26 January 2012 04:06:25 PM SANDIA_RULES_PRB C++ version Test the SANDIA_RULES library. TEST01 CHEBYSHEV1_COMPUTE computes a Gauss-Chebyshev type 1 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) / sqrt ( 1 - x^2 ) dx. CHEBYSHEV1_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 3.14159 3.14159 0 1 1 0 0 0 1 2 0 1.5708 1.5708 1 3 0 0 0 1 4 0 1.1781 1.1781 2 0 3.14159 3.14159 0 2 1 0 0 0 2 2 1.5708 1.5708 0 2 3 2.22045e-16 0 2.22045e-16 2 4 0.785398 1.1781 0.392699 2 5 3.33067e-16 0 3.33067e-16 2 6 0.392699 0.981748 0.589049 3 0 3.14159 3.14159 0 3 1 0 0 0 3 2 1.5708 1.5708 2.22045e-16 3 3 0 0 0 3 4 1.1781 1.1781 4.44089e-16 3 5 0 0 0 3 6 0.883573 0.981748 0.0981748 3 7 0 0 0 3 8 0.66268 0.859029 0.19635 4 0 3.14159 3.14159 0 4 1 -1.11022e-16 0 1.11022e-16 4 2 1.5708 1.5708 0 4 3 1.11022e-16 0 1.11022e-16 4 4 1.1781 1.1781 0 4 5 0 0 0 4 6 0.981748 0.981748 1.11022e-16 4 7 0 0 0 4 8 0.834486 0.859029 0.0245437 4 9 0 0 0 4 10 0.711767 0.773126 0.0613592 5 0 3.14159 3.14159 0 5 1 1.11022e-16 0 1.11022e-16 5 2 1.5708 1.5708 0 5 3 1.11022e-16 0 1.11022e-16 5 4 1.1781 1.1781 4.44089e-16 5 5 5.55112e-17 0 5.55112e-17 5 6 0.981748 0.981748 3.33067e-16 5 7 5.55112e-17 0 5.55112e-17 5 8 0.859029 0.859029 5.55112e-16 5 9 0 0 0 5 10 0.76699 0.773126 0.00613592 5 11 0 0 0 5 12 0.690291 0.708699 0.0184078 6 0 3.14159 3.14159 4.44089e-16 6 1 2.22045e-16 0 2.22045e-16 6 2 1.5708 1.5708 0 6 3 2.77556e-16 0 2.77556e-16 6 4 1.1781 1.1781 0 6 5 3.33067e-16 0 3.33067e-16 6 6 0.981748 0.981748 0 6 7 4.44089e-16 0 4.44089e-16 6 8 0.859029 0.859029 1.11022e-16 6 9 4.44089e-16 0 4.44089e-16 6 10 0.773126 0.773126 0 6 11 5.55112e-16 0 5.55112e-16 6 12 0.707165 0.708699 0.00153398 6 13 5.55112e-16 0 5.55112e-16 6 14 0.652709 0.658078 0.00536893 7 0 3.14159 3.14159 0 7 1 1.11022e-16 0 1.11022e-16 7 2 1.5708 1.5708 2.22045e-16 7 3 1.11022e-16 0 1.11022e-16 7 4 1.1781 1.1781 0 7 5 1.11022e-16 0 1.11022e-16 7 6 0.981748 0.981748 3.33067e-16 7 7 -1.66533e-16 0 1.66533e-16 7 8 0.859029 0.859029 3.33067e-16 7 9 -1.66533e-16 0 1.66533e-16 7 10 0.773126 0.773126 5.55112e-16 7 11 -2.77556e-16 0 2.77556e-16 7 12 0.708699 0.708699 7.77156e-16 7 13 -3.33067e-16 0 3.33067e-16 7 14 0.657694 0.658078 0.000383495 7 15 -4.996e-16 0 4.996e-16 7 16 0.615414 0.616948 0.00153398 8 0 3.14159 3.14159 0 8 1 2.22045e-16 0 2.22045e-16 8 2 1.5708 1.5708 0 8 3 1.11022e-16 0 1.11022e-16 8 4 1.1781 1.1781 0 8 5 5.55112e-17 0 5.55112e-17 8 6 0.981748 0.981748 1.11022e-16 8 7 0 0 0 8 8 0.859029 0.859029 3.33067e-16 8 9 -5.55112e-17 0 5.55112e-17 8 10 0.773126 0.773126 1.11022e-16 8 11 -5.55112e-17 0 5.55112e-17 8 12 0.708699 0.708699 0 8 13 -5.55112e-17 0 5.55112e-17 8 14 0.658078 0.658078 1.11022e-16 8 15 -5.55112e-17 0 5.55112e-17 8 16 0.616852 0.616948 9.58738e-05 8 17 0 0 0 8 18 0.582242 0.582673 0.000431432 9 0 3.14159 3.14159 0 9 1 1.66533e-16 0 1.66533e-16 9 2 1.5708 1.5708 2.22045e-16 9 3 1.11022e-16 0 1.11022e-16 9 4 1.1781 1.1781 0 9 5 5.55112e-17 0 5.55112e-17 9 6 0.981748 0.981748 1.11022e-16 9 7 -5.55112e-17 0 5.55112e-17 9 8 0.859029 0.859029 1.11022e-16 9 9 0 0 0 9 10 0.773126 0.773126 0 9 11 -1.11022e-16 0 1.11022e-16 9 12 0.708699 0.708699 0 9 13 -2.22045e-16 0 2.22045e-16 9 14 0.658078 0.658078 1.11022e-16 9 15 -3.33067e-16 0 3.33067e-16 9 16 0.616948 0.616948 0 9 17 -3.33067e-16 0 3.33067e-16 9 18 0.582649 0.582673 2.39684e-05 9 19 -4.996e-16 0 4.996e-16 9 20 0.55342 0.553539 0.000119842 10 0 3.14159 3.14159 0 10 1 1.66533e-16 0 1.66533e-16 10 2 1.5708 1.5708 2.22045e-16 10 3 3.33067e-16 0 3.33067e-16 10 4 1.1781 1.1781 2.22045e-16 10 5 3.33067e-16 0 3.33067e-16 10 6 0.981748 0.981748 1.11022e-16 10 7 4.996e-16 0 4.996e-16 10 8 0.859029 0.859029 3.33067e-16 10 9 4.996e-16 0 4.996e-16 10 10 0.773126 0.773126 1.11022e-16 10 11 5.55112e-16 0 5.55112e-16 10 12 0.708699 0.708699 2.22045e-16 10 13 5.55112e-16 0 5.55112e-16 10 14 0.658078 0.658078 3.33067e-16 10 15 6.10623e-16 0 6.10623e-16 10 16 0.616948 0.616948 3.33067e-16 10 17 6.10623e-16 0 6.10623e-16 10 18 0.582673 0.582673 4.44089e-16 10 19 6.38378e-16 0 6.38378e-16 10 20 0.553533 0.553539 5.99211e-06 10 21 6.93889e-16 0 6.93889e-16 10 22 0.528346 0.528378 3.29566e-05 TEST02 CHEBYSHEV1_COMPUTE computes a Gauss-Chebyshev type 1 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) / sqrt(1-x^2) dx. Order = 1 0 0 3.141592653589793 Order = 2 0 -0.7071067811865475 1.570796326794897 1 0.7071067811865476 1.570796326794897 Order = 3 0 -0.8660254037844387 1.047197551196598 1 0 1.047197551196598 2 0.8660254037844387 1.047197551196598 Order = 4 0 -0.9238795325112867 0.7853981633974483 1 -0.3826834323650897 0.7853981633974483 2 0.3826834323650898 0.7853981633974483 3 0.9238795325112867 0.7853981633974483 Order = 5 0 -0.9510565162951535 0.6283185307179586 1 -0.587785252292473 0.6283185307179586 2 0 0.6283185307179586 3 0.5877852522924732 0.6283185307179586 4 0.9510565162951535 0.6283185307179586 Order = 6 0 -0.9659258262890682 0.5235987755982988 1 -0.7071067811865475 0.5235987755982988 2 -0.2588190451025206 0.5235987755982988 3 0.2588190451025207 0.5235987755982988 4 0.7071067811865476 0.5235987755982988 5 0.9659258262890683 0.5235987755982988 Order = 7 0 -0.9749279121818237 0.4487989505128276 1 -0.7818314824680295 0.4487989505128276 2 -0.4338837391175581 0.4487989505128276 3 0 0.4487989505128276 4 0.4338837391175582 0.4487989505128276 5 0.7818314824680298 0.4487989505128276 6 0.9749279121818236 0.4487989505128276 Order = 8 0 -0.9807852804032304 0.3926990816987241 1 -0.8314696123025453 0.3926990816987241 2 -0.555570233019602 0.3926990816987241 3 -0.1950903220161282 0.3926990816987241 4 0.1950903220161283 0.3926990816987241 5 0.5555702330196023 0.3926990816987241 6 0.8314696123025452 0.3926990816987241 7 0.9807852804032304 0.3926990816987241 Order = 9 0 -0.9848077530122081 0.3490658503988659 1 -0.8660254037844385 0.3490658503988659 2 -0.6427876096865393 0.3490658503988659 3 -0.3420201433256685 0.3490658503988659 4 0 0.3490658503988659 5 0.3420201433256688 0.3490658503988659 6 0.6427876096865394 0.3490658503988659 7 0.8660254037844387 0.3490658503988659 8 0.984807753012208 0.3490658503988659 Order = 10 0 -0.9876883405951377 0.3141592653589793 1 -0.8910065241883678 0.3141592653589793 2 -0.7071067811865475 0.3141592653589793 3 -0.4539904997395467 0.3141592653589793 4 -0.1564344650402306 0.3141592653589793 5 0.1564344650402309 0.3141592653589793 6 0.4539904997395468 0.3141592653589793 7 0.7071067811865476 0.3141592653589793 8 0.8910065241883679 0.3141592653589793 9 0.9876883405951378 0.3141592653589793 TEST03 CHEBYSHEV2_COMPUTE computes a Gauss-Chebyshev type 2 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) * sqrt ( 1 - x^2 ) dx. CHEBYSHEV2_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 1.5708 1.5708 0 1 1 0 0 0 1 2 0 0.392699 0.392699 1 3 0 0 0 1 4 0 0.19635 0.19635 2 0 1.5708 1.5708 0 2 1 0 0 0 2 2 0.392699 0.392699 5.55112e-17 2 3 1.249e-16 0 1.249e-16 2 4 0.0981748 0.19635 0.0981748 2 5 6.59195e-17 0 6.59195e-17 2 6 0.0245437 0.122718 0.0981748 3 0 1.5708 1.5708 0 3 1 -1.11022e-16 0 1.11022e-16 3 2 0.392699 0.392699 0 3 3 0 0 0 3 4 0.19635 0.19635 0 3 5 2.77556e-17 0 2.77556e-17 3 6 0.0981748 0.122718 0.0245437 3 7 2.77556e-17 0 2.77556e-17 3 8 0.0490874 0.0859029 0.0368155 4 0 1.5708 1.5708 2.22045e-16 4 1 -5.55112e-17 0 5.55112e-17 4 2 0.392699 0.392699 0 4 3 0 0 0 4 4 0.19635 0.19635 2.77556e-17 4 5 -1.38778e-17 0 1.38778e-17 4 6 0.122718 0.122718 1.38778e-17 4 7 0 0 0 4 8 0.079767 0.0859029 0.00613592 4 9 6.93889e-18 0 6.93889e-18 4 10 0.0521553 0.0644272 0.0122718 5 0 1.5708 1.5708 0 5 1 0 0 0 5 2 0.392699 0.392699 5.55112e-17 5 3 6.93889e-17 0 6.93889e-17 5 4 0.19635 0.19635 0 5 5 4.16334e-17 0 4.16334e-17 5 6 0.122718 0.122718 4.16334e-17 5 7 1.38778e-17 0 1.38778e-17 5 8 0.0859029 0.0859029 4.16334e-17 5 9 6.93889e-18 0 6.93889e-18 5 10 0.0628932 0.0644272 0.00153398 5 11 0 0 0 5 12 0.0467864 0.0506214 0.00383495 6 0 1.5708 1.5708 0 6 1 -2.77556e-17 0 2.77556e-17 6 2 0.392699 0.392699 0 6 3 1.38778e-17 0 1.38778e-17 6 4 0.19635 0.19635 2.77556e-17 6 5 3.46945e-17 0 3.46945e-17 6 6 0.122718 0.122718 0 6 7 2.77556e-17 0 2.77556e-17 6 8 0.0859029 0.0859029 1.38778e-17 6 9 2.08167e-17 0 2.08167e-17 6 10 0.0644272 0.0644272 1.38778e-17 6 11 2.08167e-17 0 2.08167e-17 6 12 0.0502379 0.0506214 0.000383495 6 13 2.08167e-17 0 2.08167e-17 6 14 0.0399794 0.0411299 0.00115049 7 0 1.5708 1.5708 0 7 1 -1.38778e-17 0 1.38778e-17 7 2 0.392699 0.392699 5.55112e-17 7 3 6.93889e-18 0 6.93889e-18 7 4 0.19635 0.19635 0 7 5 0 0 0 7 6 0.122718 0.122718 1.38778e-17 7 7 0 0 0 7 8 0.0859029 0.0859029 0 7 9 6.93889e-18 0 6.93889e-18 7 10 0.0644272 0.0644272 1.38778e-17 7 11 -3.46945e-18 0 3.46945e-18 7 12 0.0506214 0.0506214 6.93889e-18 7 13 -3.46945e-18 0 3.46945e-18 7 14 0.041034 0.0411299 9.58738e-05 7 15 -3.46945e-18 0 3.46945e-18 7 16 0.0339393 0.0342749 0.000335558 8 0 1.5708 1.5708 2.22045e-16 8 1 6.93889e-18 0 6.93889e-18 8 2 0.392699 0.392699 0 8 3 8.32667e-17 0 8.32667e-17 8 4 0.19635 0.19635 5.55112e-17 8 5 2.77556e-17 0 2.77556e-17 8 6 0.122718 0.122718 1.38778e-17 8 7 3.81639e-17 0 3.81639e-17 8 8 0.0859029 0.0859029 1.38778e-17 8 9 3.81639e-17 0 3.81639e-17 8 10 0.0644272 0.0644272 0 8 11 2.42861e-17 0 2.42861e-17 8 12 0.0506214 0.0506214 1.38778e-17 8 13 2.77556e-17 0 2.77556e-17 8 14 0.0411299 0.0411299 6.93889e-18 8 15 2.77556e-17 0 2.77556e-17 8 16 0.0342509 0.0342749 2.39684e-05 8 17 2.25514e-17 0 2.25514e-17 8 18 0.0290378 0.0291337 9.58738e-05 9 0 1.5708 1.5708 2.22045e-16 9 1 -6.59195e-17 0 6.59195e-17 9 2 0.392699 0.392699 0 9 3 1.73472e-17 0 1.73472e-17 9 4 0.19635 0.19635 2.77556e-17 9 5 0 0 0 9 6 0.122718 0.122718 1.38778e-17 9 7 0 0 0 9 8 0.0859029 0.0859029 0 9 9 0 0 0 9 10 0.0644272 0.0644272 1.38778e-17 9 11 -3.46945e-18 0 3.46945e-18 9 12 0.0506214 0.0506214 1.38778e-17 9 13 -8.67362e-18 0 8.67362e-18 9 14 0.0411299 0.0411299 2.77556e-17 9 15 -6.93889e-18 0 6.93889e-18 9 16 0.0342749 0.0342749 2.08167e-17 9 17 -8.67362e-18 0 8.67362e-18 9 18 0.0291277 0.0291337 5.99211e-06 9 19 -5.20417e-18 0 5.20417e-18 9 20 0.0251339 0.0251609 2.69645e-05 10 0 1.5708 1.5708 2.22045e-16 10 1 -6.59195e-17 0 6.59195e-17 10 2 0.392699 0.392699 5.55112e-17 10 3 2.42861e-17 0 2.42861e-17 10 4 0.19635 0.19635 5.55112e-17 10 5 1.73472e-17 0 1.73472e-17 10 6 0.122718 0.122718 0 10 7 1.73472e-17 0 1.73472e-17 10 8 0.0859029 0.0859029 1.38778e-17 10 9 1.56125e-17 0 1.56125e-17 10 10 0.0644272 0.0644272 0 10 11 1.56125e-17 0 1.56125e-17 10 12 0.0506214 0.0506214 6.93889e-18 10 13 1.38778e-17 0 1.38778e-17 10 14 0.0411299 0.0411299 2.08167e-17 10 15 8.67362e-18 0 8.67362e-18 10 16 0.0342749 0.0342749 1.38778e-17 10 17 6.93889e-18 0 6.93889e-18 10 18 0.0291337 0.0291337 2.08167e-17 10 19 6.93889e-18 0 6.93889e-18 10 20 0.0251594 0.0251609 1.49803e-06 10 21 6.93889e-18 0 6.93889e-18 10 22 0.0220083 0.0220158 7.49014e-06 TEST04 CHEBYSHEV2_COMPUTE computes a Gauss-Chebyshev type 2 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) * sqrt(1-x^2) dx. Order = 1 0 0 1.570796326794897 Order = 2 0 -0.4999999999999998 0.7853981633974486 1 0.5000000000000001 0.7853981633974481 Order = 3 0 -0.7071067811865475 0.3926990816987243 1 0 0.7853981633974483 2 0.7071067811865476 0.392699081698724 Order = 4 0 -0.8090169943749473 0.2170787134227061 1 -0.3090169943749473 0.5683194499747424 2 0.3090169943749475 0.5683194499747423 3 0.8090169943749475 0.217078713422706 Order = 5 0 -0.8660254037844387 0.1308996938995747 1 -0.4999999999999998 0.3926990816987243 2 0 0.5235987755982988 3 0.5000000000000001 0.392699081698724 4 0.8660254037844387 0.1308996938995747 Order = 6 0 -0.900968867902419 0.08448869089158863 1 -0.6234898018587334 0.2743330560697779 2 -0.2225209339563143 0.4265764164360819 3 0.2225209339563144 0.4265764164360819 4 0.6234898018587336 0.2743330560697778 5 0.9009688679024191 0.08448869089158857 Order = 7 0 -0.9238795325112867 0.05750944903191315 1 -0.7071067811865475 0.1963495408493621 2 -0.3826834323650897 0.335189632666811 3 0 0.3926990816987241 4 0.3826834323650898 0.335189632666811 5 0.7071067811865476 0.196349540849362 6 0.9238795325112867 0.05750944903191313 Order = 8 0 -0.9396926207859083 0.04083294770910712 1 -0.7660444431189779 0.1442256007956728 2 -0.4999999999999998 0.2617993877991495 3 -0.1736481776669303 0.338540227093519 4 0.1736481776669304 0.338540227093519 5 0.5000000000000001 0.2617993877991494 6 0.7660444431189781 0.1442256007956727 7 0.9396926207859084 0.04083294770910708 Order = 9 0 -0.9510565162951535 0.02999954037160818 1 -0.8090169943749473 0.108539356711353 2 -0.587785252292473 0.2056199086476263 3 -0.3090169943749473 0.2841597249873712 4 0 0.3141592653589793 5 0.3090169943749475 0.2841597249873711 6 0.5877852522924732 0.2056199086476263 7 0.8090169943749475 0.108539356711353 8 0.9510565162951535 0.02999954037160816 Order = 10 0 -0.9594929736144974 0.02266894250185883 1 -0.8412535328311811 0.08347854093418908 2 -0.654860733945285 0.1631221774548167 3 -0.4154150130018863 0.2363135602034873 4 -0.142314838273285 0.2798149423030966 5 0.1423148382732851 0.2798149423030965 6 0.4154150130018864 0.2363135602034873 7 0.6548607339452852 0.1631221774548166 8 0.8412535328311812 0.08347854093418902 9 0.9594929736144974 0.02266894250185884 TEST05 CLENSHAW_CURTIS_COMPUTE computes a Clenshaw Curtis rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. LEGENDRE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = ORDER+1 if ORDER is odd, or N = ORDER if ORDER is even In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 2 2 0 1 1 0 0 0 1 2 0 0.666667 0.666667 1 3 0 0 0 1 4 0 0.4 0.4 2 0 2 2 0 2 1 0 0 0 2 2 2 0.666667 1.33333 2 3 0 0 0 2 4 2 0.4 1.6 3 0 2 2 0 3 1 0 0 0 3 2 0.666667 0.666667 1.11022e-16 3 3 0 0 0 3 4 0.666667 0.4 0.266667 3 5 0 0 0 3 6 0.666667 0.285714 0.380952 4 0 2 2 4.44089e-16 4 1 4.57967e-16 0 4.57967e-16 4 2 0.666667 0.666667 1.11022e-16 4 3 2.63678e-16 0 2.63678e-16 4 4 0.333333 0.4 0.0666667 4 5 9.71445e-17 0 9.71445e-17 4 6 0.25 0.285714 0.0357143 5 0 2 2 0 5 1 1.38778e-16 0 1.38778e-16 5 2 0.666667 0.666667 1.11022e-16 5 3 1.11022e-16 0 1.11022e-16 5 4 0.4 0.4 0 5 5 1.249e-16 0 1.249e-16 5 6 0.266667 0.285714 0.0190476 5 7 6.93889e-17 0 6.93889e-17 5 8 0.2 0.222222 0.0222222 6 0 2 2 0 6 1 1.38778e-16 0 1.38778e-16 6 2 0.666667 0.666667 1.11022e-16 6 3 1.38778e-16 0 1.38778e-16 6 4 0.4 0.4 0 6 5 8.32667e-17 0 8.32667e-17 6 6 0.283333 0.285714 0.00238095 6 7 6.93889e-17 0 6.93889e-17 6 8 0.2125 0.222222 0.00972222 7 0 2 2 0 7 1 3.57353e-16 0 3.57353e-16 7 2 0.666667 0.666667 1.11022e-16 7 3 1.9082e-16 0 1.9082e-16 7 4 0.4 0.4 1.11022e-16 7 5 1.21431e-16 0 1.21431e-16 7 6 0.285714 0.285714 1.11022e-16 7 7 5.20417e-17 0 5.20417e-17 7 8 0.221429 0.222222 0.000793651 7 9 2.42861e-17 0 2.42861e-17 7 10 0.178571 0.181818 0.00324675 8 0 2 2 0 8 1 1.49186e-16 0 1.49186e-16 8 2 0.666667 0.666667 1.11022e-16 8 3 1.49186e-16 0 1.49186e-16 8 4 0.4 0.4 0 8 5 1.07553e-16 0 1.07553e-16 8 6 0.285714 0.285714 5.55112e-17 8 7 9.36751e-17 0 9.36751e-17 8 8 0.222024 0.222222 0.000198413 8 9 9.36751e-17 0 9.36751e-17 8 10 0.181101 0.181818 0.000716991 9 0 2 2 0 9 1 2.56739e-16 0 2.56739e-16 9 2 0.666667 0.666667 1.11022e-16 9 3 1.8735e-16 0 1.8735e-16 9 4 0.4 0.4 5.55112e-17 9 5 1.45717e-16 0 1.45717e-16 9 6 0.285714 0.285714 5.55112e-17 9 7 1.17961e-16 0 1.17961e-16 9 8 0.222222 0.222222 5.55112e-17 9 9 7.63278e-17 0 7.63278e-17 9 10 0.181746 0.181818 7.21501e-05 9 11 6.93889e-17 0 6.93889e-17 9 12 0.153571 0.153846 0.000274725 10 0 2 2 0 10 1 3.95517e-16 0 3.95517e-16 10 2 0.666667 0.666667 1.11022e-16 10 3 2.56739e-16 0 2.56739e-16 10 4 0.4 0.4 5.55112e-17 10 5 2.56739e-16 0 2.56739e-16 10 6 0.285714 0.285714 0 10 7 1.59595e-16 0 1.59595e-16 10 8 0.222222 0.222222 2.77556e-17 10 9 1.73472e-16 0 1.73472e-16 10 10 0.181796 0.181818 2.25469e-05 10 11 1.249e-16 0 1.249e-16 10 12 0.153757 0.153846 8.87134e-05 TEST06 CLENSHAW_CURTIS_COMPUTE computes a Clenshaw Curtis rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. Order = 1 0 0 2 Order = 2 0 -1 1 1 1 1 Order = 3 0 -1 0.3333333333333334 1 0 1.333333333333333 2 1 0.3333333333333334 Order = 4 0 -1 0.1111111111111111 1 -0.4999999999999998 0.8888888888888888 2 0.5000000000000001 0.8888888888888892 3 1 0.1111111111111111 Order = 5 0 -1 0.06666666666666668 1 -0.7071067811865475 0.5333333333333333 2 0 0.7999999999999999 