03-Jun-2023 14:07:03 SANDIA_RULES_TEST MATLAB version. Test the SANDIA_RULES library. SANDIA_RULES_TEST01 CHEBYSHEV1_COMPUTE computes a Gauss-Chebyshev type 1 rule which is appropriate for integrands of the form Integral ( 0 <= x < +oo ) 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 1.923671e-16 1 2 0.000000 1.570796 1.570796e+00 1 3 0.000000 0.000000 7.212610e-49 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 6.412236e-17 3 2 1.570796 1.570796 2.220446e-16 3 3 0.000000 0.000000 2.404203e-49 3 4 1.178097 1.178097 2.220446e-16 3 5 0.000000 0.000000 9.014318e-82 3 6 0.883573 0.981748 9.817477e-02 3 7 0.000000 0.000000 3.379828e-114 3 8 0.662680 0.859029 1.963495e-01 4 0 3.141593 3.141593 0.000000e+00 4 1 0.000000 0.000000 5.551115e-17 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 9.398457e-17 5 2 1.570796 1.570796 0.000000e+00 5 3 0.000000 0.000000 5.551115e-17 5 4 1.178097 1.178097 2.220446e-16 5 5 0.000000 0.000000 4.163336e-17 5 6 0.981748 0.981748 1.110223e-16 5 7 0.000000 0.000000 2.081668e-17 5 8 0.859029 0.859029 4.440892e-16 5 9 0.000000 0.000000 9.540979e-18 5 10 0.766990 0.773126 6.135923e-03 5 11 0.000000 0.000000 3.903128e-18 5 12 0.690291 0.708699 1.840777e-02 6 0 3.141593 3.141593 0.000000e+00 6 1 0.000000 0.000000 2.220446e-16 6 2 1.570796 1.570796 2.220446e-16 6 3 0.000000 0.000000 2.775558e-16 6 4 1.178097 1.178097 2.220446e-16 6 5 0.000000 0.000000 2.775558e-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 1.385033e-16 7 2 1.570796 1.570796 2.220446e-16 7 3 0.000000 0.000000 1.249001e-16 7 4 1.178097 1.178097 0.000000e+00 7 5 0.000000 0.000000 9.280771e-17 7 6 0.981748 0.981748 1.110223e-16 7 7 -0.000000 0.000000 3.816392e-17 7 8 0.859029 0.859029 1.110223e-16 7 9 -0.000000 0.000000 1.794897e-16 7 10 0.773126 0.773126 3.330669e-16 7 11 -0.000000 0.000000 2.860261e-16 7 12 0.708699 0.708699 6.661338e-16 7 13 -0.000000 0.000000 3.248761e-16 7 14 0.657694 0.658078 3.834952e-04 7 15 -0.000000 0.000000 4.057586e-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 1.110223e-16 8 2 1.570796 1.570796 0.000000e+00 8 3 0.000000 0.000000 2.775558e-17 8 4 1.178097 1.178097 0.000000e+00 8 5 -0.000000 0.000000 5.551115e-17 8 6 0.981748 0.981748 2.220446e-16 8 7 0.000000 0.000000 0.000000e+00 8 8 0.859029 0.859029 1.110223e-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 1.942890e-16 9 2 1.570796 1.570796 2.220446e-16 9 3 0.000000 0.000000 1.942890e-16 9 4 1.178097 1.178097 2.220446e-16 9 5 0.000000 0.000000 1.942890e-16 9 6 0.981748 0.981748 2.220446e-16 9 7 0.000000 0.000000 2.498002e-16 9 8 0.859029 0.859029 3.330669e-16 9 9 0.000000 0.000000 2.081668e-16 9 10 0.773126 0.773126 3.330669e-16 9 11 0.000000 0.000000 2.081668e-16 9 12 0.708699 0.708699 3.330669e-16 9 13 0.000000 0.000000 1.734723e-16 9 14 0.658078 0.658078 3.330669e-16 9 15 0.000000 0.000000 1.526557e-16 9 16 0.616948 0.616948 4.440892e-16 9 17 0.000000 0.000000 1.318390e-16 9 18 0.582649 0.582673 2.396845e-05 9 19 0.000000 0.000000 1.075529e-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 2.498002e-16 10 2 1.570796 1.570796 0.000000e+00 10 3 0.000000 0.000000 3.330669e-16 10 4 1.178097 1.178097 0.000000e+00 10 5 0.000000 0.000000 3.608225e-16 10 6 0.981748 0.981748 0.000000e+00 10 7 0.000000 0.000000 3.608225e-16 10 8 0.859029 0.859029 2.220446e-16 10 9 0.000000 0.000000 4.163336e-16 10 10 0.773126 0.773126 1.110223e-16 10 11 0.000000 0.000000 4.440892e-16 10 12 0.708699 0.708699 0.000000e+00 10 13 0.000000 0.000000 4.718448e-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 5.828671e-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 SANDIA_RULES_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 1 0.0000000000000001 3.1415926535897931 Order = 2 1 -0.7071067811865475 1.5707963267948966 2 0.7071067811865476 1.5707963267948966 Order = 3 1 -0.8660254037844387 1.0471975511965976 2 0.0000000000000001 1.0471975511965976 3 0.8660254037844387 1.0471975511965976 Order = 4 1 -0.9238795325112867 0.7853981633974483 2 -0.3826834323650897 0.7853981633974483 3 0.3826834323650898 0.7853981633974483 4 0.9238795325112867 0.7853981633974483 Order = 5 1 -0.9510565162951535 0.6283185307179586 2 -0.5877852522924730 0.6283185307179586 3 0.0000000000000001 0.6283185307179586 4 0.5877852522924731 0.6283185307179586 5 0.9510565162951535 0.6283185307179586 Order = 6 1 -0.9659258262890682 0.5235987755982988 2 -0.7071067811865475 0.5235987755982988 3 -0.2588190451025206 0.5235987755982988 4 0.2588190451025207 0.5235987755982988 5 0.7071067811865476 0.5235987755982988 6 0.9659258262890683 0.5235987755982988 Order = 7 1 -0.9749279121818237 0.4487989505128276 2 -0.7818314824680295 0.4487989505128276 3 -0.4338837391175581 0.4487989505128276 4 0.0000000000000001 0.4487989505128276 5 0.4338837391175582 0.4487989505128276 6 0.7818314824680298 0.4487989505128276 7 0.9749279121818236 0.4487989505128276 Order = 8 1 -0.9807852804032304 0.3926990816987241 2 -0.8314696123025453 0.3926990816987241 3 -0.5555702330196020 0.3926990816987241 4 -0.1950903220161282 0.3926990816987241 5 0.1950903220161283 0.3926990816987241 6 0.5555702330196023 0.3926990816987241 7 0.8314696123025452 0.3926990816987241 8 0.9807852804032304 0.3926990816987241 Order = 9 1 -0.9848077530122080 0.3490658503988659 2 -0.8660254037844385 0.3490658503988659 3 -0.6427876096865394 0.3490658503988659 4 -0.3420201433256685 0.3490658503988659 5 0.0000000000000001 0.3490658503988659 6 0.3420201433256688 0.3490658503988659 7 0.6427876096865394 0.3490658503988659 8 0.8660254037844387 0.3490658503988659 9 0.9848077530122080 0.3490658503988659 Order = 10 1 -0.9876883405951377 0.3141592653589793 2 -0.8910065241883678 0.3141592653589793 3 -0.7071067811865475 0.3141592653589793 4 -0.4539904997395467 0.3141592653589793 5 -0.1564344650402306 0.3141592653589793 6 0.1564344650402309 0.3141592653589793 7 0.4539904997395468 0.3141592653589793 8 0.7071067811865476 0.3141592653589793 9 0.8910065241883679 0.3141592653589793 10 0.9876883405951378 0.3141592653589793 SANDIA_RULES_TEST03 CHEBYSHEV2_COMPUTE computes a Gauss-Chebyshev type 2 rule which is appropriate for integrands of the form Integral ( 0 <= x < +oo ) 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 9.618353e-17 1 2 0.000000 0.392699 3.926991e-01 1 3 0.000000 0.000000 3.606305e-49 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 6.293054e-17 3 2 0.392699 0.392699 0.000000e+00 3 3 0.000000 0.000000 1.803152e-49 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 0.000000e+00 4 1 -0.000000 0.000000 2.775558e-17 4 2 0.392699 0.392699 0.000000e+00 4 3 0.000000 0.000000 1.387779e-17 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 8.757233e-17 5 2 0.392699 0.392699 1.110223e-16 5 3 0.000000 0.000000 7.632783e-17 5 4 0.196350 0.196350 2.775558e-17 5 5 0.000000 0.000000 3.642919e-17 5 6 0.122718 0.122718 1.387779e-17 5 7 0.000000 0.000000 1.301043e-17 5 8 0.085903 0.085903 2.775558e-17 5 9 0.000000 0.000000 4.228388e-18 5 10 0.062893 0.064427 1.533981e-03 5 11 0.000000 0.000000 1.328148e-18 5 12 0.046786 0.050621 3.834952e-03 6 0 1.570796 1.570796 0.000000e+00 6 1 -0.000000 0.000000 5.551115e-17 6 2 0.392699 0.392699 0.000000e+00 6 3 0.000000 0.000000 8.673617e-19 6 4 0.196350 0.196350 2.775558e-17 6 5 0.000000 0.000000 6.938894e-18 6 6 0.122718 0.122718 2.775558e-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 1.734723e-17 6 10 0.064427 0.064427 1.387779e-17 6 11 0.000000 0.000000 1.734723e-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 3.709692e-18 7 2 0.392699 0.392699 5.551115e-17 7 3 -0.000000 0.000000 1.040834e-17 7 4 0.196350 0.196350 2.775558e-17 7 5 -0.000000 0.000000 1.474515e-17 7 6 0.122718 0.122718 2.775558e-17 7 7 -0.000000 0.000000 1.230569e-17 7 8 0.085903 0.085903 1.387779e-17 7 9 -0.000000 0.000000 8.822695e-18 7 10 0.064427 0.064427 1.387779e-17 7 11 -0.000000 0.000000 1.162637e-17 7 12 0.050621 0.050621 6.938894e-18 7 13 -0.000000 0.000000 7.489465e-18 7 14 0.041034 0.041130 9.587380e-05 7 15 -0.000000 0.000000 8.638360e-18 7 16 0.033939 0.034275 3.355583e-04 8 0 1.570796 1.570796 0.000000e+00 8 1 0.000000 0.000000 0.000000e+00 8 2 0.392699 0.392699 5.551115e-17 8 3 0.000000 0.000000 2.775558e-17 8 4 0.196350 0.196350 2.775558e-17 8 5 0.000000 0.000000 1.387779e-17 8 6 0.122718 0.122718 0.000000e+00 8 7 0.000000 0.000000 1.040834e-17 8 8 0.085903 0.085903 1.387779e-17 8 9 0.000000 0.000000 1.214306e-17 8 10 0.064427 0.064427 1.387779e-17 8 11 0.000000 0.000000 1.040834e-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 1.387779e-17 8 18 0.029038 0.029134 9.587380e-05 9 0 1.570796 1.570796 0.000000e+00 9 1 -0.000000 0.000000 1.387779e-17 9 2 0.392699 0.392699 0.000000e+00 9 3 0.000000 0.000000 6.938894e-18 9 4 0.196350 0.196350 0.000000e+00 9 5 0.000000 0.000000 3.469447e-18 9 6 0.122718 0.122718 1.387779e-17 9 7 0.000000 0.000000 6.938894e-18 9 8 0.085903 0.085903 1.387779e-17 9 9 0.000000 0.000000 6.722053e-18 9 10 0.064427 0.064427 1.387779e-17 9 11 -0.000000 0.000000 2.168404e-18 9 12 0.050621 0.050621 6.938894e-18 9 13 -0.000000 0.000000 2.114194e-18 9 14 0.041130 0.041130 2.081668e-17 9 15 -0.000000 0.000000 3.266159e-18 9 16 0.034275 0.034275 6.938894e-18 9 17 -0.000000 0.000000 2.524158e-18 9 18 0.029128 0.029134 5.992112e-06 9 19 -0.000000 0.000000 2.788432e-18 9 20 0.025134 0.025161 2.696451e-05 10 0 1.570796 1.570796 0.000000e+00 10 1 0.000000 0.000000 2.081668e-17 10 2 0.392699 0.392699 0.000000e+00 10 3 0.000000 0.000000 2.775558e-17 10 4 0.196350 0.196350 0.000000e+00 10 5 0.000000 0.000000 1.994932e-17 10 6 0.122718 0.122718 1.387779e-17 10 7 0.000000 0.000000 1.723881e-17 10 8 0.085903 0.085903 1.387779e-17 10 9 0.000000 0.000000 1.626303e-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 0.000000e+00 10 13 0.000000 0.000000 1.127570e-17 10 14 0.041130 0.041130 1.387779e-17 10 15 0.000000 0.000000 7.806256e-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 1.734723e-17 10 19 0.000000 0.000000 6.071532e-18 10 20 0.025159 0.025161 1.498028e-06 10 21 0.000000 0.000000 4.336809e-18 10 22 0.022008 0.022016 7.490141e-06 SANDIA_RULES_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 1 0.0000000000000001 1.5707963267948966 Order = 2 1 -0.4999999999999998 0.7853981633974484 2 0.5000000000000001 0.7853981633974481 Order = 3 1 -0.7071067811865475 0.3926990816987243 2 0.0000000000000001 0.7853981633974483 3 0.7071067811865476 0.3926990816987240 Order = 4 1 -0.8090169943749473 0.2170787134227061 2 -0.3090169943749473 0.5683194499747424 3 0.3090169943749475 0.5683194499747423 4 0.8090169943749475 0.2170787134227060 Order = 5 1 -0.8660254037844387 0.1308996938995747 2 -0.4999999999999998 0.3926990816987242 3 0.0000000000000001 0.5235987755982988 4 0.5000000000000001 0.3926990816987240 5 0.8660254037844387 0.1308996938995747 Order = 6 1 -0.9009688679024190 0.0844886908915886 2 -0.6234898018587335 0.2743330560697779 3 -0.2225209339563143 0.4265764164360819 4 0.2225209339563144 0.4265764164360819 5 0.6234898018587336 0.2743330560697778 6 0.9009688679024191 0.0844886908915886 Order = 7 1 -0.9238795325112867 0.0575094490319132 2 -0.7071067811865475 0.1963495408493621 3 -0.3826834323650897 0.3351896326668110 4 0.0000000000000001 0.3926990816987241 5 0.3826834323650898 0.3351896326668110 6 0.7071067811865476 0.1963495408493620 7 0.9238795325112867 0.0575094490319131 Order = 8 1 -0.9396926207859083 0.0408329477091071 2 -0.7660444431189779 0.1442256007956728 3 -0.4999999999999998 0.2617993877991495 4 -0.1736481776669303 0.3385402270935190 5 0.1736481776669304 0.3385402270935190 6 0.5000000000000001 0.2617993877991494 7 0.7660444431189780 0.1442256007956727 8 0.9396926207859084 0.0408329477091071 Order = 9 1 -0.9510565162951535 0.0299995403716082 2 -0.8090169943749473 0.1085393567113530 3 -0.5877852522924730 0.2056199086476263 4 -0.3090169943749473 0.2841597249873712 5 0.0000000000000001 0.3141592653589793 6 0.3090169943749475 0.2841597249873711 7 0.5877852522924731 0.2056199086476263 8 0.8090169943749475 0.1085393567113530 9 0.9510565162951535 0.0299995403716082 Order = 10 1 -0.9594929736144974 0.0226689425018588 2 -0.8412535328311811 0.0834785409341891 3 -0.6548607339452850 0.1631221774548166 4 -0.4154150130018863 0.2363135602034873 5 -0.1423148382732850 0.2798149423030966 6 0.1423148382732851 0.2798149423030965 7 0.4154150130018864 0.2363135602034873 8 0.6548607339452851 0.1631221774548166 9 0.8412535328311812 0.0834785409341890 10 0.9594929736144974 0.0226689425018588 SANDIA_RULES_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, 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 0.000000e+00 4 1 0.000000 0.000000 4.996004e-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 1.665335e-16 5 2 0.666667 0.666667 1.110223e-16 5 3 0.000000 0.000000 1.387779e-16 5 4 0.400000 0.400000 0.000000e+00 5 5 0.000000 0.000000 9.714451e-17 5 6 0.266667 0.285714 1.904762e-02 5 7 0.000000 0.000000 6.245005e-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 2.498002e-16 7 2 0.666667 0.666667 0.000000e+00 7 3 0.000000 0.000000 1.318390e-16 7 4 0.400000 0.400000 5.551115e-17 7 5 0.000000 0.000000 5.204170e-17 7 6 0.285714 0.285714 0.000000e+00 7 7 0.000000 0.000000 1.951564e-17 7 8 0.221429 0.222222 7.936508e-04 7 9 0.000000 0.000000 3.903128e-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 8.326673e-17 8 2 0.666667 0.666667 1.110223e-16 8 3 0.000000 0.000000 9.714451e-17 8 4 0.400000 0.400000 0.000000e+00 8 5 0.000000 0.000000 8.326673e-17 8 6 0.285714 0.285714 0.000000e+00 8 7 0.000000 0.000000 9.020562e-17 8 8 0.222024 0.222222 1.984127e-04 8 9 0.000000 0.000000 9.020562e-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 1.942890e-16 9 2 0.666667 0.666667 1.110223e-16 9 3 0.000000 0.000000 1.526557e-16 9 4 0.400000 0.400000 0.000000e+00 9 5 0.000000 0.000000 7.632783e-17 9 6 0.285714 0.285714 0.000000e+00 9 7 0.000000 0.000000 5.898060e-17 9 8 0.222222 0.222222 0.000000e+00 9 9 0.000000 0.000000 4.683753e-17 9 10 0.181746 0.181818 7.215007e-05 9 11 0.000000 0.000000 3.209238e-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 3.053113e-16 10 2 0.666667 0.666667 0.000000e+00 10 3 0.000000 0.000000 2.498002e-16 10 4 0.400000 0.400000 0.000000e+00 10 5 0.000000 0.000000 1.873501e-16 10 6 0.285714 0.285714 5.551115e-17 10 7 0.000000 0.000000 1.630640e-16 10 8 0.222222 0.222222 2.775558e-17 10 9 0.000000 0.000000 1.595946e-16 10 10 0.181796 0.181818 2.254690e-05 10 11 0.000000 0.000000 1.387779e-16 10 12 0.153757 0.153846 8.871337e-05 SANDIA_RULES_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 1 0.0000000000000000 2.0000000000000000 Order = 2 1 -1.0000000000000000 1.0000000000000000 2 1.0000000000000000 1.0000000000000000 Order = 3 1 -1.0000000000000000 0.3333333333333334 2 0.0000000000000000 1.3333333333333333 3 1.0000000000000000 0.3333333333333334 Order = 4 1 -1.0000000000000000 0.1111111111111111 2 -0.4999999999999998 0.8888888888888888 3 0.5000000000000001 0.8888888888888892 4 1.0000000000000000 0.1111111111111111 Order = 5 1 -1.0000000000000000 0.0666666666666667 2 -0.7071067811865475 0.5333333333333333 3 0.0000000000000000 0.7999999999999999 4 0.7071067811865476 0.5333333333333334 5 1.0000000000000000 0.0666666666666667 Order = 6 1 -1.0000000000000000 0.0400000000000000 2 -0.8090169943749473 0.3607430412000112 3 -0.3090169943749473 0.5992569587999889 4 0.3090169943749475 0.5992569587999887 5 0.8090169943749475 0.3607430412000113 6 1.0000000000000000 0.0400000000000000 Order = 7 1 -1.0000000000000000 0.0285714285714286 2 -0.8660254037844387 0.2539682539682539 3 -0.4999999999999998 0.4571428571428571 4 0.0000000000000000 0.5206349206349206 5 0.5000000000000001 0.4571428571428573 6 0.8660254037844387 0.2539682539682539 7 1.0000000000000000 0.0285714285714286 Order = 8 1 -1.0000000000000000 0.0204081632653061 2 -0.9009688679024190 0.1901410072182084 3 -0.6234898018587335 0.3522424237181591 4 -0.2225209339563143 0.4372084057983264 5 0.2225209339563144 0.4372084057983264 6 0.6234898018587336 0.3522424237181591 7 0.9009688679024191 0.1901410072182084 8 1.0000000000000000 0.0204081632653061 Order = 9 1 -1.0000000000000000 0.0158730158730159 2 -0.9238795325112867 0.1462186492160181 3 -0.7071067811865475 0.2793650793650794 4 -0.3826834323650897 0.3617178587204897 5 0.0000000000000000 0.3936507936507936 6 0.3826834323650898 0.3617178587204898 7 0.7071067811865476 0.2793650793650794 8 0.9238795325112867 0.1462186492160182 9 1.0000000000000000 0.0158730158730159 Order = 10 1 -1.0000000000000000 0.0123456790123457 2 -0.9396926207859083 0.1165674565720371 3 -0.7660444431189779 0.2252843233381044 4 -0.4999999999999998 0.3019400352733685 5 -0.1736481776669303 0.3438625058041442 6 0.1736481776669304 0.3438625058041442 7 0.5000000000000001 0.3019400352733687 8 0.7660444431189780 0.2252843233381044 9 0.9396926207859084 0.1165674565720372 10 1.0000000000000000 0.0123456790123457 SANDIA_RULES_TEST07 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 = 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 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 0.000000e+00 4 1 0.000000 0.000000 4.996004e-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 1.665335e-16 5 2 0.666667 0.666667 1.110223e-16 5 3 0.000000 0.000000 1.387779e-16 5 4 0.400000 0.400000 0.000000e+00 5 5 0.000000 0.000000 9.714451e-17 5 6 0.266667 0.285714 1.904762e-02 5 7 0.000000 0.000000 6.245005e-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 2.498002e-16 7 2 0.666667 0.666667 0.000000e+00 7 3 0.000000 0.000000 1.318390e-16 7 4 0.400000 0.400000 5.551115e-17 7 5 0.000000 0.000000 5.204170e-17 7 6 0.285714 0.285714 0.000000e+00 7 7 0.000000 0.000000 1.951564e-17 7 8 0.221429 0.222222 7.936508e-04 7 9 0.000000 0.000000 3.903128e-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 8.326673e-17 8 2 0.666667 0.666667 1.110223e-16 8 3 0.000000 0.000000 9.714451e-17 8 4 0.400000 0.400000 0.000000e+00 8 5 0.000000 0.000000 8.326673e-17 8 6 0.285714 0.285714 0.000000e+00 8 7 0.000000 0.000000 9.020562e-17 8 8 0.222024 0.222222 1.984127e-04 8 9 0.000000 0.000000 9.020562e-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 1.942890e-16 9 2 0.666667 0.666667 1.110223e-16 9 3 0.000000 0.000000 1.526557e-16 9 4 0.400000 0.400000 0.000000e+00 9 5 0.000000 0.000000 7.632783e-17 9 6 0.285714 0.285714 0.000000e+00 9 7 0.000000 0.000000 5.898060e-17 9 8 0.222222 0.222222 0.000000e+00 9 9 0.000000 0.000000 4.683753e-17 9 10 0.181746 0.181818 7.215007e-05 9 11 0.000000 0.000000 3.209238e-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 3.053113e-16 10 2 0.666667 0.666667 0.000000e+00 10 3 0.000000 0.000000 2.498002e-16 10 4 0.400000 0.400000 0.000000e+00 10 5 0.000000 0.000000 1.873501e-16 10 6 0.285714 0.285714 5.551115e-17 10 7 0.000000 0.000000 1.630640e-16 10 8 0.222222 0.222222 2.775558e-17 10 9 0.000000 0.000000 1.595946e-16 10 10 0.181796 0.181818 2.254690e-05 10 11 0.000000 0.000000 1.387779e-16 10 12 0.153757 0.153846 8.871337e-05 SANDIA_RULES_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 1 0.0000000000000000 2.0000000000000000 Order = 2 1 -0.4999999999999998 1.0000000000000000 2 0.5000000000000001 1.0000000000000000 Order = 3 1 -0.7071067811865475 0.6666666666666667 2 0.0000000000000000 0.6666666666666666 3 0.7071067811865476 0.6666666666666666 Order = 4 1 -0.8090169943749473 0.2175954681666807 2 -0.3090169943749473 0.7157378651666526 3 0.3090169943749475 0.5157378651666527 4 0.8090169943749475 0.4175954681666807 Order = 5 1 -0.8660254037844387 0.3111111111111111 2 -0.4999999999999998 0.4000000000000001 3 0.0000000000000000 0.5777777777777777 4 0.5000000000000001 0.4000000000000000 5 0.8660254037844387 0.3111111111111111 Order = 6 1 -0.9009688679024190 0.1278122637966723 2 -0.6234898018587335 0.4100408966736759 3 -0.2225209339563143 0.3859563633391757 4 0.2225209339563144 0.4811944585772709 5 0.6234898018587336 0.3148028014355805 6 0.9009688679024191 0.2230503590347675 Order = 7 1 -0.9238795325112867 0.1779646809620499 2 -0.7071067811865475 0.2476190476190477 3 -0.3826834323650897 0.3934638904665215 4 0.0000000000000000 0.3619047619047619 5 0.3826834323650898 0.3934638904665215 6 0.7071067811865476 0.2476190476190476 7 0.9238795325112867 0.1779646809620499 Order = 8 1 -0.9396926207859083 0.0821604539562636 2 -0.7660444431189779 0.2584662947181325 3 -0.4999999999999998 0.2706349206349207 4 -0.1736481776669303 0.3728653148176675 5 0.1736481776669304 0.3173097592621119 6 0.5000000000000001 0.3261904761904761 7 0.7660444431189780 0.2029107391625769 8 0.9396926207859084 0.1377160095118191 Order = 9 1 -0.9510565162951535 0.1147810750857218 2 -0.8090169943749473 0.1654331942222276 3 -0.5877852522924730 0.2737903534857068 4 -0.3090169943749473 0.2790112502222169 5 0.0000000000000000 0.3339682539682539 6 0.3090169943749475 0.2790112502222170 7 0.5877852522924731 0.2737903534857068 8 0.8090169943749475 0.1654331942222276 9 0.9510565162951535 0.1147810750857217 Order = 10 1 -0.9594929736144974 0.0568540271213534 2 -0.8412535328311811 0.1763122770580662 3 -0.6548607339452850 0.1949753624051099 4 -0.4154150130018863 0.2796250178823501 5 -0.1423148382732850 0.2639504872502921 6 0.1423148382732851 0.3003141236139285 7 0.4154150130018864 0.2432613815187137 8 0.6548607339452851 0.2313389987687463 9 0.8412535328311812 0.1399486406944298 10 0.9594929736144974 0.0932176634849897 SANDIA_RULES_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 0.5 1.5708 1.570796 1.110223e-15 1 1 0.5 0 0.000000 0.000000e+00 1 2 0.5 0 0.392699 3.926991e-01 1 3 0.5 0 0.000000 0.000000e+00 1 4 0.5 0 0.196350 1.963495e-01 2 0 0.5 1.5708 1.570796 1.110223e-15 2 1 0.5 0 0.000000 0.000000e+00 2 2 0.5 0.392699 0.392699 0.000000e+00 2 3 0.5 0 0.000000 0.000000e+00 2 4 0.5 0.0981748 0.196350 9.817477e-02 2 5 0.5 0 0.000000 0.000000e+00 2 6 0.5 0.0245437 0.122718 9.817477e-02 3 0 0.5 1.5708 1.570796 1.776357e-15 3 1 0.5 0 0.000000 0.000000e+00 3 2 0.5 0.392699 0.392699 4.440892e-16 3 3 0.5 0 0.000000 0.000000e+00 3 4 0.5 0.19635 0.196350 3.330669e-16 3 5 0.5 0 0.000000 0.000000e+00 3 6 0.5 0.0981748 0.122718 2.454369e-02 3 7 0.5 0 0.000000 0.000000e+00 3 8 0.5 0.0490874 0.085903 3.681554e-02 4 0 0.5 1.5708 1.570796 1.110223e-15 4 1 0.5 -5.55112e-17 0.000000 5.551115e-17 4 2 0.5 0.392699 0.392699 0.000000e+00 4 3 0.5 0 0.000000 0.000000e+00 4 4 0.5 0.19635 0.196350 8.326673e-17 4 5 0.5 0 0.000000 0.000000e+00 