0 -0.8250542504682752 0.2590735729869860 1 -0.3610180502309864 0.6101454555318188 2 0.2308286409011868 0.5011891464890378 3 0.7447173440086011 0.1380862915234592 Order = 5 ALPHA = 1.000000 BETA = 0.500000 0 -0.8757120529564132 0.1609007645163642 1 -0.5335962093870673 0.4466360854739414 2 -0.0582907697984681 0.5176668424406611 3 0.4326084003843104 0.3103351805927452 4 0.8175993274098121 0.0729555935075893 Order = 6 ALPHA = 1.000000 BETA = 0.500000 0 -0.9072453478183379 0.1060642452077330 1 -0.6461272538533950 0.3245589588523248 2 -0.2649141017150741 0.4544246037858501 3 0.1659246338882320 0.3864855437976816 4 0.5667404500118451 0.1951823756778569 5 0.8633993972645074 0.0417787392098546 Order = 7 ALPHA = 1.000000 BETA = 0.500000 0 -0.9281704237863851 0.0733394189023952 1 -0.7229723820004327 0.2389626799471577 2 -0.4138036987843730 0.3759820326953336 3 -0.0449578958653259 0.3894994287201925 4 0.3307207485144988 0.2785906683424280 5 0.6594047339183108 0.1266052232513153 6 0.8939724663908039 0.0255150146724793 Order = 8 ALPHA = 1.000000 BETA = 0.500000 0 -0.9427521789659081 0.0527087671879105 1 -0.7775490329551161 0.1792709060668855 2 -0.5232643292074005 0.3048404655795938 3 -0.2089491097631811 0.3577748934099902 4 0.1294870729403755 0.3123493200327135 5 0.4533780733947834 0.2002825508076697 6 0.7257160980084200 0.0848554733563242 7 0.9153619779765982 0.0164120900902137 Order = 9 ALPHA = 1.000000 BETA = 0.500000 0 -0.9533131723221203 0.0390998276552453 1 -0.8176042531316312 0.1371247634867023 2 -0.6055223420849569 0.2462380941383862 3 -0.3368351070820349 0.3145887170547920 4 -0.0365863464439503 0.3121710958691360 5 0.2672381326388559 0.2442199669465213 6 0.5463185334791457 0.1453933047463929 7 0.7746386880679894 0.0586443046921292 8 0.9308966361094718 0.0110143919419965 Order = 10 ALPHA = 1.000000 BETA = 0.500000 0 -0.9612042779760442 0.0297778546390378 1 -0.8478228797503713 0.1068280415109265 2 -0.6686402341966643 0.1996303237647662 3 -0.4375388507874068 0.2709981593343721 4 -0.1724237674317270 0.2937514863341334 5 0.1061646285278261 0.2614695484772919 6 0.3766418843493626 0.1896495013540224 7 0.6180513952640334 0.1070695285744137 8 0.8116864125172604 0.0416630238085870 9 0.9425275499488474 0.0076569987337497 Order = 1 ALPHA = 1.000000 BETA = 1.000000 0 0.0000000000000000 1.3333333333333333 Order = 2 ALPHA = 1.000000 BETA = 1.000000 0 -0.4472135954999579 0.6666666666666665 1 0.4472135954999580 0.6666666666666667 Order = 3 ALPHA = 1.000000 BETA = 1.000000 0 -0.6546536707079772 0.3111111111111113 1 0.0000000000000000 0.7111111111111110 2 0.6546536707079772 0.3111111111111113 Order = 4 ALPHA = 1.000000 BETA = 1.000000 0 -0.7650553239294646 0.1569499125956942 1 -0.2852315164806451 0.5097167540709727 2 0.2852315164806451 0.5097167540709727 3 0.7650553239294647 0.1569499125956939 Order = 5 ALPHA = 1.000000 BETA = 1.000000 0 -0.8302238962785669 0.0860176821228073 1 -0.4688487934707142 0.3368394607343354 2 -0.0000000000000000 0.4876190476190476 3 0.4688487934707142 0.3368394607343354 4 0.8302238962785670 0.0860176821228076 Order = 6 ALPHA = 1.000000 BETA = 1.000000 0 -0.8717401485096066 0.0505835770160819 1 -0.5917001814331423 0.2216925320225178 2 -0.2092992179024789 0.3943905576280671 3 0.2092992179024789 0.3943905576280671 4 0.5917001814331423 0.2216925320225178 5 0.8717401485096067 0.0505835770160817 Order = 7 ALPHA = 1.000000 BETA = 1.000000 0 -0.8997579954114602 0.0315162001504988 1 -0.6771862795107377 0.1486404045834101 2 -0.3631174638261782 0.3007504247445493 3 0.0000000000000000 0.3715192743764172 4 0.3631174638261782 0.3007504247445493 5 0.6771862795107377 0.1486404045834101 6 0.8997579954114602 0.0315162001504988 Order = 8 ALPHA = 1.000000 BETA = 1.000000 0 -0.9195339081664589 0.0205900956491219 1 -0.7387738651055050 0.1021477023603581 2 -0.4779249498104445 0.2253365549698578 3 -0.1652789576663870 0.3185923136873282 4 0.1652789576663870 0.3185923136873282 5 0.4779249498104445 0.2253365549698580 6 0.7387738651055050 0.1021477023603581 7 0.9195339081664589 0.0205900956491219 Order = 9 ALPHA = 1.000000 BETA = 1.000000 0 -0.9340014304080592 0.0139910561244267 1 -0.7844834736631444 0.0719828561756607 2 -0.5652353269962050 0.1687989736144601 3 -0.2957581355869394 0.2617849830242735 4 -0.0000000000000000 0.3002175954556907 5 0.2957581355869394 0.2617849830242735 6 0.5652353269962050 0.1687989736144601 7 0.7844834736631444 0.0719828561756607 8 0.9340014304080592 0.0139910561244267 Order = 10 ALPHA = 1.000000 BETA = 1.000000 0 -0.9448992722228823 0.0098254048057683 1 -0.8192793216440066 0.0519391437742078 2 -0.6328761530318606 0.1273919483477328 3 -0.3995309409653490 0.2111657418760753 4 -0.1365529328549276 0.2663444278628824 5 0.1365529328549276 0.2663444278628824 6 0.3995309409653490 0.2111657418760753 7 0.6328761530318606 0.1273919483477326 8 0.8192793216440066 0.0519391437742078 9 0.9448992722228822 0.0098254048057681 Order = 1 ALPHA = 1.000000 BETA = 2.500000 0 0.2727272727272727 1.4366613966964776 Order = 2 ALPHA = 1.000000 BETA = 2.500000 0 -0.1843075691322092 0.5823920991551215 1 0.5843075691322092 0.8542692975413559 Order = 3 ALPHA = 1.000000 BETA = 2.500000 0 -0.4423027360564286 0.2004355575312488 1 0.1860296262211931 0.7756953091585612 2 0.7299573203615513 0.4605305300066671 Order = 4 ALPHA = 1.000000 BETA = 2.500000 0 -0.5972505982938984 0.0735380497612851 1 -0.1011259685235210 0.4428338893708901 2 0.4098198869416555 0.6621803747350362 3 0.8102958103105465 0.2581090828292666 Order = 5 ALPHA = 1.000000 BETA = 2.500000 0 -0.6964392762006608 0.0296301520114946 1 -0.3026155820854005 0.2285362779908424 2 0.1412598214162333 0.5280679973185711 3 0.5539710608669561 0.4975219906041118 4 0.8593795315584275 0.1529049787714576 Order = 6 ALPHA = 1.000000 BETA = 2.500000 0 -0.7633972293043557 0.0130439061802296 1 -0.4461529248760192 0.1174255136957451 2 -0.0686784265705635 0.3490370430142563 3 0.3155805295665462 0.5014050892168356 4 0.6517131087948295 0.3603484489584897 5 0.8915801036798852 0.0954013956309209 Order = 7 ALPHA = 1.000000 BETA = 2.500000 0 -0.8105874821854350 0.0062027903657017 1 -0.5508759997785407 0.0619382418889595 2 -0.2306894365569858 0.2164618775946895 3 0.1138876346036921 0.3988213674613817 4 0.4436003431475740 0.4310114591490307 5 0.7208146570340379 0.2599688822120858 6 0.9138502837356576 0.0622567780246282 Order = 8 ALPHA = 1.000000 BETA = 2.500000 0 -0.8450352779186007 0.0031506505662605 1 -0.6291538133028907 0.0338481948002515 2 -0.3563555020358466 0.1322157770527708 3 -0.0516690747442928 0.2867379491637254 4 0.2565958341049683 0.3952631184069368 5 0.5397363757841802 0.3540223921913583 6 0.7713710662668609 0.1892132923890615 7 0.9298950072302367 0.0422100221261126 Order = 9 ALPHA = 1.000000 BETA = 2.500000 0 -0.8709215211401559 0.0016929758046626 1 -0.6889839677544539 0.0191902962920152 2 -0.4549257546478147 0.0812176110475679 3 -0.1865538397099543 0.1974051443898491 4 0.0954122417770832 0.3197898022964820 5 0.3691513199896207 0.3631243013752984 6 0.6134647393951399 0.2850633375254082 7 0.8094259398841321 0.1396134790482662 8 0.9418378189505888 0.0295644489169276 Order = 10 ALPHA = 1.000000 BETA = 2.500000 0 -0.8908519983624362 0.0009546834154140 1 -0.7356332759809727 0.0112695230304568 2 -0.5332475568814602 0.0506378825686693 3 -0.2966592827516151 0.1337845805120266 4 -0.0412830087188659 0.2429565425267210 5 0.2161975251464401 0.3233164884771634 6 0.4589489303958385 0.3198849047578979 7 0.6710962948352331 0.2279739859839844 8 0.8387626487347722 0.1045909447918312 9 0.9509675959234917 0.0212918606323119 Order = 1 ALPHA = 2.500000 BETA = 0.500000 0 -0.4000000000000000 1.9634954084936209 Order = 2 ALPHA = 2.500000 BETA = 0.500000 0 -0.6769446839322616 1.2685345647240849 1 0.1055161125036901 0.6949608437695367 Order = 3 ALPHA = 2.500000 BETA = 0.500000 0 -0.7968729949845809 0.7662999913501441 1 -0.2647662774451411 0.9816213720815016 2 0.3949726057630553 0.2155740450619749 Order = 4 ALPHA = 2.500000 BETA = 0.500000 0 -0.8602174966199764 0.4820034265555970 1 -0.4782610448380588 0.9082548004475427 2 0.0438129709574188 0.4997101315556028 3 0.5673928432278891 0.0735270499348779 Order = 5 ALPHA = 2.500000 BETA = 0.500000 0 -0.8978486665826094 0.3185512380674296 1 -0.6116098169968444 0.7438254266897734 2 -0.1979083288674991 0.6351035815608072 3 0.2615276527389886 0.2379108911216858 4 0.6766083904771950 0.0281042710539255 Order = 6 ALPHA = 2.500000 BETA = 0.500000 0 -0.9220559149817765 0.2200343192676289 1 -0.7000843065535390 0.5868443910765002 2 -0.3678575869061302 0.6464461334770021 3 0.0240992331493380 0.3830143850747027 4 0.4162703993964171 0.1152628351592500 5 0.7496281758956908 0.0118933444385382 Order = 7 ALPHA = 2.500000 BETA = 0.500000 0 -0.9385563889913962 0.1577614936540615 1 -0.7616344330284673 0.4597291058720926 2 -0.4905660544916727 0.5963414735975124 3 -0.1580287890648371 0.4638578133115204 4 0.1959098000069028 0.2221608406878788 5 0.5286792390323162 0.0581588649983059 6 0.8006672147724484 0.0054858163722489 Order = 8 ALPHA = 2.500000 BETA = 0.500000 0 -0.9503120678323927 0.1167010207835005 1 -0.8061111752639282 0.3618821131546534 2 -0.5815096611955630 0.5250491653932977 3 -0.2984865079144489 0.4881539875384043 4 0.0152678951698322 0.3092909316152765 5 0.3290736457267179 0.1289926195321416 6 0.6123079224732050 0.0307056736312334 7 0.8376646856786829 0.0027198968451134 Order = 9 ALPHA = 2.500000 BETA = 0.500000 0 -0.9589850203521829 0.0886248999318064 1 -0.8392624591522134 0.2875734496942861 2 -0.6505301580495421 0.4521871712839320 3 -0.4080752758535235 0.4761564190243817 4 -0.1315346053479575 0.3633349438080311 5 0.1566993909441265 0.2010801278076634 6 0.4333019473215542 0.0761680132420717 7 0.6759411199023827 0.0169372413862485 8 0.8653022034444986 0.0014331423151996 Order = 10 ALPHA = 2.500000 BETA = 0.500000 0 -0.9655674511738290 0.0688197946460142 1 -0.8646161230141249 0.2310686902877482 2 -0.7040250190649781 0.3857617289382685 3 -0.4947368451601082 0.4448595186308592 4 -0.2510118405448966 0.3871376587183347 5 0.0105450517531137 0.2590099865557985 6 0.2721183637403178 0.1303012691755842 7 0.5159038845260623 0.0460145772880254 8 0.7253515462755796 0.0097270000118212 9 0.8864732152715591 0.0007951842411665 Order = 1 ALPHA = 2.500000 BETA = 1.000000 0 -0.2727272727272727 1.4366613966964776 Order = 2 ALPHA = 2.500000 BETA = 1.000000 0 -0.5843075691322092 0.8542692975413559 1 0.1843075691322092 0.5823920991551215 Order = 3 ALPHA = 2.500000 BETA = 1.000000 0 -0.7299573203615513 0.4605305300066671 1 -0.1860296262211931 0.7756953091585612 2 0.4423027360564286 0.2004355575312490 Order = 4 ALPHA = 2.500000 BETA = 1.000000 0 -0.8102958103105465 0.2581090828292662 1 -0.4098198869416555 0.6621803747350362 2 0.1011259685235210 0.4428338893708898 3 0.5972505982938984 0.0735380497612851 Order = 5 ALPHA = 2.500000 BETA = 1.000000 0 -0.8593795315584275 0.1529049787714576 1 -0.5539710608669561 0.4975219906041122 2 -0.1412598214162333 0.5280679973185711 3 0.3026155820854005 0.2285362779908424 4 0.6964392762006608 0.0296301520114946 Order = 6 ALPHA = 2.500000 BETA = 1.000000 0 -0.8915801036798852 0.0954013956309209 1 -0.6517131087948296 0.3603484489584904 2 -0.3155805295665462 0.5014050892168356 3 0.0686784265705634 0.3490370430142563 4 0.4461529248760192 0.1174255136957451 5 0.7633972293043557 0.0130439061802296 Order = 7 ALPHA = 2.500000 BETA = 1.000000 0 -0.9138502837356576 0.0622567780246282 1 -0.7208146570340379 0.2599688822120858 2 -0.4436003431475740 0.4310114591490307 3 -0.1138876346036921 0.3988213674613817 4 0.2306894365569858 0.2164618775946895 5 0.5508759997785407 0.0619382418889595 6 0.8105874821854350 0.0062027903657017 Order = 8 ALPHA = 2.500000 BETA = 1.000000 0 -0.9298950072302367 0.0422100221261126 1 -0.7713710662668609 0.1892132923890619 2 -0.5397363757841802 0.3540223921913583 3 -0.2565958341049683 0.3952631184069370 4 0.0516690747442927 0.2867379491637254 5 0.3563555020358466 0.1322157770527708 6 0.6291538133028907 0.0338481948002515 7 0.8450352779186007 0.0031506505662605 Order = 9 ALPHA = 2.500000 BETA = 1.000000 0 -0.9418378189505888 0.0295644489169276 1 -0.8094259398841321 0.1396134790482662 2 -0.6134647393951399 0.2850633375254075 3 -0.3691513199896207 0.3631243013752985 4 -0.0954122417770831 0.3197898022964820 5 0.1865538397099543 0.1974051443898491 6 0.4549257546478147 0.0812176110475680 7 0.6889839677544539 0.0191902962920152 8 0.8709215211401559 0.0016929758046627 Order = 10 ALPHA = 2.500000 BETA = 1.000000 0 -0.9509675959234917 0.0212918606323124 1 -0.8387626487347722 0.1045909447918318 2 -0.6710962948352331 0.2279739859839844 3 -0.4589489303958385 0.3198849047578984 4 -0.2161975251464401 0.3233164884771634 5 0.0412830087188659 0.2429565425267210 6 0.2966592827516151 0.1337845805120266 7 0.5332475568814602 0.0506378825686693 8 0.7356332759809727 0.0112695230304568 9 0.8908519983624362 0.0009546834154140 Order = 1 ALPHA = 2.500000 BETA = 2.500000 0 0.0000000000000000 0.9817477042468103 Order = 2 ALPHA = 2.500000 BETA = 2.500000 0 -0.3535533905932737 0.4908738521234051 1 0.3535533905932737 0.4908738521234051 Order = 3 ALPHA = 2.500000 BETA = 2.500000 0 -0.5477225575051661 0.2045307717180856 1 0.0000000000000000 0.5726861608106394 2 0.5477225575051661 0.2045307717180856 Order = 4 ALPHA = 2.500000 BETA = 2.500000 0 -0.6660699417556469 0.0870080715314833 1 -0.2373833033084449 0.4038657805919217 2 0.2373833033084449 0.4038657805919217 3 0.6660699417556469 0.0870080715314835 Order = 5 ALPHA = 2.500000 BETA = 2.500000 0 -0.7434770045212192 0.0392893891317672 1 -0.4019050360892101 0.2454174450998077 2 -0.0000000000000000 0.4123340357836603 3 0.4019050360892101 0.2454174450998076 4 0.7434770045212192 0.0392893891317672 Order = 6 ALPHA = 2.500000 BETA = 2.500000 0 -0.7968373566406807 0.0189116433666931 1 -0.5196148456949133 0.1431407318203965 2 -0.1804179569647766 0.3288214769363154 3 0.1804179569647766 0.3288214769363154 4 0.5196148456949133 0.1431407318203969 5 0.7968373566406807 0.0189116433666931 Order = 7 ALPHA = 2.500000 BETA = 2.500000 0 -0.8351564541339112 0.0096585250210008 1 -0.6063141605450342 0.0834446705489847 2 -0.3186902924591673 0.2357822853526957 3 -0.0000000000000000 0.3239767424014473 4 0.3186902924591672 0.2357822853526958 5 0.6063141605450342 0.0834446705489847 6 0.8351564541339112 0.0096585250210010 Order = 8 ALPHA = 2.500000 BETA = 2.500000 0 -0.8635912403827584 0.0051997074221325 1 -0.6718232507247468 0.0494181881565223 2 -0.4260961495798479 0.1615070338519726 3 -0.1459649294626305 0.2747489226927776 4 0.1459649294626305 0.2747489226927777 5 0.4260961495798479 0.1615070338519726 6 0.6718232507247468 0.0494181881565223 7 0.8635912403827584 0.0051997074221325 Order = 9 ALPHA = 2.500000 BETA = 2.500000 0 -0.8852662998295736 0.0029319009849924 1 -0.7224309555097356 0.0299209571936803 2 -0.5107538326640052 0.1088013928543096 3 -0.2643695362367075 0.2155149137501430 4 0.0000000000000000 0.2674093746805597 5 0.2643695362367075 0.2155149137501430 6 0.5107538326640052 0.1088013928543096 7 0.7224309555097356 0.0299209571936803 8 0.8852662998295736 0.0029319009849924 Order = 10 ALPHA = 2.500000 BETA = 2.500000 0 -0.9021636877146698 0.0017216382288932 1 -0.7622868838395971 0.0185564370059903 2 -0.5784393987065429 0.0731613283971410 3 -0.3610608688185057 0.1622660968500981 4 -0.1227285554678421 0.2351683516412825 5 0.1227285554678421 0.2351683516412824 6 0.3610608688185057 0.1622660968500981 7 0.5784393987065429 0.0731613283971409 8 0.7622868838395971 0.0185564370059903 9 0.9021636877146698 0.0017216382288932 TEST19 LAGUERRE_COMPUTE computes a Gauss-Laguerre rule which is appropriate for integrands of the form Integral ( 0 <= x < +oo ) f(x) exp(-x) dx. LAGUERRE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 1.000000e+00 1.000000e+00 0.000000e+00 1 1 1.000000e+00 1.000000e+00 0.000000e+00 1 2 1.000000e+00 2.000000e+00 1.000000e+00 1 3 1.000000e+00 6.000000e+00 5.000000e+00 1 4 1.000000e+00 2.400000e+01 2.300000e+01 2 0 1.000000e+00 1.000000e+00 0.000000e+00 2 1 1.000000e+00 1.000000e+00 2.220446e-16 2 2 2.000000e+00 2.000000e+00 4.440892e-16 2 3 6.000000e+00 6.000000e+00 8.881784e-16 2 4 2.000000e+01 2.400000e+01 4.000000e+00 2 5 6.800000e+01 1.200000e+02 5.200000e+01 2 6 2.320000e+02 7.200000e+02 4.880000e+02 3 0 1.000000e+00 1.000000e+00 6.661338e-16 3 1 1.000000e+00 1.000000e+00 4.440892e-16 3 2 2.000000e+00 2.000000e+00 0.000000e+00 3 3 6.000000e+00 6.000000e+00 0.000000e+00 3 4 2.400000e+01 2.400000e+01 3.552714e-15 3 5 1.200000e+02 1.200000e+02 2.842171e-14 3 6 6.840000e+02 7.200000e+02 3.600000e+01 3 7 4.140000e+03 5.040000e+03 9.000000e+02 3 8 2.566800e+04 4.032000e+04 1.465200e+04 4 0 1.000000e+00 1.000000e+00 1.110223e-15 4 1 1.000000e+00 1.000000e+00 4.440892e-16 4 2 2.000000e+00 2.000000e+00 0.000000e+00 4 3 6.000000e+00 6.000000e+00 0.000000e+00 4 4 2.400000e+01 2.400000e+01 0.000000e+00 4 5 1.200000e+02 1.200000e+02 1.421085e-14 4 6 7.200000e+02 7.200000e+02 1.136868e-13 4 7 5.040000e+03 5.040000e+03 9.094947e-13 4 8 3.974400e+04 4.032000e+04 5.760000e+02 4 9 3.392640e+05 3.628800e+05 2.361600e+04 4 10 3.033216e+06 3.628800e+06 5.955840e+05 5 0 1.000000e+00 1.000000e+00 7.771561e-16 5 1 1.000000e+00 1.000000e+00 2.220446e-16 5 2 2.000000e+00 2.000000e+00 6.661338e-16 5 3 6.000000e+00 6.000000e+00 8.881784e-16 5 4 2.400000e+01 2.400000e+01 3.552714e-15 5 5 1.200000e+02 1.200000e+02 1.421085e-14 5 6 7.200000e+02 7.200000e+02 0.000000e+00 5 7 5.040000e+03 5.040000e+03 9.094947e-13 5 8 4.032000e+04 4.032000e+04 0.000000e+00 5 9 3.628800e+05 3.628800e+05 1.164153e-10 5 10 3.614400e+06 3.628800e+06 1.440000e+04 5 11 3.903840e+07 3.991680e+07 8.784000e+05 5 12 4.472208e+08 4.790016e+08 3.178080e+07 6 0 1.000000e+00 1.000000e+00 2.442491e-15 6 1 1.000000e+00 1.000000e+00 1.332268e-15 6 2 2.000000e+00 2.000000e+00 1.332268e-15 6 3 6.000000e+00 6.000000e+00 2.664535e-15 6 4 2.400000e+01 2.400000e+01 1.421085e-14 6 5 1.200000e+02 1.200000e+02 8.526513e-14 6 6 7.200000e+02 7.200000e+02 5.684342e-13 6 7 5.040000e+03 5.040000e+03 3.637979e-12 6 8 4.032000e+04 4.032000e+04 2.910383e-11 6 9 3.628800e+05 3.628800e+05 2.910383e-10 6 10 3.628800e+06 3.628800e+06 2.793968e-09 6 11 3.991680e+07 3.991680e+07 3.725290e-08 6 12 4.784832e+08 4.790016e+08 5.184000e+05 6 13 6.182957e+09 6.227021e+09 4.406400e+07 6 14 8.501242e+10 8.717829e+10 2.165875e+09 7 0 1.000000e+00 1.000000e+00 3.552714e-15 7 1 1.000000e+00 1.000000e+00 8.881784e-16 7 2 2.000000e+00 2.000000e+00 0.000000e+00 7 3 6.000000e+00 6.000000e+00 8.881784e-16 7 4 2.400000e+01 2.400000e+01 3.552714e-15 7 5 1.200000e+02 1.200000e+02 0.000000e+00 7 6 7.200000e+02 7.200000e+02 1.136868e-13 7 7 5.040000e+03 5.040000e+03 0.000000e+00 7 8 4.032000e+04 4.032000e+04 0.000000e+00 7 9 3.628800e+05 3.628800e+05 1.164153e-10 7 10 3.628800e+06 3.628800e+06 9.313226e-10 7 11 3.991680e+07 3.991680e+07 7.450581e-09 7 12 4.790016e+08 4.790016e+08 1.192093e-07 7 13 6.227021e+09 6.227021e+09 9.536743e-07 7 14 8.715289e+10 8.717829e+10 2.540160e+07 7 15 1.304804e+12 1.307674e+12 2.870381e+09 7 16 2.073870e+13 2.092279e+13 1.840854e+11 8 0 1.000000e+00 1.000000e+00 1.221245e-15 8 1 1.000000e+00 1.000000e+00 2.220446e-16 8 2 2.000000e+00 2.000000e+00 4.440892e-16 8 3 6.000000e+00 6.000000e+00 8.881784e-16 8 4 2.400000e+01 2.400000e+01 7.105427e-15 8 5 1.200000e+02 1.200000e+02 4.263256e-14 8 6 7.200000e+02 7.200000e+02 2.273737e-13 8 7 5.040000e+03 5.040000e+03 1.818989e-12 8 8 4.032000e+04 4.032000e+04 2.182787e-11 8 9 3.628800e+05 3.628800e+05 1.164153e-10 8 10 3.628800e+06 3.628800e+06 9.313226e-10 8 11 3.991680e+07 3.991680e+07 0.000000e+00 8 12 4.790016e+08 4.790016e+08 0.000000e+00 8 13 6.227021e+09 6.227021e+09 9.536743e-07 8 14 8.717829e+10 8.717829e+10 0.000000e+00 8 15 1.307674e+12 1.307674e+12 0.000000e+00 8 16 2.092116e+13 2.092279e+13 1.625702e+09 8 17 3.554517e+14 3.556874e+14 2.357268e+11 8 18 6.383252e+15 6.402374e+15 1.912151e+13 9 0 1.000000e+00 1.000000e+00 3.330669e-16 9 1 1.000000e+00 1.000000e+00 1.665335e-15 9 2 2.000000e+00 2.000000e+00 2.220446e-15 9 3 6.000000e+00 6.000000e+00 4.440892e-15 9 4 2.400000e+01 2.400000e+01 3.552714e-15 9 5 1.200000e+02 1.200000e+02 0.000000e+00 9 6 7.200000e+02 7.200000e+02 0.000000e+00 9 7 5.040000e+03 5.040000e+03 9.094947e-13 9 8 4.032000e+04 4.032000e+04 7.275958e-12 9 9 3.628800e+05 3.628800e+05 1.164153e-10 9 10 3.628800e+06 3.628800e+06 9.313226e-10 9 11 3.991680e+07 3.991680e+07 1.490116e-08 9 12 4.790016e+08 4.790016e+08 1.788139e-07 9 13 6.227021e+09 6.227021e+09 1.907349e-06 9 14 8.717829e+10 8.717829e+10 1.525879e-05 9 15 1.307674e+12 1.307674e+12 2.441406e-04 9 16 2.092279e+13 2.092279e+13 7.812500e-03 9 17 3.556874e+14 3.556874e+14 1.250000e-01 9 18 6.402242e+15 6.402374e+15 1.316819e+11 9 19 1.216213e+17 1.216451e+17 2.383442e+13 9 20 2.430516e+18 2.432902e+18 2.385944e+15 10 0 1.000000e+00 1.000000e+00 3.330669e-15 10 1 1.000000e+00 1.000000e+00 1.443290e-15 10 2 2.000000e+00 2.000000e+00 1.776357e-15 10 3 6.000000e+00 6.000000e+00 3.552714e-15 10 4 2.400000e+01 2.400000e+01 1.421085e-14 10 5 1.200000e+02 1.200000e+02 2.842171e-14 10 6 7.200000e+02 7.200000e+02 1.136868e-13 10 7 5.040000e+03 5.040000e+03 0.000000e+00 10 8 4.032000e+04 4.032000e+04 7.275958e-12 10 9 3.628800e+05 3.628800e+05 5.820766e-11 10 10 3.628800e+06 3.628800e+06 1.396984e-09 10 11 3.991680e+07 3.991680e+07 1.490116e-08 10 12 4.790016e+08 4.790016e+08 1.788139e-07 10 13 6.227021e+09 6.227021e+09 1.907349e-06 10 14 8.717829e+10 8.717829e+10 4.577637e-05 10 15 1.307674e+12 1.307674e+12 4.882812e-04 10 16 2.092279e+13 2.092279e+13 7.812500e-03 10 17 3.556874e+14 3.556874e+14 6.250000e-02 10 18 6.402374e+15 6.402374e+15 1.000000e+00 10 19 1.216451e+17 1.216451e+17 0.000000e+00 10 20 2.432889e+18 2.432902e+18 1.316819e+13 10 21 5.108803e+19 5.109094e+19 2.910170e+15 10 22 1.123648e+21 1.124001e+21 3.524071e+17 TEST20 LAGUERRE_COMPUTE computes a generalized Gauss-Laguerre rule which is appropriate for integrands of the form Integral ( 0 <= x < +oo ) f(x) exp(-x) dx. Order = 1 0 1.0000000000000000 1.0000000000000000 Order = 2 0 0.5857864376269050 0.8535533905932738 1 3.4142135623730949 0.1464466094067263 Order = 3 0 0.4157745567834791 0.7110930099291736 1 2.2942803602790418 0.2785177335692409 2 6.2899450829374794 0.0103892565015861 Order = 4 0 0.3225476896193922 0.6031541043416347 1 1.7457611011583467 0.3574186924377997 2 4.5366202969211278 0.0388879085150054 3 9.3950709123011329 0.0005392947055613 Order = 5 0 0.2635603197181409 0.5217556105828079 1 1.4134030591065168 0.3986668110831759 2 3.5964257710407219 0.0759424496817076 3 7.0858100058588374 0.0036117586799220 4 12.6408008442757822 0.0000233699723858 Order = 6 0 0.2228466041792606 0.4589646739499650 1 1.1889321016726229 0.4170008307721219 2 2.9927363260593141 0.1133733820740450 3 5.7751435691045110 0.0103991974531491 4 9.8374674183825892 0.0002610172028149 5 15.9828739806017026 0.0000008985479064 Order = 7 0 0.1930436765603623 0.4093189517012772 1 1.0266648953391919 0.4218312778617200 2 2.5678767449507465 0.1471263486575052 3 4.9003530845264844 0.0206335144687169 4 8.1821534445628608 0.0010740101432807 5 12.7341802917978129 0.0000158654643486 6 19.3957278622625395 0.0000000317031548 Order = 8 0 0.1702796323051010 0.3691885893416355 1 0.9037017767993799 0.4187867808143441 2 2.2510866298661312 0.1757949866371716 3 4.2667001702876588 0.0333434922612157 4 7.0459054023934655 0.0027945362352257 5 10.7585160101809958 0.0000907650877336 6 15.7406786412780040 0.0000008485746716 7 22.8631317368892653 0.0000000010480012 Order = 9 0 0.1523222277318083 0.3361264217979637 1 0.8072200227422559 0.4112139804239832 2 2.0051351556193473 0.1992875253708851 3 3.7834739733312328 0.0474605627656516 4 6.2049567778766130 0.0055996266107946 5 9.3729852516875756 0.0003052497670932 6 13.4662369110920928 0.0000065921230261 7 18.8335977889916961 0.0000000411076933 8 26.3740718909273753 0.0000000000329087 Order = 10 0 0.1377934705404924 0.3084411157650176 1 0.7294545495031703 0.4011199291552729 2 1.8083429017403159 0.2180682876118093 3 3.4014336978548996 0.0620874560986777 4 5.5524961400638038 0.0095015169751811 5 8.3301527467644974 0.0007530083885875 6 11.8437858379000662 0.0000282592334960 7 16.2792578313781036 0.0000004249313985 8 21.9965858119807613 0.0000000018395648 9 29.9206970122738909 0.0000000000009912 TEST21 LEGENDRE_COMPUTE computes a Gauss-Legendre rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. LEGENDRE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 2.000000 2.000000 0.000000e+00 1 1 0.000000 0.000000 0.000000e+00 1 2 0.000000 0.666667 6.666667e-01 1 3 0.000000 0.000000 0.000000e+00 1 4 0.000000 0.400000 4.000000e-01 2 0 2.000000 2.000000 8.881784e-16 2 1 0.000000 0.000000 0.000000e+00 2 2 0.666667 0.666667 1.110223e-16 2 3 0.000000 0.000000 0.000000e+00 2 4 0.222222 0.400000 1.777778e-01 2 5 0.000000 0.000000 0.000000e+00 2 6 0.074074 0.285714 2.116402e-01 3 0 2.000000 2.000000 4.440892e-16 3 1 0.000000 0.000000 0.000000e+00 3 2 0.666667 0.666667 2.220446e-16 3 3 0.000000 0.000000 0.000000e+00 3 4 0.400000 0.400000 5.551115e-17 3 5 0.000000 0.000000 0.000000e+00 3 6 0.240000 0.285714 4.571429e-02 3 7 0.000000 0.000000 0.000000e+00 3 8 0.144000 0.222222 7.822222e-02 4 0 2.000000 2.000000 0.000000e+00 4 1 0.000000 0.000000 5.551115e-17 4 2 0.666667 0.666667 0.000000e+00 4 3 0.000000 0.000000 0.000000e+00 4 4 0.400000 0.400000 0.000000e+00 4 5 0.000000 0.000000 0.000000e+00 4 6 0.285714 0.285714 0.000000e+00 4 7 0.000000 0.000000 0.000000e+00 4 8 0.210612 0.222222 1.160998e-02 4 9 0.000000 0.000000 0.000000e+00 4 10 0.156035 0.181818 2.578320e-02 5 0 2.000000 2.000000 0.000000e+00 5 1 0.000000 0.000000 0.000000e+00 5 2 0.666667 0.666667 1.110223e-16 5 3 0.000000 0.000000 0.000000e+00 5 4 0.400000 0.400000 0.000000e+00 5 5 0.000000 0.000000 0.000000e+00 5 6 0.285714 0.285714 0.000000e+00 5 7 0.000000 0.000000 0.000000e+00 5 8 0.222222 0.222222 2.775558e-17 5 9 0.000000 0.000000 0.000000e+00 5 10 0.178886 0.181818 2.931812e-03 5 11 0.000000 0.000000 0.000000e+00 5 12 0.145853 0.153846 7.993574e-03 6 0 2.000000 2.000000 0.000000e+00 6 1 0.000000 0.000000 8.326673e-17 6 2 0.666667 0.666667 2.220446e-16 6 3 -0.000000 0.000000 2.775558e-17 6 4 0.400000 0.400000 1.665335e-16 6 5 0.000000 0.000000 0.000000e+00 6 6 0.285714 0.285714 1.110223e-16 6 7 0.000000 0.000000 0.000000e+00 6 8 0.222222 0.222222 5.551115e-17 6 9 0.000000 0.000000 0.000000e+00 6 10 0.181818 0.181818 8.326673e-17 6 11 0.000000 0.000000 0.000000e+00 6 12 0.153108 0.153846 7.380787e-04 6 13 0.000000 0.000000 0.000000e+00 6 14 0.130949 0.133333 2.384218e-03 7 0 2.000000 2.000000 4.440892e-16 7 1 0.000000 0.000000 1.387779e-17 7 2 0.666667 0.666667 4.440892e-16 7 3 -0.000000 0.000000 4.163336e-17 7 4 0.400000 0.400000 3.330669e-16 7 5 0.000000 0.000000 0.000000e+00 7 6 0.285714 0.285714 3.330669e-16 7 7 0.000000 0.000000 1.387779e-17 7 8 0.222222 0.222222 2.775558e-16 7 9 0.000000 0.000000 1.387779e-17 7 10 0.181818 0.181818 1.942890e-16 7 11 0.000000 0.000000 0.000000e+00 7 12 0.153846 0.153846 1.665335e-16 7 13 0.000000 0.000000 0.000000e+00 7 14 0.133148 0.133333 1.854659e-04 7 15 0.000000 0.000000 0.000000e+00 7 16 0.116955 0.117647 6.923502e-04 8 0 2.000000 2.000000 0.000000e+00 8 1 -0.000000 0.000000 1.387779e-17 8 2 0.666667 0.666667 2.220446e-16 8 3 0.000000 0.000000 0.000000e+00 8 4 0.400000 0.400000 5.551115e-17 8 5 0.000000 0.000000 0.000000e+00 8 6 0.285714 0.285714 5.551115e-17 8 7 0.000000 0.000000 1.387779e-17 8 8 0.222222 0.222222 5.551115e-17 8 9 0.000000 0.000000 0.000000e+00 8 10 0.181818 0.181818 0.000000e+00 8 11 0.000000 0.000000 0.000000e+00 8 12 0.153846 0.153846 2.775558e-17 8 13 0.000000 0.000000 0.000000e+00 8 14 0.133333 0.133333 2.775558e-17 8 15 0.000000 0.000000 0.000000e+00 8 16 0.117601 0.117647 4.654831e-05 8 17 0.000000 0.000000 0.000000e+00 8 18 0.105066 0.105263 1.971362e-04 9 0 2.000000 2.000000 0.000000e+00 9 1 0.000000 0.000000 1.387779e-17 9 2 0.666667 0.666667 0.000000e+00 9 3 0.000000 0.000000 0.000000e+00 9 4 0.400000 0.400000 5.551115e-17 9 5 0.000000 0.000000 1.387779e-17 9 6 0.285714 0.285714 0.000000e+00 9 7 0.000000 0.000000 0.000000e+00 9 8 0.222222 0.222222 5.551115e-17 