3 0.7071067811865476 0.5333333333333334 4 1 0.06666666666666668 Order = 6 0 -1 0.04000000000000001 1 -0.8090169943749473 0.3607430412000112 2 -0.3090169943749473 0.5992569587999889 3 0.3090169943749475 0.5992569587999887 4 0.8090169943749475 0.3607430412000113 5 1 0.04000000000000001 Order = 7 0 -1 0.02857142857142858 1 -0.8660254037844387 0.2539682539682539 2 -0.4999999999999998 0.4571428571428571 3 0 0.5206349206349206 4 0.5000000000000001 0.4571428571428573 5 0.8660254037844387 0.253968253968254 6 1 0.02857142857142858 Order = 8 0 -1 0.02040816326530613 1 -0.900968867902419 0.1901410072182084 2 -0.6234898018587334 0.3522424237181591 3 -0.2225209339563143 0.4372084057983264 4 0.2225209339563144 0.4372084057983264 5 0.6234898018587336 0.3522424237181591 6 0.9009688679024191 0.1901410072182084 7 1 0.02040816326530613 Order = 9 0 -1 0.01587301587301588 1 -0.9238795325112867 0.1462186492160181 2 -0.7071067811865475 0.2793650793650794 3 -0.3826834323650897 0.3617178587204897 4 0 0.3936507936507936 5 0.3826834323650898 0.3617178587204898 6 0.7071067811865476 0.2793650793650794 7 0.9238795325112867 0.1462186492160182 8 1 0.01587301587301588 Order = 10 0 -1 0.01234567901234569 1 -0.9396926207859083 0.1165674565720371 2 -0.7660444431189779 0.2252843233381044 3 -0.4999999999999998 0.3019400352733685 4 -0.1736481776669303 0.3438625058041442 5 0.1736481776669304 0.3438625058041442 6 0.5000000000000001 0.3019400352733687 7 0.7660444431189781 0.2252843233381044 8 0.9396926207859084 0.1165674565720372 9 1 0.01234567901234569 TEST07 FEJER2_COMPUTE computes a Fejer Type 2 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. LEGENDRE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = ORDER+1 if ORDER is odd, or N = ORDER if ORDER is even In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 2 2 0 1 1 0 0 0 1 2 0 0.666667 0.666667 1 3 0 0 0 1 4 0 0.4 0.4 2 0 2 2 0 2 1 3.33067e-16 0 3.33067e-16 2 2 0.5 0.666667 0.166667 2 3 2.498e-16 0 2.498e-16 2 4 0.125 0.4 0.275 3 0 2 2 0 3 1 0 0 0 3 2 0.666667 0.666667 1.11022e-16 3 3 8.32667e-17 0 8.32667e-17 3 4 0.333333 0.4 0.0666667 3 5 1.11022e-16 0 1.11022e-16 3 6 0.166667 0.285714 0.119048 4 0 2 2 0 4 1 1.11022e-16 0 1.11022e-16 4 2 0.666667 0.666667 1.11022e-16 4 3 0 0 0 4 4 0.375 0.4 0.025 4 5 0 0 0 4 6 0.239583 0.285714 0.046131 5 0 2 2 0 5 1 0 0 0 5 2 0.666667 0.666667 1.11022e-16 5 3 5.55112e-17 0 5.55112e-17 5 4 0.4 0.4 1.11022e-16 5 5 0 0 0 5 6 0.275 0.285714 0.0107143 5 7 -1.38778e-17 0 1.38778e-17 5 8 0.2 0.222222 0.0222222 6 0 2 2 2.22045e-16 6 1 1.66533e-16 0 1.66533e-16 6 2 0.666667 0.666667 0 6 3 1.66533e-16 0 1.66533e-16 6 4 0.4 0.4 1.11022e-16 6 5 1.11022e-16 0 1.11022e-16 6 6 0.28125 0.285714 0.00446429 6 7 1.11022e-16 0 1.11022e-16 6 8 0.211979 0.222222 0.0102431 7 0 2 2 0 7 1 -8.32667e-17 0 8.32667e-17 7 2 0.666667 0.666667 1.11022e-16 7 3 -8.32667e-17 0 8.32667e-17 7 4 0.4 0.4 0 7 5 -5.55112e-17 0 5.55112e-17 7 6 0.285714 0.285714 5.55112e-17 7 7 -4.16334e-17 0 4.16334e-17 7 8 0.220238 0.222222 0.00198413 7 9 -4.16334e-17 0 4.16334e-17 7 10 0.176786 0.181818 0.00503247 8 0 2 2 4.44089e-16 8 1 0 0 0 8 2 0.666667 0.666667 0 8 3 1.249e-16 0 1.249e-16 8 4 0.4 0.4 1.11022e-16 8 5 1.38778e-16 0 1.38778e-16 8 6 0.285714 0.285714 0 8 7 1.11022e-16 0 1.11022e-16 8 8 0.221354 0.222222 0.000868056 8 9 1.249e-16 0 1.249e-16 8 10 0.179408 0.181818 0.0024097 9 0 2 2 0 9 1 6.93889e-17 0 6.93889e-17 9 2 0.666667 0.666667 0 9 3 1.38778e-17 0 1.38778e-17 9 4 0.4 0.4 5.55112e-17 9 5 0 0 0 9 6 0.285714 0.285714 5.55112e-17 9 7 1.38778e-17 0 1.38778e-17 9 8 0.222222 0.222222 5.55112e-17 9 9 -1.38778e-17 0 1.38778e-17 9 10 0.181424 0.181818 0.000394571 9 11 -1.38778e-17 0 1.38778e-17 9 12 0.152654 0.153846 0.00119238 10 0 2 2 0 10 1 2.77556e-17 0 2.77556e-17 10 2 0.666667 0.666667 1.11022e-16 10 3 1.38778e-16 0 1.38778e-16 10 4 0.4 0.4 1.11022e-16 10 5 6.93889e-17 0 6.93889e-17 10 6 0.285714 0.285714 0 10 7 6.93889e-17 0 6.93889e-17 10 8 0.222222 0.222222 0 10 9 2.77556e-17 0 2.77556e-17 10 10 0.181641 0.181818 0.000177557 10 11 4.16334e-17 0 4.16334e-17 10 12 0.153266 0.153846 0.000580095 TEST08 FEJER2_COMPUTE computes a Fejer Type 2 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. Order = 1 0 0 2 Order = 2 0 -0.4999999999999998 1 1 0.5000000000000001 1 Order = 3 0 -0.7071067811865475 0.6666666666666667 1 0 0.6666666666666666 2 0.7071067811865476 0.6666666666666666 Order = 4 0 -0.8090169943749473 0.4254644007500071 1 -0.3090169943749473 0.574535599249993 2 0.3090169943749475 0.574535599249993 3 0.8090169943749475 0.425464400750007 Order = 5 0 -0.8660254037844387 0.3111111111111111 1 -0.4999999999999998 0.4000000000000001 2 0 0.5777777777777777 3 0.5000000000000001 0.4 4 0.8660254037844387 0.3111111111111111 Order = 6 0 -0.900968867902419 0.2269152467244296 1 -0.6234898018587334 0.3267938603769863 2 -0.2225209339563143 0.4462908928985842 3 0.2225209339563144 0.4462908928985841 4 0.6234898018587336 0.3267938603769863 5 0.9009688679024191 0.2269152467244296 Order = 7 0 -0.9238795325112867 0.17796468096205 1 -0.7071067811865475 0.2476190476190477 2 -0.3826834323650897 0.3934638904665215 3 0 0.3619047619047619 4 0.3826834323650898 0.3934638904665215 5 0.7071067811865476 0.2476190476190476 6 0.9238795325112867 0.1779646809620499 Order = 8 0 -0.9396926207859083 0.1397697435050226 1 -0.7660444431189779 0.2063696457302284 2 -0.4999999999999998 0.3142857142857144 3 -0.1736481776669303 0.3395748964790348 4 0.1736481776669304 0.3395748964790348 5 0.5000000000000001 0.3142857142857143 6 0.7660444431189781 0.2063696457302284 7 0.9396926207859084 0.1397697435050225 Order = 9 0 -0.9510565162951535 0.1147810750857218 1 -0.8090169943749473 0.1654331942222276 2 -0.587785252292473 0.2737903534857068 3 -0.3090169943749473 0.2790112502222169 4 0 0.3339682539682539 5 0.3090169943749475 0.279011250222217 6 0.5877852522924732 0.2737903534857068 7 0.8090169943749475 0.1654331942222276 8 0.9510565162951535 0.1147810750857217 Order = 10 0 -0.9594929736144974 0.09441954173982806 1 -0.8412535328311811 0.1411354380109716 2 -0.654860733945285 0.2263866903636005 3 -0.4154150130018863 0.2530509772156453 4 -0.142314838273285 0.2850073526699546 5 0.1423148382732851 0.2850073526699544 6 0.4154150130018864 0.2530509772156453 7 0.6548607339452852 0.2263866903636005 8 0.8412535328311812 0.1411354380109716 9 0.9594929736144974 0.09441954173982806 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 1.5708 1.33227e-15 1 1 0.5 0 0 0 1 2 0.5 0 0.392699 0.392699 1 3 0.5 0 0 0 1 4 0.5 0 0.19635 0.19635 2 0 0.5 1.5708 1.5708 1.33227e-15 2 1 0.5 0 0 0 2 2 0.5 0.392699 0.392699 1.11022e-16 2 3 0.5 0 0 0 2 4 0.5 0.0981748 0.19635 0.0981748 2 5 0.5 0 0 0 2 6 0.5 0.0245437 0.122718 0.0981748 3 0 0.5 1.5708 1.5708 1.33227e-15 3 1 0.5 4.996e-16 0 4.996e-16 3 2 0.5 0.392699 0.392699 1.11022e-16 3 3 0.5 3.05311e-16 0 3.05311e-16 3 4 0.5 0.19635 0.19635 8.32667e-17 3 5 0.5 1.94289e-16 0 1.94289e-16 3 6 0.5 0.0981748 0.122718 0.0245437 3 7 0.5 1.04083e-16 0 1.04083e-16 3 8 0.5 0.0490874 0.0859029 0.0368155 4 0 0.5 1.5708 1.5708 1.11022e-15 4 1 0.5 -2.77556e-17 0 2.77556e-17 4 2 0.5 0.392699 0.392699 1.11022e-16 4 3 0.5 -1.38778e-17 0 1.38778e-17 4 4 0.5 0.19635 0.19635 5.55112e-17 4 5 0.5 0 0 0 4 6 0.5 0.122718 0.122718 1.38778e-17 4 7 0.5 0 0 0 4 8 0.5 0.079767 0.0859029 0.00613592 4 9 0.5 0 0 0 4 10 0.5 0.0521553 0.0644272 0.0122718 5 0 0.5 1.5708 1.5708 2.22045e-15 5 1 0.5 4.44089e-16 0 4.44089e-16 5 2 0.5 0.392699 0.392699 7.21645e-16 5 3 0.5 2.91434e-16 0 2.91434e-16 5 4 0.5 0.19635 0.19635 5.55112e-16 5 5 0.5 2.35922e-16 0 2.35922e-16 5 6 0.5 0.122718 0.122718 3.46945e-16 5 7 0.5 1.73472e-16 0 1.73472e-16 5 8 0.5 0.0859029 0.0859029 2.35922e-16 5 9 0.5 1.31839e-16 0 1.31839e-16 5 10 0.5 0.0628932 0.0644272 0.00153398 5 11 0.5 9.71445e-17 0 9.71445e-17 5 12 0.5 0.0467864 0.0506214 0.00383495 6 0 0.5 1.5708 1.5708 1.9984e-15 6 1 0.5 -1.38778e-17 0 1.38778e-17 6 2 0.5 0.392699 0.392699 6.10623e-16 6 3 0.5 1.38778e-17 0 1.38778e-17 6 4 0.5 0.19635 0.19635 4.16334e-16 6 5 0.5 6.93889e-18 0 6.93889e-18 6 6 0.5 0.122718 0.122718 2.498e-16 6 7 0.5 0 0 0 6 8 0.5 0.0859029 0.0859029 1.94289e-16 6 9 0.5 0 0 0 6 10 0.5 0.0644272 0.0644272 1.38778e-17 6 11 0.5 0 0 0 6 12 0.5 0.0502379 0.0506214 0.000383495 6 13 0.5 0 0 0 6 14 0.5 0.0399794 0.0411299 0.00115049 7 0 0.5 1.5708 1.5708 1.11022e-15 7 1 0.5 3.19189e-16 0 3.19189e-16 7 2 0.5 0.392699 0.392699 2.22045e-16 7 3 0.5 2.08167e-16 0 2.08167e-16 7 4 0.5 0.19635 0.19635 1.66533e-16 7 5 0.5 1.17961e-16 0 1.17961e-16 7 6 0.5 0.122718 0.122718 2.08167e-16 7 7 0.5 6.93889e-17 0 6.93889e-17 7 8 0.5 0.0859029 0.0859029 1.66533e-16 7 9 0.5 3.81639e-17 0 3.81639e-17 7 10 0.5 0.0644272 0.0644272 3.05311e-16 7 11 0.5 2.08167e-17 0 2.08167e-17 7 12 0.5 0.0506214 0.0506214 6.03684e-16 7 13 0.5 1.04083e-17 0 1.04083e-17 7 14 0.5 0.041034 0.0411299 9.58738e-05 7 15 0.5 6.93889e-18 0 6.93889e-18 7 16 0.5 0.0339393 0.0342749 0.000335558 8 0 0.5 1.5708 1.5708 6.66134e-16 8 1 0.5 5.6205e-16 0 5.6205e-16 8 2 0.5 0.392699 0.392699 5.55112e-17 8 3 0.5 4.996e-16 0 4.996e-16 8 4 0.5 0.19635 0.19635 1.11022e-16 8 5 0.5 4.4062e-16 0 4.4062e-16 8 6 0.5 0.122718 0.122718 1.66533e-16 8 7 0.5 3.85109e-16 0 3.85109e-16 8 8 0.5 0.0859029 0.0859029 1.52656e-16 8 9 0.5 3.43475e-16 0 3.43475e-16 8 10 0.5 0.0644272 0.0644272 3.19189e-16 8 11 0.5 3.08781e-16 0 3.08781e-16 8 12 0.5 0.0506214 0.0506214 6.17562e-16 8 13 0.5 2.67147e-16 0 2.67147e-16 8 14 0.5 0.0411299 0.0411299 8.32667e-17 8 15 0.5 2.32453e-16 0 2.32453e-16 8 16 0.5 0.0342509 0.0342749 2.39684e-05 8 17 0.5 2.11636e-16 0 2.11636e-16 8 18 0.5 0.0290378 0.0291337 9.58738e-05 9 0 0.5 1.5708 1.5708 4.44089e-16 9 1 0.5 -6.93889e-18 0 6.93889e-18 9 2 0.5 0.392699 0.392699 4.44089e-16 9 3 0.5 -3.46945e-18 0 3.46945e-18 9 4 0.5 0.19635 0.19635 3.33067e-16 9 5 0.5 0 0 0 9 6 0.5 0.122718 0.122718 3.60822e-16 9 7 0.5 0 0 0 9 8 0.5 0.0859029 0.0859029 3.46945e-16 9 9 0.5 0 0 0 9 10 0.5 0.0644272 0.0644272 4.71845e-16 9 11 0.5 0 0 0 9 12 0.5 0.0506214 0.0506214 7.70217e-16 9 13 0.5 0 0 0 9 14 0.5 0.0411299 0.0411299 2.08167e-16 9 15 0.5 1.73472e-18 0 1.73472e-18 9 16 0.5 0.0342749 0.0342749 6.93889e-17 9 17 0.5 0 0 0 9 18 0.5 0.0291277 0.0291337 5.99211e-06 9 19 0.5 0 0 0 9 20 0.5 0.0251339 0.0251609 2.69645e-05 10 0 0.5 1.5708 1.5708 1.55431e-15 10 1 0.5 -2.42861e-17 0 2.42861e-17 10 2 0.5 0.392699 0.392699 5.55112e-17 10 3 0.5 1.73472e-17 0 1.73472e-17 10 4 0.5 0.19635 0.19635 5.55112e-17 10 5 0.5 -3.46945e-18 0 3.46945e-18 10 6 0.5 0.122718 0.122718 1.249e-16 10 7 0.5 0 0 0 10 8 0.5 0.0859029 0.0859029 1.66533e-16 10 9 0.5 0 0 0 10 10 0.5 0.0644272 0.0644272 3.33067e-16 10 11 0.5 -1.73472e-18 0 1.73472e-18 10 12 0.5 0.0506214 0.0506214 6.45317e-16 10 13 0.5 -1.73472e-18 0 1.73472e-18 10 14 0.5 0.0411299 0.0411299 1.04083e-16 10 15 0.5 0 0 0 10 16 0.5 0.0342749 0.0342749 3.46945e-17 10 17 0.5 0 0 0 10 18 0.5 0.0291337 0.0291337 1.38778e-16 10 19 0.5 0 0 0 10 20 0.5 0.0251594 0.0251609 1.49803e-06 10 21 0.5 0 0 0 10 22 0.5 0.0220083 0.0220158 7.49014e-06 1 0 1 1.33333 1.33333 0 1 1 1 0 0 0 1 2 1 0 0.266667 0.266667 1 3 1 0 0 0 1 4 1 0 0.114286 0.114286 2 0 1 1.33333 1.33333 0 2 1 1 1.66533e-16 0 1.66533e-16 2 2 1 0.266667 0.266667 5.55112e-17 2 3 1 4.85723e-17 0 4.85723e-17 2 4 1 0.0533333 0.114286 0.0609524 2 5 1 1.04083e-17 0 1.04083e-17 2 6 1 0.0106667 0.0634921 0.0528254 3 0 1 1.33333 1.33333 4.44089e-16 3 1 1 0 0 0 3 2 1 0.266667 0.266667 2.77556e-16 3 3 1 0 0 0 3 4 1 0.114286 0.114286 1.11022e-16 3 5 1 0 0 0 3 6 1 0.0489796 0.0634921 0.0145125 3 7 1 0 0 0 3 8 1 0.0209913 0.040404 0.0194128 4 0 1 1.33333 1.33333 2.22045e-16 4 1 1 -2.08167e-16 0 2.08167e-16 4 2 1 0.266667 0.266667 5.55112e-17 4 3 1 -1.11022e-16 0 1.11022e-16 4 4 1 0.114286 0.114286 2.77556e-17 4 5 1 -6.245e-17 0 6.245e-17 4 6 1 0.0634921 0.0634921 1.38778e-17 4 7 1 -3.1225e-17 0 3.1225e-17 4 8 1 0.0368859 0.040404 0.00351817 4 9 1 -1.73472e-17 0 1.73472e-17 4 10 1 0.0215671 0.027972 0.00640488 5 0 1 1.33333 1.33333 0 5 1 1 2.91434e-16 0 2.91434e-16 5 2 1 0.266667 0.266667 0 5 3 1 2.22045e-16 0 2.22045e-16 5 4 1 0.114286 0.114286 0 5 5 1 1.59595e-16 0 1.59595e-16 5 6 1 0.0634921 0.0634921 0 5 7 1 1.249e-16 0 1.249e-16 5 8 1 0.040404 0.040404 6.93889e-18 5 9 1 8.67362e-17 0 8.67362e-17 5 10 1 0.0271109 0.027972 0.000861092 5 11 1 6.76542e-17 0 6.76542e-17 5 12 1 0.0185245 0.0205128 0.00198834 6 0 1 1.33333 1.33333 0 6 1 1 -2.08167e-16 0 2.08167e-16 6 2 1 0.266667 0.266667 5.55112e-17 6 3 1 -1.38778e-16 0 1.38778e-16 6 4 1 0.114286 0.114286 1.38778e-17 6 5 1 -9.71445e-17 0 9.71445e-17 6 6 1 0.0634921 0.0634921 1.38778e-17 6 7 1 -6.93889e-17 0 6.93889e-17 6 8 1 0.040404 0.040404 2.08167e-17 6 9 1 -4.85723e-17 0 4.85723e-17 6 10 1 0.027972 0.027972 1.49186e-16 6 11 1 -3.29597e-17 0 3.29597e-17 6 12 1 0.0203009 0.0205128 0.000211961 6 13 1 -2.42861e-17 0 2.42861e-17 6 14 1 0.0150926 0.0156863 0.000593683 7 0 1 1.33333 1.33333 2.22045e-16 7 1 1 -3.81639e-17 0 3.81639e-17 7 2 1 0.266667 0.266667 5.55112e-17 7 3 1 -3.46945e-18 0 3.46945e-18 7 4 1 0.114286 0.114286 5.55112e-17 7 5 1 0 0 0 7 6 1 0.0634921 0.0634921 2.77556e-17 7 7 1 1.73472e-18 0 1.73472e-18 7 8 1 0.040404 0.040404 2.08167e-17 7 9 1 0 0 0 7 10 1 0.027972 0.027972 1.21431e-16 7 11 1 0 0 0 7 12 1 0.0205128 0.0205128 8.32667e-17 7 13 1 0 0 0 7 14 1 0.0156339 0.0156863 5.23668e-05 7 15 1 0 0 0 7 16 1 0.0122114 0.0123839 0.000172535 8 0 1 1.33333 1.33333 1.33227e-15 8 1 1 5.20417e-17 0 5.20417e-17 8 2 1 0.266667 0.266667 3.88578e-16 8 3 1 1.38778e-17 0 1.38778e-17 8 4 1 0.114286 0.114286 2.22045e-16 8 5 1 0 0 0 8 6 1 0.0634921 0.0634921 1.38778e-16 8 7 1 1.73472e-18 0 1.73472e-18 8 8 1 0.040404 0.040404 1.04083e-16 8 9 1 1.73472e-18 0 1.73472e-18 8 10 1 0.027972 0.027972 2.11636e-16 8 11 1 0 0 0 8 12 1 0.0205128 0.0205128 1.04083e-17 8 13 1 8.67362e-19 0 8.67362e-19 8 14 1 0.0156863 0.0156863 2.42861e-17 8 15 1 0 0 0 8 16 1 0.0123709 0.0123839 1.29701e-05 8 17 1 0 0 0 8 18 1 0.00997591 0.0100251 4.91557e-05 9 0 1 1.33333 1.33333 2.22045e-16 9 1 1 6.93889e-18 0 6.93889e-18 9 2 1 0.266667 0.266667 5.55112e-17 9 3 1 0 0 0 9 4 1 0.114286 0.114286 5.55112e-17 9 5 1 3.46945e-18 0 3.46945e-18 9 6 1 0.0634921 0.0634921 2.77556e-17 9 7 1 -3.46945e-18 0 3.46945e-18 9 8 1 0.040404 0.040404 2.08167e-17 9 9 1 0 0 0 9 10 1 0.027972 0.027972 1.52656e-16 9 11 1 8.67362e-19 0 8.67362e-19 9 12 1 0.0205128 0.0205128 4.85723e-17 9 13 1 0 0 0 9 14 1 0.0156863 0.0156863 6.245e-17 9 15 1 0 0 0 9 16 1 0.0123839 0.0123839 5.0307e-17 9 17 1 8.67362e-19 0 8.67362e-19 9 18 1 0.0100218 0.0100251 3.21815e-06 9 19 1 0 0 0 9 20 1 0.00826778 0.00828157 1.37931e-05 10 0 1 1.33333 1.33333 4.44089e-16 10 1 1 -3.40006e-16 0 3.40006e-16 10 2 1 0.266667 0.266667 2.22045e-16 10 3 1 -2.56739e-16 0 2.56739e-16 10 4 1 0.114286 0.114286 1.52656e-16 10 5 1 -1.94289e-16 0 1.94289e-16 10 6 1 0.0634921 0.0634921 1.11022e-16 10 7 1 -1.66533e-16 0 1.66533e-16 10 8 1 0.040404 0.040404 8.32667e-17 10 9 1 -1.48319e-16 0 1.48319e-16 10 10 1 0.027972 0.027972 1.83881e-16 10 11 1 -1.30104e-16 0 1.30104e-16 10 12 1 0.0205128 0.0205128 3.46945e-17 10 13 1 -1.17961e-16 0 1.17961e-16 10 14 1 0.0156863 0.0156863 5.20417e-17 10 15 1 -1.04083e-16 0 1.04083e-16 10 16 1 0.0123839 0.0123839 5.72459e-17 10 17 1 -9.28077e-17 0 9.28077e-17 10 18 1 0.0100251 0.0100251 1.73472e-18 10 19 1 -8.41341e-17 0 8.41341e-17 10 20 1 0.00828077 0.00828157 7.9954e-07 10 21 1 -7.54605e-17 0 7.54605e-17 10 22 1 0.0069527 0.00695652 3.82409e-06 1 0 2.5 0.981748 0.981748 1.11022e-16 1 1 2.5 0 0 0 1 2 2.5 0 0.122718 0.122718 1 3 2.5 0 0 0 1 4 2.5 0 0.0368155 0.0368155 2 0 2.5 0.981748 0.981748 1.11022e-16 2 1 2.5 0 0 0 2 2 2.5 0.122718 0.122718 1.249e-16 2 3 2.5 0 0 0 2 4 2.5 0.0153398 0.0368155 0.0214757 2 5 2.5 0 0 0 2 6 2.5 0.00191748 0.0153398 0.0134223 3 0 2.5 0.981748 0.981748 4.44089e-16 3 1 2.5 0 0 0 3 2 2.5 0.122718 0.122718 1.38778e-17 3 3 2.5 0 0 0 3 4 2.5 0.0368155 0.0368155 2.77556e-17 3 5 2.5 0 0 0 3 6 2.5 0.0110447 0.0153398 0.00429515 3 7 2.5 0 0 0 3 8 2.5 0.0033134 0.0076699 0.00435651 4 0 2.5 0.981748 0.981748 0 4 1 2.5 1.45717e-16 0 1.45717e-16 4 2 2.5 0.122718 0.122718 6.93889e-17 4 3 2.5 6.59195e-17 0 6.59195e-17 4 4 2.5 0.0368155 0.0368155 2.77556e-17 4 5 2.5 2.94903e-17 0 2.94903e-17 4 6 2.5 0.0153398 0.0153398 3.46945e-18 4 7 2.5 1.30104e-17 0 1.30104e-17 4 8 2.5 0.00674952 