4 6 0.5 0.122718 0.122718 2.775558e-17 4 7 0.5 0 0.000000 0.000000e+00 4 8 0.5 0.079767 0.085903 6.135923e-03 4 9 0.5 0 0.000000 0.000000e+00 4 10 0.5 0.0521553 0.064427 1.227185e-02 5 0 0.5 1.5708 1.570796 1.554312e-15 5 1 0.5 7.91034e-16 0.000000 7.910339e-16 5 2 0.5 0.392699 0.392699 2.775558e-16 5 3 0.5 5.96745e-16 0.000000 5.967449e-16 5 4 0.5 0.19635 0.196350 3.608225e-16 5 5 0.5 4.44089e-16 0.000000 4.440892e-16 5 6 0.5 0.122718 0.122718 1.665335e-16 5 7 0.5 3.33067e-16 0.000000 3.330669e-16 5 8 0.5 0.0859029 0.085903 9.714451e-17 5 9 0.5 2.56739e-16 0.000000 2.567391e-16 5 10 0.5 0.0628932 0.064427 1.533981e-03 5 11 0.5 1.8735e-16 0.000000 1.873501e-16 5 12 0.5 0.0467864 0.050621 3.834952e-03 6 0 0.5 1.5708 1.570796 1.332268e-15 6 1 0.5 3.747e-16 0.000000 3.747003e-16 6 2 0.5 0.392699 0.392699 1.110223e-16 6 3 0.5 2.98372e-16 0.000000 2.983724e-16 6 4 0.5 0.19635 0.196350 1.665335e-16 6 5 0.5 2.28983e-16 0.000000 2.289835e-16 6 6 0.5 0.122718 0.122718 2.775558e-17 6 7 0.5 1.73472e-16 0.000000 1.734723e-16 6 8 0.5 0.0859029 0.085903 2.775558e-17 6 9 0.5 1.31839e-16 0.000000 1.318390e-16 6 10 0.5 0.0644272 0.064427 2.775558e-17 6 11 0.5 1.00614e-16 0.000000 1.006140e-16 6 12 0.5 0.0502379 0.050621 3.834952e-04 6 13 0.5 7.63278e-17 0.000000 7.632783e-17 6 14 0.5 0.0399794 0.041130 1.150486e-03 7 0 0.5 1.5708 1.570796 8.881784e-16 7 1 0.5 2.77556e-16 0.000000 2.775558e-16 7 2 0.5 0.392699 0.392699 2.775558e-16 7 3 0.5 1.9082e-16 0.000000 1.908196e-16 7 4 0.5 0.19635 0.196350 1.387779e-16 7 5 0.5 1.14492e-16 0.000000 1.144917e-16 7 6 0.5 0.122718 0.122718 2.081668e-16 7 7 0.5 6.34258e-17 0.000000 6.342583e-17 7 8 0.5 0.0859029 0.085903 1.804112e-16 7 9 0.5 3.44573e-17 0.000000 3.445730e-17 7 10 0.5 0.0644272 0.064427 1.387779e-16 7 11 0.5 1.87143e-17 0.000000 1.871435e-17 7 12 0.5 0.0506214 0.050621 1.040834e-16 7 13 0.5 9.42134e-18 0.000000 9.421336e-18 7 14 0.5 0.041034 0.041130 9.587380e-05 7 15 0.5 5.23898e-18 0.000000 5.238978e-18 7 16 0.5 0.0339393 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1.0842e-19 0.000000 1.084202e-19 6 14 2.5 0.00160369 0.001714 1.100516e-04 7 0 2.5 0.981748 0.981748 0.000000e+00 7 1 2.5 1.94289e-16 0.000000 1.942890e-16 7 2 2.5 0.122718 0.122718 1.942890e-16 7 3 2.5 1.17961e-16 0.000000 1.179612e-16 7 4 2.5 0.0368155 0.036816 1.110223e-16 7 5 2.5 8.12067e-17 0.000000 8.120674e-17 7 6 2.5 0.0153398 0.015340 5.551115e-17 7 7 2.5 5.69206e-17 0.000000 5.692061e-17 7 8 2.5 0.0076699 0.007670 3.989864e-17 7 9 2.5 3.87992e-17 0.000000 3.879919e-17 7 10 2.5 0.00431432 0.004314 2.688821e-17 7 11 2.5 2.68334e-17 0.000000 2.683337e-17 7 12 2.5 0.00263653 0.002637 1.951564e-17 7 13 2.5 1.84735e-17 0.000000 1.847354e-17 7 14 2.5 0.00170276 0.001714 1.098554e-05 7 15 2.5 1.26307e-17 0.000000 1.263072e-17 7 16 2.5 0.00113767 0.001168 3.079280e-05 8 0 2.5 0.981748 0.981748 0.000000e+00 8 1 2.5 1.38778e-17 0.000000 1.387779e-17 8 2 2.5 0.122718 0.122718 1.110223e-16 8 3 2.5 0 0.000000 0.000000e+00 8 4 2.5 0.0368155 0.036816 3.469447e-17 8 5 2.5 0 0.000000 0.000000e+00 8 6 2.5 0.0153398 0.015340 1.734723e-18 8 7 2.5 0 0.000000 0.000000e+00 8 8 2.5 0.0076699 0.007670 0.000000e+00 8 9 2.5 0 0.000000 0.000000e+00 8 10 2.5 0.00431432 0.004314 1.734723e-18 8 11 2.5 0 0.000000 0.000000e+00 8 12 2.5 0.00263653 0.002637 2.168404e-18 8 13 2.5 0 0.000000 0.000000e+00 8 14 2.5 0.00171374 0.001714 6.505213e-19 8 15 2.5 0 0.000000 0.000000e+00 8 16 2.5 0.00116587 0.001168 2.596582e-06 8 17 2.5 0 0.000000 0.000000e+00 8 18 2.5 0.000819157 0.000828 8.503806e-06 9 0 2.5 0.981748 0.981748 1.110223e-16 9 1 2.5 0 0.000000 0.000000e+00 9 2 2.5 0.122718 0.122718 2.775558e-17 9 3 2.5 0 0.000000 0.000000e+00 9 4 2.5 0.0368155 0.036816 6.938894e-18 9 5 2.5 0 0.000000 0.000000e+00 9 6 2.5 0.0153398 0.015340 2.775558e-17 9 7 2.5 0 0.000000 0.000000e+00 9 8 2.5 0.0076699 0.007670 1.474515e-17 9 9 2.5 0 0.000000 0.000000e+00 9 10 2.5 0.00431432 0.004314 1.127570e-17 9 11 2.5 0 0.000000 0.000000e+00 9 12 2.5 0.00263653 0.002637 7.806256e-18 9 13 2.5 0 0.000000 0.000000e+00 9 14 2.5 0.00171374 0.001714 4.553649e-18 9 15 2.5 0 0.000000 0.000000e+00 9 16 2.5 0.00116846 0.001168 3.252607e-18 9 17 2.5 0 0.000000 0.000000e+00 9 18 2.5 0.000827041 0.000828 6.196389e-07 9 19 2.5 0 0.000000 0.000000e+00 9 20 2.5 0.000602504 0.000605 2.324729e-06 10 0 2.5 0.981748 0.981748 1.110223e-16 10 1 2.5 -1.00614e-16 0.000000 1.006140e-16 10 2 2.5 0.122718 0.122718 1.665335e-16 10 3 2.5 -3.20924e-17 0.000000 3.209238e-17 10 4 2.5 0.0368155 0.036816 6.245005e-17 10 5 2.5 -1.12757e-17 0.000000 1.127570e-17 10 6 2.5 0.0153398 0.015340 8.673617e-18 10 7 2.5 -3.57787e-18 0.000000 3.577867e-18 10 8 2.5 0.0076699 0.007670 4.336809e-18 10 9 2.5 -1.0842e-18 0.000000 1.084202e-18 10 10 2.5 0.00431432 0.004314 0.000000e+00 10 11 2.5 -4.33681e-19 0.000000 4.336809e-19 10 12 2.5 0.00263653 0.002637 8.673617e-19 10 13 2.5 -1.35525e-19 0.000000 1.355253e-19 10 14 2.5 0.00171374 0.001714 2.168404e-19 10 15 2.5 -4.06576e-20 0.000000 4.065758e-20 10 16 2.5 0.00116846 0.001168 8.673617e-19 10 17 2.5 -1.35525e-20 0.000000 1.355253e-20 10 18 2.5 0.000827661 0.000828 3.252607e-19 10 19 2.5 0 0.000000 0.000000e+00 10 20 2.5 0.00060468 0.000605 1.489517e-07 10 21 2.5 0 0.000000 0.000000e+00 10 22 2.5 0.000452991 0.000454 6.303847e-07 SANDIA_RULES_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 1 0.0000000000000000 1.5707963267948968 Order = 2 ALPHA = 0.500000 1 -0.5000000000000000 0.7853981633974484 2 0.5000000000000000 0.7853981633974484 Order = 3 ALPHA = 0.500000 1 -0.7071067811865475 0.3926990816987239 2 0.0000000000000000 0.7853981633974484 3 0.7071067811865475 0.3926990816987239 Order = 4 ALPHA = 0.500000 1 -0.8090169943749475 0.2170787134227060 2 -0.3090169943749475 0.5683194499747424 3 0.3090169943749474 0.5683194499747424 4 0.8090169943749475 0.2170787134227060 Order = 5 ALPHA = 0.500000 1 -0.8660254037844387 0.1308996938995740 2 -0.5000000000000000 0.3926990816987242 3 0.0000000000000000 0.5235987755982989 4 0.5000000000000000 0.3926990816987242 5 0.8660254037844387 0.1308996938995750 Order = 6 ALPHA = 0.500000 1 -0.9009688679024191 0.0844886908915884 2 -0.6234898018587335 0.2743330560697777 3 -0.2225209339563144 0.4265764164360819 4 0.2225209339563144 0.4265764164360819 5 0.6234898018587335 0.2743330560697777 6 0.9009688679024190 0.0844886908915888 Order = 7 ALPHA = 0.500000 1 -0.9238795325112867 0.0575094490319133 2 -0.7071067811865475 0.1963495408493619 3 -0.3826834323650898 0.3351896326668111 4 0.0000000000000000 0.3926990816987242 5 0.3826834323650898 0.3351896326668108 6 0.7071067811865476 0.1963495408493624 7 0.9238795325112867 0.0575094490319133 Order = 8 ALPHA = 0.500000 1 -0.9396926207859084 0.0408329477091069 2 -0.7660444431189780 0.1442256007956728 3 -0.5000000000000000 0.2617993877991495 4 -0.1736481776669303 0.3385402270935191 5 0.1736481776669303 0.3385402270935191 6 0.5000000000000000 0.2617993877991495 7 0.7660444431189780 0.1442256007956728 8 0.9396926207859084 0.0408329477091075 Order = 9 ALPHA = 0.500000 1 -0.9510565162951536 0.0299995403716084 2 -0.8090169943749475 0.1085393567113530 3 -0.5877852522924731 0.2056199086476264 4 -0.3090169943749475 0.2841597249873712 5 0.0000000000000000 0.3141592653589794 6 0.3090169943749475 0.2841597249873712 7 0.5877852522924731 0.2056199086476264 8 0.8090169943749475 0.1085393567113530 9 0.9510565162951536 0.0299995403716084 Order = 10 ALPHA = 0.500000 1 -0.9594929736144974 0.0226689425018590 2 -0.8412535328311812 0.0834785409341889 3 -0.6548607339452851 0.1631221774548165 4 -0.4154150130018864 0.2363135602034873 5 -0.1423148382732851 0.2798149423030965 6 0.1423148382732851 0.2798149423030966 7 0.4154150130018864 0.2363135602034873 8 0.6548607339452851 0.1631221774548165 9 0.8412535328311812 0.0834785409341889 10 0.9594929736144974 0.0226689425018590 Order = 1 ALPHA = 1.000000 1 0.0000000000000000 1.3333333333333333 Order = 2 ALPHA = 1.000000 1 -0.4472135954999579 0.6666666666666665 2 0.4472135954999580 0.6666666666666667 Order = 3 ALPHA = 1.000000 1 -0.6546536707079772 0.3111111111111113 2 0.0000000000000000 0.7111111111111110 3 0.6546536707079772 0.3111111111111109 Order = 4 ALPHA = 1.000000 1 -0.7650553239294646 0.1569499125956942 2 -0.2852315164806451 0.5097167540709727 3 0.2852315164806451 0.5097167540709727 4 0.7650553239294646 0.1569499125956942 Order = 5 ALPHA = 1.000000 1 -0.8302238962785669 0.0860176821228073 2 -0.4688487934707142 0.3368394607343354 3 -0.0000000000000000 0.4876190476190476 4 0.4688487934707142 0.3368394607343354 5 0.8302238962785669 0.0860176821228073 Order = 6 ALPHA = 1.000000 1 -0.8717401485096066 0.0505835770160819 2 -0.5917001814331423 0.2216925320225178 3 -0.2092992179024789 0.3943905576280671 4 0.2092992179024789 0.3943905576280671 5 0.5917001814331423 0.2216925320225178 6 0.8717401485096067 0.0505835770160817 Order = 7 ALPHA = 1.000000 1 -0.8997579954114602 0.0315162001504988 2 -0.6771862795107377 0.1486404045834101 3 -0.3631174638261782 0.3007504247445493 4 0.0000000000000000 0.3715192743764172 5 0.3631174638261782 0.3007504247445493 6 0.6771862795107377 0.1486404045834101 7 0.8997579954114602 0.0315162001504988 Order = 8 ALPHA = 1.000000 1 -0.9195339081664589 0.0205900956491219 2 -0.7387738651055050 0.1021477023603581 3 -0.4779249498104445 0.2253365549698578 4 -0.1652789576663870 0.3185923136873282 5 0.1652789576663870 0.3185923136873282 6 0.4779249498104445 0.2253365549698580 7 0.7387738651055050 0.1021477023603581 8 0.9195339081664589 0.0205900956491219 Order = 9 ALPHA = 1.000000 1 -0.9340014304080592 0.0139910561244267 2 -0.7844834736631444 0.0719828561756607 3 -0.5652353269962050 0.1687989736144601 4 -0.2957581355869394 0.2617849830242735 5 -0.0000000000000000 0.3002175954556907 6 0.2957581355869394 0.2617849830242735 7 0.5652353269962050 0.1687989736144601 8 0.7844834736631444 0.0719828561756607 9 0.9340014304080592 0.0139910561244267 Order = 10 ALPHA = 1.000000 1 -0.9448992722228823 0.0098254048057683 2 -0.8192793216440066 0.0519391437742078 3 -0.6328761530318606 0.1273919483477328 4 -0.3995309409653490 0.2111657418760753 5 -0.1365529328549276 0.2663444278628824 6 0.1365529328549276 0.2663444278628824 7 0.3995309409653490 0.2111657418760753 8 0.6328761530318606 0.1273919483477326 9 0.8192793216440066 0.0519391437742078 10 0.9448992722228822 0.0098254048057681 Order = 1 ALPHA = 2.500000 1 0.0000000000000000 0.9817477042468105 Order = 2 ALPHA = 2.500000 1 -0.3535533905932737 0.4908738521234051 2 0.3535533905932737 0.4908738521234051 Order = 3 ALPHA = 2.500000 1 -0.5477225575051661 0.2045307717180856 2 0.0000000000000000 0.5726861608106394 3 0.5477225575051661 0.2045307717180856 Order = 4 ALPHA = 2.500000 1 -0.6660699417556469 0.0870080715314833 2 -0.2373833033084449 0.4038657805919219 3 0.2373833033084449 0.4038657805919219 4 0.6660699417556469 0.0870080715314833 Order = 5 ALPHA = 2.500000 1 -0.7434770045212192 0.0392893891317672 2 -0.4019050360892101 0.2454174450998077 3 -0.0000000000000000 0.4123340357836603 4 0.4019050360892101 0.2454174450998076 5 0.7434770045212192 0.0392893891317672 Order = 6 ALPHA = 2.500000 1 -0.7968373566406807 0.0189116433666931 2 -0.5196148456949133 0.1431407318203965 3 -0.1804179569647766 0.3288214769363154 4 0.1804179569647766 0.3288214769363154 5 0.5196148456949133 0.1431407318203969 6 0.7968373566406807 0.0189116433666931 Order = 7 ALPHA = 2.500000 1 -0.8351564541339112 0.0096585250210008 2 -0.6063141605450342 0.0834446705489847 3 -0.3186902924591673 0.2357822853526957 4 -0.0000000000000000 0.3239767424014474 5 0.3186902924591672 0.2357822853526959 6 0.6063141605450342 0.0834446705489847 7 0.8351564541339110 0.0096585250210010 Order = 8 ALPHA = 2.500000 1 -0.8635912403827584 0.0051997074221325 2 -0.6718232507247468 0.0494181881565223 3 -0.4260961495798479 0.1615070338519727 4 -0.1459649294626305 0.2747489226927776 5 0.1459649294626305 0.2747489226927778 6 0.4260961495798479 0.1615070338519727 7 0.6718232507247468 0.0494181881565223 8 0.8635912403827584 0.0051997074221325 Order = 9 ALPHA = 2.500000 1 -0.8852662998295736 0.0029319009849924 2 -0.7224309555097356 0.0299209571936803 3 -0.5107538326640052 0.1088013928543096 4 -0.2643695362367075 0.2155149137501430 5 0.0000000000000000 0.2674093746805597 6 0.2643695362367075 0.2155149137501430 7 0.5107538326640052 0.1088013928543096 8 0.7224309555097356 0.0299209571936803 9 0.8852662998295736 0.0029319009849924 Order = 10 ALPHA = 2.500000 1 -0.9021636877146698 0.0017216382288932 2 -0.7622868838395971 0.0185564370059903 3 -0.5784393987065429 0.0731613283971411 4 -0.3610608688185057 0.1622660968500981 5 -0.1227285554678421 0.2351683516412825 6 0.1227285554678421 0.2351683516412824 7 0.3610608688185057 0.1622660968500981 8 0.5784393987065429 0.0731613283971409 9 0.7622868838395971 0.0185564370059903 10 0.9021636877146698 0.0017216382288932 SANDIA_RULES_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 1.22542 1.22542 0.000000e+00 1 1 0.500000 0 0 0.000000e+00 1 2 0.500000 0 0.919063 9.190625e-01 1 3 0.500000 0 0 0.000000e+00 1 4 0.500000 0 1.60836 1.608359e+00 2 0 0.500000 1.22542 1.22542 0.000000e+00 2 1 0.500000 0 0 0.000000e+00 2 2 0.500000 0.919063 0.919063 4.440892e-16 2 3 0.500000 0 0 0.000000e+00 2 4 0.500000 0.689297 1.60836 9.190625e-01 2 5 0.500000 0 0 0.000000e+00 2 6 0.500000 0.516973 4.42299 3.906016e+00 3 0 0.500000 1.22542 1.22542 0.000000e+00 3 1 0.500000 -3.18732e-16 0 3.187317e-16 3 2 0.500000 0.919063 0.919063 6.661338e-16 3 3 0.500000 -9.99201e-16 0 9.992007e-16 3 4 0.500000 1.60836 1.60836 1.776357e-15 3 5 0.500000 -1.9984e-15 0 1.998401e-15 3 6 0.500000 2.81463 4.42299 1.608359e+00 3 7 0.500000 -3.9968e-15 0 3.996803e-15 3 8 0.500000 4.9256 16.5862 1.166061e+01 4 0 0.500000 1.22542 1.22542 0.000000e+00 4 1 0.500000 2.77556e-16 0 2.775558e-16 4 2 0.500000 0.919063 0.919063 5.551115e-16 4 3 0.500000 3.33067e-16 0 3.330669e-16 4 4 0.500000 1.60836 1.60836 1.110223e-15 4 5 0.500000 6.66134e-16 0 6.661338e-16 4 6 0.500000 4.42299 4.42299 2.664535e-15 4 7 0.500000 1.33227e-15 0 1.332268e-15 4 8 0.500000 13.3695 16.5862 3.216719e+00 4 9 0.500000 5.32907e-15 0 5.329071e-15 4 10 0.500000 40.988 78.7845 3.779645e+01 5 0 0.500000 1.22542 1.22542 1.110223e-15 5 1 0.500000 -5.95469e-16 0 5.954693e-16 5 2 0.500000 0.919063 0.919063 4.440892e-16 5 3 0.500000 -6.38378e-16 0 6.383782e-16 5 4 0.500000 1.60836 1.60836 0.000000e+00 5 5 0.500000 -3.94129e-15 0 3.941292e-15 5 6 0.500000 4.42299 4.42299 1.776357e-15 5 7 0.500000 -2.35367e-14 0 2.353673e-14 5 8 0.500000 16.5862 16.5862 7.105427e-15 5 9 0.500000 -1.29341e-13 0 1.293410e-13 5 10 0.500000 69.9385 78.7845 8.845977e+00 5 11 0.500000 -6.64913e-13 0 6.649126e-13 5 12 0.500000 304.841 453.011 1.481701e+02 6 0 0.500000 1.22542 1.22542 4.440892e-16 6 1 0.500000 3.88578e-16 0 3.885781e-16 6 2 0.500000 0.919063 0.919063 9.992007e-16 6 3 0.500000 7.77156e-16 0 7.771561e-16 6 4 0.500000 1.60836 1.60836 2.220446e-15 6 5 0.500000 1.22125e-15 0 1.221245e-15 6 6 0.500000 4.42299 4.42299 8.881784e-15 6 7 0.500000 2.33147e-15 0 2.331468e-15 6 8 0.500000 16.5862 16.5862 4.973799e-14 6 9 0.500000 7.10543e-15 0 7.105427e-15 6 10 0.500000 78.7845 78.7845 2.842171e-13 6 11 0.500000 5.68434e-14 0 5.684342e-14 6 12 0.500000 426.473 453.011 2.653793e+01 6 13 0.500000 6.25278e-13 0 6.252776e-13 6 14 0.500000 2440.82 3057.82 6.170069e+02 7 0 0.500000 1.22542 1.22542 2.220446e-16 7 1 0.500000 5.2845e-16 0 5.284496e-16 7 2 0.500000 0.919063 0.919063 6.661338e-16 7 3 0.500000 9.99201e-16 0 9.992007e-16 7 4 0.500000 1.60836 1.60836 1.110223e-15 7 5 0.500000 2.94209e-15 0 2.942091e-15 7 6 0.500000 4.42299 4.42299 1.776357e-15 7 7 0.500000 9.79772e-15 0 9.797718e-15 7 8 0.500000 16.5862 16.5862 1.065814e-14 7 9 0.500000 3.28626e-14 0 3.286260e-14 7 10 0.500000 78.7845 78.7845 1.421085e-13 7 11 0.500000 9.31061e-14 0 9.310608e-14 7 12 0.500000 453.011 453.011 1.477929e-12 7 13 0.500000 2.3645e-13 0 2.364497e-13 7 14 0.500000 2958.31 3057.82 9.951724e+01 7 15 0.500000 3.01578e-13 0 3.015782e-13 7 16 0.500000 20687.7 23698.1 3.010396e+03 8 0 0.500000 1.22542 1.22542 1.332268e-15 8 1 0.500000 7.49401e-16 0 7.494005e-16 8 2 0.500000 0.919063 0.919063 2.220446e-16 8 3 0.500000 2.05391e-15 0 2.053913e-15 8 4 0.500000 1.60836 1.60836 2.220446e-16 8 5 0.500000 6.55032e-15 0 6.550316e-15 8 6 0.500000 4.42299 4.42299 4.440892e-15 8 7 0.500000 2.59792e-14 0 2.597922e-14 8 8 0.500000 16.5862 16.5862 3.552714e-14 8 9 0.500000 1.12799e-13 0 1.127987e-13 8 10 0.500000 78.7845 78.7845 2.700062e-13 8 11 0.500000 5.54223e-13 0 5.542233e-13 8 12 0.500000 453.011 453.011 2.046363e-12 8 13 0.500000 2.64322e-12 0 2.643219e-12 8 14 0.500000 3057.82 3057.82 1.637090e-11 8 15 0.500000 1.27329e-11 0 1.273293e-11 8 16 0.500000 23300.1 23698.1 3.980690e+02 8 17 0.500000 5.09317e-11 0 5.093170e-11 8 18 0.500000 191933 207359 1.542517e+04 9 0 0.500000 1.22542 1.22542 0.000000e+00 9 1 0.500000 -1.66533e-16 0 1.665335e-16 9 2 0.500000 0.919063 0.919063 8.881784e-16 9 3 0.500000 -7.77156e-16 0 7.771561e-16 9 4 0.500000 1.60836 1.60836 1.998401e-15 9 5 0.500000 -3.9968e-15 0 3.996803e-15 9 6 0.500000 4.42299 4.42299 8.881784e-15 9 7 0.500000 -2.06501e-14 0 2.065015e-14 9 8 0.500000 16.5862 16.5862 4.973799e-14 9 9 0.500000 -1.1191e-13 0 1.119105e-13 9 10 0.500000 78.7845 78.7845 3.268497e-13 9 11 0.500000 -6.92779e-13 0 6.927792e-13 9 12 0.500000 453.011 453.011 2.103206e-12 9 13 0.500000 -4.70379e-12 0 4.703793e-12 9 14 0.500000 3057.82 3057.82 1.500666e-11 9 15 0.500000 -3.60316e-11 0 3.603162e-11 9 16 0.500000 23698.1 23698.1 1.200533e-10 9 17 0.500000 -3.13122e-10 0 3.131220e-10 9 18 0.500000 205468 207359 1.890828e+03 9 19 0.500000 -2.96188e-09 0 2.961883e-09 9 20 0.500000 1.93146e+06 2.02175e+06 9.028702e+04 10 0 0.500000 1.22542 1.22542 1.332268e-15 10 1 0.500000 7.42462e-16 0 7.424616e-16 10 2 0.500000 0.919063 0.919063 1.110223e-15 10 3 0.500000 1.08247e-15 0 1.082467e-15 10 4 0.500000 1.60836 1.60836 2.664535e-15 10 5 0.500000 2.66454e-15 0 2.664535e-15 10 6 0.500000 4.42299 4.42299 9.769963e-15 10 7 0.500000 9.10383e-15 0 9.103829e-15 10 8 0.500000 16.5862 16.5862 4.618528e-14 10 9 0.500000 3.84137e-14 0 3.841372e-14 10 10 0.500000 78.7845 78.7845 2.557954e-13 10 11 0.500000 1.59872e-13 0 1.598721e-13 10 12 0.500000 453.011 453.011 1.477929e-12 10 13 0.500000 9.66338e-13 0 9.663381e-13 10 14 0.500000 3057.82 3057.82 9.549694e-12 10 15 0.500000 7.84439e-12 0 7.844392e-12 10 16 0.500000 23698.1 23698.1 6.184564e-11 10 17 0.500000 1.00954e-10 0 1.009539e-10 10 18 0.500000 207359 207359 4.074536e-10 10 19 0.500000 1.36788e-09 0 1.367880e-09 10 20 0.500000 2.01229e+06 2.02175e+06 9.454138e+03 10 21 0.500000 1.76951e-08 0 1.769513e-08 10 22 0.500000 2.11831e+07 2.17338e+07 5.507035e+05 1 0 1.000000 1 1 0.000000e+00 1 1 1.000000 0 0 0.000000e+00 1 2 1.000000 0 1 1.000000e+00 1 3 1.000000 0 0 0.000000e+00 1 4 1.000000 0 2 2.000000e+00 2 0 1.000000 1 1 2.220446e-16 2 1 1.000000 0 0 0.000000e+00 2 2 1.000000 1 1 6.661338e-16 2 3 1.000000 0 0 0.000000e+00 2 4 1.000000 1 2 1.000000e+00 2 5 1.000000 0 0 0.000000e+00 2 6 1.000000 1 6 5.000000e+00 3 0 1.000000 1 1 4.440892e-16 3 1 1.000000 -3.7817e-16 0 3.781697e-16 3 2 1.000000 1 1 0.000000e+00 3 3 1.000000 -1.11022e-15 0 1.110223e-15 3 4 1.000000 2 2 8.881784e-16 3 5 1.000000 -2.66454e-15 0 2.664535e-15 3 6 1.000000 4 6 2.000000e+00 3 7 1.000000 -6.21725e-15 0 6.217249e-15 3 8 1.000000 8 24 1.600000e+01 4 0 1.000000 1 1 4.440892e-16 4 1 1.000000 -1.66533e-16 0 1.665335e-16 4 2 1.000000 1 1 1.110223e-16 4 3 1.000000 -1.11022e-16 0 1.110223e-16 4 4 1.000000 2 2 4.440892e-16 4 5 1.000000 0 0 0.000000e+00 4 6 1.000000 6 6 5.329071e-15 4 7 1.000000 0 0 0.000000e+00 4 8 1.000000 20 24 4.000000e+00 4 9 1.000000 0 0 0.000000e+00 4 10 1.000000 68 120 5.200000e+01 5 0 1.000000 1 1 2.220446e-16 5 1 1.000000 -7.43757e-16 0 7.437568e-16 5 2 1.000000 1 1 7.771561e-16 5 3 1.000000 -6.10623e-16 0 6.106227e-16 5 4 1.000000 2 2 2.664535e-15 5 5 1.000000 -3.33067e-16 0 3.330669e-16 5 6 1.000000 6 6 8.881784e-15 5 7 1.000000 3.33067e-15 0 3.330669e-15 5 8 1.000000 24 24 4.263256e-14 5 9 1.000000 2.45359e-14 0 2.453593e-14 5 10 1.000000 108 120 1.200000e+01 5 11 1.000000 1.70974e-13 0 1.709743e-13 5 12 1.000000 504 720 2.160000e+02 6 0 1.000000 1 1 4.440892e-16 6 1 1.000000 7.21645e-16 0 7.216450e-16 6 2 1.000000 1 1 4.440892e-16 6 3 1.000000 1.44329e-15 0 1.443290e-15 6 4 1.000000 2 2 8.881784e-16 6 5 1.000000 2.77556e-15 0 2.775558e-15 6 6 1.000000 6 6 7.105427e-15 6 7 1.000000 1.88738e-15 0 1.887379e-15 6 8 1.000000 24 24 4.973799e-14 6 9 1.000000 -4.9738e-14 0 4.973799e-14 6 10 1.000000 120 120 3.694822e-13 6 11 1.000000 -5.96856e-13 0 5.968559e-13 6 12 1.000000 684 720 3.600000e+01 6 13 1.000000 -4.88853e-12 0 4.888534e-12 6 14 1.000000 4140 5040 9.000000e+02 7 0 1.000000 1 1 2.220446e-16 7 1 1.000000 5.16302e-17 0 5.163018e-17 7 2 1.000000 1 1 2.220446e-16 7 3 1.000000 -2.77556e-16 0 2.775558e-16 7 4 1.000000 2 2 8.881784e-16 7 5 1.000000 -3.88578e-16 0 3.885781e-16 7 6 1.000000 6 6 8.881784e-16 7 7 1.000000 4.996e-15 0 4.996004e-15 7 8 1.000000 24 24 7.105427e-15 7 9 1.000000 5.94802e-14 0 5.948020e-14 7 10 1.000000 120 120 1.136868e-13 7 11 1.000000 5.07122e-13 0 5.071221e-13 7 12 1.000000 720 720 1.023182e-12 7 13 1.000000 4.27519e-12 0 4.275191e-12 7 14 1.000000 4896 5040 1.440000e+02 7 15 1.000000 3.31658e-11 0 3.316580e-11 7 16 1.000000 35712 40320 4.608000e+03 8 0 1.000000 1 1 2.220446e-16 8 1 1.000000 -5.55112e-17 0 5.551115e-17 8 2 1.000000 1 1 6.661338e-16 8 3 1.000000 -5.55112e-17 0 5.551115e-17 8 4 1.000000 2 2 1.776357e-15 8 5 1.000000 1.33227e-15 0 1.332268e-15 8 6 1.000000 6 6 5.329071e-15 8 7 1.000000 1.24345e-14 0 1.243450e-14 8 8 1.000000 24 24 1.421085e-14 8 9 1.000000 7.63833e-14 0 7.638334e-14 8 10 1.000000 120 120 4.263256e-14 8 11 1.000000 3.55271e-13 0 3.552714e-13 8 12 1.000000 720 720 3.410605e-13 8 13 1.000000 1.08002e-12 0 1.080025e-12 8 14 1.000000 5040 5040 4.547474e-12 8 15 1.000000 -9.09495e-13 0 9.094947e-13 8 16 1.000000 39744 40320 5.760000e+02 8 17 1.000000 -9.45874e-11 0 9.458745e-11 8 18 1.000000 339264 362880 2.361600e+04 9 0 1.000000 1 1 1.110223e-15 9 1 1.000000 9.15934e-16 0 9.159340e-16 9 2 1.000000 1 1 1.332268e-15 9 3 1.000000 2.33147e-15 0 2.331468e-15 9 4 1.000000 2 2 1.776357e-15 9 5 1.000000 8.54872e-15 0 8.548717e-15 9 6 1.000000 6 6 8.881784e-16 9 7 1.000000 4.13003e-14 