9 9 0.000000 0.000000 0.000000e+00 9 10 0.181818 0.181818 2.775558e-17 9 11 0.000000 0.000000 0.000000e+00 9 12 0.153846 0.153846 2.775558e-17 9 13 0.000000 0.000000 0.000000e+00 9 14 0.133333 0.133333 0.000000e+00 9 15 0.000000 0.000000 0.000000e+00 9 16 0.117647 0.117647 1.387779e-17 9 17 0.000000 0.000000 0.000000e+00 9 18 0.105251 0.105263 1.167311e-05 9 19 0.000000 0.000000 0.000000e+00 9 20 0.095183 0.095238 5.529194e-05 10 0 2.000000 2.000000 4.440892e-16 10 1 -0.000000 0.000000 5.551115e-17 10 2 0.666667 0.666667 1.110223e-16 10 3 0.000000 0.000000 6.938894e-18 10 4 0.400000 0.400000 0.000000e+00 10 5 0.000000 0.000000 6.938894e-18 10 6 0.285714 0.285714 1.110223e-16 10 7 -0.000000 0.000000 6.938894e-18 10 8 0.222222 0.222222 1.110223e-16 10 9 -0.000000 0.000000 6.938894e-18 10 10 0.181818 0.181818 1.110223e-16 10 11 0.000000 0.000000 6.938894e-18 10 12 0.153846 0.153846 8.326673e-17 10 13 0.000000 0.000000 0.000000e+00 10 14 0.133333 0.133333 8.326673e-17 10 15 0.000000 0.000000 0.000000e+00 10 16 0.117647 0.117647 6.938894e-17 10 17 0.000000 0.000000 0.000000e+00 10 18 0.105263 0.105263 5.551115e-17 10 19 0.000000 0.000000 0.000000e+00 10 20 0.095235 0.095238 2.925590e-06 10 21 0.000000 0.000000 0.000000e+00 10 22 0.086941 0.086957 1.532420e-05 TEST22 LEGENDRE_COMPUTE computes a Gauss-Legendre rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. Order = 1 0 0.0000000000000000 2.0000000000000000 Order = 2 0 -0.5773502691896258 0.9999999999999996 1 0.5773502691896258 0.9999999999999996 Order = 3 0 -0.7745966692414833 0.5555555555555558 1 0.0000000000000000 0.8888888888888888 2 0.7745966692414833 0.5555555555555558 Order = 4 0 -0.8611363115940526 0.3478548451374539 1 -0.3399810435848563 0.6521451548625461 2 0.3399810435848563 0.6521451548625461 3 0.8611363115940526 0.3478548451374539 Order = 5 0 -0.9061798459386640 0.2369268850561891 1 -0.5384693101056831 0.4786286704993665 2 0.0000000000000000 0.5688888888888889 3 0.5384693101056831 0.4786286704993665 4 0.9061798459386640 0.2369268850561891 Order = 6 0 -0.9324695142031521 0.1713244923791702 1 -0.6612093864662645 0.3607615730481386 2 -0.2386191860831969 0.4679139345726910 3 0.2386191860831969 0.4679139345726910 4 0.6612093864662645 0.3607615730481386 5 0.9324695142031521 0.1713244923791702 Order = 7 0 -0.9491079123427585 0.1294849661688699 1 -0.7415311855993945 0.2797053914892767 2 -0.4058451513773972 0.3818300505051190 3 0.0000000000000000 0.4179591836734694 4 0.4058451513773972 0.3818300505051190 5 0.7415311855993945 0.2797053914892767 6 0.9491079123427585 0.1294849661688699 Order = 8 0 -0.9602898564975362 0.1012285362903764 1 -0.7966664774136267 0.2223810344533744 2 -0.5255324099163290 0.3137066458778873 3 -0.1834346424956498 0.3626837833783618 4 0.1834346424956498 0.3626837833783618 5 0.5255324099163290 0.3137066458778873 6 0.7966664774136267 0.2223810344533744 7 0.9602898564975362 0.1012285362903764 Order = 9 0 -0.9681602395076261 0.0812743883615744 1 -0.8360311073266358 0.1806481606948574 2 -0.6133714327005904 0.2606106964029354 3 -0.3242534234038089 0.3123470770400029 4 0.0000000000000000 0.3302393550012598 5 0.3242534234038089 0.3123470770400029 6 0.6133714327005904 0.2606106964029354 7 0.8360311073266358 0.1806481606948574 8 0.9681602395076261 0.0812743883615744 Order = 10 0 -0.9739065285171717 0.0666713443086880 1 -0.8650633666889845 0.1494513491505806 2 -0.6794095682990244 0.2190863625159821 3 -0.4333953941292472 0.2692667193099965 4 -0.1488743389816312 0.2955242247147530 5 0.1488743389816312 0.2955242247147530 6 0.4333953941292472 0.2692667193099965 7 0.6794095682990244 0.2190863625159821 8 0.8650633666889845 0.1494513491505806 9 0.9739065285171717 0.0666713443086880 TEST23 LEVEL_GROWTH_TO_ORDER uses Level, Growth and Rule to determine the orders of each entry of a vector of 1D rules. Here we examine rule 1. LEVEL:0 1 2 3 4 5 6 7 8 9 10 GROWTH 0 1 5 9 17 17 33 33 33 33 65 65 1 1 2 3 4 5 6 7 8 9 10 11 2 1 3 3 5 5 7 7 9 9 11 11 3 1 3 5 7 9 11 13 15 17 19 21 4 1 3 5 9 9 17 17 17 17 33 33 5 1 5 9 17 17 33 33 33 33 65 65 6 1 3 5 9 17 33 65 129 257 513 1025 TEST23 LEVEL_GROWTH_TO_ORDER uses Level, Growth and Rule to determine the orders of each entry of a vector of 1D rules. Here we examine rule 3. LEVEL:0 1 2 3 4 5 6 7 8 9 10 GROWTH 0 1 3 7 15 15 15 31 31 31 31 31 4 1 3 3 7 7 7 15 15 15 15 15 5 1 3 7 15 15 15 31 31 31 31 31 6 1 3 7 15 31 63 127 255 511 1023 2047 TEST23 LEVEL_GROWTH_TO_ORDER uses Level, Growth and Rule to determine the orders of each entry of a vector of 1D rules. Here we examine rule 4. LEVEL:0 1 2 3 4 5 6 7 8 9 10 GROWTH 0 1 3 5 7 9 11 13 15 17 19 21 1 1 2 3 4 5 6 7 8 9 10 11 2 1 3 3 5 5 7 7 9 9 11 11 3 1 3 5 7 9 11 13 15 17 19 21 4 1 3 3 7 7 7 7 15 15 15 15 5 1 3 7 7 15 15 15 15 31 31 31 6 1 3 7 15 31 63 127 255 511 1023 2047 TEST23 LEVEL_GROWTH_TO_ORDER uses Level, Growth and Rule to determine the orders of each entry of a vector of 1D rules. Here we examine rule 11. LEVEL:0 1 2 3 4 5 6 7 8 9 10 GROWTH 0 1 3 9 9 19 19 19 19 35 35 35 4 1 3 3 9 9 9 9 9 19 19 19 5 1 3 9 9 19 19 19 19 35 35 35 6 1 3 9 19 35 35 35 35 35 35 35 TEST24 LEVEL_TO_ORDER_DEFAULT uses a default rule to determine the order of a rule from its level. RULE/LEVEL 0 1 2 3 4 5 6 7 8 9 10 1 1 3 5 9 17 33 65 129 257 513 1025 2 1 3 7 15 31 63 127 255 511 1023 2047 3 1 3 7 15 31 63 127 255 511 1023 2047 4 1 3 5 7 9 11 13 15 17 19 21 5 1 3 5 7 9 11 13 15 17 19 21 6 1 3 5 7 9 11 13 15 17 19 21 7 1 3 5 7 9 11 13 15 17 19 21 8 1 3 5 7 9 11 13 15 17 19 21 9 1 3 5 7 9 11 13 15 17 19 21 10 1 3 5 7 9 11 13 15 17 19 21 11 1 3 5 9 9 17 17 17 17 33 33 12 1 3 7 7 15 15 15 15 31 31 31 13 1 3 3 7 7 7 15 15 15 15 15 TEST25 LEVEL_TO_ORDER_EXPONENTIAL uses an exponential rule to determine the order of a rule from its level. RULE/LEVEL 0 1 2 3 4 5 6 7 8 9 10 1 1 3 5 9 17 33 65 129 257 513 1025 2 1 3 7 15 31 63 127 255 511 1023 2047 3 1 3 7 15 31 63 127 255 511 1023 2047 4 1 3 7 15 31 63 127 255 511 1023 2047 5 1 3 7 15 31 63 127 255 511 1023 2047 6 1 3 7 15 31 63 127 255 511 1023 2047 7 1 3 7 15 31 63 127 255 511 1023 2047 8 1 3 7 15 31 63 127 255 511 1023 2047 9 1 3 7 15 31 63 127 255 511 1023 2047 10 1 3 7 15 31 63 127 255 511 1023 2047 TEST26 LEVEL_TO_ORDER_EXPONENTIAL_SLOW uses a slow exponential rule to determine the order of a rule from its level. Since it is really only useful for fully nested rules, we only consider rules 11, 12 and 13. RULE/LEVEL 0 1 2 3 4 5 6 7 8 9 10 11 1 3 5 9 9 17 17 17 17 33 33 12 1 3 7 7 15 15 15 15 31 31 31 13 1 3 3 7 7 7 15 15 15 15 15 TEST27 LEVEL_TO_ORDER_LINEAR uses a linear rule to determine the order of a rule from its level. RULE/LEVEL 0 1 2 3 4 5 6 7 8 9 10 1 1 3 5 7 9 11 13 15 17 19 21 2 1 3 5 7 9 11 13 15 17 19 21 3 1 3 5 7 9 11 13 15 17 19 21 4 1 3 5 7 9 11 13 15 17 19 21 5 1 3 5 7 9 11 13 15 17 19 21 6 1 3 5 7 9 11 13 15 17 19 21 7 1 3 5 7 9 11 13 15 17 19 21 8 1 3 5 7 9 11 13 15 17 19 21 9 1 3 5 7 9 11 13 15 17 19 21 10 1 3 5 7 9 11 13 15 17 19 21 TEST28 PATTERSON_LOOKUP computes a Gauss-Patterson rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. LEGENDRE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = (3*ORDER+1)/2 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 2.000000 2.000000 0.000000e+00 1 1 0.000000 0.000000 0.000000e+00 1 2 0.000000 0.666667 6.666667e-01 1 3 0.000000 0.000000 0.000000e+00 1 4 0.000000 0.400000 4.000000e-01 3 0 2.000000 2.000000 0.000000e+00 3 1 0.000000 0.000000 0.000000e+00 3 2 0.666667 0.666667 1.110223e-16 3 3 0.000000 0.000000 0.000000e+00 3 4 0.400000 0.400000 5.551115e-17 3 5 0.000000 0.000000 0.000000e+00 3 6 0.240000 0.285714 4.571429e-02 3 7 0.000000 0.000000 0.000000e+00 3 8 0.144000 0.222222 7.822222e-02 7 0 2.000000 2.000000 0.000000e+00 7 1 -0.000000 0.000000 1.387779e-17 7 2 0.666667 0.666667 1.110223e-16 7 3 -0.000000 0.000000 4.163336e-17 7 4 0.400000 0.400000 0.000000e+00 7 5 0.000000 0.000000 0.000000e+00 7 6 0.285714 0.285714 5.551115e-17 7 7 0.000000 0.000000 1.387779e-17 7 8 0.222222 0.222222 2.775558e-17 7 9 0.000000 0.000000 0.000000e+00 7 10 0.181818 0.181818 2.775558e-17 7 11 0.000000 0.000000 0.000000e+00 7 12 0.154127 0.153846 2.806521e-04 7 13 -0.000000 0.000000 6.938894e-18 7 14 0.134081 0.133333 7.481222e-04 15 0 2.000000 2.000000 0.000000e+00 15 1 0.000000 0.000000 2.428613e-17 15 2 0.666667 0.666667 0.000000e+00 15 3 0.000000 0.000000 3.122502e-17 15 4 0.400000 0.400000 5.551115e-17 15 5 0.000000 0.000000 0.000000e+00 15 6 0.285714 0.285714 0.000000e+00 15 7 -0.000000 0.000000 6.938894e-18 15 8 0.222222 0.222222 2.775558e-17 15 9 0.000000 0.000000 1.734723e-17 15 10 0.181818 0.181818 0.000000e+00 15 11 0.000000 0.000000 3.469447e-18 15 12 0.153846 0.153846 2.775558e-17 15 13 -0.000000 0.000000 1.734723e-17 15 14 0.133333 0.133333 0.000000e+00 15 15 0.000000 0.000000 3.469447e-18 15 16 0.117647 0.117647 1.387779e-17 15 17 -0.000000 0.000000 1.734723e-18 15 18 0.105263 0.105263 0.000000e+00 15 19 -0.000000 0.000000 3.469447e-18 15 20 0.095238 0.095238 0.000000e+00 15 21 -0.000000 0.000000 1.734723e-18 15 22 0.086957 0.086957 0.000000e+00 15 23 0.000000 0.000000 0.000000e+00 15 24 0.080000 0.080000 5.394906e-09 15 25 0.000000 0.000000 0.000000e+00 15 26 0.074074 0.074074 1.752466e-08 31 0 2.000000 2.000000 0.000000e+00 31 1 0.000000 0.000000 5.637851e-18 31 2 0.666667 0.666667 1.110223e-16 31 3 0.000000 0.000000 1.821460e-17 31 4 0.400000 0.400000 5.551115e-17 31 5 0.000000 0.000000 4.336809e-19 31 6 0.285714 0.285714 0.000000e+00 31 7 0.000000 0.000000 8.673617e-19 31 8 0.222222 0.222222 0.000000e+00 31 9 -0.000000 0.000000 6.071532e-18 31 10 0.181818 0.181818 5.551115e-17 31 11 0.000000 0.000000 1.734723e-18 31 12 0.153846 0.153846 0.000000e+00 31 13 -0.000000 0.000000 3.903128e-18 31 14 0.133333 0.133333 0.000000e+00 31 15 -0.000000 0.000000 8.673617e-19 31 16 0.117647 0.117647 1.387779e-17 31 17 -0.000000 0.000000 9.540979e-18 31 18 0.105263 0.105263 1.387779e-17 31 19 -0.000000 0.000000 1.040834e-17 31 20 0.095238 0.095238 2.775558e-17 31 21 -0.000000 0.000000 1.301043e-18 31 22 0.086957 0.086957 0.000000e+00 31 23 0.000000 0.000000 4.336809e-19 31 24 0.080000 0.080000 0.000000e+00 31 25 0.000000 0.000000 8.673617e-19 31 26 0.074074 0.074074 2.775558e-17 31 27 -0.000000 0.000000 2.602085e-18 31 28 0.068966 0.068966 0.000000e+00 31 29 0.000000 0.000000 4.336809e-19 31 30 0.064516 0.064516 1.387779e-17 31 31 0.000000 0.000000 5.204170e-18 31 32 0.060606 0.060606 2.081668e-17 31 33 -0.000000 0.000000 8.673617e-19 31 34 0.057143 0.057143 0.000000e+00 31 35 -0.000000 0.000000 2.168404e-18 31 36 0.054054 0.054054 3.469447e-17 31 37 0.000000 0.000000 4.336809e-19 31 38 0.051282 0.051282 6.938894e-18 31 39 0.000000 0.000000 4.336809e-19 31 40 0.048780 0.048780 6.938894e-18 31 41 0.000000 0.000000 4.336809e-19 31 42 0.046512 0.046512 1.387779e-17 31 43 0.000000 0.000000 3.903128e-18 31 44 0.044444 0.044444 2.081668e-17 31 45 -0.000000 0.000000 8.673617e-19 31 46 0.042553 0.042553 0.000000e+00 31 47 0.000000 0.000000 0.000000e+00 31 48 0.040816 0.040816 0.000000e+00 31 49 -0.000000 0.000000 4.336809e-19 31 50 0.039216 0.039216 6.938894e-18 63 0 2.000000 2.000000 2.220446e-16 63 1 -0.000000 0.000000 9.269929e-18 63 2 0.666667 0.666667 1.110223e-16 63 3 -0.000000 0.000000 9.757820e-19 63 4 0.400000 0.400000 5.551115e-17 63 5 0.000000 0.000000 5.654114e-17 63 6 0.285714 0.285714 1.110223e-16 63 7 -0.000000 0.000000 1.203464e-17 63 8 0.222222 0.222222 0.000000e+00 63 9 0.000000 0.000000 8.348357e-18 63 10 0.181818 0.181818 8.326673e-17 63 11 -0.000000 0.000000 1.181780e-17 63 12 0.153846 0.153846 2.775558e-17 63 13 0.000000 0.000000 1.051676e-17 63 14 0.133333 0.133333 2.775558e-17 63 15 -0.000000 0.000000 8.077306e-18 63 16 0.117647 0.117647 1.387779e-17 63 17 0.000000 0.000000 1.680513e-18 63 18 0.105263 0.105263 2.775558e-17 63 19 -0.000000 0.000000 3.686287e-18 63 20 0.095238 0.095238 1.387779e-17 63 21 0.000000 0.000000 1.008308e-17 63 22 0.086957 0.086957 0.000000e+00 63 23 -0.000000 0.000000 2.656295e-18 63 24 0.080000 0.080000 2.775558e-17 63 25 -0.000000 0.000000 2.276825e-18 63 26 0.074074 0.074074 1.387779e-17 63 27 -0.000000 0.000000 5.421011e-19 63 28 0.068966 0.068966 1.387779e-17 63 29 0.000000 0.000000 1.463673e-18 63 30 0.064516 0.064516 1.387779e-17 63 31 -0.000000 0.000000 5.421011e-19 63 32 0.060606 0.060606 1.387779e-17 63 33 -0.000000 0.000000 4.336809e-19 63 34 0.057143 0.057143 1.387779e-17 63 35 0.000000 0.000000 4.499439e-18 63 36 0.054054 0.054054 1.387779e-17 63 37 -0.000000 0.000000 5.421011e-19 63 38 0.051282 0.051282 1.387779e-17 63 39 -0.000000 0.000000 9.757820e-19 63 40 0.048780 0.048780 1.387779e-17 63 41 0.000000 0.000000 8.673617e-19 63 42 0.046512 0.046512 1.387779e-17 63 43 0.000000 0.000000 0.000000e+00 63 44 0.044444 0.044444 0.000000e+00 63 45 0.000000 0.000000 2.818926e-18 63 46 0.042553 0.042553 1.387779e-17 63 47 -0.000000 0.000000 3.740497e-18 63 48 0.040816 0.040816 2.081668e-17 63 49 0.000000 0.000000 3.794708e-19 63 50 0.039216 0.039216 2.775558e-17 63 51 -0.000000 0.000000 1.734723e-18 63 52 0.037736 0.037736 0.000000e+00 63 53 -0.000000 0.000000 7.589415e-19 63 54 0.036364 0.036364 2.081668e-17 63 55 0.000000 0.000000 1.192622e-18 63 56 0.035088 0.035088 1.387779e-17 63 57 0.000000 0.000000 1.572093e-18 63 58 0.033898 0.033898 0.000000e+00 63 59 -0.000000 0.000000 9.215718e-19 63 60 0.032787 0.032787 3.469447e-17 63 61 0.000000 0.000000 1.626303e-19 63 62 0.031746 0.031746 2.081668e-17 63 63 0.000000 0.000000 1.788934e-18 63 64 0.030769 0.030769 1.387779e-17 63 65 -0.000000 0.000000 5.963112e-19 63 66 0.029851 0.029851 1.387779e-17 63 67 0.000000 0.000000 5.963112e-19 63 68 0.028986 0.028986 1.734723e-17 63 69 0.000000 0.000000 2.168404e-19 63 70 0.028169 0.028169 1.387779e-17 63 71 -0.000000 0.000000 2.168404e-19 63 72 0.027397 0.027397 2.081668e-17 63 73 -0.000000 0.000000 7.589415e-19 63 74 0.026667 0.026667 2.428613e-17 63 75 0.000000 0.000000 0.000000e+00 63 76 0.025974 0.025974 1.734723e-17 63 77 0.000000 0.000000 1.029992e-18 63 78 0.025316 0.025316 2.081668e-17 63 79 -0.000000 0.000000 1.626303e-19 63 80 0.024691 0.024691 2.081668e-17 63 81 0.000000 0.000000 2.710505e-19 63 82 0.024096 0.024096 1.387779e-17 63 83 -0.000000 0.000000 5.421011e-19 63 84 0.023529 0.023529 1.387779e-17 63 85 -0.000000 0.000000 7.047314e-19 63 86 0.022989 0.022989 2.428613e-17 63 87 -0.000000 0.000000 1.138412e-18 63 88 0.022472 0.022472 1.734723e-17 63 89 0.000000 0.000000 1.680513e-18 63 90 0.021978 0.021978 1.734723e-17 63 91 -0.000000 0.000000 4.878910e-19 63 92 0.021505 0.021505 1.734723e-17 63 93 -0.000000 0.000000 5.421011e-19 63 94 0.021053 0.021053 1.734723e-17 63 95 0.000000 0.000000 2.168404e-19 63 96 0.020619 0.020619 1.387779e-17 63 97 -0.000000 0.000000 6.505213e-19 63 98 0.020202 0.020202 1.734723e-17 TEST29 R8COL_TOL_UNDEX produces index vectors which create a sorted list of the tolerably unique columns of an R8COL, and a map from the original R8COL to the (implicit) R8COL of sorted tolerably unique elements. The unsorted R8COL (transposed): Row: 1 2 3 Col 1: 1.900000 0.000000 10.000000 2: 2.000000 6.000000 10.000000 3: 4.000000 8.000000 12.000000 4: 1.000000 5.000000 9.000000 5: 3.000000 7.000000 11.000000 6: 2.000000 6.000000 0.000000 7: 2.000000 0.000000 10.100000 8: 2.000000 0.100000 10.000000 9: 3.000000 4.000000 18.000000 10: 1.900000 8.000000 10.000000 11: 0.000000 0.000000 0.000000 12: 0.000000 6.000000 10.000000 13: 2.100000 0.000000 10.000000 14: 2.000000 6.000000 10.000000 15: 3.000000 7.000000 11.000000 16: 2.000000 0.000000 10.000000 17: 2.000000 0.000000 10.000000 18: 2.000000 6.000000 10.000000 19: 1.000000 5.000000 9.000000 20: 2.000000 0.000000 10.100000 21: 1.000000 5.000000 9.100000 22: 1.000000 5.100000 9.000000 Using tolerance = 0.250000 Number of tolerably unique columns is 10 XDNU points to the representative for each item. UNDX selects the representatives. I XDNU UNDX 0 3 10 1 6 11 2 9 18 3 2 0 4 8 9 5 5 5 6 3 13 7 3 8 8 7 14 9 4 2 10 0 11 1 12 3 13 6 14 8 15 3 16 3 17 6 18 2 19 3 20 2 21 2 The tolerably unique R8COL (transposed): Row: 1 2 3 Col 1: 0.000000 0.000000 0.000000 2: 0.000000 6.000000 10.000000 3: 1.000000 5.000000 9.000000 4: 1.900000 0.000000 10.000000 5: 1.900000 8.000000 10.000000 6: 2.000000 6.000000 0.000000 7: 2.000000 6.000000 10.000000 8: 3.000000 4.000000 18.000000 9: 3.000000 7.000000 11.000000 10: 4.000000 8.000000 12.000000 TEST30 R8VEC_SORT_HEAP_INDEX_A creates an ascending sort index for a R8VEC. The unsorted array: 0: 0.218418 1: 0.956318 2: 0.829509 3: 0.561695 4: 0.415307 5: 0.066119 6: 0.257578 7: 0.109957 8: 0.043829 9: 0.633966 10: 0.061727 11: 0.449539 12: 0.401306 13: 0.754673 14: 0.797287 15: 0.001838 16: 0.897504 17: 0.350752 18: 0.094545 19: 0.013617 The index vector: 0: 15 1: 19 2: 8 3: 10 4: 5 5: 18 6: 7 7: 0 8: 6 9: 17 10: 12 11: 4 12: 11 13: 3 14: 9 15: 13 16: 14 17: 2 18: 16 19: 1 The sorted array A(INDX(:)): 0: 0.001838 1: 0.013617 2: 0.043829 3: 0.061727 4: 0.066119 5: 0.094545 6: 0.109957 7: 0.218418 8: 0.257578 9: 0.350752 10: 0.401306 11: 0.415307 12: 0.449539 13: 0.561695 14: 0.633966 15: 0.754673 16: 0.797287 17: 0.829509 18: 0.897504 19: 0.956318 TEST01 CHEBYSHEV1_COMPUTE_NP computes a Gauss-Chebyshev type 1 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) / sqrt ( 1 - x^2 ) dx. CHEBYSHEV1_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 3.141593 3.141593 0.000000e+00 1 1 0.000000 0.000000 0.000000e+00 1 2 0.000000 1.570796 1.570796e+00 1 3 0.000000 0.000000 0.000000e+00 1 4 0.000000 1.178097 1.178097e+00 2 0 3.141593 3.141593 0.000000e+00 2 1 0.000000 0.000000 0.000000e+00 2 2 1.570796 1.570796 0.000000e+00 2 3 0.000000 0.000000 2.220446e-16 2 4 0.785398 1.178097 3.926991e-01 2 5 0.000000 0.000000 2.220446e-16 2 6 0.392699 0.981748 5.890486e-01 3 0 3.141593 3.141593 0.000000e+00 3 1 0.000000 0.000000 0.000000e+00 3 2 1.570796 1.570796 2.220446e-16 3 3 0.000000 0.000000 0.000000e+00 3 4 1.178097 1.178097 2.220446e-16 3 5 0.000000 0.000000 0.000000e+00 3 6 0.883573 0.981748 9.817477e-02 3 7 0.000000 0.000000 0.000000e+00 3 8 0.662680 0.859029 1.963495e-01 4 0 3.141593 3.141593 0.000000e+00 4 1 -0.000000 0.000000 1.110223e-16 4 2 1.570796 1.570796 0.000000e+00 4 3 0.000000 0.000000 1.110223e-16 4 4 1.178097 1.178097 0.000000e+00 4 5 0.000000 0.000000 0.000000e+00 4 6 0.981748 0.981748 1.110223e-16 4 7 0.000000 0.000000 0.000000e+00 4 8 0.834486 0.859029 2.454369e-02 4 9 0.000000 0.000000 0.000000e+00 4 10 0.711767 0.773126 6.135923e-02 5 0 3.141593 3.141593 0.000000e+00 5 1 0.000000 0.000000 1.110223e-16 5 2 1.570796 1.570796 0.000000e+00 5 3 0.000000 0.000000 0.000000e+00 5 4 1.178097 1.178097 2.220446e-16 5 5 0.000000 0.000000 0.000000e+00 5 6 0.981748 0.981748 2.220446e-16 5 7 0.000000 0.000000 0.000000e+00 5 8 0.859029 0.859029 4.440892e-16 5 9 0.000000 0.000000 0.000000e+00 5 10 0.766990 0.773126 6.135923e-03 5 11 0.000000 0.000000 0.000000e+00 5 12 0.690291 0.708699 1.840777e-02 6 0 3.141593 3.141593 4.440892e-16 6 1 0.000000 0.000000 2.220446e-16 6 2 1.570796 1.570796 0.000000e+00 6 3 0.000000 0.000000 2.220446e-16 6 4 1.178097 1.178097 0.000000e+00 6 5 0.000000 0.000000 2.220446e-16 6 6 0.981748 0.981748 1.110223e-16 6 7 0.000000 0.000000 3.885781e-16 6 8 0.859029 0.859029 3.330669e-16 6 9 0.000000 0.000000 3.885781e-16 6 10 0.773126 0.773126 1.110223e-16 6 11 0.000000 0.000000 4.996004e-16 6 12 0.707165 0.708699 1.533981e-03 6 13 0.000000 0.000000 5.551115e-16 6 14 0.652709 0.658078 5.368933e-03 7 0 3.141593 3.141593 0.000000e+00 7 1 0.000000 0.000000 1.110223e-16 7 2 1.570796 1.570796 2.220446e-16 7 3 0.000000 0.000000 1.110223e-16 7 4 1.178097 1.178097 0.000000e+00 7 5 0.000000 0.000000 5.551115e-17 7 6 0.981748 0.981748 1.110223e-16 7 7 -0.000000 0.000000 5.551115e-17 7 8 0.859029 0.859029 1.110223e-16 7 9 -0.000000 0.000000 1.665335e-16 7 10 0.773126 0.773126 3.330669e-16 7 11 -0.000000 0.000000 2.775558e-16 7 12 0.708699 0.708699 6.661338e-16 7 13 -0.000000 0.000000 3.330669e-16 7 14 0.657694 0.658078 3.834952e-04 7 15 -0.000000 0.000000 4.440892e-16 7 16 0.615414 0.616948 1.533981e-03 8 0 3.141593 3.141593 0.000000e+00 8 1 0.000000 0.000000 2.220446e-16 8 2 1.570796 1.570796 0.000000e+00 8 3 0.000000 0.000000 5.551115e-17 8 4 1.178097 1.178097 0.000000e+00 8 5 0.000000 0.000000 0.000000e+00 8 6 0.981748 0.981748 1.110223e-16 8 7 0.000000 0.000000 0.000000e+00 8 8 0.859029 0.859029 3.330669e-16 8 9 -0.000000 0.000000 1.110223e-16 8 10 0.773126 0.773126 0.000000e+00 8 11 -0.000000 0.000000 5.551115e-17 8 12 0.708699 0.708699 0.000000e+00 8 13 -0.000000 0.000000 5.551115e-17 8 14 0.658078 0.658078 1.110223e-16 8 15 -0.000000 0.000000 5.551115e-17 8 16 0.616852 0.616948 9.587380e-05 8 17 -0.000000 0.000000 5.551115e-17 8 18 0.582242 0.582673 4.314321e-04 9 0 3.141593 3.141593 0.000000e+00 9 1 0.000000 0.000000 1.665335e-16 9 2 1.570796 1.570796 2.220446e-16 9 3 0.000000 0.000000 2.220446e-16 9 4 1.178097 1.178097 2.220446e-16 9 5 0.000000 0.000000 2.220446e-16 9 6 0.981748 0.981748 1.110223e-16 9 7 0.000000 0.000000 2.220446e-16 9 8 0.859029 0.859029 3.330669e-16 9 9 0.000000 0.000000 1.665335e-16 9 10 0.773126 0.773126 2.220446e-16 9 11 0.000000 0.000000 2.220446e-16 9 12 0.708699 0.708699 2.220446e-16 9 13 0.000000 0.000000 1.665335e-16 9 14 0.658078 0.658078 3.330669e-16 9 15 0.000000 0.000000 1.665335e-16 9 16 0.616948 0.616948 4.440892e-16 9 17 0.000000 0.000000 1.110223e-16 9 18 0.582649 0.582673 2.396845e-05 9 19 0.000000 0.000000 1.110223e-16 9 20 0.553420 0.553539 1.198422e-04 10 0 3.141593 3.141593 0.000000e+00 10 1 0.000000 0.000000 1.665335e-16 10 2 1.570796 1.570796 2.220446e-16 10 3 0.000000 0.000000 3.330669e-16 10 4 1.178097 1.178097 2.220446e-16 10 5 0.000000 0.000000 3.885781e-16 10 6 0.981748 0.981748 0.000000e+00 10 7 0.000000 0.000000 3.330669e-16 10 8 0.859029 0.859029 2.220446e-16 10 9 0.000000 0.000000 4.440892e-16 10 10 0.773126 0.773126 0.000000e+00 10 11 0.000000 0.000000 4.440892e-16 10 12 0.708699 0.708699 0.000000e+00 10 13 0.000000 0.000000 4.996004e-16 10 14 0.658078 0.658078 1.110223e-16 10 15 0.000000 0.000000 5.551115e-16 10 16 0.616948 0.616948 1.110223e-16 10 17 0.000000 0.000000 6.106227e-16 10 18 0.582673 0.582673 2.220446e-16 10 19 0.000000 0.000000 6.106227e-16 10 20 0.553533 0.553539 5.992112e-06 10 21 0.000000 0.000000 6.383782e-16 10 22 0.528346 0.528378 3.295662e-05 TEST02 CHEBYSHEV1_COMPUTE_NP computes a Gauss-Chebyshev type 1 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) / sqrt(1-x^2) dx. Order = 1 0 0.0000000000000000 3.1415926535897931 Order = 2 0 -0.7071067811865475 1.5707963267948966 1 0.7071067811865476 1.5707963267948966 Order = 3 0 -0.8660254037844387 1.0471975511965976 1 0.0000000000000000 1.0471975511965976 2 0.8660254037844387 1.0471975511965976 Order = 4 0 -0.9238795325112867 0.7853981633974483 1 -0.3826834323650897 0.7853981633974483 2 0.3826834323650898 0.7853981633974483 3 0.9238795325112867 0.7853981633974483 Order = 5 0 -0.9510565162951535 0.6283185307179586 1 -0.5877852522924730 0.6283185307179586 2 0.0000000000000000 0.6283185307179586 3 0.5877852522924731 0.6283185307179586 4 0.9510565162951535 0.6283185307179586 Order = 6 0 -0.9659258262890682 0.5235987755982988 1 -0.7071067811865475 0.5235987755982988 2 -0.2588190451025206 0.5235987755982988 3 0.2588190451025207 0.5235987755982988 4 0.7071067811865476 0.5235987755982988 5 0.9659258262890683 0.5235987755982988 Order = 7 0 -0.9749279121818237 0.4487989505128276 1 -0.7818314824680295 0.4487989505128276 2 -0.4338837391175581 0.4487989505128276 3 0.0000000000000000 0.4487989505128276 4 0.4338837391175582 0.4487989505128276 5 0.7818314824680298 0.4487989505128276 6 0.9749279121818236 0.4487989505128276 Order = 8 0 -0.9807852804032304 0.3926990816987241 1 -0.8314696123025453 0.3926990816987241 2 -0.5555702330196020 0.3926990816987241 3 -0.1950903220161282 0.3926990816987241 4 0.1950903220161283 0.3926990816987241 5 0.5555702330196023 0.3926990816987241 6 0.8314696123025452 0.3926990816987241 7 0.9807852804032304 0.3926990816987241 Order = 9 0 -0.9848077530122080 0.3490658503988659 1 -0.8660254037844385 0.3490658503988659 2 -0.6427876096865394 0.3490658503988659 3 -0.3420201433256685 0.3490658503988659 4 0.0000000000000000 0.3490658503988659 5 0.3420201433256688 0.3490658503988659 6 0.6427876096865394 0.3490658503988659 7 0.8660254037844387 0.3490658503988659 8 0.9848077530122080 0.3490658503988659 Order = 10 0 -0.9876883405951377 0.3141592653589793 1 -0.8910065241883678 0.3141592653589793 2 -0.7071067811865475 0.3141592653589793 3 -0.4539904997395467 0.3141592653589793 4 -0.1564344650402306 0.3141592653589793 5 0.1564344650402309 0.3141592653589793 6 0.4539904997395468 0.3141592653589793 7 0.7071067811865476 0.3141592653589793 8 0.8910065241883679 0.3141592653589793 9 0.9876883405951378 0.3141592653589793 TEST03 CHEBYSHEV2_COMPUTE_NP computes a Gauss-Chebyshev type 2 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) * sqrt ( 1 - x^2 ) dx. CHEBYSHEV2_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 1.570796 1.570796 0.000000e+00 1 1 0.000000 0.000000 0.000000e+00 1 2 0.000000 0.392699 3.926991e-01 1 3 0.000000 0.000000 0.000000e+00 1 4 0.000000 0.196350 1.963495e-01 2 0 1.570796 1.570796 0.000000e+00 2 1 0.000000 0.000000 1.110223e-16 2 2 0.392699 0.392699 1.110223e-16 2 3 0.000000 0.000000 1.526557e-16 2 4 0.098175 0.196350 9.817477e-02 2 5 0.000000 0.000000 7.285839e-17 2 6 0.024544 0.122718 9.817477e-02 3 0 1.570796 1.570796 0.000000e+00 3 1 -0.000000 0.000000 1.110223e-16 3 2 0.392699 0.392699 0.000000e+00 3 3 0.000000 0.000000 0.000000e+00 3 4 0.196350 0.196350 0.000000e+00 3 5 0.000000 0.000000 1.387779e-17 3 6 0.098175 0.122718 2.454369e-02 3 7 0.000000 0.000000 1.387779e-17 3 8 0.049087 0.085903 3.681554e-02 4 0 1.570796 1.570796 2.220446e-16 4 1 -0.000000 0.000000 5.551115e-17 4 2 0.392699 0.392699 0.000000e+00 4 3 0.000000 0.000000 0.000000e+00 4 4 0.196350 0.196350 2.775558e-17 4 5 0.000000 0.000000 1.387779e-17 4 6 0.122718 0.122718 1.387779e-17 4 7 0.000000 0.000000 2.775558e-17 4 8 0.079767 0.085903 6.135923e-03 4 9 0.000000 0.000000 2.775558e-17 4 10 0.052155 0.064427 1.227185e-02 5 0 1.570796 1.570796 0.000000e+00 5 1 0.000000 0.000000 5.551115e-17 5 2 0.392699 0.392699 5.551115e-17 5 3 0.000000 0.000000 9.714451e-17 5 4 0.196350 0.196350 2.775558e-17 5 5 0.000000 0.000000 4.163336e-17 5 6 0.122718 0.122718 1.387779e-17 5 7 0.000000 0.000000 1.387779e-17 5 8 0.085903 0.085903 1.387779e-17 5 9 0.000000 0.000000 6.938894e-18 5 10 0.062893 0.064427 1.533981e-03 5 11 0.000000 0.000000 0.000000e+00 5 12 0.046786 0.050621 3.834952e-03 6 0 1.570796 1.570796 0.000000e+00 6 1 -0.000000 0.000000 8.326673e-17 6 2 0.392699 0.392699 0.000000e+00 6 3 -0.000000 0.000000 1.387779e-17 6 4 0.196350 0.196350 2.775558e-17 6 5 0.000000 0.000000 1.387779e-17 6 6 0.122718 0.122718 1.387779e-17 6 7 0.000000 0.000000 1.387779e-17 6 8 0.085903 0.085903 1.387779e-17 6 9 0.000000 0.000000 2.081668e-17 6 10 0.064427 0.064427 1.387779e-17 6 11 0.000000 0.000000 2.081668e-17 6 12 0.050238 0.050621 3.834952e-04 6 13 0.000000 0.000000 2.775558e-17 6 14 0.039979 0.041130 1.150486e-03 7 0 1.570796 1.570796 0.000000e+00 7 1 -0.000000 0.000000 1.387779e-17 7 2 0.392699 0.392699 5.551115e-17 7 3 -0.000000 0.000000 2.081668e-17 7 4 0.196350 0.196350 0.000000e+00 7 5 -0.000000 0.000000 2.081668e-17 7 6 0.122718 0.122718 2.775558e-17 7 7 -0.000000 0.000000 1.387779e-17 7 8 0.085903 0.085903 0.000000e+00 7 9 -0.000000 0.000000 1.387779e-17 7 10 0.064427 0.064427 1.387779e-17 7 11 -0.000000 0.000000 1.387779e-17 7 12 0.050621 0.050621 0.000000e+00 7 13 -0.000000 0.000000 6.938894e-18 7 14 0.041034 0.041130 9.587380e-05 7 15 -0.000000 0.000000 6.938894e-18 7 16 0.033939 0.034275 3.355583e-04 8 0 1.570796 1.570796 2.220446e-16 8 1 0.000000 0.000000 4.857226e-17 8 2 0.392699 0.392699 0.000000e+00 8 3 0.000000 0.000000 4.163336e-17 8 4 0.196350 0.196350 0.000000e+00 8 5 0.000000 0.000000 1.387779e-17 8 6 0.122718 0.122718 1.387779e-17 8 7 0.000000 0.000000 1.734723e-17 8 8 0.085903 0.085903 1.387779e-17 8 9 0.000000 0.000000 2.081668e-17 8 10 0.064427 0.064427 0.000000e+00 8 11 0.000000 0.000000 1.387779e-17 8 12 0.050621 0.050621 6.938894e-18 