0.0076699 0.000920388 4 9 2.5 5.63785e-18 0 5.63785e-18 4 10 2.5 0.00299126 0.00431432 0.00132306 5 0 2.5 0.981748 0.981748 3.33067e-16 5 1 2.5 -3.46945e-17 0 3.46945e-17 5 2 2.5 0.122718 0.122718 1.52656e-16 5 3 2.5 -6.93889e-18 0 6.93889e-18 5 4 2.5 0.0368155 0.0368155 6.245e-17 5 5 2.5 -1.73472e-18 0 1.73472e-18 5 6 2.5 0.0153398 0.0153398 1.56125e-17 5 7 2.5 0 0 0 5 8 2.5 0.0076699 0.0076699 1.04083e-17 5 9 2.5 0 0 0 5 10 2.5 0.00410888 0.00431432 0.000205444 5 11 2.5 0 0 0 5 12 2.5 0.0022501 0.00263653 0.00038643 6 0 2.5 0.981748 0.981748 2.22045e-16 6 1 2.5 1.83881e-16 0 1.83881e-16 6 2 2.5 0.122718 0.122718 6.93889e-17 6 3 2.5 4.16334e-17 0 4.16334e-17 6 4 2.5 0.0368155 0.0368155 3.46945e-17 6 5 2.5 1.30104e-17 0 1.30104e-17 6 6 2.5 0.0153398 0.0153398 0 6 7 2.5 3.90313e-18 0 3.90313e-18 6 8 2.5 0.0076699 0.0076699 1.99493e-17 6 9 2.5 8.67362e-19 0 8.67362e-19 6 10 2.5 0.00431432 0.00431432 1.99493e-17 6 11 2.5 2.1684e-19 0 2.1684e-19 6 12 2.5 0.00258945 0.00263653 4.70809e-05 6 13 2.5 0 0 0 6 14 2.5 0.00160369 0.00171374 0.000110052 7 0 2.5 0.981748 0.981748 1.11022e-16 7 1 2.5 1.78677e-16 0 1.78677e-16 7 2 2.5 0.122718 0.122718 2.08167e-16 7 3 2.5 1.04083e-16 0 1.04083e-16 7 4 2.5 0.0368155 0.0368155 1.31839e-16 7 5 2.5 7.02563e-17 0 7.02563e-17 7 6 2.5 0.0153398 0.0153398 6.245e-17 7 7 2.5 4.85723e-17 0 4.85723e-17 7 8 2.5 0.0076699 0.0076699 2.60209e-17 7 9 2.5 3.40439e-17 0 3.40439e-17 7 10 2.5 0.00431432 0.00431432 1.21431e-17 7 11 2.5 2.38524e-17 0 2.38524e-17 7 12 2.5 0.00263653 0.00263653 1.69136e-17 7 13 2.5 1.64799e-17 0 1.64799e-17 7 14 2.5 0.00170276 0.00171374 1.09855e-05 7 15 2.5 1.1601e-17 0 1.1601e-17 7 16 2.5 0.00113767 0.00116846 3.07928e-05 8 0 2.5 0.981748 0.981748 1.11022e-16 8 1 2.5 2.60209e-17 0 2.60209e-17 8 2 2.5 0.122718 0.122718 9.71445e-17 8 3 2.5 4.33681e-19 0 4.33681e-19 8 4 2.5 0.0368155 0.0368155 3.46945e-17 8 5 2.5 -4.33681e-19 0 4.33681e-19 8 6 2.5 0.0153398 0.0153398 0 8 7 2.5 4.33681e-19 0 4.33681e-19 8 8 2.5 0.0076699 0.0076699 2.08167e-17 8 9 2.5 2.1684e-19 0 2.1684e-19 8 10 2.5 0.00431432 0.00431432 2.1684e-17 8 11 2.5 0 0 0 8 12 2.5 0.00263653 0.00263653 6.07153e-18 8 13 2.5 0 0 0 8 14 2.5 0.00171374 0.00171374 2.1684e-19 8 15 2.5 0 0 0 8 16 2.5 0.00116587 0.00116846 2.59658e-06 8 17 2.5 0 0 0 8 18 2.5 0.000819157 0.000827661 8.50381e-06 9 0 2.5 0.981748 0.981748 1.11022e-16 9 1 2.5 1.64799e-17 0 1.64799e-17 9 2 2.5 0.122718 0.122718 2.77556e-17 9 3 2.5 1.73472e-18 0 1.73472e-18 9 4 2.5 0.0368155 0.0368155 0 9 5 2.5 -8.67362e-19 0 8.67362e-19 9 6 2.5 0.0153398 0.0153398 2.60209e-17 9 7 2.5 -2.1684e-19 0 2.1684e-19 9 8 2.5 0.0076699 0.0076699 3.55618e-17 9 9 2.5 0 0 0 9 10 2.5 0.00431432 0.00431432 2.94903e-17 9 11 2.5 -1.0842e-19 0 1.0842e-19 9 12 2.5 0.00263653 0.00263653 1.12757e-17 9 13 2.5 0 0 0 9 14 2.5 0.00171374 0.00171374 3.68629e-18 9 15 2.5 -5.42101e-20 0 5.42101e-20 9 16 2.5 0.00116846 0.00116846 6.50521e-19 9 17 2.5 0 0 0 9 18 2.5 0.000827041 0.000827661 6.19639e-07 9 19 2.5 0 0 0 9 20 2.5 0.000602504 0.000604829 2.32473e-06 10 0 2.5 0.981748 0.981748 4.44089e-16 10 1 2.5 -1.32706e-16 0 1.32706e-16 10 2 2.5 0.122718 0.122718 1.66533e-16 10 3 2.5 -3.49113e-17 0 3.49113e-17 10 4 2.5 0.0368155 0.0368155 6.93889e-17 10 5 2.5 -1.12757e-17 0 1.12757e-17 10 6 2.5 0.0153398 0.0153398 1.21431e-17 10 7 2.5 -4.01155e-18 0 4.01155e-18 10 8 2.5 0.0076699 0.0076699 1.47451e-17 10 9 2.5 -1.40946e-18 0 1.40946e-18 10 10 2.5 0.00431432 0.00431432 1.82146e-17 10 11 2.5 -3.25261e-19 0 3.25261e-19 10 12 2.5 0.00263653 0.00263653 6.50521e-18 10 13 2.5 -3.25261e-19 0 3.25261e-19 10 14 2.5 0.00171374 0.00171374 0 10 15 2.5 -1.0842e-19 0 1.0842e-19 10 16 2.5 0.00116846 0.00116846 2.81893e-18 10 17 2.5 0 0 0 10 18 2.5 0.000827661 0.000827661 4.22839e-18 10 19 2.5 2.71051e-20 0 2.71051e-20 10 20 2.5 0.00060468 0.000604829 1.48952e-07 10 21 2.5 2.71051e-20 0 2.71051e-20 10 22 2.5 0.000452991 0.000453622 6.30385e-07 TEST10 GEGENBAUER_COMPUTE computes a generalized Gauss-Gegenbauer rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) (1-x^2)^alpha dx. Order = 1 ALPHA = 0.5 0 0 1.570796326794897 Order = 2 ALPHA = 0.5 0 -0.5 0.7853981633974484 1 0.5 0.7853981633974484 Order = 3 ALPHA = 0.5 0 -0.7071067811865475 0.3926990816987239 1 0 0.7853981633974484 2 0.7071067811865476 0.3926990816987245 Order = 4 ALPHA = 0.5 0 -0.8090169943749475 0.217078713422706 1 -0.3090169943749475 0.5683194499747424 2 0.3090169943749474 0.5683194499747424 3 0.8090169943749475 0.217078713422706 Order = 5 ALPHA = 0.5 0 -0.8660254037844387 0.130899693899574 1 -0.5 0.3926990816987242 2 0 0.5235987755982989 3 0.5 0.3926990816987242 4 0.8660254037844387 0.1308996938995745 Order = 6 ALPHA = 0.5 0 -0.9009688679024191 0.08448869089158841 1 -0.6234898018587335 0.2743330560697777 2 -0.2225209339563144 0.4265764164360819 3 0.2225209339563144 0.4265764164360819 4 0.6234898018587335 0.2743330560697777 5 0.9009688679024191 0.08448869089158841 Order = 7 ALPHA = 0.5 0 -0.9238795325112867 0.05750944903191331 1 -0.7071067811865475 0.1963495408493619 2 -0.3826834323650898 0.3351896326668111 3 0 0.3926990816987242 4 0.3826834323650898 0.3351896326668108 5 0.7071067811865476 0.1963495408493624 6 0.9238795325112867 0.05750944903191331 Order = 8 ALPHA = 0.5 0 -0.9396926207859084 0.04083294770910693 1 -0.766044443118978 0.1442256007956728 2 -0.5 0.2617993877991495 3 -0.1736481776669303 0.3385402270935191 4 0.1736481776669303 0.3385402270935191 5 0.5 0.2617993877991495 6 0.766044443118978 0.1442256007956728 7 0.9396926207859084 0.04083294770910754 Order = 9 ALPHA = 0.5 0 -0.9510565162951536 0.02999954037160841 1 -0.8090169943749475 0.108539356711353 2 -0.5877852522924731 0.2056199086476264 3 -0.3090169943749475 0.2841597249873712 4 0 0.3141592653589794 5 0.3090169943749475 0.2841597249873712 6 0.5877852522924731 0.2056199086476264 7 0.8090169943749475 0.108539356711353 8 0.9510565162951536 0.02999954037160841 Order = 10 ALPHA = 0.5 0 -0.9594929736144974 0.02266894250185901 1 -0.8412535328311812 0.08347854093418892 2 -0.6548607339452851 0.1631221774548165 3 -0.4154150130018864 0.2363135602034873 4 -0.1423148382732851 0.2798149423030965 5 0.1423148382732851 0.2798149423030966 6 0.4154150130018864 0.2363135602034873 7 0.6548607339452851 0.1631221774548165 8 0.8412535328311812 0.08347854093418892 9 0.9594929736144974 0.02266894250185901 Order = 1 ALPHA = 1 0 0 1.333333333333333 Order = 2 ALPHA = 1 0 -0.4472135954999579 0.6666666666666665 1 0.447213595499958 0.6666666666666667 Order = 3 ALPHA = 1 0 -0.6546536707079772 0.3111111111111113 1 0 0.711111111111111 2 0.6546536707079772 0.3111111111111113 Order = 4 ALPHA = 1 0 -0.7650553239294646 0.1569499125956942 1 -0.2852315164806451 0.5097167540709727 2 0.2852315164806451 0.5097167540709727 3 0.7650553239294647 0.1569499125956939 Order = 5 ALPHA = 1 0 -0.8302238962785669 0.08601768212280729 1 -0.4688487934707142 0.3368394607343354 2 -2.407412430484045e-35 0.4876190476190476 3 0.4688487934707142 0.3368394607343354 4 0.830223896278567 0.08601768212280764 Order = 6 ALPHA = 1 0 -0.8717401485096066 0.05058357701608194 1 -0.5917001814331423 0.2216925320225178 2 -0.2092992179024789 0.3943905576280671 3 0.2092992179024789 0.3943905576280671 4 0.5917001814331423 0.2216925320225178 5 0.8717401485096067 0.05058357701608171 Order = 7 ALPHA = 1 0 -0.8997579954114602 0.03151620015049876 1 -0.6771862795107377 0.1486404045834101 2 -0.3631174638261782 0.3007504247445493 3 0 0.3715192743764172 4 0.3631174638261782 0.3007504247445493 5 0.6771862795107377 0.1486404045834101 6 0.8997579954114602 0.03151620015049876 Order = 8 ALPHA = 1 0 -0.9195339081664589 0.02059009564912187 1 -0.738773865105505 0.1021477023603581 2 -0.4779249498104445 0.2253365549698578 3 -0.165278957666387 0.3185923136873282 4 0.165278957666387 0.3185923136873282 5 0.4779249498104445 0.225336554969858 6 0.738773865105505 0.1021477023603581 7 0.9195339081664589 0.02059009564912187 Order = 9 ALPHA = 1 0 -0.9340014304080592 0.01399105612442671 1 -0.7844834736631444 0.07198285617566075 2 -0.565235326996205 0.1687989736144601 3 -0.2957581355869394 0.2617849830242735 4 -3.673419846319648e-40 0.3002175954556907 5 0.2957581355869394 0.2617849830242735 6 0.565235326996205 0.1687989736144601 7 0.7844834736631444 0.07198285617566075 8 0.9340014304080592 0.01399105612442671 Order = 10 ALPHA = 1 0 -0.9448992722228823 0.009825404805768293 1 -0.8192793216440066 0.05193914377420781 2 -0.6328761530318606 0.1273919483477328 3 -0.399530940965349 0.2111657418760753 4 -0.1365529328549276 0.2663444278628824 5 0.1365529328549276 0.2663444278628824 6 0.399530940965349 0.2111657418760753 7 0.6328761530318606 0.1273919483477326 8 0.8192793216440066 0.05193914377420781 9 0.9448992722228822 0.009825404805768065 Order = 1 ALPHA = 2.5 0 0 0.9817477042468103 Order = 2 ALPHA = 2.5 0 -0.3535533905932737 0.4908738521234051 1 0.3535533905932737 0.4908738521234051 Order = 3 ALPHA = 2.5 0 -0.5477225575051661 0.2045307717180856 1 7.703719777548943e-34 0.5726861608106394 2 0.5477225575051661 0.2045307717180856 Order = 4 ALPHA = 2.5 0 -0.6660699417556469 0.08700807153148325 1 -0.2373833033084449 0.4038657805919217 2 0.2373833033084449 0.4038657805919217 3 0.6660699417556469 0.08700807153148347 Order = 5 ALPHA = 2.5 0 -0.7434770045212192 0.03928938913176718 1 -0.4019050360892101 0.2454174450998077 2 -1.540743955509789e-33 0.4123340357836603 3 0.4019050360892101 0.2454174450998076 4 0.7434770045212192 0.03928938913176718 Order = 6 ALPHA = 2.5 0 -0.7968373566406807 0.01891164336669311 1 -0.5196148456949133 0.1431407318203965 2 -0.1804179569647766 0.3288214769363154 3 0.1804179569647766 0.3288214769363154 4 0.5196148456949133 0.1431407318203969 5 0.7968373566406807 0.01891164336669311 Order = 7 ALPHA = 2.5 0 -0.8351564541339112 0.009658525021000812 1 -0.6063141605450342 0.08344467054898472 2 -0.3186902924591673 0.2357822853526957 3 -1.469367938527859e-39 0.3239767424014473 4 0.3186902924591672 0.2357822853526958 5 0.6063141605450342 0.08344467054898472 6 0.8351564541339112 0.009658525021000983 Order = 8 ALPHA = 2.5 0 -0.8635912403827584 0.005199707422132487 1 -0.6718232507247468 0.04941818815652229 2 -0.4260961495798479 0.1615070338519726 3 -0.1459649294626305 0.2747489226927776 4 0.1459649294626305 0.2747489226927777 5 0.4260961495798479 0.1615070338519726 6 0.6718232507247468 0.04941818815652229 7 0.8635912403827584 0.005199707422132487 Order = 9 ALPHA = 2.5 0 -0.8852662998295736 0.002931900984992394 1 -0.7224309555097356 0.02992095719368033 2 -0.5107538326640052 0.1088013928543096 3 -0.2643695362367075 0.215514913750143 4 0 0.2674093746805597 5 0.2643695362367075 0.215514913750143 6 0.5107538326640052 0.1088013928543096 7 0.7224309555097356 0.02992095719368033 8 0.8852662998295736 0.002931900984992394 Order = 10 ALPHA = 2.5 0 -0.9021636877146698 0.00172163822889316 1 -0.7622868838395971 0.01855643700599032 2 -0.5784393987065429 0.07316132839714104 3 -0.3610608688185057 0.1622660968500981 4 -0.1227285554678421 0.2351683516412825 5 0.1227285554678421 0.2351683516412824 6 0.3610608688185057 0.1622660968500981 7 0.5784393987065429 0.07316132839714087 8 0.7622868838395971 0.01855643700599032 9 0.9021636877146698 0.00172163822889316 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 0.5 1.22542 1.22542 0 1 1 0.5 0 0 0 1 2 0.5 0 0.919063 0.919063 1 3 0.5 0 0 0 1 4 0.5 0 1.60836 1.60836 2 0 0.5 1.22542 1.22542 0 2 1 0.5 0 0 0 2 2 0.5 0.919063 0.919063 4.44089e-16 2 3 0.5 0 0 0 2 4 0.5 0.689297 1.60836 0.919063 2 5 0.5 0 0 0 2 6 0.5 0.516973 4.42299 3.90602 3 0 0.5 1.22542 1.22542 0 3 1 0.5 -3.33067e-16 0 3.33067e-16 3 2 0.5 0.919063 0.919063 6.66134e-16 3 3 0.5 -9.99201e-16 0 9.99201e-16 3 4 0.5 1.60836 1.60836 1.9984e-15 3 5 0.5 -1.9984e-15 0 1.9984e-15 3 6 0.5 2.81463 4.42299 1.60836 3 7 0.5 -4.44089e-15 0 4.44089e-15 3 8 0.5 4.9256 16.5862 11.6606 4 0 0.5 1.22542 1.22542 0 4 1 0.5 2.498e-16 0 2.498e-16 4 2 0.5 0.919063 0.919063 5.55112e-16 4 3 0.5 2.22045e-16 0 2.22045e-16 4 4 0.5 1.60836 1.60836 1.33227e-15 4 5 0.5 6.66134e-16 0 6.66134e-16 4 6 0.5 4.42299 4.42299 3.55271e-15 4 7 0.5 1.33227e-15 0 1.33227e-15 4 8 0.5 13.3695 16.5862 3.21672 4 9 0.5 3.55271e-15 0 3.55271e-15 4 10 0.5 40.988 78.7845 37.7964 5 0 0.5 1.22542 1.22542 1.11022e-15 5 1 0.5 -5.48173e-16 0 5.48173e-16 5 2 0.5 0.919063 0.919063 4.44089e-16 5 3 0.5 -6.66134e-16 0 6.66134e-16 5 4 0.5 1.60836 1.60836 2.22045e-16 5 5 0.5 -4.10783e-15 0 4.10783e-15 5 6 0.5 4.42299 4.42299 8.88178e-16 5 7 0.5 -2.53131e-14 0 2.53131e-14 5 8 0.5 16.5862 16.5862 3.55271e-15 5 9 0.5 -1.38556e-13 0 1.38556e-13 5 10 0.5 69.9385 78.7845 8.84598 5 11 0.5 -6.96332e-13 0 6.96332e-13 5 12 0.5 304.841 453.011 148.17 6 0 0.5 1.22542 1.22542 4.44089e-16 6 1 0.5 3.98986e-16 0 3.98986e-16 6 2 0.5 0.919063 0.919063 9.99201e-16 6 3 0.5 6.52256e-16 0 6.52256e-16 6 4 0.5 1.60836 1.60836 1.9984e-15 6 5 0.5 1.38778e-15 0 1.38778e-15 6 6 0.5 4.42299 4.42299 8.88178e-15 6 7 0.5 1.77636e-15 0 1.77636e-15 6 8 0.5 16.5862 16.5862 4.9738e-14 6 9 0.5 8.88178e-15 0 8.88178e-15 6 10 0.5 78.7845 78.7845 2.70006e-13 6 11 0.5 5.68434e-14 0 5.68434e-14 6 12 0.5 426.473 453.011 26.5379 6 13 0.5 8.52651e-13 0 8.52651e-13 6 14 0.5 2440.82 3057.82 617.007 7 0 0.5 1.22542 1.22542 2.22045e-16 7 1 0.5 5.39499e-16 0 5.39499e-16 7 2 0.5 0.919063 0.919063 6.66134e-16 7 3 0.5 1.04777e-15 0 1.04777e-15 7 4 0.5 1.60836 1.60836 1.11022e-15 7 5 0.5 2.91434e-15 0 2.91434e-15 7 6 0.5 4.42299 4.42299 1.77636e-15 7 7 0.5 8.88178e-15 0 8.88178e-15 7 8 0.5 16.5862 16.5862 3.55271e-15 7 9 0.5 3.01981e-14 0 3.01981e-14 7 10 0.5 78.7845 78.7845 1.27898e-13 7 11 0.5 1.13687e-13 0 1.13687e-13 7 12 0.5 453.011 453.011 1.36424e-12 7 13 0.5 2.84217e-13 0 2.84217e-13 7 14 0.5 2958.31 3057.82 99.5172 7 15 0.5 0 0 0 7 16 0.5 20687.7 23698.1 3010.4 8 0 0.5 1.22542 1.22542 1.55431e-15 8 1 0.5 6.95841e-16 0 6.95841e-16 8 2 0.5 0.919063 0.919063 2.22045e-16 8 3 0.5 2.02789e-15 0 2.02789e-15 8 4 0.5 1.60836 1.60836 2.22045e-16 8 5 0.5 6.16868e-15 0 6.16868e-15 8 6 0.5 4.42299 4.42299 4.44089e-15 8 7 0.5 2.53686e-14 0 2.53686e-14 8 8 0.5 16.5862 16.5862 3.55271e-14 8 9 0.5 1.06581e-13 0 1.06581e-13 8 10 0.5 78.7845 78.7845 2.70006e-13 8 11 0.5 5.47118e-13 0 5.47118e-13 8 12 0.5 453.011 453.011 2.10321e-12 8 13 0.5 2.44427e-12 0 2.44427e-12 8 14 0.5 3057.82 3057.82 1.68257e-11 8 15 0.5 1.27329e-11 0 1.27329e-11 8 16 0.5 23300.1 23698.1 398.069 8 17 0.5 5.09317e-11 0 5.09317e-11 8 18 0.5 191933 207359 15425.2 9 0 0.5 1.22542 1.22542 2.22045e-16 9 1 0.5 -1.59703e-16 0 1.59703e-16 9 2 0.5 0.919063 0.919063 8.88178e-16 9 3 0.5 -7.80626e-16 0 7.80626e-16 9 4 0.5 1.60836 1.60836 2.22045e-15 9 5 0.5 -3.93088e-15 0 3.93088e-15 9 6 0.5 4.42299 4.42299 9.76996e-15 9 7 0.5 -1.94844e-14 0 1.94844e-14 9 8 0.5 16.5862 16.5862 4.9738e-14 9 9 0.5 -1.10134e-13 0 1.10134e-13 9 10 0.5 78.7845 78.7845 3.12639e-13 9 11 0.5 -6.82121e-13 0 6.82121e-13 9 12 0.5 453.011 453.011 2.04636e-12 9 13 0.5 -4.718e-12 0 4.718e-12 9 14 0.5 3057.82 3057.82 1.40972e-11 9 15 0.5 -3.50155e-11 0 3.50155e-11 9 16 0.5 23698.1 23698.1 1.09139e-10 9 17 0.5 -2.98314e-10 0 2.98314e-10 9 18 0.5 205468 207359 1890.83 9 19 0.5 -2.85218e-09 0 2.85218e-09 9 20 0.5 1.93146e+06 2.02175e+06 90287 10 0 0.5 1.22542 1.22542 1.11022e-15 10 1 0.5 7.70027e-16 0 7.70027e-16 10 2 0.5 0.919063 0.919063 9.99201e-16 10 3 0.5 9.876e-16 0 9.876e-16 10 4 0.5 1.60836 1.60836 2.44249e-15 10 5 0.5 2.52402e-15 0 2.52402e-15 10 6 0.5 4.42299 4.42299 9.76996e-15 10 7 0.5 8.25034e-15 0 8.25034e-15 10 8 0.5 16.5862 16.5862 4.61853e-14 10 9 0.5 3.4639e-14 0 3.4639e-14 10 10 0.5 78.7845 78.7845 2.55795e-13 10 11 0.5 1.3145e-13 0 1.3145e-13 10 12 0.5 453.011 453.011 1.47793e-12 10 13 0.5 7.95808e-13 0 7.95808e-13 10 14 0.5 3057.82 3057.82 9.09495e-12 10 15 0.5 7.27596e-12 0 7.27596e-12 10 16 0.5 23698.1 23698.1 6.18456e-11 10 17 0.5 9.45874e-11 0 9.45874e-11 10 18 0.5 207359 207359 4.07454e-10 10 19 0.5 1.36788e-09 0 1.36788e-09 10 20 0.5 2.01229e+06 2.02175e+06 9454.14 10 21 0.5 1.81608e-08 0 1.81608e-08 10 22 0.5 2.11831e+07 2.17338e+07 550704 1 0 1 1 1 0 1 1 1 0 0 0 1 2 1 0 1 1 1 3 1 0 0 0 1 4 1 0 2 2 2 0 1 1 1 2.22045e-16 2 1 1 0 0 0 2 2 1 1 1 6.66134e-16 2 3 1 0 0 0 2 4 1 1 2 1 2 5 1 0 0 0 2 6 1 1 6 5 3 0 1 1 1 2.22045e-16 3 1 1 -3.88578e-16 0 3.88578e-16 3 2 1 1 1 0 3 3 1 -1.11022e-15 0 1.11022e-15 3 4 1 2 2 6.66134e-16 3 5 1 -2.44249e-15 0 