0 4.130030e-14 9 8 1.000000 24 24 1.421085e-14 9 9 1.000000 2.12275e-13 0 2.122746e-13 9 10 1.000000 120 120 8.526513e-14 9 11 1.000000 1.10489e-12 0 1.104894e-12 9 12 1.000000 720 720 3.410605e-13 9 13 1.000000 5.07328e-12 0 5.073275e-12 9 14 1.000000 5040 5040 2.728484e-12 9 15 1.000000 1.23492e-11 0 1.234923e-11 9 16 1.000000 40320 40320 8.003553e-11 9 17 1.000000 -1.51374e-10 0 1.513740e-10 9 18 1.000000 360000 362880 2.880000e+03 9 19 1.000000 -3.4305e-09 0 3.430500e-09 9 20 1.000000 3.4848e+06 3.6288e+06 1.440000e+05 10 0 1.000000 1 1 1.110223e-16 10 1 1.000000 6.10623e-16 0 6.106227e-16 10 2 1.000000 1 1 4.440892e-16 10 3 1.000000 9.4369e-16 0 9.436896e-16 10 4 1.000000 2 2 2.664535e-15 10 5 1.000000 2.22045e-15 0 2.220446e-15 10 6 1.000000 6 6 1.776357e-14 10 7 1.000000 5.55112e-15 0 5.551115e-15 10 8 1.000000 24 24 1.065814e-13 10 9 1.000000 1.78746e-14 0 1.787459e-14 10 10 1.000000 120 120 6.963319e-13 10 11 1.000000 1.35003e-13 0 1.350031e-13 10 12 1.000000 720 720 4.888534e-12 10 13 1.000000 5.68434e-13 0 5.684342e-13 10 14 1.000000 5040 5040 3.728928e-11 10 15 1.000000 1.13687e-12 0 1.136868e-12 10 16 1.000000 40320 40320 2.983143e-10 10 17 1.000000 -5.16138e-11 0 5.161382e-11 10 18 1.000000 362880 362880 2.619345e-09 10 19 1.000000 -1.4843e-09 0 1.484295e-09 10 20 1.000000 3.6144e+06 3.6288e+06 1.440000e+04 10 21 1.000000 -2.7474e-08 0 2.747402e-08 10 22 1.000000 3.90384e+07 3.99168e+07 8.784000e+05 1 0 2.500000 0.919063 0.919063 0.000000e+00 1 1 2.500000 0 0 0.000000e+00 1 2 2.500000 0 1.60836 1.608359e+00 1 3 2.500000 0 0 0.000000e+00 1 4 2.500000 0 4.42299 4.422988e+00 2 0 2.500000 0.919063 0.919063 2.220446e-16 2 1 2.500000 0 0 0.000000e+00 2 2 2.500000 1.60836 1.60836 8.881784e-16 2 3 2.500000 0 0 0.000000e+00 2 4 2.500000 2.81463 4.42299 1.608359e+00 2 5 2.500000 0 0 0.000000e+00 2 6 2.500000 4.9256 16.5862 1.166061e+01 3 0 2.500000 0.919063 0.919063 0.000000e+00 3 1 2.500000 1.04355e-17 0 1.043554e-17 3 2 2.500000 1.60836 1.60836 2.220446e-16 3 3 2.500000 1.01747e-50 0 1.017468e-50 3 4 2.500000 4.42299 4.42299 8.881784e-16 3 5 2.500000 9.92033e-84 0 9.920331e-84 3 6 2.500000 12.1632 16.5862 4.422988e+00 3 7 2.500000 9.67234e-117 0 9.672343e-117 3 8 2.500000 33.4488 78.7845 4.533563e+01 4 0 2.500000 0.919063 0.919063 3.330669e-16 4 1 2.500000 -7.77156e-16 0 7.771561e-16 4 2 2.500000 1.60836 1.60836 8.881784e-16 4 3 2.500000 -1.33227e-15 0 1.332268e-15 4 4 2.500000 4.42299 4.42299 1.776357e-15 4 5 2.500000 -2.66454e-15 0 2.664535e-15 4 6 2.500000 16.5862 16.5862 0.000000e+00 4 7 2.500000 -5.32907e-15 0 5.329071e-15 4 8 2.500000 69.9385 78.7845 8.845977e+00 4 9 2.500000 -1.42109e-14 0 1.421085e-14 4 10 2.500000 304.841 453.011 1.481701e+02 5 0 2.500000 0.919063 0.919063 2.220446e-16 5 1 2.500000 -7.62759e-17 0 7.627588e-17 5 2 2.500000 1.60836 1.60836 4.440892e-16 5 3 2.500000 -1.66533e-16 0 1.665335e-16 5 4 2.500000 4.42299 4.42299 2.664535e-15 5 5 2.500000 8.88178e-16 0 8.881784e-16 5 6 2.500000 16.5862 16.5862 1.776357e-14 5 7 2.500000 1.68754e-14 0 1.687539e-14 5 8 2.500000 78.7845 78.7845 1.136868e-13 5 9 2.500000 1.35003e-13 0 1.350031e-13 5 10 2.500000 419.838 453.011 3.317241e+01 5 11 2.500000 1.00542e-12 0 1.005418e-12 5 12 2.500000 2336.32 3057.82 7.215000e+02 6 0 2.500000 0.919063 0.919063 2.220446e-16 6 1 2.500000 7.77156e-16 0 7.771561e-16 6 2 2.500000 1.60836 1.60836 8.881784e-16 6 3 2.500000 2.33147e-15 0 2.331468e-15 6 4 2.500000 4.42299 4.42299 8.881784e-16 6 5 2.500000 1.02141e-14 0 1.021405e-14 6 6 2.500000 16.5862 16.5862 7.105427e-15 6 7 2.500000 5.31797e-14 0 5.317968e-14 6 8 2.500000 78.7845 78.7845 8.526513e-14 6 9 2.500000 3.2685e-13 0 3.268497e-13 6 10 2.500000 453.011 453.011 8.526513e-13 6 11 2.500000 2.21689e-12 0 2.216893e-12 6 12 2.500000 2958.31 3057.82 9.951724e+01 6 13 2.500000 1.59162e-11 0 1.591616e-11 6 14 2.500000 20687.7 23698.1 3.010396e+03 7 0 2.500000 0.919063 0.919063 4.440892e-16 7 1 2.500000 -4.62956e-16 0 4.629560e-16 7 2 2.500000 1.60836 1.60836 1.332268e-15 7 3 2.500000 -6.66134e-16 0 6.661338e-16 7 4 2.500000 4.42299 4.42299 8.881784e-16 7 5 2.500000 3.77476e-15 0 3.774758e-15 7 6 2.500000 16.5862 16.5862 1.776357e-14 7 7 2.500000 2.90878e-14 0 2.908784e-14 7 8 2.500000 78.7845 78.7845 1.847411e-13 7 9 2.500000 1.01918e-13 0 1.019185e-13 7 10 2.500000 453.011 453.011 1.648459e-12 7 11 2.500000 -1.72307e-13 0 1.723066e-13 7 12 2.500000 3057.82 3057.82 1.364242e-11 7 13 2.500000 -8.57492e-12 0 8.574919e-12 7 14 2.500000 23225.4 23698.1 4.727069e+02 7 15 2.500000 -1.19514e-10 0 1.195137e-10 7 16 2.500000 189750 207359 1.760833e+04 8 0 2.500000 0.919063 0.919063 0.000000e+00 8 1 2.500000 8.32667e-17 0 8.326673e-17 8 2 2.500000 1.60836 1.60836 6.661338e-16 8 3 2.500000 -6.66134e-16 0 6.661338e-16 8 4 2.500000 4.42299 4.42299 8.881784e-16 8 5 2.500000 -7.32747e-15 0 7.327472e-15 8 6 2.500000 16.5862 16.5862 0.000000e+00 8 7 2.500000 -5.77316e-14 0 5.773160e-14 8 8 2.500000 78.7845 78.7845 2.842171e-14 8 9 2.500000 -4.12115e-13 0 4.121148e-13 8 10 2.500000 453.011 453.011 4.547474e-13 8 11 2.500000 -3.2685e-12 0 3.268497e-12 8 12 2.500000 3057.82 3057.82 4.092726e-12 8 13 2.500000 -2.93312e-11 0 2.933120e-11 8 14 2.500000 23698.1 23698.1 3.637979e-11 8 15 2.500000 -2.81943e-10 0 2.819434e-10 8 16 2.500000 205468 207359 1.890828e+03 8 17 2.500000 -2.82307e-09 0 2.823072e-09 8 18 2.500000 1.93146e+06 2.02175e+06 9.028702e+04 9 0 2.500000 0.919063 0.919063 0.000000e+00 9 1 2.500000 8.88178e-16 0 8.881784e-16 9 2 2.500000 1.60836 1.60836 8.881784e-16 9 3 2.500000 2.10942e-15 0 2.109424e-15 9 4 2.500000 4.42299 4.42299 5.329071e-15 9 5 2.500000 5.32907e-15 0 5.329071e-15 9 6 2.500000 16.5862 16.5862 3.197442e-14 9 7 2.500000 2.66454e-14 0 2.664535e-14 9 8 2.500000 78.7845 78.7845 1.989520e-13 9 9 2.500000 1.74083e-13 0 1.740830e-13 9 10 2.500000 453.011 453.011 1.080025e-12 9 11 2.500000 1.19371e-12 0 1.193712e-12 9 12 2.500000 3057.82 3057.82 6.366463e-12 9 13 2.500000 9.77707e-12 0 9.777068e-12 9 14 2.500000 23698.1 23698.1 2.182787e-11 9 15 2.500000 9.28821e-11 0 9.288215e-11 9 16 2.500000 207359 207359 1.455192e-10 9 17 2.500000 1.12777e-09 0 1.127773e-09 9 18 2.500000 2.01087e+06 2.02175e+06 1.087226e+04 9 19 2.500000 1.2662e-08 0 1.266199e-08 9 20 2.500000 2.11168e+07 2.17338e+07 6.170007e+05 10 0 2.500000 0.919063 0.919063 5.551115e-16 10 1 2.500000 0 0 0.000000e+00 10 2 2.500000 1.60836 1.60836 8.881784e-16 10 3 2.500000 -5.55112e-17 0 5.551115e-17 10 4 2.500000 4.42299 4.42299 1.776357e-15 10 5 2.500000 -4.44089e-16 0 4.440892e-16 10 6 2.500000 16.5862 16.5862 7.105427e-15 10 7 2.500000 1.11022e-14 0 1.110223e-14 10 8 2.500000 78.7845 78.7845 1.136868e-13 10 9 2.500000 1.9984e-13 0 1.998401e-13 10 10 2.500000 453.011 453.011 1.136868e-12 10 11 2.500000 2.38742e-12 0 2.387424e-12 10 12 2.500000 3057.82 3057.82 1.045919e-11 10 13 2.500000 2.47837e-11 0 2.478373e-11 10 14 2.500000 23698.1 23698.1 9.458745e-11 10 15 2.500000 2.38288e-10 0 2.382876e-10 10 16 2.500000 207359 207359 8.731149e-10 10 17 2.500000 2.17915e-09 0 2.179149e-09 10 18 2.500000 2.02175e+06 2.02175e+06 7.450581e-09 10 19 2.500000 1.90339e-08 0 1.903391e-08 10 20 2.500000 2.16794e+07 2.17338e+07 5.436129e+04 10 21 2.500000 1.49012e-07 0 1.490116e-07 10 22 2.500000 2.51607e+08 2.55372e+08 3.764519e+06 SANDIA_RULES_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 1 0.0000000000000000 1.2254167024651776 Order = 2 ALPHA = 0.500000 1 -0.8660254037844385 0.6127083512325888 2 0.8660254037844385 0.6127083512325888 Order = 3 ALPHA = 0.500000 1 -1.3228756555322951 0.2625892933853953 2 0.0000000000000001 0.7002381156943873 3 1.3228756555322949 0.2625892933853950 Order = 4 ALPHA = 0.500000 1 -1.7529619663678662 0.0747721865343164 2 -0.6535475074298001 0.5379361646982722 3 0.6535475074298001 0.5379361646982725 4 1.7529619663678662 0.0747721865343164 Order = 5 ALPHA = 0.500000 1 -2.0995981508797588 0.0206908527402406 2 -1.0448385544294867 0.3373854564216620 3 -0.0000000000000003 0.5092640841413719 4 1.0448385544294874 0.3373854564216615 5 2.0995981508797574 0.0206908527402406 Order = 6 ALPHA = 0.500000 1 -2.4311960068148712 0.0047584322858768 2 -1.4282643308502345 0.1432946705182553 3 -0.5471261076464521 0.4646552484284568 4 0.5471261076464523 0.4646552484284568 5 1.4282643308502347 0.1432946705182556 6 2.4311960068148721 0.0047584322858768 Order = 7 ALPHA = 0.500000 1 -2.7198800885562906 0.0011062894019685 2 -1.7473607788965215 0.0556473312506609 3 -0.8938582730216025 0.3522490969234104 4 0.0000000000000002 0.4074112673130981 5 0.8938582730216027 0.3522490969234107 6 1.7473607788965222 0.0556473312506609 7 2.7198800885562906 0.0011062894019685 Order = 8 ALPHA = 0.500000 1 -2.9990789683433174 0.0002288084584739 2 -2.0574394184774687 0.0178757746392672 3 -1.2417383409431890 0.1866121206001916 4 -0.4801606747408061 0.4079916475346554 5 0.4801606747408064 0.4079916475346549 6 1.2417383409431892 0.1866121206001921 7 2.0574394184774696 0.0178757746392672 8 2.9990789683433166 0.0002288084584739 Order = 9 ALPHA = 0.500000 1 -3.2511523261341324 0.0000482442834952 2 -2.3313221193007125 0.0055757541036437 3 -1.5374164086847442 0.0887579748998607 4 -0.7945417010067838 0.3467847917084952 5 0.0000000000000002 0.3430831724741884 6 0.7945417010067843 0.3467847917084951 7 1.5374164086847442 0.0887579748998605 8 2.3313221193007121 0.0055757541036437 9 3.2511523261341315 0.0000482442834952 Order = 10 ALPHA = 0.500000 1 -3.4966058807476776 0.0000093473340834 2 -2.5983971495446250 0.0015363564424025 3 -1.8279918123652750 0.0351763431437458 4 -1.1149053705666441 0.2117439807373518 5 -0.4330259998733385 0.3642423235750056 6 0.4330259998733390 0.3642423235750059 7 1.1149053705666441 0.2117439807373522 8 1.8279918123652759 0.0351763431437458 9 2.5983971495446245 0.0015363564424026 10 3.4966058807476781 0.0000093473340834 Order = 1 ALPHA = 1.000000 1 0.0000000000000000 1.0000000000000000 Order = 2 ALPHA = 1.000000 1 -0.9999999999999998 0.4999999999999999 2 0.9999999999999998 0.4999999999999999 Order = 3 ALPHA = 1.000000 1 -1.4142135623730949 0.2500000000000002 2 0.0000000000000001 0.5000000000000001 3 1.4142135623730947 0.2499999999999999 Order = 4 ALPHA = 1.000000 1 -1.8477590650225741 0.0732233047033630 2 -0.7653668647301797 0.4267766952966369 3 0.7653668647301797 0.4267766952966366 4 1.8477590650225741 0.0732233047033630 Order = 5 ALPHA = 1.000000 1 -2.1753277471610746 0.0223290993692602 2 -1.1260325006104939 0.3110042339640734 3 -0.0000000000000001 0.3333333333333333 4 1.1260325006104941 0.3110042339640728 5 2.1753277471610750 0.0223290993692602 Order = 6 ALPHA = 1.000000 1 -2.5079762923395998 0.0051946282507931 2 -1.5146882056314563 0.1392588667846203 3 -0.6448058287449634 0.3555465049645866 4 0.6448058287449638 0.3555465049645871 5 1.5146882056314572 0.1392588667846204 6 2.5079762923395985 0.0051946282507931 Order = 7 ALPHA = 1.000000 1 -2.7854569612800750 0.0012954668385735 2 -1.8180779106881744 0.0591781927275504 3 -0.9673790505919011 0.3145263404338762 4 0.0000000000000000 0.2499999999999997 5 0.9673790505919010 0.3145263404338764 6 1.8180779106881746 0.0591781927275503 7 2.7854569612800750 0.0012954668385735 Order = 8 ALPHA = 1.000000 1 -3.0651379923750790 0.0002696473527807 2 -2.1299343409882678 0.0194439542575027 3 -1.3212725309936428 0.1787093462188998 4 -0.5679328213965031 0.3015770521708169 5 0.5679328213965031 0.3015770521708167 6 1.3212725309936417 0.1787093462188998 7 2.1299343409882683 0.0194439542575027 8 3.0651379923750781 0.0002696473527807 Order = 9 ALPHA = 1.000000 1 -3.3096667978337639 0.0000600630994212 2 -2.3939880433471457 0.0064714248102269 3 -1.6036318179826308 0.0928661670384225 4 -0.8621437977399313 0.3006023450519296 5 0.0000000000000005 0.2000000000000001 6 0.8621437977399317 0.3006023450519298 7 1.6036318179826310 0.0928661670384228 8 2.3939880433471470 0.0064714248102269 9 3.3096667978337626 0.0000600630994212 Order = 10 ALPHA = 1.000000 1 -3.5553903926679840 0.0000116849861929 2 -2.6619184821964113 0.0018058793399610 3 -1.8964244701650330 0.0379712248408538 4 -1.1888662915174757 0.1993334055415881 5 -0.5133812615572766 0.2608778052914038 6 0.5133812615572765 0.2608778052914043 7 1.1888662915174761 0.1993334055415882 8 1.8964244701650328 0.0379712248408538 9 2.6619184821964117 0.0018058793399610 10 3.5553903926679813 0.0000116849861929 Order = 1 ALPHA = 2.500000 1 0.0000000000000000 0.9190625268488832 Order = 2 ALPHA = 2.500000 1 -1.3228756555322951 0.4595312634244415 2 1.3228756555322951 0.4595312634244415 Order = 3 ALPHA = 2.500000 1 -1.6583123951777001 0.2924289858155537 2 0.0000000000000000 0.3342045552177758 3 1.6583123951777001 0.2924289858155537 Order = 4 ALPHA = 2.500000 1 -2.0995981508797579 0.0912117426015990 2 -1.0448385544294874 0.3683195208228430 3 1.0448385544294869 0.3683195208228425 4 2.0995981508797579 0.0912117426015990 Order = 5 ALPHA = 2.500000 1 -2.3846365914125593 0.0341953454149221 2 -1.3466656329231439 0.3362147032847794 3 0.0000000000000001 0.1782424294494806 4 1.3466656329231437 0.3362147032847794 5 2.3846365914125598 0.0341953454149221 Order = 6 ALPHA = 2.500000 1 -2.7198800885562910 0.0081840498746596 2 -1.7473607788965220 0.1699063099275072 3 -0.8938582730216031 0.2814409036222746 4 0.8938582730216029 0.2814409036222749 5 1.7473607788965220 0.1699063099275075 6 2.7198800885562910 0.0081840498746597 Order = 7 ALPHA = 2.500000 1 -2.9702203625084236 0.0024098131356743 2 -2.0150214027494169 0.0871896206077823 3 -1.1693928957378272 0.3136447466969385 4 0.0000000000000001 0.1125741659680929 5 1.1693928957378263 0.3136447466969383 6 2.0150214027494182 0.0871896206077823 7 2.9702203625084227 0.0024098131356743 Order = 8 ALPHA = 2.500000 1 -3.2511523261341324 0.0005099416639457 2 -2.3313221193007139 0.0303045738441492 3 -1.5374164086847446 0.2097927175810473 4 -0.7945417010067843 0.2189240303352995 5 0.7945417010067850 0.2189240303352991 6 1.5374164086847439 0.2097927175810477 7 2.3313221193007139 0.0303045738441492 8 3.2511523261341324 0.0005099416639457 Order = 9 ALPHA = 2.500000 1 -3.4765341365933922 0.0001303236536236 2 -2.5712913745972621 0.0117476928703869 3 -1.7891453520001157 0.1310178979856828 4 -1.0493474035985140 0.2774791172736721 5 -0.0000000000000003 0.0783124632821516 6 1.0493474035985142 0.2774791172736726 7 1.7891453520001153 0.1310178979856830 8 2.5712913745972616 0.0117476928703870 9 3.4765341365933926 0.0001303236536236 Order = 10 ALPHA = 2.500000 1 -3.7229793194118361 0.0000256325065578 2 -2.8414549096381929 0.0033881904305600 3 -2.0875400348754183 0.0579439549110724 4 -1.3912394012659239 0.2236074413923169 5 -0.7226261238582253 0.1745660441839348 6 0.7226261238582262 0.1745660441839347 7 1.3912394012659242 0.2236074413923169 8 2.0875400348754178 0.0579439549110723 9 2.8414549096381942 0.0033881904305600 10 3.7229793194118344 0.0000256325065578 SANDIA_RULES_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 5.000000e-01 8.862269e-01 8.862269e-01 0.000000e+00 1 1 5.000000e-01 1.329340e+00 1.329340e+00 0.000000e+00 1 2 5.000000e-01 1.994011e+00 3.323351e+00 1.329340e+00 1 3 5.000000e-01 2.991016e+00 1.163173e+01 8.640713e+00 1 4 5.000000e-01 4.486524e+00 5.234278e+01 4.785625e+01 2 0 5.000000e-01 8.862269e-01 8.862269e-01 3.330669e-16 2 1 5.000000e-01 1.329340e+00 1.329340e+00 4.440892e-16 2 2 5.000000e-01 3.323351e+00 3.323351e+00 2.220446e-15 2 3 5.000000e-01 1.163173e+01 1.163173e+01 1.065814e-14 2 4 5.000000e-01 4.569608e+01 5.234278e+01 6.646702e+00 2 5 5.000000e-01 1.848614e+02 2.878853e+02 1.030239e+02 2 6 5.000000e-01 7.529467e+02 1.871254e+03 1.118308e+03 3 0 5.000000e-01 8.862269e-01 8.862269e-01 3.330669e-16 3 1 5.000000e-01 1.329340e+00 1.329340e+00 0.000000e+00 3 2 5.000000e-01 3.323351e+00 3.323351e+00 8.881784e-16 3 3 5.000000e-01 1.163173e+01 1.163173e+01 7.105427e-15 3 4 5.000000e-01 5.234278e+01 5.234278e+01 2.842171e-14 3 5 5.000000e-01 2.878853e+02 2.878853e+02 2.273737e-13 3 6 5.000000e-01 1.801464e+03 1.871254e+03 6.979037e+01 3 7 5.000000e-01 1.204538e+04 1.403441e+04 1.989026e+03 3 8 5.000000e-01 8.296657e+04 1.192925e+05 3.632589e+04 4 0 5.000000e-01 8.862269e-01 8.862269e-01 6.661338e-16 4 1 5.000000e-01 1.329340e+00 1.329340e+00 1.332268e-15 4 2 5.000000e-01 3.323351e+00 3.323351e+00 2.664535e-15 4 3 5.000000e-01 1.163173e+01 1.163173e+01 5.329071e-15 4 4 5.000000e-01 5.234278e+01 5.234278e+01 3.552714e-14 4 5 5.000000e-01 2.878853e+02 2.878853e+02 2.273737e-13 4 6 5.000000e-01 1.871254e+03 1.871254e+03 2.501110e-12 4 7 5.000000e-01 1.403441e+04 1.403441e+04 2.182787e-11 4 8 5.000000e-01 1.180362e+05 1.192925e+05 1.256227e+03 4 9 5.000000e-01 1.076120e+06 1.133278e+06 5.715831e+04 4 10 5.000000e-01 1.031564e+07 1.189942e+07 1.583788e+06 5 0 5.000000e-01 8.862269e-01 8.862269e-01 1.110223e-16 5 1 5.000000e-01 1.329340e+00 1.329340e+00 4.440892e-16 5 2 5.000000e-01 3.323351e+00 3.323351e+00 1.332268e-15 5 3 5.000000e-01 1.163173e+01 1.163173e+01 3.552714e-15 5 4 5.000000e-01 5.234278e+01 5.234278e+01 2.842171e-14 5 5 5.000000e-01 2.878853e+02 2.878853e+02 2.273737e-13 5 6 5.000000e-01 1.871254e+03 1.871254e+03 1.818989e-12 5 7 5.000000e-01 1.403441e+04 1.403441e+04 1.091394e-11 5 8 5.000000e-01 1.192925e+05 1.192925e+05 7.275958e-11 5 9 5.000000e-01 1.133278e+06 1.133278e+06 6.984919e-10 5 10 5.000000e-01 1.186488e+07 1.189942e+07 3.454623e+04 5 11 5.000000e-01 1.345460e+08 1.368434e+08 2.297325e+06 5 12 5.000000e-01 1.620549e+09 1.710542e+09 8.999294e+07 6 0 5.000000e-01 8.862269e-01 8.862269e-01 3.330669e-16 6 1 5.000000e-01 1.329340e+00 1.329340e+00 8.881784e-16 6 2 5.000000e-01 3.323351e+00 3.323351e+00 2.664535e-15 6 3 5.000000e-01 1.163173e+01 1.163173e+01 8.881784e-15 6 4 5.000000e-01 5.234278e+01 5.234278e+01 3.552714e-14 6 5 5.000000e-01 2.878853e+02 2.878853e+02 1.136868e-13 6 6 5.000000e-01 1.871254e+03 1.871254e+03 2.273737e-13 6 7 5.000000e-01 1.403441e+04 1.403441e+04 3.637979e-12 6 8 5.000000e-01 1.192925e+05 1.192925e+05 7.275958e-11 6 9 5.000000e-01 1.133278e+06 1.133278e+06 1.164153e-09 6 10 5.000000e-01 1.189942e+07 1.189942e+07 1.676381e-08 6 11 5.000000e-01 1.368434e+08 1.368434e+08 1.788139e-07 6 12 5.000000e-01 1.709195e+09 1.710542e+09 1.347303e+06 6 13 5.000000e-01 2.296904e+10 2.309232e+10 1.232782e+08 6 14 5.000000e-01 3.283483e+11 3.348386e+11 6.490296e+09 7 0 5.000000e-01 8.862269e-01 8.862269e-01 1.110223e-16 7 1 5.000000e-01 1.329340e+00 1.329340e+00 4.440892e-16 7 2 5.000000e-01 3.323351e+00 3.323351e+00 3.108624e-15 7 3 5.000000e-01 1.163173e+01 1.163173e+01 1.776357e-14 7 4 5.000000e-01 5.234278e+01 5.234278e+01 9.947598e-14 7 5 5.000000e-01 2.878853e+02 2.878853e+02 6.252776e-13 7 6 5.000000e-01 1.871254e+03 1.871254e+03 3.865352e-12 7 7 5.000000e-01 1.403441e+04 1.403441e+04 2.546585e-11 7 8 5.000000e-01 1.192925e+05 1.192925e+05 1.600711e-10 7 9 5.000000e-01 1.133278e+06 1.133278e+06 1.164153e-09 7 10 5.000000e-01 1.189942e+07 1.189942e+07 5.587935e-09 7 11 5.000000e-01 1.368434e+08 1.368434e+08 8.940697e-08 7 12 5.000000e-01 1.710542e+09 1.710542e+09 7.152557e-07 7 13 5.000000e-01 2.309232e+10 2.309232e+10 7.629395e-06 7 14 5.000000e-01 3.347679e+11 3.348386e+11 7.073341e+07 7 15 5.000000e-01 5.181475e+12 5.189998e+12 8.523376e+09 7 16 5.000000e-01 8.505429e+13 8.563497e+13 5.806860e+11 8 0 5.000000e-01 8.862269e-01 8.862269e-01 1.110223e-16 8 1 5.000000e-01 1.329340e+00 1.329340e+00 0.000000e+00 8 2 5.000000e-01 3.323351e+00 3.323351e+00 8.881784e-16 8 3 5.000000e-01 1.163173e+01 1.163173e+01 0.000000e+00 8 4 5.000000e-01 5.234278e+01 5.234278e+01 2.842171e-14 8 5 5.000000e-01 2.878853e+02 2.878853e+02 2.273737e-13 8 6 5.000000e-01 1.871254e+03 1.871254e+03 2.501110e-12 8 7 5.000000e-01 1.403441e+04 1.403441e+04 2.546585e-11 8 8 5.000000e-01 1.192925e+05 1.192925e+05 2.764864e-10 8 9 5.000000e-01 1.133278e+06 1.133278e+06 2.561137e-09 8 10 5.000000e-01 1.189942e+07 1.189942e+07 3.166497e-08 8 11 5.000000e-01 1.368434e+08 1.368434e+08 3.576279e-07 8 12 5.000000e-01 1.710542e+09 1.710542e+09 5.006790e-06 8 13 5.000000e-01 2.309232e+10 2.309232e+10 6.484985e-05 8 14 5.000000e-01 3.348386e+11 3.348386e+11 1.037598e-03 8 15 5.000000e-01 5.189998e+12 5.189998e+12 1.562500e-02 8 16 5.000000e-01 8.563016e+13 8.563497e+13 4.809872e+09 8 17 5.000000e-01 1.497874e+15 1.498612e+15 7.383154e+11 8 18 5.000000e-01 2.766111e+16 2.772432e+16 6.321014e+13 9 0 5.000000e-01 8.862269e-01 8.862269e-01 2.220446e-16 9 1 5.000000e-01 1.329340e+00 1.329340e+00 2.220446e-16 9 2 5.000000e-01 3.323351e+00 3.323351e+00 8.881784e-16 9 3 5.000000e-01 1.163173e+01 1.163173e+01 7.105427e-15 9 4 5.000000e-01 5.234278e+01 5.234278e+01 4.263256e-14 9 5 5.000000e-01 2.878853e+02 2.878853e+02 3.410605e-13 9 6 5.000000e-01 1.871254e+03 1.871254e+03 2.273737e-12 9 7 5.000000e-01 1.403441e+04 1.403441e+04 2.182787e-11 9 8 5.000000e-01 1.192925e+05 1.192925e+05 1.891749e-10 9 9 5.000000e-01 1.133278e+06 1.133278e+06 1.629815e-09 9 10 5.000000e-01 1.189942e+07 1.189942e+07 1.303852e-08 9 11 5.000000e-01 1.368434e+08 1.368434e+08 2.980232e-08 9 12 5.000000e-01 1.710542e+09 1.710542e+09 0.000000e+00 9 13 5.000000e-01 2.309232e+10 2.309232e+10 1.525879e-05 9 14 5.000000e-01 3.348386e+11 3.348386e+11 2.441406e-04 9 15 5.000000e-01 5.189998e+12 5.189998e+12 6.835938e-03 9 16 5.000000e-01 8.563497e+13 8.563497e+13 1.406250e-01 9 17 5.000000e-01 1.498612e+15 1.498612e+15 2.750000e+00 9 18 5.000000e-01 2.772391e+16 2.772432e+16 4.112441e+11 9 19 5.000000e-01 5.405460e+17 5.406243e+17 7.834199e+13 9 20 5.000000e-01 1.107456e+19 1.108280e+19 8.233929e+15 10 0 5.000000e-01 8.862269e-01 8.862269e-01 0.000000e+00 10 1 5.000000e-01 1.329340e+00 1.329340e+00 0.000000e+00 10 2 5.000000e-01 3.323351e+00 3.323351e+00 1.776357e-15 10 3 5.000000e-01 1.163173e+01 1.163173e+01 1.065814e-14 10 4 5.000000e-01 5.234278e+01 5.234278e+01 4.263256e-14 10 5 5.000000e-01 2.878853e+02 2.878853e+02 2.273737e-13 10 6 5.000000e-01 1.871254e+03 1.871254e+03 6.821210e-13 10 7 5.000000e-01 1.403441e+04 1.403441e+04 0.000000e+00 10 8 5.000000e-01 1.192925e+05 1.192925e+05 1.455192e-11 10 9 5.000000e-01 1.133278e+06 1.133278e+06 4.656613e-10 10 10 5.000000e-01 1.189942e+07 1.189942e+07 3.725290e-09 10 11 5.000000e-01 1.368434e+08 1.368434e+08 8.940697e-08 10 12 5.000000e-01 1.710542e+09 1.710542e+09 7.152557e-07 10 13 5.000000e-01 2.309232e+10 2.309232e+10 3.814697e-06 10 14 5.000000e-01 3.348386e+11 3.348386e+11 1.220703e-04 