8 13 0.000000 0.000000 1.387779e-17 8 14 0.041130 0.041130 6.938894e-18 8 15 0.000000 0.000000 1.734723e-17 8 16 0.034251 0.034275 2.396845e-05 8 17 0.000000 0.000000 1.561251e-17 8 18 0.029038 0.029134 9.587380e-05 9 0 1.570796 1.570796 2.220446e-16 9 1 -0.000000 0.000000 5.204170e-17 9 2 0.392699 0.392699 0.000000e+00 9 3 0.000000 0.000000 1.734723e-17 9 4 0.196350 0.196350 2.775558e-17 9 5 0.000000 0.000000 0.000000e+00 9 6 0.122718 0.122718 1.387779e-17 9 7 0.000000 0.000000 3.469447e-18 9 8 0.085903 0.085903 0.000000e+00 9 9 0.000000 0.000000 1.387779e-17 9 10 0.064427 0.064427 0.000000e+00 9 11 0.000000 0.000000 0.000000e+00 9 12 0.050621 0.050621 1.387779e-17 9 13 -0.000000 0.000000 1.734723e-18 9 14 0.041130 0.041130 2.081668e-17 9 15 -0.000000 0.000000 3.469447e-18 9 16 0.034275 0.034275 1.387779e-17 9 17 -0.000000 0.000000 3.469447e-18 9 18 0.029128 0.029134 5.992112e-06 9 19 -0.000000 0.000000 1.734723e-18 9 20 0.025134 0.025161 2.696451e-05 10 0 1.570796 1.570796 2.220446e-16 10 1 -0.000000 0.000000 1.040834e-17 10 2 0.392699 0.392699 5.551115e-17 10 3 0.000000 0.000000 1.387779e-17 10 4 0.196350 0.196350 0.000000e+00 10 5 0.000000 0.000000 1.040834e-17 10 6 0.122718 0.122718 1.387779e-17 10 7 0.000000 0.000000 1.387779e-17 10 8 0.085903 0.085903 1.387779e-17 10 9 0.000000 0.000000 1.387779e-17 10 10 0.064427 0.064427 0.000000e+00 10 11 0.000000 0.000000 1.387779e-17 10 12 0.050621 0.050621 6.938894e-18 10 13 0.000000 0.000000 1.040834e-17 10 14 0.041130 0.041130 1.387779e-17 10 15 0.000000 0.000000 8.673617e-18 10 16 0.034275 0.034275 6.938894e-18 10 17 0.000000 0.000000 6.938894e-18 10 18 0.029134 0.029134 1.040834e-17 10 19 0.000000 0.000000 5.204170e-18 10 20 0.025159 0.025161 1.498028e-06 10 21 0.000000 0.000000 5.204170e-18 10 22 0.022008 0.022016 7.490141e-06 TEST04 CHEBYSHEV2_COMPUTE_NP computes a Gauss-Chebyshev type 2 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) * sqrt(1-x^2) dx. Order = 1 0 0.0000000000000000 1.5707963267948966 Order = 2 0 -0.4999999999999998 0.7853981633974484 1 0.5000000000000001 0.7853981633974481 Order = 3 0 -0.7071067811865475 0.3926990816987243 1 0.0000000000000000 0.7853981633974483 2 0.7071067811865476 0.3926990816987240 Order = 4 0 -0.8090169943749473 0.2170787134227061 1 -0.3090169943749473 0.5683194499747424 2 0.3090169943749475 0.5683194499747423 3 0.8090169943749475 0.2170787134227060 Order = 5 0 -0.8660254037844387 0.1308996938995747 1 -0.4999999999999998 0.3926990816987242 2 0.0000000000000000 0.5235987755982988 3 0.5000000000000001 0.3926990816987240 4 0.8660254037844387 0.1308996938995747 Order = 6 0 -0.9009688679024190 0.0844886908915886 1 -0.6234898018587335 0.2743330560697779 2 -0.2225209339563143 0.4265764164360819 3 0.2225209339563144 0.4265764164360819 4 0.6234898018587336 0.2743330560697778 5 0.9009688679024191 0.0844886908915886 Order = 7 0 -0.9238795325112867 0.0575094490319132 1 -0.7071067811865475 0.1963495408493621 2 -0.3826834323650897 0.3351896326668110 3 0.0000000000000000 0.3926990816987241 4 0.3826834323650898 0.3351896326668110 5 0.7071067811865476 0.1963495408493620 6 0.9238795325112867 0.0575094490319131 Order = 8 0 -0.9396926207859083 0.0408329477091071 1 -0.7660444431189779 0.1442256007956728 2 -0.4999999999999998 0.2617993877991495 3 -0.1736481776669303 0.3385402270935190 4 0.1736481776669304 0.3385402270935190 5 0.5000000000000001 0.2617993877991494 6 0.7660444431189780 0.1442256007956727 7 0.9396926207859084 0.0408329477091071 Order = 9 0 -0.9510565162951535 0.0299995403716082 1 -0.8090169943749473 0.1085393567113530 2 -0.5877852522924730 0.2056199086476263 3 -0.3090169943749473 0.2841597249873712 4 0.0000000000000000 0.3141592653589793 5 0.3090169943749475 0.2841597249873711 6 0.5877852522924731 0.2056199086476263 7 0.8090169943749475 0.1085393567113530 8 0.9510565162951535 0.0299995403716082 Order = 10 0 -0.9594929736144974 0.0226689425018588 1 -0.8412535328311811 0.0834785409341891 2 -0.6548607339452850 0.1631221774548166 3 -0.4154150130018863 0.2363135602034873 4 -0.1423148382732850 0.2798149423030966 5 0.1423148382732851 0.2798149423030965 6 0.4154150130018864 0.2363135602034873 7 0.6548607339452851 0.1631221774548166 8 0.8412535328311812 0.0834785409341890 9 0.9594929736144974 0.0226689425018588 TEST05 CLENSHAW_CURTIS_COMPUTE_NP computes a Clenshaw Curtis rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. LEGENDRE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = ORDER+1 if ORDER is odd, or N = ORDER if ORDER is even In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 2.000000 2.000000 0.000000e+00 1 1 0.000000 0.000000 0.000000e+00 1 2 0.000000 0.666667 6.666667e-01 1 3 0.000000 0.000000 0.000000e+00 1 4 0.000000 0.400000 4.000000e-01 2 0 2.000000 2.000000 0.000000e+00 2 1 0.000000 0.000000 0.000000e+00 2 2 2.000000 0.666667 1.333333e+00 2 3 0.000000 0.000000 0.000000e+00 2 4 2.000000 0.400000 1.600000e+00 3 0 2.000000 2.000000 0.000000e+00 3 1 0.000000 0.000000 0.000000e+00 3 2 0.666667 0.666667 1.110223e-16 3 3 0.000000 0.000000 0.000000e+00 3 4 0.666667 0.400000 2.666667e-01 3 5 0.000000 0.000000 0.000000e+00 3 6 0.666667 0.285714 3.809524e-01 4 0 2.000000 2.000000 4.440892e-16 4 1 0.000000 0.000000 4.579670e-16 4 2 0.666667 0.666667 1.110223e-16 4 3 0.000000 0.000000 2.636780e-16 4 4 0.333333 0.400000 6.666667e-02 4 5 0.000000 0.000000 9.714451e-17 4 6 0.250000 0.285714 3.571429e-02 5 0 2.000000 2.000000 0.000000e+00 5 1 0.000000 0.000000 1.387779e-16 5 2 0.666667 0.666667 1.110223e-16 5 3 0.000000 0.000000 1.110223e-16 5 4 0.400000 0.400000 0.000000e+00 5 5 0.000000 0.000000 8.326673e-17 5 6 0.266667 0.285714 1.904762e-02 5 7 0.000000 0.000000 5.551115e-17 5 8 0.200000 0.222222 2.222222e-02 6 0 2.000000 2.000000 0.000000e+00 6 1 0.000000 0.000000 1.387779e-16 6 2 0.666667 0.666667 1.110223e-16 6 3 0.000000 0.000000 1.665335e-16 6 4 0.400000 0.400000 0.000000e+00 6 5 0.000000 0.000000 1.110223e-16 6 6 0.283333 0.285714 2.380952e-03 6 7 0.000000 0.000000 1.110223e-16 6 8 0.212500 0.222222 9.722222e-03 7 0 2.000000 2.000000 0.000000e+00 7 1 0.000000 0.000000 2.740863e-16 7 2 0.666667 0.666667 0.000000e+00 7 3 0.000000 0.000000 1.353084e-16 7 4 0.400000 0.400000 0.000000e+00 7 5 0.000000 0.000000 5.204170e-17 7 6 0.285714 0.285714 0.000000e+00 7 7 0.000000 0.000000 1.040834e-17 7 8 0.221429 0.222222 7.936508e-04 7 9 -0.000000 0.000000 3.469447e-18 7 10 0.178571 0.181818 3.246753e-03 8 0 2.000000 2.000000 0.000000e+00 8 1 0.000000 0.000000 1.491862e-16 8 2 0.666667 0.666667 1.110223e-16 8 3 0.000000 0.000000 9.367507e-17 8 4 0.400000 0.400000 0.000000e+00 8 5 0.000000 0.000000 7.979728e-17 8 6 0.285714 0.285714 0.000000e+00 8 7 0.000000 0.000000 7.979728e-17 8 8 0.222024 0.222222 1.984127e-04 8 9 0.000000 0.000000 7.979728e-17 8 10 0.181101 0.181818 7.169913e-04 9 0 2.000000 2.000000 0.000000e+00 9 1 0.000000 0.000000 2.012279e-16 9 2 0.666667 0.666667 1.110223e-16 9 3 0.000000 0.000000 1.318390e-16 9 4 0.400000 0.400000 0.000000e+00 9 5 0.000000 0.000000 1.040834e-16 9 6 0.285714 0.285714 0.000000e+00 9 7 0.000000 0.000000 7.632783e-17 9 8 0.222222 0.222222 0.000000e+00 9 9 0.000000 0.000000 4.857226e-17 9 10 0.181746 0.181818 7.215007e-05 9 11 0.000000 0.000000 4.163336e-17 9 12 0.153571 0.153846 2.747253e-04 10 0 2.000000 2.000000 0.000000e+00 10 1 0.000000 0.000000 3.816392e-16 10 2 0.666667 0.666667 1.110223e-16 10 3 0.000000 0.000000 2.567391e-16 10 4 0.400000 0.400000 5.551115e-17 10 5 0.000000 0.000000 1.873501e-16 10 6 0.285714 0.285714 5.551115e-17 10 7 0.000000 0.000000 1.457168e-16 10 8 0.222222 0.222222 0.000000e+00 10 9 0.000000 0.000000 1.595946e-16 10 10 0.181796 0.181818 2.254690e-05 10 11 0.000000 0.000000 1.526557e-16 10 12 0.153757 0.153846 8.871337e-05 TEST06 CLENSHAW_CURTIS_COMPUTE_NP computes a Clenshaw Curtis rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. Order = 1 0 0.0000000000000000 2.0000000000000000 Order = 2 0 -1.0000000000000000 1.0000000000000000 1 1.0000000000000000 1.0000000000000000 Order = 3 0 -1.0000000000000000 0.3333333333333334 1 0.0000000000000000 1.3333333333333333 2 1.0000000000000000 0.3333333333333334 Order = 4 0 -1.0000000000000000 0.1111111111111111 1 -0.4999999999999998 0.8888888888888888 2 0.5000000000000001 0.8888888888888892 3 1.0000000000000000 0.1111111111111111 Order = 5 0 -1.0000000000000000 0.0666666666666667 1 -0.7071067811865475 0.5333333333333333 2 0.0000000000000000 0.7999999999999999 3 0.7071067811865476 0.5333333333333334 4 1.0000000000000000 0.0666666666666667 Order = 6 0 -1.0000000000000000 0.0400000000000000 1 -0.8090169943749473 0.3607430412000112 2 -0.3090169943749473 0.5992569587999889 3 0.3090169943749475 0.5992569587999887 4 0.8090169943749475 0.3607430412000113 5 1.0000000000000000 0.0400000000000000 Order = 7 0 -1.0000000000000000 0.0285714285714286 1 -0.8660254037844387 0.2539682539682539 2 -0.4999999999999998 0.4571428571428571 3 0.0000000000000000 0.5206349206349206 4 0.5000000000000001 0.4571428571428573 5 0.8660254037844387 0.2539682539682539 6 1.0000000000000000 0.0285714285714286 Order = 8 0 -1.0000000000000000 0.0204081632653061 1 -0.9009688679024190 0.1901410072182084 2 -0.6234898018587335 0.3522424237181591 3 -0.2225209339563143 0.4372084057983264 4 0.2225209339563144 0.4372084057983264 5 0.6234898018587336 0.3522424237181591 6 0.9009688679024191 0.1901410072182084 7 1.0000000000000000 0.0204081632653061 Order = 9 0 -1.0000000000000000 0.0158730158730159 1 -0.9238795325112867 0.1462186492160181 2 -0.7071067811865475 0.2793650793650794 3 -0.3826834323650897 0.3617178587204897 4 0.0000000000000000 0.3936507936507936 5 0.3826834323650898 0.3617178587204898 6 0.7071067811865476 0.2793650793650794 7 0.9238795325112867 0.1462186492160182 8 1.0000000000000000 0.0158730158730159 Order = 10 0 -1.0000000000000000 0.0123456790123457 1 -0.9396926207859083 0.1165674565720371 2 -0.7660444431189779 0.2252843233381044 3 -0.4999999999999998 0.3019400352733685 4 -0.1736481776669303 0.3438625058041442 5 0.1736481776669304 0.3438625058041442 6 0.5000000000000001 0.3019400352733687 7 0.7660444431189780 0.2252843233381044 8 0.9396926207859084 0.1165674565720372 9 1.0000000000000000 0.0123456790123457 TEST07 FEJER2_COMPUTE_NP computes a Fejer Type 2 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. LEGENDRE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = ORDER+1 if ORDER is odd, or N = ORDER if ORDER is even In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 2.000000 2.000000 0.000000e+00 1 1 0.000000 0.000000 0.000000e+00 1 2 0.000000 0.666667 6.666667e-01 1 3 0.000000 0.000000 0.000000e+00 1 4 0.000000 0.400000 4.000000e-01 2 0 2.000000 2.000000 0.000000e+00 2 1 0.000000 0.000000 3.330669e-16 2 2 0.500000 0.666667 1.666667e-01 2 3 0.000000 0.000000 2.498002e-16 2 4 0.125000 0.400000 2.750000e-01 3 0 2.000000 2.000000 0.000000e+00 3 1 0.000000 0.000000 0.000000e+00 3 2 0.666667 0.666667 1.110223e-16 3 3 0.000000 0.000000 8.326673e-17 3 4 0.333333 0.400000 6.666667e-02 3 5 0.000000 0.000000 6.938894e-17 3 6 0.166667 0.285714 1.190476e-01 4 0 2.000000 2.000000 0.000000e+00 4 1 0.000000 0.000000 1.110223e-16 4 2 0.666667 0.666667 1.110223e-16 4 3 0.000000 0.000000 5.551115e-17 4 4 0.375000 0.400000 2.500000e-02 4 5 0.000000 0.000000 5.551115e-17 4 6 0.239583 0.285714 4.613095e-02 5 0 2.000000 2.000000 0.000000e+00 5 1 0.000000 0.000000 1.110223e-16 5 2 0.666667 0.666667 0.000000e+00 5 3 0.000000 0.000000 1.110223e-16 5 4 0.400000 0.400000 0.000000e+00 5 5 0.000000 0.000000 2.775558e-17 5 6 0.275000 0.285714 1.071429e-02 5 7 0.000000 0.000000 1.387779e-17 5 8 0.200000 0.222222 2.222222e-02 6 0 2.000000 2.000000 2.220446e-16 6 1 0.000000 0.000000 5.551115e-17 6 2 0.666667 0.666667 0.000000e+00 6 3 0.000000 0.000000 1.110223e-16 6 4 0.400000 0.400000 1.110223e-16 6 5 0.000000 0.000000 8.326673e-17 6 6 0.281250 0.285714 4.464286e-03 6 7 0.000000 0.000000 1.110223e-16 6 8 0.211979 0.222222 1.024306e-02 7 0 2.000000 2.000000 0.000000e+00 7 1 -0.000000 0.000000 8.326673e-17 7 2 0.666667 0.666667 1.110223e-16 7 3 0.000000 0.000000 0.000000e+00 7 4 0.400000 0.400000 0.000000e+00 7 5 -0.000000 0.000000 2.775558e-17 7 6 0.285714 0.285714 0.000000e+00 7 7 0.000000 0.000000 0.000000e+00 7 8 0.220238 0.222222 1.984127e-03 7 9 -0.000000 0.000000 1.387779e-17 7 10 0.176786 0.181818 5.032468e-03 8 0 2.000000 2.000000 4.440892e-16 8 1 -0.000000 0.000000 2.775558e-17 8 2 0.666667 0.666667 0.000000e+00 8 3 0.000000 0.000000 1.110223e-16 8 4 0.400000 0.400000 5.551115e-17 8 5 0.000000 0.000000 8.326673e-17 8 6 0.285714 0.285714 0.000000e+00 8 7 0.000000 0.000000 8.326673e-17 8 8 0.221354 0.222222 8.680556e-04 8 9 0.000000 0.000000 9.714451e-17 8 10 0.179408 0.181818 2.409700e-03 9 0 2.000000 2.000000 0.000000e+00 9 1 -0.000000 0.000000 1.387779e-17 9 2 0.666667 0.666667 0.000000e+00 9 3 -0.000000 0.000000 1.387779e-17 9 4 0.400000 0.400000 5.551115e-17 9 5 -0.000000 0.000000 1.387779e-17 9 6 0.285714 0.285714 0.000000e+00 9 7 0.000000 0.000000 0.000000e+00 9 8 0.222222 0.222222 2.775558e-17 9 9 -0.000000 0.000000 2.775558e-17 9 10 0.181424 0.181818 3.945707e-04 9 11 -0.000000 0.000000 1.387779e-17 9 12 0.152654 0.153846 1.192384e-03 10 0 2.000000 2.000000 2.220446e-16 10 1 0.000000 0.000000 0.000000e+00 10 2 0.666667 0.666667 0.000000e+00 10 3 0.000000 0.000000 9.714451e-17 10 4 0.400000 0.400000 1.110223e-16 10 5 0.000000 0.000000 1.387779e-17 10 6 0.285714 0.285714 0.000000e+00 10 7 0.000000 0.000000 4.163336e-17 10 8 0.222222 0.222222 5.551115e-17 10 9 0.000000 0.000000 4.163336e-17 10 10 0.181641 0.181818 1.775568e-04 10 11 0.000000 0.000000 4.163336e-17 10 12 0.153266 0.153846 5.800948e-04 TEST08 FEJER2_COMPUTE_NP computes a Fejer Type 2 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. Order = 1 0 0.0000000000000000 2.0000000000000000 Order = 2 0 -0.4999999999999998 1.0000000000000000 1 0.5000000000000001 1.0000000000000000 Order = 3 0 -0.7071067811865475 0.6666666666666667 1 0.0000000000000000 0.6666666666666666 2 0.7071067811865476 0.6666666666666666 Order = 4 0 -0.8090169943749473 0.4254644007500071 1 -0.3090169943749473 0.5745355992499930 2 0.3090169943749475 0.5745355992499930 3 0.8090169943749475 0.4254644007500070 Order = 5 0 -0.8660254037844387 0.3111111111111111 1 -0.4999999999999998 0.4000000000000001 2 0.0000000000000000 0.5777777777777777 3 0.5000000000000001 0.4000000000000000 4 0.8660254037844387 0.3111111111111111 Order = 6 0 -0.9009688679024190 0.2269152467244296 1 -0.6234898018587335 0.3267938603769863 2 -0.2225209339563143 0.4462908928985842 3 0.2225209339563144 0.4462908928985841 4 0.6234898018587336 0.3267938603769863 5 0.9009688679024191 0.2269152467244296 Order = 7 0 -0.9238795325112867 0.1779646809620499 1 -0.7071067811865475 0.2476190476190477 2 -0.3826834323650897 0.3934638904665215 3 0.0000000000000000 0.3619047619047619 4 0.3826834323650898 0.3934638904665215 5 0.7071067811865476 0.2476190476190476 6 0.9238795325112867 0.1779646809620499 Order = 8 0 -0.9396926207859083 0.1397697435050226 1 -0.7660444431189779 0.2063696457302284 2 -0.4999999999999998 0.3142857142857144 3 -0.1736481776669303 0.3395748964790348 4 0.1736481776669304 0.3395748964790348 5 0.5000000000000001 0.3142857142857143 6 0.7660444431189780 0.2063696457302284 7 0.9396926207859084 0.1397697435050225 Order = 9 0 -0.9510565162951535 0.1147810750857218 1 -0.8090169943749473 0.1654331942222276 2 -0.5877852522924730 0.2737903534857068 3 -0.3090169943749473 0.2790112502222169 4 0.0000000000000000 0.3339682539682539 5 0.3090169943749475 0.2790112502222170 6 0.5877852522924731 0.2737903534857068 7 0.8090169943749475 0.1654331942222276 8 0.9510565162951535 0.1147810750857217 Order = 10 0 -0.9594929736144974 0.0944195417398281 1 -0.8412535328311811 0.1411354380109716 2 -0.6548607339452850 0.2263866903636005 3 -0.4154150130018863 0.2530509772156453 4 -0.1423148382732850 0.2850073526699546 5 0.1423148382732851 0.2850073526699544 6 0.4154150130018864 0.2530509772156453 7 0.6548607339452851 0.2263866903636005 8 0.8412535328311812 0.1411354380109716 9 0.9594929736144974 0.0944195417398281 TEST09 GEGENBAUER_COMPUTE_NP computes a generalized Gauss-Gegenbauer rule which is appropriate for integrands of the form Integral ( 0 <= x < +oo ) f(x) (1-x^2)^alpha dx. GEGENBAUER_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Alpha Estimate Exact Error 1 0 0.500000 1.570796 1.570796 1.110223e-15 1 1 0.500000 0.000000 0.000000 0.000000e+00 1 2 0.500000 0.000000 0.392699 3.926991e-01 1 3 0.500000 0.000000 0.000000 0.000000e+00 1 4 0.500000 0.000000 0.196350 1.963495e-01 2 0 0.500000 1.570796 1.570796 1.110223e-15 2 1 0.500000 0.000000 0.000000 0.000000e+00 2 2 0.500000 0.392699 0.392699 0.000000e+00 2 3 0.500000 0.000000 0.000000 0.000000e+00 2 4 0.500000 0.098175 0.196350 9.817477e-02 2 5 0.500000 0.000000 0.000000 0.000000e+00 2 6 0.500000 0.024544 0.122718 9.817477e-02 3 0 0.500000 1.570796 1.570796 1.110223e-15 3 1 0.500000 0.000000 0.000000 4.996004e-16 3 2 0.500000 0.392699 0.392699 0.000000e+00 3 3 0.500000 0.000000 0.000000 3.053113e-16 3 4 0.500000 0.196350 0.196350 5.551115e-17 3 5 0.500000 0.000000 0.000000 1.804112e-16 3 6 0.500000 0.098175 0.122718 2.454369e-02 3 7 0.500000 0.000000 0.000000 9.714451e-17 3 8 0.500000 0.049087 0.085903 3.681554e-02 4 0 0.500000 1.570796 1.570796 8.881784e-16 4 1 0.500000 -0.000000 0.000000 2.775558e-17 4 2 0.500000 0.392699 0.392699 0.000000e+00 4 3 0.500000 -0.000000 0.000000 1.387779e-17 4 4 0.500000 0.196350 0.196350 2.775558e-17 4 5 0.500000 0.000000 0.000000 0.000000e+00 4 6 0.500000 0.122718 0.122718 2.775558e-17 4 7 0.500000 0.000000 0.000000 0.000000e+00 4 8 0.500000 0.079767 0.085903 6.135923e-03 4 9 0.500000 0.000000 0.000000 0.000000e+00 4 10 0.500000 0.052155 0.064427 1.227185e-02 5 0 0.500000 1.570796 1.570796 1.998401e-15 5 1 0.500000 0.000000 0.000000 4.440892e-16 5 2 0.500000 0.392699 0.392699 6.106227e-16 5 3 0.500000 0.000000 0.000000 2.914335e-16 5 4 0.500000 0.196350 0.196350 5.273559e-16 5 5 0.500000 0.000000 0.000000 2.359224e-16 5 6 0.500000 0.122718 0.122718 3.469447e-16 5 7 0.500000 0.000000 0.000000 1.734723e-16 5 8 0.500000 0.085903 0.085903 2.498002e-16 5 9 0.500000 0.000000 0.000000 1.318390e-16 5 10 0.500000 0.062893 0.064427 1.533981e-03 5 11 0.500000 0.000000 0.000000 9.714451e-17 5 12 0.500000 0.046786 0.050621 3.834952e-03 6 0 0.500000 1.570796 1.570796 1.776357e-15 6 1 0.500000 -0.000000 0.000000 1.387779e-17 6 2 0.500000 0.392699 0.392699 4.996004e-16 6 3 0.500000 0.000000 0.000000 1.387779e-17 6 4 0.500000 0.196350 0.196350 3.608225e-16 6 5 0.500000 -0.000000 0.000000 6.938894e-18 6 6 0.500000 0.122718 0.122718 2.220446e-16 6 7 0.500000 0.000000 0.000000 0.000000e+00 6 8 0.500000 0.085903 0.085903 1.804112e-16 6 9 0.500000 0.000000 0.000000 0.000000e+00 6 10 0.500000 0.064427 0.064427 4.163336e-17 6 11 0.500000 0.000000 0.000000 0.000000e+00 6 12 0.500000 0.050238 0.050621 3.834952e-04 6 13 0.500000 0.000000 0.000000 0.000000e+00 6 14 0.500000 0.039979 0.041130 1.150486e-03 7 0 0.500000 1.570796 1.570796 8.881784e-16 7 1 0.500000 0.000000 0.000000 3.191891e-16 7 2 0.500000 0.392699 0.392699 3.330669e-16 7 3 0.500000 0.000000 0.000000 2.081668e-16 7 4 0.500000 0.196350 0.196350 1.942890e-16 7 5 0.500000 0.000000 0.000000 1.179612e-16 7 6 0.500000 0.122718 0.122718 2.220446e-16 7 7 0.500000 0.000000 0.000000 6.245005e-17 7 8 0.500000 0.085903 0.085903 1.804112e-16 7 9 0.500000 0.000000 0.000000 3.469447e-17 7 10 0.500000 0.064427 0.064427 3.191891e-16 7 11 0.500000 0.000000 0.000000 2.081668e-17 7 12 0.500000 0.050621 0.050621 6.036838e-16 7 13 0.500000 0.000000 0.000000 1.040834e-17 7 14 0.500000 0.041034 0.041130 9.587380e-05 7 15 0.500000 0.000000 0.000000 3.469447e-18 7 16 0.500000 0.033939 0.034275 3.355583e-04 8 0 0.500000 1.570796 1.570796 4.440892e-16 8 1 0.500000 0.000000 0.000000 5.620504e-16 8 2 0.500000 0.392699 0.392699 1.665335e-16 8 3 0.500000 0.000000 0.000000 5.065393e-16 8 4 0.500000 0.196350 0.196350 1.387779e-16 8 5 0.500000 0.000000 0.000000 4.406198e-16 8 6 0.500000 0.122718 0.122718 1.804112e-16 8 7 0.500000 0.000000 0.000000 3.851086e-16 8 8 0.500000 0.085903 0.085903 1.665335e-16 8 9 0.500000 0.000000 0.000000 3.434752e-16 8 10 0.500000 0.064427 0.064427 3.330669e-16 8 11 0.500000 0.000000 0.000000 3.053113e-16 8 12 0.500000 0.050621 0.050621 6.383782e-16 8 13 0.500000 0.000000 0.000000 2.671474e-16 8 14 0.500000 0.041130 0.041130 9.020562e-17 8 15 0.500000 0.000000 0.000000 2.324529e-16 8 16 0.500000 0.034251 0.034275 2.396845e-05 8 17 0.500000 0.000000 0.000000 2.099015e-16 8 18 0.500000 0.029038 0.029134 9.587380e-05 9 0 0.500000 1.570796 1.570796 2.220446e-16 9 1 0.500000 -0.000000 0.000000 6.938894e-18 9 2 0.500000 0.392699 0.392699 5.551115e-16 9 3 0.500000 -0.000000 0.000000 3.469447e-18 9 4 0.500000 0.196350 0.196350 3.608225e-16 9 5 0.500000 -0.000000 0.000000 6.938894e-18 9 6 0.500000 0.122718 0.122718 3.885781e-16 9 7 0.500000 -0.000000 0.000000 3.469447e-18 9 8 0.500000 0.085903 0.085903 3.608225e-16 9 9 0.500000 0.000000 0.000000 0.000000e+00 9 10 0.500000 0.064427 0.064427 4.996004e-16 9 11 0.500000 0.000000 0.000000 0.000000e+00 9 12 0.500000 0.050621 0.050621 7.840950e-16 9 13 0.500000 -0.000000 0.000000 1.734723e-18 9 14 0.500000 0.041130 0.041130 2.220446e-16 9 15 0.500000 0.000000 0.000000 0.000000e+00 9 16 0.500000 0.034275 0.034275 8.326673e-17 9 17 0.500000 0.000000 0.000000 0.000000e+00 9 18 0.500000 0.029128 0.029134 5.992112e-06 9 19 0.500000 0.000000 0.000000 0.000000e+00 9 20 0.500000 0.025134 0.025161 2.696451e-05 10 0 0.500000 1.570796 1.570796 1.332268e-15 10 1 0.500000 -0.000000 0.000000 2.428613e-17 10 2 0.500000 0.392699 0.392699 5.551115e-17 10 3 0.500000 0.000000 0.000000 1.734723e-17 10 4 0.500000 0.196350 0.196350 1.110223e-16 10 5 0.500000 0.000000 0.000000 3.469447e-18 10 6 0.500000 0.122718 0.122718 1.665335e-16 10 7 0.500000 0.000000 0.000000 0.000000e+00 10 8 0.500000 0.085903 0.085903 1.804112e-16 10 9 0.500000 0.000000 0.000000 0.000000e+00 10 10 0.500000 0.064427 0.064427 3.469447e-16 10 11 0.500000 0.000000 0.000000 0.000000e+00 10 12 0.500000 0.050621 0.050621 6.661338e-16 10 13 0.500000 -0.000000 0.000000 1.734723e-18 10 14 0.500000 0.041130 0.041130 1.179612e-16 10 15 0.500000 0.000000 0.000000 0.000000e+00 10 16 0.500000 0.034275 0.034275 2.081668e-17 10 17 0.500000 0.000000 0.000000 0.000000e+00 10 18 0.500000 0.029134 0.029134 1.491862e-16 10 19 0.500000 0.000000 0.000000 1.734723e-18 10 20 0.500000 0.025159 0.025161 1.498028e-06 10 21 0.500000 0.000000 0.000000 0.000000e+00 10 22 0.500000 0.022008 0.022016 7.490141e-06 1 0 1.000000 1.333333 1.333333 0.000000e+00 1 1 1.000000 0.000000 0.000000 0.000000e+00 1 2 1.000000 0.000000 0.266667 2.666667e-01 1 3 1.000000 0.000000 0.000000 0.000000e+00 1 4 1.000000 0.000000 0.114286 1.142857e-01 2 0 1.000000 1.333333 1.333333 0.000000e+00 2 1 1.000000 0.000000 0.000000 1.665335e-16 2 2 1.000000 0.266667 0.266667 5.551115e-17 2 3 1.000000 0.000000 0.000000 4.857226e-17 2 4 1.000000 0.053333 0.114286 6.095238e-02 2 5 1.000000 0.000000 0.000000 1.040834e-17 2 6 1.000000 0.010667 0.063492 5.282540e-02 3 0 1.000000 1.333333 1.333333 4.440892e-16 3 1 1.000000 0.000000 0.000000 0.000000e+00 3 2 1.000000 0.266667 0.266667 2.775558e-16 3 3 1.000000 0.000000 0.000000 0.000000e+00 3 4 1.000000 0.114286 0.114286 1.110223e-16 3 5 1.000000 0.000000 0.000000 0.000000e+00 3 6 1.000000 0.048980 0.063492 1.451247e-02 3 7 1.000000 0.000000 0.000000 0.000000e+00 3 8 1.000000 0.020991 0.040404 1.941279e-02 4 0 1.000000 1.333333 1.333333 2.220446e-16 4 1 1.000000 -0.000000 0.000000 2.081668e-16 4 2 1.000000 0.266667 0.266667 5.551115e-17 4 3 1.000000 -0.000000 0.000000 9.714451e-17 4 4 1.000000 0.114286 0.114286 1.387779e-17 4 5 1.000000 -0.000000 0.000000 4.163336e-17 4 6 1.000000 0.063492 0.063492 1.387779e-17 4 7 1.000000 -0.000000 0.000000 2.428613e-17 4 8 1.000000 0.036886 0.040404 3.518175e-03 4 9 1.000000 -0.000000 0.000000 8.673617e-18 4 10 1.000000 0.021567 0.027972 6.404883e-03 5 0 1.000000 1.333333 1.333333 0.000000e+00 5 1 1.000000 0.000000 0.000000 2.914335e-16 5 2 1.000000 0.266667 0.266667 0.000000e+00 5 3 1.000000 0.000000 0.000000 2.220446e-16 5 4 1.000000 0.114286 0.114286 0.000000e+00 5 5 1.000000 0.000000 0.000000 1.595946e-16 5 6 1.000000 0.063492 0.063492 0.000000e+00 5 7 1.000000 0.000000 0.000000 1.179612e-16 5 8 1.000000 0.040404 0.040404 6.938894e-18 5 9 1.000000 0.000000 0.000000 8.673617e-17 5 10 1.000000 0.027111 0.027972 8.610918e-04 5 11 1.000000 0.000000 0.000000 6.071532e-17 5 12 1.000000 0.018524 0.020513 1.988339e-03 6 0 1.000000 1.333333 1.333333 0.000000e+00 6 1 1.000000 -0.000000 0.000000 2.081668e-16 6 2 1.000000 0.266667 0.266667 5.551115e-17 6 3 1.000000 -0.000000 0.000000 1.387779e-16 6 4 1.000000 0.114286 0.114286 0.000000e+00 6 5 1.000000 -0.000000 0.000000 1.075529e-16 6 6 1.000000 0.063492 0.063492 0.000000e+00 6 7 1.000000 -0.000000 0.000000 7.632783e-17 6 8 1.000000 0.040404 0.040404 6.938894e-18 6 9 1.000000 -0.000000 0.000000 5.030698e-17 6 10 1.000000 0.027972 0.027972 1.422473e-16 6 11 1.000000 -0.000000 0.000000 3.469447e-17 6 12 1.000000 0.020301 0.020513 2.119611e-04 6 13 1.000000 -0.000000 0.000000 2.428613e-17 6 14 1.000000 0.015093 0.015686 5.936828e-04 7 0 1.000000 1.333333 1.333333 2.220446e-16 7 1 1.000000 -0.000000 0.000000 3.816392e-17 7 2 1.000000 0.266667 0.266667 5.551115e-17 7 3 1.000000 -0.000000 0.000000 3.469447e-18 7 4 1.000000 0.114286 0.114286 5.551115e-17 7 5 1.000000 0.000000 0.000000 0.000000e+00 7 6 1.000000 0.063492 0.063492 2.775558e-17 7 7 1.000000 0.000000 0.000000 1.734723e-18 7 8 1.000000 0.040404 0.040404 2.081668e-17 7 9 1.000000 0.000000 0.000000 0.000000e+00 7 10 1.000000 0.027972 0.027972 1.249001e-16 7 11 1.000000 0.000000 0.000000 0.000000e+00 7 12 1.000000 0.020513 0.020513 7.632783e-17 7 13 1.000000 0.000000 0.000000 0.000000e+00 7 14 1.000000 0.015634 0.015686 5.236685e-05 7 15 1.000000 0.000000 0.000000 0.000000e+00 7 16 1.000000 0.012211 0.012384 1.725350e-04 8 0 1.000000 1.333333 1.333333 1.332268e-15 8 1 1.000000 0.000000 0.000000 5.204170e-17 8 2 1.000000 0.266667 0.266667 3.885781e-16 8 3 1.000000 -0.000000 0.000000 3.469447e-18 8 4 1.000000 0.114286 0.114286 2.220446e-16 8 5 1.000000 0.000000 0.000000 0.000000e+00 8 6 1.000000 0.063492 0.063492 1.526557e-16 8 7 1.000000 -0.000000 0.000000 1.734723e-18 8 8 1.000000 0.040404 0.040404 1.110223e-16 8 9 1.000000 0.000000 0.000000 1.734723e-18 8 10 1.000000 0.027972 0.027972 2.116363e-16 8 11 1.000000 -0.000000 0.000000 1.734723e-18 8 12 1.000000 0.020513 0.020513 6.938894e-18 8 13 1.000000 0.000000 0.000000 0.000000e+00 8 14 1.000000 0.015686 0.015686 2.428613e-17 8 15 1.000000 0.000000 0.000000 0.000000e+00 8 16 1.000000 0.012371 0.012384 1.297012e-05 8 17 1.000000 0.000000 0.000000 0.000000e+00 8 18 1.000000 0.009976 0.010025 4.915565e-05 9 0 1.000000 1.333333 1.333333 2.220446e-16 9 1 1.000000 0.000000 0.000000 6.938894e-18 9 2 1.000000 0.266667 0.266667 5.551115e-17 9 3 1.000000 -0.000000 0.000000 1.734723e-18 9 4 1.000000 0.114286 0.114286 5.551115e-17 9 5 1.000000 0.000000 0.000000 1.734723e-18 9 6 1.000000 0.063492 0.063492 2.775558e-17 9 7 1.000000 0.000000 0.000000 0.000000e+00 9 8 1.000000 0.040404 0.040404 3.469447e-17 9 9 1.000000 0.000000 0.000000 0.000000e+00 9 10 1.000000 0.027972 0.027972 1.561251e-16 9 11 1.000000 0.000000 0.000000 8.673617e-19 9 12 1.000000 0.020513 0.020513 4.857226e-17 9 13 1.000000 0.000000 0.000000 0.000000e+00 9 14 1.000000 0.015686 0.015686 5.551115e-17 9 15 1.000000 0.000000 0.000000 0.000000e+00 9 16 1.000000 0.012384 0.012384 5.377643e-17 9 17 1.000000 0.000000 0.000000 0.000000e+00 9 18 1.000000 0.010022 0.010025 3.218149e-06 9 19 1.000000 0.000000 0.000000 0.000000e+00 9 20 1.000000 0.008268 0.008282 1.379312e-05 10 0 1.000000 1.333333 1.333333 4.440892e-16 10 1 1.000000 -0.000000 0.000000 3.400058e-16 10 2 1.000000 0.266667 0.266667 2.220446e-16 10 3 1.000000 -0.000000 0.000000 2.463307e-16 10 4 1.000000 0.114286 0.114286 1.526557e-16 10 5 1.000000 -0.000000 0.000000 1.942890e-16 10 6 1.000000 0.063492 0.063492 1.110223e-16 10 7 1.000000 -0.000000 0.000000 1.674008e-16 10 8 1.000000 0.040404 0.040404 