2.44249e-15 3 6 1 4 6 2 3 7 1 -5.77316e-15 0 5.77316e-15 3 8 1 8 24 16 4 0 1 1 1 4.44089e-16 4 1 1 -1.94289e-16 0 1.94289e-16 4 2 1 1 1 1.11022e-16 4 3 1 -1.11022e-16 0 1.11022e-16 4 4 1 2 2 8.88178e-16 4 5 1 0 0 0 4 6 1 6 6 7.10543e-15 4 7 1 0 0 0 4 8 1 20 24 4 4 9 1 0 0 0 4 10 1 68 120 52 5 0 1 1 1 0 5 1 1 -7.35523e-16 0 7.35523e-16 5 2 1 1 1 7.77156e-16 5 3 1 -7.49401e-16 0 7.49401e-16 5 4 1 2 2 2.88658e-15 5 5 1 -4.44089e-16 0 4.44089e-16 5 6 1 6 6 1.06581e-14 5 7 1 2.66454e-15 0 2.66454e-15 5 8 1 24 24 4.9738e-14 5 9 1 2.4869e-14 0 2.4869e-14 5 10 1 108 120 12 5 11 1 1.27898e-13 0 1.27898e-13 5 12 1 504 720 216 6 0 1 1 1 4.44089e-16 6 1 1 7.33788e-16 0 7.33788e-16 6 2 1 1 1 4.44089e-16 6 3 1 1.3739e-15 0 1.3739e-15 6 4 1 2 2 8.88178e-16 6 5 1 2.9976e-15 0 2.9976e-15 6 6 1 6 6 7.10543e-15 6 7 1 3.55271e-15 0 3.55271e-15 6 8 1 24 24 4.9738e-14 6 9 1 -4.26326e-14 0 4.26326e-14 6 10 1 120 120 3.69482e-13 6 11 1 -5.40012e-13 0 5.40012e-13 6 12 1 684 720 36 6 13 1 -4.54747e-12 0 4.54747e-12 6 14 1 4140 5040 900 7 0 1 1 1 1.11022e-16 7 1 1 3.20924e-17 0 3.20924e-17 7 2 1 1 1 4.44089e-16 7 3 1 -2.32453e-16 0 2.32453e-16 7 4 1 2 2 8.88178e-16 7 5 1 -2.22045e-16 0 2.22045e-16 7 6 1 6 6 1.77636e-15 7 7 1 5.10703e-15 0 5.10703e-15 7 8 1 24 24 3.55271e-15 7 9 1 6.21725e-14 0 6.21725e-14 7 10 1 120 120 7.10543e-14 7 11 1 5.40012e-13 0 5.40012e-13 7 12 1 720 720 1.02318e-12 7 13 1 4.20641e-12 0 4.20641e-12 7 14 1 4896 5040 144 7 15 1 3.27418e-11 0 3.27418e-11 7 16 1 35712 40320 4608 8 0 1 1 1 3.33067e-16 8 1 1 -7.84962e-17 0 7.84962e-17 8 2 1 1 1 6.66134e-16 8 3 1 -1.82146e-17 0 1.82146e-17 8 4 1 2 2 1.55431e-15 8 5 1 1.23512e-15 0 1.23512e-15 8 6 1 6 6 5.32907e-15 8 7 1 1.37668e-14 0 1.37668e-14 8 8 1 24 24 1.06581e-14 8 9 1 7.81597e-14 0 7.81597e-14 8 10 1 120 120 5.68434e-14 8 11 1 3.76588e-13 0 3.76588e-13 8 12 1 720 720 4.54747e-13 8 13 1 1.25056e-12 0 1.25056e-12 8 14 1 5040 5040 5.45697e-12 8 15 1 0 0 0 8 16 1 39744 40320 576 8 17 1 -8.00355e-11 0 8.00355e-11 8 18 1 339264 362880 23616 9 0 1 1 1 8.88178e-16 9 1 1 9.23605e-16 0 9.23605e-16 9 2 1 1 1 1.33227e-15 9 3 1 2.23259e-15 0 2.23259e-15 9 4 1 2 2 1.33227e-15 9 5 1 8.3579e-15 0 8.3579e-15 9 6 1 6 6 8.88178e-16 9 7 1 3.9746e-14 0 3.9746e-14 9 8 1 24 24 1.77636e-14 9 9 1 2.05613e-13 0 2.05613e-13 9 10 1 120 120 1.27898e-13 9 11 1 1.09068e-12 0 1.09068e-12 9 12 1 720 720 5.68434e-13 9 13 1 4.94538e-12 0 4.94538e-12 9 14 1 5040 5040 0 9 15 1 1.00044e-11 0 1.00044e-11 9 16 1 40320 40320 5.09317e-11 9 17 1 -1.67347e-10 0 1.67347e-10 9 18 1 360000 362880 2880 9 19 1 -3.49246e-09 0 3.49246e-09 9 20 1 3.4848e+06 3.6288e+06 144000 10 0 1 1 1 3.33067e-16 10 1 1 6.07905e-16 0 6.07905e-16 10 2 1 1 1 4.44089e-16 10 3 1 9.83588e-16 0 9.83588e-16 10 4 1 2 2 3.10862e-15 10 5 1 2.25167e-15 0 2.25167e-15 10 6 1 6 6 1.77636e-14 10 7 1 5.67602e-15 0 5.67602e-15 10 8 1 24 24 1.10134e-13 10 9 1 1.35447e-14 0 1.35447e-14 10 10 1 120 120 6.96332e-13 10 11 1 8.17124e-14 0 8.17124e-14 10 12 1 720 720 4.88853e-12 10 13 1 3.12639e-13 0 3.12639e-13 10 14 1 5040 5040 3.72893e-11 10 15 1 -1.81899e-12 0 1.81899e-12 10 16 1 40320 40320 2.98314e-10 10 17 1 -7.27596e-11 0 7.27596e-11 10 18 1 362880 362880 2.61934e-09 10 19 1 -1.62981e-09 0 1.62981e-09 10 20 1 3.6144e+06 3.6288e+06 14400 10 21 1 -2.8871e-08 0 2.8871e-08 10 22 1 3.90384e+07 3.99168e+07 878400 1 0 2.5 0.919063 0.919063 0 1 1 2.5 0 0 0 1 2 2.5 0 1.60836 1.60836 1 3 2.5 0 0 0 1 4 2.5 0 4.42299 4.42299 2 0 2.5 0.919063 0.919063 2.22045e-16 2 1 2.5 0 0 0 2 2 2.5 1.60836 1.60836 8.88178e-16 2 3 2.5 0 0 0 2 4 2.5 2.81463 4.42299 1.60836 2 5 2.5 0 0 0 2 6 2.5 4.9256 16.5862 11.6606 3 0 2.5 0.919063 0.919063 0 3 1 2.5 0 0 0 3 2 2.5 1.60836 1.60836 2.22045e-16 3 3 2.5 0 0 0 3 4 2.5 4.42299 4.42299 1.77636e-15 3 5 2.5 0 0 0 3 6 2.5 12.1632 16.5862 4.42299 3 7 2.5 0 0 0 3 8 2.5 33.4488 78.7845 45.3356 4 0 2.5 0.919063 0.919063 3.33067e-16 4 1 2.5 -7.77156e-16 0 7.77156e-16 4 2 2.5 1.60836 1.60836 6.66134e-16 4 3 2.5 -1.44329e-15 0 1.44329e-15 4 4 2.5 4.42299 4.42299 8.88178e-16 4 5 2.5 -3.10862e-15 0 3.10862e-15 4 6 2.5 16.5862 16.5862 7.10543e-15 4 7 2.5 -1.06581e-14 0 1.06581e-14 4 8 2.5 69.9385 78.7845 8.84598 4 9 2.5 -1.42109e-14 0 1.42109e-14 4 10 2.5 304.841 453.011 148.17 5 0 2.5 0.919063 0.919063 3.33067e-16 5 1 2.5 -5.55112e-17 0 5.55112e-17 5 2 2.5 1.60836 1.60836 2.22045e-16 5 3 2.5 -2.22045e-16 0 2.22045e-16 5 4 2.5 4.42299 4.42299 1.77636e-15 5 5 2.5 8.88178e-16 0 8.88178e-16 5 6 2.5 16.5862 16.5862 1.77636e-14 5 7 2.5 1.59872e-14 0 1.59872e-14 5 8 2.5 78.7845 78.7845 1.27898e-13 5 9 2.5 1.13687e-13 0 1.13687e-13 5 10 2.5 419.838 453.011 33.1724 5 11 2.5 8.52651e-13 0 8.52651e-13 5 12 2.5 2336.32 3057.82 721.5 6 0 2.5 0.919063 0.919063 2.22045e-16 6 1 2.5 8.18789e-16 0 8.18789e-16 6 2 2.5 1.60836 1.60836 1.11022e-15 6 3 2.5 2.44249e-15 0 2.44249e-15 6 4 2.5 4.42299 4.42299 8.88178e-16 6 5 2.5 1.06581e-14 0 1.06581e-14 6 6 2.5 16.5862 16.5862 7.10543e-15 6 7 2.5 5.32907e-14 0 5.32907e-14 6 8 2.5 78.7845 78.7845 8.52651e-14 6 9 2.5 3.2685e-13 0 3.2685e-13 6 10 2.5 453.011 453.011 8.52651e-13 6 11 2.5 2.27374e-12 0 2.27374e-12 6 12 2.5 2958.31 3057.82 99.5172 6 13 2.5 1.59162e-11 0 1.59162e-11 6 14 2.5 20687.7 23698.1 3010.4 7 0 2.5 0.919063 0.919063 4.44089e-16 7 1 2.5 -5.44703e-16 0 5.44703e-16 7 2 2.5 1.60836 1.60836 1.55431e-15 7 3 2.5 -5.13478e-16 0 5.13478e-16 7 4 2.5 4.42299 4.42299 1.77636e-15 7 5 2.5 3.77476e-15 0 3.77476e-15 7 6 2.5 16.5862 16.5862 1.42109e-14 7 7 2.5 3.4639e-14 0 3.4639e-14 7 8 2.5 78.7845 78.7845 1.56319e-13 7 9 2.5 1.27898e-13 0 1.27898e-13 7 10 2.5 453.011 453.011 1.36424e-12 7 11 2.5 -1.13687e-13 0 1.13687e-13 7 12 2.5 3057.82 3057.82 1.18234e-11 7 13 2.5 -8.18545e-12 0 8.18545e-12 7 14 2.5 23225.4 23698.1 472.707 7 15 2.5 -1.09139e-10 0 1.09139e-10 7 16 2.5 189750 207359 17608.3 8 0 2.5 0.919063 0.919063 1.11022e-16 8 1 2.5 5.35596e-17 0 5.35596e-17 8 2 2.5 1.60836 1.60836 8.88178e-16 8 3 2.5 -9.12465e-16 0 9.12465e-16 8 4 2.5 4.42299 4.42299 8.88178e-16 8 5 2.5 -8.04912e-15 0 8.04912e-15 8 6 2.5 16.5862 16.5862 3.55271e-15 8 7 2.5 -5.39568e-14 0 5.39568e-14 8 8 2.5 78.7845 78.7845 2.84217e-14 8 9 2.5 -3.97904e-13 0 3.97904e-13 8 10 2.5 453.011 453.011 4.54747e-13 8 11 2.5 -3.15481e-12 0 3.15481e-12 8 12 2.5 3057.82 3057.82 4.54747e-12 8 13 2.5 -2.95586e-11 0 2.95586e-11 8 14 2.5 23698.1 23698.1 4.36557e-11 8 15 2.5 -2.76486e-10 0 2.76486e-10 8 16 2.5 205468 207359 1890.83 8 17 2.5 -2.82307e-09 0 2.82307e-09 8 18 2.5 1.93146e+06 2.02175e+06 90287 9 0 2.5 0.919063 0.919063 1.11022e-16 9 1 2.5 8.48713e-16 0 8.48713e-16 9 2 2.5 1.60836 1.60836 1.11022e-15 9 3 2.5 1.99927e-15 0 1.99927e-15 9 4 2.5 4.42299 4.42299 6.21725e-15 9 5 2.5 5.80092e-15 0 5.80092e-15 9 6 2.5 16.5862 16.5862 3.19744e-14 9 7 2.5 2.9976e-14 0 2.9976e-14 9 8 2.5 78.7845 78.7845 1.84741e-13 9 9 2.5 1.91847e-13 0 1.91847e-13 9 10 2.5 453.011 453.011 1.13687e-12 9 11 2.5 1.50635e-12 0 1.50635e-12 9 12 2.5 3057.82 3057.82 5.91172e-12 9 13 2.5 1.25056e-11 0 1.25056e-11 9 14 2.5 23698.1 23698.1 2.18279e-11 9 15 2.5 1.09139e-10 0 1.09139e-10 9 16 2.5 207359 207359 1.45519e-10 9 17 2.5 1.22236e-09 0 1.22236e-09 9 18 2.5 2.01087e+06 2.02175e+06 10872.3 9 19 2.5 1.44355e-08 0 1.44355e-08 9 20 2.5 2.11168e+07 2.17338e+07 617001 10 0 2.5 0.919063 0.919063 5.55112e-16 10 1 2.5 3.22279e-17 0 3.22279e-17 10 2 2.5 1.60836 1.60836 1.11022e-15 10 3 2.5 -1.60896e-16 0 1.60896e-16 10 4 2.5 4.42299 4.42299 1.77636e-15 10 5 2.5 -1.06165e-15 0 1.06165e-15 10 6 2.5 16.5862 16.5862 1.06581e-14 10 7 2.5 4.21885e-15 0 4.21885e-15 10 8 2.5 78.7845 78.7845 1.42109e-13 10 9 2.5 1.63869e-13 0 1.63869e-13 10 10 2.5 453.011 453.011 1.3074e-12 10 11 2.5 2.01084e-12 0 2.01084e-12 10 12 2.5 3057.82 3057.82 1.18234e-11 10 13 2.5 2.22826e-11 0 2.22826e-11 10 14 2.5 23698.1 23698.1 1.05501e-10 10 15 2.5 2.21917e-10 0 2.21917e-10 10 16 2.5 207359 207359 9.31323e-10 10 17 2.5 1.96451e-09 0 1.96451e-09 10 18 2.5 2.02175e+06 2.02175e+06 8.14907e-09 10 19 2.5 1.6531e-08 0 1.6531e-08 10 20 2.5 2.16794e+07 2.17338e+07 54361.3 10 21 2.5 1.19209e-07 0 1.19209e-07 10 22 2.5 2.51607e+08 2.55372e+08 3.76452e+06 TEST12 GEN_HERMITE_COMPUTE computes a generalized Gauss-Hermite rule which is appropriate for integrands of the form Integral ( -oo < x < +oo ) f(x) x^alpha exp(-x*x) dx. Order = 1 ALPHA = 0.5 0 0 1.225416702465178 Order = 2 ALPHA = 0.5 0 -0.8660254037844385 0.6127083512325888 1 0.8660254037844385 0.6127083512325888 Order = 3 ALPHA = 0.5 0 -1.322875655532295 0.2625892933853953 1 9.974659986866641e-17 0.7002381156943873 2 1.322875655532295 0.262589293385395 Order = 4 ALPHA = 0.5 0 -1.752961966367866 0.07477218653431637 1 -0.6535475074298001 0.5379361646982722 2 0.6535475074298001 0.5379361646982725 3 1.752961966367866 0.0747721865343164 Order = 5 ALPHA = 0.5 0 -2.099598150879759 0.0206908527402406 1 -1.044838554429487 0.337385456421662 2 -2.563766590867021e-16 0.5092640841413719 3 1.044838554429487 0.3373854564216615 4 2.099598150879757 0.02069085274024055 Order = 6 ALPHA = 0.5 0 -2.431196006814871 0.004758432285876815 1 -1.428264330850235 0.1432946705182553 2 -0.5471261076464521 0.4646552484284568 3 0.5471261076464523 0.4646552484284568 4 1.428264330850235 0.1432946705182556 5 2.431196006814872 0.004758432285876798 Order = 7 ALPHA = 0.5 0 -2.719880088556291 0.00110628940196846 1 -1.747360778896522 0.05564733125066088 2 -0.8938582730216025 0.3522490969234104 3 1.730010352426829e-16 0.4074112673130981 4 0.8938582730216027 0.3522490969234107 5 1.747360778896522 0.05564733125066092 6 2.719880088556291 0.00110628940196846 Order = 8 ALPHA = 0.5 0 -2.999078968343317 0.0002288084584739132 1 -2.057439418477469 0.01787577463926715 2 -1.241738340943189 0.1866121206001916 3 -0.4801606747408061 0.4079916475346554 4 0.4801606747408064 0.4079916475346549 5 1.241738340943189 0.1866121206001921 6 2.05743941847747 0.01787577463926723 7 2.999078968343317 0.0002288084584739144 Order = 9 ALPHA = 0.5 0 -3.251152326134132 4.824428349517051e-05 1 -2.331322119300713 0.005575754103643749 2 -1.537416408684744 0.0887579748998607 3 -0.7945417010067838 0.3467847917084952 4 2.193238645380863e-16 0.3430831724741884 5 0.7945417010067843 0.3467847917084951 6 1.537416408684744 0.08875797489986054 7 2.331322119300712 0.005575754103643714 8 3.251152326134132 4.824428349517031e-05 Order = 10 ALPHA = 0.5 0 -3.496605880747678 9.347334083394694e-06 1 -2.598397149544625 0.001536356442402549 2 -1.827991812365275 0.03517634314374581 3 -1.114905370566644 0.2117439807373518 4 -0.4330259998733385 0.3642423235750056 5 0.433025999873339 0.3642423235750059 6 1.114905370566644 0.2117439807373522 7 1.827991812365276 0.03517634314374581 8 2.598397149544625 0.00153635644240255 9 3.496605880747678 9.347334083394745e-06 Order = 1 ALPHA = 1 0 0 1 Order = 2 ALPHA = 1 0 -0.9999999999999998 0.4999999999999999 1 0.9999999999999998 0.4999999999999999 Order = 3 ALPHA = 1 0 -1.414213562373095 0.2500000000000002 1 1.318389841742373e-16 0.5000000000000001 2 1.414213562373095 0.2499999999999999 Order = 4 ALPHA = 1 0 -1.847759065022574 0.07322330470336304 1 -0.7653668647301797 0.4267766952966369 2 0.7653668647301797 0.4267766952966366 3 1.847759065022574 0.07322330470336304 Order = 5 ALPHA = 1 0 -2.175327747161075 0.0223290993692602 1 -1.126032500610494 0.3110042339640734 2 -1.496023277898329e-16 0.3333333333333333 3 1.126032500610494 0.3110042339640728 4 2.175327747161075 0.02232909936926018 Order = 6 ALPHA = 1 0 -2.5079762923396 0.005194628250793071 1 -1.514688205631456 0.1392588667846203 2 -0.6448058287449634 0.3555465049645866 3 0.6448058287449638 0.3555465049645871 4 1.514688205631457 0.1392588667846204 5 2.507976292339599 0.005194628250793074 Order = 7 ALPHA = 1 0 -2.785456961280075 0.001295466838573473 1 -1.818077910688174 0.05917819272755037 2 -0.9673790505919011 0.3145263404338762 3 3.998727457032224e-17 0.2499999999999997 4 0.967379050591901 0.3145263404338764 5 1.818077910688175 0.05917819272755026 6 2.785456961280075 0.00129546683857348 Order = 8 ALPHA = 1 0 -3.065137992375079 0.0002696473527806644 1 -2.129934340988268 0.01944395425750265 2 -1.321272530993643 0.1787093462188998 3 -0.5679328213965031 0.3015770521708169 4 0.5679328213965031 0.3015770521708167 5 1.321272530993642 0.1787093462188998 6 2.129934340988268 0.01944395425750272 7 3.065137992375078 0.0002696473527806649 Order = 9 ALPHA = 1 0 -3.309666797833764 6.006309942116473e-05 1 -2.393988043347146 0.00647142481022687 2 -1.603631817982631 0.09286616703842253 3 -0.8621437977399313 0.3006023450519296 4 5.341886282576138e-16 0.2000000000000001 5 0.8621437977399317 0.3006023450519298 6 1.603631817982631 0.09286616703842276 7 2.393988043347147 0.006471424810226918 8 3.309666797833763 6.006309942116449e-05 Order = 10 ALPHA = 1 0 -3.555390392667984 1.168498619288802e-05 1 -2.661918482196411 0.001805879339961025 2 -1.896424470165033 0.03797122484085379 3 -1.188866291517476 0.1993334055415881 4 -0.5133812615572766 0.2608778052914038 5 0.5133812615572765 0.2608778052914043 6 1.188866291517476 0.1993334055415882 7 1.896424470165033 0.03797122484085385 8 2.661918482196412 0.001805879339961023 9 3.555390392667981 1.168498619288812e-05 Order = 1 ALPHA = 2.5 0 0 0.9190625268488832 Order = 2 ALPHA = 2.5 0 -1.322875655532295 0.4595312634244415 1 1.322875655532295 0.4595312634244415 Order = 3 ALPHA = 2.5 0 -1.6583123951777 0.2924289858155537 1 3.122502256758253e-17 0.3342045552177758 2 1.6583123951777 0.2924289858155537 Order = 4 ALPHA = 2.5 0 -2.099598150879758 0.09121174260159903 1 -1.044838554429487 0.368319520822843 2 1.044838554429487 0.3683195208228425 3 2.099598150879758 0.09121174260159901 Order = 5 ALPHA = 2.5 0 -2.384636591412559 0.03419534541492211 1 -1.346665632923144 0.3362147032847794 2 1.170800878753932e-16 0.1782424294494806 3 1.346665632923144 0.3362147032847794 4 2.38463659141256 0.03419534541492211 Order = 6 ALPHA = 2.5 0 -2.719880088556291 0.008184049874659632 1 -1.747360778896522 0.1699063099275072 2 -0.8938582730216031 0.2814409036222746 3 0.8938582730216029 0.2814409036222749 4 1.747360778896522 0.1699063099275075 5 2.719880088556291 0.008184049874659666 Order = 7 ALPHA = 2.5 0 -2.970220362508424 0.002409813135674279 1 -2.015021402749417 0.08718962060778231 2 -1.169392895737827 0.3136447466969385 3 7.8959598226121e-17 0.1125741659680929 4 1.169392895737826 0.3136447466969383 5 2.015021402749418 0.08718962060778225 6 2.970220362508423 0.002409813135674278 Order = 8 ALPHA = 2.5 0 -3.251152326134132 0.0005099416639456907 1 -2.331322119300714 0.03030457384414924 2 -1.537416408684745 0.2097927175810473 3 -0.7945417010067843 0.2189240303352995 4 0.794541701006785 0.2189240303352991 5 1.537416408684744 0.2097927175810477 6 2.331322119300714 0.03030457384414915 7 3.251152326134132 0.0005099416639456854 Order = 9 ALPHA = 2.5 0 -3.476534136593392 0.0001303236536236376 1 -2.571291374597262 0.0117476928703869 2 -1.789145352000116 0.1310178979856828 3 -1.049347403598514 0.2774791172736721 4 -3.404766541673904e-16 0.07831246328215158 5 1.049347403598514 0.2774791172736726 6 1.789145352000115 0.131017897985683 7 2.571291374597262 0.01174769287038695 8 3.476534136593393 0.0001303236536236379 Order = 10 ALPHA = 2.5 0 -3.722979319411836 2.563250655784923e-05 1 -2.841454909638193 0.003388190430560018 2 -2.087540034875418 0.05794395491107242 3 -1.391239401265924 0.2236074413923169 4 -0.7226261238582253 0.1745660441839348 5 0.7226261238582262 0.1745660441839347 6 1.391239401265924 0.2236074413923169 7 2.087540034875418 0.05794395491107234 8 2.841454909638194 0.00338819043056003 9 3.722979319411834 2.563250655784949e-05 TEST13 GEN_LAGUERRE_COMPUTE computes a generalized Gauss-Laguerre rule which is appropriate for integrands of the form Integral ( 0 <= x < +oo ) f(x) x^alpha exp(-x) dx. GEN_LAGUERRE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Alpha Estimate Exact Error 1 0 0.5 0.886227 0.886227 0 1 1 0.5 1.32934 1.32934 0 1 2 0.5 1.99401 3.32335 1.32934 1 3 0.5 2.99102 11.6317 8.64071 1 4 0.5 4.48652 52.3428 47.8563 2 0 0.5 0.886227 0.886227 3.33067e-16 2 1 0.5 1.32934 1.32934 4.44089e-16 2 2 0.5 3.32335 3.32335 2.22045e-15 2 3 0.5 11.6317 11.6317 1.06581e-14 2 4 0.5 45.6961 52.3428 6.6467 2 5 0.5 184.861 287.885 103.024 2 6 0.5 752.947 1871.25 1118.31 3 0 0.5 0.886227 0.886227 2.22045e-16 3 1 0.5 1.32934 1.32934 0 3 2 0.5 3.32335 3.32335 8.88178e-16 3 