10 15 5.000000e-01 5.189998e+12 5.189998e+12 2.929688e-03 10 16 5.000000e-01 8.563497e+13 8.563497e+13 7.812500e-02 10 17 5.000000e-01 1.498612e+15 1.498612e+15 1.250000e+00 10 18 5.000000e-01 2.772432e+16 2.772432e+16 3.600000e+01 10 19 5.000000e-01 5.406243e+17 5.406243e+17 1.216000e+03 10 20 5.000000e-01 1.108275e+19 1.108280e+19 4.318063e+13 10 21 5.000000e-01 2.382702e+20 2.382802e+20 9.996315e+15 10 22 5.000000e-01 5.360038e+21 5.361304e+21 1.265462e+18 1 0 1.000000e+00 1.000000e+00 1.000000e+00 0.000000e+00 1 1 1.000000e+00 2.000000e+00 2.000000e+00 0.000000e+00 1 2 1.000000e+00 4.000000e+00 6.000000e+00 2.000000e+00 1 3 1.000000e+00 8.000000e+00 2.400000e+01 1.600000e+01 1 4 1.000000e+00 1.600000e+01 1.200000e+02 1.040000e+02 2 0 1.000000e+00 1.000000e+00 1.000000e+00 2.220446e-16 2 1 1.000000e+00 2.000000e+00 2.000000e+00 0.000000e+00 2 2 1.000000e+00 6.000000e+00 6.000000e+00 0.000000e+00 2 3 1.000000e+00 2.400000e+01 2.400000e+01 3.552714e-15 2 4 1.000000e+00 1.080000e+02 1.200000e+02 1.200000e+01 2 5 1.000000e+00 5.040000e+02 7.200000e+02 2.160000e+02 2 6 1.000000e+00 2.376000e+03 5.040000e+03 2.664000e+03 3 0 1.000000e+00 1.000000e+00 1.000000e+00 4.440892e-16 3 1 1.000000e+00 2.000000e+00 2.000000e+00 1.332268e-15 3 2 1.000000e+00 6.000000e+00 6.000000e+00 6.217249e-15 3 3 1.000000e+00 2.400000e+01 2.400000e+01 2.842171e-14 3 4 1.000000e+00 1.200000e+02 1.200000e+02 1.705303e-13 3 5 1.000000e+00 7.200000e+02 7.200000e+02 1.364242e-12 3 6 1.000000e+00 4.896000e+03 5.040000e+03 1.440000e+02 3 7 1.000000e+00 3.571200e+04 4.032000e+04 4.608000e+03 3 8 1.000000e+00 2.695680e+05 3.628800e+05 9.331200e+04 4 0 1.000000e+00 1.000000e+00 1.000000e+00 0.000000e+00 4 1 1.000000e+00 2.000000e+00 2.000000e+00 2.220446e-16 4 2 1.000000e+00 6.000000e+00 6.000000e+00 1.776357e-15 4 3 1.000000e+00 2.400000e+01 2.400000e+01 1.421085e-14 4 4 1.000000e+00 1.200000e+02 1.200000e+02 8.526513e-14 4 5 1.000000e+00 7.200000e+02 7.200000e+02 5.684342e-13 4 6 1.000000e+00 5.040000e+03 5.040000e+03 6.366463e-12 4 7 1.000000e+00 4.032000e+04 4.032000e+04 5.820766e-11 4 8 1.000000e+00 3.600000e+05 3.628800e+05 2.880000e+03 4 9 1.000000e+00 3.484800e+06 3.628800e+06 1.440000e+05 4 10 1.000000e+00 3.556800e+07 3.991680e+07 4.348800e+06 5 0 1.000000e+00 1.000000e+00 1.000000e+00 0.000000e+00 5 1 1.000000e+00 2.000000e+00 2.000000e+00 2.220446e-16 5 2 1.000000e+00 6.000000e+00 6.000000e+00 2.664535e-15 5 3 1.000000e+00 2.400000e+01 2.400000e+01 2.131628e-14 5 4 1.000000e+00 1.200000e+02 1.200000e+02 1.989520e-13 5 5 1.000000e+00 7.200000e+02 7.200000e+02 1.705303e-12 5 6 1.000000e+00 5.040000e+03 5.040000e+03 1.455192e-11 5 7 1.000000e+00 4.032000e+04 4.032000e+04 1.382432e-10 5 8 1.000000e+00 3.628800e+05 3.628800e+05 1.280569e-09 5 9 1.000000e+00 3.628800e+06 3.628800e+06 1.117587e-08 5 10 1.000000e+00 3.983040e+07 3.991680e+07 8.640000e+04 5 11 1.000000e+00 4.727808e+08 4.790016e+08 6.220800e+06 5 12 1.000000e+00 5.964710e+09 6.227021e+09 2.623104e+08 6 0 1.000000e+00 1.000000e+00 1.000000e+00 4.440892e-16 6 1 1.000000e+00 2.000000e+00 2.000000e+00 8.881784e-16 6 2 1.000000e+00 6.000000e+00 6.000000e+00 5.329071e-15 6 3 1.000000e+00 2.400000e+01 2.400000e+01 2.842171e-14 6 4 1.000000e+00 1.200000e+02 1.200000e+02 1.421085e-13 6 5 1.000000e+00 7.200000e+02 7.200000e+02 1.023182e-12 6 6 1.000000e+00 5.040000e+03 5.040000e+03 7.275958e-12 6 7 1.000000e+00 4.032000e+04 4.032000e+04 5.093170e-11 6 8 1.000000e+00 3.628800e+05 3.628800e+05 4.074536e-10 6 9 1.000000e+00 3.628800e+06 3.628800e+06 3.725290e-09 6 10 1.000000e+00 3.991680e+07 3.991680e+07 3.725290e-08 6 11 1.000000e+00 4.790016e+08 4.790016e+08 3.576279e-07 6 12 1.000000e+00 6.223392e+09 6.227021e+09 3.628800e+06 6 13 1.000000e+00 8.682267e+10 8.717829e+10 3.556224e+08 6 14 1.000000e+00 1.287709e+12 1.307674e+12 1.996566e+10 7 0 1.000000e+00 1.000000e+00 1.000000e+00 0.000000e+00 7 1 1.000000e+00 2.000000e+00 2.000000e+00 0.000000e+00 7 2 1.000000e+00 6.000000e+00 6.000000e+00 1.776357e-15 7 3 1.000000e+00 2.400000e+01 2.400000e+01 7.105427e-15 7 4 1.000000e+00 1.200000e+02 1.200000e+02 5.684342e-14 7 5 1.000000e+00 7.200000e+02 7.200000e+02 7.958079e-13 7 6 1.000000e+00 5.040000e+03 5.040000e+03 6.366463e-12 7 7 1.000000e+00 4.032000e+04 4.032000e+04 6.548362e-11 7 8 1.000000e+00 3.628800e+05 3.628800e+05 6.402843e-10 7 9 1.000000e+00 3.628800e+06 3.628800e+06 8.381903e-09 7 10 1.000000e+00 3.991680e+07 3.991680e+07 8.940697e-08 7 11 1.000000e+00 4.790016e+08 4.790016e+08 9.536743e-07 7 12 1.000000e+00 6.227021e+09 6.227021e+09 1.144409e-05 7 13 1.000000e+00 8.717829e+10 8.717829e+10 1.373291e-04 7 14 1.000000e+00 1.307471e+12 1.307674e+12 2.032128e+08 7 15 1.000000e+00 2.089678e+13 2.092279e+13 2.601124e+10 7 16 1.000000e+00 3.538114e+14 3.556874e+14 1.876061e+12 8 0 1.000000e+00 1.000000e+00 1.000000e+00 2.220446e-16 8 1 1.000000e+00 2.000000e+00 2.000000e+00 4.440892e-16 8 2 1.000000e+00 6.000000e+00 6.000000e+00 1.776357e-15 8 3 1.000000e+00 2.400000e+01 2.400000e+01 1.421085e-14 8 4 1.000000e+00 1.200000e+02 1.200000e+02 1.136868e-13 8 5 1.000000e+00 7.200000e+02 7.200000e+02 9.094947e-13 8 6 1.000000e+00 5.040000e+03 5.040000e+03 1.000444e-11 8 7 1.000000e+00 4.032000e+04 4.032000e+04 9.458745e-11 8 8 1.000000e+00 3.628800e+05 3.628800e+05 9.895302e-10 8 9 1.000000e+00 3.628800e+06 3.628800e+06 9.313226e-09 8 10 1.000000e+00 3.991680e+07 3.991680e+07 9.685755e-08 8 11 1.000000e+00 4.790016e+08 4.790016e+08 8.940697e-07 8 12 1.000000e+00 6.227021e+09 6.227021e+09 6.675720e-06 8 13 1.000000e+00 8.717829e+10 8.717829e+10 0.000000e+00 8 14 1.000000e+00 1.307674e+12 1.307674e+12 1.464844e-03 8 15 1.000000e+00 2.092279e+13 2.092279e+13 4.296875e-02 8 16 1.000000e+00 3.556728e+14 3.556874e+14 1.463132e+10 8 17 1.000000e+00 6.400003e+15 6.402374e+15 2.370274e+12 8 18 1.000000e+00 1.214315e+17 1.216451e+17 2.135880e+14 9 0 1.000000e+00 1.000000e+00 1.000000e+00 2.220446e-16 9 1 1.000000e+00 2.000000e+00 2.000000e+00 6.661338e-16 9 2 1.000000e+00 6.000000e+00 6.000000e+00 8.881784e-16 9 3 1.000000e+00 2.400000e+01 2.400000e+01 3.552714e-15 9 4 1.000000e+00 1.200000e+02 1.200000e+02 2.842171e-14 9 5 1.000000e+00 7.200000e+02 7.200000e+02 3.410605e-13 9 6 1.000000e+00 5.040000e+03 5.040000e+03 2.728484e-12 9 7 1.000000e+00 4.032000e+04 4.032000e+04 1.455192e-11 9 8 1.000000e+00 3.628800e+05 3.628800e+05 1.746230e-10 9 9 1.000000e+00 3.628800e+06 3.628800e+06 1.862645e-09 9 10 1.000000e+00 3.991680e+07 3.991680e+07 2.980232e-08 9 11 1.000000e+00 4.790016e+08 4.790016e+08 3.576279e-07 9 12 1.000000e+00 6.227021e+09 6.227021e+09 5.722046e-06 9 13 1.000000e+00 8.717829e+10 8.717829e+10 9.155273e-05 9 14 1.000000e+00 1.307674e+12 1.307674e+12 1.220703e-03 9 15 1.000000e+00 2.092279e+13 2.092279e+13 1.171875e-02 9 16 1.000000e+00 3.556874e+14 3.556874e+14 6.250000e-02 9 17 1.000000e+00 6.402374e+15 6.402374e+15 3.000000e+00 9 18 1.000000e+00 1.216438e+17 1.216451e+17 1.316819e+12 9 19 1.000000e+00 2.432639e+18 2.432902e+18 2.633638e+14 9 20 1.000000e+00 5.106195e+19 5.109094e+19 2.899635e+16 10 0 1.000000e+00 1.000000e+00 1.000000e+00 2.220446e-16 10 1 1.000000e+00 2.000000e+00 2.000000e+00 0.000000e+00 10 2 1.000000e+00 6.000000e+00 6.000000e+00 1.776357e-15 10 3 1.000000e+00 2.400000e+01 2.400000e+01 1.065814e-14 10 4 1.000000e+00 1.200000e+02 1.200000e+02 9.947598e-14 10 5 1.000000e+00 7.200000e+02 7.200000e+02 1.023182e-12 10 6 1.000000e+00 5.040000e+03 5.040000e+03 8.185452e-12 10 7 1.000000e+00 4.032000e+04 4.032000e+04 8.003553e-11 10 8 1.000000e+00 3.628800e+05 3.628800e+05 6.402843e-10 10 9 1.000000e+00 3.628800e+06 3.628800e+06 6.519258e-09 10 10 1.000000e+00 3.991680e+07 3.991680e+07 5.215406e-08 10 11 1.000000e+00 4.790016e+08 4.790016e+08 4.768372e-07 10 12 1.000000e+00 6.227021e+09 6.227021e+09 3.814697e-06 10 13 1.000000e+00 8.717829e+10 8.717829e+10 3.051758e-05 10 14 1.000000e+00 1.307674e+12 1.307674e+12 2.441406e-04 10 15 1.000000e+00 2.092279e+13 2.092279e+13 1.171875e-02 10 16 1.000000e+00 3.556874e+14 3.556874e+14 4.375000e-01 10 17 1.000000e+00 6.402374e+15 6.402374e+15 1.300000e+01 10 18 1.000000e+00 1.216451e+17 1.216451e+17 3.360000e+02 10 19 1.000000e+00 2.432902e+18 2.432902e+18 9.216000e+03 10 20 1.000000e+00 5.109080e+19 5.109094e+19 1.448501e+14 10 21 1.000000e+00 1.123966e+21 1.124001e+21 3.505372e+16 10 22 1.000000e+00 2.584739e+22 2.585202e+22 4.630278e+18 1 0 2.500000e+00 3.323351e+00 3.323351e+00 4.440892e-16 1 1 2.500000e+00 1.163173e+01 1.163173e+01 1.776357e-15 1 2 2.500000e+00 4.071105e+01 5.234278e+01 1.163173e+01 1 3 2.500000e+00 1.424887e+02 2.878853e+02 1.453966e+02 1 4 2.500000e+00 4.987104e+02 1.871254e+03 1.372544e+03 2 0 2.500000e+00 3.323351e+00 3.323351e+00 8.881784e-16 2 1 2.500000e+00 1.163173e+01 1.163173e+01 3.552714e-15 2 2 2.500000e+00 5.234278e+01 5.234278e+01 2.131628e-14 2 3 2.500000e+00 2.878853e+02 2.878853e+02 1.705303e-13 2 4 2.500000e+00 1.766569e+03 1.871254e+03 1.046856e+02 2 5 2.500000e+00 1.136493e+04 1.403441e+04 2.669482e+03 2 6 2.500000e+00 7.446087e+04 1.192925e+05 4.483159e+04 3 0 2.500000e+00 3.323351e+00 3.323351e+00 8.881784e-16 3 1 2.500000e+00 1.163173e+01 1.163173e+01 3.552714e-15 3 2 2.500000e+00 5.234278e+01 5.234278e+01 2.131628e-14 3 3 2.500000e+00 2.878853e+02 2.878853e+02 1.705303e-13 3 4 2.500000e+00 1.871254e+03 1.871254e+03 1.364242e-12 3 5 2.500000e+00 1.403441e+04 1.403441e+04 1.455192e-11 3 6 2.500000e+00 1.175652e+05 1.192925e+05 1.727312e+03 3 7 2.500000e+00 1.059868e+06 1.133278e+06 7.341075e+04 3 8 2.500000e+00 9.974334e+06 1.189942e+07 1.925089e+06 4 0 2.500000e+00 3.323351e+00 3.323351e+00 8.881784e-16 4 1 2.500000e+00 1.163173e+01 1.163173e+01 1.776357e-15 4 2 2.500000e+00 5.234278e+01 5.234278e+01 0.000000e+00 4 3 2.500000e+00 2.878853e+02 2.878853e+02 0.000000e+00 4 4 2.500000e+00 1.871254e+03 1.871254e+03 4.547474e-13 4 5 2.500000e+00 1.403441e+04 1.403441e+04 1.091394e-11 4 6 2.500000e+00 1.192925e+05 1.192925e+05 1.455192e-10 4 7 2.500000e+00 1.133278e+06 1.133278e+06 2.095476e-09 4 8 2.500000e+00 1.185451e+07 1.189942e+07 4.491010e+04 4 9 2.500000e+00 1.339916e+08 1.368434e+08 2.851792e+06 4 10 2.500000e+00 1.603083e+09 1.710542e+09 1.074586e+08 5 0 2.500000e+00 3.323351e+00 3.323351e+00 8.881784e-16 5 1 2.500000e+00 1.163173e+01 1.163173e+01 3.552714e-15 5 2 2.500000e+00 5.234278e+01 5.234278e+01 2.842171e-14 5 3 2.500000e+00 2.878853e+02 2.878853e+02 2.273737e-13 5 4 2.500000e+00 1.871254e+03 1.871254e+03 1.591616e-12 5 5 2.500000e+00 1.403441e+04 1.403441e+04 1.091394e-11 5 6 2.500000e+00 1.192925e+05 1.192925e+05 1.309672e-10 5 7 2.500000e+00 1.133278e+06 1.133278e+06 1.862645e-09 5 8 2.500000e+00 1.189942e+07 1.189942e+07 2.235174e-08 5 9 2.500000e+00 1.368434e+08 1.368434e+08 3.576279e-07 5 10 2.500000e+00 1.708858e+09 1.710542e+09 1.684129e+06 5 11 2.500000e+00 2.294327e+10 2.309232e+10 1.490454e+08 5 12 2.500000e+00 3.272146e+11 3.348386e+11 7.624051e+09 6 0 2.500000e+00 3.323351e+00 3.323351e+00 4.440892e-16 6 1 2.500000e+00 1.163173e+01 1.163173e+01 0.000000e+00 6 2 2.500000e+00 5.234278e+01 5.234278e+01 0.000000e+00 6 3 2.500000e+00 2.878853e+02 2.878853e+02 0.000000e+00 6 4 2.500000e+00 1.871254e+03 1.871254e+03 6.821210e-13 6 5 2.500000e+00 1.403441e+04 1.403441e+04 5.456968e-12 6 6 2.500000e+00 1.192925e+05 1.192925e+05 8.731149e-11 6 7 2.500000e+00 1.133278e+06 1.133278e+06 1.164153e-09 6 8 2.500000e+00 1.189942e+07 1.189942e+07 1.490116e-08 6 9 2.500000e+00 1.368434e+08 1.368434e+08 2.682209e-07 6 10 2.500000e+00 1.710542e+09 1.710542e+09 3.099442e-06 6 11 2.500000e+00 2.309232e+10 2.309232e+10 4.577637e-05 6 12 2.500000e+00 3.347527e+11 3.348386e+11 8.589057e+07 6 13 2.500000e+00 5.179906e+12 5.189998e+12 1.009214e+10 6 14 2.500000e+00 8.496252e+13 8.563497e+13 6.724588e+11 7 0 2.500000e+00 3.323351e+00 3.323351e+00 1.776357e-15 7 1 2.500000e+00 1.163173e+01 1.163173e+01 7.105427e-15 7 2 2.500000e+00 5.234278e+01 5.234278e+01 4.263256e-14 7 3 2.500000e+00 2.878853e+02 2.878853e+02 1.705303e-13 7 4 2.500000e+00 1.871254e+03 1.871254e+03 6.821210e-13 7 5 2.500000e+00 1.403441e+04 1.403441e+04 3.637979e-12 7 6 2.500000e+00 1.192925e+05 1.192925e+05 0.000000e+00 7 7 2.500000e+00 1.133278e+06 1.133278e+06 6.984919e-10 7 8 2.500000e+00 1.189942e+07 1.189942e+07 1.490116e-08 7 9 2.500000e+00 1.368434e+08 1.368434e+08 2.682209e-07 7 10 2.500000e+00 1.710542e+09 1.710542e+09 4.529953e-06 7 11 2.500000e+00 2.309232e+10 2.309232e+10 7.247925e-05 7 12 2.500000e+00 3.348386e+11 3.348386e+11 1.098633e-03 7 13 2.500000e+00 5.189998e+12 5.189998e+12 1.953125e-02 7 14 2.500000e+00 8.562926e+13 8.563497e+13 5.711723e+09 7 15 2.500000e+00 1.497752e+15 1.498612e+15 8.596143e+11 7 16 2.500000e+00 2.765202e+16 2.772432e+16 7.230756e+13 8 0 2.500000e+00 3.323351e+00 3.323351e+00 8.881784e-16 8 1 2.500000e+00 1.163173e+01 1.163173e+01 3.552714e-15 8 2 2.500000e+00 5.234278e+01 5.234278e+01 1.421085e-14 8 3 2.500000e+00 2.878853e+02 2.878853e+02 0.000000e+00 8 4 2.500000e+00 1.871254e+03 1.871254e+03 2.273737e-13 8 5 2.500000e+00 1.403441e+04 1.403441e+04 0.000000e+00 8 6 2.500000e+00 1.192925e+05 1.192925e+05 1.455192e-11 8 7 2.500000e+00 1.133278e+06 1.133278e+06 2.328306e-10 8 8 2.500000e+00 1.189942e+07 1.189942e+07 1.862645e-09 8 9 2.500000e+00 1.368434e+08 1.368434e+08 5.960464e-08 8 10 2.500000e+00 1.710542e+09 1.710542e+09 9.536743e-07 8 11 2.500000e+00 2.309232e+10 2.309232e+10 1.907349e-05 8 12 2.500000e+00 3.348386e+11 3.348386e+11 2.441406e-04 8 13 2.500000e+00 5.189998e+12 5.189998e+12 4.882812e-03 8 14 2.500000e+00 8.563497e+13 8.563497e+13 1.250000e-01 8 15 2.500000e+00 1.498612e+15 1.498612e+15 3.250000e+00 8 16 2.500000e+00 2.772384e+16 2.772432e+16 4.797847e+11 8 17 2.500000e+00 5.405343e+17 5.406243e+17 8.995964e+13 8 18 2.500000e+00 1.107348e+19 1.108280e+19 9.320178e+15 9 0 2.500000e+00 3.323351e+00 3.323351e+00 1.776357e-15 9 1 2.500000e+00 1.163173e+01 1.163173e+01 7.105427e-15 9 2 2.500000e+00 5.234278e+01 5.234278e+01 1.421085e-14 9 3 2.500000e+00 2.878853e+02 2.878853e+02 0.000000e+00 9 4 2.500000e+00 1.871254e+03 1.871254e+03 9.094947e-13 9 5 2.500000e+00 1.403441e+04 1.403441e+04 3.637979e-12 9 6 2.500000e+00 1.192925e+05 1.192925e+05 2.910383e-11 9 7 2.500000e+00 1.133278e+06 1.133278e+06 2.328306e-10 9 8 2.500000e+00 1.189942e+07 1.189942e+07 1.862645e-09 9 9 2.500000e+00 1.368434e+08 1.368434e+08 0.000000e+00 9 10 2.500000e+00 1.710542e+09 1.710542e+09 2.384186e-07 9 11 2.500000e+00 2.309232e+10 2.309232e+10 3.814697e-06 9 12 2.500000e+00 3.348386e+11 3.348386e+11 6.103516e-05 9 13 2.500000e+00 5.189998e+12 5.189998e+12 0.000000e+00 9 14 2.500000e+00 8.563497e+13 8.563497e+13 4.687500e-02 9 15 2.500000e+00 1.498612e+15 1.498612e+15 2.750000e+00 9 16 2.500000e+00 2.772432e+16 2.772432e+16 7.200000e+01 9 17 2.500000e+00 5.406243e+17 5.406243e+17 1.792000e+03 9 18 2.500000e+00 1.108275e+19 1.108280e+19 4.965772e+13 9 19 2.500000e+00 2.382688e+20 2.382802e+20 1.134679e+16 9 20 2.500000e+00 5.359884e+21 5.361304e+21 1.419416e+18 10 0 2.500000e+00 3.323351e+00 3.323351e+00 1.776357e-15 10 1 2.500000e+00 1.163173e+01 1.163173e+01 8.881784e-15 10 2 2.500000e+00 5.234278e+01 5.234278e+01 5.684342e-14 10 3 2.500000e+00 2.878853e+02 2.878853e+02 3.979039e-13 10 4 2.500000e+00 1.871254e+03 1.871254e+03 2.728484e-12 10 5 2.500000e+00 1.403441e+04 1.403441e+04 1.818989e-11 10 6 2.500000e+00 1.192925e+05 1.192925e+05 1.600711e-10 10 7 2.500000e+00 1.133278e+06 1.133278e+06 1.629815e-09 10 8 2.500000e+00 1.189942e+07 1.189942e+07 1.676381e-08 10 9 2.500000e+00 1.368434e+08 1.368434e+08 2.682209e-07 10 10 2.500000e+00 1.710542e+09 1.710542e+09 3.576279e-06 10 11 2.500000e+00 2.309232e+10 2.309232e+10 5.722046e-05 10 12 2.500000e+00 3.348386e+11 3.348386e+11 9.765625e-04 10 13 2.500000e+00 5.189998e+12 5.189998e+12 1.757812e-02 10 14 2.500000e+00 8.563497e+13 8.563497e+13 3.281250e-01 10 15 2.500000e+00 1.498612e+15 1.498612e+15 6.750000e+00 10 16 2.500000e+00 2.772432e+16 2.772432e+16 1.400000e+02 10 17 2.500000e+00 5.406243e+17 5.406243e+17 2.688000e+03 10 18 2.500000e+00 1.108280e+19 1.108280e+19 5.529600e+04 10 19 2.500000e+00 2.382802e+20 2.382802e+20 1.212416e+06 10 20 2.500000e+00 5.361297e+21 5.361304e+21 6.207215e+15 10 21 2.500000e+00 1.259889e+23 1.259906e+23 1.697673e+18 10 22 2.500000e+00 3.086518e+24 3.086771e+24 2.522504e+20 SANDIA_RULES_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 1 1.5000000000000000 0.8862269254527581 Order = 2 ALPHA = 0.500000 1 0.9188611699158105 0.7233630235462752 2 4.0811388300841891 0.1628639019064825 Order = 3 ALPHA = 0.500000 1 0.6663259077023710 0.5671862778403116 2 2.8007750541502561 0.3053717688445466 3 7.0328990381473719 0.0136688787679001 Order = 4 ALPHA = 0.500000 1 0.5235260767382686 0.4530087465586073 2 2.1566487632690938 0.3816169601718006 3 5.1373875461767122 0.0507946275722407 4 10.1824376138159263 0.0008065911501100 Order = 5 ALPHA = 0.500000 1 0.4313988071478522 0.3704505700074587 2 1.7597536984236979 0.4125843737694527 3 4.1044653628283161 0.0977798200531806 4 7.7467037795425577 0.0053734153411720 5 13.4576783520575773 0.0000387462814939 Order = 6 ALPHA = 0.500000 1 0.3669498773083705 0.3094240968362603 2 1.4885342923104525 0.4177521497070222 3 3.4340079684240710 0.1432858732209771 4 6.3490679256803793 0.0153324910226339 5 10.5404698584483416 0.0004306911960439 6 16.8209700778283846 0.0000016234698211 Order = 7 ALPHA = 0.500000 1 0.3193036339206293 0.2631245143958913 2 1.2907586229591517 0.4091418694141027 3 2.9583744586966509 0.1821177320927163 4 5.4090315972444358 0.0300533243012710 5 8.8040795780567738 0.0017608941175401 6 13.4685357432514810 0.0000285294712212 7 20.2499163658708774 0.0000000616600154 Order = 8 ALPHA = 0.500000 1 0.2826336481165983 0.2271393619524710 2 1.1398738015816119 0.3935945428036152 3 2.6015248434060290 0.2129089708672288 4 4.7241145375277904 0.0478774832031382 5 7.6052562992316135 0.0045425174747626 6 11.4171820765458296 0.0001624046001853 7 16.4994107976558126 0.0000016423774138 8 23.7300039959347124 0.0000000021739431 Order = 9 ALPHA = 0.500000 1 0.2535325549744193 0.1985712548680196 2 1.0208442777203901 0.3749207846631705 3 2.3230960770224649 0.2360748210008252 4 4.1993506006572927 0.0670961050032043 5 6.7139743166150296 0.0090085088966443 6 9.9720091595393505 0.0005426607386359 7 14.1540536712780511 0.0000127053668791 8 19.6119028191659481 0.0000000848430924 9 27.2512365230270603 0.0000000000722865 Order = 10 ALPHA = 0.500000 1 0.2298729805186566 0.1754708150466599 2 0.9244815469866562 0.3552233888020720 3 2.0994104627087977 0.2526835596756785 4 3.7828808737072888 0.0863561026953326 5 6.0199180277014612 0.0151097780348608 6 8.8803475979967086 0.0013282156283636 7 12.4748324048362065 0.0000541878002117 8 16.9908472935425578 0.0000008737475869 9 22.7910028949489458 0.0000000040196999 10 30.8064059170527180 0.0000000000022922 Order = 1 ALPHA = 1.000000 1 2.0000000000000000 1.0000000000000000 Order = 2 ALPHA = 1.000000 1 1.2679491924311226 0.7886751345948131 2 4.7320508075688776 0.2113248654051871 Order = 3 ALPHA = 1.000000 1 0.9358222275240875 0.5886814810396593 2 3.3054072893322788 0.3912160592223105 3 7.7587704831436346 0.0201024597380306 Order = 4 ALPHA = 1.000000 1 0.7432919279814317 0.4468705932187766 2 2.5716350076462793 0.4776357723638680 3 5.7311787516890984 0.0741777847310521 4 10.9538943126831878 0.0013158496863032 Order = 5 ALPHA = 1.000000 1 0.6170308532782706 0.3480145400233489 2 2.1129659585785245 0.5022806741324930 3 4.6108331510175313 0.1409159194944725 4 8.3990669712048351 0.0087198930261000 5 14.2601030659208359 0.0000689733235856 Order = 6 ALPHA = 1.000000 1 0.5276681217111289 0.2776501420298752 2 1.7962998096434091 0.4939105830503541 3 3.8766415204769120 0.2030042967437302 4 6.9188165667047228 0.0246688203691898 5 11.2346104290831139 0.0007630427674635 6 17.6459635523807101 0.0000031150393875 Order = 7 ALPHA = 1.000000 1 0.4610242198049947 0.2262105963831890 2 1.5635861896542627 0.4697087077412661 3 3.3520505025367338 0.2531225162324008 4 5.9162972490205386 0.0478031088842371 5 9.4206993830215939 0.0031004597560254 6 14.1941655480074633 0.0000544846688934 7 21.0921769079544141 0.0000001263339882 Order = 8 ALPHA = 1.000000 1 0.4093835732031860 0.1876325414057240 2 1.3849631848031410 0.4389853607311423 3 2.9562545561688638 0.2899960707813130 4 5.1819431010400727 0.0751413846166973 5 8.1617096881458160 0.0079326466487073 6 12.0700551268371559 0.0003086421368133 7 17.2497355261489957 0.0000033489582098 8 24.5859552436527800 0.0000000047213928 Order = 9 ALPHA = 1.000000 1 0.3681784529417419 0.1580309235007823 2 1.2433579621404844 0.4065825794555729 3 2.6460338413842086 0.3149824836650970 4 4.6168825146350621 0.1037755199516432 5 7.2217865393965708 0.0155776377242784 6 10.5673208077418668 0.0010248719698508 7 14.8359145152609333 0.0000258002113872 8 20.3821819854492468 0.0000001833559550 9 28.1183433810498862 0.0000000001654329 Order = 10 ALPHA = 1.000000 1 0.3345286763247526 0.1348565545313779 2 1.1282533558766390 0.3749243411054598 3 2.3958699247473088 0.3302471308042158 4 4.1668409879287678 0.1315288846413064 5 6.4873530313808105 0.0258420642117172 6 9.4283548133356057 0.0024896457958960 7 13.1017235803677980 0.0001094884764267 8 17.6964875668462227 0.0000018812754142 9 23.5777870883601501 0.0000000091526850 10 31.6828009748319346 0.0000000000055013 Order = 1 ALPHA = 2.500000 1 3.5000000000000000 3.3233509704478421 Order = 2 ALPHA = 2.500000 1 2.3786796564403576 2.4449968210461086 2 6.6213203435596419 0.8783541494017333 Order = 3 ALPHA = 2.500000 1 1.8248198030758735 1.6404916909909482 2 4.8137462264876980 1.5764206102831910 3 9.8614339704364280 0.1064386691737022 Order = 4 ALPHA = 2.500000 1 1.4862411426653499 1.1090225785851486 2 3.8376839879001001 1.8203241249078206 3 7.4820583668989809 0.3853099204909315 4 13.1940165025355665 0.0086943464639431 Order = 5 ALPHA = 2.500000 1 1.2558461409184427 0.7693461876473180 2 3.2070406266149090 1.7891436571378665 3 6.1239082233284110 0.7075497394015858 4 10.3161258663913511 0.0567556714589862 5 16.5970791427468889 0.0005557148020873 Order = 6 ALPHA = 2.500000 1 1.0882635378141627 0.5488342282458472 2 2.7608555713855099 1.6358288552513585 3 5.2130824709557162 0.9757018876753600 4 8.6088345005370002 0.1568919688876181 5 13.2736722693042068 0.0060640457339780 6 20.0552916500034115 0.0000299846536813 Order = 7 ALPHA = 2.500000 1 0.9606221295119317 0.4018644652334340 2 2.4266771014277411 1.4440235761293296 3 4.5506549675827692 1.1578194454504420 4 7.4324177726246079 0.2949071223080977 5 11.2446775658018723 0.0242175011439700 6 16.3271095162749234 0.0005174322318932 7 23.5578409467761603 0.0000014279506744 Order = 8 ALPHA = 2.500000 1 0.8600416237603872 0.3011867253184345 2 2.1662435289922470 1.2529637803437597 3 4.0437426773491252 1.2583457960796316 4 6.5589225384279022 0.4473442363684211 5 9.8172555052257060 0.0605825502790592 6 13.9994797507150555 0.0028904817699157 7 19.4574474318146287 0.0000373385286657 8 27.0968669437149465 0.0000000617599561 Order = 9 ALPHA = 2.500000 1 0.7786800215607389 0.2304206696494764 2 1.9571902235047749 1.0787334348681767 3 3.6417887950645969 1.2944983831865118 4 5.8793297220349796 0.5941550811279991 5 8.7400288949175451 0.1158251816851087 6 12.3346755858879824 0.0094316286919319 7 16.8507961274953786 0.0002842237871894 8 22.6510256983860714 0.0000023649776431 9 30.6664849311479344 0.0000000024738037 Order = 10 ALPHA = 2.500000 1 0.7114760698983040 0.1795029442066140 2 1.7854778061294854 0.9263170003438098 3 3.3144812176022080 1.2847675181016529 4 5.3330492524705857 0.7226374668786899 5 7.8907634025314408 0.1864748527585193 6 11.0616739762852898 0.0224379062050689 7 14.9610223243985168 0.0011891427230148 8 19.7822180787733295 0.0000240042379159 9 25.8976644597361130 0.0000001348994981 10 34.2621734121747323 0.0000000000930604 SANDIA_RULES_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 1.772454e+00 1.772454e+00 2.220446e-16 1 1 0.000000e+00 0.000000e+00 0.000000e+00 1 2 0.000000e+00 8.862269e-01 8.862269e-01 1 3 0.000000e+00 0.000000e+00 0.000000e+00 1 4 0.000000e+00 1.329340e+00 1.329340e+00 2 0 1.772454e+00 1.772454e+00 2.220446e-16 2 1 0.000000e+00 0.000000e+00 0.000000e+00 2 2 8.862269e-01 8.862269e-01 3.330669e-16 2 3 0.000000e+00 0.000000e+00 0.000000e+00 2 4 4.431135e-01 1.329340e+00 8.862269e-01 2 5 0.000000e+00 0.000000e+00 0.000000e+00 2 6 2.215567e-01 3.323351e+00 3.101794e+00 3 0 1.772454e+00 1.772454e+00 2.220446e-16 3 1 1.665335e-16 0.000000e+00 1.665335e-16 3 2 8.862269e-01 8.862269e-01 1.221245e-15 3 3 3.330669e-16 0.000000e+00 3.330669e-16 3 4 1.329340e+00 1.329340e+00 2.664535e-15 3 5 4.440892e-16 0.000000e+00 4.440892e-16 3 6 1.994011e+00 3.323351e+00 1.329340e+00 3 7 6.661338e-16 0.000000e+00 6.661338e-16 3 8 2.991016e+00 1.163173e+01 8.640713e+00 4 0 1.772454e+00 1.772454e+00 4.440892e-16 4 1 -3.330669e-16 0.000000e+00 3.330669e-16 4 2 8.862269e-01 8.862269e-01 3.330669e-16 4 3 7.494005e-16 0.000000e+00 7.494005e-16 4 4 1.329340e+00 1.329340e+00 1.332268e-15 4 5 2.664535e-15 0.000000e+00 2.664535e-15 4 6 3.323351e+00 3.323351e+00 5.773160e-15 4 7 9.769963e-15 0.000000e+00 9.769963e-15 4 8 8.973048e+00 1.163173e+01 2.658681e+00 4 9 3.019807e-14 0.000000e+00 3.019807e-14 4 10 2.442663e+01 5.234278e+01 2.791615e+01 5 0 1.772454e+00 1.772454e+00 1.332268e-15 5 1 -2.289835e-16 0.000000e+00 2.289835e-16 5 2 8.862269e-01 8.862269e-01 4.440892e-16 5 3 -4.996004e-16 0.000000e+00 4.996004e-16 5 4 1.329340e+00 1.329340e+00 6.661338e-16 5 5 -2.498002e-15 0.000000e+00 2.498002e-15 5 6 3.323351e+00 3.323351e+00 2.664535e-15 5 7 -1.132427e-14 0.000000e+00 1.132427e-14 5 8 1.163173e+01 1.163173e+01 8.881784e-15 5 9 -4.723999e-14 0.000000e+00 4.723999e-14 5 10 4.569608e+01 5.234278e+01 6.646702e+00 5 11 -1.981471e-13 0.000000e+00 1.981471e-13 5 12 1.848614e+02 2.878853e+02 1.030239e+02 6 0 1.772454e+00 1.772454e+00 8.881784e-16 6 1 -7.494005e-16 0.000000e+00 7.494005e-16 6 2 8.862269e-01 8.862269e-01 6.661338e-16 6 3 -1.165734e-15 0.000000e+00 1.165734e-15 6 4 1.329340e+00 1.329340e+00 1.776357e-15 6 5 -3.108624e-15 0.000000e+00 3.108624e-15 6 6 3.323351e+00 3.323351e+00 3.108624e-15 6 7 -1.221245e-14 0.000000e+00 1.221245e-14 6 8 1.163173e+01 1.163173e+01 0.000000e+00 6 9 -6.039613e-14 0.000000e+00 6.039613e-14 6 10 5.234278e+01 5.234278e+01 5.684342e-14 6 11 -3.055334e-13 0.000000e+00 3.055334e-13 6 12 2.679452e+02 2.878853e+02 1.994011e+01 6 13 -1.477929e-12 0.000000e+00 1.477929e-12 6 14 1.442542e+03 1.871254e+03 4.287123e+02 7 0 1.772454e+00 1.772454e+00 1.332268e-15 7 1 5.273559e-16 0.000000e+00 5.273559e-16 7 2 8.862269e-01 8.862269e-01 5.551115e-16 7 3 5.828671e-16 0.000000e+00 5.828671e-16 7 4 1.329340e+00 1.329340e+00 1.332268e-15 7 5 1.637579e-15 0.000000e+00 1.637579e-15 7 6 3.323351e+00 3.323351e+00 3.996803e-15 7 7 5.814793e-15 0.000000e+00 5.814793e-15 7 8 1.163173e+01 1.163173e+01 1.243450e-14 7 9 2.675637e-14 0.000000e+00 2.675637e-14 7 10 5.234278e+01 5.234278e+01 3.552714e-14 7 11 1.341566e-13 0.000000e+00 1.341566e-13 7 12 2.878853e+02 2.878853e+02 2.273737e-13 7 13 7.452407e-13 0.000000e+00 7.452407e-13 7 14 1.801464e+03 1.871254e+03 6.979037e+01 7 15 4.781738e-12 0.000000e+00 4.781738e-12 7 16 1.204538e+04 1.403441e+04 1.989026e+03 8 0 1.772454e+00 1.772454e+00 8.881784e-16 8 1 -6.106227e-16 0.000000e+00 6.106227e-16 8 2 8.862269e-01 8.862269e-01 4.440892e-16 8 3 -3.330669e-16 0.000000e+00 3.330669e-16 8 4 1.329340e+00 1.329340e+00 1.332268e-15 8 5 2.775558e-16 0.000000e+00 2.775558e-16 8 6 3.323351e+00 3.323351e+00 8.437695e-15 8 7 3.996803e-15 0.000000e+00 3.996803e-15 8 8 1.163173e+01 1.163173e+01 4.618528e-14 8 9 1.953993e-14 0.000000e+00 1.953993e-14 8 10 5.234278e+01 5.234278e+01 2.486900e-13 8 11 8.171241e-14 0.000000e+00 8.171241e-14 8 12 2.878853e+02 2.878853e+02 1.364242e-12 8 13 2.415845e-13 0.000000e+00 2.415845e-13 8 14 1.871254e+03 1.871254e+03 8.640200e-12 8 15 0.000000e+00 0.000000e+00 0.000000e+00 8 16 1.375525e+04 1.403441e+04 2.791615e+02 8 17 -7.275958e-12 0.000000e+00 7.275958e-12 8 18 1.091031e+05 1.192925e+05 1.018939e+04 9 0 1.772454e+00 1.772454e+00 1.332268e-15 9 1 6.661338e-16 0.000000e+00 6.661338e-16 9 2 8.862269e-01 8.862269e-01 1.665335e-15 9 3 6.661338e-16 0.000000e+00 6.661338e-16 9 4 1.329340e+00 1.329340e+00 3.552714e-15 9 5 9.436896e-16 0.000000e+00 9.436896e-16 9 6 3.323351e+00 3.323351e+00 1.154632e-14 9 7 3.330669e-15 0.000000e+00 3.330669e-15 9 8 1.163173e+01 1.163173e+01 3.907985e-14 9 9 1.865175e-14 0.000000e+00 1.865175e-14 9 10 5.234278e+01 5.234278e+01 1.563194e-13 9 11 8.704149e-14 0.000000e+00 8.704149e-14 9 12 2.878853e+02 2.878853e+02 7.389644e-13 9 13 3.623768e-13 0.000000e+00 3.623768e-13 9 14 1.871254e+03 1.871254e+03 3.183231e-12 9 15 5.044853e-13 0.000000e+00 5.044853e-13 9 16 1.403441e+04 1.403441e+04 7.275958e-12 9 17 -1.352163e-11 0.000000e+00 1.352163e-11 9 18 1.180362e+05 1.192925e+05 1.256227e+03 9 19 -2.980016e-10 0.000000e+00 2.980016e-10 9 20 1.076120e+06 1.133278e+06 5.715831e+04 10 0 1.772454e+00 1.772454e+00 4.440892e-16 10 1 -7.008283e-16 0.000000e+00 7.008283e-16 10 2 8.862269e-01 8.862269e-01 1.110223e-16 10 3 -1.915135e-15 0.000000e+00 1.915135e-15 10 4 1.329340e+00 1.329340e+00 1.998401e-15 10 5 -4.107825e-15 0.000000e+00 4.107825e-15 10 6 3.323351e+00 3.323351e+00 1.021405e-14 10 7 -5.551115e-15 0.000000e+00 5.551115e-15 10 8 1.163173e+01 1.163173e+01 5.684342e-14 10 9 2.459144e-14 0.000000e+00 2.459144e-14 10 10 5.234278e+01 5.234278e+01 3.197442e-13 10 11 3.819167e-13 0.000000e+00 3.819167e-13 10 12 2.878853e+02 2.878853e+02 1.818989e-12 10 13 3.282707e-12 0.000000e+00 3.282707e-12 10 14 1.871254e+03 1.871254e+03 1.091394e-11 10 15 2.501110e-11 0.000000e+00 2.501110e-11 10 16 1.403441e+04 1.403441e+04 6.184564e-11 10 17 1.761009e-10 0.000000e+00 1.761009e-10 10 18 1.192925e+05 1.192925e+05 3.492460e-10 10 19 1.156877e-09 0.000000e+00 1.156877e-09 10 20 1.126997e+06 1.133278e+06 6.281133e+03 10 21 7.101335e-09 0.000000e+00 7.101335e-09 10 22 1.155082e+07 1.189942e+07 3.486029e+05 SANDIA_RULES_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 1 0.0000000000000000 1.7724538509055161 Order = 2 1 -0.7071067811865475 0.8862269254527578 2 0.7071067811865475 0.8862269254527578 Order = 3 1 -1.2247448713915894 0.2954089751509195 2 0.0000000000000000 1.1816359006036770 3 1.2247448713915894 0.2954089751509196 Order = 4 1 -1.6506801238857838 0.0813128354472451 2 -0.5246476232752904 0.8049140900055129 3 0.5246476232752902 0.8049140900055121 4 1.6506801238857842 0.0813128354472453 Order = 5 1 -2.0201828704560856 0.0199532420590460 2 -0.9585724646138184 0.3936193231522413 3 0.0000000000000000 0.9453087204829430 4 0.9585724646138187 0.3936193231522411 5 2.0201828704560856 0.0199532420590459 Order = 6 1 -2.3506049736744927 0.0045300099055089 2 -1.3358490740136961 0.1570673203228569 3 -0.4360774119276162 0.7246295952243929 4 0.4360774119276160 0.7246295952243927 5 1.3358490740136963 0.1570673203228566 6 2.3506049736744932 0.0045300099055088 Order = 7 1 -2.6519613568352338 0.0009717812450995 2 -1.6735516287674714 0.0545155828191271 3 -0.8162878828589647 0.4256072526101277 4 0.0000000000000000 0.8102646175568082 5 0.8162878828589646 0.4256072526101282 6 1.6735516287674717 0.0545155828191271 7 2.6519613568352338 0.0009717812450995 Order = 8 1 -2.9306374202572432 0.0001996040722114 2 -1.9816567566958418 0.0170779830074134 3 -1.1571937124467804 0.2078023258148922 4 -0.3811869902073223 0.6611470125582418 5 0.3811869902073220 0.6611470125582410 6 1.1571937124467799 0.2078023258148921 7 1.9816567566958421 0.0170779830074135 8 2.9306374202572432 0.0001996040722114 Order = 9 1 -3.1909932017815272 0.0000396069772633 2 -2.2665805845318432 0.0049436242755370 3 -1.4685532892166682 0.0884745273943768 4 -0.7235510187528373 0.4326515590025557 5 0.0000000000000000 0.7202352156060512 6 0.7235510187528379 0.4326515590025563 7 1.4685532892166684 0.0884745273943768 8 2.2665805845318436 0.0049436242755370 9 3.1909932017815268 0.0000396069772633 Order = 10 1 -3.4361591188377378 0.0000076404328552 2 -2.5327316742327879 0.0013436457467812 3 -1.7566836492998807 0.0338743944554812 4 -1.0366108297895136 0.2401386110823150 5 -0.3429013272237045 0.6108626337353257 6 0.3429013272237046 0.6108626337353257 7 1.0366108297895125 0.2401386110823148 8 1.7566836492998799 0.0338743944554811 9 2.5327316742327906 0.0013436457467812 10 3.4361591188377378 0.0000076404328552 SANDIA_RULES_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 0.5000 0.5000 1.570796 1.570796 1.110223e-15 1 1 0.5000 0.5000 0.000000 0.000000 0.000000e+00 1 2 0.5000 0.5000 0.000000 0.392699 3.926991e-01 1 3 0.5000 0.5000 0.000000 0.000000 0.000000e+00 1 4 0.5000 0.5000 0.000000 0.196350 1.963495e-01 2 0 0.5000 0.5000 1.570796 1.570796 1.110223e-15 2 1 0.5000 0.5000 0.000000 0.000000 0.000000e+00 2 2 0.5000 0.5000 0.392699 0.392699 1.665335e-16 2 3 0.5000 0.5000 0.000000 0.000000 0.000000e+00 2 4 0.5000 0.5000 0.098175 0.196350 9.817477e-02 2 5 0.5000 0.5000 0.000000 0.000000 0.000000e+00 2 6 0.5000 0.5000 0.024544 0.122718 9.817477e-02 3 0 0.5000 0.5000 1.570796 1.570796 8.881784e-16 3 1 0.5000 0.5000 -0.000000 0.000000 3.923162e-16 3 2 0.5000 0.5000 0.392699 0.392699 1.665335e-16 3 3 0.5000 0.5000 -0.000000 0.000000 2.498002e-16 3 4 0.5000 0.5000 0.196350 0.196350 2.220446e-16 3 5 0.5000 0.5000 -0.000000 0.000000 1.526557e-16 3 6 0.5000 0.5000 0.098175 0.122718 2.454369e-02 3 7 0.5000 0.5000 -0.000000 0.000000 8.326673e-17 3 8 0.5000 0.5000 0.049087 0.085903 3.681554e-02 4 0 0.5000 0.5000 1.570796 1.570796 0.000000e+00 4 1 0.5000 0.5000 0.000000 0.000000 3.330669e-16 4 2 0.5000 0.5000 0.392699 0.392699 2.775558e-16 4 3 0.5000 0.5000 0.000000 0.000000 1.665335e-16 4 4 0.5000 0.5000 0.196350 0.196350 1.665335e-16 4 5 0.5000 0.5000 0.000000 0.000000 1.249001e-16 4 6 0.5000 0.5000 0.122718 0.122718 1.804112e-16 4 7 0.5000 0.5000 0.000000 0.000000 1.179612e-16 4 8 0.5000 0.5000 0.079767 0.085903 6.135923e-03 4 9 0.5000 0.5000 0.000000 0.000000 9.714451e-17 4 10 0.5000 0.5000 0.052155 0.064427 1.227185e-02 5 0 0.5000 0.5000 1.570796 1.570796 1.110223e-15 5 1 0.5000 0.5000 -0.000000 0.000000 2.602664e-16 5 2 0.5000 0.5000 0.392699 0.392699 1.110223e-16 5 3 0.5000 0.5000 -0.000000 0.000000 2.012279e-16 5 4 0.5000 0.5000 0.196350 0.196350 1.110223e-16 5 5 0.5000 0.5000 -0.000000 0.000000 1.578598e-16 5 6 0.5000 0.5000 0.122718 0.122718 1.804112e-16 5 7 0.5000 0.5000 -0.000000 0.000000 1.262011e-16 5 8 0.5000 0.5000 0.085903 0.085903 1.942890e-16 5 9 0.5000 0.5000 -0.000000 0.000000 9.746978e-17 5 10 0.5000 0.5000 0.062893 0.064427 1.533981e-03 5 11 0.5000 0.5000 -0.000000 0.000000 7.987860e-17 5 12 0.5000 0.5000 0.046786 0.050621 3.834952e-03 6 0 0.5000 0.5000 1.570796 1.570796 8.881784e-16 6 1 0.5000 0.5000 -0.000000 0.000000 4.163336e-17 6 2 0.5000 0.5000 0.392699 0.392699 5.551115e-17 6 3 0.5000 0.5000 -0.000000 0.000000 1.708703e-16 6 4 0.5000 0.5000 0.196350 0.196350 2.220446e-16 6 5 0.5000 0.5000 -0.000000 0.000000 1.491862e-16 6 6 0.5000 0.5000 0.122718 0.122718 2.081668e-16 6 7 0.5000 0.5000 -0.000000 0.000000 1.040834e-16 6 8 0.5000 0.5000 0.085903 0.085903 2.081668e-16 6 9 0.5000 0.5000 -0.000000 0.000000 8.673617e-17 6 10 0.5000 0.5000 0.064427 0.064427 1.942890e-16 6 11 0.5000 0.5000 -0.000000 0.000000 6.245005e-17 6 12 0.5000 0.5000 0.050238 0.050621 3.834952e-04 6 13 0.5000 0.5000 -0.000000 0.000000 4.510281e-17 6 14 0.5000 0.5000 0.039979 0.041130 1.150486e-03 7 0 0.5000 0.5000 1.570796 1.570796 4.440892e-16 7 1 0.5000 0.5000 0.000000 0.000000 5.861423e-17 7 2 0.5000 0.5000 0.392699 0.392699 3.885781e-16 7 3 0.5000 0.5000 -0.000000 0.000000 7.285839e-17 7 4 0.5000 0.5000 0.196350 0.196350 1.110223e-16 7 5 0.5000 0.5000 -0.000000 0.000000 1.643650e-16 7 6 0.5000 0.5000 0.122718 0.122718 5.551115e-17 7 7 0.5000 0.5000 -0.000000 0.000000 1.885970e-16 7 8 0.5000 0.5000 0.085903 0.085903 4.163336e-17 7 9 0.5000 0.5000 -0.000000 0.000000 1.977720e-16 7 10 0.5000 0.5000 0.064427 0.064427 0.000000e+00 7 11 0.5000 0.5000 -0.000000 0.000000 1.946160e-16 7 12 0.5000 0.5000 0.050621 0.050621 4.163336e-17 7 13 0.5000 0.5000 -0.000000 0.000000 1.935341e-16 7 14 0.5000 0.5000 0.041034 0.041130 9.587380e-05 7 15 0.5000 0.5000 -0.000000 0.000000 1.855794e-16 7 16 0.5000 0.5000 0.033939 0.034275 3.355583e-04 8 0 0.5000 0.5000 1.570796 1.570796 0.000000e+00 8 1 0.5000 0.5000 -0.000000 0.000000 1.387779e-16 8 2 0.5000 0.5000 0.392699 0.392699 2.775558e-16 8 3 0.5000 0.5000 -0.000000 0.000000 2.220446e-16 8 4 0.5000 0.5000 0.196350 0.196350 5.551115e-17 8 5 0.5000 0.5000 -0.000000 0.000000 1.734723e-16 8 6 0.5000 0.5000 0.122718 0.122718 9.714451e-17 8 7 0.5000 0.5000 -0.000000 0.000000 1.075529e-16 8 8 0.5000 0.5000 0.085903 0.085903 1.249001e-16 8 9 0.5000 0.5000 -0.000000 0.000000 5.377643e-17 8 10 0.5000 0.5000 0.064427 0.064427 1.387779e-16 8 11 0.5000 0.5000 -0.000000 0.000000 2.602085e-17 8 12 0.5000 0.5000 0.050621 0.050621 9.714451e-17 8 13 0.5000 0.5000 0.000000 0.000000 6.938894e-18 8 14 0.5000 0.5000 0.041130 0.041130 1.179612e-16 8 15 0.5000 0.5000 0.000000 0.000000 2.255141e-17 8 16 0.5000 0.5000 0.034251 0.034275 2.396845e-05 8 17 0.5000 0.5000 0.000000 0.000000 3.122502e-17 8 18 0.5000 0.5000 0.029038 0.029134 9.587380e-05 9 0 0.5000 0.5000 1.570796 1.570796 4.440892e-16 9 1 0.5000 0.5000 0.000000 0.000000 2.914335e-16 9 2 0.5000 0.5000 0.392699 0.392699 3.885781e-16 9 3 0.5000 0.5000 -0.000000 0.000000 3.469447e-17 9 4 0.5000 0.5000 0.196350 0.196350 1.110223e-16 9 5 0.5000 0.5000 -0.000000 0.000000 7.112366e-17 9 6 0.5000 0.5000 0.122718 0.122718 8.326673e-17 9 7 0.5000 0.5000 -0.000000 0.000000 7.285839e-17 9 8 0.5000 0.5000 0.085903 0.085903 2.775558e-17 9 9 0.5000 0.5000 -0.000000 0.000000 7.676151e-17 9 10 0.5000 0.5000 0.064427 0.064427 2.775558e-17 9 11 0.5000 0.5000 -0.000000 0.000000 6.938894e-17 9 12 0.5000 0.5000 0.050621 0.050621 1.387779e-17 9 13 0.5000 0.5000 -0.000000 0.000000 6.760001e-17 9 14 0.5000 0.5000 0.041130 0.041130 6.938894e-18 9 15 0.5000 0.5000 -0.000000 0.000000 6.104058e-17 9 16 0.5000 0.5000 0.034275 0.034275 2.775558e-17 9 17 0.5000 0.5000 -0.000000 0.000000 5.824537e-17 9 18 0.5000 0.5000 0.029128 0.029134 5.992112e-06 9 19 0.5000 0.5000 -0.000000 0.000000 5.296158e-17 9 20 0.5000 0.5000 0.025134 0.025161 2.696451e-05 10 0 0.5000 0.5000 1.570796 1.570796 4.440892e-16 10 1 0.5000 0.5000 -0.000000 0.000000 2.012279e-16 10 2 0.5000 0.5000 0.392699 0.392699 3.330669e-16 10 3 0.5000 0.5000 -0.000000 0.000000 1.040834e-16 10 4 0.5000 0.5000 0.196350 0.196350 0.000000e+00 10 5 0.5000 0.5000 -0.000000 0.000000 7.242471e-17 10 6 0.5000 0.5000 0.122718 0.122718 4.163336e-17 10 7 0.5000 0.5000 -0.000000 0.000000 5.464379e-17 10 8 0.5000 0.5000 0.085903 0.085903 2.775558e-17 10 9 0.5000 0.5000 -0.000000 0.000000 4.380177e-17 10 10 0.5000 0.5000 0.064427 0.064427 2.775558e-17 10 11 0.5000 0.5000 -0.000000 0.000000 2.949030e-17 10 12 0.5000 0.5000 0.050621 0.050621 1.387779e-17 10 13 0.5000 0.5000 -0.000000 0.000000 2.688821e-17 10 14 0.5000 0.5000 0.041130 0.041130 3.469447e-17 10 15 0.5000 0.5000 -0.000000 0.000000 2.688821e-17 10 16 0.5000 0.5000 0.034275 0.034275 1.387779e-17 10 17 0.5000 0.5000 -0.000000 0.000000 2.255141e-17 10 18 0.5000 0.5000 0.029134 0.029134 1.734723e-17 10 19 0.5000 0.5000 -0.000000 0.000000 2.341877e-17 10 20 0.5000 0.5000 0.025159 0.025161 1.498028e-06 10 21 0.5000 0.5000 -0.000000 0.000000 2.428613e-17 10 22 0.5000 0.5000 0.022008 0.022016 7.490141e-06 1 0 0.5000 1.0000 1.508494 1.508494 2.220446e-16 1 1 0.5000 1.0000 0.215499 0.215499 2.775558e-17 1 2 0.5000 1.0000 0.030786 0.359165 3.283797e-01 1 3 0.5000 1.0000 0.004398 0.111015 1.066168e-01 1 4 0.5000 1.0000 0.000628 0.174308 1.736799e-01 2 0 0.5000 1.0000 1.508494 1.508494 2.220446e-16 2 1 0.5000 1.0000 0.215499 0.215499 2.775558e-17 2 2 0.5000 1.0000 0.359165 0.359165 5.551115e-17 2 3 0.5000 1.0000 0.111015 0.111015 1.387779e-17 2 4 0.5000 1.0000 0.096371 0.174308 7.793713e-02 2 5 0.5000 1.0000 0.041071 0.070828 2.975781e-02 2 6 0.5000 1.0000 0.027910 0.106701 7.879089e-02 3 0 0.5000 1.0000 1.508494 1.508494 2.220446e-16 3 1 0.5000 1.0000 0.215499 0.215499 2.775558e-17 3 2 0.5000 1.0000 0.359165 0.359165 1.110223e-16 3 3 0.5000 1.0000 0.111015 0.111015 9.714451e-17 3 4 0.5000 1.0000 0.174308 0.174308 5.551115e-17 3 5 0.5000 1.0000 0.070828 0.070828 8.326673e-17 3 6 0.5000 1.0000 0.087742 0.106701 1.895882e-02 3 7 0.5000 1.0000 0.042566 0.050350 7.783096e-03 3 8 0.5000 1.0000 0.044985 0.073531 2.854635e-02 4 0 0.5000 1.0000 1.508494 1.508494 6.661338e-16 4 1 0.5000 1.0000 0.215499 0.215499 2.775558e-17 4 2 0.5000 1.0000 0.359165 0.359165 1.665335e-16 4 3 0.5000 1.0000 0.111015 0.111015 9.714451e-17 4 4 0.5000 1.0000 0.174308 0.174308 8.326673e-17 4 5 0.5000 1.0000 0.070828 0.070828 6.938894e-17 4 6 0.5000 1.0000 0.106701 0.106701 9.714451e-17 4 7 0.5000 1.0000 0.050350 0.050350 5.551115e-17 4 8 0.5000 1.0000 0.068871 0.073531 4.660378e-03 4 9 0.5000 1.0000 0.036229 0.038223 1.994258e-03 4 10 0.5000 1.0000 0.045135 0.054471 9.336981e-03 5 0 0.5000 1.0000 1.508494 1.508494 1.110223e-15 5 1 0.5000 1.0000 0.215499 0.215499 2.775558e-17 5 2 0.5000 1.0000 0.359165 0.359165 3.330669e-16 5 3 0.5000 1.0000 0.111015 0.111015 1.526557e-16 5 4 0.5000 1.0000 0.174308 0.174308 2.220446e-16 5 5 0.5000 1.0000 0.070828 0.070828 1.804112e-16 5 6 0.5000 1.0000 0.106701 0.106701 1.942890e-16 5 7 0.5000 1.0000 0.050350 0.050350 1.873501e-16 5 8 0.5000 1.0000 0.073531 0.073531 1.387779e-16 5 9 0.5000 1.0000 0.038223 0.038223 1.804112e-16 5 10 0.5000 1.0000 0.053320 0.054471 1.151817e-03 5 11 0.5000 1.0000 0.029824 0.030331 5.063543e-04 5 12 0.5000 1.0000 0.039487 0.042369 2.882239e-03 6 0 0.5000 1.0000 1.508494 1.508494 1.110223e-15 6 1 0.5000 1.0000 0.215499 0.215499 2.498002e-16 6 2 0.5000 1.0000 0.359165 0.359165 4.440892e-16 6 3 0.5000 1.0000 0.111015 0.111015 2.914335e-16 6 4 0.5000 1.0000 0.174308 0.174308 3.608225e-16 6 5 0.5000 1.0000 0.070828 0.070828 2.359224e-16 6 6 0.5000 1.0000 0.106701 0.106701 2.498002e-16 6 7 0.5000 1.0000 0.050350 0.050350 1.526557e-16 6 8 0.5000 1.0000 0.073531 0.073531 2.220446e-16 6 9 0.5000 1.0000 0.038223 0.038223 1.040834e-16 6 10 0.5000 1.0000 0.054471 0.054471 1.595946e-16 6 11 0.5000 1.0000 0.030331 0.030331 9.367507e-17 6 12 0.5000 1.0000 0.042084 0.042369 2.855761e-04 6 13 0.5000 1.0000 0.024721 0.024849 1.279463e-04 6 14 0.5000 1.0000 0.033277 0.034135 8.572079e-04 7 0 0.5000 1.0000 1.508494 1.508494 2.220446e-16 7 1 0.5000 1.0000 0.215499 0.215499 1.942890e-16 7 2 0.5000 1.0000 0.359165 0.359165 5.551115e-17 7 3 0.5000 1.0000 0.111015 0.111015 2.359224e-16 7 4 0.5000 1.0000 0.174308 0.174308 1.110223e-16 7 5 0.5000 1.0000 0.070828 0.070828 1.665335e-16 7 6 0.5000 1.0000 0.106701 0.106701 5.551115e-17 7 7 0.5000 1.0000 0.050350 0.050350 1.457168e-16 7 8 0.5000 1.0000 0.073531 0.073531 4.163336e-17 7 9 0.5000 1.0000 0.038223 0.038223 1.457168e-16 7 10 0.5000 1.0000 0.054471 0.054471 3.469447e-17 7 11 0.5000 1.0000 0.030331 0.030331 1.075529e-16 7 12 0.5000 1.0000 0.042369 0.042369 4.163336e-17 7 13 0.5000 1.0000 0.024849 0.024849 7.979728e-17 7 14 0.5000 1.0000 0.034064 0.034135 7.094710e-05 7 15 0.5000 1.0000 0.020822 0.020854 3.223679e-05 7 16 0.5000 1.0000 0.027992 0.028240 2.484045e-04 8 0 0.5000 1.0000 1.508494 1.508494 4.440892e-16 8 1 0.5000 1.0000 0.215499 0.215499 1.942890e-16 8 2 0.5000 1.0000 0.359165 0.359165 1.110223e-16 8 3 0.5000 1.0000 0.111015 0.111015 5.551115e-17 8 4 0.5000 1.0000 0.174308 0.174308 1.110223e-16 8 5 0.5000 1.0000 0.070828 0.070828 1.110223e-16 8 6 0.5000 1.0000 0.106701 0.106701 2.775558e-17 8 7 0.5000 1.0000 0.050350 0.050350 7.632783e-17 8 8 0.5000 1.0000 0.073531 0.073531 1.387779e-17 8 9 0.5000 1.0000 0.038223 0.038223 4.163336e-17 8 10 0.5000 1.0000 0.054471 0.054471 1.387779e-17 8 11 0.5000 1.0000 0.030331 0.030331 5.204170e-17 8 12 0.5000 1.0000 0.042369 0.042369 0.000000e+00 8 13 0.5000 1.0000 0.024849 0.024849 4.163336e-17 8 14 0.5000 1.0000 0.034135 0.034135 1.387779e-17 8 15 0.5000 1.0000 0.020854 0.020854 3.122502e-17 8 16 0.5000 1.0000 0.028223 0.028240 1.764972e-05 8 17 0.5000 1.0000 0.017827 0.017835 8.107236e-06 8 18 0.5000 1.0000 0.023783 0.023854 7.061626e-05 9 0 0.5000 1.0000 1.508494 1.508494 4.440892e-16 9 1 0.5000 1.0000 0.215499 0.215499 3.053113e-16 9 2 0.5000 1.0000 0.359165 0.359165 5.551115e-17 9 3 0.5000 1.0000 0.111015 0.111015 5.551115e-17 9 4 0.5000 1.0000 0.174308 0.174308 8.326673e-17 9 5 0.5000 1.0000 0.070828 0.070828 1.387779e-17 9 6 0.5000 1.0000 0.106701 0.106701 8.326673e-17 9 7 0.5000 1.0000 0.050350 0.050350 2.081668e-17 9 8 0.5000 1.0000 0.073531 0.073531 9.714451e-17 9 9 0.5000 1.0000 0.038223 0.038223 6.938894e-18 9 10 0.5000 