8.326673e-17 10 9 1.000000 -0.000000 0.000000 1.465841e-16 10 10 1.000000 0.027972 0.027972 1.838807e-16 10 11 1.000000 -0.000000 0.000000 1.292369e-16 10 12 1.000000 0.020513 0.020513 3.469447e-17 10 13 1.000000 -0.000000 0.000000 1.179612e-16 10 14 1.000000 0.015686 0.015686 5.204170e-17 10 15 1.000000 -0.000000 0.000000 1.049508e-16 10 16 1.000000 0.012384 0.012384 5.724587e-17 10 17 1.000000 -0.000000 0.000000 9.540979e-17 10 18 1.000000 0.010025 0.010025 3.469447e-18 10 19 1.000000 -0.000000 0.000000 8.630249e-17 10 20 1.000000 0.008281 0.008282 7.995402e-07 10 21 1.000000 -0.000000 0.000000 7.676151e-17 10 22 1.000000 0.006953 0.006957 3.824087e-06 1 0 2.500000 0.981748 0.981748 1.110223e-16 1 1 2.500000 0.000000 0.000000 0.000000e+00 1 2 2.500000 0.000000 0.122718 1.227185e-01 1 3 2.500000 0.000000 0.000000 0.000000e+00 1 4 2.500000 0.000000 0.036816 3.681554e-02 2 0 2.500000 0.981748 0.981748 1.110223e-16 2 1 2.500000 0.000000 0.000000 0.000000e+00 2 2 2.500000 0.122718 0.122718 1.249001e-16 2 3 2.500000 0.000000 0.000000 0.000000e+00 2 4 2.500000 0.015340 0.036816 2.147573e-02 2 5 2.500000 0.000000 0.000000 0.000000e+00 2 6 2.500000 0.001917 0.015340 1.342233e-02 3 0 2.500000 0.981748 0.981748 4.440892e-16 3 1 2.500000 0.000000 0.000000 0.000000e+00 3 2 2.500000 0.122718 0.122718 1.387779e-17 3 3 2.500000 0.000000 0.000000 0.000000e+00 3 4 2.500000 0.036816 0.036816 2.081668e-17 3 5 2.500000 0.000000 0.000000 0.000000e+00 3 6 2.500000 0.011045 0.015340 4.295146e-03 3 7 2.500000 0.000000 0.000000 0.000000e+00 3 8 2.500000 0.003313 0.007670 4.356505e-03 4 0 2.500000 0.981748 0.981748 0.000000e+00 4 1 2.500000 0.000000 0.000000 1.457168e-16 4 2 2.500000 0.122718 0.122718 6.938894e-17 4 3 2.500000 0.000000 0.000000 6.591949e-17 4 4 2.500000 0.036816 0.036816 3.469447e-17 4 5 2.500000 0.000000 0.000000 2.949030e-17 4 6 2.500000 0.015340 0.015340 1.734723e-18 4 7 2.500000 0.000000 0.000000 1.301043e-17 4 8 2.500000 0.006750 0.007670 9.203885e-04 4 9 2.500000 0.000000 0.000000 5.637851e-18 4 10 2.500000 0.002991 0.004314 1.323058e-03 5 0 2.500000 0.981748 0.981748 3.330669e-16 5 1 2.500000 -0.000000 0.000000 3.469447e-17 5 2 2.500000 0.122718 0.122718 1.526557e-16 5 3 2.500000 -0.000000 0.000000 6.938894e-18 5 4 2.500000 0.036816 0.036816 6.245005e-17 5 5 2.500000 -0.000000 0.000000 1.734723e-18 5 6 2.500000 0.015340 0.015340 1.387779e-17 5 7 2.500000 0.000000 0.000000 0.000000e+00 5 8 2.500000 0.007670 0.007670 1.040834e-17 5 9 2.500000 0.000000 0.000000 0.000000e+00 5 10 2.500000 0.004109 0.004314 2.054439e-04 5 11 2.500000 0.000000 0.000000 0.000000e+00 5 12 2.500000 0.002250 0.002637 3.864301e-04 6 0 2.500000 0.981748 0.981748 2.220446e-16 6 1 2.500000 0.000000 0.000000 1.838807e-16 6 2 2.500000 0.122718 0.122718 6.938894e-17 6 3 2.500000 0.000000 0.000000 4.510281e-17 6 4 2.500000 0.036816 0.036816 2.775558e-17 6 5 2.500000 0.000000 0.000000 1.214306e-17 6 6 2.500000 0.015340 0.015340 0.000000e+00 6 7 2.500000 0.000000 0.000000 3.469447e-18 6 8 2.500000 0.007670 0.007670 2.081668e-17 6 9 2.500000 0.000000 0.000000 8.673617e-19 6 10 2.500000 0.004314 0.004314 2.081668e-17 6 11 2.500000 0.000000 0.000000 2.168404e-19 6 12 2.500000 0.002589 0.002637 4.708088e-05 6 13 2.500000 0.000000 0.000000 0.000000e+00 6 14 2.500000 0.001604 0.001714 1.100516e-04 7 0 2.500000 0.981748 0.981748 1.110223e-16 7 1 2.500000 0.000000 0.000000 1.786765e-16 7 2 2.500000 0.122718 0.122718 2.081668e-16 7 3 2.500000 0.000000 0.000000 1.040834e-16 7 4 2.500000 0.036816 0.036816 1.318390e-16 7 5 2.500000 0.000000 0.000000 6.938894e-17 7 6 2.500000 0.015340 0.015340 6.591949e-17 7 7 2.500000 0.000000 0.000000 4.813858e-17 7 8 2.500000 0.007670 0.007670 2.775558e-17 7 9 2.500000 0.000000 0.000000 3.382711e-17 7 10 2.500000 0.004314 0.004314 1.301043e-17 7 11 2.500000 0.000000 0.000000 2.363561e-17 7 12 2.500000 0.002637 0.002637 1.691355e-17 7 13 2.500000 0.000000 0.000000 1.647987e-17 7 14 2.500000 0.001703 0.001714 1.098554e-05 7 15 2.500000 0.000000 0.000000 1.149254e-17 7 16 2.500000 0.001138 0.001168 3.079280e-05 8 0 2.500000 0.981748 0.981748 1.110223e-16 8 1 2.500000 0.000000 0.000000 2.602085e-17 8 2 2.500000 0.122718 0.122718 9.714451e-17 8 3 2.500000 0.000000 0.000000 4.336809e-19 8 4 2.500000 0.036816 0.036816 3.469447e-17 8 5 2.500000 0.000000 0.000000 2.168404e-18 8 6 2.500000 0.015340 0.015340 0.000000e+00 8 7 2.500000 0.000000 0.000000 2.168404e-19 8 8 2.500000 0.007670 0.007670 1.994932e-17 8 9 2.500000 0.000000 0.000000 0.000000e+00 8 10 2.500000 0.004314 0.004314 2.081668e-17 8 11 2.500000 0.000000 0.000000 0.000000e+00 8 12 2.500000 0.002637 0.002637 6.071532e-18 8 13 2.500000 0.000000 0.000000 0.000000e+00 8 14 2.500000 0.001714 0.001714 2.168404e-19 8 15 2.500000 0.000000 0.000000 0.000000e+00 8 16 2.500000 0.001166 0.001168 2.596582e-06 8 17 2.500000 0.000000 0.000000 0.000000e+00 8 18 2.500000 0.000819 0.000828 8.503806e-06 9 0 2.500000 0.981748 0.981748 1.110223e-16 9 1 2.500000 0.000000 0.000000 1.647987e-17 9 2 2.500000 0.122718 0.122718 2.775558e-17 9 3 2.500000 0.000000 0.000000 2.168404e-18 9 4 2.500000 0.036816 0.036816 0.000000e+00 9 5 2.500000 -0.000000 0.000000 6.505213e-19 9 6 2.500000 0.015340 0.015340 2.775558e-17 9 7 2.500000 0.000000 0.000000 4.336809e-19 9 8 2.500000 0.007670 0.007670 3.642919e-17 9 9 2.500000 0.000000 0.000000 0.000000e+00 9 10 2.500000 0.004314 0.004314 3.122502e-17 9 11 2.500000 -0.000000 0.000000 1.084202e-19 9 12 2.500000 0.002637 0.002637 1.214306e-17 9 13 2.500000 0.000000 0.000000 0.000000e+00 9 14 2.500000 0.001714 0.001714 4.119968e-18 9 15 2.500000 0.000000 0.000000 0.000000e+00 9 16 2.500000 0.001168 0.001168 2.168404e-19 9 17 2.500000 -0.000000 0.000000 5.421011e-20 9 18 2.500000 0.000827 0.000828 6.196389e-07 9 19 2.500000 0.000000 0.000000 0.000000e+00 9 20 2.500000 0.000603 0.000605 2.324729e-06 10 0 2.500000 0.981748 0.981748 4.440892e-16 10 1 2.500000 -0.000000 0.000000 1.327063e-16 10 2 2.500000 0.122718 0.122718 1.665335e-16 10 3 2.500000 -0.000000 0.000000 3.491131e-17 10 4 2.500000 0.036816 0.036816 6.938894e-17 10 5 2.500000 -0.000000 0.000000 1.214306e-17 10 6 2.500000 0.015340 0.015340 1.040834e-17 10 7 2.500000 -0.000000 0.000000 4.228388e-18 10 8 2.500000 0.007670 0.007670 1.474515e-17 10 9 2.500000 -0.000000 0.000000 1.409463e-18 10 10 2.500000 0.004314 0.004314 1.821460e-17 10 11 2.500000 -0.000000 0.000000 3.252607e-19 10 12 2.500000 0.002637 0.002637 6.071532e-18 10 13 2.500000 -0.000000 0.000000 1.626303e-19 10 14 2.500000 0.001714 0.001714 2.168404e-19 10 15 2.500000 -0.000000 0.000000 1.626303e-19 10 16 2.500000 0.001168 0.001168 3.035766e-18 10 17 2.500000 0.000000 0.000000 5.421011e-20 10 18 2.500000 0.000828 0.000828 4.011548e-18 10 19 2.500000 -0.000000 0.000000 2.710505e-20 10 20 2.500000 0.000605 0.000605 1.489517e-07 10 21 2.500000 0.000000 0.000000 2.710505e-20 10 22 2.500000 0.000453 0.000454 6.303847e-07 TEST10 GEGENBAUER_COMPUTE_NP computes a generalized Gauss-Gegenbauer rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) (1-x^2)^alpha dx. Order = 1 ALPHA = 0.500000 0 0.0000000000000000 1.5707963267948968 Order = 2 ALPHA = 0.500000 0 -0.5000000000000000 0.7853981633974484 1 0.5000000000000000 0.7853981633974484 Order = 3 ALPHA = 0.500000 0 -0.7071067811865475 0.3926990816987239 1 0.0000000000000000 0.7853981633974484 2 0.7071067811865476 0.3926990816987245 Order = 4 ALPHA = 0.500000 0 -0.8090169943749475 0.2170787134227060 1 -0.3090169943749475 0.5683194499747424 2 0.3090169943749474 0.5683194499747424 3 0.8090169943749475 0.2170787134227060 Order = 5 ALPHA = 0.500000 0 -0.8660254037844387 0.1308996938995740 1 -0.5000000000000000 0.3926990816987242 2 0.0000000000000000 0.5235987755982989 3 0.5000000000000000 0.3926990816987242 4 0.8660254037844387 0.1308996938995745 Order = 6 ALPHA = 0.500000 0 -0.9009688679024191 0.0844886908915884 1 -0.6234898018587335 0.2743330560697777 2 -0.2225209339563144 0.4265764164360819 3 0.2225209339563144 0.4265764164360819 4 0.6234898018587335 0.2743330560697777 5 0.9009688679024191 0.0844886908915884 Order = 7 ALPHA = 0.500000 0 -0.9238795325112867 0.0575094490319133 1 -0.7071067811865475 0.1963495408493619 2 -0.3826834323650898 0.3351896326668111 3 0.0000000000000000 0.3926990816987242 4 0.3826834323650898 0.3351896326668108 5 0.7071067811865476 0.1963495408493624 6 0.9238795325112867 0.0575094490319133 Order = 8 ALPHA = 0.500000 0 -0.9396926207859084 0.0408329477091069 1 -0.7660444431189780 0.1442256007956728 2 -0.5000000000000000 0.2617993877991495 3 -0.1736481776669303 0.3385402270935191 4 0.1736481776669303 0.3385402270935191 5 0.5000000000000000 0.2617993877991495 6 0.7660444431189780 0.1442256007956728 7 0.9396926207859084 0.0408329477091075 Order = 9 ALPHA = 0.500000 0 -0.9510565162951536 0.0299995403716084 1 -0.8090169943749475 0.1085393567113530 2 -0.5877852522924731 0.2056199086476264 3 -0.3090169943749475 0.2841597249873712 4 0.0000000000000000 0.3141592653589794 5 0.3090169943749475 0.2841597249873712 6 0.5877852522924731 0.2056199086476264 7 0.8090169943749475 0.1085393567113530 8 0.9510565162951536 0.0299995403716084 Order = 10 ALPHA = 0.500000 0 -0.9594929736144974 0.0226689425018590 1 -0.8412535328311812 0.0834785409341889 2 -0.6548607339452851 0.1631221774548165 3 -0.4154150130018864 0.2363135602034873 4 -0.1423148382732851 0.2798149423030965 5 0.1423148382732851 0.2798149423030966 6 0.4154150130018864 0.2363135602034873 7 0.6548607339452851 0.1631221774548165 8 0.8412535328311812 0.0834785409341889 9 0.9594929736144974 0.0226689425018590 Order = 1 ALPHA = 1.000000 0 0.0000000000000000 1.3333333333333333 Order = 2 ALPHA = 1.000000 0 -0.4472135954999579 0.6666666666666665 1 0.4472135954999580 0.6666666666666667 Order = 3 ALPHA = 1.000000 0 -0.6546536707079772 0.3111111111111113 1 0.0000000000000000 0.7111111111111110 2 0.6546536707079772 0.3111111111111113 Order = 4 ALPHA = 1.000000 0 -0.7650553239294646 0.1569499125956942 1 -0.2852315164806451 0.5097167540709727 2 0.2852315164806451 0.5097167540709727 3 0.7650553239294647 0.1569499125956939 Order = 5 ALPHA = 1.000000 0 -0.8302238962785669 0.0860176821228073 1 -0.4688487934707142 0.3368394607343354 2 -0.0000000000000000 0.4876190476190476 3 0.4688487934707142 0.3368394607343354 4 0.8302238962785670 0.0860176821228076 Order = 6 ALPHA = 1.000000 0 -0.8717401485096066 0.0505835770160819 1 -0.5917001814331423 0.2216925320225178 2 -0.2092992179024789 0.3943905576280671 3 0.2092992179024789 0.3943905576280671 4 0.5917001814331423 0.2216925320225178 5 0.8717401485096067 0.0505835770160817 Order = 7 ALPHA = 1.000000 0 -0.8997579954114602 0.0315162001504988 1 -0.6771862795107377 0.1486404045834101 2 -0.3631174638261782 0.3007504247445493 3 0.0000000000000000 0.3715192743764172 4 0.3631174638261782 0.3007504247445493 5 0.6771862795107377 0.1486404045834101 6 0.8997579954114602 0.0315162001504988 Order = 8 ALPHA = 1.000000 0 -0.9195339081664589 0.0205900956491219 1 -0.7387738651055050 0.1021477023603581 2 -0.4779249498104445 0.2253365549698578 3 -0.1652789576663870 0.3185923136873282 4 0.1652789576663870 0.3185923136873282 5 0.4779249498104445 0.2253365549698580 6 0.7387738651055050 0.1021477023603581 7 0.9195339081664589 0.0205900956491219 Order = 9 ALPHA = 1.000000 0 -0.9340014304080592 0.0139910561244267 1 -0.7844834736631444 0.0719828561756607 2 -0.5652353269962050 0.1687989736144601 3 -0.2957581355869394 0.2617849830242735 4 -0.0000000000000000 0.3002175954556907 5 0.2957581355869394 0.2617849830242735 6 0.5652353269962050 0.1687989736144601 7 0.7844834736631444 0.0719828561756607 8 0.9340014304080592 0.0139910561244267 Order = 10 ALPHA = 1.000000 0 -0.9448992722228823 0.0098254048057683 1 -0.8192793216440066 0.0519391437742078 2 -0.6328761530318606 0.1273919483477328 3 -0.3995309409653490 0.2111657418760753 4 -0.1365529328549276 0.2663444278628824 5 0.1365529328549276 0.2663444278628824 6 0.3995309409653490 0.2111657418760753 7 0.6328761530318606 0.1273919483477326 8 0.8192793216440066 0.0519391437742078 9 0.9448992722228822 0.0098254048057681 Order = 1 ALPHA = 2.500000 0 0.0000000000000000 0.9817477042468103 Order = 2 ALPHA = 2.500000 0 -0.3535533905932737 0.4908738521234051 1 0.3535533905932737 0.4908738521234051 Order = 3 ALPHA = 2.500000 0 -0.5477225575051661 0.2045307717180856 1 0.0000000000000000 0.5726861608106394 2 0.5477225575051661 0.2045307717180856 Order = 4 ALPHA = 2.500000 0 -0.6660699417556469 0.0870080715314833 1 -0.2373833033084449 0.4038657805919217 2 0.2373833033084449 0.4038657805919217 3 0.6660699417556469 0.0870080715314835 Order = 5 ALPHA = 2.500000 0 -0.7434770045212192 0.0392893891317672 1 -0.4019050360892101 0.2454174450998077 2 -0.0000000000000000 0.4123340357836603 3 0.4019050360892101 0.2454174450998076 4 0.7434770045212192 0.0392893891317672 Order = 6 ALPHA = 2.500000 0 -0.7968373566406807 0.0189116433666931 1 -0.5196148456949133 0.1431407318203965 2 -0.1804179569647766 0.3288214769363154 3 0.1804179569647766 0.3288214769363154 4 0.5196148456949133 0.1431407318203969 5 0.7968373566406807 0.0189116433666931 Order = 7 ALPHA = 2.500000 0 -0.8351564541339112 0.0096585250210008 1 -0.6063141605450342 0.0834446705489847 2 -0.3186902924591673 0.2357822853526957 3 -0.0000000000000000 0.3239767424014473 4 0.3186902924591672 0.2357822853526958 5 0.6063141605450342 0.0834446705489847 6 0.8351564541339112 0.0096585250210010 Order = 8 ALPHA = 2.500000 0 -0.8635912403827584 0.0051997074221325 1 -0.6718232507247468 0.0494181881565223 2 -0.4260961495798479 0.1615070338519726 3 -0.1459649294626305 0.2747489226927776 4 0.1459649294626305 0.2747489226927777 5 0.4260961495798479 0.1615070338519726 6 0.6718232507247468 0.0494181881565223 7 0.8635912403827584 0.0051997074221325 Order = 9 ALPHA = 2.500000 0 -0.8852662998295736 0.0029319009849924 1 -0.7224309555097356 0.0299209571936803 2 -0.5107538326640052 0.1088013928543096 3 -0.2643695362367075 0.2155149137501430 4 0.0000000000000000 0.2674093746805597 5 0.2643695362367075 0.2155149137501430 6 0.5107538326640052 0.1088013928543096 7 0.7224309555097356 0.0299209571936803 8 0.8852662998295736 0.0029319009849924 Order = 10 ALPHA = 2.500000 0 -0.9021636877146698 0.0017216382288932 1 -0.7622868838395971 0.0185564370059903 2 -0.5784393987065429 0.0731613283971410 3 -0.3610608688185057 0.1622660968500981 4 -0.1227285554678421 0.2351683516412825 5 0.1227285554678421 0.2351683516412824 6 0.3610608688185057 0.1622660968500981 7 0.5784393987065429 0.0731613283971409 8 0.7622868838395971 0.0185564370059903 9 0.9021636877146698 0.0017216382288932 TEST11 GEN_HERMITE_COMPUTE_NP computes a generalized Gauss-Hermite rule which is appropriate for integrands of the form Integral ( -oo < x < +oo ) f(x) x^alpha exp(-x*x) dx. GEN_HERMITE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Alpha Estimate Exact Error 1 0 0.500000 1.225417e+00 1.225417e+00 0.000000e+00 1 1 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 1 2 0.500000 0.000000e+00 9.190625e-01 9.190625e-01 1 3 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 1 4 0.500000 0.000000e+00 1.608359e+00 1.608359e+00 2 0 0.500000 1.225417e+00 1.225417e+00 0.000000e+00 2 1 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 2 2 0.500000 9.190625e-01 9.190625e-01 1.110223e-16 2 3 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 2 4 0.500000 6.892969e-01 1.608359e+00 9.190625e-01 2 5 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 2 6 0.500000 5.169727e-01 4.422988e+00 3.906016e+00 3 0 0.500000 1.225417e+00 1.225417e+00 0.000000e+00 3 1 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 3 2 0.500000 9.190625e-01 9.190625e-01 1.110223e-16 3 3 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 3 4 0.500000 1.608359e+00 1.608359e+00 2.220446e-16 3 5 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 3 6 0.500000 2.814629e+00 4.422988e+00 1.608359e+00 3 7 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 3 8 0.500000 4.925601e+00 1.658621e+01 1.166061e+01 4 0 0.500000 1.225417e+00 1.225417e+00 2.220446e-16 4 1 0.500000 -2.775558e-17 0.000000e+00 2.775558e-17 4 2 0.500000 9.190625e-01 9.190625e-01 2.220446e-16 4 3 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 4 4 0.500000 1.608359e+00 1.608359e+00 4.440892e-16 4 5 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 4 6 0.500000 4.422988e+00 4.422988e+00 8.881784e-16 4 7 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 4 8 0.500000 1.336949e+01 1.658621e+01 3.216719e+00 4 9 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 4 10 0.500000 4.098803e+01 7.878448e+01 3.779645e+01 5 0 0.500000 1.225417e+00 1.225417e+00 0.000000e+00 5 1 0.500000 -1.387779e-17 0.000000e+00 1.387779e-17 5 2 0.500000 9.190625e-01 9.190625e-01 0.000000e+00 5 3 0.500000 -5.551115e-17 0.000000e+00 5.551115e-17 5 4 0.500000 1.608359e+00 1.608359e+00 0.000000e+00 5 5 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 5 6 0.500000 4.422988e+00 4.422988e+00 8.881784e-16 5 7 0.500000 -4.440892e-16 0.000000e+00 4.440892e-16 5 8 0.500000 1.658621e+01 1.658621e+01 3.552714e-15 5 9 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 5 10 0.500000 6.993850e+01 7.878448e+01 8.845977e+00 5 11 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 5 12 0.500000 3.048407e+02 4.530108e+02 1.481701e+02 6 0 0.500000 1.225417e+00 1.225417e+00 6.661338e-16 6 1 0.500000 3.295975e-17 0.000000e+00 3.295975e-17 6 2 0.500000 9.190625e-01 9.190625e-01 3.330669e-16 6 3 0.500000 1.387779e-17 0.000000e+00 1.387779e-17 6 4 0.500000 1.608359e+00 1.608359e+00 0.000000e+00 6 5 0.500000 5.551115e-17 0.000000e+00 5.551115e-17 6 6 0.500000 4.422988e+00 4.422988e+00 0.000000e+00 6 7 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 6 8 0.500000 1.658621e+01 1.658621e+01 0.000000e+00 6 9 0.500000 -1.776357e-15 0.000000e+00 1.776357e-15 6 10 0.500000 7.878448e+01 7.878448e+01 1.421085e-14 6 11 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 6 12 0.500000 4.264728e+02 4.530108e+02 2.653793e+01 6 13 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 6 14 0.500000 2.440816e+03 3.057823e+03 6.170069e+02 7 0 0.500000 1.225417e+00 1.225417e+00 0.000000e+00 7 1 0.500000 5.204170e-18 0.000000e+00 5.204170e-18 7 2 0.500000 9.190625e-01 9.190625e-01 4.440892e-16 7 3 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 7 4 0.500000 1.608359e+00 1.608359e+00 6.661338e-16 7 5 0.500000 5.551115e-17 0.000000e+00 5.551115e-17 7 6 0.500000 4.422988e+00 4.422988e+00 8.881784e-16 7 7 0.500000 2.220446e-16 0.000000e+00 2.220446e-16 7 8 0.500000 1.658621e+01 1.658621e+01 7.105427e-15 7 9 0.500000 1.776357e-15 0.000000e+00 1.776357e-15 7 10 0.500000 7.878448e+01 7.878448e+01 5.684342e-14 7 11 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 7 12 0.500000 4.530108e+02 4.530108e+02 4.547474e-13 7 13 0.500000 -5.684342e-14 0.000000e+00 5.684342e-14 7 14 0.500000 2.958305e+03 3.057823e+03 9.951724e+01 7 15 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 7 16 0.500000 2.068773e+04 2.369813e+04 3.010396e+03 8 0 0.500000 1.225417e+00 1.225417e+00 2.220446e-16 8 1 0.500000 2.049142e-17 0.000000e+00 2.049142e-17 8 2 0.500000 9.190625e-01 9.190625e-01 1.110223e-16 8 3 0.500000 -2.081668e-17 0.000000e+00 2.081668e-17 8 4 0.500000 1.608359e+00 1.608359e+00 4.440892e-16 8 5 0.500000 -4.857226e-17 0.000000e+00 4.857226e-17 8 6 0.500000 4.422988e+00 4.422988e+00 8.881784e-16 8 7 0.500000 5.551115e-17 0.000000e+00 5.551115e-17 8 8 0.500000 1.658621e+01 1.658621e+01 0.000000e+00 8 9 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 8 10 0.500000 7.878448e+01 7.878448e+01 0.000000e+00 8 11 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 8 12 0.500000 4.530108e+02 4.530108e+02 0.000000e+00 8 13 0.500000 5.684342e-14 0.000000e+00 5.684342e-14 8 14 0.500000 3.057823e+03 3.057823e+03 0.000000e+00 8 15 0.500000 4.547474e-13 0.000000e+00 4.547474e-13 8 16 0.500000 2.330006e+04 2.369813e+04 3.980690e+02 8 17 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 8 18 0.500000 1.919334e+05 2.073586e+05 1.542517e+04 9 0 0.500000 1.225417e+00 1.225417e+00 0.000000e+00 9 1 0.500000 -1.862117e-17 0.000000e+00 1.862117e-17 9 2 0.500000 9.190625e-01 9.190625e-01 1.110223e-16 9 3 0.500000 7.264155e-17 0.000000e+00 7.264155e-17 9 4 0.500000 1.608359e+00 1.608359e+00 0.000000e+00 9 5 0.500000 3.122502e-17 0.000000e+00 3.122502e-17 9 6 0.500000 4.422988e+00 4.422988e+00 8.881784e-16 9 7 0.500000 -3.608225e-16 0.000000e+00 3.608225e-16 9 8 0.500000 1.658621e+01 1.658621e+01 3.552714e-15 9 9 0.500000 -1.554312e-15 0.000000e+00 1.554312e-15 9 10 0.500000 7.878448e+01 7.878448e+01 2.842171e-14 9 11 0.500000 -7.105427e-15 0.000000e+00 7.105427e-15 9 12 0.500000 4.530108e+02 4.530108e+02 1.136868e-13 9 13 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 9 14 0.500000 3.057823e+03 3.057823e+03 4.547474e-13 9 15 0.500000 4.547474e-13 0.000000e+00 4.547474e-13 9 16 0.500000 2.369813e+04 2.369813e+04 3.637979e-12 9 17 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 9 18 0.500000 2.054678e+05 2.073586e+05 1.890828e+03 9 19 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 9 20 0.500000 1.931459e+06 2.021746e+06 9.028702e+04 10 0 0.500000 1.225417e+00 1.225417e+00 2.220446e-16 10 1 0.500000 3.276323e-17 0.000000e+00 3.276323e-17 10 2 0.500000 9.190625e-01 9.190625e-01 4.440892e-16 10 3 0.500000 -2.916504e-17 0.000000e+00 2.916504e-17 10 4 0.500000 1.608359e+00 1.608359e+00 4.440892e-16 10 5 0.500000 2.168404e-17 0.000000e+00 2.168404e-17 10 6 0.500000 4.422988e+00 4.422988e+00 0.000000e+00 10 7 0.500000 -2.775558e-17 0.000000e+00 2.775558e-17 10 8 0.500000 1.658621e+01 1.658621e+01 3.552714e-15 10 9 0.500000 -2.220446e-16 0.000000e+00 2.220446e-16 10 10 0.500000 7.878448e+01 7.878448e+01 0.000000e+00 10 11 0.500000 -3.552714e-15 0.000000e+00 3.552714e-15 10 12 0.500000 4.530108e+02 4.530108e+02 0.000000e+00 10 13 0.500000 2.842171e-14 0.000000e+00 2.842171e-14 10 14 0.500000 3.057823e+03 3.057823e+03 1.364242e-12 10 15 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 10 16 0.500000 2.369813e+04 2.369813e+04 1.091394e-11 10 17 0.500000 -1.818989e-12 0.000000e+00 1.818989e-12 10 18 0.500000 2.073586e+05 2.073586e+05 1.455192e-10 10 19 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 10 20 0.500000 2.012292e+06 2.021746e+06 9.454138e+03 10 21 0.500000 0.000000e+00 0.000000e+00 0.000000e+00 10 22 0.500000 2.118307e+07 2.173377e+07 5.507035e+05 1 0 1.000000 1.000000e+00 1.000000e+00 0.000000e+00 1 1 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 1 2 1.000000 0.000000e+00 1.000000e+00 1.000000e+00 1 3 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 1 4 1.000000 0.000000e+00 2.000000e+00 2.000000e+00 2 0 1.000000 1.000000e+00 1.000000e+00 0.000000e+00 2 1 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 2 2 1.000000 1.000000e+00 1.000000e+00 0.000000e+00 2 3 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 2 4 1.000000 1.000000e+00 2.000000e+00 1.000000e+00 2 5 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 2 6 1.000000 1.000000e+00 6.000000e+00 5.000000e+00 3 0 1.000000 1.000000e+00 1.000000e+00 0.000000e+00 3 1 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 3 2 1.000000 1.000000e+00 1.000000e+00 2.220446e-16 3 3 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 3 4 1.000000 2.000000e+00 2.000000e+00 4.440892e-16 3 5 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 3 6 1.000000 4.000000e+00 6.000000e+00 2.000000e+00 3 7 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 3 8 1.000000 8.000000e+00 2.400000e+01 1.600000e+01 4 0 1.000000 1.000000e+00 1.000000e+00 0.000000e+00 4 1 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 4 2 1.000000 1.000000e+00 1.000000e+00 2.220446e-16 4 3 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 4 4 1.000000 2.000000e+00 2.000000e+00 4.440892e-16 4 5 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 4 6 1.000000 6.000000e+00 6.000000e+00 8.881784e-16 4 7 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 4 8 1.000000 2.000000e+01 2.400000e+01 4.000000e+00 4 9 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 4 10 1.000000 6.800000e+01 1.200000e+02 5.200000e+01 5 0 1.000000 1.000000e+00 1.000000e+00 1.110223e-16 5 1 1.000000 6.938894e-18 0.000000e+00 6.938894e-18 5 2 1.000000 1.000000e+00 1.000000e+00 3.330669e-16 5 3 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 5 4 1.000000 2.000000e+00 2.000000e+00 2.220446e-16 5 5 1.000000 -2.220446e-16 0.000000e+00 2.220446e-16 5 6 1.000000 6.000000e+00 6.000000e+00 0.000000e+00 5 7 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 5 8 1.000000 2.400000e+01 2.400000e+01 3.552714e-15 5 9 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 5 10 1.000000 1.080000e+02 1.200000e+02 1.200000e+01 5 11 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 5 12 1.000000 5.040000e+02 7.200000e+02 2.160000e+02 6 0 1.000000 1.000000e+00 1.000000e+00 4.440892e-16 6 1 1.000000 -3.469447e-18 0.000000e+00 3.469447e-18 6 2 1.000000 1.000000e+00 1.000000e+00 2.220446e-16 6 3 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 6 4 1.000000 2.000000e+00 2.000000e+00 0.000000e+00 6 5 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 6 6 1.000000 6.000000e+00 6.000000e+00 0.000000e+00 6 7 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 6 8 1.000000 2.400000e+01 2.400000e+01 3.552714e-15 6 9 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 6 10 1.000000 1.200000e+02 1.200000e+02 0.000000e+00 6 11 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 6 12 1.000000 6.840000e+02 7.200000e+02 3.600000e+01 6 13 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 6 14 1.000000 4.140000e+03 5.040000e+03 9.000000e+02 7 0 1.000000 1.000000e+00 1.000000e+00 0.000000e+00 7 1 1.000000 2.602085e-18 0.000000e+00 2.602085e-18 7 2 1.000000 1.000000e+00 1.000000e+00 0.000000e+00 7 3 1.000000 4.510281e-17 0.000000e+00 4.510281e-17 7 4 1.000000 2.000000e+00 2.000000e+00 0.000000e+00 7 5 1.000000 5.551115e-17 0.000000e+00 5.551115e-17 7 6 1.000000 6.000000e+00 6.000000e+00 0.000000e+00 7 7 1.000000 4.440892e-16 0.000000e+00 4.440892e-16 7 8 1.000000 2.400000e+01 2.400000e+01 7.105427e-15 7 9 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 7 10 1.000000 1.200000e+02 1.200000e+02 8.526513e-14 7 11 1.000000 -1.421085e-14 0.000000e+00 1.421085e-14 7 12 1.000000 7.200000e+02 7.200000e+02 7.958079e-13 7 13 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 7 14 1.000000 4.896000e+03 5.040000e+03 1.440000e+02 7 15 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 7 16 1.000000 3.571200e+04 4.032000e+04 4.608000e+03 8 0 1.000000 1.000000e+00 1.000000e+00 8.881784e-16 8 1 1.000000 5.009014e-17 0.000000e+00 5.009014e-17 8 2 1.000000 1.000000e+00 1.000000e+00 4.440892e-16 8 3 1.000000 -1.474515e-17 0.000000e+00 1.474515e-17 8 4 1.000000 2.000000e+00 2.000000e+00 4.440892e-16 8 5 1.000000 6.938894e-17 0.000000e+00 6.938894e-17 8 6 1.000000 6.000000e+00 6.000000e+00 3.552714e-15 8 7 1.000000 -3.330669e-16 0.000000e+00 3.330669e-16 8 8 1.000000 2.400000e+01 2.400000e+01 1.065814e-14 8 9 1.000000 1.776357e-15 0.000000e+00 1.776357e-15 8 10 1.000000 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1.000000 1.200000e+02 1.200000e+02 0.000000e+00 9 11 1.000000 1.421085e-14 0.000000e+00 1.421085e-14 9 12 1.000000 7.200000e+02 7.200000e+02 1.136868e-13 9 13 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 9 14 1.000000 5.040000e+03 5.040000e+03 0.000000e+00 9 15 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 9 16 1.000000 4.032000e+04 4.032000e+04 7.275958e-12 9 17 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 9 18 1.000000 3.600000e+05 3.628800e+05 2.880000e+03 9 19 1.000000 5.820766e-11 0.000000e+00 5.820766e-11 9 20 1.000000 3.484800e+06 3.628800e+06 1.440000e+05 10 0 1.000000 1.000000e+00 1.000000e+00 7.771561e-16 10 1 1.000000 5.192651e-17 0.000000e+00 5.192651e-17 10 2 1.000000 1.000000e+00 1.000000e+00 3.330669e-16 10 3 1.000000 -1.561251e-17 0.000000e+00 1.561251e-17 10 4 1.000000 2.000000e+00 2.000000e+00 2.220446e-16 10 5 1.000000 3.469447e-18 0.000000e+00 3.469447e-18 10 6 1.000000 6.000000e+00 6.000000e+00 1.776357e-15 10 7 1.000000 1.249001e-16 0.000000e+00 1.249001e-16 10 8 1.000000 