3 0.5 11.6317 11.6317 8.88178e-15 3 4 0.5 52.3428 52.3428 4.9738e-14 3 5 0.5 287.885 287.885 2.84217e-13 3 6 0.5 1801.46 1871.25 69.7904 3 7 0.5 12045.4 14034.4 1989.03 3 8 0.5 82966.6 119292 36325.9 4 0 0.5 0.886227 0.886227 6.66134e-16 4 1 0.5 1.32934 1.32934 1.33227e-15 4 2 0.5 3.32335 3.32335 2.66454e-15 4 3 0.5 11.6317 11.6317 7.10543e-15 4 4 0.5 52.3428 52.3428 2.13163e-14 4 5 0.5 287.885 287.885 1.7053e-13 4 6 0.5 1871.25 1871.25 2.04636e-12 4 7 0.5 14034.4 14034.4 2.18279e-11 4 8 0.5 118036 119292 1256.23 4 9 0.5 1.07612e+06 1.13328e+06 57158.3 4 10 0.5 1.03156e+07 1.18994e+07 1.58379e+06 5 0 0.5 0.886227 0.886227 2.22045e-16 5 1 0.5 1.32934 1.32934 4.44089e-16 5 2 0.5 3.32335 3.32335 1.33227e-15 5 3 0.5 11.6317 11.6317 3.55271e-15 5 4 0.5 52.3428 52.3428 2.84217e-14 5 5 0.5 287.885 287.885 1.7053e-13 5 6 0.5 1871.25 1871.25 1.36424e-12 5 7 0.5 14034.4 14034.4 9.09495e-12 5 8 0.5 119292 119292 8.73115e-11 5 9 0.5 1.13328e+06 1.13328e+06 4.65661e-10 5 10 0.5 1.18649e+07 1.18994e+07 34546.2 5 11 0.5 1.34546e+08 1.36843e+08 2.29732e+06 5 12 0.5 1.62055e+09 1.71054e+09 8.99929e+07 6 0 0.5 0.886227 0.886227 3.33067e-16 6 1 0.5 1.32934 1.32934 6.66134e-16 6 2 0.5 3.32335 3.32335 2.66454e-15 6 3 0.5 11.6317 11.6317 8.88178e-15 6 4 0.5 52.3428 52.3428 3.55271e-14 6 5 0.5 287.885 287.885 1.13687e-13 6 6 0.5 1871.25 1871.25 0 6 7 0.5 14034.4 14034.4 5.45697e-12 6 8 0.5 119292 119292 8.73115e-11 6 9 0.5 1.13328e+06 1.13328e+06 1.39698e-09 6 10 0.5 1.18994e+07 1.18994e+07 1.67638e-08 6 11 0.5 1.36843e+08 1.36843e+08 5.66244e-07 6 12 0.5 1.70919e+09 1.71054e+09 1.3473e+06 6 13 0.5 2.2969e+10 2.30923e+10 1.23278e+08 6 14 0.5 3.28348e+11 3.34839e+11 6.4903e+09 7 0 0.5 0.886227 0.886227 1.11022e-16 7 1 0.5 1.32934 1.32934 8.88178e-16 7 2 0.5 3.32335 3.32335 2.66454e-15 7 3 0.5 11.6317 11.6317 1.42109e-14 7 4 0.5 52.3428 52.3428 9.9476e-14 7 5 0.5 287.885 287.885 5.68434e-13 7 6 0.5 1871.25 1871.25 3.63798e-12 7 7 0.5 14034.4 14034.4 2.18279e-11 7 8 0.5 119292 119292 1.45519e-10 7 9 0.5 1.13328e+06 1.13328e+06 1.16415e-09 7 10 0.5 1.18994e+07 1.18994e+07 7.45058e-09 7 11 0.5 1.36843e+08 1.36843e+08 3.27826e-07 7 12 0.5 1.71054e+09 1.71054e+09 1.90735e-06 7 13 0.5 2.30923e+10 2.30923e+10 0.000106812 7 14 0.5 3.34768e+11 3.34839e+11 7.07334e+07 7 15 0.5 5.18148e+12 5.19e+12 8.52338e+09 7 16 0.5 8.50543e+13 8.5635e+13 5.80686e+11 8 0 0.5 0.886227 0.886227 0 8 1 0.5 1.32934 1.32934 0 8 2 0.5 3.32335 3.32335 8.88178e-16 8 3 0.5 11.6317 11.6317 1.77636e-15 8 4 0.5 52.3428 52.3428 3.55271e-14 8 5 0.5 287.885 287.885 2.84217e-13 8 6 0.5 1871.25 1871.25 2.95586e-12 8 7 0.5 14034.4 14034.4 2.72848e-11 8 8 0.5 119292 119292 2.47383e-10 8 9 0.5 1.13328e+06 1.13328e+06 2.56114e-09 8 10 0.5 1.18994e+07 1.18994e+07 3.1665e-08 8 11 0.5 1.36843e+08 1.36843e+08 7.15256e-07 8 12 0.5 1.71054e+09 1.71054e+09 3.33786e-06 8 13 0.5 2.30923e+10 2.30923e+10 0.000175476 8 14 0.5 3.34839e+11 3.34839e+11 0.000366211 8 15 0.5 5.19e+12 5.19e+12 0.00195312 8 16 0.5 8.56302e+13 8.5635e+13 4.80987e+09 8 17 0.5 1.49787e+15 1.49861e+15 7.38315e+11 8 18 0.5 2.76611e+16 2.77243e+16 6.32101e+13 9 0 0.5 0.886227 0.886227 1.11022e-16 9 1 0.5 1.32934 1.32934 2.22045e-16 9 2 0.5 3.32335 3.32335 0 9 3 0.5 11.6317 11.6317 7.10543e-15 9 4 0.5 52.3428 52.3428 4.26326e-14 9 5 0.5 287.885 287.885 4.54747e-13 9 6 0.5 1871.25 1871.25 3.18323e-12 9 7 0.5 14034.4 14034.4 2.36469e-11 9 8 0.5 119292 119292 2.03727e-10 9 9 0.5 1.13328e+06 1.13328e+06 1.86265e-09 9 10 0.5 1.18994e+07 1.18994e+07 1.49012e-08 9 11 0.5 1.36843e+08 1.36843e+08 4.76837e-07 9 12 0.5 1.71054e+09 1.71054e+09 9.53674e-07 9 13 0.5 2.30923e+10 2.30923e+10 9.53674e-05 9 14 0.5 3.34839e+11 3.34839e+11 0.00164795 9 15 0.5 5.19e+12 5.19e+12 0.0234375 9 16 0.5 8.5635e+13 8.5635e+13 0.015625 9 17 0.5 1.49861e+15 1.49861e+15 6.25 9 18 0.5 2.77239e+16 2.77243e+16 4.11244e+11 9 19 0.5 5.40546e+17 5.40624e+17 7.8342e+13 9 20 0.5 1.10746e+19 1.10828e+19 8.23393e+15 10 0 0.5 0.886227 0.886227 0 10 1 0.5 1.32934 1.32934 2.22045e-16 10 2 0.5 3.32335 3.32335 1.77636e-15 10 3 0.5 11.6317 11.6317 1.06581e-14 10 4 0.5 52.3428 52.3428 5.68434e-14 10 5 0.5 287.885 287.885 2.27374e-13 10 6 0.5 1871.25 1871.25 1.36424e-12 10 7 0.5 14034.4 14034.4 5.45697e-12 10 8 0.5 119292 119292 0 10 9 0.5 1.13328e+06 1.13328e+06 0 10 10 0.5 1.18994e+07 1.18994e+07 0 10 11 0.5 1.36843e+08 1.36843e+08 3.57628e-07 10 12 0.5 1.71054e+09 1.71054e+09 1.19209e-06 10 13 0.5 2.30923e+10 2.30923e+10 0.000114441 10 14 0.5 3.34839e+11 3.34839e+11 0.00109863 10 15 0.5 5.19e+12 5.19e+12 0.0126953 10 16 0.5 8.5635e+13 8.5635e+13 0.25 10 17 0.5 1.49861e+15 1.49861e+15 1.5 10 18 0.5 2.77243e+16 2.77243e+16 40 10 19 0.5 5.40624e+17 5.40624e+17 896 10 20 0.5 1.10828e+19 1.10828e+19 4.31806e+13 10 21 0.5 2.3827e+20 2.3828e+20 9.99632e+15 10 22 0.5 5.36004e+21 5.3613e+21 1.26546e+18 1 0 1 1 1 0 1 1 1 2 2 0 1 2 1 4 6 2 1 3 1 8 24 16 1 4 1 16 120 104 2 0 1 1 1 2.22045e-16 2 1 1 2 2 0 2 2 1 6 6 0 2 3 1 24 24 3.55271e-15 2 4 1 108 120 12 2 5 1 504 720 216 2 6 1 2376 5040 2664 3 0 1 1 1 4.44089e-16 3 1 1 2 2 1.33227e-15 3 2 1 6 6 5.32907e-15 3 3 1 24 24 2.84217e-14 3 4 1 120 120 1.7053e-13 3 5 1 720 720 1.13687e-12 3 6 1 4896 5040 144 3 7 1 35712 40320 4608 3 8 1 269568 362880 93312 4 0 1 1 1 0 4 1 1 2 2 0 4 2 1 6 6 1.77636e-15 4 3 1 24 24 1.06581e-14 4 4 1 120 120 9.9476e-14 4 5 1 720 720 9.09495e-13 4 6 1 5040 5040 8.18545e-12 4 7 1 40320 40320 8.00355e-11 4 8 1 360000 362880 2880 4 9 1 3.4848e+06 3.6288e+06 144000 4 10 1 3.5568e+07 3.99168e+07 4.3488e+06 5 0 1 1 1 0 5 1 1 2 2 2.22045e-16 5 2 1 6 6 2.66454e-15 5 3 1 24 24 2.4869e-14 5 4 1 120 120 1.84741e-13 5 5 1 720 720 1.81899e-12 5 6 1 5040 5040 1.54614e-11 5 7 1 40320 40320 1.52795e-10 5 8 1 362880 362880 1.39698e-09 5 9 1 3.6288e+06 3.6288e+06 1.21072e-08 5 10 1 3.98304e+07 3.99168e+07 86400 5 11 1 4.72781e+08 4.79002e+08 6.2208e+06 5 12 1 5.96471e+09 6.22702e+09 2.6231e+08 6 0 1 1 1 2.22045e-16 6 1 1 2 2 8.88178e-16 6 2 1 6 6 6.21725e-15 6 3 1 24 24 3.19744e-14 6 4 1 120 120 1.56319e-13 6 5 1 720 720 1.02318e-12 6 6 1 5040 5040 7.27596e-12 6 7 1 40320 40320 4.36557e-11 6 8 1 362880 362880 3.49246e-10 6 9 1 3.6288e+06 3.6288e+06 2.79397e-09 6 10 1 3.99168e+07 3.99168e+07 7.45058e-09 6 11 1 4.79002e+08 4.79002e+08 2.6226e-06 6 12 1 6.22339e+09 6.22702e+09 3.6288e+06 6 13 1 8.68227e+10 8.71783e+10 3.55622e+08 6 14 1 1.28771e+12 1.30767e+12 1.99657e+10 7 0 1 1 1 1.11022e-16 7 1 1 2 2 0 7 2 1 6 6 1.77636e-15 7 3 1 24 24 7.10543e-15 7 4 1 120 120 7.10543e-14 7 5 1 720 720 6.82121e-13 7 6 1 5040 5040 5.45697e-12 7 7 1 40320 40320 5.82077e-11 7 8 1 362880 362880 5.82077e-10 7 9 1 3.6288e+06 3.6288e+06 6.98492e-09 7 10 1 3.99168e+07 3.99168e+07 5.21541e-08 7 11 1 4.79002e+08 4.79002e+08 3.21865e-06 7 12 1 6.22702e+09 6.22702e+09 2.09808e-05 7 13 1 8.71783e+10 8.71783e+10 0.000274658 7 14 1 1.30747e+12 1.30767e+12 2.03213e+08 7 15 1 2.08968e+13 2.09228e+13 2.60112e+10 7 16 1 3.53811e+14 3.55687e+14 1.87606e+12 8 0 1 1 1 3.33067e-16 8 1 1 2 2 6.66134e-16 8 2 1 6 6 8.88178e-16 8 3 1 24 24 1.06581e-14 8 4 1 120 120 7.10543e-14 8 5 1 720 720 1.02318e-12 8 6 1 5040 5040 1.00044e-11 8 7 1 40320 40320 1.01863e-10 8 8 1 362880 362880 9.8953e-10 8 9 1 3.6288e+06 3.6288e+06 1.02445e-08 8 10 1 3.99168e+07 3.99168e+07 1.2666e-07 8 11 1 4.79002e+08 4.79002e+08 1.49012e-06 8 12 1 6.22702e+09 6.22702e+09 6.67572e-06 8 13 1 8.71783e+10 8.71783e+10 0.000167847 8 14 1 1.30767e+12 1.30767e+12 0.00366211 8 15 1 2.09228e+13 2.09228e+13 0 8 16 1 3.55673e+14 3.55687e+14 1.46313e+10 8 17 1 6.4e+15 6.40237e+15 2.37027e+12 8 18 1 1.21432e+17 1.21645e+17 2.13588e+14 9 0 1 1 1 1.11022e-16 9 1 1 2 2 8.88178e-16 9 2 1 6 6 0 9 3 1 24 24 3.55271e-15 9 4 1 120 120 2.84217e-14 9 5 1 720 720 2.27374e-13 9 6 1 5040 5040 1.81899e-12 9 7 1 40320 40320 7.27596e-12 9 8 1 362880 362880 5.82077e-11 9 9 1 3.6288e+06 3.6288e+06 1.39698e-09 9 10 1 3.99168e+07 3.99168e+07 7.45058e-09 9 11 1 4.79002e+08 4.79002e+08 2.74181e-06 9 12 1 6.22702e+09 6.22702e+09 1.81198e-05 9 13 1 8.71783e+10 8.71783e+10 0.000244141 9 14 1 1.30767e+12 1.30767e+12 0.00317383 9 15 1 2.09228e+13 2.09228e+13 0.0390625 9 16 1 3.55687e+14 3.55687e+14 0.5625 9 17 1 6.40237e+15 6.40237e+15 5 9 18 1 1.21644e+17 1.21645e+17 1.31682e+12 9 19 1 2.43264e+18 2.4329e+18 2.63364e+14 9 20 1 5.10619e+19 5.10909e+19 2.89964e+16 10 0 1 1 1 4.44089e-16 10 1 1 2 2 4.44089e-16 10 2 1 6 6 1.77636e-15 10 3 1 24 24 1.42109e-14 10 4 1 120 120 8.52651e-14 10 5 1 720 720 6.82121e-13 10 6 1 5040 5040 6.36646e-12 10 7 1 40320 40320 7.27596e-11 10 8 1 362880 362880 6.40284e-10 10 9 1 3.6288e+06 3.6288e+06 6.0536e-09 10 10 1 3.99168e+07 3.99168e+07 2.23517e-08 10 11 1 4.79002e+08 4.79002e+08 2.86102e-06 10 12 1 6.22702e+09 6.22702e+09 1.62125e-05 10 13 1 8.71783e+10 8.71783e+10 0.000183105 10 14 1 1.30767e+12 1.30767e+12 0.00170898 10 15 1 2.09228e+13 2.09228e+13 0.0664062 10 16 1 3.55687e+14 3.55687e+14 1.125 10 17 1 6.40237e+15 6.40237e+15 6 10 18 1 1.21645e+17 1.21645e+17 432 10 19 1 2.4329e+18 2.4329e+18 9216 10 20 1 5.10908e+19 5.10909e+19 1.4485e+14 10 21 1 1.12397e+21 1.124e+21 3.50537e+16 10 22 1 2.58474e+22 2.5852e+22 4.63028e+18 1 0 2.5 3.32335 3.32335 4.44089e-16 1 1 2.5 11.6317 11.6317 1.77636e-15 1 2 2.5 40.711 52.3428 11.6317 1 3 2.5 142.489 287.885 145.397 1 4 2.5 498.71 1871.25 1372.54 2 0 2.5 3.32335 3.32335 8.88178e-16 2 1 2.5 11.6317 11.6317 3.55271e-15 2 2 2.5 52.3428 52.3428 2.84217e-14 2 3 2.5 287.885 287.885 2.27374e-13 2 4 2.5 1766.57 1871.25 104.686 2 5 2.5 11364.9 14034.4 2669.48 2 6 2.5 74460.9 119292 44831.6 3 0 2.5 3.32335 3.32335 1.33227e-15 3 1 2.5 11.6317 11.6317 3.55271e-15 3 2 2.5 52.3428 52.3428 2.84217e-14 3 3 2.5 287.885 287.885 1.7053e-13 3 4 2.5 1871.25 1871.25 1.36424e-12 3 5 2.5 14034.4 14034.4 1.27329e-11 3 6 2.5 117565 119292 1727.31 3 7 2.5 1.05987e+06 1.13328e+06 73410.7 3 8 2.5 9.97433e+06 1.18994e+07 1.92509e+06 4 0 2.5 3.32335 3.32335 1.33227e-15 4 1 2.5 11.6317 11.6317 1.77636e-15 4 2 2.5 52.3428 52.3428 0 4 3 2.5 287.885 287.885 1.13687e-13 4 4 2.5 1871.25 1871.25 1.13687e-12 4 5 2.5 14034.4 14034.4 1.27329e-11 4 6 2.5 119292 119292 1.74623e-10 4 7 2.5 1.13328e+06 1.13328e+06 2.32831e-09 4 8 2.5 1.18545e+07 1.18994e+07 44910.1 4 9 2.5 1.33992e+08 1.36843e+08 2.85179e+06 4 10 2.5 1.60308e+09 1.71054e+09 1.07459e+08 5 0 2.5 3.32335 3.32335 1.33227e-15 5 1 2.5 11.6317 11.6317 5.32907e-15 5 2 2.5 52.3428 52.3428 1.42109e-14 5 3 2.5 287.885 287.885 1.7053e-13 5 4 2.5 1871.25 1871.25 9.09495e-13 5 5 2.5 14034.4 14034.4 1.09139e-11 5 6 2.5 119292 119292 1.16415e-10 5 7 2.5 1.13328e+06 1.13328e+06 1.39698e-09 5 8 2.5 1.18994e+07 1.18994e+07 2.04891e-08 5 9 2.5 1.36843e+08 1.36843e+08 5.96046e-08 5 10 2.5 1.70886e+09 1.71054e+09 1.68413e+06 5 11 2.5 2.29433e+10 2.30923e+10 1.49045e+08 5 12 2.5 3.27215e+11 3.34839e+11 7.62405e+09 6 0 2.5 3.32335 3.32335 4.44089e-16 6 1 2.5 11.6317 11.6317 1.77636e-15 6 2 2.5 52.3428 52.3428 0 6 3 2.5 287.885 287.885 5.68434e-14 6 4 2.5 1871.25 1871.25 2.27374e-13 6 5 2.5 14034.4 14034.4 3.63798e-12 6 6 2.5 119292 119292 4.36557e-11 6 7 2.5 1.13328e+06 1.13328e+06 6.98492e-10 6 8 2.5 1.18994e+07 1.18994e+07 1.30385e-08 6 9 2.5 1.36843e+08 1.36843e+08 1.78814e-07 6 10 2.5 1.71054e+09 1.71054e+09 3.57628e-06 6 11 2.5 2.30923e+10 2.30923e+10 7.24792e-05 6 12 2.5 3.34753e+11 3.34839e+11 8.58906e+07 6 13 2.5 5.17991e+12 5.19e+12 1.00921e+10 6 14 2.5 8.49625e+13 8.5635e+13 6.72459e+11 7 0 2.5 3.32335 3.32335 1.77636e-15 7 1 2.5 11.6317 11.6317 1.06581e-14 7 2 2.5 52.3428 52.3428 4.26326e-14 7 3 2.5 287.885 287.885 2.84217e-13 7 4 2.5 1871.25 1871.25 1.59162e-12 7 5 2.5 14034.4 14034.4 5.45697e-12 7 6 2.5 119292 119292 1.45519e-11 7 7 2.5 1.13328e+06 1.13328e+06 4.65661e-10 7 8 2.5 1.18994e+07 1.18994e+07 1.30385e-08 7 9 2.5 1.36843e+08 1.36843e+08 8.9407e-08 7 10 2.5 1.71054e+09 1.71054e+09 5.72205e-06 7 11 2.5 2.30923e+10 2.30923e+10 3.43323e-05 7 12 2.5 3.34839e+11 3.34839e+11 0.00250244 7 13 2.5 5.19e+12 5.19e+12 0.0371094 7 14 2.5 8.56293e+13 8.5635e+13 5.71172e+09 7 15 2.5 1.49775e+15 1.49861e+15 8.59614e+11 7 16 2.5 2.7652e+16 2.77243e+16 7.23076e+13 8 0 2.5 3.32335 3.32335 1.33227e-15 8 1 2.5 11.6317 11.6317 3.55271e-15 8 2 2.5 52.3428 52.3428 1.42109e-14 8 3 2.5 287.885 287.885 5.68434e-14 8 4 2.5 1871.25 1871.25 2.27374e-13 8 5 2.5 14034.4 14034.4 0 8 6 2.5 119292 119292 1.45519e-11 8 7 2.5 1.13328e+06 1.13328e+06 0 8 8 2.5 1.18994e+07 1.18994e+07 1.86265e-09 8 9 2.5 1.36843e+08 1.36843e+08 2.98023e-07 8 10 2.5 1.71054e+09 1.71054e+09 2.38419e-06 8 11 2.5 2.30923e+10 2.30923e+10 8.7738e-05 8 12 2.5 3.34839e+11 3.34839e+11 0.00164795 8 13 2.5 5.19e+12 5.19e+12 0.0224609 8 14 2.5 8.5635e+13 8.5635e+13 0.015625 8 15 2.5 1.49861e+15 1.49861e+15 6.25 8 16 2.5 2.77238e+16 2.77243e+16 4.79785e+11 8 17 2.5 5.40534e+17 5.40624e+17 8.99596e+13 8 18 2.5 1.10735e+19 1.10828e+19 9.32018e+15 9 0 2.5 3.32335 3.32335 1.33227e-15 9 1 2.5 11.6317 11.6317 7.10543e-15 9 2 2.5 52.3428 52.3428 1.42109e-14 9 3 2.5 287.885 287.885 5.68434e-14 9 4 2.5 1871.25 1871.25 2.27374e-13 9 5 2.5 14034.4 14034.4 5.45697e-12 9 6 2.5 119292 119292 4.36557e-11 9 7 2.5 1.13328e+06 1.13328e+06 0 9 8 2.5 1.18994e+07 1.18994e+07 1.86265e-09 9 9 2.5 1.36843e+08 1.36843e+08 3.57628e-07 9 10 2.5 1.71054e+09 1.71054e+09 1.43051e-06 9 11 2.5 2.30923e+10 2.30923e+10 0.000102997 9 12 2.5 3.34839e+11 3.34839e+11 0.00146484 9 13 2.5 5.19e+12 5.19e+12 0.0195312 9 14 2.5 8.5635e+13 8.5635e+13 0.03125 9 15 2.5 1.49861e+15 1.49861e+15 6.75 9 16 2.5 2.77243e+16 2.77243e+16 164 9 17 2.5 5.40624e+17 5.40624e+17 2304 9 18 2.5 1.10827e+19 1.10828e+19 4.96577e+13 9 19 2.5 2.38269e+20 2.3828e+20 1.13468e+16 9 20 2.5 5.35988e+21 5.3613e+21 1.41942e+18 10 0 2.5 3.32335 3.32335 1.77636e-15 10 1 2.5 11.6317 11.6317 7.10543e-15 10 2 2.5 52.3428 52.3428 5.68434e-14 10 3 2.5 287.885 287.885 3.41061e-13 10 4 2.5 1871.25 1871.25 2.50111e-12 10 5 2.5 14034.4 14034.4 1.63709e-11 10 6 2.5 119292 119292 1.45519e-10 10 7 2.5 1.13328e+06 1.13328e+06 1.62981e-09 10 8 2.5 1.18994e+07 1.18994e+07 1.49012e-08 10 9 2.5 1.36843e+08 1.36843e+08 1.78814e-07 10 10 2.5 1.71054e+09 1.71054e+09 4.52995e-06 10 11 2.5 2.30923e+10 2.30923e+10 5.34058e-05 10 12 2.5 3.34839e+11 3.34839e+11 0.0022583 10 13 2.5 5.19e+12 5.19e+12 0.0322266 10 14 2.5 8.5635e+13 8.5635e+13 0.171875 10 15 2.5 1.49861e+15 1.49861e+15 10 10 16 2.5 2.77243e+16 2.77243e+16 212 10 17 2.5 5.40624e+17 5.40624e+17 2944 10 18 2.5 1.10828e+19 1.10828e+19 6144 10 19 2.5 2.3828e+20 2.3828e+20 360448 10 20 2.5 5.3613e+21 5.3613e+21 6.20722e+15 10 21 2.5 1.25989e+23 1.25991e+23 1.69767e+18 10 22 2.5 3.08652e+24 3.08677e+24 2.5225e+20 TEST14 GEN_LAGUERRE_COMPUTE computes a generalized Gauss-Laguerre rule which is appropriate for integrands of the form Integral ( 0 <= x < +oo ) f(x) x^alpha exp(-x) dx. Order = 1 ALPHA = 0.5 0 1.5 0.8862269254527581 Order = 2 ALPHA = 0.5 0 0.9188611699158105 0.7233630235462752 1 4.081138830084189 0.1628639019064825 Order = 3 ALPHA = 0.5 0 0.666325907702371 0.5671862778403116 1 2.800775054150256 0.3053717688445466 2 7.032899038147372 0.01366887876790012 Order = 4 ALPHA = 0.5 0 0.5235260767382686 0.4530087465586073 1 2.156648763269094 0.3816169601718006 2 5.137387546176712 0.05079462757224074 3 10.18243761381593 0.0008065911501100318 Order = 5 ALPHA = 0.5 0 0.4313988071478522 0.3704505700074587 1 1.759753698423698 0.4125843737694527 2 4.104465362828316 0.09777982005318063 3 7.746703779542558 0.005373415341171988 4 13.45767835205758 3.874628149393578e-05 Order = 6 ALPHA = 0.5 0 0.3669498773083705 0.3094240968362603 1 1.488534292310453 0.4177521497070222 2 3.434007968424071 0.1432858732209771 3 6.349067925680379 0.01533249102263385 4 10.54046985844834 0.0004306911960439409 5 16.82097007782838 1.623469821074067e-06 Order = 7 ALPHA = 0.5 0 0.3193036339206293 0.2631245143958913 1 1.290758622959152 0.4091418694141027 2 2.958374458696651 0.1821177320927163 3 5.409031597244436 0.03005332430127098 4 8.804079578056774 0.001760894117540066 5 13.46853574325148 2.852947122115974e-05 6 20.24991636587088 6.166001541039146e-08 Order = 8 ALPHA = 0.5 0 0.2826336481165983 0.227139361952471 1 1.139873801581612 0.3935945428036152 2 2.601524843406029 0.2129089708672288 3 4.72411453752779 0.04787748320313817 4 7.605256299231614 0.004542517474762631 5 11.41718207654583 0.0001624046001853252 6 16.49941079765581 1.642377413806097e-06 7 