1.0000 0.054471 0.054471 9.020562e-17 9 11 0.5000 1.0000 0.030331 0.030331 2.428613e-17 9 12 0.5000 1.0000 0.042369 0.042369 9.714451e-17 9 13 0.5000 1.0000 0.024849 0.024849 3.122502e-17 9 14 0.5000 1.0000 0.034135 0.034135 7.632783e-17 9 15 0.5000 1.0000 0.020854 0.020854 2.428613e-17 9 16 0.5000 1.0000 0.028240 0.028240 7.632783e-17 9 17 0.5000 1.0000 0.017835 0.017835 4.163336e-17 9 18 0.5000 1.0000 0.023849 0.023854 4.394996e-06 9 19 0.5000 1.0000 0.015484 0.015487 2.036322e-06 9 20 0.5000 1.0000 0.020468 0.020487 1.978095e-05 10 0 0.5000 1.0000 1.508494 1.508494 6.661338e-16 10 1 0.5000 1.0000 0.215499 0.215499 5.273559e-16 10 2 0.5000 1.0000 0.359165 0.359165 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1.436661 8.881784e-16 10 1 2.5000 1.0000 -0.391817 -0.391817 2.775558e-16 10 2 2.5000 1.0000 0.311444 0.311444 1.665335e-16 10 3 2.5000 1.0000 -0.166773 -0.166773 5.551115e-17 10 4 2.5000 1.0000 0.139352 0.139352 0.000000e+00 10 5 2.5000 1.0000 -0.092223 -0.092223 6.938894e-17 10 6 2.5000 1.0000 0.079533 0.079533 4.163336e-17 10 7 2.5000 1.0000 -0.058490 -0.058490 2.775558e-17 10 8 2.5000 1.0000 0.051557 0.051557 3.469447e-17 10 9 2.5000 1.0000 -0.040390 -0.040390 2.081668e-17 10 10 2.5000 1.0000 0.036179 0.036179 4.857226e-17 10 11 2.5000 1.0000 -0.029559 -0.029559 6.245005e-17 10 12 2.5000 1.0000 0.026807 0.026807 5.204170e-17 10 13 2.5000 1.0000 -0.022567 -0.022567 5.204170e-17 10 14 2.5000 1.0000 0.020667 0.020667 6.938894e-17 10 15 2.5000 1.0000 -0.017792 -0.017792 5.898060e-17 10 16 2.5000 1.0000 0.016424 0.016424 5.898060e-17 10 17 2.5000 1.0000 -0.014386 -0.014386 5.724587e-17 10 18 2.5000 1.0000 0.013368 0.013368 6.071532e-17 10 19 2.5000 1.0000 -0.011872 -0.011872 6.071532e-17 10 20 2.5000 1.0000 0.011094 0.011094 3.490698e-07 10 21 2.5000 1.0000 -0.009964 -0.009964 4.486793e-07 10 22 2.5000 1.0000 0.009354 0.009356 1.863779e-06 1 0 2.5000 2.5000 0.981748 0.981748 1.110223e-16 1 1 2.5000 2.5000 0.000000 0.000000 0.000000e+00 1 2 2.5000 2.5000 0.000000 0.122718 1.227185e-01 1 3 2.5000 2.5000 0.000000 0.000000 0.000000e+00 1 4 2.5000 2.5000 0.000000 0.036816 3.681554e-02 2 0 2.5000 2.5000 0.981748 0.981748 1.110223e-16 2 1 2.5000 2.5000 0.000000 0.000000 0.000000e+00 2 2 2.5000 2.5000 0.122718 0.122718 1.387779e-16 2 3 2.5000 2.5000 0.000000 0.000000 0.000000e+00 2 4 2.5000 2.5000 0.015340 0.036816 2.147573e-02 2 5 2.5000 2.5000 0.000000 0.000000 0.000000e+00 2 6 2.5000 2.5000 0.001917 0.015340 1.342233e-02 3 0 2.5000 2.5000 0.981748 0.981748 3.330669e-16 3 1 2.5000 2.5000 0.000000 0.000000 1.081030e-16 3 2 2.5000 2.5000 0.122718 0.122718 2.775558e-17 3 3 2.5000 2.5000 0.000000 0.000000 4.671189e-50 3 4 2.5000 2.5000 0.036816 0.036816 6.938894e-18 3 5 2.5000 2.5000 -0.000000 0.000000 8.673617e-18 3 6 2.5000 2.5000 0.011045 0.015340 4.295146e-03 3 7 2.5000 2.5000 -0.000000 0.000000 4.770490e-18 3 8 2.5000 2.5000 0.003313 0.007670 4.356505e-03 4 0 2.5000 2.5000 0.981748 0.981748 1.110223e-16 4 1 2.5000 2.5000 -0.000000 0.000000 5.551115e-17 4 2 2.5000 2.5000 0.122718 0.122718 5.551115e-17 4 3 2.5000 2.5000 0.000000 0.000000 1.040834e-17 4 4 2.5000 2.5000 0.036816 0.036816 2.081668e-17 4 5 2.5000 2.5000 0.000000 0.000000 5.204170e-18 4 6 2.5000 2.5000 0.015340 0.015340 8.673617e-18 4 7 2.5000 2.5000 0.000000 0.000000 2.602085e-18 4 8 2.5000 2.5000 0.006750 0.007670 9.203885e-04 4 9 2.5000 2.5000 0.000000 0.000000 8.673617e-19 4 10 2.5000 2.5000 0.002991 0.004314 1.323058e-03 5 0 2.5000 2.5000 0.981748 0.981748 5.551115e-16 5 1 2.5000 2.5000 0.000000 0.000000 2.942069e-16 5 2 2.5000 2.5000 0.122718 0.122718 2.775558e-17 5 3 2.5000 2.5000 0.000000 0.000000 6.245005e-17 5 4 2.5000 2.5000 0.036816 0.036816 6.938894e-18 5 5 2.5000 2.5000 0.000000 0.000000 2.038300e-17 5 6 2.5000 2.5000 0.015340 0.015340 1.387779e-17 5 7 2.5000 2.5000 0.000000 0.000000 7.101524e-18 5 8 2.5000 2.5000 0.007670 0.007670 3.469447e-18 5 9 2.5000 2.5000 0.000000 0.000000 2.778268e-18 5 10 2.5000 2.5000 0.004109 0.004314 2.054439e-04 5 11 2.5000 2.5000 0.000000 0.000000 1.116389e-18 5 12 2.5000 2.5000 0.002250 0.002637 3.864301e-04 6 0 2.5000 2.5000 0.981748 0.981748 1.332268e-15 6 1 2.5000 2.5000 0.000000 0.000000 2.151057e-16 6 2 2.5000 2.5000 0.122718 0.122718 4.163336e-17 6 3 2.5000 2.5000 0.000000 0.000000 4.683753e-17 6 4 2.5000 2.5000 0.036816 0.036816 4.163336e-17 6 5 2.5000 2.5000 0.000000 0.000000 1.387779e-17 6 6 2.5000 2.5000 0.015340 0.015340 1.734723e-17 6 7 2.5000 2.5000 0.000000 0.000000 8.673617e-18 6 8 2.5000 2.5000 0.007670 0.007670 1.734723e-17 6 9 2.5000 2.5000 0.000000 0.000000 6.071532e-18 6 10 2.5000 2.5000 0.004314 0.004314 1.214306e-17 6 11 2.5000 2.5000 0.000000 0.000000 4.987330e-18 6 12 2.5000 2.5000 0.002589 0.002637 4.708088e-05 6 13 2.5000 2.5000 0.000000 0.000000 3.794708e-18 6 14 2.5000 2.5000 0.001604 0.001714 1.100516e-04 7 0 2.5000 2.5000 0.981748 0.981748 3.330669e-16 7 1 2.5000 2.5000 -0.000000 0.000000 3.203330e-17 7 2 2.5000 2.5000 0.122718 0.122718 1.387779e-17 7 3 2.5000 2.5000 -0.000000 0.000000 1.561251e-17 7 4 2.5000 2.5000 0.036816 0.036816 6.938894e-18 7 5 2.5000 2.5000 -0.000000 0.000000 4.770490e-18 7 6 2.5000 2.5000 0.015340 0.015340 8.673617e-18 7 7 2.5000 2.5000 -0.000000 0.000000 1.368805e-18 7 8 2.5000 2.5000 0.007670 0.007670 2.602085e-18 7 9 2.5000 2.5000 -0.000000 0.000000 3.083200e-19 7 10 2.5000 2.5000 0.004314 0.004314 3.469447e-18 7 11 2.5000 2.5000 -0.000000 0.000000 1.047144e-19 7 12 2.5000 2.5000 0.002637 0.002637 3.903128e-18 7 13 2.5000 2.5000 -0.000000 0.000000 3.868558e-20 7 14 2.5000 2.5000 0.001703 0.001714 1.098554e-05 7 15 2.5000 2.5000 -0.000000 0.000000 2.305683e-19 7 16 2.5000 2.5000 0.001138 0.001168 3.079280e-05 8 0 2.5000 2.5000 0.981748 0.981748 9.992007e-16 8 1 2.5000 2.5000 0.000000 0.000000 2.081668e-16 8 2 2.5000 2.5000 0.122718 0.122718 6.938894e-17 8 3 2.5000 2.5000 0.000000 0.000000 7.632783e-17 8 4 2.5000 2.5000 0.036816 0.036816 6.938894e-18 8 5 2.5000 2.5000 0.000000 0.000000 4.163336e-17 8 6 2.5000 2.5000 0.015340 0.015340 1.387779e-17 8 7 2.5000 2.5000 0.000000 0.000000 2.688821e-17 8 8 2.5000 2.5000 0.007670 0.007670 1.734723e-18 8 9 2.5000 2.5000 0.000000 0.000000 1.951564e-17 8 10 2.5000 2.5000 0.004314 0.004314 8.673617e-19 8 11 2.5000 2.5000 0.000000 0.000000 1.474515e-17 8 12 2.5000 2.5000 0.002637 0.002637 4.336809e-19 8 13 2.5000 2.5000 0.000000 0.000000 1.138412e-17 8 14 2.5000 2.5000 0.001714 0.001714 4.336809e-19 8 15 2.5000 2.5000 0.000000 0.000000 9.161508e-18 8 16 2.5000 2.5000 0.001166 0.001168 2.596582e-06 8 17 2.5000 2.5000 0.000000 0.000000 7.155734e-18 8 18 2.5000 2.5000 0.000819 0.000828 8.503806e-06 9 0 2.5000 2.5000 0.981748 0.981748 1.110223e-16 9 1 2.5000 2.5000 -0.000000 0.000000 5.551115e-17 9 2 2.5000 2.5000 0.122718 0.122718 2.775558e-17 9 3 2.5000 2.5000 -0.000000 0.000000 1.561251e-17 9 4 2.5000 2.5000 0.036816 0.036816 2.081668e-17 9 5 2.5000 2.5000 -0.000000 0.000000 1.344411e-17 9 6 2.5000 2.5000 0.015340 0.015340 4.510281e-17 9 7 2.5000 2.5000 -0.000000 0.000000 1.387779e-17 9 8 2.5000 2.5000 0.007670 0.007670 2.515349e-17 9 9 2.5000 2.5000 -0.000000 0.000000 1.214306e-17 9 10 2.5000 2.5000 0.004314 0.004314 1.647987e-17 9 11 2.5000 2.5000 -0.000000 0.000000 9.757820e-18 9 12 2.5000 2.5000 0.002637 0.002637 1.084202e-17 9 13 2.5000 2.5000 -0.000000 0.000000 7.670730e-18 9 14 2.5000 2.5000 0.001714 0.001714 5.854692e-18 9 15 2.5000 2.5000 -0.000000 0.000000 5.834363e-18 9 16 2.5000 2.5000 0.001168 0.001168 3.903128e-18 9 17 2.5000 2.5000 -0.000000 0.000000 4.460687e-18 9 18 2.5000 2.5000 0.000827 0.000828 6.196389e-07 9 19 2.5000 2.5000 -0.000000 0.000000 3.375056e-18 9 20 2.5000 2.5000 0.000603 0.000605 2.324729e-06 10 0 2.5000 2.5000 0.981748 0.981748 1.110223e-16 10 1 2.5000 2.5000 0.000000 0.000000 1.387779e-17 10 2 2.5000 2.5000 0.122718 0.122718 2.775558e-17 10 3 2.5000 2.5000 0.000000 0.000000 1.821460e-17 10 4 2.5000 2.5000 0.036816 0.036816 3.469447e-17 10 5 2.5000 2.5000 -0.000000 0.000000 8.673617e-19 10 6 2.5000 2.5000 0.015340 0.015340 3.469447e-18 10 7 2.5000 2.5000 -0.000000 0.000000 1.951564e-18 10 8 2.5000 2.5000 0.007670 0.007670 6.071532e-18 10 9 2.5000 2.5000 0.000000 0.000000 0.000000e+00 10 10 2.5000 2.5000 0.004314 0.004314 3.469447e-18 10 11 2.5000 2.5000 0.000000 0.000000 7.047314e-19 10 12 2.5000 2.5000 0.002637 0.002637 2.168404e-18 10 13 2.5000 2.5000 0.000000 0.000000 9.554532e-19 10 14 2.5000 2.5000 0.001714 0.001714 2.168404e-18 10 15 2.5000 2.5000 0.000000 0.000000 1.043545e-18 10 16 2.5000 2.5000 0.001168 0.001168 1.301043e-18 10 17 2.5000 2.5000 0.000000 0.000000 8.673617e-19 10 18 2.5000 2.5000 0.000828 0.000828 1.084202e-18 10 19 2.5000 2.5000 0.000000 0.000000 7.589415e-19 10 20 2.5000 2.5000 0.000605 0.000605 1.489517e-07 10 21 2.5000 2.5000 0.000000 0.000000 6.505213e-19 10 22 2.5000 2.5000 0.000453 0.000454 6.303847e-07 SANDIA_RULES_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 1 0.0000000000000000 1.5707963267948968 Order = 2 ALPHA = 0.500000 BETA = 0.500000 1 -0.4999999999999999 0.7853981633974484 2 0.4999999999999999 0.7853981633974484 Order = 3 ALPHA = 0.500000 BETA = 0.500000 1 -0.7071067811865475 0.3926990816987245 2 0.0000000000000001 0.7853981633974486 3 0.7071067811865474 0.3926990816987239 Order = 4 ALPHA = 0.500000 BETA = 0.500000 1 -0.8090169943749475 0.2170787134227061 2 -0.3090169943749473 0.5683194499747424 3 0.3090169943749472 0.5683194499747432 4 0.8090169943749477 0.2170787134227062 Order = 5 ALPHA = 0.500000 BETA = 0.500000 1 -0.8660254037844389 0.1308996938995749 2 -0.4999999999999998 0.3926990816987244 3 0.0000000000000001 0.5235987755982987 4 0.4999999999999998 0.3926990816987242 5 0.8660254037844388 0.1308996938995747 Order = 6 ALPHA = 0.500000 BETA = 0.500000 1 -0.9009688679024188 0.0844886908915887 2 -0.6234898018587335 0.2743330560697781 3 -0.2225209339563142 0.4265764164360816 4 0.2225209339563143 0.4265764164360817 5 0.6234898018587332 0.2743330560697784 6 0.9009688679024188 0.0844886908915885 Order = 7 ALPHA = 0.500000 BETA = 0.500000 1 -0.9238795325112868 0.0575094490319133 2 -0.7071067811865476 0.1963495408493622 3 -0.3826834323650896 0.3351896326668111 4 0.0000000000000000 0.3926990816987249 5 0.3826834323650899 0.3351896326668110 6 0.7071067811865475 0.1963495408493624 7 0.9238795325112863 0.0575094490319132 Order = 8 ALPHA = 0.500000 BETA = 0.500000 1 -0.9396926207859084 0.0408329477091071 2 -0.7660444431189782 0.1442256007956730 3 -0.4999999999999999 0.2617993877991496 4 -0.1736481776669302 0.3385402270935193 5 0.1736481776669302 0.3385402270935190 6 0.5000000000000000 0.2617993877991501 7 0.7660444431189779 0.1442256007956725 8 0.9396926207859086 0.0408329477091071 Order = 9 ALPHA = 0.500000 BETA = 0.500000 1 -0.9510565162951536 0.0299995403716082 2 -0.8090169943749472 0.1085393567113534 3 -0.5877852522924730 0.2056199086476264 4 -0.3090169943749472 0.2841597249873707 5 0.0000000000000001 0.3141592653589796 6 0.3090169943749471 0.2841597249873716 7 0.5877852522924728 0.2056199086476266 8 0.8090169943749472 0.1085393567113536 9 0.9510565162951536 0.0299995403716080 Order = 10 ALPHA = 0.500000 BETA = 0.500000 1 -0.9594929736144974 0.0226689425018589 2 -0.8412535328311809 0.0834785409341892 3 -0.6548607339452849 0.1631221774548168 4 -0.4154150130018863 0.2363135602034877 5 -0.1423148382732851 0.2798149423030964 6 0.1423148382732851 0.2798149423030961 7 0.4154150130018863 0.2363135602034874 8 0.6548607339452848 0.1631221774548172 9 0.8412535328311812 0.0834785409341888 10 0.9594929736144973 0.0226689425018589 Order = 1 ALPHA = 0.500000 BETA = 1.000000 1 0.1428571428571428 1.5084944665313014 Order = 2 ALPHA = 0.500000 BETA = 1.000000 1 -0.3785434359039294 0.6707847483124826 2 0.5603616177221111 0.8377097182188189 Order = 3 ALPHA = 0.500000 BETA = 1.000000 1 -0.6191289006136985 0.2886624303931599 2 0.0828374856848100 0.7711842727220678 3 0.7362914149288884 0.4486477634160737 Order = 4 ALPHA = 0.500000 BETA = 1.000000 1 -0.7447173440086010 0.1380862915234591 2 -0.2308286409011867 0.5011891464890383 3 0.3610180502309863 0.6101454555318188 4 0.8250542504682752 0.2590735729869861 Order = 5 ALPHA = 0.500000 BETA = 1.000000 1 -0.8175993274098119 0.0729555935075893 2 -0.4326084003843105 0.3103351805927456 3 0.0582907697984680 0.5176668424406614 4 0.5335962093870672 0.4466360854739412 5 0.8757120529564133 0.1609007645163651 Order = 6 ALPHA = 0.500000 BETA = 1.000000 1 -0.8633993972645073 0.0417787392098550 2 -0.5667404500118454 0.1951823756778568 3 -0.1659246338882321 0.3864855437976809 4 0.2649141017150740 0.4544246037858506 5 0.6461272538533950 0.3245589588523247 6 0.9072453478183379 0.1060642452077325 Order = 7 ALPHA = 0.500000 BETA = 1.000000 1 -0.8939724663908035 0.0255150146724792 2 -0.6594047339183104 0.1266052232513152 3 -0.3307207485144990 0.2785906683424275 4 0.0449578958653256 0.3894994287201929 5 0.4138036987843730 0.3759820326953337 6 0.7229723820004327 0.2389626799471576 7 0.9281704237863850 0.0733394189023954 Order = 8 ALPHA = 0.500000 BETA = 1.000000 1 -0.9153619779765985 0.0164120900902138 2 -0.7257160980084201 0.0848554733563241 3 -0.4533780733947833 0.2002825508076703 4 -0.1294870729403755 0.3123493200327140 5 0.2089491097631813 0.3577748934099902 6 0.5232643292074005 0.3048404655795939 7 0.7775490329551161 0.1792709060668855 8 0.9427521789659081 0.0527087671879101 Order = 9 ALPHA = 0.500000 BETA = 1.000000 1 -0.9308966361094720 0.0110143919419964 2 -0.7746386880679893 0.0586443046921294 3 -0.5463185334791455 0.1453933047463930 4 -0.2672381326388558 0.2442199669465208 5 0.0365863464439503 0.3121710958691362 6 0.3368351070820351 0.3145887170547924 7 0.6055223420849569 0.2462380941383865 8 0.8176042531316312 0.1371247634867023 9 0.9533131723221203 0.0390998276552451 Order = 10 ALPHA = 0.500000 BETA = 1.000000 1 -0.9425275499488472 0.0076569987337497 2 -0.8116864125172604 0.0416630238085872 3 -0.6180513952640334 0.1070695285744134 4 -0.3766418843493625 0.1896495013540221 5 -0.1061646285278262 0.2614695484772913 6 0.1724237674317270 0.2937514863341338 7 0.4375388507874067 0.2709981593343714 8 0.6686402341966643 0.1996303237647672 9 0.8478228797503714 0.1068280415109269 10 0.9612042779760440 0.0297778546390379 Order = 1 ALPHA = 0.500000 BETA = 2.500000 1 0.4000000000000000 1.9634954084936209 Order = 2 ALPHA = 0.500000 BETA = 2.500000 1 -0.1055161125036901 0.6949608437695366 2 0.6769446839322615 1.2685345647240844 Order = 3 ALPHA = 0.500000 BETA = 2.500000 1 -0.3949726057630552 0.2155740450619748 2 0.2647662774451410 0.9816213720815014 3 0.7968729949845808 0.7662999913501441 Order = 4 ALPHA = 0.500000 BETA = 2.500000 1 -0.5673928432278891 0.0735270499348779 2 -0.0438129709574190 0.4997101315556031 3 0.4782610448380588 0.9082548004475428 4 0.8602174966199764 0.4820034265555980 Order = 5 ALPHA = 0.500000 BETA = 2.500000 1 -0.6766083904771948 0.0281042710539254 2 -0.2615276527389885 0.2379108911216861 3 0.1979083288674991 0.6351035815608072 4 0.6116098169968440 0.7438254266897733 5 0.8978486665826094 0.3185512380674297 Order = 6 ALPHA = 0.500000 BETA = 2.500000 1 -0.7496281758956908 0.0118933444385382 2 -0.4162703993964169 0.1152628351592501 3 -0.0240992331493379 0.3830143850747024 4 0.3678575869061303 0.6464461334770024 5 0.7000843065535390 0.5868443910764994 6 0.9220559149817764 0.2200343192676292 Order = 7 ALPHA = 0.500000 BETA = 2.500000 1 -0.8006672147724485 0.0054858163722489 2 -0.5286792390323160 0.0581588649983062 3 -0.1959098000069027 0.2221608406878784 4 0.1580287890648373 0.4638578133115197 5 0.4905660544916725 0.5963414735975121 6 0.7616344330284672 0.4597291058720924 7 0.9385563889913959 0.1577614936540629 Order = 8 ALPHA = 0.500000 BETA = 2.500000 1 -0.8376646856786830 0.0027198968451133 2 -0.6123079224732046 0.0307056736312336 3 -0.3290736457267182 0.1289926195321413 4 -0.0152678951698322 0.3092909316152767 5 0.2984865079144489 0.4881539875384042 6 0.5815096611955630 0.5250491653932975 7 0.8061111752639282 0.3618821131546536 8 0.9503120678323930 0.1167010207835007 Order = 9 ALPHA = 0.500000 BETA = 2.500000 1 -0.8653022034444988 0.0014331423151996 2 -0.6759411199023829 0.0169372413862485 3 -0.4333019473215539 0.0761680132420716 4 -0.1566993909441266 0.2010801278076632 5 0.1315346053479577 0.3633349438080313 6 0.4080752758535235 0.4761564190243807 7 0.6505301580495418 0.4521871712839325 8 0.8392624591522133 0.2875734496942869 9 0.9589850203521831 0.0886248999318063 Order = 10 ALPHA = 0.500000 BETA = 2.500000 1 -0.8864732152715593 0.0007951842411665 2 -0.7253515462755796 0.0097270000118211 3 -0.5159038845260621 0.0460145772880253 4 -0.2721183637403177 0.1303012691755840 5 -0.0105450517531136 0.2590099865557985 6 0.2510118405448967 0.3871376587183351 7 0.4947368451601080 0.4448595186308594 8 0.7040250190649779 0.3857617289382688 9 0.8646161230141249 0.2310686902877489 10 0.9655674511738290 0.0688197946460142 Order = 1 ALPHA = 1.000000 BETA = 0.500000 1 -0.1428571428571428 1.5084944665313014 Order = 2 ALPHA = 1.000000 BETA = 0.500000 1 -0.5603616177221111 0.8377097182188189 2 0.3785434359039294 0.6707847483124826 Order = 3 ALPHA = 1.000000 BETA = 0.500000 1 -0.7362914149288884 0.4486477634160737 2 -0.0828374856848100 0.7711842727220678 3 0.6191289006136985 0.2886624303931599 Order = 4 ALPHA = 1.000000 BETA = 0.500000 1 -0.8250542504682752 0.2590735729869861 2 -0.3610180502309863 0.6101454555318188 3 0.2308286409011867 0.5011891464890383 4 0.7447173440086010 0.1380862915234591 Order = 5 ALPHA = 1.000000 BETA = 0.500000 1 -0.8757120529564133 0.1609007645163651 2 -0.5335962093870672 0.4466360854739412 3 -0.0582907697984680 0.5176668424406614 4 0.4326084003843105 0.3103351805927456 5 0.8175993274098119 0.0729555935075893 Order = 6 ALPHA = 1.000000 BETA = 0.500000 1 -0.9072453478183379 0.1060642452077325 2 -0.6461272538533950 0.3245589588523247 3 -0.2649141017150740 0.4544246037858506 4 0.1659246338882321 0.3864855437976809 5 0.5667404500118454 0.1951823756778568 6 0.8633993972645073 0.0417787392098550 Order = 7 ALPHA = 1.000000 BETA = 0.500000 1 -0.9281704237863850 0.0733394189023954 2 -0.7229723820004327 0.2389626799471576 3 -0.4138036987843730 0.3759820326953337 4 -0.0449578958653256 0.3894994287201929 5 0.3307207485144990 0.2785906683424275 6 0.6594047339183104 0.1266052232513152 7 0.8939724663908035 0.0255150146724792 Order = 8 ALPHA = 1.000000 BETA = 0.500000 1 -0.9427521789659081 0.0527087671879101 2 -0.7775490329551161 0.1792709060668855 3 -0.5232643292074005 0.3048404655795939 4 -0.2089491097631813 0.3577748934099902 5 0.1294870729403755 0.3123493200327140 6 0.4533780733947833 0.2002825508076703 7 0.7257160980084201 0.0848554733563241 8 0.9153619779765985 0.0164120900902138 Order = 9 ALPHA = 1.000000 BETA = 0.500000 1 -0.9533131723221203 0.0390998276552451 2 -0.8176042531316312 0.1371247634867023 3 -0.6055223420849569 0.2462380941383865 4 -0.3368351070820351 0.3145887170547924 5 -0.0365863464439503 0.3121710958691362 6 0.2672381326388558 0.2442199669465208 7 0.5463185334791455 0.1453933047463930 8 0.7746386880679893 0.0586443046921294 9 0.9308966361094720 0.0110143919419964 Order = 10 ALPHA = 1.000000 BETA = 0.500000 1 -0.9612042779760440 0.0297778546390379 2 -0.8478228797503714 0.1068280415109269 3 -0.6686402341966643 0.1996303237647672 4 -0.4375388507874067 0.2709981593343714 5 -0.1724237674317270 0.2937514863341338 6 0.1061646285278262 0.2614695484772913 7 0.3766418843493625 0.1896495013540221 8 0.6180513952640334 0.1070695285744134 9 0.8116864125172604 0.0416630238085872 10 0.9425275499488472 0.0076569987337497 Order = 1 ALPHA = 1.000000 BETA = 1.000000 1 0.0000000000000000 1.3333333333333333 Order = 2 ALPHA = 1.000000 BETA = 1.000000 1 -0.4472135954999578 0.6666666666666665 2 0.4472135954999578 0.6666666666666665 Order = 3 ALPHA = 1.000000 BETA = 1.000000 1 -0.6546536707079771 0.3111111111111109 2 0.0000000000000001 0.7111111111111111 3 0.6546536707079770 0.3111111111111108 Order = 4 ALPHA = 1.000000 BETA = 1.000000 1 -0.7650553239294646 0.1569499125956939 2 -0.2852315164806450 0.5097167540709726 3 0.2852315164806451 0.5097167540709729 4 0.7650553239294645 0.1569499125956942 Order = 5 ALPHA = 1.000000 BETA = 1.000000 1 -0.8302238962785671 0.0860176821228074 2 -0.4688487934707142 0.3368394607343353 3 0.0000000000000002 0.4876190476190478 4 0.4688487934707142 0.3368394607343351 5 0.8302238962785669 0.0860176821228075 Order = 6 ALPHA = 1.000000 BETA = 1.000000 1 -0.8717401485096067 0.0505835770160818 2 -0.5917001814331422 0.2216925320225175 3 -0.2092992179024789 0.3943905576280672 4 0.2092992179024789 0.3943905576280671 5 0.5917001814331422 0.2216925320225176 6 0.8717401485096065 0.0505835770160817 Order = 7 ALPHA = 1.000000 BETA = 1.000000 1 -0.8997579954114602 0.0315162001504987 2 -0.6771862795107376 0.1486404045834097 3 -0.3631174638261783 0.3007504247445495 4 0.0000000000000002 0.3715192743764173 5 0.3631174638261781 0.3007504247445493 6 0.6771862795107377 0.1486404045834098 7 0.8997579954114601 0.0315162001504987 Order = 8 ALPHA = 1.000000 BETA = 1.000000 1 -0.9195339081664586 0.0205900956491218 2 -0.7387738651055049 0.1021477023603584 3 -0.4779249498104446 0.2253365549698581 4 -0.1652789576663870 0.3185923136873287 5 0.1652789576663869 0.3185923136873286 6 0.4779249498104445 0.2253365549698582 7 0.7387738651055049 0.1021477023603586 8 0.9195339081664586 0.0205900956491218 Order = 9 ALPHA = 1.000000 BETA = 1.000000 1 -0.9340014304080593 0.0139910561244267 2 -0.7844834736631444 0.0719828561756609 3 -0.5652353269962052 0.1687989736144601 4 -0.2957581355869395 0.2617849830242734 5 0.0000000000000003 0.3002175954556909 6 0.2957581355869395 0.2617849830242732 7 0.5652353269962052 0.1687989736144599 8 0.7844834736631442 0.0719828561756609 9 0.9340014304080595 0.0139910561244267 Order = 10 ALPHA = 1.000000 BETA = 1.000000 1 -0.9448992722228819 0.0098254048057681 2 -0.8192793216440070 0.0519391437742078 3 -0.6328761530318606 0.1273919483477327 4 -0.3995309409653490 0.2111657418760753 5 -0.1365529328549277 0.2663444278628823 6 0.1365529328549273 0.2663444278628825 7 0.3995309409653491 0.2111657418760754 8 0.6328761530318608 0.1273919483477327 9 0.8192793216440072 0.0519391437742081 10 0.9448992722228821 0.0098254048057682 Order = 1 ALPHA = 1.000000 BETA = 2.500000 1 0.2727272727272727 1.4366613966964779 Order = 2 ALPHA = 1.000000 BETA = 2.500000 1 -0.1843075691322091 0.5823920991551214 