2.400000e+01 2.400000e+01 1.065814e-14 10 9 1.000000 8.881784e-16 0.000000e+00 8.881784e-16 10 10 1.000000 1.200000e+02 1.200000e+02 7.105427e-14 10 11 1.000000 -3.552714e-15 0.000000e+00 3.552714e-15 10 12 1.000000 7.200000e+02 7.200000e+02 4.547474e-13 10 13 1.000000 -2.842171e-14 0.000000e+00 2.842171e-14 10 14 1.000000 5.040000e+03 5.040000e+03 1.818989e-12 10 15 1.000000 -4.547474e-13 0.000000e+00 4.547474e-13 10 16 1.000000 4.032000e+04 4.032000e+04 2.910383e-11 10 17 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 10 18 1.000000 3.628800e+05 3.628800e+05 1.164153e-10 10 19 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 10 20 1.000000 3.614400e+06 3.628800e+06 1.440000e+04 10 21 1.000000 0.000000e+00 0.000000e+00 0.000000e+00 10 22 1.000000 3.903840e+07 3.991680e+07 8.784000e+05 1 0 2.500000 9.190625e-01 9.190625e-01 0.000000e+00 1 1 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 1 2 2.500000 0.000000e+00 1.608359e+00 1.608359e+00 1 3 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 1 4 2.500000 0.000000e+00 4.422988e+00 4.422988e+00 2 0 2.500000 9.190625e-01 9.190625e-01 0.000000e+00 2 1 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 2 2 2.500000 1.608359e+00 1.608359e+00 2.220446e-16 2 3 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 2 4 2.500000 2.814629e+00 4.422988e+00 1.608359e+00 2 5 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 2 6 2.500000 4.925601e+00 1.658621e+01 1.166061e+01 3 0 2.500000 9.190625e-01 9.190625e-01 0.000000e+00 3 1 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 3 2 2.500000 1.608359e+00 1.608359e+00 0.000000e+00 3 3 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 3 4 2.500000 4.422988e+00 4.422988e+00 0.000000e+00 3 5 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 3 6 2.500000 1.216322e+01 1.658621e+01 4.422988e+00 3 7 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 3 8 2.500000 3.344885e+01 7.878448e+01 4.533563e+01 4 0 2.500000 9.190625e-01 9.190625e-01 1.110223e-16 4 1 2.500000 -5.551115e-17 0.000000e+00 5.551115e-17 4 2 2.500000 1.608359e+00 1.608359e+00 2.220446e-16 4 3 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 4 4 2.500000 4.422988e+00 4.422988e+00 8.881784e-16 4 5 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 4 6 2.500000 1.658621e+01 1.658621e+01 3.552714e-15 4 7 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 4 8 2.500000 6.993850e+01 7.878448e+01 8.845977e+00 4 9 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 4 10 2.500000 3.048407e+02 4.530108e+02 1.481701e+02 5 0 2.500000 9.190625e-01 9.190625e-01 1.110223e-16 5 1 2.500000 4.163336e-17 0.000000e+00 4.163336e-17 5 2 2.500000 1.608359e+00 1.608359e+00 2.220446e-16 5 3 2.500000 1.110223e-16 0.000000e+00 1.110223e-16 5 4 2.500000 4.422988e+00 4.422988e+00 0.000000e+00 5 5 2.500000 -4.440892e-16 0.000000e+00 4.440892e-16 5 6 2.500000 1.658621e+01 1.658621e+01 0.000000e+00 5 7 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 5 8 2.500000 7.878448e+01 7.878448e+01 1.421085e-14 5 9 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 5 10 2.500000 4.198384e+02 4.530108e+02 3.317241e+01 5 11 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 5 12 2.500000 2.336323e+03 3.057823e+03 7.215000e+02 6 0 2.500000 9.190625e-01 9.190625e-01 4.440892e-16 6 1 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 6 2 2.500000 1.608359e+00 1.608359e+00 6.661338e-16 6 3 2.500000 -8.326673e-17 0.000000e+00 8.326673e-17 6 4 2.500000 4.422988e+00 4.422988e+00 1.776357e-15 6 5 2.500000 2.220446e-16 0.000000e+00 2.220446e-16 6 6 2.500000 1.658621e+01 1.658621e+01 7.105427e-15 6 7 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 6 8 2.500000 7.878448e+01 7.878448e+01 5.684342e-14 6 9 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 6 10 2.500000 4.530108e+02 4.530108e+02 4.547474e-13 6 11 2.500000 5.684342e-14 0.000000e+00 5.684342e-14 6 12 2.500000 2.958305e+03 3.057823e+03 9.951724e+01 6 13 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 6 14 2.500000 2.068773e+04 2.369813e+04 3.010396e+03 7 0 2.500000 9.190625e-01 9.190625e-01 1.110223e-16 7 1 2.500000 -1.994932e-17 0.000000e+00 1.994932e-17 7 2 2.500000 1.608359e+00 1.608359e+00 6.661338e-16 7 3 2.500000 -9.714451e-17 0.000000e+00 9.714451e-17 7 4 2.500000 4.422988e+00 4.422988e+00 8.881784e-16 7 5 2.500000 1.110223e-16 0.000000e+00 1.110223e-16 7 6 2.500000 1.658621e+01 1.658621e+01 3.552714e-15 7 7 2.500000 -8.881784e-16 0.000000e+00 8.881784e-16 7 8 2.500000 7.878448e+01 7.878448e+01 2.842171e-14 7 9 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 7 10 2.500000 4.530108e+02 4.530108e+02 2.273737e-13 7 11 2.500000 5.684342e-14 0.000000e+00 5.684342e-14 7 12 2.500000 3.057823e+03 3.057823e+03 2.273737e-12 7 13 2.500000 4.547474e-13 0.000000e+00 4.547474e-13 7 14 2.500000 2.322542e+04 2.369813e+04 4.727069e+02 7 15 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 7 16 2.500000 1.897503e+05 2.073586e+05 1.760833e+04 8 0 2.500000 9.190625e-01 9.190625e-01 1.110223e-16 8 1 2.500000 5.876376e-17 0.000000e+00 5.876376e-17 8 2 2.500000 1.608359e+00 1.608359e+00 2.220446e-16 8 3 2.500000 -2.428613e-17 0.000000e+00 2.428613e-17 8 4 2.500000 4.422988e+00 4.422988e+00 8.881784e-16 8 5 2.500000 5.551115e-17 0.000000e+00 5.551115e-17 8 6 2.500000 1.658621e+01 1.658621e+01 0.000000e+00 8 7 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 8 8 2.500000 7.878448e+01 7.878448e+01 1.421085e-14 8 9 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 8 10 2.500000 4.530108e+02 4.530108e+02 5.684342e-14 8 11 2.500000 -5.684342e-14 0.000000e+00 5.684342e-14 8 12 2.500000 3.057823e+03 3.057823e+03 4.547474e-13 8 13 2.500000 -4.547474e-13 0.000000e+00 4.547474e-13 8 14 2.500000 2.369813e+04 2.369813e+04 0.000000e+00 8 15 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 8 16 2.500000 2.054678e+05 2.073586e+05 1.890828e+03 8 17 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 8 18 2.500000 1.931459e+06 2.021746e+06 9.028702e+04 9 0 2.500000 9.190625e-01 9.190625e-01 1.110223e-16 9 1 2.500000 -3.778445e-17 0.000000e+00 3.778445e-17 9 2 2.500000 1.608359e+00 1.608359e+00 4.440892e-16 9 3 2.500000 -6.071532e-18 0.000000e+00 6.071532e-18 9 4 2.500000 4.422988e+00 4.422988e+00 0.000000e+00 9 5 2.500000 2.636780e-16 0.000000e+00 2.636780e-16 9 6 2.500000 1.658621e+01 1.658621e+01 3.552714e-15 9 7 2.500000 -2.553513e-15 0.000000e+00 2.553513e-15 9 8 2.500000 7.878448e+01 7.878448e+01 2.842171e-14 9 9 2.500000 7.105427e-15 0.000000e+00 7.105427e-15 9 10 2.500000 4.530108e+02 4.530108e+02 1.136868e-13 9 11 2.500000 -5.684342e-14 0.000000e+00 5.684342e-14 9 12 2.500000 3.057823e+03 3.057823e+03 9.094947e-13 9 13 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 9 14 2.500000 2.369813e+04 2.369813e+04 3.637979e-12 9 15 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 9 16 2.500000 2.073586e+05 2.073586e+05 0.000000e+00 9 17 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 9 18 2.500000 2.010874e+06 2.021746e+06 1.087226e+04 9 19 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 9 20 2.500000 2.111677e+07 2.173377e+07 6.170007e+05 10 0 2.500000 9.190625e-01 9.190625e-01 5.551115e-16 10 1 2.500000 1.299687e-17 0.000000e+00 1.299687e-17 10 2 2.500000 1.608359e+00 1.608359e+00 2.220446e-16 10 3 2.500000 1.426810e-16 0.000000e+00 1.426810e-16 10 4 2.500000 4.422988e+00 4.422988e+00 8.881784e-16 10 5 2.500000 -4.163336e-16 0.000000e+00 4.163336e-16 10 6 2.500000 1.658621e+01 1.658621e+01 7.105427e-15 10 7 2.500000 -4.440892e-16 0.000000e+00 4.440892e-16 10 8 2.500000 7.878448e+01 7.878448e+01 4.263256e-14 10 9 2.500000 5.329071e-15 0.000000e+00 5.329071e-15 10 10 2.500000 4.530108e+02 4.530108e+02 1.136868e-13 10 11 2.500000 7.105427e-14 0.000000e+00 7.105427e-14 10 12 2.500000 3.057823e+03 3.057823e+03 4.547474e-13 10 13 2.500000 4.547474e-13 0.000000e+00 4.547474e-13 10 14 2.500000 2.369813e+04 2.369813e+04 0.000000e+00 10 15 2.500000 1.818989e-12 0.000000e+00 1.818989e-12 10 16 2.500000 2.073586e+05 2.073586e+05 2.910383e-11 10 17 2.500000 -1.455192e-11 0.000000e+00 1.455192e-11 10 18 2.500000 2.021746e+06 2.021746e+06 0.000000e+00 10 19 2.500000 0.000000e+00 0.000000e+00 0.000000e+00 10 20 2.500000 2.167941e+07 2.173377e+07 5.436129e+04 10 21 2.500000 3.725290e-09 0.000000e+00 3.725290e-09 10 22 2.500000 2.516073e+08 2.553718e+08 3.764519e+06 TEST12 GEN_HERMITE_COMPUTE_NP computes a generalized Gauss-Hermite rule which is appropriate for integrands of the form Integral ( -oo < x < +oo ) f(x) x^alpha exp(-x*x) dx. Order = 1 ALPHA = 0.500000 0 0.0000000000000000 1.2254167024651776 Order = 2 ALPHA = 0.500000 0 -0.8660254037844386 0.6127083512325888 1 0.8660254037844386 0.6127083512325888 Order = 3 ALPHA = 0.500000 0 -1.3228756555322954 0.2625892933853952 1 0.0000000000000000 0.7002381156943872 2 1.3228756555322954 0.2625892933853952 Order = 4 ALPHA = 0.500000 0 -1.7529619663678659 0.0747721865343165 1 -0.6535475074298002 0.5379361646982722 2 0.6535475074298002 0.5379361646982722 3 1.7529619663678659 0.0747721865343165 Order = 5 ALPHA = 0.500000 0 -2.0995981508797583 0.0206908527402406 1 -1.0448385544294869 0.3373854564216621 2 0.0000000000000000 0.5092640841413724 3 1.0448385544294869 0.3373854564216621 4 2.0995981508797583 0.0206908527402406 Order = 6 ALPHA = 0.500000 0 -2.4311960068148712 0.0047584322858768 1 -1.4282643308502343 0.1432946705182554 2 -0.5471261076464519 0.4646552484284570 3 0.5471261076464519 0.4646552484284570 4 1.4282643308502343 0.1432946705182554 5 2.4311960068148712 0.0047584322858768 Order = 7 ALPHA = 0.500000 0 -2.7198800885562915 0.0011062894019685 1 -1.7473607788965217 0.0556473312506609 2 -0.8938582730216025 0.3522490969234108 3 0.0000000000000000 0.4074112673130975 4 0.8938582730216025 0.3522490969234108 5 1.7473607788965217 0.0556473312506609 6 2.7198800885562915 0.0011062894019685 Order = 8 ALPHA = 0.500000 0 -2.9990789683433161 0.0002288084584739 1 -2.0574394184774678 0.0178757746392672 2 -1.2417383409431890 0.1866121206001918 3 -0.4801606747408061 0.4079916475346560 4 0.4801606747408061 0.4079916475346560 5 1.2417383409431890 0.1866121206001918 6 2.0574394184774678 0.0178757746392672 7 2.9990789683433161 0.0002288084584739 Order = 9 ALPHA = 0.500000 0 -3.2511523261341324 0.0000482442834952 1 -2.3313221193007134 0.0055757541036437 2 -1.5374164086847444 0.0887579748998606 3 -0.7945417010067842 0.3467847917084955 4 0.0000000000000000 0.3430831724741876 5 0.7945417010067842 0.3467847917084955 6 1.5374164086847444 0.0887579748998606 7 2.3313221193007134 0.0055757541036437 8 3.2511523261341324 0.0000482442834952 Order = 10 ALPHA = 0.500000 0 -3.4966058807476776 0.0000093473340834 1 -2.5983971495446241 0.0015363564424026 2 -1.8279918123652743 0.0351763431437458 3 -1.1149053705666438 0.2117439807373521 4 -0.4330259998733385 0.3642423235750050 5 0.4330259998733385 0.3642423235750050 6 1.1149053705666438 0.2117439807373521 7 1.8279918123652743 0.0351763431437458 8 2.5983971495446241 0.0015363564424026 9 3.4966058807476776 0.0000093473340834 Order = 1 ALPHA = 1.000000 0 0.0000000000000000 1.0000000000000000 Order = 2 ALPHA = 1.000000 0 -1.0000000000000000 0.5000000000000000 1 1.0000000000000000 0.5000000000000000 Order = 3 ALPHA = 1.000000 0 -1.4142135623730951 0.2500000000000000 1 0.0000000000000000 0.5000000000000000 2 1.4142135623730951 0.2500000000000000 Order = 4 ALPHA = 1.000000 0 -1.8477590650225735 0.0732233047033631 1 -0.7653668647301796 0.4267766952966369 2 0.7653668647301796 0.4267766952966369 3 1.8477590650225735 0.0732233047033631 Order = 5 ALPHA = 1.000000 0 -2.1753277471610750 0.0223290993692602 1 -1.1260325006104943 0.3110042339640730 2 0.0000000000000000 0.3333333333333336 3 1.1260325006104943 0.3110042339640730 4 2.1753277471610750 0.0223290993692602 Order = 6 ALPHA = 1.000000 0 -2.5079762923395985 0.0051946282507931 1 -1.5146882056314566 0.1392588667846204 2 -0.6448058287449634 0.3555465049645868 3 0.6448058287449634 0.3555465049645868 4 1.5146882056314566 0.1392588667846204 5 2.5079762923395985 0.0051946282507931 Order = 7 ALPHA = 1.000000 0 -2.7854569612800755 0.0012954668385735 1 -1.8180779106881748 0.0591781927275502 2 -0.9673790505919010 0.3145263404338763 3 0.0000000000000000 0.2499999999999999 4 0.9673790505919010 0.3145263404338763 5 1.8180779106881748 0.0591781927275502 6 2.7854569612800755 0.0012954668385735 Order = 8 ALPHA = 1.000000 0 -3.0651379923750794 0.0002696473527807 1 -2.1299343409882683 0.0194439542575027 2 -1.3212725309936428 0.1787093462188998 3 -0.5679328213965030 0.3015770521708174 4 0.5679328213965030 0.3015770521708174 5 1.3212725309936428 0.1787093462188998 6 2.1299343409882683 0.0194439542575027 7 3.0651379923750794 0.0002696473527807 Order = 9 ALPHA = 1.000000 0 -3.3096667978337626 0.0000600630994212 1 -2.3939880433471465 0.0064714248102269 2 -1.6036318179826310 0.0928661670384225 3 -0.8621437977399313 0.3006023450519293 4 0.0000000000000000 0.2000000000000004 5 0.8621437977399313 0.3006023450519293 6 1.6036318179826310 0.0928661670384225 7 2.3939880433471465 0.0064714248102269 8 3.3096667978337626 0.0000600630994212 Order = 10 ALPHA = 1.000000 0 -3.5553903926679813 0.0000116849861929 1 -2.6619184821964095 0.0018058793399610 2 -1.8964244701650319 0.0379712248408538 3 -1.1888662915174761 0.1993334055415880 4 -0.5133812615572767 0.2608778052914040 5 0.5133812615572767 0.2608778052914040 6 1.1888662915174761 0.1993334055415880 7 1.8964244701650319 0.0379712248408538 8 2.6619184821964095 0.0018058793399610 9 3.5553903926679813 0.0000116849861929 Order = 1 ALPHA = 2.500000 0 0.0000000000000000 0.9190625268488832 Order = 2 ALPHA = 2.500000 0 -1.3228756555322954 0.4595312634244416 1 1.3228756555322954 0.4595312634244416 Order = 3 ALPHA = 2.500000 0 -1.6583123951776999 0.2924289858155538 1 0.0000000000000000 0.3342045552177757 2 1.6583123951776999 0.2924289858155538 Order = 4 ALPHA = 2.500000 0 -2.0995981508797583 0.0912117426015989 1 -1.0448385544294869 0.3683195208228427 2 1.0448385544294869 0.3683195208228427 3 2.0995981508797583 0.0912117426015989 Order = 5 ALPHA = 2.500000 0 -2.3846365914125589 0.0341953454149221 1 -1.3466656329231439 0.3362147032847793 2 0.0000000000000000 0.1782424294494804 3 1.3466656329231439 0.3362147032847793 4 2.3846365914125589 0.0341953454149221 Order = 6 ALPHA = 2.500000 0 -2.7198800885562915 0.0081840498746597 1 -1.7473607788965217 0.1699063099275073 2 -0.8938582730216025 0.2814409036222749 3 0.8938582730216025 0.2814409036222749 4 1.7473607788965217 0.1699063099275073 5 2.7198800885562915 0.0081840498746597 Order = 7 ALPHA = 2.500000 0 -2.9702203625084231 0.0024098131356743 1 -2.0150214027494173 0.0871896206077823 2 -1.1693928957378275 0.3136447466969384 3 0.0000000000000000 0.1125741659680933 4 1.1693928957378275 0.3136447466969384 5 2.0150214027494173 0.0871896206077823 6 2.9702203625084231 0.0024098131356743 Order = 8 ALPHA = 2.500000 0 -3.2511523261341324 0.0005099416639457 1 -2.3313221193007134 0.0303045738441493 2 -1.5374164086847444 0.2097927175810472 3 -0.7945417010067842 0.2189240303352995 4 0.7945417010067842 0.2189240303352995 5 1.5374164086847444 0.2097927175810472 6 2.3313221193007134 0.0303045738441493 7 3.2511523261341324 0.0005099416639457 Order = 9 ALPHA = 2.500000 0 -3.4765341365933926 0.0001303236536236 1 -2.5712913745972612 0.0117476928703869 2 -1.7891453520001153 0.1310178979856828 3 -1.0493474035985140 0.2774791172736727 4 0.0000000000000000 0.0783124632821511 5 1.0493474035985140 0.2774791172736727 6 1.7891453520001153 0.1310178979856828 7 2.5712913745972612 0.0117476928703869 8 3.4765341365933926 0.0001303236536236 Order = 10 ALPHA = 2.500000 0 -3.7229793194118348 0.0000256325065578 1 -2.8414549096381942 0.0033881904305600 2 -2.0875400348754183 0.0579439549110723 3 -1.3912394012659237 0.2236074413923168 4 -0.7226261238582256 0.1745660441839349 5 0.7226261238582256 0.1745660441839349 6 1.3912394012659237 0.2236074413923168 7 2.0875400348754183 0.0579439549110723 8 2.8414549096381942 0.0033881904305600 9 3.7229793194118348 0.0000256325065578 TEST13 GEN_LAGUERRE_COMPUTE_NP computes a generalized Gauss-Laguerre rule which is appropriate for integrands of the form Integral ( 0 <= x < +oo ) f(x) x^alpha exp(-x) dx. GEN_LAGUERRE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Alpha Estimate Exact Error 1 0 0.500000 8.862269e-01 8.862269e-01 0.000000e+00 1 1 0.500000 1.329340e+00 1.329340e+00 0.000000e+00 1 2 0.500000 1.994011e+00 3.323351e+00 1.329340e+00 1 3 0.500000 2.991016e+00 1.163173e+01 8.640713e+00 1 4 0.500000 4.486524e+00 5.234278e+01 4.785625e+01 2 0 0.500000 8.862269e-01 8.862269e-01 1.110223e-16 2 1 0.500000 1.329340e+00 1.329340e+00 0.000000e+00 2 2 0.500000 3.323351e+00 3.323351e+00 0.000000e+00 2 3 0.500000 1.163173e+01 1.163173e+01 0.000000e+00 2 4 0.500000 4.569608e+01 5.234278e+01 6.646702e+00 2 5 0.500000 1.848614e+02 2.878853e+02 1.030239e+02 2 6 0.500000 7.529467e+02 1.871254e+03 1.118308e+03 3 0 0.500000 8.862269e-01 8.862269e-01 0.000000e+00 3 1 0.500000 1.329340e+00 1.329340e+00 2.220446e-16 3 2 0.500000 3.323351e+00 3.323351e+00 4.440892e-16 3 3 0.500000 1.163173e+01 1.163173e+01 3.552714e-15 3 4 0.500000 5.234278e+01 5.234278e+01 2.131628e-14 3 5 0.500000 2.878853e+02 2.878853e+02 1.136868e-13 3 6 0.500000 1.801464e+03 1.871254e+03 6.979037e+01 3 7 0.500000 1.204538e+04 1.403441e+04 1.989026e+03 3 8 0.500000 8.296657e+04 1.192925e+05 3.632589e+04 4 0 0.500000 8.862269e-01 8.862269e-01 0.000000e+00 4 1 0.500000 1.329340e+00 1.329340e+00 0.000000e+00 4 2 0.500000 3.323351e+00 3.323351e+00 4.440892e-16 4 3 0.500000 1.163173e+01 1.163173e+01 0.000000e+00 4 4 0.500000 5.234278e+01 5.234278e+01 7.105427e-15 4 5 0.500000 2.878853e+02 2.878853e+02 5.684342e-14 4 6 0.500000 1.871254e+03 1.871254e+03 0.000000e+00 4 7 0.500000 1.403441e+04 1.403441e+04 1.818989e-12 4 8 0.500000 1.180362e+05 1.192925e+05 1.256227e+03 4 9 0.500000 1.076120e+06 1.133278e+06 5.715831e+04 4 10 0.500000 1.031564e+07 1.189942e+07 1.583788e+06 5 0 0.500000 8.862269e-01 8.862269e-01 9.992007e-16 5 1 0.500000 1.329340e+00 1.329340e+00 4.440892e-16 5 2 0.500000 3.323351e+00 3.323351e+00 0.000000e+00 5 3 0.500000 1.163173e+01 1.163173e+01 0.000000e+00 5 4 0.500000 5.234278e+01 5.234278e+01 7.105427e-15 5 5 0.500000 2.878853e+02 2.878853e+02 5.684342e-14 5 6 0.500000 1.871254e+03 1.871254e+03 6.821210e-13 5 7 0.500000 1.403441e+04 1.403441e+04 7.275958e-12 5 8 0.500000 1.192925e+05 1.192925e+05 8.731149e-11 5 9 0.500000 1.133278e+06 1.133278e+06 1.164153e-09 5 10 0.500000 1.186488e+07 1.189942e+07 3.454623e+04 5 11 0.500000 1.345460e+08 1.368434e+08 2.297325e+06 5 12 0.500000 1.620549e+09 1.710542e+09 8.999294e+07 6 0 0.500000 8.862269e-01 8.862269e-01 4.440892e-16 6 1 0.500000 1.329340e+00 1.329340e+00 0.000000e+00 6 2 0.500000 3.323351e+00 3.323351e+00 0.000000e+00 6 3 0.500000 1.163173e+01 1.163173e+01 1.776357e-15 6 4 0.500000 5.234278e+01 5.234278e+01 0.000000e+00 6 5 0.500000 2.878853e+02 2.878853e+02 0.000000e+00 6 6 0.500000 1.871254e+03 1.871254e+03 2.273737e-13 6 7 0.500000 1.403441e+04 1.403441e+04 3.637979e-12 6 8 0.500000 1.192925e+05 1.192925e+05 5.820766e-11 6 9 0.500000 1.133278e+06 1.133278e+06 6.984919e-10 6 10 0.500000 1.189942e+07 1.189942e+07 9.313226e-09 6 11 0.500000 1.368434e+08 1.368434e+08 2.384186e-07 6 12 0.500000 1.709195e+09 1.710542e+09 1.347303e+06 6 13 0.500000 2.296904e+10 2.309232e+10 1.232782e+08 6 14 0.500000 3.283483e+11 3.348386e+11 6.490296e+09 7 0 0.500000 8.862269e-01 8.862269e-01 1.110223e-16 7 1 0.500000 1.329340e+00 1.329340e+00 8.881784e-16 7 2 0.500000 3.323351e+00 3.323351e+00 8.881784e-16 7 3 0.500000 1.163173e+01 1.163173e+01 0.000000e+00 7 4 0.500000 5.234278e+01 5.234278e+01 7.105427e-15 7 5 0.500000 2.878853e+02 2.878853e+02 0.000000e+00 7 6 0.500000 1.871254e+03 1.871254e+03 4.547474e-13 7 7 0.500000 1.403441e+04 1.403441e+04 1.818989e-12 7 8 0.500000 1.192925e+05 1.192925e+05 0.000000e+00 7 9 0.500000 1.133278e+06 1.133278e+06 0.000000e+00 7 10 0.500000 1.189942e+07 1.189942e+07 0.000000e+00 7 11 0.500000 1.368434e+08 1.368434e+08 2.980232e-07 7 12 0.500000 1.710542e+09 1.710542e+09 1.668930e-06 7 13 0.500000 2.309232e+10 2.309232e+10 9.918213e-05 7 14 0.500000 3.347679e+11 3.348386e+11 7.073341e+07 7 15 0.500000 5.181475e+12 5.189998e+12 8.523376e+09 7 16 0.500000 8.505429e+13 8.563497e+13 5.806860e+11 8 0 0.500000 8.862269e-01 8.862269e-01 2.220446e-16 8 1 0.500000 1.329340e+00 1.329340e+00 4.440892e-16 8 2 0.500000 3.323351e+00 3.323351e+00 8.881784e-16 8 3 0.500000 1.163173e+01 1.163173e+01 0.000000e+00 8 4 0.500000 5.234278e+01 5.234278e+01 7.105427e-15 8 5 0.500000 2.878853e+02 2.878853e+02 0.000000e+00 8 6 0.500000 1.871254e+03 1.871254e+03 2.273737e-13 8 7 0.500000 1.403441e+04 1.403441e+04 3.637979e-12 8 8 0.500000 1.192925e+05 1.192925e+05 2.910383e-11 8 9 0.500000 1.133278e+06 1.133278e+06 0.000000e+00 8 10 0.500000 1.189942e+07 1.189942e+07 1.862645e-09 8 11 0.500000 1.368434e+08 1.368434e+08 3.874302e-07 8 12 0.500000 1.710542e+09 1.710542e+09 2.384186e-07 8 13 0.500000 2.309232e+10 2.309232e+10 1.182556e-04 8 14 0.500000 3.348386e+11 3.348386e+11 1.220703e-03 8 15 0.500000 5.189998e+12 5.189998e+12 1.367188e-02 8 16 0.500000 8.563016e+13 8.563497e+13 4.809872e+09 8 17 0.500000 1.497874e+15 1.498612e+15 7.383154e+11 8 18 0.500000 2.766111e+16 2.772432e+16 6.321014e+13 9 0 0.500000 8.862269e-01 8.862269e-01 2.220446e-16 9 1 0.500000 1.329340e+00 1.329340e+00 2.220446e-16 9 2 0.500000 3.323351e+00 3.323351e+00 1.776357e-15 9 3 0.500000 1.163173e+01 1.163173e+01 5.329071e-15 9 4 0.500000 5.234278e+01 5.234278e+01 2.131628e-14 9 5 0.500000 2.878853e+02 2.878853e+02 5.684342e-14 9 6 0.500000 1.871254e+03 1.871254e+03 6.821210e-13 9 7 0.500000 1.403441e+04 1.403441e+04 3.637979e-12 9 8 0.500000 1.192925e+05 1.192925e+05 4.365575e-11 9 9 0.500000 1.133278e+06 1.133278e+06 2.328306e-10 9 10 0.500000 1.189942e+07 1.189942e+07 3.725290e-09 9 11 0.500000 1.368434e+08 1.368434e+08 3.278255e-07 9 12 0.500000 1.710542e+09 1.710542e+09 1.430511e-06 9 13 0.500000 2.309232e+10 2.309232e+10 1.029968e-04 9 14 0.500000 3.348386e+11 3.348386e+11 1.281738e-03 9 15 0.500000 5.189998e+12 5.189998e+12 1.562500e-02 9 16 0.500000 8.563497e+13 8.563497e+13 1.718750e-01 9 17 0.500000 1.498612e+15 1.498612e+15 3.500000e+00 9 18 0.500000 2.772391e+16 2.772432e+16 4.112441e+11 9 19 0.500000 5.405460e+17 5.406243e+17 7.834199e+13 9 20 0.500000 1.107456e+19 1.108280e+19 8.233929e+15 10 0 0.500000 8.862269e-01 8.862269e-01 3.108624e-15 10 1 0.500000 1.329340e+00 1.329340e+00 1.332268e-15 10 2 0.500000 3.323351e+00 3.323351e+00 1.332268e-15 10 3 0.500000 1.163173e+01 1.163173e+01 3.552714e-15 10 4 0.500000 5.234278e+01 5.234278e+01 1.421085e-14 10 5 0.500000 2.878853e+02 2.878853e+02 1.136868e-13 10 6 0.500000 1.871254e+03 1.871254e+03 2.273737e-13 10 7 0.500000 1.403441e+04 1.403441e+04 3.637979e-12 10 8 0.500000 1.192925e+05 1.192925e+05 1.455192e-11 10 9 0.500000 1.133278e+06 1.133278e+06 2.328306e-10 10 10 0.500000 1.189942e+07 1.189942e+07 0.000000e+00 10 11 0.500000 1.368434e+08 1.368434e+08 3.576279e-07 10 12 0.500000 1.710542e+09 1.710542e+09 1.430511e-06 10 13 0.500000 2.309232e+10 2.309232e+10 9.918213e-05 10 14 0.500000 3.348386e+11 3.348386e+11 1.403809e-03 10 15 0.500000 5.189998e+12 5.189998e+12 1.757812e-02 10 16 0.500000 8.563497e+13 8.563497e+13 1.250000e-01 10 17 0.500000 1.498612e+15 1.498612e+15 4.250000e+00 10 18 0.500000 2.772432e+16 2.772432e+16 1.000000e+02 10 19 0.500000 5.406243e+17 5.406243e+17 5.760000e+02 10 20 0.500000 1.108275e+19 1.108280e+19 4.318063e+13 10 21 0.500000 2.382702e+20 2.382802e+20 9.996315e+15 10 22 0.500000 5.360038e+21 5.361304e+21 1.265462e+18 1 0 1.000000 1.000000e+00 1.000000e+00 0.000000e+00 1 1 1.000000 2.000000e+00 2.000000e+00 0.000000e+00 1 2 1.000000 4.000000e+00 6.000000e+00 2.000000e+00 1 3 1.000000 8.000000e+00 2.400000e+01 1.600000e+01 1 4 1.000000 1.600000e+01 1.200000e+02 1.040000e+02 2 0 1.000000 1.000000e+00 1.000000e+00 4.440892e-16 2 1 1.000000 2.000000e+00 2.000000e+00 4.440892e-16 2 2 1.000000 6.000000e+00 6.000000e+00 8.881784e-16 2 3 1.000000 2.400000e+01 2.400000e+01 0.000000e+00 2 4 1.000000 1.080000e+02 1.200000e+02 1.200000e+01 2 5 1.000000 5.040000e+02 7.200000e+02 2.160000e+02 2 6 1.000000 2.376000e+03 5.040000e+03 2.664000e+03 3 0 1.000000 1.000000e+00 1.000000e+00 0.000000e+00 3 1 1.000000 2.000000e+00 2.000000e+00 2.220446e-16 3 2 1.000000 6.000000e+00 6.000000e+00 0.000000e+00 3 3 1.000000 2.400000e+01 2.400000e+01 7.105427e-15 3 4 1.000000 1.200000e+02 1.200000e+02 5.684342e-14 3 5 1.000000 7.200000e+02 7.200000e+02 4.547474e-13 3 6 1.000000 4.896000e+03 5.040000e+03 1.440000e+02 3 7 1.000000 3.571200e+04 4.032000e+04 4.608000e+03 3 8 1.000000 2.695680e+05 3.628800e+05 9.331200e+04 4 0 1.000000 1.000000e+00 1.000000e+00 1.110223e-16 4 1 1.000000 2.000000e+00 2.000000e+00 2.220446e-16 4 2 1.000000 6.000000e+00 6.000000e+00 8.881784e-16 4 3 1.000000 2.400000e+01 2.400000e+01 0.000000e+00 4 4 1.000000 1.200000e+02 1.200000e+02 0.000000e+00 4 5 1.000000 7.200000e+02 7.200000e+02 0.000000e+00 4 6 1.000000 5.040000e+03 5.040000e+03 1.818989e-12 4 7 1.000000 4.032000e+04 4.032000e+04 1.455192e-11 4 8 1.000000 3.600000e+05 3.628800e+05 2.880000e+03 4 9 1.000000 3.484800e+06 3.628800e+06 1.440000e+05 4 10 1.000000 3.556800e+07 3.991680e+07 4.348800e+06 5 0 1.000000 1.000000e+00 1.000000e+00 0.000000e+00 5 1 1.000000 2.000000e+00 2.000000e+00 2.220446e-16 5 2 1.000000 6.000000e+00 6.000000e+00 0.000000e+00 5 3 1.000000 2.400000e+01 2.400000e+01 0.000000e+00 5 4 1.000000 1.200000e+02 1.200000e+02 1.421085e-14 5 5 1.000000 7.200000e+02 7.200000e+02 1.136868e-13 5 6 1.000000 5.040000e+03 5.040000e+03 0.000000e+00 5 7 1.000000 4.032000e+04 4.032000e+04 0.000000e+00 5 8 1.000000 3.628800e+05 3.628800e+05 5.820766e-11 5 9 1.000000 3.628800e+06 3.628800e+06 9.313226e-10 5 10 1.000000 3.983040e+07 3.991680e+07 8.640000e+04 5 11 1.000000 4.727808e+08 4.790016e+08 6.220800e+06 5 12 1.000000 5.964710e+09 6.227021e+09 2.623104e+08 6 0 1.000000 1.000000e+00 1.000000e+00 1.110223e-15 6 1 1.000000 2.000000e+00 2.000000e+00 8.881784e-16 6 2 1.000000 6.000000e+00 6.000000e+00 0.000000e+00 6 3 1.000000 2.400000e+01 2.400000e+01 0.000000e+00 6 4 1.000000 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1.453966e+02 1 4 2.500000 4.987104e+02 1.871254e+03 1.372544e+03 2 0 2.500000 3.323351e+00 3.323351e+00 4.440892e-16 2 1 2.500000 1.163173e+01 1.163173e+01 0.000000e+00 2 2 2.500000 5.234278e+01 5.234278e+01 0.000000e+00 2 3 2.500000 2.878853e+02 2.878853e+02 5.684342e-14 2 4 2.500000 1.766569e+03 1.871254e+03 1.046856e+02 2 5 2.500000 1.136493e+04 1.403441e+04 2.669482e+03 2 6 2.500000 7.446087e+04 1.192925e+05 4.483159e+04 3 0 2.500000 3.323351e+00 3.323351e+00 2.220446e-15 3 1 2.500000 1.163173e+01 1.163173e+01 1.776357e-15 3 2 2.500000 5.234278e+01 5.234278e+01 7.105427e-15 3 3 2.500000 2.878853e+02 2.878853e+02 5.684342e-14 3 4 2.500000 1.871254e+03 1.871254e+03 4.547474e-13 3 5 2.500000 1.403441e+04 1.403441e+04 1.818989e-12 3 6 2.500000 1.175652e+05 1.192925e+05 1.727312e+03 3 7 2.500000 1.059868e+06 1.133278e+06 7.341075e+04 3 8 2.500000 9.974334e+06 1.189942e+07 1.925089e+06 4 0 2.500000 3.323351e+00 3.323351e+00 2.664535e-15 4 1 2.500000 1.163173e+01 1.163173e+01 5.329071e-15 4 2 2.500000 5.234278e+01 5.234278e+01 1.421085e-14 4 3 2.500000 2.878853e+02 2.878853e+02 5.684342e-14 4 4 2.500000 1.871254e+03 1.871254e+03 6.821210e-13 4 5 2.500000 1.403441e+04 1.403441e+04 5.456968e-12 4 6 2.500000 1.192925e+05 1.192925e+05 2.910383e-11 4 7 2.500000 1.133278e+06 1.133278e+06 2.328306e-10 4 8 2.500000 1.185451e+07 1.189942e+07 4.491010e+04 4 9 2.500000 1.339916e+08 1.368434e+08 2.851792e+06 4 10 2.500000 1.603083e+09 1.710542e+09 1.074586e+08 5 0 2.500000 3.323351e+00 3.323351e+00 1.332268e-15 5 1 2.500000 1.163173e+01 1.163173e+01 5.329071e-15 5 2 2.500000 5.234278e+01 5.234278e+01 2.842171e-14 5 3 2.500000 2.878853e+02 2.878853e+02 1.705303e-13 5 4 2.500000 1.871254e+03 1.871254e+03 9.094947e-13 5 5 2.500000 1.403441e+04 1.403441e+04 5.456968e-12 5 6 2.500000 1.192925e+05 1.192925e+05 4.365575e-11 5 7 2.500000 1.133278e+06 1.133278e+06 0.000000e+00 5 8 2.500000 1.189942e+07 1.189942e+07 