23.73000399593471 2.173943126630911e-09 Order = 9 ALPHA = 0.5 0 0.2535325549744193 0.1985712548680196 1 1.02084427772039 0.3749207846631705 2 2.323096077022465 0.2360748210008252 3 4.199350600657293 0.06709610500320433 4 6.71397431661503 0.009008508896644308 5 9.972009159539351 0.0005426607386359281 6 14.15405367127805 1.270536687910834e-05 7 19.61190281916595 8.484309239668581e-08 8 27.25123652302706 7.228647164396506e-11 Order = 10 ALPHA = 0.5 0 0.2298729805186566 0.1754708150466599 1 0.9244815469866562 0.355223388802072 2 2.099410462708798 0.2526835596756785 3 3.782880873707289 0.08635610269533257 4 6.019918027701461 0.01510977803486079 5 8.880347597996709 0.001328215628363565 6 12.47483240483621 5.418780021170343e-05 7 16.99084729354256 8.737475869187116e-07 8 22.79100289494895 4.019699886939779e-09 9 30.80640591705272 2.292221530204704e-12 Order = 1 ALPHA = 1 0 2 1 Order = 2 ALPHA = 1 0 1.267949192431123 0.7886751345948131 1 4.732050807568878 0.2113248654051871 Order = 3 ALPHA = 1 0 0.9358222275240875 0.5886814810396593 1 3.305407289332279 0.3912160592223105 2 7.758770483143635 0.02010245973803056 Order = 4 ALPHA = 1 0 0.7432919279814317 0.4468705932187766 1 2.571635007646279 0.477635772363868 2 5.731178751689098 0.07417778473105212 3 10.95389431268319 0.00131584968630324 Order = 5 ALPHA = 1 0 0.6170308532782706 0.3480145400233489 1 2.112965958578525 0.502280674132493 2 4.610833151017531 0.1409159194944725 3 8.399066971204835 0.00871989302609999 4 14.26010306592084 6.89733235856401e-05 Order = 6 ALPHA = 1 0 0.5276681217111289 0.2776501420298752 1 1.796299809643409 0.4939105830503541 2 3.876641520476912 0.2030042967437302 3 6.918816566704723 0.0246688203691898 4 11.23461042908311 0.000763042767463531 5 17.64596355238071 3.115039387527555e-06 Order = 7 ALPHA = 1 0 0.4610242198049947 0.226210596383189 1 1.563586189654263 0.4697087077412661 2 3.352050502536734 0.2531225162324008 3 5.916297249020539 0.04780310888423712 4 9.420699383021594 0.003100459756025395 5 14.19416554800746 5.448466889341361e-05 6 21.09217690795441 1.26333988153628e-07 Order = 8 ALPHA = 1 0 0.409383573203186 0.187632541405724 1 1.384963184803141 0.4389853607311423 2 2.956254556168864 0.289996070781313 3 5.181943101040073 0.07514138461669727 4 8.161709688145816 0.007932646648707327 5 12.07005512683716 0.0003086421368133028 6 17.249735526149 3.348958209797085e-06 7 24.58595524365278 4.721392823193254e-09 Order = 9 ALPHA = 1 0 0.3681784529417419 0.1580309235007823 1 1.243357962140484 0.4065825794555729 2 2.646033841384209 0.314982483665097 3 4.616882514635062 0.1037755199516432 4 7.221786539396571 0.01557763772427843 5 10.56732080774187 0.001024871969850758 6 14.83591451526093 2.580021138717233e-05 7 20.38218198544925 1.833559550172429e-07 8 28.11834338104989 1.654329455969927e-10 Order = 10 ALPHA = 1 0 0.3345286763247526 0.1348565545313779 1 1.128253355876639 0.3749243411054598 2 2.395869924747309 0.3302471308042158 3 4.166840987928768 0.1315288846413064 4 6.487353031380811 0.02584206421171715 5 9.428354813335606 0.002489645795895956 6 13.1017235803678 0.0001094884764266961 7 17.69648756684622 1.881275414181711e-06 8 23.57778708836015 9.152685028575581e-09 9 31.68280097483193 5.501303496178695e-12 Order = 1 ALPHA = 2.5 0 3.5 3.323350970447842 Order = 2 ALPHA = 2.5 0 2.378679656440358 2.444996821046109 1 6.621320343559642 0.8783541494017333 Order = 3 ALPHA = 2.5 0 1.824819803075874 1.640491690990948 1 4.813746226487698 1.576420610283191 2 9.861433970436428 0.1064386691737022 Order = 4 ALPHA = 2.5 0 1.48624114266535 1.109022578585149 1 3.8376839879001 1.820324124907821 2 7.482058366898981 0.3853099204909315 3 13.19401650253557 0.008694346463943117 Order = 5 ALPHA = 2.5 0 1.255846140918443 0.769346187647318 1 3.207040626614909 1.789143657137866 2 6.123908223328411 0.7075497394015858 3 10.31612586639135 0.05675567145898618 4 16.59707914274689 0.000555714802087273 Order = 6 ALPHA = 2.5 0 1.088263537814163 0.5488342282458472 1 2.76085557138551 1.635828855251358 2 5.213082470955716 0.97570188767536 3 8.608834500537 0.1568919688876181 4 13.27367226930421 0.006064045733978006 5 20.05529165000341 2.99846536813126e-05 Order = 7 ALPHA = 2.5 0 0.9606221295119317 0.401864465233434 1 2.426677101427741 1.44402357612933 2 4.550654967582769 1.157819445450442 3 7.432417772624608 0.2949071223080977 4 11.24467756580187 0.02421750114397002 5 16.32710951627492 0.0005174322318932411 6 23.55784094677616 1.427950674360879e-06 Order = 8 ALPHA = 2.5 0 0.8600416237603872 0.3011867253184345 1 2.166243528992247 1.25296378034376 2 4.043742677349125 1.258345796079632 3 6.558922538427902 0.4473442363684211 4 9.817255505225706 0.06058255027905918 5 13.99947975071506 0.002890481769915709 6 19.45744743181463 3.733852866568666e-05 7 27.09686694371495 6.175995605353352e-08 Order = 9 ALPHA = 2.5 0 0.7786800215607389 0.2304206696494764 1 1.957190223504775 1.078733434868177 2 3.641788795064597 1.294498383186512 3 5.87932972203498 0.5941550811279991 4 8.740028894917545 0.1158251816851087 5 12.33467558588798 0.00943162869193187 6 16.85079612749538 0.00028422378718938 7 22.65102569838607 2.364977643120573e-06 8 30.66648493114793 2.473803674819386e-09 Order = 10 ALPHA = 2.5 0 0.711476069898304 0.179502944206614 1 1.785477806129485 0.9263170003438098 2 3.314481217602208 1.284767518101653 3 5.333049252470586 0.7226374668786899 4 7.890763402531441 0.1864748527585193 5 11.06167397628529 0.02243790620506889 6 14.96102232439852 0.001189142723014832 7 19.78221807877333 2.400423791594193e-05 8 25.89766445973611 1.348994981145529e-07 9 34.26217341217473 9.306038978295321e-11 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^d. A rule of order N should be exact for monomials x^d up to degree D = 2*N-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. N D Estimate Exact Error 1 0 1.77245 1.77245 2.22045e-16 1 1 0 0 0 1 2 0 0.886227 0.886227 1 3 0 0 0 1 4 0 1.32934 1.32934 2 0 1.77245 1.77245 2.22045e-16 2 1 0 0 0 2 2 0.886227 0.886227 3.33067e-16 2 3 0 0 0 2 4 0.443113 1.32934 0.886227 2 5 0 0 0 2 6 0.221557 3.32335 3.10179 3 0 1.77245 1.77245 2.22045e-16 3 1 5.55112e-17 0 5.55112e-17 3 2 0.886227 0.886227 1.22125e-15 3 3 3.33067e-16 0 3.33067e-16 3 4 1.32934 1.32934 2.66454e-15 3 5 4.44089e-16 0 4.44089e-16 3 6 1.99401 3.32335 1.32934 3 7 6.66134e-16 0 6.66134e-16 3 8 2.99102 11.6317 8.64071 4 0 1.77245 1.77245 4.44089e-16 4 1 -3.33067e-16 0 3.33067e-16 4 2 0.886227 0.886227 3.33067e-16 4 3 7.21645e-16 0 7.21645e-16 4 4 1.32934 1.32934 1.33227e-15 4 5 2.88658e-15 0 2.88658e-15 4 6 3.32335 3.32335 5.32907e-15 4 7 1.06581e-14 0 1.06581e-14 4 8 8.97305 11.6317 2.65868 4 9 3.28626e-14 0 3.28626e-14 4 10 24.4266 52.3428 27.9161 5 0 1.77245 1.77245 1.33227e-15 5 1 -2.77556e-17 0 2.77556e-17 5 2 0.886227 0.886227 4.44089e-16 5 3 -4.71845e-16 0 4.71845e-16 5 4 1.32934 1.32934 4.44089e-16 5 5 -2.55351e-15 0 2.55351e-15 5 6 3.32335 3.32335 1.33227e-15 5 7 -1.15463e-14 0 1.15463e-14 5 8 11.6317 11.6317 1.77636e-15 5 9 -4.79616e-14 0 4.79616e-14 5 10 45.6961 52.3428 6.6467 5 11 -1.98952e-13 0 1.98952e-13 5 12 184.861 287.885 103.024 6 0 1.77245 1.77245 6.66134e-16 6 1 -7.45931e-16 0 7.45931e-16 6 2 0.886227 0.886227 7.77156e-16 6 3 -1.19349e-15 0 1.19349e-15 6 4 1.32934 1.32934 1.77636e-15 6 5 -3.16414e-15 0 3.16414e-15 6 6 3.32335 3.32335 3.10862e-15 6 7 -1.26565e-14 0 1.26565e-14 6 8 11.6317 11.6317 0 6 9 -6.03961e-14 0 6.03961e-14 6 10 52.3428 52.3428 6.39488e-14 6 11 -3.12639e-13 0 3.12639e-13 6 12 267.945 287.885 19.9401 6 13 -1.53477e-12 0 1.53477e-12 6 14 1442.54 1871.25 428.712 7 0 1.77245 1.77245 1.33227e-15 7 1 3.47378e-16 0 3.47378e-16 7 2 0.886227 0.886227 6.66134e-16 7 3 5.44703e-16 0 5.44703e-16 7 4 1.32934 1.32934 1.33227e-15 7 5 1.7486e-15 0 1.7486e-15 7 6 3.32335 3.32335 3.9968e-15 7 7 6.43929e-15 0 6.43929e-15 7 8 11.6317 11.6317 1.24345e-14 7 9 2.93099e-14 0 2.93099e-14 7 10 52.3428 52.3428 5.68434e-14 7 11 1.42109e-13 0 1.42109e-13 7 12 287.885 287.885 3.41061e-13 7 13 7.38964e-13 0 7.38964e-13 7 14 1801.46 1871.25 69.7904 7 15 5.00222e-12 0 5.00222e-12 7 16 12045.4 14034.4 1989.03 8 0 1.77245 1.77245 8.88178e-16 8 1 -6.66893e-16 0 6.66893e-16 8 2 0.886227 0.886227 4.44089e-16 8 3 -3.66027e-16 0 3.66027e-16 8 4 1.32934 1.32934 1.33227e-15 8 5 2.56739e-16 0 2.56739e-16 8 6 3.32335 3.32335 7.99361e-15 8 7 3.83027e-15 0 3.83027e-15 8 8 11.6317 11.6317 4.44089e-14 8 9 1.95399e-14 0 1.95399e-14 8 10 52.3428 52.3428 2.27374e-13 8 11 7.81597e-14 0 7.81597e-14 8 12 287.885 287.885 1.25056e-12 8 13 1.7053e-13 0 1.7053e-13 8 14 1871.25 1871.25 8.18545e-12 8 15 0 0 0 8 16 13755.2 14034.4 279.161 8 17 -1.09139e-11 0 1.09139e-11 8 18 109103 119292 10189.4 9 0 1.77245 1.77245 1.33227e-15 9 1 4.80871e-16 0 4.80871e-16 9 2 0.886227 0.886227 1.55431e-15 9 3 6.39462e-16 0 6.39462e-16 9 4 1.32934 1.32934 3.77476e-15 9 5 8.50015e-16 0 8.50015e-16 9 6 3.32335 3.32335 1.15463e-14 9 7 2.69229e-15 0 2.69229e-15 9 8 11.6317 11.6317 3.90799e-14 9 9 1.33227e-14 0 1.33227e-14 9 10 52.3428 52.3428 1.63425e-13 9 11 7.4607e-14 0 7.4607e-14 9 12 287.885 287.885 6.82121e-13 9 13 1.7053e-13 0 1.7053e-13 9 14 1871.25 1871.25 2.95586e-12 9 15 -4.54747e-13 0 4.54747e-13 9 16 14034.4 14034.4 3.63798e-12 9 17 -2.72848e-11 0 2.72848e-11 9 18 118036 119292 1256.23 9 19 -4.65661e-10 0 4.65661e-10 9 20 1.07612e+06 1.13328e+06 57158.3 10 0 1.77245 1.77245 6.66134e-16 10 1 -7.40117e-16 0 7.40117e-16 10 2 0.886227 0.886227 1.11022e-16 10 3 -1.93498e-15 0 1.93498e-15 10 4 1.32934 1.32934 2.22045e-15 10 5 -4.02109e-15 0 4.02109e-15 10 6 3.32335 3.32335 1.11022e-14 10 7 -5.72459e-15 0 5.72459e-15 10 8 11.6317 11.6317 6.03961e-14 10 9 2.40918e-14 0 2.40918e-14 10 10 52.3428 52.3428 3.2685e-13 10 11 3.78364e-13 0 3.78364e-13 10 12 287.885 287.885 1.81899e-12 10 13 3.36797e-12 0 3.36797e-12 10 14 1871.25 1871.25 1.11413e-11 10 15 2.55795e-11 0 2.55795e-11 10 16 14034.4 14034.4 6.18456e-11 10 17 1.81899e-10 0 1.81899e-10 10 18 119292 119292 3.49246e-10 10 19 1.19326e-09 0 1.19326e-09 10 20 1.127e+06 1.13328e+06 6281.13 10 21 6.98492e-09 0 6.98492e-09 10 22 1.15508e+07 1.18994e+07 348603 TEST16 HERMITE_COMPUTE computes a Gauss-Hermite rule which is appropriate for integrands of the form Integral ( -oo < x < +oo ) f(x) exp(-x*x) dx. Order = 1 0 0 1.772453850905516 Order = 2 0 -0.7071067811865475 0.8862269254527578 1 0.7071067811865475 0.8862269254527578 Order = 3 0 -1.224744871391589 0.2954089751509195 1 -9.862844991098402e-17 1.181635900603677 2 1.224744871391589 0.2954089751509196 Order = 4 0 -1.650680123885784 0.08131283544724513 1 -0.5246476232752904 0.8049140900055129 2 0.5246476232752902 0.8049140900055121 3 1.650680123885784 0.08131283544724525 Order = 5 0 -2.020182870456086 0.01995324205904597 1 -0.9585724646138184 0.3936193231522413 2 2.402579402021668e-16 0.945308720482943 3 0.9585724646138187 0.3936193231522411 4 2.020182870456086 0.01995324205904589 Order = 6 0 -2.350604973674493 0.004530009905508857 1 -1.335849074013696 0.1570673203228569 2 -0.4360774119276162 0.7246295952243929 3 0.436077411927616 0.7246295952243927 4 1.335849074013696 0.1570673203228566 5 2.350604973674493 0.004530009905508823 Order = 7 0 -2.651961356835234 0.0009717812450995172 1 -1.673551628767471 0.05451558281912706 2 -0.8162878828589647 0.4256072526101277 3 -1.059786349168287e-16 0.8102646175568082 4 0.8162878828589646 0.4256072526101282 5 1.673551628767472 0.05451558281912713 6 2.651961356835234 0.0009717812450995191 Order = 8 0 -2.930637420257243 0.0001996040722113679 1 -1.981656756695842 0.01707798300741343 2 -1.15719371244678 0.2078023258148922 3 -0.3811869902073223 0.6611470125582418 4 0.381186990207322 0.661147012558241 5 1.15719371244678 0.2078023258148921 6 1.981656756695842 0.01707798300741346 7 2.930637420257243 0.0001996040722113677 Order = 9 0 -3.190993201781527 3.960697726326444e-05 1 -2.266580584531843 0.004943624275536957 2 -1.468553289216668 0.08847452739437681 3 -0.7235510187528373 0.4326515590025557 4 -2.361558093483325e-16 0.7202352156060512 5 0.7235510187528379 0.4326515590025563 6 1.468553289216668 0.08847452739437676 7 2.266580584531844 0.004943624275536958 8 3.190993201781527 3.960697726326444e-05 Order = 10 0 -3.436159118837738 7.640432855232631e-06 1 -2.532731674232788 0.001343645746781224 2 -1.756683649299881 0.03387439445548121 3 -1.036610829789514 0.240138611082315 4 -0.3429013272237045 0.6108626337353257 5 0.3429013272237046 0.6108626337353257 6 1.036610829789512 0.2401386110823148 7 1.75668364929988 0.03387439445548109 8 2.532731674232791 0.001343645746781228 9 3.436159118837738 7.640432855232614e-06 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 0.5 0.5 1.5708 1.5708 1.33227e-15 1 1 0.5 0.5 0 0 0 1 2 0.5 0.5 0 0.392699 0.392699 1 3 0.5 0.5 0 0 0 1 4 0.5 0.5 0 0.19635 0.19635 2 0 0.5 0.5 1.5708 1.5708 1.33227e-15 2 1 0.5 0.5 0 0 0 2 2 0.5 0.5 0.392699 0.392699 2.77556e-16 2 3 0.5 0.5 0 0 0 2 4 0.5 0.5 0.0981748 0.19635 0.0981748 2 5 0.5 0.5 0 0 0 2 6 0.5 0.5 0.0245437 0.122718 0.0981748 3 0 0.5 0.5 1.5708 1.5708 1.11022e-15 3 1 0.5 0.5 -3.88578e-16 0 3.88578e-16 3 2 0.5 0.5 0.392699 0.392699 2.77556e-16 3 3 0.5 0.5 -2.498e-16 0 2.498e-16 3 4 0.5 0.5 0.19635 0.19635 2.22045e-16 3 5 0.5 0.5 -1.38778e-16 0 1.38778e-16 3 6 0.5 0.5 0.0981748 0.122718 0.0245437 3 7 0.5 0.5 -8.32667e-17 0 8.32667e-17 3 8 0.5 0.5 0.0490874 0.0859029 0.0368155 4 0 0.5 0.5 1.5708 1.5708 2.22045e-16 4 1 0.5 0.5 3.60822e-16 0 3.60822e-16 4 2 0.5 0.5 0.392699 0.392699 1.66533e-16 4 3 0.5 0.5 1.80411e-16 0 1.80411e-16 4 4 0.5 0.5 0.19635 0.19635 1.66533e-16 4 5 0.5 0.5 1.66533e-16 0 1.66533e-16 4 6 0.5 0.5 0.122718 0.122718 1.80411e-16 4 7 0.5 0.5 1.45717e-16 0 1.45717e-16 4 8 0.5 0.5 0.079767 0.0859029 0.00613592 4 9 0.5 0.5 1.17961e-16 0 1.17961e-16 4 10 0.5 0.5 0.0521553 0.0644272 0.0122718 5 0 0.5 0.5 1.5708 1.5708 1.55431e-15 5 1 0.5 0.5 -2.35922e-16 0 2.35922e-16 5 2 0.5 0.5 0.392699 0.392699 5.55112e-17 5 3 0.5 0.5 -1.80411e-16 0 1.80411e-16 5 4 0.5 0.5 0.19635 0.19635 1.11022e-16 5 5 0.5 0.5 -1.249e-16 0 1.249e-16 5 6 0.5 0.5 0.122718 0.122718 1.66533e-16 5 7 0.5 0.5 -9.71445e-17 0 9.71445e-17 5 8 0.5 0.5 0.0859029 0.0859029 1.66533e-16 5 9 0.5 0.5 -9.02056e-17 0 9.02056e-17 5 10 0.5 0.5 0.0628932 0.0644272 0.00153398 5 11 0.5 0.5 -6.59195e-17 0 6.59195e-17 5 12 0.5 0.5 0.0467864 0.0506214 0.00383495 6 0 0.5 0.5 1.5708 1.5708 1.11022e-15 6 1 0.5 0.5 -1.38778e-17 0 1.38778e-17 6 2 0.5 0.5 0.392699 0.392699 5.55112e-17 6 3 0.5 0.5 -1.52656e-16 0 1.52656e-16 6 4 0.5 0.5 0.19635 0.19635 2.22045e-16 6 5 0.5 0.5 -1.38778e-16 0 1.38778e-16 6 6 0.5 0.5 0.122718 0.122718 2.08167e-16 6 7 0.5 0.5 -1.04083e-16 0 1.04083e-16 6 8 0.5 0.5 0.0859029 0.0859029 2.22045e-16 6 9 0.5 0.5 -7.63278e-17 0 7.63278e-17 6 10 0.5 0.5 0.0644272 0.0644272 4.16334e-17 6 11 0.5 0.5 -5.89806e-17 0 5.89806e-17 6 12 0.5 0.5 0.0502379 0.0506214 0.000383495 6 13 0.5 0.5 -4.85723e-17 0 4.85723e-17 6 14 0.5 0.5 0.0399794 0.0411299 0.00115049 7 0 0.5 0.5 1.5708 1.5708 0 7 1 0.5 0.5 7.63278e-17 0 7.63278e-17 7 2 0.5 0.5 0.392699 0.392699 2.22045e-16 7 3 0.5 0.5 -5.55112e-17 0 5.55112e-17 7 4 0.5 0.5 0.19635 0.19635 1.11022e-16 7 5 0.5 0.5 -1.59595e-16 0 1.59595e-16 7 6 0.5 0.5 0.122718 0.122718 8.32667e-17 7 7 0.5 0.5 -1.8735e-16 0 1.8735e-16 7 8 0.5 0.5 0.0859029 0.0859029 2.77556e-17 7 9 0.5 0.5 -1.94289e-16 0 1.94289e-16 7 10 0.5 0.5 0.0644272 0.0644272 1.80411e-16 7 11 0.5 0.5 -1.9082e-16 0 1.9082e-16 7 12 0.5 0.5 0.0506214 0.0506214 4.51028e-16 7 13 0.5 0.5 -1.8735e-16 0 1.8735e-16 7 14 0.5 0.5 0.041034 0.0411299 9.58738e-05 7 15 0.5 0.5 -1.76942e-16 0 1.76942e-16 7 16 0.5 0.5 0.0339393 0.0342749 0.000335558 8 0 0.5 0.5 1.5708 1.5708 0 8 1 0.5 0.5 -1.80411e-16 0 1.80411e-16 8 2 0.5 0.5 0.392699 0.392699 2.22045e-16 8 3 0.5 0.5 -2.01228e-16 0 2.01228e-16 8 4 0.5 0.5 0.19635 0.19635 8.32667e-17 8 5 0.5 0.5 -1.70003e-16 0 1.70003e-16 8 6 0.5 0.5 0.122718 0.122718 8.32667e-17 8 7 0.5 0.5 -1.14492e-16 0 1.14492e-16 8 8 0.5 0.5 0.0859029 0.0859029 9.71445e-17 8 9 0.5 0.5 -5.89806e-17 0 5.89806e-17 8 10 0.5 0.5 0.0644272 0.0644272 2.63678e-16 8 11 0.5 0.5 -1.73472e-17 0 1.73472e-17 8 12 0.5 0.5 0.0506214 0.0506214 5.75928e-16 8 13 0.5 0.5 6.93889e-18 0 6.93889e-18 8 14 0.5 0.5 0.0411299 0.0411299 5.55112e-17 8 15 0.5 0.5 2.08167e-17 0 2.08167e-17 8 16 0.5 0.5 0.0342509 0.0342749 2.39684e-05 8 17 0.5 0.5 3.81639e-17 0 3.81639e-17 8 18 0.5 0.5 