2 0.5843075691322090 0.8542692975413564 Order = 3 ALPHA = 1.000000 BETA = 2.500000 1 -0.4423027360564286 0.2004355575312489 2 0.1860296262211932 0.7756953091585610 3 0.7299573203615513 0.4605305300066668 Order = 4 ALPHA = 1.000000 BETA = 2.500000 1 -0.5972505982938984 0.0735380497612852 2 -0.1011259685235210 0.4428338893708899 3 0.4098198869416554 0.6621803747350364 4 0.8102958103105465 0.2581090828292663 Order = 5 ALPHA = 1.000000 BETA = 2.500000 1 -0.6964392762006608 0.0296301520114946 2 -0.3026155820854005 0.2285362779908424 3 0.1412598214162332 0.5280679973185715 4 0.5539710608669561 0.4975219906041121 5 0.8593795315584275 0.1529049787714574 Order = 6 ALPHA = 1.000000 BETA = 2.500000 1 -0.7633972293043557 0.0130439061802296 2 -0.4461529248760190 0.1174255136957449 3 -0.0686784265705634 0.3490370430142557 4 0.3155805295665461 0.5014050892168358 5 0.6517131087948295 0.3603484489584900 6 0.8915801036798848 0.0954013956309210 Order = 7 ALPHA = 1.000000 BETA = 2.500000 1 -0.8105874821854354 0.0062027903657016 2 -0.5508759997785407 0.0619382418889596 3 -0.2306894365569856 0.2164618775946896 4 0.1138876346036921 0.3988213674613816 5 0.4436003431475740 0.4310114591490306 6 0.7208146570340379 0.2599688822120861 7 0.9138502837356578 0.0622567780246286 Order = 8 ALPHA = 1.000000 BETA = 2.500000 1 -0.8450352779186010 0.0031506505662605 2 -0.6291538133028906 0.0338481948002515 3 -0.3563555020358466 0.1322157770527710 4 -0.0516690747442928 0.2867379491637258 5 0.2565958341049685 0.3952631184069375 6 0.5397363757841801 0.3540223921913581 7 0.7713710662668611 0.1892132923890619 8 0.9298950072302367 0.0422100221261125 Order = 9 ALPHA = 1.000000 BETA = 2.500000 1 -0.8709215211401555 0.0016929758046626 2 -0.6889839677544535 0.0191902962920151 3 -0.4549257546478145 0.0812176110475681 4 -0.1865538397099543 0.1974051443898488 5 0.0954122417770833 0.3197898022964819 6 0.3691513199896206 0.3631243013752991 7 0.6134647393951399 0.2850633375254082 8 0.8094259398841321 0.1396134790482666 9 0.9418378189505883 0.0295644489169278 Order = 10 ALPHA = 1.000000 BETA = 2.500000 1 -0.8908519983624360 0.0009546834154140 2 -0.7356332759809726 0.0112695230304569 3 -0.5332475568814603 0.0506378825686695 4 -0.2966592827516153 0.1337845805120262 5 -0.0412830087188659 0.2429565425267207 6 0.2161975251464398 0.3233164884771637 7 0.4589489303958385 0.3198849047578982 8 0.6710962948352331 0.2279739859839845 9 0.8387626487347724 0.1045909447918315 10 0.9509675959234919 0.0212918606323121 Order = 1 ALPHA = 2.500000 BETA = 0.500000 1 -0.4000000000000000 1.9634954084936209 Order = 2 ALPHA = 2.500000 BETA = 0.500000 1 -0.6769446839322615 1.2685345647240844 2 0.1055161125036901 0.6949608437695366 Order = 3 ALPHA = 2.500000 BETA = 0.500000 1 -0.7968729949845808 0.7662999913501441 2 -0.2647662774451410 0.9816213720815014 3 0.3949726057630552 0.2155740450619748 Order = 4 ALPHA = 2.500000 BETA = 0.500000 1 -0.8602174966199764 0.4820034265555980 2 -0.4782610448380588 0.9082548004475428 3 0.0438129709574190 0.4997101315556031 4 0.5673928432278891 0.0735270499348779 Order = 5 ALPHA = 2.500000 BETA = 0.500000 1 -0.8978486665826094 0.3185512380674297 2 -0.6116098169968440 0.7438254266897733 3 -0.1979083288674991 0.6351035815608072 4 0.2615276527389885 0.2379108911216861 5 0.6766083904771948 0.0281042710539254 Order = 6 ALPHA = 2.500000 BETA = 0.500000 1 -0.9220559149817764 0.2200343192676292 2 -0.7000843065535390 0.5868443910764994 3 -0.3678575869061303 0.6464461334770024 4 0.0240992331493379 0.3830143850747024 5 0.4162703993964169 0.1152628351592501 6 0.7496281758956908 0.0118933444385382 Order = 7 ALPHA = 2.500000 BETA = 0.500000 1 -0.9385563889913959 0.1577614936540629 2 -0.7616344330284672 0.4597291058720924 3 -0.4905660544916725 0.5963414735975121 4 -0.1580287890648373 0.4638578133115197 5 0.1959098000069027 0.2221608406878784 6 0.5286792390323160 0.0581588649983062 7 0.8006672147724485 0.0054858163722489 Order = 8 ALPHA = 2.500000 BETA = 0.500000 1 -0.9503120678323930 0.1167010207835007 2 -0.8061111752639282 0.3618821131546536 3 -0.5815096611955630 0.5250491653932975 4 -0.2984865079144489 0.4881539875384042 5 0.0152678951698322 0.3092909316152767 6 0.3290736457267182 0.1289926195321413 7 0.6123079224732046 0.0307056736312336 8 0.8376646856786830 0.0027198968451133 Order = 9 ALPHA = 2.500000 BETA = 0.500000 1 -0.9589850203521831 0.0886248999318063 2 -0.8392624591522133 0.2875734496942869 3 -0.6505301580495418 0.4521871712839325 4 -0.4080752758535235 0.4761564190243807 5 -0.1315346053479577 0.3633349438080313 6 0.1566993909441266 0.2010801278076632 7 0.4333019473215539 0.0761680132420716 8 0.6759411199023829 0.0169372413862485 9 0.8653022034444988 0.0014331423151996 Order = 10 ALPHA = 2.500000 BETA = 0.500000 1 -0.9655674511738290 0.0688197946460142 2 -0.8646161230141249 0.2310686902877489 3 -0.7040250190649779 0.3857617289382688 4 -0.4947368451601080 0.4448595186308594 5 -0.2510118405448967 0.3871376587183351 6 0.0105450517531136 0.2590099865557985 7 0.2721183637403177 0.1303012691755840 8 0.5159038845260621 0.0460145772880253 9 0.7253515462755796 0.0097270000118211 10 0.8864732152715593 0.0007951842411665 Order = 1 ALPHA = 2.500000 BETA = 1.000000 1 -0.2727272727272727 1.4366613966964779 Order = 2 ALPHA = 2.500000 BETA = 1.000000 1 -0.5843075691322090 0.8542692975413564 2 0.1843075691322091 0.5823920991551214 Order = 3 ALPHA = 2.500000 BETA = 1.000000 1 -0.7299573203615513 0.4605305300066668 2 -0.1860296262211932 0.7756953091585610 3 0.4423027360564286 0.2004355575312489 Order = 4 ALPHA = 2.500000 BETA = 1.000000 1 -0.8102958103105465 0.2581090828292663 2 -0.4098198869416554 0.6621803747350364 3 0.1011259685235210 0.4428338893708899 4 0.5972505982938984 0.0735380497612852 Order = 5 ALPHA = 2.500000 BETA = 1.000000 1 -0.8593795315584275 0.1529049787714574 2 -0.5539710608669561 0.4975219906041121 3 -0.1412598214162332 0.5280679973185715 4 0.3026155820854005 0.2285362779908424 5 0.6964392762006608 0.0296301520114946 Order = 6 ALPHA = 2.500000 BETA = 1.000000 1 -0.8915801036798848 0.0954013956309210 2 -0.6517131087948295 0.3603484489584900 3 -0.3155805295665461 0.5014050892168358 4 0.0686784265705634 0.3490370430142557 5 0.4461529248760190 0.1174255136957449 6 0.7633972293043557 0.0130439061802296 Order = 7 ALPHA = 2.500000 BETA = 1.000000 1 -0.9138502837356578 0.0622567780246286 2 -0.7208146570340379 0.2599688822120861 3 -0.4436003431475740 0.4310114591490306 4 -0.1138876346036921 0.3988213674613816 5 0.2306894365569856 0.2164618775946896 6 0.5508759997785407 0.0619382418889596 7 0.8105874821854354 0.0062027903657016 Order = 8 ALPHA = 2.500000 BETA = 1.000000 1 -0.9298950072302367 0.0422100221261125 2 -0.7713710662668611 0.1892132923890619 3 -0.5397363757841801 0.3540223921913581 4 -0.2565958341049685 0.3952631184069375 5 0.0516690747442928 0.2867379491637258 6 0.3563555020358466 0.1322157770527710 7 0.6291538133028906 0.0338481948002515 8 0.8450352779186010 0.0031506505662605 Order = 9 ALPHA = 2.500000 BETA = 1.000000 1 -0.9418378189505883 0.0295644489169278 2 -0.8094259398841321 0.1396134790482666 3 -0.6134647393951399 0.2850633375254082 4 -0.3691513199896206 0.3631243013752991 5 -0.0954122417770833 0.3197898022964819 6 0.1865538397099543 0.1974051443898488 7 0.4549257546478145 0.0812176110475681 8 0.6889839677544535 0.0191902962920151 9 0.8709215211401555 0.0016929758046626 Order = 10 ALPHA = 2.500000 BETA = 1.000000 1 -0.9509675959234919 0.0212918606323121 2 -0.8387626487347724 0.1045909447918315 3 -0.6710962948352331 0.2279739859839845 4 -0.4589489303958385 0.3198849047578982 5 -0.2161975251464398 0.3233164884771637 6 0.0412830087188659 0.2429565425267207 7 0.2966592827516153 0.1337845805120262 8 0.5332475568814603 0.0506378825686695 9 0.7356332759809726 0.0112695230304569 10 0.8908519983624360 0.0009546834154140 Order = 1 ALPHA = 2.500000 BETA = 2.500000 1 0.0000000000000000 0.9817477042468103 Order = 2 ALPHA = 2.500000 BETA = 2.500000 1 -0.3535533905932737 0.4908738521234051 2 0.3535533905932737 0.4908738521234051 Order = 3 ALPHA = 2.500000 BETA = 2.500000 1 -0.5477225575051663 0.2045307717180854 2 0.0000000000000000 0.5726861608106395 3 0.5477225575051661 0.2045307717180856 Order = 4 ALPHA = 2.500000 BETA = 2.500000 1 -0.6660699417556469 0.0870080715314834 2 -0.2373833033084450 0.4038657805919218 3 0.2373833033084450 0.4038657805919215 4 0.6660699417556469 0.0870080715314834 Order = 5 ALPHA = 2.500000 BETA = 2.500000 1 -0.7434770045212192 0.0392893891317672 2 -0.4019050360892100 0.2454174450998076 3 0.0000000000000001 0.4123340357836606 4 0.4019050360892101 0.2454174450998080 5 0.7434770045212191 0.0392893891317673 Order = 6 ALPHA = 2.500000 BETA = 2.500000 1 -0.7968373566406803 0.0189116433666931 2 -0.5196148456949135 0.1431407318203966 3 -0.1804179569647765 0.3288214769363159 4 0.1804179569647765 0.3288214769363157 5 0.5196148456949131 0.1431407318203972 6 0.7968373566406806 0.0189116433666931 Order = 7 ALPHA = 2.500000 BETA = 2.500000 1 -0.8351564541339109 0.0096585250210010 2 -0.6063141605450342 0.0834446705489849 3 -0.3186902924591671 0.2357822853526958 4 0.0000000000000001 0.3239767424014474 5 0.3186902924591672 0.2357822853526957 6 0.6063141605450342 0.0834446705489848 7 0.8351564541339108 0.0096585250210010 Order = 8 ALPHA = 2.500000 BETA = 2.500000 1 -0.8635912403827581 0.0051997074221325 2 -0.6718232507247469 0.0494181881565223 3 -0.4260961495798481 0.1615070338519728 4 -0.1459649294626306 0.2747489226927776 5 0.1459649294626304 0.2747489226927781 6 0.4260961495798480 0.1615070338519730 7 0.6718232507247468 0.0494181881565224 8 0.8635912403827586 0.0051997074221325 Order = 9 ALPHA = 2.500000 BETA = 2.500000 1 -0.8852662998295733 0.0029319009849924 2 -0.7224309555097358 0.0299209571936803 3 -0.5107538326640052 0.1088013928543096 4 -0.2643695362367074 0.2155149137501429 5 -0.0000000000000000 0.2674093746805600 6 0.2643695362367075 0.2155149137501428 7 0.5107538326640052 0.1088013928543095 8 0.7224309555097355 0.0299209571936804 9 0.8852662998295734 0.0029319009849924 Order = 10 ALPHA = 2.500000 BETA = 2.500000 1 -0.9021636877146701 0.0017216382288931 2 -0.7622868838395971 0.0185564370059904 3 -0.5784393987065429 0.0731613283971410 4 -0.3610608688185056 0.1622660968500983 5 -0.1227285554678421 0.2351683516412827 6 0.1227285554678420 0.2351683516412817 7 0.3610608688185061 0.1622660968500987 8 0.5784393987065428 0.0731613283971409 9 0.7622868838395969 0.0185564370059904 10 0.9021636877146699 0.0017216382288932 SANDIA_RULES_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 0.000000e+00 2 2 2.000000e+00 2.000000e+00 0.000000e+00 2 3 6.000000e+00 6.000000e+00 0.000000e+00 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 0.000000e+00 3 1 1.000000e+00 1.000000e+00 0.000000e+00 3 2 2.000000e+00 2.000000e+00 4.440892e-16 3 3 6.000000e+00 6.000000e+00 3.552714e-15 3 4 2.400000e+01 2.400000e+01 2.842171e-14 3 5 1.200000e+02 1.200000e+02 2.131628e-13 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 4.440892e-16 4 1 1.000000e+00 1.000000e+00 4.440892e-16 4 2 2.000000e+00 2.000000e+00 8.881784e-16 4 3 6.000000e+00 6.000000e+00 8.881784e-16 4 4 2.400000e+01 2.400000e+01 7.105427e-15 4 5 1.200000e+02 1.200000e+02 8.526513e-14 4 6 7.200000e+02 7.200000e+02 9.094947e-13 4 7 5.040000e+03 5.040000e+03 1.091394e-11 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 2.220446e-16 5 1 1.000000e+00 1.000000e+00 0.000000e+00 5 2 2.000000e+00 2.000000e+00 8.881784e-16 5 3 6.000000e+00 6.000000e+00 5.329071e-15 5 4 2.400000e+01 2.400000e+01 3.552714e-14 5 5 1.200000e+02 1.200000e+02 1.989520e-13 5 6 7.200000e+02 7.200000e+02 1.364242e-12 5 7 5.040000e+03 5.040000e+03 9.094947e-12 5 8 4.032000e+04 4.032000e+04 6.548362e-11 5 9 3.628800e+05 3.628800e+05 5.238689e-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 1.110223e-16 6 1 1.000000e+00 1.000000e+00 4.440892e-16 6 2 2.000000e+00 2.000000e+00 1.776357e-15 6 3 6.000000e+00 6.000000e+00 7.105427e-15 6 4 2.400000e+01 2.400000e+01 3.552714e-14 6 5 1.200000e+02 1.200000e+02 1.563194e-13 6 6 7.200000e+02 7.200000e+02 6.821210e-13 6 7 5.040000e+03 5.040000e+03 3.637979e-12 6 8 4.032000e+04 4.032000e+04 0.000000e+00 6 9 3.628800e+05 3.628800e+05 1.746230e-10 6 10 3.628800e+06 3.628800e+06 3.725290e-09 6 11 3.991680e+07 3.991680e+07 5.960464e-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 0.000000e+00 7 1 1.000000e+00 1.000000e+00 2.220446e-16 7 2 2.000000e+00 2.000000e+00 8.881784e-16 7 3 6.000000e+00 6.000000e+00 1.776357e-15 7 4 2.400000e+01 2.400000e+01 3.552714e-15 7 5 1.200000e+02 1.200000e+02 2.842171e-14 7 6 7.200000e+02 7.200000e+02 2.273737e-13 7 7 5.040000e+03 5.040000e+03 1.818989e-12 7 8 4.032000e+04 4.032000e+04 1.455192e-11 7 9 3.628800e+05 3.628800e+05 2.328306e-10 7 10 3.628800e+06 3.628800e+06 2.793968e-09 7 11 3.991680e+07 3.991680e+07 4.470348e-08 7 12 4.790016e+08 4.790016e+08 7.152557e-07 7 13 6.227021e+09 6.227021e+09 9.536743e-06 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 0.000000e+00 8 1 1.000000e+00 1.000000e+00 2.220446e-16 8 2 2.000000e+00 2.000000e+00 8.881784e-16 8 3 6.000000e+00 6.000000e+00 4.440892e-15 8 4 2.400000e+01 2.400000e+01 2.486900e-14 8 5 1.200000e+02 1.200000e+02 1.136868e-13 8 6 7.200000e+02 7.200000e+02 6.821210e-13 8 7 5.040000e+03 5.040000e+03 3.637979e-12 8 8 4.032000e+04 4.032000e+04 2.910383e-11 8 9 3.628800e+05 3.628800e+05 3.492460e-10 8 10 3.628800e+06 3.628800e+06 3.725290e-09 8 11 3.991680e+07 3.991680e+07 5.960464e-08 8 12 4.790016e+08 4.790016e+08 8.940697e-07 8 13 6.227021e+09 6.227021e+09 1.335144e-05 8 14 8.717829e+10 8.717829e+10 2.136230e-04 8 15 1.307674e+12 1.307674e+12 3.417969e-03 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 6.661338e-16 9 1 1.000000e+00 1.000000e+00 6.661338e-16 9 2 2.000000e+00 2.000000e+00 1.776357e-15 9 3 6.000000e+00 6.000000e+00 5.329071e-15 9 4 2.400000e+01 2.400000e+01 2.131628e-14 9 5 1.200000e+02 1.200000e+02 7.105427e-14 9 6 7.200000e+02 7.200000e+02 2.273737e-13 9 7 5.040000e+03 5.040000e+03 0.000000e+00 9 8 4.032000e+04 4.032000e+04 0.000000e+00 9 9 3.628800e+05 3.628800e+05 0.000000e+00 9 10 3.628800e+06 3.628800e+06 0.000000e+00 9 11 3.991680e+07 3.991680e+07 0.000000e+00 9 12 4.790016e+08 4.790016e+08 0.000000e+00 9 13 6.227021e+09 6.227021e+09 0.000000e+00 9 14 8.717829e+10 8.717829e+10 3.051758e-05 9 15 1.307674e+12 1.307674e+12 9.765625e-04 9 16 2.092279e+13 2.092279e+13 2.343750e-02 9 17 3.556874e+14 3.556874e+14 5.000000e-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 4.440892e-16 10 1 1.000000e+00 1.000000e+00 8.881784e-16 10 2 2.000000e+00 2.000000e+00 2.220446e-15 10 3 6.000000e+00 6.000000e+00 8.881784e-15 10 4 2.400000e+01 2.400000e+01 3.552714e-14 10 5 1.200000e+02 1.200000e+02 1.421085e-13 10 6 7.200000e+02 7.200000e+02 4.547474e-13 10 7 5.040000e+03 5.040000e+03 1.818989e-12 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 3.725290e-08 10 12 4.790016e+08 4.790016e+08 7.152557e-07 10 13 6.227021e+09 6.227021e+09 1.335144e-05 10 14 8.717829e+10 8.717829e+10 2.136230e-04 10 15 1.307674e+12 1.307674e+12 3.662109e-03 10 16 2.092279e+13 2.092279e+13 5.859375e-02 10 17 3.556874e+14 3.556874e+14 8.750000e-01 10 18 6.402374e+15 6.402374e+15 1.400000e+01 10 19 1.216451e+17 1.216451e+17 1.920000e+02 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 SANDIA_RULES_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 1 1.0000000000000000 1.0000000000000000 Order = 2 1 0.5857864376269051 0.8535533905932737 2 3.4142135623730949 0.1464466094067262 Order = 3 1 0.4157745567834789 0.7110930099291729 2 2.2942803602790414 0.2785177335692409 3 6.2899450829374803 0.0103892565015861 Order = 4 1 0.3225476896193923 0.6031541043416337 2 1.7457611011583465 0.3574186924377999 3 4.5366202969211278 0.0388879085150054 4 9.3950709123011364 0.0005392947055613 Order = 5 1 0.2635603197181410 0.5217556105828089 2 1.4134030591065174 0.3986668110831760 3 3.5964257710407215 0.0759424496817075 4 7.0858100058588347 0.0036117586799220 5 12.6408008442757840 0.0000233699723858 Order = 6 1 0.2228466041792610 0.4589646739499636 2 1.1889321016726233 0.4170008307721207 3 2.9927363260593141 0.1133733820740452 4 5.7751435691045119 0.0103991974531491 5 9.8374674183825874 0.0002610172028149 6 15.9828739806017026 0.0000008985479064 Order = 7 1 0.1930436765603621 0.4093189517012737 2 1.0266648953391915 0.4218312778617199 3 2.5678767449507451 0.1471263486575055 4 4.9003530845264844 0.0206335144687169 5 8.1821534445628590 0.0010740101432807 6 12.7341802917978146 0.0000158654643486 7 19.3957278622625431 0.0000000317031548 Order = 8 1 0.1702796323051008 0.3691885893416376 2 0.9037017767993794 0.4187867808143426 3 2.2510866298661294 0.1757949866371721 4 4.2667001702876570 0.0333434922612156 5 7.0459054023934637 0.0027945362352257 6 10.7585160101809922 0.0000907650877336 7 15.7406786412780040 0.0000008485746716 8 22.8631317368892653 0.0000000010480012 Order = 9 1 0.1523222277318080 0.3361264217979629 2 0.8072200227422562 0.4112139804239849 3 2.0051351556193495 0.1992875253708856 4 3.7834739733312355 0.0474605627656515 5 6.2049567778766113 0.0055996266107946 6 9.3729852516875738 0.0003052497670932 7 13.4662369110920928 0.0000065921230261 8 18.8335977889916890 0.0000000411076933 9 26.3740718909273753 0.0000000000329087 Order = 10 1 0.1377934705404923 0.3084411157650204 2 0.7294545495031706 0.4011199291552736 3 1.8083429017403159 0.2180682876118095 4 3.4014336978548978 0.0620874560986779 5 5.5524961400638011 0.0095015169751812 6 8.3301527467644956 0.0007530083885875 7 11.8437858379000680 0.0000282592334960 8 16.2792578313781071 0.0000004249313985 9 21.9965858119807578 0.0000000018395648 10 29.9206970122738909 0.0000000000009912 SANDIA_RULES_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 4.440892e-16 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 0.666667 0.666667 3.330669e-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 2.043119e-16 3 2 0.666667 0.666667 3.330669e-16 3 3 -0.000000 0.000000 2.220446e-16 3 4 0.400000 0.400000 3.885781e-16 3 5 -0.000000 0.000000 1.665335e-16 3 6 0.240000 0.285714 4.571429e-02 3 7 -0.000000 0.000000 1.249001e-16 3 8 0.144000 0.222222 7.822222e-02 4 0 2.000000 2.000000 1.332268e-15 4 1 -0.000000 0.000000 7.771561e-16 4 2 0.666667 0.666667 8.881784e-16 4 3 -0.000000 0.000000 4.440892e-16 4 4 0.400000 0.400000 6.661338e-16 4 5 -0.000000 0.000000 3.608225e-16 4 6 0.285714 0.285714 5.551115e-16 4 7 -0.000000 0.000000 3.053113e-16 4 8 0.210612 0.222222 1.160998e-02 4 9 -0.000000 0.000000 2.359224e-16 4 10 0.156035 0.181818 2.578320e-02 5 0 2.000000 2.000000 1.332268e-15 5 1 0.000000 0.000000 2.172126e-17 5 2 0.666667 0.666667 4.440892e-16 5 3 -0.000000 0.000000 1.387779e-16 5 4 0.400000 0.400000 1.110223e-16 5 5 -0.000000 0.000000 2.081668e-16 5 6 0.285714 0.285714 5.551115e-17 5 7 -0.000000 0.000000 2.506675e-16 5 8 0.222222 0.222222 2.775558e-17 5 9 -0.000000 0.000000 2.450297e-16 5 10 0.178886 0.181818 2.931812e-03 5 11 -0.000000 0.000000 2.620517e-16 5 12 0.145853 0.153846 7.993574e-03 6 0 2.000000 2.000000 6.661338e-16 6 1 0.000000 0.000000 2.636780e-16 6 2 0.666667 0.666667 5.551115e-16 6 3 0.000000 0.000000 2.012279e-16 6 4 0.400000 0.400000 5.551115e-16 6 5 0.000000 0.000000 1.942890e-16 6 6 0.285714 0.285714 6.661338e-16 6 7 0.000000 0.000000 1.804112e-16 6 8 0.222222 0.222222 6.106227e-16 6 9 0.000000 0.000000 1.665335e-16 6 10 0.181818 0.181818 5.551115e-16 6 11 0.000000 0.000000 1.526557e-16 6 12 0.153108 0.153846 7.380787e-04 6 13 0.000000 0.000000 1.387779e-16 6 14 0.130949 0.133333 2.384218e-03 7 0 2.000000 2.000000 1.776357e-15 7 1 0.000000 0.000000 9.002180e-16 7 2 0.666667 0.666667 3.330669e-16 7 3 0.000000 0.000000 4.406198e-16 7 4 0.400000 0.400000 0.000000e+00 7 5 0.000000 0.000000 2.324529e-16 7 6 0.285714 0.285714 0.000000e+00 7 7 0.000000 0.000000 7.990570e-17 7 8 0.222222 0.222222 1.110223e-16 7 9 0.000000 0.000000 7.494548e-18 7 10 0.181818 0.181818 1.665335e-16 7 11 -0.000000 0.000000 2.581079e-17 7 12 0.153846 0.153846 1.942890e-16 7 13 -0.000000 0.000000 7.019913e-17 7 14 0.133148 0.133333 1.854659e-04 7 15 -0.000000 0.000000 7.529985e-17 7 16 0.116955 0.117647 6.923502e-04 8 0 2.000000 2.000000 1.332268e-15 8 1 -0.000000 0.000000 5.551115e-16 8 2 0.666667 0.666667 5.551115e-16 8 3 -0.000000 0.000000 2.775558e-16 8 4 0.400000 0.400000 4.440892e-16 8 5 -0.000000 0.000000 2.220446e-16 8 6 0.285714 0.285714 2.220446e-16 8 7 -0.000000 0.000000 2.289835e-16 8 8 0.222222 0.222222 1.387779e-16 8 9 -0.000000 0.000000 2.081668e-16 8 10 0.181818 0.181818 5.551115e-17 8 11 -0.000000 0.000000 1.942890e-16 8 12 0.153846 0.153846 0.000000e+00 8 13 -0.000000 0.000000 1.873501e-16 8 14 0.133333 0.133333 2.775558e-17 8 15 -0.000000 0.000000 1.804112e-16 8 16 0.117601 0.117647 4.654831e-05 8 17 -0.000000 0.000000 1.665335e-16 8 18 0.105066 0.105263 1.971362e-04 9 0 2.000000 2.000000 2.664535e-15 9 1 0.000000 0.000000 3.053113e-16 9 2 0.666667 0.666667 1.332268e-15 9 3 0.000000 0.000000 2.914335e-16 9 4 0.400000 0.400000 7.771561e-16 9 5 0.000000 0.000000 1.387779e-17 9 6 0.285714 0.285714 4.440892e-16 9 7 -0.000000 0.000000 1.457168e-16 9 8 0.222222 0.222222 2.498002e-16 9 9 -0.000000 0.000000 2.363561e-16 9 10 0.181818 0.181818 1.110223e-16 9 11 -0.000000 0.000000 2.834104e-16 9 12 0.153846 0.153846 0.000000e+00 9 13 -0.000000 0.000000 2.975051e-16 9 14 0.133333 0.133333 8.326673e-17 9 15 -0.000000 0.000000 3.002698e-16 9 16 0.117647 0.117647 1.387779e-16 9 17 -0.000000 0.000000 3.021807e-16 9 18 0.105251 0.105263 1.167311e-05 9 19 -0.000000 0.000000 2.995312e-16 9 20 0.095183 0.095238 5.529194e-05 10 0 2.000000 2.000000 1.332268e-15 10 1 0.000000 0.000000 2.775558e-16 10 2 0.666667 0.666667 2.220446e-16 10 3 0.000000 0.000000 4.267420e-16 10 4 0.400000 0.400000 2.775558e-16 10 5 0.000000 0.000000 4.041906e-16 10 6 0.285714 0.285714 2.220446e-16 10 7 0.000000 0.000000 3.053113e-16 10 8 0.222222 0.222222 1.942890e-16 10 9 0.000000 0.000000 2.255141e-16 10 10 0.181818 0.181818 1.110223e-16 10 11 0.000000 0.000000 1.491862e-16 10 12 0.153846 0.153846 5.551115e-17 10 13 0.000000 0.000000 9.020562e-17 10 14 0.133333 0.133333 5.551115e-17 10 15 0.000000 0.000000 5.551115e-17 10 16 0.117647 0.117647 0.000000e+00 10 17 0.000000 0.000000 2.428613e-17 10 18 0.105263 0.105263 1.387779e-17 10 19 -0.000000 0.000000 3.469447e-18 10 20 0.095235 0.095238 