0.000000e+00 5 9 2.500000 1.368434e+08 1.368434e+08 3.576279e-07 5 10 2.500000 1.708858e+09 1.710542e+09 1.684129e+06 5 11 2.500000 2.294327e+10 2.309232e+10 1.490454e+08 5 12 2.500000 3.272146e+11 3.348386e+11 7.624051e+09 6 0 2.500000 3.323351e+00 3.323351e+00 2.220446e-15 6 1 2.500000 1.163173e+01 1.163173e+01 7.105427e-15 6 2 2.500000 5.234278e+01 5.234278e+01 4.263256e-14 6 3 2.500000 2.878853e+02 2.878853e+02 2.273737e-13 6 4 2.500000 1.871254e+03 1.871254e+03 9.094947e-13 6 5 2.500000 1.403441e+04 1.403441e+04 5.456968e-12 6 6 2.500000 1.192925e+05 1.192925e+05 2.910383e-11 6 7 2.500000 1.133278e+06 1.133278e+06 0.000000e+00 6 8 2.500000 1.189942e+07 1.189942e+07 1.862645e-09 6 9 2.500000 1.368434e+08 1.368434e+08 2.980232e-07 6 10 2.500000 1.710542e+09 1.710542e+09 2.145767e-06 6 11 2.500000 2.309232e+10 2.309232e+10 9.918213e-05 6 12 2.500000 3.347527e+11 3.348386e+11 8.589057e+07 6 13 2.500000 5.179906e+12 5.189998e+12 1.009214e+10 6 14 2.500000 8.496252e+13 8.563497e+13 6.724588e+11 7 0 2.500000 3.323351e+00 3.323351e+00 1.776357e-15 7 1 2.500000 1.163173e+01 1.163173e+01 5.329071e-15 7 2 2.500000 5.234278e+01 5.234278e+01 1.421085e-14 7 3 2.500000 2.878853e+02 2.878853e+02 5.684342e-14 7 4 2.500000 1.871254e+03 1.871254e+03 0.000000e+00 7 5 2.500000 1.403441e+04 1.403441e+04 1.818989e-12 7 6 2.500000 1.192925e+05 1.192925e+05 0.000000e+00 7 7 2.500000 1.133278e+06 1.133278e+06 2.328306e-10 7 8 2.500000 1.189942e+07 1.189942e+07 1.862645e-09 7 9 2.500000 1.368434e+08 1.368434e+08 3.576279e-07 7 10 2.500000 1.710542e+09 1.710542e+09 1.430511e-06 7 11 2.500000 2.309232e+10 2.309232e+10 1.068115e-04 7 12 2.500000 3.348386e+11 3.348386e+11 1.342773e-03 7 13 2.500000 5.189998e+12 5.189998e+12 1.660156e-02 7 14 2.500000 8.562926e+13 8.563497e+13 5.711723e+09 7 15 2.500000 1.497752e+15 1.498612e+15 8.596143e+11 7 16 2.500000 2.765202e+16 2.772432e+16 7.230756e+13 8 0 2.500000 3.323351e+00 3.323351e+00 3.996803e-15 8 1 2.500000 1.163173e+01 1.163173e+01 1.065814e-14 8 2 2.500000 5.234278e+01 5.234278e+01 2.131628e-14 8 3 2.500000 2.878853e+02 2.878853e+02 0.000000e+00 8 4 2.500000 1.871254e+03 1.871254e+03 0.000000e+00 8 5 2.500000 1.403441e+04 1.403441e+04 1.818989e-12 8 6 2.500000 1.192925e+05 1.192925e+05 4.365575e-11 8 7 2.500000 1.133278e+06 1.133278e+06 6.984919e-10 8 8 2.500000 1.189942e+07 1.189942e+07 9.313226e-09 8 9 2.500000 1.368434e+08 1.368434e+08 5.066395e-07 8 10 2.500000 1.710542e+09 1.710542e+09 9.536743e-07 8 11 2.500000 2.309232e+10 2.309232e+10 1.449585e-04 8 12 2.500000 3.348386e+11 3.348386e+11 6.713867e-04 8 13 2.500000 5.189998e+12 5.189998e+12 3.906250e-03 8 14 2.500000 8.563497e+13 8.563497e+13 3.750000e-01 8 15 2.500000 1.498612e+15 1.498612e+15 1.000000e+00 8 16 2.500000 2.772384e+16 2.772432e+16 4.797847e+11 8 17 2.500000 5.405343e+17 5.406243e+17 8.995964e+13 8 18 2.500000 1.107348e+19 1.108280e+19 9.320178e+15 9 0 2.500000 3.323351e+00 3.323351e+00 2.220446e-15 9 1 2.500000 1.163173e+01 1.163173e+01 7.105427e-15 9 2 2.500000 5.234278e+01 5.234278e+01 2.842171e-14 9 3 2.500000 2.878853e+02 2.878853e+02 5.684342e-14 9 4 2.500000 1.871254e+03 1.871254e+03 2.273737e-13 9 5 2.500000 1.403441e+04 1.403441e+04 0.000000e+00 9 6 2.500000 1.192925e+05 1.192925e+05 0.000000e+00 9 7 2.500000 1.133278e+06 1.133278e+06 2.328306e-10 9 8 2.500000 1.189942e+07 1.189942e+07 1.862645e-09 9 9 2.500000 1.368434e+08 1.368434e+08 3.576279e-07 9 10 2.500000 1.710542e+09 1.710542e+09 9.536743e-07 9 11 2.500000 2.309232e+10 2.309232e+10 1.144409e-04 9 12 2.500000 3.348386e+11 3.348386e+11 1.220703e-03 9 13 2.500000 5.189998e+12 5.189998e+12 1.464844e-02 9 14 2.500000 8.563497e+13 8.563497e+13 1.562500e-01 9 15 2.500000 1.498612e+15 1.498612e+15 3.500000e+00 9 16 2.500000 2.772432e+16 2.772432e+16 8.000000e+01 9 17 2.500000 5.406243e+17 5.406243e+17 2.560000e+02 9 18 2.500000 1.108275e+19 1.108280e+19 4.965772e+13 9 19 2.500000 2.382688e+20 2.382802e+20 1.134679e+16 9 20 2.500000 5.359884e+21 5.361304e+21 1.419416e+18 10 0 2.500000 3.323351e+00 3.323351e+00 4.440892e-15 10 1 2.500000 1.163173e+01 1.163173e+01 1.065814e-14 10 2 2.500000 5.234278e+01 5.234278e+01 3.552714e-14 10 3 2.500000 2.878853e+02 2.878853e+02 1.705303e-13 10 4 2.500000 1.871254e+03 1.871254e+03 6.821210e-13 10 5 2.500000 1.403441e+04 1.403441e+04 5.456968e-12 10 6 2.500000 1.192925e+05 1.192925e+05 0.000000e+00 10 7 2.500000 1.133278e+06 1.133278e+06 2.328306e-10 10 8 2.500000 1.189942e+07 1.189942e+07 3.725290e-09 10 9 2.500000 1.368434e+08 1.368434e+08 4.172325e-07 10 10 2.500000 1.710542e+09 1.710542e+09 2.384186e-07 10 11 2.500000 2.309232e+10 2.309232e+10 1.258850e-04 10 12 2.500000 3.348386e+11 3.348386e+11 1.037598e-03 10 13 2.500000 5.189998e+12 5.189998e+12 9.765625e-03 10 14 2.500000 8.563497e+13 8.563497e+13 2.656250e-01 10 15 2.500000 1.498612e+15 1.498612e+15 1.250000e+00 10 16 2.500000 2.772432e+16 2.772432e+16 4.000000e+01 10 17 2.500000 5.406243e+17 5.406243e+17 5.760000e+02 10 18 2.500000 1.108280e+19 1.108280e+19 8.192000e+04 10 19 2.500000 2.382802e+20 2.382802e+20 1.212416e+06 10 20 2.500000 5.361297e+21 5.361304e+21 6.207215e+15 10 21 2.500000 1.259889e+23 1.259906e+23 1.697673e+18 10 22 2.500000 3.086518e+24 3.086771e+24 2.522504e+20 TEST14 GEN_LAGUERRE_COMPUTE_NP computes a generalized Gauss-Laguerre rule which is appropriate for integrands of the form Integral ( 0 <= x < +oo ) f(x) x^alpha exp(-x) dx. Order = 1 ALPHA = 0.500000 0 1.5000000000000000 0.8862269254527581 Order = 2 ALPHA = 0.500000 0 0.9188611699158102 0.7233630235462755 1 4.0811388300841900 0.1628639019064825 Order = 3 ALPHA = 0.500000 0 0.6663259077023709 0.5671862778403113 1 2.8007750541502565 0.3053717688445466 2 7.0328990381473728 0.0136688787679001 Order = 4 ALPHA = 0.500000 0 0.5235260767382691 0.4530087465586076 1 2.1566487632690943 0.3816169601717996 2 5.1373875461767113 0.0507946275722408 3 10.1824376138159245 0.0008065911501100 Order = 5 ALPHA = 0.500000 0 0.4313988071478514 0.3704505700074577 1 1.7597536984236963 0.4125843737694528 2 4.1044653628283152 0.0977798200531807 3 7.7467037795425568 0.0053734153411720 4 13.4576783520575809 0.0000387462814939 Order = 6 ALPHA = 0.500000 0 0.3669498773083708 0.3094240968362596 1 1.4885342923104523 0.4177521497070224 2 3.4340079684240714 0.1432858732209768 3 6.3490679256803793 0.0153324910226338 4 10.5404698584483434 0.0004306911960439 5 16.8209700778283811 0.0000016234698211 Order = 7 ALPHA = 0.500000 0 0.3193036339206300 0.2631245143958917 1 1.2907586229591526 0.4091418694141027 2 2.9583744586966501 0.1821177320927161 3 5.4090315972444332 0.0300533243012710 4 8.8040795780567755 0.0017608941175401 5 13.4685357432514810 0.0000285294712212 6 20.2499163658708774 0.0000000616600154 Order = 8 ALPHA = 0.500000 0 0.2826336481165992 0.2271393619524718 1 1.1398738015816137 0.3935945428036146 2 2.6015248434060294 0.2129089708672283 3 4.7241145375277904 0.0478774832031382 4 7.6052562992316135 0.0045425174747626 5 11.4171820765458296 0.0001624046001853 6 16.4994107976558162 0.0000016423774138 7 23.7300039959347089 0.0000000021739431 Order = 9 ALPHA = 0.500000 0 0.2535325549744191 0.1985712548680198 1 1.0208442777203899 0.3749207846631700 2 2.3230960770224658 0.2360748210008255 3 4.1993506006572927 0.0670961050032043 4 6.7139743166150287 0.0090085088966443 5 9.9720091595393487 0.0005426607386359 6 14.1540536712780458 0.0000127053668791 7 19.6119028191659481 0.0000000848430924 8 27.2512365230270603 0.0000000000722865 Order = 10 ALPHA = 0.500000 0 0.2298729805186562 0.1754708150466581 1 0.9244815469866572 0.3552233888020710 2 2.0994104627087982 0.2526835596756779 3 3.7828808737072901 0.0863561026953326 4 6.0199180277014612 0.0151097780348608 5 8.8803475979967086 0.0013282156283636 6 12.4748324048362047 0.0000541878002117 7 16.9908472935425543 0.0000008737475869 8 22.7910028949489458 0.0000000040196999 9 30.8064059170527216 0.0000000000022922 Order = 1 ALPHA = 1.000000 0 2.0000000000000000 1.0000000000000000 Order = 2 ALPHA = 1.000000 0 1.2679491924311228 0.7886751345948125 1 4.7320508075688776 0.2113248654051871 Order = 3 ALPHA = 1.000000 0 0.9358222275240878 0.5886814810396594 1 3.3054072893322788 0.3912160592223101 2 7.7587704831436328 0.0201024597380305 Order = 4 ALPHA = 1.000000 0 0.7432919279814316 0.4468705932187761 1 2.5716350076462784 0.4776357723638682 2 5.7311787516890993 0.0741777847310522 3 10.9538943126831914 0.0013158496863032 Order = 5 ALPHA = 1.000000 0 0.6170308532782703 0.3480145400233487 1 2.1129659585785241 0.5022806741324930 2 4.6108331510175322 0.1409159194944726 3 8.3990669712048422 0.0087198930261000 4 14.2601030659208305 0.0000689733235856 Order = 6 ALPHA = 1.000000 0 0.5276681217111292 0.2776501420298742 1 1.7962998096434089 0.4939105830503539 2 3.8766415204769125 0.2030042967437299 3 6.9188165667047219 0.0246688203691898 4 11.2346104290831157 0.0007630427674635 5 17.6459635523807137 0.0000031150393875 Order = 7 ALPHA = 1.000000 0 0.4610242198049946 0.2262105963831890 1 1.5635861896542627 0.4697087077412672 2 3.3520505025367342 0.2531225162324006 3 5.9162972490205394 0.0478031088842371 4 9.4206993830215904 0.0031004597560254 5 14.1941655480074633 0.0000544846688934 6 21.0921769079544177 0.0000001263339882 Order = 8 ALPHA = 1.000000 0 0.4093835732031851 0.1876325414057237 1 1.3849631848031396 0.4389853607311435 2 2.9562545561688625 0.2899960707813131 3 5.1819431010400709 0.0751413846166974 4 8.1617096881458178 0.0079326466487073 5 12.0700551268371541 0.0003086421368133 6 17.2497355261489886 0.0000033489582098 7 24.5859552436527800 0.0000000047213928 Order = 9 ALPHA = 1.000000 0 0.3681784529417413 0.1580309235007803 1 1.2433579621404816 0.4065825794555731 2 2.6460338413842073 0.3149824836650978 3 4.6168825146350603 0.1037755199516432 4 7.2217865393965708 0.0155776377242784 5 10.5673208077418668 0.0010248719698508 6 14.8359145152609297 0.0000258002113872 7 20.3821819854492503 0.0000001833559550 8 28.1183433810498933 0.0000000001654329 Order = 10 ALPHA = 1.000000 0 0.3345286763247525 0.1348565545313774 1 1.1282533558766366 0.3749243411054584 2 2.3958699247473065 0.3302471308042159 3 4.1668409879287669 0.1315288846413067 4 6.4873530313808114 0.0258420642117171 5 9.4283548133356057 0.0024896457958959 6 13.1017235803678016 0.0001094884764267 7 17.6964875668462263 0.0000018812754142 8 23.5777870883601537 0.0000000091526850 9 31.6828009748319381 0.0000000000055013 Order = 1 ALPHA = 2.500000 0 3.5000000000000000 3.3233509704478426 Order = 2 ALPHA = 2.500000 0 2.3786796564403576 2.4449968210461095 1 6.6213203435596428 0.8783541494017335 Order = 3 ALPHA = 2.500000 0 1.8248198030758733 1.6404916909909513 1 4.8137462264876980 1.5764206102831910 2 9.8614339704364280 0.1064386691737023 Order = 4 ALPHA = 2.500000 0 1.4862411426653499 1.1090225785851453 1 3.8376839879001006 1.8203241249078195 2 7.4820583668989809 0.3853099204909317 3 13.1940165025355700 0.0086943464639431 Order = 5 ALPHA = 2.500000 0 1.2558461409184438 0.7693461876473185 1 3.2070406266149099 1.7891436571378652 2 6.1239082233284119 0.7075497394015841 3 10.3161258663913475 0.0567556714589862 4 16.5970791427468853 0.0005557148020873 Order = 6 ALPHA = 2.500000 0 1.0882635378141619 0.5488342282458465 1 2.7608555713855094 1.6358288552513571 2 5.2130824709557162 0.9757018876753594 3 8.6088345005369984 0.1568919688876181 4 13.2736722693042051 0.0060640457339780 5 20.0552916500034115 0.0000299846536813 Order = 7 ALPHA = 2.500000 0 0.9606221295119329 0.4018644652334328 1 2.4266771014277406 1.4440235761293305 2 4.5506549675827701 1.1578194454504422 3 7.4324177726246079 0.2949071223080978 4 11.2446775658018687 0.0242175011439701 5 16.3271095162749198 0.0005174322318932 6 23.5578409467761567 0.0000014279506744 Order = 8 ALPHA = 2.500000 0 0.8600416237603892 0.3011867253184344 1 2.1662435289922479 1.2529637803437648 2 4.0437426773491278 1.2583457960796296 3 6.5589225384279040 0.4473442363684208 4 9.8172555052257042 0.0605825502790592 5 13.9994797507150555 0.0028904817699157 6 19.4574474318146251 0.0000373385286657 7 27.0968669437149430 0.0000000617599561 Order = 9 ALPHA = 2.500000 0 0.7786800215607390 0.2304206696494756 1 1.9571902235047764 1.0787334348681785 2 3.6417887950645977 1.2944983831865144 3 5.8793297220349769 0.5941550811279992 4 8.7400288949175451 0.1158251816851087 5 12.3346755858879824 0.0094316286919319 6 16.8507961274953821 0.0002842237871894 7 22.6510256983860536 0.0000023649776431 8 30.6664849311479486 0.0000000024738037 Order = 10 ALPHA = 2.500000 0 0.7114760698983055 0.1795029442066154 1 1.7854778061294867 0.9263170003438104 2 3.3144812176022094 1.2847675181016542 3 5.3330492524705839 0.7226374668786890 4 7.8907634025314399 0.1864748527585193 5 11.0616739762852863 0.0224379062050689 6 14.9610223243985114 0.0011891427230148 7 19.7822180787733188 0.0000240042379159 8 25.8976644597361130 0.0000001348994981 9 34.2621734121747465 0.0000000000930604 TEST15 HERMITE_COMPUTE_NP computes a Gauss-Hermite rule which is appropriate for integrands of the form Integral ( -oo < x < +oo ) f(x) exp(-x*x) dx. HERMITE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 1.772454 1.772454 5.515810e-12 1 1 0.000000 0.000000 0.000000e+00 1 2 0.000000 0.886227 8.862269e-01 1 3 0.000000 0.000000 0.000000e+00 1 4 0.000000 1.329340 1.329340e+00 2 0 1.772454 1.772454 5.516254e-12 2 1 0.000000 0.000000 0.000000e+00 2 2 0.886227 0.886227 2.758349e-12 2 3 0.000000 0.000000 0.000000e+00 2 4 0.443113 1.329340 8.862269e-01 2 5 0.000000 0.000000 0.000000e+00 2 6 0.221557 3.323351 3.101794e+00 3 0 1.772454 1.772454 5.515366e-12 3 1 0.000000 0.000000 0.000000e+00 3 2 0.886227 0.886227 2.757683e-12 3 3 0.000000 0.000000 0.000000e+00 3 4 1.329340 1.329340 4.136913e-12 3 5 0.000000 0.000000 0.000000e+00 3 6 1.994011 3.323351 1.329340e+00 3 7 0.000000 0.000000 0.000000e+00 3 8 2.991016 11.631728 8.640713e+00 4 0 1.772454 1.772454 5.516032e-12 4 1 -0.000000 0.000000 2.775558e-17 4 2 0.886227 0.886227 2.758127e-12 4 3 -0.000000 0.000000 5.551115e-17 4 4 1.329340 1.329340 4.137357e-12 4 5 -0.000000 0.000000 1.110223e-16 4 6 3.323351 3.323351 1.034284e-11 4 7 0.000000 0.000000 0.000000e+00 4 8 8.973048 11.631728 2.658681e+00 4 9 0.000000 0.000000 0.000000e+00 4 10 24.426630 52.342778 2.791615e+01 5 0 1.772454 1.772454 5.515588e-12 5 1 0.000000 0.000000 0.000000e+00 5 2 0.886227 0.886227 2.757572e-12 5 3 0.000000 0.000000 0.000000e+00 5 4 1.329340 1.329340 4.136691e-12 5 5 0.000000 0.000000 0.000000e+00 5 6 3.323351 3.323351 1.034062e-11 5 7 0.000000 0.000000 0.000000e+00 5 8 11.631728 11.631728 3.619327e-11 5 9 0.000000 0.000000 0.000000e+00 5 10 45.696076 52.342778 6.646702e+00 5 11 0.000000 0.000000 0.000000e+00 5 12 184.861398 287.885278 1.030239e+02 6 0 1.772454 1.772454 5.516254e-12 6 1 -0.000000 0.000000 3.469447e-17 6 2 0.886227 0.886227 2.758238e-12 6 3 -0.000000 0.000000 2.775558e-17 6 4 1.329340 1.329340 4.137579e-12 6 5 -0.000000 0.000000 5.551115e-17 6 6 3.323351 3.323351 1.034328e-11 6 7 0.000000 0.000000 0.000000e+00 6 8 11.631728 11.631728 3.620215e-11 6 9 0.000000 0.000000 0.000000e+00 6 10 52.342778 52.342778 1.629061e-10 6 11 0.000000 0.000000 0.000000e+00 6 12 267.945172 287.885278 1.994011e+01 6 13 0.000000 0.000000 0.000000e+00 6 14 1442.542031 1871.254306 4.287123e+02 7 0 1.772454 1.772454 5.516698e-12 7 1 -0.000000 0.000000 8.239937e-18 7 2 0.886227 0.886227 2.758571e-12 7 3 0.000000 0.000000 3.469447e-18 7 4 1.329340 1.329340 4.138467e-12 7 5 -0.000000 0.000000 2.775558e-17 7 6 3.323351 3.323351 1.034550e-11 7 7 -0.000000 0.000000 2.220446e-16 7 8 11.631728 11.631728 3.620571e-11 7 9 -0.000000 0.000000 8.881784e-16 7 10 52.342778 52.342778 1.628848e-10 7 11 0.000000 0.000000 0.000000e+00 7 12 287.885278 287.885278 8.957386e-10 7 13 0.000000 0.000000 0.000000e+00 7 14 1801.463935 1871.254306 6.979037e+01 7 15 0.000000 0.000000 0.000000e+00 7 16 12045.381738 14034.407293 1.989026e+03 8 0 1.772454 1.772454 5.516032e-12 8 1 0.000000 0.000000 3.816392e-17 8 2 0.886227 0.886227 2.758682e-12 8 3 -0.000000 0.000000 1.387779e-17 8 4 1.329340 1.329340 4.138245e-12 8 5 0.000000 0.000000 2.289835e-16 8 6 3.323351 3.323351 1.034461e-11 8 7 -0.000000 0.000000 1.665335e-16 8 8 11.631728 11.631728 3.620393e-11 8 9 0.000000 0.000000 0.000000e+00 8 10 52.342778 52.342778 1.629132e-10 8 11 -0.000000 0.000000 3.552714e-15 8 12 287.885278 287.885278 8.959091e-10 8 13 0.000000 0.000000 0.000000e+00 8 14 1871.254306 1871.254306 5.823495e-09 8 15 0.000000 0.000000 0.000000e+00 8 16 13755.245812 14034.407293 2.791615e+02 8 17 0.000000 0.000000 0.000000e+00 8 18 109103.067919 119292.461995 1.018939e+04 9 0 1.772454 1.772454 5.515588e-12 9 1 -0.000000 0.000000 2.049142e-17 9 2 0.886227 0.886227 2.757905e-12 9 3 -0.000000 0.000000 6.743738e-17 9 4 1.329340 1.329340 4.137135e-12 9 5 -0.000000 0.000000 5.204170e-18 9 6 3.323351 3.323351 1.034373e-11 9 7 0.000000 0.000000 2.775558e-16 9 8 11.631728 11.631728 3.620748e-11 9 9 0.000000 0.000000 6.661338e-16 9 10 52.342778 52.342778 1.629630e-10 9 11 0.000000 0.000000 0.000000e+00 9 12 287.885278 287.885278 8.964207e-10 9 13 0.000000 0.000000 0.000000e+00 9 14 1871.254306 1871.254306 5.828042e-09 9 15 -0.000000 0.000000 2.273737e-13 9 16 14034.407293 14034.407293 4.372123e-08 9 17 0.000000 0.000000 0.000000e+00 9 18 118036.235327 119292.461995 1.256227e+03 9 19 0.000000 0.000000 0.000000e+00 9 20 1076120.075605 1133278.388949 5.715831e+04 10 0 1.772454 1.772454 5.515810e-12 10 1 0.000000 0.000000 2.482823e-17 10 2 0.886227 0.886227 2.757794e-12 10 3 -0.000000 0.000000 2.921925e-17 10 4 1.329340 1.329340 4.136913e-12 10 5 0.000000 0.000000 6.938894e-18 10 6 3.323351 3.323351 1.034151e-11 10 7 -0.000000 0.000000 1.526557e-16 10 8 11.631728 11.631728 3.619149e-11 10 9 0.000000 0.000000 5.551115e-16 10 10 52.342778 52.342778 1.628422e-10 10 11 0.000000 0.000000 0.000000e+00 10 12 287.885278 287.885278 8.955112e-10 10 13 -0.000000 0.000000 5.684342e-14 10 14 1871.254306 1871.254306 5.820084e-09 10 15 -0.000000 0.000000 1.136868e-13 10 16 14034.407293 14034.407293 4.364483e-08 10 17 0.000000 0.000000 1.818989e-12 10 18 119292.461994 119292.461995 3.709138e-07 10 19 0.000000 0.000000 0.000000e+00 10 20 1126997.255611 1133278.388949 6.281133e+03 10 21 0.000000 0.000000 0.000000e+00 10 22 11550820.183881 11899423.083962 3.486029e+05 TEST16 HERMITE_COMPUTE_NP computes a Gauss-Hermite rule which is appropriate for integrands of the form Integral ( -oo < x < +oo ) f(x) exp(-x*x) dx. Order = 1 0 0.0000000000000000 1.7724538509000001 Order = 2 0 -0.7071067811865475 0.8862269254499998 1 0.7071067811865475 0.8862269254499998 Order = 3 0 -1.2247448713915889 0.2954089751500001 1 0.0000000000000000 1.1816359006000001 2 1.2247448713915889 0.2954089751500001 Order = 4 0 -1.6506801238857847 0.0813128354469921 1 -0.5246476232752904 0.8049140900030078 2 0.5246476232752904 0.8049140900030078 3 1.6506801238857847 0.0813128354469921 Order = 5 0 -2.0201828704560856 0.0199532420589838 1 -0.9585724646138185 0.3936193231510163 2 0.0000000000000000 0.9453087204800000 3 0.9585724646138185 0.3936193231510163 4 2.0201828704560856 0.0199532420589838 Order = 6 0 -2.3506049736744923 0.0045300099054947 1 -1.3358490740136970 0.1570673203223678 2 -0.4360774119276165 0.7246295952221373 3 0.4360774119276165 0.7246295952221373 4 1.3358490740136970 0.1570673203223678 5 2.3506049736744923 0.0045300099054947 Order = 7 0 -2.6519613568352334 0.0009717812450965 1 -1.6735516287674714 0.0545155828189573 2 -0.8162878828589647 0.4256072526088032 3 0.0000000000000000 0.8102646175542857 4 0.8162878828589647 0.4256072526088032 5 1.6735516287674714 0.0545155828189573 6 2.6519613568352334 0.0009717812450965 Order = 8 0 -2.9306374202572441 0.0001996040722107 1 -1.9816567566958430 0.0170779830073603 2 -1.1571937124467802 0.2078023258142450 3 -0.3811869902073221 0.6611470125561838 4 0.3811869902073221 0.6611470125561838 5 1.1571937124467802 0.2078023258142450 6 1.9816567566958430 0.0170779830073603 7 2.9306374202572441 0.0001996040722107 Order = 9 0 -3.1909932017815272 0.0000396069772631 1 -2.2665805845318432 0.0049436242755216 2 -1.4685532892166679 0.0884745273941013 3 -0.7235510187528376 0.4326515590012093 4 0.0000000000000000 0.7202352156038097 5 0.7235510187528376 0.4326515590012093 6 1.4685532892166679 0.0884745273941013 7 2.2665805845318432 0.0049436242755216 8 3.1909932017815272 0.0000396069772631 Order = 10 0 -3.4361591188377378 0.0000076404328552 1 -2.5327316742327897 0.0013436457467771 2 -1.7566836492998819 0.0338743944553756 3 -1.0366108297895136 0.2401386110815674 4 -0.3429013272237046 0.6108626337334246 5 0.3429013272237046 0.6108626337334246 6 1.0366108297895136 0.2401386110815674 7 1.7566836492998819 0.0338743944553756 8 2.5327316742327897 0.0013436457467771 9 3.4361591188377378 0.0000076404328552 TEST17 JACOBI_COMPUTE_NP computes a Gauss-Jacobi rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) (1-x)^alpha (1+x)^beta dx. JACOBI_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Alpha Beta Estimate Exact Error 1 0 0.500000 0.500000 1.570796 1.570796 1.110223e-15 1 1 0.500000 0.500000 0.000000 0.000000 0.000000e+00 1 2 0.500000 0.500000 0.000000 0.392699 3.926991e-01 1 3 0.500000 0.500000 0.000000 0.000000 0.000000e+00 1 4 0.500000 0.500000 0.000000 0.196350 1.963495e-01 2 0 0.500000 0.500000 1.570796 1.570796 1.110223e-15 2 1 0.500000 0.500000 0.000000 0.000000 0.000000e+00 2 2 0.500000 0.500000 0.392699 0.392699 0.000000e+00 2 3 0.500000 0.500000 0.000000 0.000000 0.000000e+00 2 4 0.500000 0.500000 0.098175 0.196350 9.817477e-02 2 5 0.500000 0.500000 0.000000 0.000000 0.000000e+00 2 6 0.500000 0.500000 0.024544 0.122718 9.817477e-02 3 0 0.500000 0.500000 1.570796 1.570796 1.110223e-15 3 1 0.500000 0.500000 0.000000 0.000000 4.996004e-16 3 2 0.500000 0.500000 0.392699 0.392699 0.000000e+00 3 3 0.500000 0.500000 0.000000 0.000000 3.053113e-16 3 4 0.500000 0.500000 0.196350 0.196350 8.326673e-17 3 5 0.500000 0.500000 0.000000 0.000000 1.804112e-16 3 6 0.500000 0.500000 0.098175 0.122718 2.454369e-02 3 7 0.500000 0.500000 0.000000 0.000000 9.714451e-17 3 8 0.500000 0.500000 0.049087 0.085903 3.681554e-02 4 0 0.500000 0.500000 1.570796 1.570796 8.881784e-16 4 1 0.500000 0.500000 -0.000000 0.000000 2.775558e-17 4 2 0.500000 0.500000 0.392699 0.392699 0.000000e+00 4 3 0.500000 0.500000 -0.000000 0.000000 1.387779e-17 4 4 0.500000 0.500000 0.196350 0.196350 5.551115e-17 4 5 0.500000 0.500000 0.000000 0.000000 0.000000e+00 4 6 0.500000 0.500000 0.122718 0.122718 1.387779e-17 4 7 0.500000 0.500000 0.000000 0.000000 0.000000e+00 4 8 0.500000 0.500000 0.079767 0.085903 6.135923e-03 4 9 0.500000 0.500000 0.000000 0.000000 0.000000e+00 4 10 0.500000 0.500000 0.052155 0.064427 1.227185e-02 5 0 0.500000 0.500000 1.570796 1.570796 1.998401e-15 5 1 0.500000 0.500000 0.000000 0.000000 4.440892e-16 5 2 0.500000 0.500000 0.392699 0.392699 6.106227e-16 5 3 0.500000 0.500000 0.000000 0.000000 2.914335e-16 5 4 0.500000 0.500000 0.196350 0.196350 5.551115e-16 5 5 0.500000 0.500000 0.000000 0.000000 2.359224e-16 5 6 0.500000 0.500000 0.122718 0.122718 3.608225e-16 5 7 0.500000 0.500000 0.000000 0.000000 1.734723e-16 5 8 0.500000 0.500000 0.085903 0.085903 2.498002e-16 5 9 0.500000 0.500000 0.000000 0.000000 1.318390e-16 5 10 0.500000 0.500000 0.062893 0.064427 1.533981e-03 5 11 0.500000 0.500000 0.000000 0.000000 9.714451e-17 5 12 0.500000 0.500000 0.046786 0.050621 3.834952e-03 6 0 0.500000 0.500000 1.570796 1.570796 1.776357e-15 6 1 0.500000 0.500000 -0.000000 0.000000 1.387779e-17 6 2 0.500000 0.500000 0.392699 0.392699 4.996004e-16 6 3 0.500000 0.500000 0.000000 0.000000 1.387779e-17 6 4 0.500000 0.500000 0.196350 0.196350 3.885781e-16 6 5 0.500000 0.500000 -0.000000 0.000000 6.938894e-18 6 6 0.500000 0.500000 0.122718 0.122718 2.359224e-16 6 7 0.500000 0.500000 0.000000 0.000000 0.000000e+00 6 8 0.500000 0.500000 0.085903 0.085903 1.804112e-16 6 9 0.500000 0.500000 0.000000 0.000000 0.000000e+00 6 10 0.500000 0.500000 0.064427 0.064427 4.163336e-17 6 11 0.500000 0.500000 0.000000 0.000000 0.000000e+00 6 12 0.500000 0.500000 0.050238 0.050621 3.834952e-04 6 13 0.500000 0.500000 0.000000 0.000000 0.000000e+00 6 14 0.500000 0.500000 0.039979 0.041130 1.150486e-03 7 0 0.500000 0.500000 1.570796 1.570796 8.881784e-16 7 1 0.500000 0.500000 0.000000 0.000000 3.191891e-16 7 2 0.500000 0.500000 0.392699 0.392699 3.330669e-16 7 3 0.500000 0.500000 0.000000 0.000000 2.081668e-16 7 4 0.500000 0.500000 0.196350 0.196350 1.665335e-16 7 5 0.500000 0.500000 0.000000 0.000000 1.179612e-16 7 6 0.500000 0.500000 0.122718 0.122718 2.081668e-16 7 7 0.500000 0.500000 0.000000 0.000000 6.245005e-17 7 8 0.500000 0.500000 0.085903 0.085903 1.804112e-16 7 9 0.500000 0.500000 0.000000 0.000000 3.469447e-17 7 10 0.500000 0.500000 0.064427 0.064427 3.191891e-16 7 11 0.500000 0.500000 0.000000 0.000000 2.081668e-17 7 12 0.500000 0.500000 0.050621 0.050621 5.967449e-16 7 13 0.500000 0.500000 0.000000 0.000000 1.040834e-17 7 14 0.500000 0.500000 0.041034 0.041130 9.587380e-05 7 15 0.500000 0.500000 0.000000 0.000000 3.469447e-18 7 16 0.500000 0.500000 0.033939 0.034275 3.355583e-04 8 0 0.500000 0.500000 1.570796 1.570796 4.440892e-16 8 1 0.500000 0.500000 0.000000 0.000000 5.620504e-16 8 2 0.500000 0.500000 0.392699 0.392699 1.665335e-16 8 3 0.500000 0.500000 0.000000 0.000000 5.065393e-16 8 4 0.500000 0.500000 0.196350 0.196350 1.110223e-16 8 5 0.500000 0.500000 0.000000 0.000000 4.406198e-16 8 6 0.500000 0.500000 0.122718 0.122718 1.665335e-16 8 7 0.500000 0.500000 0.000000 0.000000 3.851086e-16 8 8 0.500000 0.500000 0.085903 0.085903 1.665335e-16 8 9 0.500000 0.500000 0.000000 0.000000 3.434752e-16 8 10 0.500000 0.500000 0.064427 0.064427 3.330669e-16 8 11 0.500000 0.500000 0.000000 0.000000 3.053113e-16 8 12 0.500000 0.500000 0.050621 0.050621 6.314393e-16 8 13 0.500000 0.500000 0.000000 0.000000 2.671474e-16 8 14 0.500000 0.500000 0.041130 0.041130 8.326673e-17 8 15 0.500000 0.500000 0.000000 0.000000 2.324529e-16 8 16 0.500000 0.500000 0.034251 0.034275 2.396845e-05 8 17 0.500000 0.500000 0.000000 0.000000 2.099015e-16 8 18 0.500000 0.500000 0.029038 0.029134 9.587380e-05 9 0 0.500000 0.500000 1.570796 1.570796 2.220446e-16 9 1 0.500000 0.500000 -0.000000 0.000000 6.938894e-18 9 2 0.500000 0.500000 0.392699 0.392699 5.551115e-16 9 3 0.500000 0.500000 -0.000000 0.000000 3.469447e-18 9 4 0.500000 0.500000 0.196350 0.196350 3.330669e-16 9 5 0.500000 0.500000 -0.000000 0.000000 6.938894e-18 9 6 0.500000 0.500000 0.122718 0.122718 3.747003e-16 9 7 0.500000 0.500000 -0.000000 0.000000 3.469447e-18 9 8 0.500000 0.500000 0.085903 0.085903 3.608225e-16 9 9 0.500000 0.500000 0.000000 0.000000 0.000000e+00 9 10 0.500000 0.500000 0.064427 0.064427 4.996004e-16 9 11 0.500000 0.500000 0.000000 0.000000 0.000000e+00 9 12 0.500000 0.500000 0.050621 0.050621 7.771561e-16 9 13 0.500000 0.500000 -0.000000 0.000000 1.734723e-18 9 14 0.500000 0.500000 0.041130 0.041130 2.151057e-16 9 15 0.500000 0.500000 0.000000 0.000000 0.000000e+00 9 16 0.500000 0.500000 0.034275 0.034275 7.632783e-17 9 17 0.500000 0.500000 0.000000 0.000000 0.000000e+00 9 18 0.500000 0.500000 0.029128 0.029134 5.992112e-06 9 19 0.500000 0.500000 0.000000 0.000000 0.000000e+00 9 20 0.500000 0.500000 0.025134 0.025161 2.696451e-05 10 0 0.500000 0.500000 1.570796 1.570796 1.332268e-15 10 1 0.500000 0.500000 -0.000000 0.000000 2.428613e-17 10 2 0.500000 0.500000 0.392699 0.392699 5.551115e-17 10 3 0.500000 0.500000 0.000000 0.000000 1.734723e-17 10 4 0.500000 0.500000 0.196350 0.196350 8.326673e-17 10 5 0.500000 0.500000 0.000000 0.000000 3.469447e-18 10 6 0.500000 0.500000 0.122718 0.122718 1.526557e-16 10 7 0.500000 0.500000 0.000000 0.000000 0.000000e+00 10 8 0.500000 0.500000 0.085903 0.085903 1.804112e-16 10 9 0.500000 0.500000 0.000000 0.000000 0.000000e+00 10 10 0.500000 0.500000 0.064427 0.064427 3.469447e-16 10 11 0.500000 0.500000 0.000000 0.000000 0.000000e+00 10 12 0.500000 0.500000 0.050621 0.050621 6.591949e-16 10 13 0.500000 0.500000 -0.000000 0.000000 1.734723e-18 10 14 0.500000 0.500000 0.041130 0.041130 1.110223e-16 10 15 0.500000 0.500000 0.000000 0.000000 0.000000e+00 10 16 0.500000 0.500000 0.034275 0.034275 2.775558e-17 10 17 0.500000 0.500000 0.000000 