0.0290378 0.0291337 9.58738e-05 9 0 0.5 0.5 1.5708 1.5708 0 9 1 0.5 0.5 2.87964e-16 0 2.87964e-16 9 2 0.5 0.5 0.392699 0.392699 3.33067e-16 9 3 0.5 0.5 -3.46945e-17 0 3.46945e-17 9 4 0.5 0.5 0.19635 0.19635 1.11022e-16 9 5 0.5 0.5 -7.28584e-17 0 7.28584e-17 9 6 0.5 0.5 0.122718 0.122718 6.93889e-17 9 7 0.5 0.5 -7.63278e-17 0 7.63278e-17 9 8 0.5 0.5 0.0859029 0.0859029 1.38778e-17 9 9 0.5 0.5 -7.28584e-17 0 7.28584e-17 9 10 0.5 0.5 0.0644272 0.0644272 1.66533e-16 9 11 0.5 0.5 -6.93889e-17 0 6.93889e-17 9 12 0.5 0.5 0.0506214 0.0506214 4.64906e-16 9 13 0.5 0.5 -6.76542e-17 0 6.76542e-17 9 14 0.5 0.5 0.0411299 0.0411299 6.93889e-17 9 15 0.5 0.5 -6.07153e-17 0 6.07153e-17 9 16 0.5 0.5 0.0342749 0.0342749 1.94289e-16 9 17 0.5 0.5 -5.89806e-17 0 5.89806e-17 9 18 0.5 0.5 0.0291277 0.0291337 5.99211e-06 9 19 0.5 0.5 -5.20417e-17 0 5.20417e-17 9 20 0.5 0.5 0.0251339 0.0251609 2.69645e-05 10 0 0.5 0.5 1.5708 1.5708 4.44089e-16 10 1 0.5 0.5 -2.22045e-16 0 2.22045e-16 10 2 0.5 0.5 0.392699 0.392699 2.22045e-16 10 3 0.5 0.5 -1.21431e-16 0 1.21431e-16 10 4 0.5 0.5 0.19635 0.19635 2.77556e-17 10 5 0.5 0.5 -7.63278e-17 0 7.63278e-17 10 6 0.5 0.5 0.122718 0.122718 0 10 7 0.5 0.5 -5.20417e-17 0 5.20417e-17 10 8 0.5 0.5 0.0859029 0.0859029 0 10 9 0.5 0.5 -3.81639e-17 0 3.81639e-17 10 10 0.5 0.5 0.0644272 0.0644272 1.80411e-16 10 11 0.5 0.5 -2.77556e-17 0 2.77556e-17 10 12 0.5 0.5 0.0506214 0.0506214 4.92661e-16 10 13 0.5 0.5 -2.25514e-17 0 2.25514e-17 10 14 0.5 0.5 0.0411299 0.0411299 2.77556e-17 10 15 0.5 0.5 -1.73472e-17 0 1.73472e-17 10 16 0.5 0.5 0.0342749 0.0342749 1.52656e-16 10 17 0.5 0.5 -1.73472e-17 0 1.73472e-17 10 18 0.5 0.5 0.0291337 0.0291337 3.46945e-17 10 19 0.5 0.5 -1.56125e-17 0 1.56125e-17 10 20 0.5 0.5 0.0251594 0.0251609 1.49803e-06 10 21 0.5 0.5 -1.73472e-17 0 1.73472e-17 10 22 0.5 0.5 0.0220083 0.0220158 7.49014e-06 1 0 0.5 1 1.50849 1.50849 4.44089e-16 1 1 0.5 1 0.215499 0.215499 0 1 2 0.5 1 0.0307856 0.359165 0.32838 1 3 0.5 1 0.00439794 0.111015 0.106617 1 4 0.5 1 0.000628278 0.174308 0.17368 2 0 0.5 1 1.50849 1.50849 4.44089e-16 2 1 0.5 1 0.215499 0.215499 0 2 2 0.5 1 0.359165 0.359165 5.55112e-17 2 3 0.5 1 0.111015 0.111015 8.32667e-17 2 4 0.5 1 0.0963711 0.174308 0.0779371 2 5 0.5 1 0.0410706 0.0708284 0.0297578 2 6 0.5 1 0.0279097 0.106701 0.0787909 3 0 0.5 1 1.50849 1.50849 4.44089e-16 3 1 0.5 1 0.215499 0.215499 5.55112e-17 3 2 0.5 1 0.359165 0.359165 1.11022e-16 3 3 0.5 1 0.111015 0.111015 0 3 4 0.5 1 0.174308 0.174308 0 3 5 0.5 1 0.0708284 0.0708284 2.77556e-17 3 6 0.5 1 0.0877418 0.106701 0.0189588 3 7 0.5 1 0.0425665 0.0503496 0.0077831 3 8 0.5 1 0.044985 0.0735313 0.0285464 4 0 0.5 1 1.50849 1.50849 4.44089e-16 4 1 0.5 1 0.215499 0.215499 5.55112e-17 4 2 0.5 1 0.359165 0.359165 1.66533e-16 4 3 0.5 1 0.111015 0.111015 1.52656e-16 4 4 0.5 1 0.174308 0.174308 1.66533e-16 4 5 0.5 1 0.0708284 0.0708284 1.11022e-16 4 6 0.5 1 0.106701 0.106701 8.32667e-17 4 7 0.5 1 0.0503496 0.0503496 8.32667e-17 4 8 0.5 1 0.068871 0.0735313 0.00466038 4 9 0.5 1 0.0362285 0.0382228 0.00199426 4 10 0.5 1 0.0451345 0.0544715 0.00933698 5 0 0.5 1 1.50849 1.50849 8.88178e-16 5 1 0.5 1 0.215499 0.215499 0 5 2 0.5 1 0.359165 0.359165 3.88578e-16 5 3 0.5 1 0.111015 0.111015 2.22045e-16 5 4 0.5 1 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-0.0403895 -0.0403895 4.85723e-17 6 10 2.5 1 0.0361793 0.0361793 2.56739e-16 6 11 2.5 1 -0.029559 -0.029559 1.38778e-16 6 12 2.5 1 0.0267044 0.0268067 0.000102303 6 13 2.5 1 -0.0224459 -0.0225667 0.000120784 6 14 2.5 1 0.0203315 0.0206669 0.000335346 7 0 2.5 1 1.43666 1.43666 2.22045e-16 7 1 2.5 1 -0.391817 -0.391817 5.55112e-17 7 2 2.5 1 0.311444 0.311444 1.11022e-16 7 3 2.5 1 -0.166773 -0.166773 5.55112e-17 7 4 2.5 1 0.139352 0.139352 5.55112e-17 7 5 2.5 1 -0.0922233 -0.0922233 2.77556e-17 7 6 2.5 1 0.0795329 0.0795329 1.11022e-16 7 7 2.5 1 -0.0584903 -0.0584903 7.63278e-17 7 8 2.5 1 0.0515572 0.0515572 7.63278e-17 7 9 2.5 1 -0.0403895 -0.0403895 7.63278e-17 7 10 2.5 1 0.0361793 0.0361793 1.11022e-16 7 11 2.5 1 -0.029559 -0.029559 1.04083e-17 7 12 2.5 1 0.0268067 0.0268067 1.49186e-16 7 13 2.5 1 -0.0225667 -0.0225667 6.59195e-17 7 14 2.5 1 0.0206424 0.0206669 2.44522e-05 7 15 2.5 1 -0.0177618 -0.0177915 2.97189e-05 7 16 2.5 1 0.0163311 0.0164239 9.28462e-05 8 0 2.5 1 1.43666 1.43666 8.88178e-16 8 1 2.5 1 -0.391817 -0.391817 1.11022e-16 8 2 2.5 1 0.311444 0.311444 0 8 3 2.5 1 -0.166773 -0.166773 8.32667e-17 8 4 2.5 1 0.139352 0.139352 1.38778e-16 8 5 2.5 1 -0.0922233 -0.0922233 1.11022e-16 8 6 2.5 1 0.0795329 0.0795329 1.66533e-16 8 7 2.5 1 -0.0584903 -0.0584903 1.249e-16 8 8 2.5 1 0.0515572 0.0515572 1.11022e-16 8 9 2.5 1 -0.0403895 -0.0403895 1.249e-16 8 10 2.5 1 0.0361793 0.0361793 6.93889e-17 8 11 2.5 1 -0.029559 -0.029559 2.42861e-17 8 12 2.5 1 0.0268067 0.0268067 1.73472e-16 8 13 2.5 1 -0.0225667 -0.0225667 7.97973e-17 8 14 2.5 1 0.0206669 0.0206669 1.73472e-16 8 15 2.5 1 -0.0177915 -0.0177915 1.249e-16 8 16 2.5 1 0.016418 0.0164239 5.89662e-06 8 17 2.5 1 -0.0143787 -0.0143861 7.33122e-06 8 18 2.5 1 0.0133428 0.0133683 2.54338e-05 9 0 2.5 1 1.43666 1.43666 6.66134e-16 9 1 2.5 1 -0.391817 -0.391817 3.33067e-16 9 2 2.5 1 0.311444 0.311444 1.11022e-16 9 3 2.5 1 -0.166773 -0.166773 1.66533e-16 9 4 2.5 1 0.139352 0.139352 1.11022e-16 9 5 2.5 1 -0.0922233 -0.0922233 6.93889e-17 9 6 2.5 1 0.0795329 0.0795329 4.16334e-17 9 7 2.5 1 -0.0584903 -0.0584903 5.55112e-17 9 8 2.5 1 0.0515572 0.0515572 6.93889e-18 9 9 2.5 1 -0.0403895 -0.0403895 2.08167e-17 9 10 2.5 1 0.0361793 0.0361793 2.08167e-16 9 11 2.5 1 -0.029559 -0.029559 8.67362e-17 9 12 2.5 1 0.0268067 0.0268067 4.51028e-17 9 13 2.5 1 -0.0225667 -0.0225667 3.46945e-17 9 14 2.5 1 0.0206669 0.0206669 4.51028e-17 9 15 2.5 1 -0.0177915 -0.0177915 3.46945e-18 9 16 2.5 1 0.0164239 0.0164239 1.31839e-16 9 17 2.5 1 -0.0143861 -0.0143861 7.80626e-17 9 18 2.5 1 0.0133668 0.0133683 1.4312e-06 9 19 2.5 1 -0.0118706 -0.0118724 1.81219e-06 9 20 2.5 1 0.0110872 0.0110941 6.90871e-06 10 0 2.5 1 1.43666 1.43666 6.66134e-16 10 1 2.5 1 -0.391817 -0.391817 2.77556e-16 10 2 2.5 1 0.311444 0.311444 1.66533e-16 10 3 2.5 1 -0.166773 -0.166773 8.32667e-17 10 4 2.5 1 0.139352 0.139352 0 10 5 2.5 1 -0.0922233 -0.0922233 1.38778e-17 10 6 2.5 1 0.0795329 0.0795329 2.77556e-17 10 7 2.5 1 -0.0584903 -0.0584903 3.46945e-17 10 8 2.5 1 0.0515572 0.0515572 4.16334e-17 10 9 2.5 1 -0.0403895 -0.0403895 6.93889e-17 10 10 2.5 1 0.0361793 0.0361793 1.17961e-16 10 11 2.5 1 -0.029559 -0.029559 1.04083e-17 10 12 2.5 1 0.0268067 0.0268067 1.42247e-16 10 13 2.5 1 -0.0225667 -0.0225667 6.59195e-17 10 14 2.5 1 0.0206669 0.0206669 1.59595e-16 10 15 2.5 1 -0.0177915 -0.0177915 1.14492e-16 10 16 2.5 1 0.0164239 0.0164239 3.46945e-18 10 17 2.5 1 -0.0143861 -0.0143861 4.85723e-17 10 18 2.5 1 0.0133683 0.0133683 7.63278e-17 10 19 2.5 1 -0.0118724 -0.0118724 5.20417e-17 10 20 2.5 1 0.0110937 0.0110941 3.4907e-07 10 21 2.5 1 -0.00996383 -0.00996428 4.48679e-07 10 22 2.5 1 0.0093537 0.00935557 1.86378e-06 1 0 2.5 2.5 0.981748 0.981748 1.11022e-16 1 1 2.5 2.5 0 0 0 1 2 2.5 2.5 0 0.122718 0.122718 1 3 2.5 2.5 0 0 0 1 4 2.5 2.5 0 0.0368155 0.0368155 2 0 2.5 2.5 0.981748 0.981748 1.11022e-16 2 1 2.5 2.5 0 0 0 2 2 2.5 2.5 0.122718 0.122718 1.249e-16 2 3 2.5 2.5 0 0 0 2 4 2.5 2.5 0.0153398 0.0368155 0.0214757 2 5 2.5 2.5 0 0 0 2 6 2.5 2.5 0.00191748 0.0153398 0.0134223 3 0 2.5 2.5 0.981748 0.981748 3.33067e-16 3 1 2.5 2.5 1.11022e-16 0 1.11022e-16 3 2 2.5 2.5 0.122718 0.122718 1.38778e-17 3 3 2.5 2.5 -6.93889e-18 0 6.93889e-18 3 4 2.5 2.5 0.0368155 0.0368155 6.93889e-18 3 5 2.5 2.5 -1.04083e-17 0 1.04083e-17 3 6 2.5 2.5 0.0110447 0.0153398 0.00429515 3 7 2.5 2.5 -6.50521e-18 0 6.50521e-18 3 8 2.5 2.5 0.0033134 0.0076699 0.00435651 4 0 2.5 2.5 0.981748 0.981748 1.11022e-16 4 1 2.5 2.5 -6.93889e-17 0 6.93889e-17 4 2 2.5 2.5 0.122718 0.122718 4.16334e-17 4 3 2.5 2.5 1.04083e-17 0 1.04083e-17 4 4 2.5 2.5 0.0368155 0.0368155 6.93889e-18 4 5 2.5 2.5 5.20417e-18 0 5.20417e-18 4 6 2.5 2.5 0.0153398 0.0153398 1.38778e-17 4 7 2.5 2.5 2.60209e-18 0 2.60209e-18 4 8 2.5 2.5 0.00674952 0.0076699 0.000920388 4 9 2.5 2.5 1.30104e-18 0 1.30104e-18 4 10 2.5 2.5 0.00299126 0.00431432 0.00132306 5 0 2.5 2.5 0.981748 0.981748 5.55112e-16 5 1 2.5 2.5 2.91434e-16 0 2.91434e-16 5 2 2.5 2.5 0.122718 0.122718 0 5 3 2.5 2.5 5.89806e-17 0 5.89806e-17 5 4 2.5 2.5 0.0368155 0.0368155 0 5 5 2.5 2.5 2.42861e-17 0 2.42861e-17 5 6 2.5 2.5 0.0153398 0.0153398 1.73472e-17 5 7 2.5 2.5 1.04083e-17 0 1.04083e-17 5 8 2.5 2.5 0.0076699 0.0076699 2.68882e-17 5 9 2.5 2.5 3.46945e-18 0 3.46945e-18 5 10 2.5 2.5 0.00410888 0.00431432 0.000205444 5 11 2.5 2.5 2.1684e-18 0 2.1684e-18 5 12 2.5 2.5 0.0022501 0.00263653 0.00038643 6 0 2.5 2.5 0.981748 0.981748 1.33227e-15 6 1 2.5 2.5 2.28983e-16 0 2.28983e-16 6 2 2.5 2.5 0.122718 0.122718 2.77556e-17 6 3 2.5 2.5 5.0307e-17 0 5.0307e-17 6 4 2.5 2.5 0.0368155 0.0368155 4.16334e-17 6 5 2.5 2.5 1.38778e-17 0 1.38778e-17 6 6 2.5 2.5 0.0153398 0.0153398 1.04083e-17 6 7 2.5 2.5 7.37257e-18 0 7.37257e-18 6 8 2.5 2.5 0.0076699 0.0076699 5.20417e-18 6 9 2.5 2.5 6.07153e-18 0 6.07153e-18 6 10 2.5 2.5 0.00431432 0.00431432 8.67362e-18 6 11 2.5 2.5 4.77049e-18 0 4.77049e-18 6 12 2.5 2.5 0.00258945 0.00263653 4.70809e-05 6 13 2.5 2.5 3.25261e-18 0 3.25261e-18 6 14 2.5 2.5 0.00160369 0.00171374 0.000110052 7 0 2.5 2.5 0.981748 0.981748 3.33067e-16 7 1 2.5 2.5 -1.21431e-17 0 1.21431e-17 7 2 2.5 2.5 0.122718 0.122718 0 7 3 2.5 2.5 -1.47451e-17 0 1.47451e-17 7 4 2.5 2.5 0.0368155 0.0368155 0 7 5 2.5 2.5 -3.46945e-18 0 3.46945e-18 7 6 2.5 2.5 0.0153398 0.0153398 1.04083e-17 7 7 2.5 2.5 -8.67362e-19 0 8.67362e-19 7 8 2.5 2.5 0.0076699 0.0076699 2.08167e-17 7 9 2.5 2.5 1.0842e-18 0 1.0842e-18 7 10 2.5 2.5 0.00431432 0.00431432 1.64799e-17 7 11 2.5 2.5 6.50521e-19 0 6.50521e-19 7 12 2.5 2.5 0.00263653 0.00263653 8.67362e-19 7 13 2.5 2.5 4.33681e-19 0 4.33681e-19 7 14 2.5 2.5 0.00170276 0.00171374 1.09855e-05 7 15 2.5 2.5 4.33681e-19 0 4.33681e-19 7 16 2.5 2.5 0.00113767 0.00116846 3.07928e-05 8 0 2.5 2.5 0.981748 0.981748 9.99201e-16 8 1 2.5 2.5 1.96891e-16 0 1.96891e-16 8 2 2.5 2.5 0.122718 0.122718 9.71445e-17 8 3 2.5 2.5 8.0231e-17 0 8.0231e-17 8 4 2.5 2.5 0.0368155 0.0368155 2.08167e-17 8 5 2.5 2.5 3.98986e-17 0 3.98986e-17 8 6 2.5 2.5 0.0153398 0.0153398 1.73472e-17 8 7 2.5 2.5 2.77556e-17 0 2.77556e-17 8 8 2.5 2.5 0.0076699 0.0076699 2.42861e-17 8 9 2.5 2.5 1.92988e-17 0 1.92988e-17 8 10 2.5 2.5 0.00431432 0.00431432 2.08167e-17 8 11 2.5 2.5 1.45283e-17 0 1.45283e-17 8 12 2.5 2.5 0.00263653 0.00263653 4.77049e-18 8 13 2.5 2.5 1.11673e-17 0 1.11673e-17 8 14 2.5 2.5 0.00171374 0.00171374 1.0842e-18 8 15 2.5 2.5 8.5652e-18 0 8.5652e-18 8 16 2.5 2.5 0.00116587 0.00116846 2.59658e-06 8 17 2.5 2.5 6.83047e-18 0 6.83047e-18 8 18 2.5 2.5 0.000819157 0.000827661 8.50381e-06 9 0 2.5 2.5 0.981748 0.981748 2.22045e-16 9 1 2.5 2.5 -4.81386e-17 0 4.81386e-17 9 2 2.5 2.5 0.122718 0.122718 1.38778e-17 9 3 2.5 2.5 -1.73472e-17 0 1.73472e-17 9 4 2.5 2.5 0.0368155 0.0368155 3.46945e-17 9 5 2.5 2.5 -1.77809e-17 0 1.77809e-17 9 6 2.5 2.5 0.0153398 0.0153398 4.68375e-17 9 7 2.5 2.5 -1.60462e-17 0 1.60462e-17 9 8 2.5 2.5 0.0076699 0.0076699 4.59702e-17 9 9 2.5 2.5 -1.36609e-17 0 1.36609e-17 9 10 2.5 2.5 0.00431432 0.00431432 3.64292e-17 9 11 2.5 2.5 -1.05168e-17 0 1.05168e-17 9 12 2.5 2.5 0.00263653 0.00263653 1.51788e-17 9 13 2.5 2.5 -8.13152e-18 0 8.13152e-18 9 14 2.5 2.5 0.00171374 0.00171374 5.85469e-18 9 15 2.5 2.5 -6.07153e-18 0 6.07153e-18 9 16 2.5 2.5 0.00116846 0.00116846 4.33681e-19 9 17 2.5 2.5 -4.66207e-18 0 4.66207e-18 9 18 2.5 2.5 0.000827041 0.000827661 6.19639e-07 9 19 2.5 2.5 -3.52366e-18 0 3.52366e-18 9 20 2.5 2.5 0.000602504 0.000604829 2.32473e-06 10 0 2.5 2.5 0.981748 0.981748 1.11022e-16 10 1 2.5 2.5 1.47451e-17 0 1.47451e-17 10 2 2.5 2.5 0.122718 0.122718 1.38778e-17 10 3 2.5 2.5 1.60462e-17 0 1.60462e-17 10 4 2.5 2.5 0.0368155 0.0368155 2.08167e-17 10 5 2.5 2.5 -1.73472e-18 0 1.73472e-18 10 6 2.5 2.5 0.0153398 0.0153398 0 10 7 2.5 2.5 -1.51788e-18 0 1.51788e-18 10 8 2.5 2.5 0.0076699 0.0076699 1.56125e-17 10 9 2.5 2.5 3.25261e-19 0 3.25261e-19 10 10 2.5 2.5 0.00431432 0.00431432 1.64799e-17 10 11 2.5 2.5 8.67362e-19 0 8.67362e-19 10 12 2.5 2.5 0.00263653 0.00263653 2.60209e-18 10 13 2.5 2.5 1.30104e-18 0 1.30104e-18 10 14 2.5 2.5 0.00171374 0.00171374 2.81893e-18 10 15 2.5 2.5 1.19262e-18 0 1.19262e-18 10 16 2.5 2.5 0.00116846 0.00116846 4.98733e-18 10 17 2.5 2.5 1.19262e-18 0 1.19262e-18 10 18 2.5 2.5 0.000827661 0.000827661 2.81893e-18 10 19 2.5 2.5 9.48677e-19 0 9.48677e-19 10 20 2.5 2.5 0.00060468 0.000604829 1.48952e-07 10 21 2.5 2.5 7.86047e-19 0 7.86047e-19 10 22 2.5 2.5 0.000452991 0.000453622 6.30385e-07 TEST18 JACOBI_COMPUTE computes a Gauss-Jacobi rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) (1-x)^alpha (1+x)^beta dx. N = 1 ALPHA = 0.5 BETA = 0.5 0 0 1.570796326794897 N = 2 ALPHA = 0.5 BETA = 0.5 0 -0.4999999999999999 0.7853981633974484 1 0.4999999999999999 0.7853981633974484 N = 3 ALPHA = 0.5 BETA = 0.5 0 -0.7071067811865475 0.3926990816987245 1 6.591949208711867e-17 0.7853981633974486 2 0.7071067811865474 0.3926990816987239 N = 4 ALPHA = 0.5 BETA = 0.5 0 -0.8090169943749475 0.2170787134227061 1 -0.3090169943749473 0.5683194499747424 2 0.3090169943749472 0.5683194499747432 3 0.8090169943749477 0.2170787134227062 N = 5 ALPHA = 0.5 BETA = 0.5 0 -0.8660254037844389 0.1308996938995749 1 -0.4999999999999998 0.3926990816987244 2 5.952490290336006e-17 0.5235987755982987 3 0.4999999999999998 0.3926990816987242 4 0.8660254037844388 0.1308996938995747 N = 6 ALPHA = 0.5 BETA = 0.5 0 -0.9009688679024188 0.0844886908915887 1 -0.6234898018587335 0.2743330560697781 2 -0.2225209339563142 0.4265764164360816 3 0.2225209339563143 0.4265764164360817 4 0.6234898018587332 0.2743330560697784 5 0.9009688679024188 0.08448869089158853 N = 7 ALPHA = 0.5 BETA = 0.5 0 -0.9238795325112868 0.05750944903191328 1 -0.7071067811865476 0.1963495408493622 2 -0.3826834323650896 0.3351896326668111 3 7.901929723605659e-18 0.3926990816987249 4 0.3826834323650899 0.335189632666811 5 0.7071067811865475 0.1963495408493624 6 0.9238795325112863 0.0575094490319132 N = 8 ALPHA = 0.5 BETA = 0.5 0 -0.9396926207859084 0.04083294770910714 1 -0.7660444431189782 0.144225600795673 2 -0.4999999999999999 0.2617993877991496 3 -0.1736481776669302 0.3385402270935193 4 0.1736481776669302 0.338540227093519 5 0.5 0.2617993877991501 6 0.7660444431189779 0.1442256007956725 7 0.9396926207859086 0.04083294770910712 N = 9 ALPHA = 0.5 BETA = 0.5 0 -0.9510565162951536 0.02999954037160819 1 -0.8090169943749472 0.1085393567113534 2 -0.587785252292473 0.2056199086476264 3 -0.3090169943749472 0.2841597249873707 4 5.567534423109432e-17 0.3141592653589796 5 0.3090169943749471 0.2841597249873716 6 0.5877852522924728 0.2056199086476266 7 0.8090169943749472 0.1085393567113536 8 0.9510565162951536 0.02999954037160805 N = 10 ALPHA = 0.5 BETA = 0.5 0 -0.9594929736144974 0.02266894250185894 1 -0.8412535328311809 0.08347854093418919 2 -0.6548607339452849 0.1631221774548168 3 -0.4154150130018863 0.2363135602034877 4 -0.1423148382732851 0.2798149423030964 5 0.1423148382732851 0.2798149423030961 6 0.4154150130018863 0.2363135602034874 7 0.6548607339452848 0.1631221774548172 8 0.8412535328311812 0.08347854093418883 9 0.9594929736144973 0.02266894250185892 N = 1 ALPHA = 0.5 BETA = 1 0 0.1428571428571428 1.508494466531301 N = 2 ALPHA = 0.5 BETA = 1 0 -0.3785434359039294 0.6707847483124826 1 0.5603616177221111 0.8377097182188189 N = 3 ALPHA = 0.5 BETA = 1 0 -0.6191289006136985 0.2886624303931599 1 0.08283748568481 0.7711842727220678 2 0.7362914149288884 0.4486477634160737 N = 4 ALPHA = 0.5 BETA = 1 0 -0.744717344008601 0.1380862915234591 1 -0.2308286409011867 0.5011891464890383 2 0.3610180502309863 0.6101454555318188 3 0.8250542504682752 0.2590735729869861 N = 5 ALPHA = 0.5 BETA = 1 0 -0.8175993274098119 0.07295559350758933 1 -0.4326084003843105 0.3103351805927456 2 0.05829076979846798 0.5176668424406614 3 0.5335962093870672 0.4466360854739412 4 0.8757120529564133 0.1609007645163651 N = 6 ALPHA = 0.5 BETA = 1 0 -0.8633993972645073 0.04177873920985499 1 -0.5667404500118454 0.1951823756778568 2 -0.1659246338882321 0.3864855437976809 3 0.264914101715074 0.4544246037858506 4 0.646127253853395 0.3245589588523247 5 0.9072453478183379 0.1060642452077325 