2.925590e-06 10 21 -0.000000 0.000000 2.081668e-17 10 22 0.086941 0.086957 1.532420e-05 SANDIA_RULES_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 1 0.0000000000000000 2.0000000000000004 Order = 2 1 -0.5773502691896256 1.0000000000000000 2 0.5773502691896256 1.0000000000000000 Order = 3 1 -0.7745966692414833 0.5555555555555556 2 0.0000000000000000 0.8888888888888895 3 0.7745966692414832 0.5555555555555554 Order = 4 1 -0.8611363115940527 0.3478548451374547 2 -0.3399810435848563 0.6521451548625466 3 0.3399810435848563 0.6521451548625458 4 0.8611363115940526 0.3478548451374541 Order = 5 1 -0.9061798459386641 0.2369268850561892 2 -0.5384693101056830 0.4786286704993669 3 -0.0000000000000001 0.5688888888888890 4 0.5384693101056831 0.4786286704993672 5 0.9061798459386639 0.2369268850561891 Order = 6 1 -0.9324695142031522 0.1713244923791705 2 -0.6612093864662647 0.3607615730481384 3 -0.2386191860831970 0.4679139345726904 4 0.2386191860831969 0.4679139345726910 5 0.6612093864662647 0.3607615730481382 6 0.9324695142031522 0.1713244923791708 Order = 7 1 -0.9491079123427585 0.1294849661688697 2 -0.7415311855993943 0.2797053914892765 3 -0.4058451513773971 0.3818300505051193 4 0.0000000000000003 0.4179591836734696 5 0.4058451513773971 0.3818300505051192 6 0.7415311855993943 0.2797053914892776 7 0.9491079123427584 0.1294849661688697 Order = 8 1 -0.9602898564975365 0.1012285362903760 2 -0.7966664774136270 0.2223810344533743 3 -0.5255324099163290 0.3137066458778873 4 -0.1834346424956498 0.3626837833783622 5 0.1834346424956496 0.3626837833783619 6 0.5255324099163292 0.3137066458778869 7 0.7966664774136268 0.2223810344533742 8 0.9602898564975364 0.1012285362903759 Order = 9 1 -0.9681602395076260 0.0812743883615746 2 -0.8360311073266360 0.1806481606948576 3 -0.6133714327005900 0.2606106964029357 4 -0.3242534234038094 0.3123470770400033 5 -0.0000000000000004 0.3302393550012602 6 0.3242534234038092 0.3123470770400024 7 0.6133714327005907 0.2606106964029365 8 0.8360311073266359 0.1806481606948582 9 0.9681602395076259 0.0812743883615742 Order = 10 1 -0.9739065285171715 0.0666713443086885 2 -0.8650633666889844 0.1494513491505800 3 -0.6794095682990244 0.2190863625159823 4 -0.4333953941292472 0.2692667193099961 5 -0.1488743389816311 0.2955242247147523 6 0.1488743389816314 0.2955242247147531 7 0.4333953941292472 0.2692667193099947 8 0.6794095682990243 0.2190863625159819 9 0.8650633666889843 0.1494513491505811 10 0.9739065285171714 0.0666713443086885 SANDIA_RULES_TEST23 LEVEL_GROWTH_TO_ORDER uses level, rule and growth information to determine the order of a 1D rule. We are examining 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 SANDIA_RULES_TEST23 LEVEL_GROWTH_TO_ORDER uses level, rule and growth information to determine the order of a 1D rule. We are examining 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 SANDIA_RULES_TEST23 LEVEL_GROWTH_TO_ORDER uses level, rule and growth information to determine the order of a 1D rule. We are examining 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 SANDIA_RULES_TEST23 LEVEL_GROWTH_TO_ORDER uses level, rule and growth information to determine the order of a 1D rule. We are examining rule 11 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 SANDIA_RULES_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 14 1 5 9 17 17 33 33 33 33 65 65 15 1 7 15 15 31 31 31 31 63 63 63 16 1 3 7 15 15 15 31 31 31 31 31 SANDIA_RULES_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 SANDIA_RULES_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 through 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 SANDIA_RULES_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 SANDIA_RULES_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 0.000000e+00 7 2 0.666667 0.666667 0.000000e+00 7 3 0.000000 0.000000 0.000000e+00 7 4 0.400000 0.400000 0.000000e+00 7 5 -0.000000 0.000000 2.602085e-18 7 6 0.285714 0.285714 0.000000e+00 7 7 0.000000 0.000000 2.168404e-19 7 8 0.222222 0.222222 0.000000e+00 7 9 0.000000 0.000000 3.876023e-18 7 10 0.181818 0.181818 2.775558e-17 7 11 -0.000000 0.000000 6.430674e-18 7 12 0.154127 0.153846 2.806521e-04 7 13 0.000000 0.000000 2.046432e-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 0.000000e+00 15 2 0.666667 0.666667 1.110223e-16 15 3 0.000000 0.000000 0.000000e+00 15 4 0.400000 0.400000 0.000000e+00 15 5 0.000000 0.000000 5.637851e-18 15 6 0.285714 0.285714 0.000000e+00 15 7 0.000000 0.000000 6.071532e-18 15 8 0.222222 0.222222 5.551115e-17 15 9 0.000000 0.000000 2.046432e-18 15 10 0.181818 0.181818 0.000000e+00 15 11 0.000000 0.000000 1.473837e-18 15 12 0.153846 0.153846 0.000000e+00 15 13 0.000000 0.000000 1.627150e-18 15 14 0.133333 0.133333 0.000000e+00 15 15 0.000000 0.000000 7.349070e-19 15 16 0.117647 0.117647 0.000000e+00 15 17 0.000000 0.000000 5.768030e-19 15 18 0.105263 0.105263 0.000000e+00 15 19 -0.000000 0.000000 5.992791e-19 15 20 0.095238 0.095238 0.000000e+00 15 21 0.000000 0.000000 4.985351e-19 15 22 0.086957 0.086957 0.000000e+00 15 23 0.000000 0.000000 1.549280e-18 15 24 0.080000 0.080000 5.394906e-09 15 25 -0.000000 0.000000 6.417947e-19 15 26 0.074074 0.074074 1.752466e-08 31 0 2.000000 2.000000 0.000000e+00 31 1 -0.000000 0.000000 1.387779e-17 31 2 0.666667 0.666667 1.110223e-16 31 3 0.000000 0.000000 6.938894e-18 31 4 0.400000 0.400000 0.000000e+00 31 5 0.000000 0.000000 3.469447e-18 31 6 0.285714 0.285714 0.000000e+00 31 7 0.000000 0.000000 3.469447e-18 31 8 0.222222 0.222222 5.551115e-17 31 9 0.000000 0.000000 0.000000e+00 31 10 0.181818 0.181818 0.000000e+00 31 11 -0.000000 0.000000 6.938894e-18 31 12 0.153846 0.153846 0.000000e+00 31 13 0.000000 0.000000 0.000000e+00 31 14 0.133333 0.133333 0.000000e+00 31 15 0.000000 0.000000 1.734723e-18 31 16 0.117647 0.117647 1.387779e-17 31 17 0.000000 0.000000 1.734723e-18 31 18 0.105263 0.105263 0.000000e+00 31 19 0.000000 0.000000 1.734723e-18 31 20 0.095238 0.095238 0.000000e+00 31 21 0.000000 0.000000 0.000000e+00 31 22 0.086957 0.086957 1.387779e-17 31 23 0.000000 0.000000 4.336809e-19 31 24 0.080000 0.080000 0.000000e+00 31 25 0.000000 0.000000 3.469447e-18 31 26 0.074074 0.074074 0.000000e+00 31 27 -0.000000 0.000000 2.385245e-18 31 28 0.068966 0.068966 1.387779e-17 31 29 0.000000 0.000000 6.505213e-19 31 30 0.064516 0.064516 0.000000e+00 31 31 -0.000000 0.000000 8.673617e-19 31 32 0.060606 0.060606 1.387779e-17 31 33 -0.000000 0.000000 2.276825e-18 31 34 0.057143 0.057143 6.938894e-18 31 35 0.000000 0.000000 1.084202e-19 31 36 0.054054 0.054054 1.387779e-17 31 37 0.000000 0.000000 1.734723e-18 31 38 0.051282 0.051282 6.938894e-18 31 39 -0.000000 0.000000 1.084202e-19 31 40 0.048780 0.048780 1.387779e-17 31 41 0.000000 0.000000 3.252607e-19 31 42 0.046512 0.046512 1.387779e-17 31 43 -0.000000 0.000000 6.505213e-19 31 44 0.044444 0.044444 1.387779e-17 31 45 0.000000 0.000000 9.486769e-19 31 46 0.042553 0.042553 1.387779e-17 31 47 0.000000 0.000000 6.505213e-19 31 48 0.040816 0.040816 6.938894e-18 31 49 0.000000 0.000000 3.252607e-19 31 50 0.039216 0.039216 6.938894e-18 63 0 2.000000 2.000000 0.000000e+00 63 1 -0.000000 0.000000 6.505213e-18 63 2 0.666667 0.666667 1.110223e-16 63 3 0.000000 0.000000 2.602085e-18 63 4 0.400000 0.400000 0.000000e+00 63 5 0.000000 0.000000 5.421011e-18 63 6 0.285714 0.285714 5.551115e-17 63 7 -0.000000 0.000000 7.589415e-18 63 8 0.222222 0.222222 0.000000e+00 63 9 -0.000000 0.000000 1.951564e-18 63 10 0.181818 0.181818 0.000000e+00 63 11 -0.000000 0.000000 1.301043e-18 63 12 0.153846 0.153846 0.000000e+00 63 13 -0.000000 0.000000 4.336809e-19 63 14 0.133333 0.133333 2.775558e-17 63 15 0.000000 0.000000 8.673617e-19 63 16 0.117647 0.117647 0.000000e+00 63 17 0.000000 0.000000 0.000000e+00 63 18 0.105263 0.105263 2.775558e-17 63 19 0.000000 0.000000 8.673617e-19 63 20 0.095238 0.095238 1.387779e-17 63 21 0.000000 0.000000 1.734723e-18 63 22 0.086957 0.086957 1.387779e-17 63 23 -0.000000 0.000000 8.673617e-19 63 24 0.080000 0.080000 1.387779e-17 63 25 -0.000000 0.000000 4.336809e-19 63 26 0.074074 0.074074 0.000000e+00 63 27 0.000000 0.000000 8.673617e-19 63 28 0.068966 0.068966 0.000000e+00 63 29 0.000000 0.000000 2.168404e-18 63 30 0.064516 0.064516 0.000000e+00 63 31 -0.000000 0.000000 8.673617e-19 63 32 0.060606 0.060606 6.938894e-18 63 33 -0.000000 0.000000 4.336809e-19 63 34 0.057143 0.057143 1.387779e-17 63 35 0.000000 0.000000 1.734723e-18 63 36 0.054054 0.054054 1.387779e-17 63 37 0.000000 0.000000 8.673617e-19 63 38 0.051282 0.051282 1.387779e-17 63 39 -0.000000 0.000000 1.734723e-18 63 40 0.048780 0.048780 1.387779e-17 63 41 0.000000 0.000000 0.000000e+00 63 42 0.046512 0.046512 1.387779e-17 63 43 0.000000 0.000000 0.000000e+00 63 44 0.044444 0.044444 2.081668e-17 63 45 -0.000000 0.000000 4.336809e-19 63 46 0.042553 0.042553 6.938894e-18 63 47 0.000000 0.000000 4.336809e-19 63 48 0.040816 0.040816 1.387779e-17 63 49 0.000000 0.000000 0.000000e+00 63 50 0.039216 0.039216 1.387779e-17 63 51 -0.000000 0.000000 8.673617e-19 63 52 0.037736 0.037736 1.387779e-17 63 53 0.000000 0.000000 4.336809e-19 63 54 0.036364 0.036364 1.387779e-17 63 55 0.000000 0.000000 1.301043e-18 63 56 0.035088 0.035088 1.387779e-17 63 57 0.000000 0.000000 4.336809e-19 63 58 0.033898 0.033898 1.387779e-17 63 59 0.000000 0.000000 0.000000e+00 63 60 0.032787 0.032787 2.081668e-17 63 61 0.000000 0.000000 0.000000e+00 63 62 0.031746 0.031746 2.081668e-17 63 63 -0.000000 0.000000 4.336809e-19 63 64 0.030769 0.030769 1.734723e-17 63 65 -0.000000 0.000000 8.673617e-19 63 66 0.029851 0.029851 1.734723e-17 63 67 0.000000 0.000000 0.000000e+00 63 68 0.028986 0.028986 1.734723e-17 63 69 0.000000 0.000000 0.000000e+00 63 70 0.028169 0.028169 1.734723e-17 63 71 0.000000 0.000000 4.336809e-19 63 72 0.027397 0.027397 1.387779e-17 63 73 0.000000 0.000000 0.000000e+00 63 74 0.026667 0.026667 1.734723e-17 63 75 -0.000000 0.000000 4.336809e-19 63 76 0.025974 0.025974 2.081668e-17 63 77 0.000000 0.000000 0.000000e+00 63 78 0.025316 0.025316 1.734723e-17 63 79 0.000000 0.000000 4.336809e-19 63 80 0.024691 0.024691 1.734723e-17 63 81 0.000000 0.000000 0.000000e+00 63 82 0.024096 0.024096 1.734723e-17 63 83 0.000000 0.000000 0.000000e+00 63 84 0.023529 0.023529 2.081668e-17 63 85 0.000000 0.000000 0.000000e+00 63 86 0.022989 0.022989 2.081668e-17 63 87 0.000000 0.000000 0.000000e+00 63 88 0.022472 0.022472 1.387779e-17 63 89 0.000000 0.000000 0.000000e+00 63 90 0.021978 0.021978 1.734723e-17 63 91 -0.000000 0.000000 4.336809e-19 63 92 0.021505 0.021505 1.734723e-17 63 93 0.000000 0.000000 0.000000e+00 63 94 0.021053 0.021053 1.734723e-17 63 95 0.000000 0.000000 0.000000e+00 63 96 0.020619 0.020619 1.734723e-17 63 97 0.000000 0.000000 0.000000e+00 63 98 0.020202 0.020202 1.734723e-17 SANDIA_RULES_TEST29 R8COL_TOL_UNDEX produces index vectors which create a sorted list of the tolerably unique columns of an (unsorted) R8COL, and a map from the original R8COL to the (implicit) R8COL of sorted 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 Tolerance for equality = 0.250000 Number of unique entries in X is 10 XDNU points to the representative for each item. UNDX selects the representatives.. 1 4 11 2 7 12 3 10 19 4 3 1 5 9 10 6 6 6 7 4 14 8 4 9 9 8 15 10 5 3 11 1 12 2 13 4 14 7 15 9 16 4 17 4 18 7 19 3 20 4 21 3 22 3 The 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 SANDIA_RULES_TEST30 R8VEC_SORT_HEAP_INDEX_A creates an ascending sort index for a R8VEC. The unsorted array: 1 0.218418 2 0.956318 3 0.829509 4 0.561695 5 0.415307 6 0.066119 7 0.257578 8 0.109957 9 0.043829 10 0.633966 11 0.061727 12 0.449539 13 0.401306 14 0.754673 15 0.797287 16 0.001838 17 0.897504 18 0.350752 19 0.094545 20 0.013617 The index vector: 1: 16 2: 20 3: 9 4: 11 5: 6 6: 19 7: 8 8: 1 9: 7 10: 18 11: 13 12: 5 13: 12 14: 4 15: 10 16: 14 17: 15 18: 3 19: 17 20: 2 The sorted array A(INDX(:)): 1 0.001838 2 0.013617 3 0.043829 4 0.061727 5 0.066119 6 0.094545 7 0.109957 8 0.218418 9 0.257578 10 0.350752 11 0.401306 12 0.415307 13 0.449539 14 0.561695 15 0.633966 16 0.754673 17 0.797287 18 0.829509 19 0.897504 20 0.956318 SANDIA_RULES_TEST31: HCE_COMPUTE returns a quadrature rule for piecewise Hermite cubic splines which are based on equally spaced function and derivative data. Here we compute a rule of order N = 22 I X(I) W(I) 1 -1.000000 0.100000 2 -1.000000 0.003333 3 -0.800000 0.200000 4 -0.800000 0.000000 5 -0.600000 0.200000 6 -0.600000 0.000000 7 -0.400000 0.200000 8 -0.400000 -0.000000 9 -0.200000 0.200000 10 -0.200000 0.000000 11 0.000000 0.200000 12 0.000000 0.000000 13 0.200000 0.200000 14 0.200000 -0.000000 15 0.400000 0.200000 16 0.400000 0.000000 17 0.600000 0.200000 18 0.600000 0.000000 19 0.800000 0.200000 20 0.800000 0.000000 21 1.000000 0.100000 22 1.000000 -0.003333 SANDIA_RULES_TEST32: HCC_COMPUTE returns a quadrature rule for piecewise Hermite cubic splines which are based on Chebyshev-spaced function and derivative data. Here we compute a rule of order N = 22 I X(I) W(I) 1 -1.000000 0.024472 2 -1.000000 0.000200 3 -0.951057 0.095492 4 -0.951057 0.001482 5 -0.809017 0.181636 6 -0.809017 0.002397 7 -0.587785 0.250000 8 -0.587785 0.002397 9 -0.309017 0.293893 10 -0.309017 0.001482 11 0.000000 0.309017 12 0.000000 0.000000 13 0.309017 0.293893 14 0.309017 -0.001482 15 0.587785 0.250000 16 0.587785 -0.002397 17 0.809017 0.181636 18 0.809017 -0.002397 19 0.951057 0.095492 20 0.951057 -0.001482 21 0.000000 0.024472 22 0.000000 -0.000200 SANDIA_RULES_TEST33: HC_COMPUTE_WEIGHTS_FROM_POINTS returns quadrature weights for a Hermite cubic spline given the points. Random spacing Number of points N = 11 Interval = [0.218418,4.154423] Q = -6.753379 Q (exact) = -6.753379 SANDIA_RULES_TEST34 HERMITE_INTERPOLANT computes the Hermite interpolant to data. Here, f(x) is the Runge function and the data is evaluated at equally spaced points. As N increases, the maximum error grows. N Max | F(X) - H(F(X)) | nd = 6 3 0.530048 nd = 10 5 0.223403 nd = 14 7 0.390862 nd = 18 9 1.13792 nd = 22 11 3.83581 nd = 26 13 13.9642 nd = 30 15 53.4249 SANDIA_RULES_TEST35 HERMITE_INTERPOLANT computes the Hermite interpolant to data. Here, f(x) is the Runge function and the data is evaluated at Chebyshev spaced points. As N increases, the maximum error decreases. N Max | F(X) - H(F(X)) | 3 0.530048 5 0.325238 7 0.180448 9 0.0950617 11 0.0480707 13 0.0237723 15 0.0114442 TEST36: HERMITE_INTERPOLANT_RULE is given a set of N abscissas for a Hermite interpolant and returns N pairs of quadrature weights for function and derivative values at the abscissas. Observe behavior of quadrature weights for increasing N We are working in 0.000000 <= X <= 1.000000 I X W(F(X)) W(F'(X)) 1 0 -6.66667 -1.5 2 0.5 5.33333 -2.66667 3 1 2.33333 -0.333333 I X W(F(X)) W(F'(X)) 1 0 -951.678 -64.4722 2 0.25 -3269.21 -656.356 3 0.5 1361.6 -1009.6 4 0.75 2571.38 -333.511 5 1 288.915 -16.5056 I X W(F(X)) W(F'(X)) 1 0 -128378 -4699.38 2 0.166667 -1.66705e+06 -122484 3 0.333333 -3.20273e+06 -582226 4 0.5 1.03003e+06 -823715 5 0.666667 3.08218e+06 -382601 6 0.833333 848082 -52005.7 7 1 37868.7 -1254.21 I X W(F(X)) W(F'(X)) 1 0 -1.79706e+07 -435044 2 0.125 -5.10033e+08 -2.16079e+07 3 0.25 -2.89076e+09 -2.12806e+08 4 0.375 -4.09812e+09 -7.05643e+08 5 0.5 1.13487e+09 -9.37409e+08 6 0.625 4.30251e+09 -5.20585e+08 7 0.75 1.85366e+09 -1.14797e+08 8 0.875 2.20612e+08 -8.37644e+06 9 1 5.23433e+06 -118268 I X W(F(X)) W(F'(X)) 1 0 -2.60271e+09 -4.61959e+07 2 0.1 -1.2973e+11 -3.74681e+09 3 0.2 -1.43117e+12 -6.31919e+10 4 0.3 -5.24034e+12 -3.8283e+11 5 0.4 -6.11016e+12 -1.01787e+12 6 0.5 1.53665e+12 -1.29255e+12 7 0.6 6.73925e+12 -8.01827e+11 8 0.7 3.82127e+12 -2.36405e+11 9 0.8 7.64733e+11 -3.02886e+10 10 0.9 5.13565e+10 -1.37316e+09 11 1 7.52948e+08 -1.26888e+07 Use the rule with N = 11 to estimate integrals. Points are equally spaced. We are working in -5.000000 <= X <= 5.000000 I X W(F(X)) W(F'(X)) 1 -5 61.2405 10.2063 2 -4 4469.25 1177.08 3 -3 67539.6 26832.2 4 -2 253916 187109 5 -1 -31806.8 372881 6 0 -588350 3.95812e-09 7 1 -31806.8 -372881 8 2 253916 -187109 9 3 67539.6 -26832.2 10 4 4469.25 -1177.08 11 5 61.2405 -10.2063 Estimate integral of 1 = 1014.6 Estimate integral of X = -6.72418e-0814.6 Estimate integral of X^2 = 83.333314.6 Estimate integral of 1/(1+x^2) = -68509.514.6 Use the rule with N = 11 to estimate integrals. Points are Chebyshev spaced. We are working in -5.000000 <= X <= 5.000000 I X W(F(X)) W(F'(X)) 1 -5 114869 8510.62 2 -4.75528 -66202.1 34839.6 3 -4.04508 -14974.8 36848.8 4 -2.93893 -9014.04 37252 5 -1.54508 -14785.4 26848 6 0 -19775 -2.1937e-09 7 1.54508 -14785.4 -26848 8 2.93893 -9014.04 -37252 9 4.04508 -14974.8 -36848.8 10 4.75528 -66202.1 -34839.6 11 5 114869 -8510.62 Estimate integral of 1 = 1014.6 Estimate integral of X = 5.74728e-0714.6 Estimate integral of X^2 = 83.333314.6 Estimate integral of 1/(1+x^2) = -6276.9214.6 Use the rule with N = 11 to estimate integrals. Points are Legendre spaced. We are working in -5.000000 <= X <= 5.000000 I X W(F(X)) W(F'(X)) 1 -4.89114 7380.63 802.295 2 -4.43531 16678.5 9418.7 3 -3.65076 22800.7 32858.7 4 -2.59548 12958.6 64690.5 5 -1.34772 -29369 65052.6 6 0 -60888.7 6.97764e-09 7 1.34772 -29369 -65052.6 8 2.59548 12958.6 -64690.5 9 3.65076 22800.7 -32858.7 10 4.43531 16678.5 -9418.7 11 4.89114 7380.63 -802.295 Estimate integral of 1 = 1014.6 Estimate integral of X = -8.44634e-0814.6 Estimate integral of X^2 = 83.333314.6 Estimate integral of 1/(1+x^2) = -14818.514.6 Approximate integral of 1/(1+x^2) with increasing N. Points are Chebyshev spaced. We are working in -5.000000 <= X <= 5.000000 N Estimate Error 2 0.754438 1.99236 3 3.71302 0.966216 4 -0.0431064 2.78991 5 14.1017 11.3549 6 18.3363 15.5895 7 -73.5805 76.3273 8 -152.386 155.132 9 669.848 667.102 10 1482.59 1479.84 11 -6276.92 6279.66 Compute integral estimates for monomials X^0 through X^15. Use N = 5, 9, 13, 17, 21 point rules. Points are Chebyshev spaced. We are working in -1 <= X <= 1 Estimates are made using N = 5 F(X) Integral Estimate Error X^ 0 2 2 1.59872e-14 X^ 1 0 -2.02616e-14 2.02616e-14 X^ 2 0.666667 0.666667 2.4647e-14 X^ 3 0 -2.53131e-14 2.53131e-14 X^ 4 0.4 0.4 2.76446e-14 X^ 5 0 -2.70894e-14 2.70894e-14 X^ 6 0.285714 2.84217e-14 0.285714 X^ 7 0 -2.79776e-14 2.79776e-14 X^ 8 0.222222 2.88658e-14 0.222222 X^ 9 0 -2.88658e-14 2.88658e-14 X^ 10 0.181818 0.433333 0.251515 X^ 11 0 -3.06422e-14 3.06422e-14 X^ 12 0.153846 1.2 1.04615 X^ 13 0 -3.28626e-14 3.28626e-14 X^ 14 0.133333 2.19167 2.05833 X^ 15 0 -3.37508e-14 3.37508e-14 X^ 15 1.5708 1.67407 0.103278 Estimates are made using N = 9 F(X) Integral Estimate Error X^ 0 2 2 1.9736e-10 X^ 1 0 -1.26562e-10 1.26562e-10 X^ 2 0.666667 0.666667 9.05137e-11 X^ 3 0 -4.84022e-11 4.84022e-11 X^ 4 0.4 0.4 2.74895e-11 X^ 5 0 3.60956e-12 3.60956e-12 X^ 6 0.285714 0.285714 1.93349e-11 X^ 7 0 4.40252e-11 4.40252e-11 X^ 8 0.222222 0.222222 5.73487e-11 X^ 9 0 7.78471e-11 7.78471e-11 X^ 10 0.181818 -8.94147e-11 0.181818 X^ 11 0 1.07207e-10 1.07207e-10 X^ 12 0.153846 -1.17439e-10 0.153846 X^ 13 0 1.33241e-10 1.33241e-10 X^ 14 0.133333 -1.41938e-10 0.133333 X^ 15 0 1.56092e-10 1.56092e-10 X^ 15 1.5708 1.65263 0.081833 Estimates are made using N = 13 F(X) Integral Estimate Error X^ 0 2 2 4.87512e-06 X^ 1 0 4.75841e-06 4.75841e-06 X^ 2 0.666667 0.666662 4.67273e-06 X^ 3 0 4.58645e-06 4.58645e-06 X^ 4 0.4 0.399995 4.53048e-06 X^ 5 0 4.45541e-06 4.45541e-06 X^ 6 0.285714 0.28571 4.40483e-06 X^ 7 0 4.33392e-06 4.33392e-06 X^ 8 0.222222 0.222218 4.28563e-06 X^ 9 0 4.21644e-06 4.21644e-06 X^ 10 0.181818 0.181814 4.1714e-06 X^ 11 0 4.10356e-06 4.10356e-06 X^ 12 0.153846 0.153842 4.0633e-06 X^ 13 0 3.99714e-06 3.99714e-06 X^ 14 0.133333 -3.96241e-06 0.133337 X^ 15 0 3.89834e-06 3.89834e-06 X^ 15 1.5708 1.63535 0.0645579 Estimates are made using N = 17 F(X) Integral Estimate Error X^ 0 2 1.98227 0.017734 X^ 1 0 0.0180356 0.0180356 X^ 2 0.666667 0.649341 0.0173256 X^ 3 0 0.0173782 0.0173782 X^ 4 0.4 0.383488 0.0165125 X^ 5 0 0.0164767 0.0164767 X^ 6 0.285714 0.270158 0.0155562 X^ 7 0 0.0154844 0.0154844 X^ 8 0.222222 0.207667 0.0145549 X^ 9 0 0.0144685 0.0144685 X^ 10 0.181818 0.168265 0.0135532 X^ 11 0 0.0134627 0.0134627 X^ 12 0.153846 0.141271 0.0125755 X^ 13 0 0.0124876 0.0124876 X^ 14 0.133333 0.121696 0.0116373 X^ 15 0 0.0115569 0.0115569 X^ 15 1.5708 1.61447 0.0436754 Estimates are made using N = 21 F(X) Integral Estimate Error X^ 0 2 419.428 417.428 X^ 1 0 -414.724 414.724 X^ 2 0.666667 409.468 408.802 X^ 3 0 -408.313 408.313 X^ 4 0.4 403.991 403.591 X^ 5 0 -403.73 403.73 X^ 6 0.285714 399.72 399.434 X^ 7 0 -399.86 399.86 X^ 8 0.222222 396 395.778 X^ 9 0 -396.406 396.406 X^ 10 0.181818 392.655 392.473 X^ 11 0 -393.281 393.281 X^ 12 0.153846 389.634 389.48 X^ 13 0 -390.458 390.458 X^ 14 0.133333 386.918 386.785 X^ 15 0 -387.922 387.922 X^ 15 1.5708 213.153 211.582 sandia_rules_test37(): HERMITE_GENZ_KEISTER_LOOKUP_WEIGHTS looks up weights for Genz-Keister quadrature rules for the Hermite weight function. This test simply checks that, for each rule, the quadrature weights correctly sum to sqrt(pi). Index Order Sum of Weights 1 1 1.77245 2 3 1.77245 3 9 1.77245 4 19 1.77245 5 35 1.77245 6 37 1.77245 7 41 1.77245 8 43 1.77245 Correct sum: 1.77245 SANDIA_RULES_TEST38 HERMITE_INTERPOLANT sets up the Hermite interpolant. HERMITE_INTERPOLANT_VALUE evaluates it. Consider data for y=sin(x) at x=0,1,2,3,4. In the following table, there should be perfect agreement between F and H, and F' and H' at the data points X = 0, 1, 2, 3, and 4. In between, H and H' approximate F and F'. I X(I) F(X(I)) H(X(I)) F'(X(I)) H'(X(I)) 1 0 0 0 1 1 2 0.25 0.247404 0.247407 0.968912 0.968921 3 0.5 0.479426 0.479428 0.877583 0.877575 4 0.75 0.681639 0.681639 0.731689 0.731683 5 1 0.841471 0.841471 0.540302 0.540302 6 1.25 0.948985 0.948985 0.315322 0.315324 7 1.5 0.997495 0.997495 0.0707372 0.0707368 8 1.75 0.983986 0.983986 -0.178246 -0.178247 9 2 0.909297 0.909297 -0.416147 -0.416147 10 2.25 0.778073 0.778073 -0.628174 -0.628172 11 2.5 0.598472 0.598473 -0.801144 -0.801143 12 2.75 0.381661 0.381661 -0.924302 -0.924304 13 3 0.14112 0.14112 -0.989992 -0.989992 14 3.25 -0.108195 -0.108194 -0.99413 -0.994124 15 3.5 -0.350783 -0.350781 -0.936457 -0.936451 16 3.75 -0.571561 -0.571559 -0.820559 -0.820567 17 4 -0.756802 -0.756802 -0.653644 -0.653644 sandia_rules_test(): Normal end of execution. 03-Jun-2023 14:07:13