0.000000 0.000000e+00 10 18 0.500000 0.500000 0.029134 0.029134 1.457168e-16 10 19 0.500000 0.500000 0.000000 0.000000 1.734723e-18 10 20 0.500000 0.500000 0.025159 0.025161 1.498028e-06 10 21 0.500000 0.500000 0.000000 0.000000 0.000000e+00 10 22 0.500000 0.500000 0.022008 0.022016 7.490141e-06 1 0 0.500000 1.000000 1.508494 1.508494 2.220446e-16 1 1 0.500000 1.000000 0.215499 0.215499 2.775558e-17 1 2 0.500000 1.000000 0.030786 0.359165 3.283797e-01 1 3 0.500000 1.000000 0.004398 0.111015 1.066168e-01 1 4 0.500000 1.000000 0.000628 0.174308 1.736799e-01 2 0 0.500000 1.000000 1.508494 1.508494 6.661338e-16 2 1 0.500000 1.000000 0.215499 0.215499 2.498002e-16 2 2 0.500000 1.000000 0.359165 0.359165 1.665335e-16 2 3 0.500000 1.000000 0.111015 0.111015 2.775558e-17 2 4 0.500000 1.000000 0.096371 0.174308 7.793713e-02 2 5 0.500000 1.000000 0.041071 0.070828 2.975781e-02 2 6 0.500000 1.000000 0.027910 0.106701 7.879089e-02 3 0 0.500000 1.000000 1.508494 1.508494 2.220446e-16 3 1 0.500000 1.000000 0.215499 0.215499 2.498002e-16 3 2 0.500000 1.000000 0.359165 0.359165 5.551115e-17 3 3 0.500000 1.000000 0.111015 0.111015 1.110223e-16 3 4 0.500000 1.000000 0.174308 0.174308 2.775558e-17 3 5 0.500000 1.000000 0.070828 0.070828 6.938894e-17 3 6 0.500000 1.000000 0.087742 0.106701 1.895882e-02 3 7 0.500000 1.000000 0.042566 0.050350 7.783096e-03 3 8 0.500000 1.000000 0.044985 0.073531 2.854635e-02 4 0 0.500000 1.000000 1.508494 1.508494 0.000000e+00 4 1 0.500000 1.000000 0.215499 0.215499 2.220446e-16 4 2 0.500000 1.000000 0.359165 0.359165 5.551115e-17 4 3 0.500000 1.000000 0.111015 0.111015 1.942890e-16 4 4 0.500000 1.000000 0.174308 0.174308 0.000000e+00 4 5 0.500000 1.000000 0.070828 0.070828 9.714451e-17 4 6 0.500000 1.000000 0.106701 0.106701 4.163336e-17 4 7 0.500000 1.000000 0.050350 0.050350 4.857226e-17 4 8 0.500000 1.000000 0.068871 0.073531 4.660378e-03 4 9 0.500000 1.000000 0.036229 0.038223 1.994258e-03 4 10 0.500000 1.000000 0.045135 0.054471 9.336981e-03 5 0 0.500000 1.000000 1.508494 1.508494 2.220446e-16 5 1 0.500000 1.000000 0.215499 0.215499 4.440892e-16 5 2 0.500000 1.000000 0.359165 0.359165 3.885781e-16 5 3 0.500000 1.000000 0.111015 0.111015 3.469447e-16 5 4 0.500000 1.000000 0.174308 0.174308 2.498002e-16 5 5 0.500000 1.000000 0.070828 0.070828 2.914335e-16 5 6 0.500000 1.000000 0.106701 0.106701 2.498002e-16 5 7 0.500000 1.000000 0.050350 0.050350 2.359224e-16 5 8 0.500000 1.000000 0.073531 0.073531 1.804112e-16 5 9 0.500000 1.000000 0.038223 0.038223 1.873501e-16 5 10 0.500000 1.000000 0.053320 0.054471 1.151817e-03 5 11 0.500000 1.000000 0.029824 0.030331 5.063543e-04 5 12 0.500000 1.000000 0.039487 0.042369 2.882239e-03 6 0 0.500000 1.000000 1.508494 1.508494 2.220446e-16 6 1 0.500000 1.000000 0.215499 0.215499 1.387779e-16 6 2 0.500000 1.000000 0.359165 0.359165 1.110223e-16 6 3 0.500000 1.000000 0.111015 0.111015 0.000000e+00 6 4 0.500000 1.000000 0.174308 0.174308 1.387779e-16 6 5 0.500000 1.000000 0.070828 0.070828 1.387779e-17 6 6 0.500000 1.000000 0.106701 0.106701 6.938894e-17 6 7 0.500000 1.000000 0.050350 0.050350 2.081668e-17 6 8 0.500000 1.000000 0.073531 0.073531 9.714451e-17 6 9 0.500000 1.000000 0.038223 0.038223 1.387779e-17 6 10 0.500000 1.000000 0.054471 0.054471 1.387779e-16 6 11 0.500000 1.000000 0.030331 0.030331 3.469447e-17 6 12 0.500000 1.000000 0.042084 0.042369 2.855761e-04 6 13 0.500000 1.000000 0.024721 0.024849 1.279463e-04 6 14 0.500000 1.000000 0.033277 0.034135 8.572079e-04 7 0 0.500000 1.000000 1.508494 1.508494 2.220446e-16 7 1 0.500000 1.000000 0.215499 0.215499 8.326673e-17 7 2 0.500000 1.000000 0.359165 0.359165 5.551115e-17 7 3 0.500000 1.000000 0.111015 0.111015 2.775558e-17 7 4 0.500000 1.000000 0.174308 0.174308 0.000000e+00 7 5 0.500000 1.000000 0.070828 0.070828 0.000000e+00 7 6 0.500000 1.000000 0.106701 0.106701 2.775558e-17 7 7 0.500000 1.000000 0.050350 0.050350 1.387779e-17 7 8 0.500000 1.000000 0.073531 0.073531 1.387779e-17 7 9 0.500000 1.000000 0.038223 0.038223 6.938894e-18 7 10 0.500000 1.000000 0.054471 0.054471 7.632783e-17 7 11 0.500000 1.000000 0.030331 0.030331 4.857226e-17 7 12 0.500000 1.000000 0.042369 0.042369 3.677614e-16 7 13 0.500000 1.000000 0.024849 0.024849 7.979728e-17 7 14 0.500000 1.000000 0.034064 0.034135 7.094710e-05 7 15 0.500000 1.000000 0.020822 0.020854 3.223679e-05 7 16 0.500000 1.000000 0.027992 0.028240 2.484045e-04 8 0 0.500000 1.000000 1.508494 1.508494 6.661338e-16 8 1 0.500000 1.000000 0.215499 0.215499 1.942890e-16 8 2 0.500000 1.000000 0.359165 0.359165 3.885781e-16 8 3 0.500000 1.000000 0.111015 0.111015 6.938894e-17 8 4 0.500000 1.000000 0.174308 0.174308 2.498002e-16 8 5 0.500000 1.000000 0.070828 0.070828 6.938894e-17 8 6 0.500000 1.000000 0.106701 0.106701 2.359224e-16 8 7 0.500000 1.000000 0.050350 0.050350 6.938894e-17 8 8 0.500000 1.000000 0.073531 0.073531 1.526557e-16 8 9 0.500000 1.000000 0.038223 0.038223 5.551115e-17 8 10 0.500000 1.000000 0.054471 0.054471 6.938894e-17 8 11 0.500000 1.000000 0.030331 0.030331 8.326673e-17 8 12 0.500000 1.000000 0.042369 0.042369 2.428613e-16 8 13 0.500000 1.000000 0.024849 0.024849 1.075529e-16 8 14 0.500000 1.000000 0.034135 0.034135 1.040834e-16 8 15 0.500000 1.000000 0.020854 0.020854 3.469447e-17 8 16 0.500000 1.000000 0.028223 0.028240 1.764972e-05 8 17 0.500000 1.000000 0.017827 0.017835 8.107236e-06 8 18 0.500000 1.000000 0.023783 0.023854 7.061626e-05 9 0 0.500000 1.000000 1.508494 1.508494 2.220446e-16 9 1 0.500000 1.000000 0.215499 0.215499 1.942890e-16 9 2 0.500000 1.000000 0.359165 0.359165 5.551115e-17 9 3 0.500000 1.000000 0.111015 0.111015 2.498002e-16 9 4 0.500000 1.000000 0.174308 0.174308 5.551115e-17 9 5 0.500000 1.000000 0.070828 0.070828 1.665335e-16 9 6 0.500000 1.000000 0.106701 0.106701 0.000000e+00 9 7 0.500000 1.000000 0.050350 0.050350 1.249001e-16 9 8 0.500000 1.000000 0.073531 0.073531 5.551115e-17 9 9 0.500000 1.000000 0.038223 0.038223 1.110223e-16 9 10 0.500000 1.000000 0.054471 0.054471 1.110223e-16 9 11 0.500000 1.000000 0.030331 0.030331 6.245005e-17 9 12 0.500000 1.000000 0.042369 0.042369 4.093947e-16 9 13 0.500000 1.000000 0.024849 0.024849 1.387779e-17 9 14 0.500000 1.000000 0.034135 0.034135 4.163336e-17 9 15 0.500000 1.000000 0.020854 0.020854 1.387779e-16 9 16 0.500000 1.000000 0.028240 0.028240 8.673617e-17 9 17 0.500000 1.000000 0.017835 0.017835 4.163336e-17 9 18 0.500000 1.000000 0.023849 0.023854 4.394996e-06 9 19 0.500000 1.000000 0.015484 0.015487 2.036322e-06 9 20 0.500000 1.000000 0.020468 0.020487 1.978095e-05 10 0 0.500000 1.000000 1.508494 1.508494 2.220446e-16 10 1 0.500000 1.000000 0.215499 0.215499 5.551115e-17 10 2 0.500000 1.000000 0.359165 0.359165 0.000000e+00 10 3 0.500000 1.000000 0.111015 0.111015 1.387779e-17 10 4 0.500000 1.000000 0.174308 0.174308 0.000000e+00 10 5 0.500000 1.000000 0.070828 0.070828 1.387779e-17 10 6 0.500000 1.000000 0.106701 0.106701 9.714451e-17 10 7 0.500000 1.000000 0.050350 0.050350 4.163336e-17 10 8 0.500000 1.000000 0.073531 0.073531 5.551115e-17 10 9 0.500000 1.000000 0.038223 0.038223 4.163336e-17 10 10 0.500000 1.000000 0.054471 0.054471 1.387779e-17 10 11 0.500000 1.000000 0.030331 0.030331 6.938894e-17 10 12 0.500000 1.000000 0.042369 0.042369 3.053113e-16 10 13 0.500000 1.000000 0.024849 0.024849 1.006140e-16 10 14 0.500000 1.000000 0.034135 0.034135 6.938894e-17 10 15 0.500000 1.000000 0.020854 0.020854 4.163336e-17 10 16 0.500000 1.000000 0.028240 0.028240 1.769418e-16 10 17 0.500000 1.000000 0.017835 0.017835 1.353084e-16 10 18 0.500000 1.000000 0.023854 0.023854 3.122502e-17 10 19 0.500000 1.000000 0.015487 0.015487 7.285839e-17 10 20 0.500000 1.000000 0.020486 0.020487 1.095179e-06 10 21 0.500000 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2.500000 1.000000 0.026807 0.026807 2.567391e-16 10 13 2.500000 1.000000 -0.022567 -0.022567 1.630640e-16 10 14 2.500000 1.000000 0.020667 0.020667 2.463307e-16 10 15 2.500000 1.000000 -0.017792 -0.017792 1.942890e-16 10 16 2.500000 1.000000 0.016424 0.016424 7.632783e-17 10 17 2.500000 1.000000 -0.014386 -0.014386 1.144917e-16 10 18 2.500000 1.000000 0.013368 0.013368 1.335737e-16 10 19 2.500000 1.000000 -0.011872 -0.011872 1.058181e-16 10 20 2.500000 1.000000 0.011094 0.011094 3.490698e-07 10 21 2.500000 1.000000 -0.009964 -0.009964 4.486793e-07 10 22 2.500000 1.000000 0.009354 0.009356 1.863779e-06 1 0 2.500000 2.500000 0.981748 0.981748 1.110223e-16 1 1 2.500000 2.500000 0.000000 0.000000 0.000000e+00 1 2 2.500000 2.500000 0.000000 0.122718 1.227185e-01 1 3 2.500000 2.500000 0.000000 0.000000 0.000000e+00 1 4 2.500000 2.500000 0.000000 0.036816 3.681554e-02 2 0 2.500000 2.500000 0.981748 0.981748 1.110223e-16 2 1 2.500000 2.500000 0.000000 0.000000 0.000000e+00 2 2 2.500000 2.500000 0.122718 0.122718 1.249001e-16 2 3 2.500000 2.500000 0.000000 0.000000 0.000000e+00 2 4 2.500000 2.500000 0.015340 0.036816 2.147573e-02 2 5 2.500000 2.500000 0.000000 0.000000 0.000000e+00 2 6 2.500000 2.500000 0.001917 0.015340 1.342233e-02 3 0 2.500000 2.500000 0.981748 0.981748 4.440892e-16 3 1 2.500000 2.500000 0.000000 0.000000 0.000000e+00 3 2 2.500000 2.500000 0.122718 0.122718 1.387779e-17 3 3 2.500000 2.500000 0.000000 0.000000 0.000000e+00 3 4 2.500000 2.500000 0.036816 0.036816 1.387779e-17 3 5 2.500000 2.500000 0.000000 0.000000 0.000000e+00 3 6 2.500000 2.500000 0.011045 0.015340 4.295146e-03 3 7 2.500000 2.500000 0.000000 0.000000 0.000000e+00 3 8 2.500000 2.500000 0.003313 0.007670 4.356505e-03 4 0 2.500000 2.500000 0.981748 0.981748 0.000000e+00 4 1 2.500000 2.500000 0.000000 0.000000 1.457168e-16 4 2 2.500000 2.500000 0.122718 0.122718 6.938894e-17 4 3 2.500000 2.500000 0.000000 0.000000 6.591949e-17 4 4 2.500000 2.500000 0.036816 0.036816 2.775558e-17 4 5 2.500000 2.500000 0.000000 0.000000 2.949030e-17 4 6 2.500000 2.500000 0.015340 0.015340 5.204170e-18 4 7 2.500000 2.500000 0.000000 0.000000 1.301043e-17 4 8 2.500000 2.500000 0.006750 0.007670 9.203885e-04 4 9 2.500000 2.500000 0.000000 0.000000 5.637851e-18 4 10 2.500000 2.500000 0.002991 0.004314 1.323058e-03 5 0 2.500000 2.500000 0.981748 0.981748 3.330669e-16 5 1 2.500000 2.500000 -0.000000 0.000000 3.469447e-17 5 2 2.500000 2.500000 0.122718 0.122718 1.526557e-16 5 3 2.500000 2.500000 -0.000000 0.000000 6.938894e-18 5 4 2.500000 2.500000 0.036816 0.036816 5.551115e-17 5 5 2.500000 2.500000 -0.000000 0.000000 1.734723e-18 5 6 2.500000 2.500000 0.015340 0.015340 1.040834e-17 5 7 2.500000 2.500000 0.000000 0.000000 0.000000e+00 5 8 2.500000 2.500000 0.007670 0.007670 1.040834e-17 5 9 2.500000 2.500000 0.000000 0.000000 0.000000e+00 5 10 2.500000 2.500000 0.004109 0.004314 2.054439e-04 5 11 2.500000 2.500000 0.000000 0.000000 0.000000e+00 5 12 2.500000 2.500000 0.002250 0.002637 3.864301e-04 6 0 2.500000 2.500000 0.981748 0.981748 2.220446e-16 6 1 2.500000 2.500000 0.000000 0.000000 1.838807e-16 6 2 2.500000 2.500000 0.122718 0.122718 6.938894e-17 6 3 2.500000 2.500000 0.000000 0.000000 4.510281e-17 6 4 2.500000 2.500000 0.036816 0.036816 2.081668e-17 6 5 2.500000 2.500000 0.000000 0.000000 1.214306e-17 6 6 2.500000 2.500000 0.015340 0.015340 3.469447e-18 6 7 2.500000 2.500000 0.000000 0.000000 3.469447e-18 6 8 2.500000 2.500000 0.007670 0.007670 2.081668e-17 6 9 2.500000 2.500000 0.000000 0.000000 8.673617e-19 6 10 2.500000 2.500000 0.004314 0.004314 2.081668e-17 6 11 2.500000 2.500000 0.000000 0.000000 2.168404e-19 6 12 2.500000 2.500000 0.002589 0.002637 4.708088e-05 6 13 2.500000 2.500000 0.000000 0.000000 0.000000e+00 6 14 2.500000 2.500000 0.001604 0.001714 1.100516e-04 7 0 2.500000 2.500000 0.981748 0.981748 1.110223e-16 7 1 2.500000 2.500000 0.000000 0.000000 1.786765e-16 7 2 2.500000 2.500000 0.122718 0.122718 2.081668e-16 7 3 2.500000 2.500000 0.000000 0.000000 1.040834e-16 7 4 2.500000 2.500000 0.036816 0.036816 1.249001e-16 7 5 2.500000 2.500000 0.000000 0.000000 6.938894e-17 7 6 2.500000 2.500000 0.015340 0.015340 6.245005e-17 7 7 2.500000 2.500000 0.000000 0.000000 4.813858e-17 7 8 2.500000 2.500000 0.007670 0.007670 2.775558e-17 7 9 2.500000 2.500000 0.000000 0.000000 3.382711e-17 7 10 2.500000 2.500000 0.004314 0.004314 1.301043e-17 7 11 2.500000 2.500000 0.000000 0.000000 2.363561e-17 7 12 2.500000 2.500000 0.002637 0.002637 1.734723e-17 7 13 2.500000 2.500000 0.000000 0.000000 1.647987e-17 7 14 2.500000 2.500000 0.001703 0.001714 1.098554e-05 7 15 2.500000 2.500000 0.000000 0.000000 1.149254e-17 7 16 2.500000 2.500000 0.001138 0.001168 3.079280e-05 8 0 2.500000 2.500000 0.981748 0.981748 1.110223e-16 8 1 2.500000 2.500000 0.000000 0.000000 2.602085e-17 8 2 2.500000 2.500000 0.122718 0.122718 9.714451e-17 8 3 2.500000 2.500000 0.000000 0.000000 4.336809e-19 8 4 2.500000 2.500000 0.036816 0.036816 2.775558e-17 8 5 2.500000 2.500000 0.000000 0.000000 2.168404e-18 8 6 2.500000 2.500000 0.015340 0.015340 3.469447e-18 8 7 2.500000 2.500000 0.000000 0.000000 2.168404e-19 8 8 2.500000 2.500000 0.007670 0.007670 1.994932e-17 8 9 2.500000 2.500000 0.000000 0.000000 0.000000e+00 8 10 2.500000 2.500000 0.004314 0.004314 2.081668e-17 8 11 2.500000 2.500000 0.000000 0.000000 0.000000e+00 8 12 2.500000 2.500000 0.002637 0.002637 5.637851e-18 8 13 2.500000 2.500000 0.000000 0.000000 0.000000e+00 8 14 2.500000 2.500000 0.001714 0.001714 2.168404e-19 8 15 2.500000 2.500000 0.000000 0.000000 0.000000e+00 8 16 2.500000 2.500000 0.001166 0.001168 2.596582e-06 8 17 2.500000 2.500000 0.000000 0.000000 0.000000e+00 8 18 2.500000 2.500000 0.000819 0.000828 8.503806e-06 9 0 2.500000 2.500000 0.981748 0.981748 1.110223e-16 9 1 2.500000 2.500000 0.000000 0.000000 1.647987e-17 9 2 2.500000 2.500000 0.122718 0.122718 2.775558e-17 9 3 2.500000 2.500000 0.000000 0.000000 2.168404e-18 9 4 2.500000 2.500000 0.036816 0.036816 6.938894e-18 9 5 2.500000 2.500000 -0.000000 0.000000 6.505213e-19 9 6 2.500000 2.500000 0.015340 0.015340 3.122502e-17 9 7 2.500000 2.500000 0.000000 0.000000 4.336809e-19 9 8 2.500000 2.500000 0.007670 0.007670 3.642919e-17 9 9 2.500000 2.500000 0.000000 0.000000 0.000000e+00 9 10 2.500000 2.500000 0.004314 0.004314 3.122502e-17 9 11 2.500000 2.500000 -0.000000 0.000000 1.084202e-19 9 12 2.500000 2.500000 0.002637 0.002637 1.170938e-17 9 13 2.500000 2.500000 0.000000 0.000000 0.000000e+00 9 14 2.500000 2.500000 0.001714 0.001714 4.119968e-18 9 15 2.500000 2.500000 0.000000 0.000000 0.000000e+00 9 16 2.500000 2.500000 0.001168 0.001168 2.168404e-19 9 17 2.500000 2.500000 -0.000000 0.000000 5.421011e-20 9 18 2.500000 2.500000 0.000827 0.000828 6.196389e-07 9 19 2.500000 2.500000 0.000000 0.000000 0.000000e+00 9 20 2.500000 2.500000 0.000603 0.000605 2.324729e-06 10 0 2.500000 2.500000 0.981748 0.981748 4.440892e-16 10 1 2.500000 2.500000 -0.000000 0.000000 1.327063e-16 10 2 2.500000 2.500000 0.122718 0.122718 1.665335e-16 10 3 2.500000 2.500000 -0.000000 0.000000 3.491131e-17 10 4 2.500000 2.500000 0.036816 0.036816 6.245005e-17 10 5 2.500000 2.500000 -0.000000 0.000000 1.214306e-17 10 6 2.500000 2.500000 0.015340 0.015340 6.938894e-18 10 7 2.500000 2.500000 -0.000000 0.000000 4.228388e-18 10 8 2.500000 2.500000 0.007670 0.007670 1.474515e-17 10 9 2.500000 2.500000 -0.000000 0.000000 1.409463e-18 10 10 2.500000 2.500000 0.004314 0.004314 1.821460e-17 10 11 2.500000 2.500000 -0.000000 0.000000 3.252607e-19 10 12 2.500000 2.500000 0.002637 0.002637 5.637851e-18 10 13 2.500000 2.500000 -0.000000 0.000000 1.626303e-19 10 14 2.500000 2.500000 0.001714 0.001714 2.168404e-19 10 15 2.500000 2.500000 -0.000000 0.000000 1.626303e-19 10 16 2.500000 2.500000 0.001168 0.001168 3.035766e-18 10 17 2.500000 2.500000 0.000000 0.000000 5.421011e-20 10 18 2.500000 2.500000 0.000828 0.000828 4.119968e-18 10 19 2.500000 2.500000 -0.000000 0.000000 2.710505e-20 10 20 2.500000 2.500000 0.000605 0.000605 1.489517e-07 10 21 2.500000 2.500000 0.000000 0.000000 2.710505e-20 10 22 2.500000 2.500000 0.000453 0.000454 6.303847e-07 TEST18 JACOBI_COMPUTE_NP computes a Gauss-Jacobi rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) (1-x)^alpha (1+x)^beta dx. Order = 1 ALPHA = 0.500000 BETA = 0.500000 0 0.0000000000000000 1.5707963267948968 Order = 2 ALPHA = 0.500000 BETA = 0.500000 0 -0.5000000000000000 0.7853981633974484 1 0.5000000000000000 0.7853981633974484 Order = 3 ALPHA = 0.500000 BETA = 0.500000 0 -0.7071067811865475 0.3926990816987239 1 0.0000000000000000 0.7853981633974484 2 0.7071067811865476 0.3926990816987245 Order = 4 ALPHA = 0.500000 BETA = 0.500000 0 -0.8090169943749475 0.2170787134227060 1 -0.3090169943749475 0.5683194499747424 2 0.3090169943749474 0.5683194499747424 3 0.8090169943749475 0.2170787134227060 Order = 5 ALPHA = 0.500000 BETA = 0.500000 0 -0.8660254037844387 0.1308996938995740 1 -0.5000000000000000 0.3926990816987242 2 0.0000000000000000 0.5235987755982989 3 0.5000000000000000 0.3926990816987242 4 0.8660254037844387 0.1308996938995745 Order = 6 ALPHA = 0.500000 BETA = 0.500000 0 -0.9009688679024191 0.0844886908915884 1 -0.6234898018587335 0.2743330560697777 2 -0.2225209339563144 0.4265764164360819 3 0.2225209339563144 0.4265764164360819 4 0.6234898018587335 0.2743330560697777 5 0.9009688679024191 0.0844886908915884 Order = 7 ALPHA = 0.500000 BETA = 0.500000 0 -0.9238795325112867 0.0575094490319133 1 -0.7071067811865475 0.1963495408493619 2 -0.3826834323650898 0.3351896326668111 3 0.0000000000000000 0.3926990816987242 4 0.3826834323650898 0.3351896326668108 5 0.7071067811865476 0.1963495408493624 6 0.9238795325112867 0.0575094490319133 Order = 8 ALPHA = 0.500000 BETA = 0.500000 0 -0.9396926207859084 0.0408329477091069 1 -0.7660444431189780 0.1442256007956728 2 -0.5000000000000000 0.2617993877991495 3 -0.1736481776669303 0.3385402270935191 4 0.1736481776669303 0.3385402270935191 5 0.5000000000000000 0.2617993877991495 6 0.7660444431189780 0.1442256007956728 7 0.9396926207859084 0.0408329477091075 Order = 9 ALPHA = 0.500000 BETA = 0.500000 0 -0.9510565162951536 0.0299995403716084 1 -0.8090169943749475 0.1085393567113530 2 -0.5877852522924731 0.2056199086476264 3 -0.3090169943749475 0.2841597249873712 4 0.0000000000000000 0.3141592653589794 5 0.3090169943749475 0.2841597249873712 6 0.5877852522924731 0.2056199086476264 7 0.8090169943749475 0.1085393567113530 8 0.9510565162951536 0.0299995403716084 Order = 10 ALPHA = 0.500000 BETA = 0.500000 0 -0.9594929736144974 0.0226689425018590 1 -0.8412535328311812 0.0834785409341889 2 -0.6548607339452851 0.1631221774548165 3 -0.4154150130018864 0.2363135602034873 4 -0.1423148382732851 0.2798149423030965 5 0.1423148382732851 0.2798149423030966 6 0.4154150130018864 0.2363135602034873 7 0.6548607339452851 0.1631221774548165 8 0.8412535328311812 0.0834785409341889 9 0.9594929736144974 0.0226689425018590 Order = 1 ALPHA = 0.500000 BETA = 1.000000 0 0.1428571428571428 1.5084944665313014 Order = 2 ALPHA = 0.500000 BETA = 1.000000 0 -0.3785434359039293 0.6707847483124826 1 0.5603616177221111 0.8377097182188185 Order = 3 ALPHA = 0.500000 BETA = 1.000000 0 -0.6191289006136984 0.2886624303931601 1 0.0828374856848099 0.7711842727220675 2 0.7362914149288884 0.4486477634160737 Order = 4 ALPHA = 0.500000 BETA = 1.000000 0 -0.7447173440086010 0.1380862915234588 1 -0.2308286409011868 0.5011891464890379 2 0.3610180502309864 0.6101454555318188 3 0.8250542504682752 0.2590735729869860 Order = 5 ALPHA = 0.500000 BETA = 1.000000 0 -0.8175993274098121 0.0729555935075893 1 -0.4326084003843104 0.3103351805927452 2 0.0582907697984681 0.5176668424406611 3 0.5335962093870673 0.4466360854739414 4 0.8757120529564132 0.1609007645163642 Order = 6 ALPHA = 0.500000 BETA = 1.000000 0 -0.8633993972645075 0.0417787392098552 1 -0.5667404500118451 0.1951823756778569 2 -0.1659246338882320 0.3864855437976816 3 0.2649141017150741 0.4544246037858499 4 0.6461272538533950 0.3245589588523248 5 0.9072453478183379 0.1060642452077330 Order = 7 ALPHA = 0.500000 BETA = 1.000000 0 -0.8939724663908039 0.0255150146724793 1 -0.6594047339183108 0.1266052232513150 2 -0.3307207485144988 0.2785906683424280 3 0.0449578958653259 0.3894994287201925 4 0.4138036987843730 0.3759820326953335 5 0.7229723820004327 0.2389626799471577 6 0.9281704237863851 0.0733394189023952 Order = 8 ALPHA = 0.500000 BETA = 1.000000 0 -0.9153619779765982 0.0164120900902137 1 -0.7257160980084200 0.0848554733563242 2 -0.4533780733947834 0.2002825508076700 3 -0.1294870729403755 0.3123493200327135 4 0.2089491097631811 0.3577748934099902 5 0.5232643292074005 0.3048404655795938 6 0.7775490329551161 0.1792709060668855 7 0.9427521789659081 0.0527087671879100 Order = 9 ALPHA = 0.500000 BETA = 1.000000 0 -0.9308966361094718 0.0110143919419965 1 -0.7746386880679894 0.0586443046921291 2 -0.5463185334791457 0.1453933047463929 3 -0.2672381326388559 0.2442199669465212 4 0.0365863464439503 0.3121710958691359 5 0.3368351070820349 0.3145887170547920 6 0.6055223420849569 0.2462380941383862 7 0.8176042531316312 0.1371247634867023 8 0.9533131723221203 0.0390998276552453 Order = 10 ALPHA = 0.500000 BETA = 1.000000 0 -0.9425275499488474 0.0076569987337496 1 -0.8116864125172604 0.0416630238085871 2 -0.6180513952640334 0.1070695285744137 3 -0.3766418843493626 0.1896495013540224 4 -0.1061646285278261 0.2614695484772918 5 0.1724237674317270 0.2937514863341334 6 0.4375388507874068 0.2709981593343723 7 0.6686402341966643 0.1996303237647666 8 0.8478228797503713 0.1068280415109269 9 0.9612042779760442 0.0297778546390378 Order = 1 ALPHA = 0.500000 BETA = 2.500000 0 0.4000000000000000 1.9634954084936209 Order = 2 ALPHA = 0.500000 BETA = 2.500000 0 -0.1055161125036901 0.6949608437695367 1 0.6769446839322616 1.2685345647240831 Order = 3 ALPHA = 0.500000 BETA = 2.500000 0 -0.3949726057630553 0.2155740450619749 1 0.2647662774451411 0.9816213720815016 2 0.7968729949845809 0.7662999913501441 Order = 4 ALPHA = 0.500000 BETA = 2.500000 0 -0.5673928432278891 0.0735270499348779 1 -0.0438129709574188 0.4997101315556028 2 0.4782610448380588 0.9082548004475427 3 0.8602174966199764 0.4820034265555977 Order = 5 ALPHA = 0.500000 BETA = 2.500000 0 -0.6766083904771950 0.0281042710539255 1 -0.2615276527389886 0.2379108911216858 2 0.1979083288674991 0.6351035815608072 3 0.6116098169968442 0.7438254266897738 4 0.8978486665826095 0.3185512380674287 Order = 6 ALPHA = 0.500000 BETA = 2.500000 0 -0.7496281758956908 0.0118933444385382 1 -0.4162703993964171 0.1152628351592500 2 -0.0240992331493380 0.3830143850747026 3 0.3678575869061302 0.6464461334770021 4 0.7000843065535390 0.5868443910765002 5 0.9220559149817765 0.2200343192676289 Order = 7 ALPHA = 0.500000 BETA = 2.500000 0 -0.8006672147724484 0.0054858163722489 1 -0.5286792390323163 0.0581588649983062 2 -0.1959098000069028 0.2221608406878788 3 0.1580287890648371 0.4638578133115204 4 0.4905660544916727 0.5963414735975124 5 0.7616344330284673 0.4597291058720926 6 0.9385563889913962 0.1577614936540615 Order = 8 ALPHA = 0.500000 BETA = 2.500000 0 -0.8376646856786829 0.0027198968451134 1 -0.6123079224732051 0.0307056736312335 2 -0.3290736457267179 0.1289926195321417 3 -0.0152678951698322 0.3092909316152765 4 0.2984865079144489 0.4881539875384043 5 0.5815096611955630 0.5250491653932977 6 0.8061111752639282 0.3618821131546534 7 0.9503120678323927 0.1167010207835005 Order = 9 ALPHA = 0.500000 BETA = 2.500000 0 -0.8653022034444986 0.0014331423151996 1 -0.6759411199023827 0.0169372413862485 2 -0.4333019473215542 0.0761680132420717 3 -0.1566993909441265 0.2010801278076634 4 0.1315346053479575 0.3633349438080311 5 0.4080752758535235 0.4761564190243817 6 0.6505301580495421 0.4521871712839320 7 0.8392624591522134 0.2875734496942861 8 0.9589850203521829 0.0886248999318051 Order = 10 ALPHA = 0.500000 BETA = 2.500000 0 -0.8864732152715590 0.0007951842411665 1 -0.7253515462755796 0.0097270000118212 2 -0.5159038845260623 0.0460145772880254 3 -0.2721183637403178 0.1303012691755842 4 -0.0105450517531137 0.2590099865557985 5 0.2510118405448966 0.3871376587183347 6 0.4947368451601082 0.4448595186308595 7 0.7040250190649781 0.3857617289382685 8 0.8646161230141249 0.2310686902877482 9 0.9655674511738290 0.0688197946460142 Order = 1 ALPHA = 1.000000 BETA = 0.500000 0 -0.1428571428571428 1.5084944665313014 Order = 2 ALPHA = 1.000000 BETA = 0.500000 0 -0.5603616177221111 0.8377097182188190 1 0.3785434359039293 0.6707847483124826 Order = 3 ALPHA = 1.000000 BETA = 0.500000 0 -0.7362914149288884 0.4486477634160737 1 -0.0828374856848099 0.7711842727220676 2 0.6191289006136984 0.2886624303931601 Order = 4 ALPHA = 1.000000 BETA = 0.500000 0 -0.8250542504682752 0.2590735729869860 1 -0.3610180502309864 0.6101454555318188 2 0.2308286409011868 0.5011891464890378 3 0.7447173440086011 0.1380862915234592 Order = 5 ALPHA = 1.000000 BETA = 0.500000 0 -0.8757120529564132 0.1609007645163642 1 -0.5335962093870673 0.4466360854739414 2 -0.0582907697984681 0.5176668424406611 3 0.4326084003843104 0.3103351805927452 4 0.8175993274098121 0.0729555935075893 Order = 6 ALPHA = 1.000000 BETA = 0.500000 0 -0.9072453478183379 0.1060642452077330 1 -0.6461272538533950 0.3245589588523248 2 -0.2649141017150741 0.4544246037858501 3 0.1659246338882320 0.3864855437976816 4 0.5667404500118451 0.1951823756778569 5 0.8633993972645074 0.0417787392098546 Order = 7 ALPHA = 1.000000 BETA = 0.500000 0 -0.9281704237863851 0.0733394189023952 1 -0.7229723820004327 0.2389626799471577 2 -0.4138036987843730 0.3759820326953336 3 -0.0449578958653259 0.3894994287201925 4 0.3307207485144988 0.2785906683424280 5 0.6594047339183108 0.1266052232513153 6 0.8939724663908039 0.0255150146724793 Order = 8 ALPHA = 1.000000 BETA = 0.500000 0 -0.9427521789659081 0.0527087671879105 1 -0.7775490329551161 0.1792709060668855 2 -0.5232643292074005 0.3048404655795938 3 -0.2089491097631811 0.3577748934099902 4 0.1294870729403755 0.3123493200327135 5 0.4533780733947834 0.2002825508076697 6 0.7257160980084200 0.0848554733563242 7 0.9153619779765982 0.0164120900902137 Order = 9 ALPHA = 1.000000 BETA = 0.500000 0 -0.9533131723221203 0.0390998276552453 1 -0.8176042531316312 0.1371247634867023 2 -0.6055223420849569 0.2462380941383862 3 -0.3368351070820349 0.3145887170547920 4 -0.0365863464439503 0.3121710958691360 5 0.2672381326388559 0.2442199669465213 6 0.5463185334791457 0.1453933047463929 7 0.7746386880679894 0.0586443046921292 8 0.9308966361094718 0.0110143919419965 Order = 10 ALPHA = 1.000000 BETA = 0.500000 0 -0.9612042779760442 0.0297778546390378 1 -0.8478228797503713 0.1068280415109265 2 -0.6686402341966643 0.1996303237647662 3 -0.4375388507874068 0.2709981593343721 4 -0.1724237674317270 0.2937514863341334 5 0.1061646285278261 0.2614695484772919 6 0.3766418843493626 0.1896495013540224 7 0.6180513952640334 0.1070695285744137 8 0.8116864125172604 0.0416630238085870 9 0.9425275499488474 0.0076569987337497 Order = 1 ALPHA = 1.000000 BETA = 1.000000 0 0.0000000000000000 1.3333333333333333 Order = 2 ALPHA = 1.000000 BETA = 1.000000 0 -0.4472135954999579 0.6666666666666665 1 0.4472135954999580 0.6666666666666667 Order = 3 ALPHA = 1.000000 BETA = 1.000000 0 -0.6546536707079772 0.3111111111111113 1 0.0000000000000000 0.7111111111111110 2 0.6546536707079772 0.3111111111111113 Order = 4 ALPHA = 1.000000 BETA = 1.000000 0 -0.7650553239294646 0.1569499125956942 1 -0.2852315164806451 0.5097167540709727 2 0.2852315164806451 0.5097167540709727 3 0.7650553239294647 0.1569499125956939 Order = 5 ALPHA = 1.000000 BETA = 1.000000 0 -0.8302238962785669 0.0860176821228073 1 -0.4688487934707142 0.3368394607343354 2 -0.0000000000000000 0.4876190476190476 3 0.4688487934707142 0.3368394607343354 4 0.8302238962785670 0.0860176821228076 Order = 6 ALPHA = 1.000000 BETA = 1.000000 0 -0.8717401485096066 0.0505835770160819 1 -0.5917001814331423 0.2216925320225178 2 -0.2092992179024789 0.3943905576280671 3 0.2092992179024789 0.3943905576280671 4 0.5917001814331423 0.2216925320225178 5 0.8717401485096067 0.0505835770160817 Order = 7 ALPHA = 1.000000 BETA = 1.000000 0 -0.8997579954114602 0.0315162001504988 1 -0.6771862795107377 0.1486404045834101 2 -0.3631174638261782 0.3007504247445493 3 0.0000000000000000 0.3715192743764172 4 0.3631174638261782 0.3007504247445493 5 0.6771862795107377 0.1486404045834101 6 0.8997579954114602 0.0315162001504988 Order = 8 ALPHA = 1.000000 BETA = 1.000000 0 -0.9195339081664589 0.0205900956491219 1 -0.7387738651055050 0.1021477023603581 2 -0.4779249498104445 0.2253365549698578 3 -0.1652789576663870 0.3185923136873282 4 0.1652789576663870 0.3185923136873282 5 0.4779249498104445 0.2253365549698580 6 0.7387738651055050 0.1021477023603581 7 0.9195339081664589 0.0205900956491219 Order = 9 ALPHA = 1.000000 BETA = 1.000000 0 -0.9340014304080592 0.0139910561244267 1 -0.7844834736631444 0.0719828561756607 2 -0.5652353269962050 0.1687989736144601 3 -0.2957581355869394 0.2617849830242735 4 -0.0000000000000000 0.3002175954556907 5 0.2957581355869394 0.2617849830242735 6 0.5652353269962050 0.1687989736144601 7 0.7844834736631444 0.0719828561756607 8 0.9340014304080592 0.0139910561244267 Order = 10 ALPHA = 1.000000 BETA = 1.000000 0 -0.9448992722228823 0.0098254048057683 1 -0.8192793216440066 0.0519391437742078 2 -0.6328761530318606 0.1273919483477328 3 -0.3995309409653490 0.2111657418760753 4 -0.1365529328549276 0.2663444278628824 5 0.1365529328549276 0.2663444278628824 6 0.3995309409653490 0.2111657418760753 7 0.6328761530318606 0.1273919483477326 8 0.8192793216440066 0.0519391437742078 9 0.9448992722228822 0.0098254048057681 Order = 1 ALPHA = 1.000000 BETA = 2.500000 0 0.2727272727272727 1.4366613966964776 Order = 2 ALPHA = 1.000000 BETA = 2.500000 0 -0.1843075691322092 0.5823920991551215 1 0.5843075691322092 0.8542692975413559 Order = 3 ALPHA = 1.000000 BETA = 2.500000 0 -0.4423027360564286 0.2004355575312488 1 0.1860296262211931 0.7756953091585612 2 0.7299573203615513 0.4605305300066671 Order = 4 ALPHA = 1.000000 BETA = 2.500000 0 -0.5972505982938984 0.0735380497612851 1 -0.1011259685235210 0.4428338893708901 2 0.4098198869416555 0.6621803747350362 3 0.8102958103105465 0.2581090828292666 Order = 5 ALPHA = 1.000000 BETA = 2.500000 0 -0.6964392762006608 0.0296301520114946 1 -0.3026155820854005 0.2285362779908424 2 0.1412598214162333 0.5280679973185711 3 0.5539710608669561 0.4975219906041118 4 0.8593795315584275 0.1529049787714576 Order = 6 ALPHA = 1.000000 BETA = 2.500000 0 -0.7633972293043557 0.0130439061802296 1 -0.4461529248760192 0.1174255136957451 2 -0.0686784265705635 0.3490370430142563 3 0.3155805295665462 0.5014050892168356 4 0.6517131087948295 0.3603484489584897 5 0.8915801036798852 0.0954013956309209 Order = 7 ALPHA = 1.000000 BETA = 2.500000 0 -0.8105874821854350 0.0062027903657017 1 -0.5508759997785407 0.0619382418889595 2 -0.2306894365569858 0.2164618775946895 3 0.1138876346036921 0.3988213674613817 4 0.4436003431475740 0.4310114591490307 5 0.7208146570340379 0.2599688822120858 6 0.9138502837356576 0.0622567780246282 Order = 8 ALPHA = 1.000000 BETA = 2.500000 0 -0.8450352779186007 0.0031506505662605 1 -0.6291538133028907 0.0338481948002515 2 -0.3563555020358466 0.1322157770527708 3 -0.0516690747442928 0.2867379491637254 4 0.2565958341049683 0.3952631184069368 5 0.5397363757841802 0.3540223921913583 6 0.7713710662668609 0.1892132923890615 7 0.9298950072302367 0.0422100221261126 Order = 9 ALPHA = 1.000000 BETA = 2.500000 0 -0.8709215211401559 0.0016929758046626 1 -0.6889839677544539 0.0191902962920152 2 -0.4549257546478147 0.0812176110475679 3 -0.1865538397099543 0.1974051443898491 4 0.0954122417770832 0.3197898022964820 5 0.3691513199896207 0.3631243013752984 6 0.6134647393951399 0.2850633375254082 7 0.8094259398841321 0.1396134790482662 8 0.9418378189505888 0.0295644489169276 Order = 10 ALPHA = 1.000000 BETA = 2.500000 0 -0.8908519983624362 0.0009546834154140 1 -0.7356332759809727 0.0112695230304568 2 -0.5332475568814602 0.0506378825686693 3 -0.2966592827516151 0.1337845805120266 4 -0.0412830087188659 0.2429565425267210 5 0.2161975251464401 0.3233164884771634 6 0.4589489303958385 0.3198849047578979 7 0.6710962948352331 0.2279739859839844 8 0.8387626487347722 0.1045909447918312 9 0.9509675959234917 0.0212918606323119 Order = 1 ALPHA = 2.500000 BETA = 0.500000 0 -0.4000000000000000 1.9634954084936209 Order = 2 ALPHA = 2.500000 BETA = 0.500000 0 -0.6769446839322616 1.2685345647240849 1 0.1055161125036901 0.6949608437695367 Order = 3 ALPHA = 2.500000 BETA = 0.500000 0 -0.7968729949845809 0.7662999913501441 1 -0.2647662774451411 0.9816213720815016 2 0.3949726057630553 0.2155740450619749 Order = 4 ALPHA = 2.500000 BETA = 0.500000 0 -0.8602174966199764 0.4820034265555970 1 -0.4782610448380588 0.9082548004475427 2 0.0438129709574188 0.4997101315556028 3 0.5673928432278891 0.0735270499348779 Order = 5 ALPHA = 2.500000 BETA = 0.500000 0 -0.8978486665826094 0.3185512380674296 1 -0.6116098169968444 0.7438254266897734 2 -0.1979083288674991 0.6351035815608072 3 0.2615276527389886 0.2379108911216858 4 0.6766083904771950 0.0281042710539255 Order = 6 ALPHA = 2.500000 BETA = 0.500000 0 -0.9220559149817765 0.2200343192676289 1 -0.7000843065535390 0.5868443910765002 2 -0.3678575869061302 0.6464461334770021 3 0.0240992331493380 0.3830143850747027 4 0.4162703993964171 0.1152628351592500 5 0.7496281758956908 0.0118933444385382 Order = 7 ALPHA = 2.500000 BETA = 0.500000 0 -0.9385563889913962 0.1577614936540615 1 -0.7616344330284673 0.4597291058720926 2 -0.4905660544916727 0.5963414735975124 3 -0.1580287890648371 0.4638578133115204 4 0.1959098000069028 0.2221608406878788 5 0.5286792390323162 0.0581588649983059 6 0.8006672147724484 0.0054858163722489 Order = 8 ALPHA = 2.500000 BETA = 0.500000 0 -0.9503120678323927 0.1167010207835005 1 -0.8061111752639282 0.3618821131546534 2 -0.5815096611955630 0.5250491653932977 3 -0.2984865079144489 0.4881539875384043 4 0.0152678951698322 0.3092909316152765 5 0.3290736457267179 0.1289926195321416 6 0.6123079224732050 0.0307056736312334 7 0.8376646856786829 0.0027198968451134 Order = 9 ALPHA = 2.500000 BETA = 0.500000 0 -0.9589850203521829 0.0886248999318064 1 -0.8392624591522134 0.2875734496942861 2 -0.6505301580495421 0.4521871712839320 3 -0.4080752758535235 0.4761564190243817 4 -0.1315346053479575 0.3633349438080311 5 0.1566993909441265 0.2010801278076634 6 0.4333019473215542 0.0761680132420717 7 0.6759411199023827 0.0169372413862485 8 0.8653022034444986 0.0014331423151996 Order = 10 ALPHA = 2.500000 BETA = 0.500000 0 -0.9655674511738290 0.0688197946460142 1 -0.8646161230141249 0.2310686902877482 2 -0.7040250190649781 0.3857617289382685 3 -0.4947368451601082 0.4448595186308592 4 -0.2510118405448966 0.3871376587183347 5 0.0105450517531137 0.2590099865557985 6 0.2721183637403178 0.1303012691755842 7 0.5159038845260623 0.0460145772880254 8 0.7253515462755796 0.0097270000118212 9 0.8864732152715591 0.0007951842411665 Order = 1 ALPHA = 2.500000 BETA = 1.000000 0 -0.2727272727272727 1.4366613966964776 Order = 2 ALPHA = 2.500000 BETA = 1.000000 0 -0.5843075691322092 0.8542692975413559 1 0.1843075691322092 0.5823920991551215 Order = 3 ALPHA = 2.500000 BETA = 1.000000 0 -0.7299573203615513 0.4605305300066671 1 -0.1860296262211931 0.7756953091585612 2 0.4423027360564286 0.2004355575312490 Order = 4 ALPHA = 2.500000 BETA = 1.000000 0 -0.8102958103105465 0.2581090828292662 1 -0.4098198869416555 0.6621803747350362 2 0.1011259685235210 0.4428338893708898 3 0.5972505982938984 0.0735380497612851 Order = 5 ALPHA = 2.500000 BETA = 1.000000 0 -0.8593795315584275 0.1529049787714576 1 -0.5539710608669561 0.4975219906041122 2 -0.1412598214162333 0.5280679973185711 3 0.3026155820854005 0.2285362779908424 4 0.6964392762006608 0.0296301520114946 Order = 6 ALPHA = 2.500000 BETA = 1.000000 0 -0.8915801036798852 0.0954013956309209 1 -0.6517131087948296 0.3603484489584904 2 -0.3155805295665462 0.5014050892168356 3 0.0686784265705634 0.3490370430142563 4 0.4461529248760192 0.1174255136957451 5 0.7633972293043557 0.0130439061802296 Order = 7 ALPHA = 2.500000 BETA = 1.000000 0 -0.9138502837356576 0.0622567780246282 1 -0.7208146570340379 0.2599688822120858 2 -0.4436003431475740 0.4310114591490307 3 -0.1138876346036921 0.3988213674613817 4 0.2306894365569858 0.2164618775946895 5 0.5508759997785407 0.0619382418889595 6 0.8105874821854350 0.0062027903657017 Order = 8 ALPHA = 2.500000 BETA = 1.000000 0 -0.9298950072302367 0.0422100221261126 1 -0.7713710662668609 0.1892132923890619 2 -0.5397363757841802 0.3540223921913583 3 -0.2565958341049683 0.3952631184069370 4 0.0516690747442927 0.2867379491637254 5 0.3563555020358466 0.1322157770527708 6 0.6291538133028907 0.0338481948002515 7 0.8450352779186007 0.0031506505662605 Order = 9 ALPHA = 2.500000 BETA = 1.000000 0 -0.9418378189505888 0.0295644489169276 1 -0.8094259398841321 0.1396134790482662 2 -0.6134647393951399 0.2850633375254075 3 -0.3691513199896207 0.3631243013752985 4 -0.0954122417770831 0.3197898022964820 5 0.1865538397099543 0.1974051443898491 6 0.4549257546478147 0.0812176110475680 7 0.6889839677544539 0.0191902962920152 8 0.8709215211401559 0.0016929758046627 Order = 10 ALPHA = 2.500000 BETA = 1.000000 0 -0.9509675959234917 0.0212918606323124 1 -0.8387626487347722 0.1045909447918318 2 -0.6710962948352331 0.2279739859839844 3 -0.4589489303958385 0.3198849047578984 4 -0.2161975251464401 0.3233164884771634 5 0.0412830087188659 0.2429565425267210 6 0.2966592827516151 0.1337845805120266 7 0.5332475568814602 0.0506378825686693 8 0.7356332759809727 0.0112695230304568 9 0.8908519983624362 0.0009546834154140 Order = 1 ALPHA = 2.500000 BETA = 2.500000 0 0.0000000000000000 0.9817477042468103 Order = 2 ALPHA = 2.500000 BETA = 2.500000 0 -0.3535533905932737 0.4908738521234051 1 0.3535533905932737 0.4908738521234051 Order = 3 ALPHA = 2.500000 BETA = 2.500000 0 -0.5477225575051661 0.2045307717180856 1 0.0000000000000000 0.5726861608106394 2 0.5477225575051661 0.2045307717180856 Order = 4 ALPHA = 2.500000 BETA = 2.500000 0 -0.6660699417556469 0.0870080715314833 1 -0.2373833033084449 0.4038657805919217 2 0.2373833033084449 0.4038657805919217 3 0.6660699417556469 0.0870080715314835 Order = 5 ALPHA = 2.500000 BETA = 2.500000 0 -0.7434770045212192 0.0392893891317672 1 -0.4019050360892101 0.2454174450998077 2 -0.0000000000000000 0.4123340357836603 3 0.4019050360892101 0.2454174450998076 4 0.7434770045212192 0.0392893891317672 Order = 6 ALPHA = 2.500000 BETA = 2.500000 0 -0.7968373566406807 0.0189116433666931 1 -0.5196148456949133 0.1431407318203965 2 -0.1804179569647766 0.3288214769363154 3 0.1804179569647766 0.3288214769363154 4 0.5196148456949133 0.1431407318203969 5 0.7968373566406807 0.0189116433666931 Order = 7 ALPHA = 2.500000 BETA = 2.500000 0 -0.8351564541339112 0.0096585250210008 1 -0.6063141605450342 0.0834446705489847 2 -0.3186902924591673 0.2357822853526957 3 -0.0000000000000000 0.3239767424014473 4 0.3186902924591672 0.2357822853526958 5 0.6063141605450342 0.0834446705489847 6 0.8351564541339112 0.0096585250210010 Order = 8 ALPHA = 2.500000 BETA = 2.500000 0 -0.8635912403827584 0.0051997074221325 1 -0.6718232507247468 0.0494181881565223 2 -0.4260961495798479 0.1615070338519726 3 -0.1459649294626305 0.2747489226927776 4 0.1459649294626305 0.2747489226927777 5 0.4260961495798479 0.1615070338519726 6 0.6718232507247468 0.0494181881565223 7 0.8635912403827584 0.0051997074221325 Order = 9 ALPHA = 2.500000 BETA = 2.500000 0 -0.8852662998295736 0.0029319009849924 1 -0.7224309555097356 0.0299209571936803 2 -0.5107538326640052 0.1088013928543096 3 -0.2643695362367075 0.2155149137501430 4 0.0000000000000000 0.2674093746805597 5 0.2643695362367075 0.2155149137501430 6 0.5107538326640052 0.1088013928543096 7 0.7224309555097356 0.0299209571936803 8 0.8852662998295736 0.0029319009849924 Order = 10 ALPHA = 2.500000 BETA = 2.500000 0 -0.9021636877146698 0.0017216382288932 1 -0.7622868838395971 0.0185564370059903 2 -0.5784393987065429 0.0731613283971410 3 -0.3610608688185057 0.1622660968500981 4 -0.1227285554678421 0.2351683516412825 5 0.1227285554678421 0.2351683516412824 6 0.3610608688185057 0.1622660968500981 7 0.5784393987065429 0.0731613283971409 8 0.7622868838395971 0.0185564370059903 9 0.9021636877146698 0.0017216382288932 TEST19 LAGUERRE_COMPUTE_NP computes a Gauss-Laguerre rule which is appropriate for integrands of the form Integral ( 0 <= x < +oo ) f(x) exp(-x) dx. LAGUERRE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 1.000000e+00 1.000000e+00 0.000000e+00 1 1 1.000000e+00 1.000000e+00 0.000000e+00 1 2 1.000000e+00 2.000000e+00 1.000000e+00 1 3 1.000000e+00 6.000000e+00 5.000000e+00 1 4 1.000000e+00 2.400000e+01 2.300000e+01 2 0 1.000000e+00 1.000000e+00 0.000000e+00 2 1 1.000000e+00 1.000000e+00 2.220446e-16 2 2 2.000000e+00 2.000000e+00 4.440892e-16 2 3 6.000000e+00 6.000000e+00 8.881784e-16 2 4 2.000000e+01 2.400000e+01 4.000000e+00 2 5 6.800000e+01 1.200000e+02 5.200000e+01 2 6 2.320000e+02 7.200000e+02 4.880000e+02 3 0 1.000000e+00 1.000000e+00 6.661338e-16 3 1 1.000000e+00 1.000000e+00 4.440892e-16 3 2 2.000000e+00 2.000000e+00 0.000000e+00 3 3 6.000000e+00 6.000000e+00 0.000000e+00 3 4 2.400000e+01 2.400000e+01 3.552714e-15 3 5 1.200000e+02 1.200000e+02 2.842171e-14 3 6 6.840000e+02 7.200000e+02 3.600000e+01 3 7 4.140000e+03 5.040000e+03 9.000000e+02 3 8 2.566800e+04 4.032000e+04 1.465200e+04 4 0 1.000000e+00 1.000000e+00 1.110223e-15 4 1 1.000000e+00 1.000000e+00 4.440892e-16 4 2 2.000000e+00 2.000000e+00 0.000000e+00 4 3 6.000000e+00 6.000000e+00 0.000000e+00 4 4 2.400000e+01 2.400000e+01 0.000000e+00 4 5 1.200000e+02 1.200000e+02 1.421085e-14 4 6 7.200000e+02 7.200000e+02 1.136868e-13 4 7 5.040000e+03 5.040000e+03 9.094947e-13 4 8 3.974400e+04 4.032000e+04 5.760000e+02 4 9 3.392640e+05 3.628800e+05 2.361600e+04 4 10 3.033216e+06 3.628800e+06 5.955840e+05 5 0 1.000000e+00 1.000000e+00 7.771561e-16 5 1 1.000000e+00 1.000000e+00 2.220446e-16 5 2 2.000000e+00 2.000000e+00 6.661338e-16 5 3 6.000000e+00 6.000000e+00 8.881784e-16 5 4 2.400000e+01 2.400000e+01 3.552714e-15 5 5 1.200000e+02 1.200000e+02 1.421085e-14 5 6 7.200000e+02 7.200000e+02 0.000000e+00 5 7 5.040000e+03 5.040000e+03 9.094947e-13 5 8 4.032000e+04 4.032000e+04 0.000000e+00 5 9 3.628800e+05 3.628800e+05 1.164153e-10 5 10 3.614400e+06 3.628800e+06 1.440000e+04 5 11 3.903840e+07 3.991680e+07 8.784000e+05 5 12 4.472208e+08 4.790016e+08 3.178080e+07 6 0 1.000000e+00 1.000000e+00 2.442491e-15 6 1 1.000000e+00 1.000000e+00 1.332268e-15 6 2 2.000000e+00 2.000000e+00 1.332268e-15 6 3 6.000000e+00 6.000000e+00 2.664535e-15 6 4 2.400000e+01 2.400000e+01 1.421085e-14 6 5 1.200000e+02 1.200000e+02 8.526513e-14 6 6 7.200000e+02 7.200000e+02 5.684342e-13 6 7 5.040000e+03 5.040000e+03 3.637979e-12 6 8 4.032000e+04 4.032000e+04 2.910383e-11 6 9 3.628800e+05 3.628800e+05 2.910383e-10 6 10 3.628800e+06 3.628800e+06 2.793968e-09 6 11 3.991680e+07 3.991680e+07 3.725290e-08 6 12 4.784832e+08 4.790016e+08 5.184000e+05 6 13 6.182957e+09 6.227021e+09 4.406400e+07 6 14 8.501242e+10 8.717829e+10 2.165875e+09 7 0 1.000000e+00 1.000000e+00 3.552714e-15 7 1 1.000000e+00 1.000000e+00 8.881784e-16 7 2 2.000000e+00 2.000000e+00 0.000000e+00 7 3 6.000000e+00 6.000000e+00 8.881784e-16 7 4 2.400000e+01 2.400000e+01 3.552714e-15 7 5 1.200000e+02 1.200000e+02 0.000000e+00 7 6 7.200000e+02 7.200000e+02 1.136868e-13 7 7 5.040000e+03 5.040000e+03 0.000000e+00 7 8 4.032000e+04 4.032000e+04 0.000000e+00 7 9 3.628800e+05 3.628800e+05 1.164153e-10 7 10 3.628800e+06 3.628800e+06 9.313226e-10 7 11 3.991680e+07 3.991680e+07 7.450581e-09 7 12 4.790016e+08 4.790016e+08 1.192093e-07 7 13 6.227021e+09 6.227021e+09 9.536743e-07 7 14 8.715289e+10 8.717829e+10 2.540160e+07 7 15 1.304804e+12 1.307674e+12 2.870381e+09 7 16 2.073870e+13 2.092279e+13 1.840854e+11 8 0 1.000000e+00 1.000000e+00 1.221245e-15 8 1 1.000000e+00 1.000000e+00 2.220446e-16 8 2 2.000000e+00 2.000000e+00 4.440892e-16 8 3 6.000000e+00 6.000000e+00 8.881784e-16 8 4 2.400000e+01 2.400000e+01 7.105427e-15 8 5 1.200000e+02 1.200000e+02 4.263256e-14 8 6 7.200000e+02 7.200000e+02 2.273737e-13 8 7 5.040000e+03 5.040000e+03 1.818989e-12 8 8 4.032000e+04 4.032000e+04 2.182787e-11 8 9 3.628800e+05 3.628800e+05 1.164153e-10 8 10 3.628800e+06 3.628800e+06 9.313226e-10 8 11 3.991680e+07 3.991680e+07 0.000000e+00 8 12 4.790016e+08 4.790016e+08 0.000000e+00 8 13 6.227021e+09 6.227021e+09 9.536743e-07 8 14 8.717829e+10 8.717829e+10 0.000000e+00 8 15 1.307674e+12 1.307674e+12 0.000000e+00 8 16 2.092116e+13 2.092279e+13 1.625702e+09 8 17 3.554517e+14 3.556874e+14 2.357268e+11 8 18 6.383252e+15 6.402374e+15 1.912151e+13 9 0 1.000000e+00 1.000000e+00 3.330669e-16 9 1 1.000000e+00 1.000000e+00 1.665335e-15 9 2 2.000000e+00 2.000000e+00 2.220446e-15 9 3 6.000000e+00 6.000000e+00 4.440892e-15 9 4 2.400000e+01 2.400000e+01 3.552714e-15 9 5 1.200000e+02 1.200000e+02 0.000000e+00 9 6 7.200000e+02 7.200000e+02 0.000000e+00 9 7 5.040000e+03 5.040000e+03 9.094947e-13 9 8 4.032000e+04 4.032000e+04 7.275958e-12 9 9 3.628800e+05 3.628800e+05 1.164153e-10 9 10 3.628800e+06 3.628800e+06 9.313226e-10 9 11 3.991680e+07 3.991680e+07 1.490116e-08 9 12 4.790016e+08 4.790016e+08 1.788139e-07 9 13 6.227021e+09 6.227021e+09 1.907349e-06 9 14 8.717829e+10 8.717829e+10 1.525879e-05 9 15 1.307674e+12 1.307674e+12 2.441406e-04 9 16 2.092279e+13 2.092279e+13 7.812500e-03 9 17 3.556874e+14 3.556874e+14 1.250000e-01 9 18 6.402242e+15 6.402374e+15 1.316819e+11 9 19 1.216213e+17 1.216451e+17 2.383442e+13 9 20 2.430516e+18 2.432902e+18 2.385944e+15 10 0 1.000000e+00 1.000000e+00 3.330669e-15 10 1 1.000000e+00 1.000000e+00 1.443290e-15 10 2 2.000000e+00 2.000000e+00 1.776357e-15 10 3 6.000000e+00 6.000000e+00 3.552714e-15 10 4 2.400000e+01 2.400000e+01 1.421085e-14 10 5 1.200000e+02 1.200000e+02 2.842171e-14 10 6 7.200000e+02 7.200000e+02 1.136868e-13 10 7 5.040000e+03 5.040000e+03 0.000000e+00 10 8 4.032000e+04 4.032000e+04 7.275958e-12 10 9 3.628800e+05 3.628800e+05 5.820766e-11 10 10 3.628800e+06 3.628800e+06 1.396984e-09 10 11 3.991680e+07 3.991680e+07 1.490116e-08 10 12 4.790016e+08 4.790016e+08 1.788139e-07 10 13 6.227021e+09 6.227021e+09 1.907349e-06 10 14 8.717829e+10 8.717829e+10 4.577637e-05 10 15 1.307674e+12 1.307674e+12 4.882812e-04 10 16 2.092279e+13 2.092279e+13 7.812500e-03 10 17 3.556874e+14 3.556874e+14 6.250000e-02 10 18 6.402374e+15 6.402374e+15 1.000000e+00 10 19 1.216451e+17 1.216451e+17 0.000000e+00 10 20 2.432889e+18 2.432902e+18 1.316819e+13 10 21 5.108803e+19 5.109094e+19 2.910170e+15 10 22 1.123648e+21 1.124001e+21 3.524071e+17 TEST20 LAGUERRE_COMPUTE_NP computes a generalized Gauss-Laguerre rule which is appropriate for integrands of the form Integral ( 0 <= x < +oo ) f(x) exp(-x) dx. Order = 1 0 1.0000000000000000 1.0000000000000000 Order = 2 0 0.5857864376269050 0.8535533905932738 1 3.4142135623730949 0.1464466094067263 Order = 3 0 0.4157745567834791 0.7110930099291736 1 2.2942803602790418 0.2785177335692409 2 6.2899450829374794 0.0103892565015861 Order = 4 0 0.3225476896193922 0.6031541043416347 1 1.7457611011583467 0.3574186924377997 2 4.5366202969211278 0.0388879085150054 3 9.3950709123011329 0.0005392947055613 Order = 5 0 0.2635603197181409 0.5217556105828079 1 1.4134030591065168 0.3986668110831759 2 3.5964257710407219 0.0759424496817076 3 7.0858100058588374 0.0036117586799220 4 12.6408008442757822 0.0000233699723858 Order = 6 0 0.2228466041792606 0.4589646739499650 1 1.1889321016726229 0.4170008307721219 2 2.9927363260593141 0.1133733820740450 3 5.7751435691045110 0.0103991974531491 4 9.8374674183825892 0.0002610172028149 5 15.9828739806017026 0.0000008985479064 Order = 7 0 0.1930436765603623 0.4093189517012772 1 1.0266648953391919 0.4218312778617200 2 2.5678767449507465 0.1471263486575052 3 4.9003530845264844 0.0206335144687169 4 8.1821534445628608 0.0010740101432807 5 12.7341802917978129 0.0000158654643486 6 19.3957278622625395 0.0000000317031548 Order = 8 0 0.1702796323051010 0.3691885893416355 1 0.9037017767993799 0.4187867808143441 2 2.2510866298661312 0.1757949866371716 3 4.2667001702876588 0.0333434922612157 4 7.0459054023934655 0.0027945362352257 5 10.7585160101809958 0.0000907650877336 6 15.7406786412780040 0.0000008485746716 7 22.8631317368892653 0.0000000010480012 Order = 9 0 0.1523222277318083 0.3361264217979637 1 0.8072200227422559 0.4112139804239832 2 2.0051351556193473 0.1992875253708851 3 3.7834739733312328 0.0474605627656516 4 6.2049567778766130 0.0055996266107946 5 9.3729852516875756 0.0003052497670932 6 13.4662369110920928 0.0000065921230261 7 18.8335977889916961 0.0000000411076933 8 26.3740718909273753 0.0000000000329087 Order = 10 0 0.1377934705404924 0.3084411157650176 1 0.7294545495031703 0.4011199291552729 2 1.8083429017403159 0.2180682876118093 3 3.4014336978548996 0.0620874560986777 4 5.5524961400638038 0.0095015169751811 5 8.3301527467644974 0.0007530083885875 6 11.8437858379000662 0.0000282592334960 7 16.2792578313781036 0.0000004249313985 8 21.9965858119807613 0.0000000018395648 9 29.9206970122738909 0.0000000000009912 TEST21 LEGENDRE_COMPUTE_NP computes a Gauss-Legendre rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. LEGENDRE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 2.000000 2.000000 0.000000e+00 1 1 0.000000 0.000000 0.000000e+00 1 2 0.000000 0.666667 6.666667e-01 1 3 0.000000 0.000000 0.000000e+00 1 4 0.000000 0.400000 4.000000e-01 2 0 2.000000 2.000000 8.881784e-16 2 1 0.000000 0.000000 0.000000e+00 2 2 0.666667 0.666667 1.110223e-16 2 3 0.000000 0.000000 0.000000e+00 2 4 0.222222 0.400000 1.777778e-01 2 5 0.000000 0.000000 0.000000e+00 2 6 0.074074 0.285714 2.116402e-01 3 0 2.000000 2.000000 4.440892e-16 3 1 0.000000 0.000000 0.000000e+00 3 2 0.666667 0.666667 2.220446e-16 3 3 0.000000 0.000000 0.000000e+00 3 4 0.400000 0.400000 5.551115e-17 3 5 0.000000 0.000000 0.000000e+00 3 6 0.240000 0.285714 4.571429e-02 3 7 0.000000 0.000000 0.000000e+00 3 8 0.144000 0.222222 7.822222e-02 4 0 2.000000 2.000000 0.000000e+00 4 1 0.000000 0.000000 5.551115e-17 4 2 0.666667 0.666667 0.000000e+00 4 3 0.000000 0.000000 0.000000e+00 4 4 0.400000 0.400000 0.000000e+00 4 5 0.000000 0.000000 0.000000e+00 4 6 0.285714 0.285714 0.000000e+00 4 7 0.000000 0.000000 0.000000e+00 4 8 0.210612 0.222222 1.160998e-02 4 9 0.000000 0.000000 0.000000e+00 4 10 0.156035 0.181818 2.578320e-02 5 0 2.000000 2.000000 0.000000e+00 5 1 0.000000 0.000000 0.000000e+00 5 2 0.666667 0.666667 1.110223e-16 5 3 0.000000 0.000000 0.000000e+00 5 4 0.400000 0.400000 0.000000e+00 5 5 0.000000 0.000000 0.000000e+00 5 6 0.285714 0.285714 0.000000e+00 5 7 0.000000 0.000000 0.000000e+00 5 8 0.222222 0.222222 2.775558e-17 5 9 0.000000 0.000000 0.000000e+00 5 10 0.178886 0.181818 2.931812e-03 5 11 0.000000 0.000000 0.000000e+00 5 12 0.145853 0.153846 7.993574e-03 6 0 2.000000 2.000000 0.000000e+00 6 1 0.000000 0.000000 8.326673e-17 6 2 0.666667 0.666667 2.220446e-16 6 3 -0.000000 0.000000 2.775558e-17 6 4 0.400000 0.400000 1.665335e-16 6 5 0.000000 0.000000 0.000000e+00 6 6 0.285714 0.285714 1.110223e-16 6 7 0.000000 0.000000 0.000000e+00 6 8 0.222222 0.222222 5.551115e-17 6 9 0.000000 0.000000 0.000000e+00 6 10 0.181818 0.181818 8.326673e-17 6 11 0.000000 0.000000 0.000000e+00 6 12 0.153108 0.153846 7.380787e-04 6 13 0.000000 0.000000 0.000000e+00 6 14 0.130949 0.133333 2.384218e-03 7 0 2.000000 2.000000 4.440892e-16 7 1 0.000000 0.000000 1.387779e-17 7 2 0.666667 0.666667 4.440892e-16 7 3 -0.000000 0.000000 4.163336e-17 7 4 0.400000 0.400000 3.330669e-16 7 5 0.000000 0.000000 0.000000e+00 7 6 0.285714 0.285714 3.330669e-16 7 7 0.000000 0.000000 1.387779e-17 7 8 0.222222 0.222222 2.775558e-16 7 9 0.000000 0.000000 1.387779e-17 7 10 0.181818 0.181818 1.942890e-16 7 11 0.000000 0.000000 0.000000e+00 7 12 0.153846 0.153846 1.665335e-16 7 13 0.000000 0.000000 0.000000e+00 7 14 0.133148 0.133333 1.854659e-04 7 15 0.000000 0.000000 0.000000e+00 7 16 0.116955 0.117647 6.923502e-04 8 0 2.000000 2.000000 0.000000e+00 8 1 -0.000000 0.000000 1.387779e-17 8 2 0.666667 0.666667 2.220446e-16 8 3 0.000000 0.000000 0.000000e+00 8 4 0.400000 0.400000 5.551115e-17 8 5 0.000000 0.000000 0.000000e+00 8 6 0.285714 0.285714 5.551115e-17 8 7 0.000000 0.000000 1.387779e-17 8 8 0.222222 0.222222 5.551115e-17 8 9 0.000000 0.000000 0.000000e+00 8 10 0.181818 0.181818 0.000000e+00 8 11 0.000000 0.000000 0.000000e+00 8 12 0.153846 0.153846 2.775558e-17 8 13 0.000000 0.000000 0.000000e+00 8 14 0.133333 0.133333 2.775558e-17 8 15 0.000000 0.000000 0.000000e+00 8 16 0.117601 0.117647 4.654831e-05 8 17 0.000000 0.000000 0.000000e+00 8 18 0.105066 0.105263 1.971362e-04 9 0 2.000000 2.000000 0.000000e+00 9 1 0.000000 0.000000 1.387779e-17 9 2 0.666667 0.666667 0.000000e+00 9 3 0.000000 0.000000 0.000000e+00 9 4 0.400000 0.400000 5.551115e-17 9 5 0.000000 0.000000 1.387779e-17 9 6 0.285714 0.285714 0.000000e+00 9 7 0.000000 0.000000 0.000000e+00 9 8 0.222222 0.222222 5.551115e-17 9 9 0.000000 0.000000 0.000000e+00 9 10 0.181818 0.181818 2.775558e-17 9 11 0.000000 0.000000 0.000000e+00 9 12 0.153846 0.153846 2.775558e-17 9 13 0.000000 0.000000 0.000000e+00 9 14 0.133333 0.133333 0.000000e+00 9 15 0.000000 0.000000 0.000000e+00 9 16 0.117647 0.117647 1.387779e-17 9 17 0.000000 0.000000 0.000000e+00 9 18 0.105251 0.105263 1.167311e-05 9 19 0.000000 0.000000 0.000000e+00 9 20 0.095183 0.095238 5.529194e-05 10 0 2.000000 2.000000 4.440892e-16 10 1 -0.000000 0.000000 5.551115e-17 10 2 0.666667 0.666667 1.110223e-16 10 3 0.000000 0.000000 6.938894e-18 10 4 0.400000 0.400000 0.000000e+00 10 5 0.000000 0.000000 6.938894e-18 10 6 0.285714 0.285714 1.110223e-16 10 7 -0.000000 0.000000 6.938894e-18 10 8 0.222222 0.222222 1.110223e-16 10 9 -0.000000 0.000000 6.938894e-18 10 10 0.181818 0.181818 1.110223e-16 10 11 0.000000 0.000000 6.938894e-18 10 12 0.153846 0.153846 8.326673e-17 10 13 0.000000 0.000000 0.000000e+00 10 14 0.133333 0.133333 8.326673e-17 10 15 0.000000 0.000000 0.000000e+00 10 16 0.117647 0.117647 6.938894e-17 10 17 0.000000 0.000000 0.000000e+00 10 18 0.105263 0.105263 5.551115e-17 10 19 0.000000 0.000000 0.000000e+00 10 20 0.095235 0.095238 2.925590e-06 10 21 0.000000 0.000000 0.000000e+00 10 22 0.086941 0.086957 1.532420e-05 TEST22 LEGENDRE_COMPUTE_NP computes a Gauss-Legendre rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. Order = 1 0 0.0000000000000000 2.0000000000000000 Order = 2 0 -0.5773502691896258 0.9999999999999996 1 0.5773502691896258 0.9999999999999996 Order = 3 0 -0.7745966692414833 0.5555555555555558 1 0.0000000000000000 0.8888888888888888 2 0.7745966692414833 0.5555555555555558 Order = 4 0 -0.8611363115940526 0.3478548451374539 1 -0.3399810435848563 0.6521451548625461 2 0.3399810435848563 0.6521451548625461 3 0.8611363115940526 0.3478548451374539 Order = 5 0 -0.9061798459386640 0.2369268850561891 1 -0.5384693101056831 0.4786286704993665 2 0.0000000000000000 0.5688888888888889 3 0.5384693101056831 0.4786286704993665 4 0.9061798459386640 0.2369268850561891 Order = 6 0 -0.9324695142031521 0.1713244923791702 1 -0.6612093864662645 0.3607615730481386 2 -0.2386191860831969 0.4679139345726910 3 0.2386191860831969 0.4679139345726910 4 0.6612093864662645 0.3607615730481386 5 0.9324695142031521 0.1713244923791702 Order = 7 0 -0.9491079123427585 0.1294849661688699 1 -0.7415311855993945 0.2797053914892767 2 -0.4058451513773972 0.3818300505051190 3 0.0000000000000000 0.4179591836734694 4 0.4058451513773972 0.3818300505051190 5 0.7415311855993945 0.2797053914892767 6 0.9491079123427585 0.1294849661688699 Order = 8 0 -0.9602898564975362 0.1012285362903764 1 -0.7966664774136267 0.2223810344533744 2 -0.5255324099163290 0.3137066458778873 3 -0.1834346424956498 0.3626837833783618 4 0.1834346424956498 0.3626837833783618 5 0.5255324099163290 0.3137066458778873 6 0.7966664774136267 0.2223810344533744 7 0.9602898564975362 0.1012285362903764 Order = 9 0 -0.9681602395076261 0.0812743883615744 1 -0.8360311073266358 0.1806481606948574 2 -0.6133714327005904 0.2606106964029354 3 -0.3242534234038089 0.3123470770400029 4 0.0000000000000000 0.3302393550012598 5 0.3242534234038089 0.3123470770400029 6 0.6133714327005904 0.2606106964029354 7 0.8360311073266358 0.1806481606948574 8 0.9681602395076261 0.0812743883615744 Order = 10 0 -0.9739065285171717 0.0666713443086880 1 -0.8650633666889845 0.1494513491505806 2 -0.6794095682990244 0.2190863625159821 3 -0.4333953941292472 0.2692667193099965 4 -0.1488743389816312 0.2955242247147530 5 0.1488743389816312 0.2955242247147530 6 0.4333953941292472 0.2692667193099965 7 0.6794095682990244 0.2190863625159821 8 0.8650633666889845 0.1494513491505806 9 0.9739065285171717 0.0666713443086880 SANDIA_RULES_PRB Normal end of execution. 20 January 2017 08:58:13 PM