N = 7 ALPHA = 0.5 BETA = 1 0 -0.8939724663908035 0.02551501467247916 1 -0.6594047339183104 0.1266052232513152 2 -0.330720748514499 0.2785906683424275 3 0.04495789586532556 0.3894994287201929 4 0.413803698784373 0.3759820326953337 5 0.7229723820004327 0.2389626799471576 6 0.928170423786385 0.07333941890239537 N = 8 ALPHA = 0.5 BETA = 1 0 -0.9153619779765985 0.0164120900902138 1 -0.7257160980084201 0.08485547335632405 2 -0.4533780733947833 0.2002825508076703 3 -0.1294870729403755 0.312349320032714 4 0.2089491097631813 0.3577748934099902 5 0.5232643292074005 0.3048404655795939 6 0.7775490329551161 0.1792709060668855 7 0.9427521789659081 0.05270876718791008 N = 9 ALPHA = 0.5 BETA = 1 0 -0.930896636109472 0.01101439194199641 1 -0.7746386880679893 0.0586443046921294 2 -0.5463185334791455 0.145393304746393 3 -0.2672381326388558 0.2442199669465208 4 0.03658634644395028 0.3121710958691362 5 0.3368351070820351 0.3145887170547924 6 0.6055223420849569 0.2462380941383865 7 0.8176042531316312 0.1371247634867023 8 0.9533131723221203 0.03909982765524507 N = 10 ALPHA = 0.5 BETA = 1 0 -0.9425275499488472 0.00765699873374967 1 -0.8116864125172604 0.04166302380858723 2 -0.6180513952640334 0.1070695285744134 3 -0.3766418843493625 0.1896495013540221 4 -0.1061646285278262 0.2614695484772913 5 0.172423767431727 0.2937514863341338 6 0.4375388507874067 0.2709981593343714 7 0.6686402341966643 0.1996303237647672 8 0.8478228797503714 0.1068280415109269 9 0.961204277976044 0.02977785463903795 N = 1 ALPHA = 0.5 BETA = 2.5 0 0.4 1.963495408493621 N = 2 ALPHA = 0.5 BETA = 2.5 0 -0.1055161125036901 0.6949608437695366 1 0.6769446839322615 1.268534564724084 N = 3 ALPHA = 0.5 BETA = 2.5 0 -0.3949726057630552 0.2155740450619748 1 0.264766277445141 0.9816213720815014 2 0.7968729949845808 0.7662999913501441 N = 4 ALPHA = 0.5 BETA = 2.5 0 -0.5673928432278891 0.07352704993487792 1 -0.04381297095741903 0.4997101315556031 2 0.4782610448380588 0.9082548004475428 3 0.8602174966199764 0.482003426555598 N = 5 ALPHA = 0.5 BETA = 2.5 0 -0.6766083904771948 0.02810427105392544 1 -0.2615276527389885 0.2379108911216861 2 0.1979083288674991 0.6351035815608072 3 0.611609816996844 0.7438254266897733 4 0.8978486665826094 0.3185512380674297 N = 6 ALPHA = 0.5 BETA = 2.5 0 -0.7496281758956908 0.01189334443853825 1 -0.4162703993964169 0.1152628351592501 2 -0.02409923314933786 0.3830143850747024 3 0.3678575869061303 0.6464461334770024 4 0.700084306553539 0.5868443910764994 5 0.9220559149817764 0.2200343192676292 N = 7 ALPHA = 0.5 BETA = 2.5 0 -0.8006672147724485 0.005485816372248914 1 -0.528679239032316 0.05815886499830619 2 -0.1959098000069027 0.2221608406878784 3 0.1580287890648373 0.4638578133115197 4 0.4905660544916725 0.5963414735975121 5 0.7616344330284672 0.4597291058720924 6 0.9385563889913959 0.1577614936540629 N = 8 ALPHA = 0.5 BETA = 2.5 0 -0.837664685678683 0.002719896845113349 1 -0.6123079224732046 0.03070567363123355 2 -0.3290736457267182 0.1289926195321413 3 -0.01526789516983224 0.3092909316152767 4 0.2984865079144489 0.4881539875384042 5 0.581509661195563 0.5250491653932975 6 0.8061111752639282 0.3618821131546536 7 0.950312067832393 0.1167010207835007 N = 9 ALPHA = 0.5 BETA = 2.5 0 -0.8653022034444988 0.001433142315199593 1 -0.6759411199023829 0.01693724138624853 2 -0.4333019473215539 0.07616801324207163 3 -0.1566993909441266 0.2010801278076632 4 0.1315346053479577 0.3633349438080313 5 0.4080752758535235 0.4761564190243807 6 0.6505301580495418 0.4521871712839325 7 0.8392624591522133 0.2875734496942869 8 0.9589850203521831 0.08862489993180625 N = 10 ALPHA = 0.5 BETA = 2.5 0 -0.8864732152715593 0.00079518424116648 1 -0.7253515462755796 0.009727000011821089 2 -0.5159038845260621 0.04601457728802532 3 -0.2721183637403177 0.130301269175584 4 -0.01054505175311357 0.2590099865557985 5 0.2510118405448967 0.3871376587183351 6 0.494736845160108 0.4448595186308594 7 0.7040250190649779 0.3857617289382688 8 0.8646161230141249 0.2310686902877489 9 0.965567451173829 0.06881979464601419 N = 1 ALPHA = 1 BETA = 0.5 0 -0.1428571428571428 1.508494466531301 N = 2 ALPHA = 1 BETA = 0.5 0 -0.5603616177221111 0.8377097182188189 1 0.3785434359039294 0.6707847483124826 N = 3 ALPHA = 1 BETA = 0.5 0 -0.7362914149288884 0.4486477634160737 1 -0.08283748568481 0.7711842727220678 2 0.6191289006136985 0.2886624303931599 N = 4 ALPHA = 1 BETA = 0.5 0 -0.8250542504682752 0.2590735729869861 1 -0.3610180502309863 0.6101454555318188 2 0.2308286409011867 0.5011891464890383 3 0.744717344008601 0.1380862915234591 N = 5 ALPHA = 1 BETA = 0.5 0 -0.8757120529564133 0.1609007645163651 1 -0.5335962093870672 0.4466360854739412 2 -0.05829076979846798 0.5176668424406614 3 0.4326084003843105 0.3103351805927456 4 0.8175993274098119 0.07295559350758933 N = 6 ALPHA = 1 BETA = 0.5 0 -0.9072453478183379 0.1060642452077325 1 -0.646127253853395 0.3245589588523247 2 -0.264914101715074 0.4544246037858506 3 0.1659246338882321 0.3864855437976809 4 0.5667404500118454 0.1951823756778568 5 0.8633993972645073 0.04177873920985499 N = 7 ALPHA = 1 BETA = 0.5 0 -0.928170423786385 0.07333941890239537 1 -0.7229723820004327 0.2389626799471576 2 -0.413803698784373 0.3759820326953337 3 -0.04495789586532556 0.3894994287201929 4 0.330720748514499 0.2785906683424275 5 0.6594047339183104 0.1266052232513152 6 0.8939724663908035 0.02551501467247916 N = 8 ALPHA = 1 BETA = 0.5 0 -0.9427521789659081 0.05270876718791008 1 -0.7775490329551161 0.1792709060668855 2 -0.5232643292074005 0.3048404655795939 3 -0.2089491097631813 0.3577748934099902 4 0.1294870729403755 0.312349320032714 5 0.4533780733947833 0.2002825508076703 6 0.7257160980084201 0.08485547335632405 7 0.9153619779765985 0.0164120900902138 N = 9 ALPHA = 1 BETA = 0.5 0 -0.9533131723221203 0.03909982765524507 1 -0.8176042531316312 0.1371247634867023 2 -0.6055223420849569 0.2462380941383865 3 -0.3368351070820351 0.3145887170547924 4 -0.03658634644395028 0.3121710958691362 5 0.2672381326388558 0.2442199669465208 6 0.5463185334791455 0.145393304746393 7 0.7746386880679893 0.0586443046921294 8 0.930896636109472 0.01101439194199641 N = 10 ALPHA = 1 BETA = 0.5 0 -0.961204277976044 0.02977785463903795 1 -0.8478228797503714 0.1068280415109269 2 -0.6686402341966643 0.1996303237647672 3 -0.4375388507874067 0.2709981593343714 4 -0.172423767431727 0.2937514863341338 5 0.1061646285278262 0.2614695484772913 6 0.3766418843493625 0.1896495013540221 7 0.6180513952640334 0.1070695285744134 8 0.8116864125172604 0.04166302380858723 9 0.9425275499488472 0.00765699873374967 N = 1 ALPHA = 1 BETA = 1 0 0 1.333333333333333 N = 2 ALPHA = 1 BETA = 1 0 -0.4472135954999578 0.6666666666666665 1 0.4472135954999578 0.6666666666666665 N = 3 ALPHA = 1 BETA = 1 0 -0.6546536707079771 0.3111111111111109 1 6.982261990806649e-17 0.7111111111111111 2 0.654653670707977 0.3111111111111108 N = 4 ALPHA = 1 BETA = 1 0 -0.7650553239294646 0.1569499125956939 1 -0.285231516480645 0.5097167540709726 2 0.2852315164806451 0.5097167540709729 3 0.7650553239294645 0.1569499125956942 N = 5 ALPHA = 1 BETA = 1 0 -0.8302238962785671 0.08601768212280743 1 -0.4688487934707142 0.3368394607343353 2 1.630695671821295e-16 0.4876190476190478 3 0.4688487934707142 0.3368394607343351 4 0.8302238962785669 0.08601768212280747 N = 6 ALPHA = 1 BETA = 1 0 -0.8717401485096067 0.05058357701608177 1 -0.5917001814331422 0.2216925320225175 2 -0.2092992179024789 0.3943905576280672 3 0.2092992179024789 0.3943905576280671 4 0.5917001814331422 0.2216925320225176 5 0.8717401485096065 0.05058357701608172 N = 7 ALPHA = 1 BETA = 1 0 -0.8997579954114602 0.03151620015049866 1 -0.6771862795107376 0.1486404045834097 2 -0.3631174638261783 0.3007504247445495 3 2.454652867869912e-16 0.3715192743764173 4 0.3631174638261781 0.3007504247445493 5 0.6771862795107377 0.1486404045834098 6 0.8997579954114601 0.03151620015049871 N = 8 ALPHA = 1 BETA = 1 0 -0.9195339081664586 0.02059009564912185 1 -0.7387738651055049 0.1021477023603584 2 -0.4779249498104446 0.2253365549698581 3 -0.165278957666387 0.3185923136873287 4 0.1652789576663869 0.3185923136873286 5 0.4779249498104445 0.2253365549698582 6 0.7387738651055049 0.1021477023603586 7 0.9195339081664586 0.02059009564912179 N = 9 ALPHA = 1 BETA = 1 0 -0.9340014304080593 0.01399105612442672 1 -0.7844834736631444 0.07198285617566094 2 -0.5652353269962052 0.1687989736144601 3 -0.2957581355869395 0.2617849830242734 4 2.636198166955511e-16 0.3002175954556909 5 0.2957581355869395 0.2617849830242732 6 0.5652353269962052 0.1687989736144599 7 0.7844834736631442 0.07198285617566089 8 0.9340014304080595 0.01399105612442669 N = 10 ALPHA = 1 BETA = 1 0 -0.9448992722228819 0.00982540480576815 1 -0.819279321644007 0.05193914377420777 2 -0.6328761530318606 0.1273919483477327 3 -0.399530940965349 0.2111657418760753 4 -0.1365529328549277 0.2663444278628823 5 0.1365529328549273 0.2663444278628825 6 0.3995309409653491 0.2111657418760754 7 0.6328761530318608 0.1273919483477327 8 0.8192793216440072 0.05193914377420812 9 0.9448992722228821 0.009825404805768231 N = 1 ALPHA = 1 BETA = 2.5 0 0.2727272727272727 1.436661396696478 N = 2 ALPHA = 1 BETA = 2.5 0 -0.1843075691322091 0.5823920991551214 1 0.584307569132209 0.8542692975413564 N = 3 ALPHA = 1 BETA = 2.5 0 -0.4423027360564286 0.2004355575312489 1 0.1860296262211932 0.775695309158561 2 0.7299573203615513 0.4605305300066668 N = 4 ALPHA = 1 BETA = 2.5 0 -0.5972505982938984 0.07353804976128515 1 -0.101125968523521 0.4428338893708899 2 0.4098198869416554 0.6621803747350364 3 0.8102958103105465 0.2581090828292663 N = 5 ALPHA = 1 BETA = 2.5 0 -0.6964392762006608 0.02963015201149462 1 -0.3026155820854005 0.2285362779908424 2 0.1412598214162332 0.5280679973185715 3 0.5539710608669561 0.4975219906041121 4 0.8593795315584275 0.1529049787714574 N = 6 ALPHA = 1 BETA = 2.5 0 -0.7633972293043557 0.0130439061802296 1 -0.446152924876019 0.1174255136957449 2 -0.06867842657056338 0.3490370430142557 3 0.3155805295665461 0.5014050892168358 4 0.6517131087948295 0.36034844895849 5 0.8915801036798848 0.09540139563092104 N = 7 ALPHA = 1 BETA = 2.5 0 -0.8105874821854354 0.00620279036570164 1 -0.5508759997785407 0.06193824188895962 2 -0.2306894365569856 0.2164618775946896 3 0.1138876346036921 0.3988213674613816 4 0.443600343147574 0.4310114591490306 5 0.7208146570340379 0.2599688822120861 6 0.9138502837356578 0.06225677802462855 N = 8 ALPHA = 1 BETA = 2.5 0 -0.845035277918601 0.003150650566260482 1 -0.6291538133028906 0.03384819480025154 2 -0.3563555020358466 0.132215777052771 3 -0.05166907474429276 0.2867379491637258 4 0.2565958341049685 0.3952631184069375 5 0.5397363757841801 0.3540223921913581 6 0.7713710662668611 0.1892132923890619 7 0.9298950072302367 0.04221002212611247 N = 9 ALPHA = 1 BETA = 2.5 0 -0.8709215211401555 0.001692975804662645 1 -0.6889839677544535 0.01919029629201514 2 -0.4549257546478145 0.08121761104756811 3 -0.1865538397099543 0.1974051443898488 4 0.09541224177708327 0.3197898022964819 5 0.3691513199896206 0.3631243013752991 6 0.6134647393951399 0.2850633375254082 7 0.8094259398841321 0.1396134790482666 8 0.9418378189505883 0.02956444891692782 N = 10 ALPHA = 1 BETA = 2.5 0 -0.890851998362436 0.0009546834154139908 1 -0.7356332759809726 0.01126952303045689 2 -0.5332475568814603 0.05063788256866946 3 -0.2966592827516153 0.1337845805120262 4 -0.04128300871886588 0.2429565425267207 5 0.2161975251464398 0.3233164884771637 6 0.4589489303958385 0.3198849047578982 7 0.6710962948352331 0.2279739859839845 8 0.8387626487347724 0.1045909447918315 9 0.9509675959234919 0.02129186063231215 N = 1 ALPHA = 2.5 BETA = 0.5 0 -0.4 1.963495408493621 N = 2 ALPHA = 2.5 BETA = 0.5 0 -0.6769446839322615 1.268534564724084 1 0.1055161125036901 0.6949608437695366 N = 3 ALPHA = 2.5 BETA = 0.5 0 -0.7968729949845808 0.7662999913501441 1 -0.264766277445141 0.9816213720815014 2 0.3949726057630552 0.2155740450619748 N = 4 ALPHA = 2.5 BETA = 0.5 0 -0.8602174966199764 0.482003426555598 1 -0.4782610448380588 0.9082548004475428 2 0.04381297095741903 0.4997101315556031 3 0.5673928432278891 0.07352704993487792 N = 5 ALPHA = 2.5 BETA = 0.5 0 -0.8978486665826094 0.3185512380674297 1 -0.611609816996844 0.7438254266897733 2 -0.1979083288674991 0.6351035815608072 3 0.2615276527389885 0.2379108911216861 4 0.6766083904771948 0.02810427105392544 N = 6 ALPHA = 2.5 BETA = 0.5 0 -0.9220559149817764 0.2200343192676292 1 -0.700084306553539 0.5868443910764994 2 -0.3678575869061303 0.6464461334770024 3 0.02409923314933786 0.3830143850747024 4 0.4162703993964169 0.1152628351592501 5 0.7496281758956908 0.01189334443853825 N = 7 ALPHA = 2.5 BETA = 0.5 0 -0.9385563889913959 0.1577614936540629 1 -0.7616344330284672 0.4597291058720924 2 -0.4905660544916725 0.5963414735975121 3 -0.1580287890648373 0.4638578133115197 4 0.1959098000069027 0.2221608406878784 5 0.528679239032316 0.05815886499830619 6 0.8006672147724485 0.005485816372248914 N = 8 ALPHA = 2.5 BETA = 0.5 0 -0.950312067832393 0.1167010207835007 1 -0.8061111752639282 0.3618821131546536 2 -0.581509661195563 0.5250491653932975 3 -0.2984865079144489 0.4881539875384042 4 0.01526789516983224 0.3092909316152767 5 0.3290736457267182 0.1289926195321413 6 0.6123079224732046 0.03070567363123355 7 0.837664685678683 0.002719896845113349 N = 9 ALPHA = 2.5 BETA = 0.5 0 -0.9589850203521831 0.08862489993180625 1 -0.8392624591522133 0.2875734496942869 2 -0.6505301580495418 0.4521871712839325 3 -0.4080752758535235 0.4761564190243807 4 -0.1315346053479577 0.3633349438080313 5 0.1566993909441266 0.2010801278076632 6 0.4333019473215539 0.07616801324207163 7 0.6759411199023829 0.01693724138624853 8 0.8653022034444988 0.001433142315199593 N = 10 ALPHA = 2.5 BETA = 0.5 0 -0.965567451173829 0.06881979464601419 1 -0.8646161230141249 0.2310686902877489 2 -0.7040250190649779 0.3857617289382688 3 -0.494736845160108 0.4448595186308594 4 -0.2510118405448967 0.3871376587183351 5 0.01054505175311357 0.2590099865557985 6 0.2721183637403177 0.130301269175584 7 0.5159038845260621 0.04601457728802532 8 0.7253515462755796 0.009727000011821089 9 0.8864732152715593 0.00079518424116648 N = 1 ALPHA = 2.5 BETA = 1 0 -0.2727272727272727 1.436661396696478 N = 2 ALPHA = 2.5 BETA = 1 0 -0.584307569132209 0.8542692975413564 1 0.1843075691322091 0.5823920991551214 N = 3 ALPHA = 2.5 BETA = 1 0 -0.7299573203615513 0.4605305300066668 1 -0.1860296262211932 0.775695309158561 2 0.4423027360564286 0.2004355575312489 N = 4 ALPHA = 2.5 BETA = 1 0 -0.8102958103105465 0.2581090828292663 1 -0.4098198869416554 0.6621803747350364 2 0.101125968523521 0.4428338893708899 3 0.5972505982938984 0.07353804976128515 N = 5 ALPHA = 2.5 BETA = 1 0 -0.8593795315584275 0.1529049787714574 1 -0.5539710608669561 0.4975219906041121 2 -0.1412598214162332 0.5280679973185715 3 0.3026155820854005 0.2285362779908424 4 0.6964392762006608 0.02963015201149462 N = 6 ALPHA = 2.5 BETA = 1 0 -0.8915801036798848 0.09540139563092104 1 -0.6517131087948295 0.36034844895849 2 -0.3155805295665461 0.5014050892168358 3 0.06867842657056338 0.3490370430142557 4 0.446152924876019 0.1174255136957449 5 0.7633972293043557 0.0130439061802296 N = 7 ALPHA = 2.5 BETA = 1 0 -0.9138502837356578 0.06225677802462855 1 -0.7208146570340379 0.2599688822120861 2 -0.443600343147574 0.4310114591490306 3 -0.1138876346036921 0.3988213674613816 4 0.2306894365569856 0.2164618775946896 5 0.5508759997785407 0.06193824188895962 6 0.8105874821854354 0.00620279036570164 N = 8 ALPHA = 2.5 BETA = 1 0 -0.9298950072302367 0.04221002212611247 1 -0.7713710662668611 0.1892132923890619 2 -0.5397363757841801 0.3540223921913581 3 -0.2565958341049685 0.3952631184069375 4 0.05166907474429276 0.2867379491637258 5 0.3563555020358466 0.132215777052771 6 0.6291538133028906 0.03384819480025154 7 0.845035277918601 0.003150650566260482 N = 9 ALPHA = 2.5 BETA = 1 0 -0.9418378189505883 0.02956444891692782 1 -0.8094259398841321 0.1396134790482666 2 -0.6134647393951399 0.2850633375254082 3 -0.3691513199896206 0.3631243013752991 4 -0.09541224177708327 0.3197898022964819 5 0.1865538397099543 0.1974051443898488 6 0.4549257546478145 0.08121761104756811 7 0.6889839677544535 0.01919029629201514 8 0.8709215211401555 0.001692975804662645 N = 10 ALPHA = 2.5 BETA = 1 0 -0.9509675959234919 0.02129186063231215 1 -0.8387626487347724 0.1045909447918315 2 -0.6710962948352331 0.2279739859839845 3 -0.4589489303958385 0.3198849047578982 4 -0.2161975251464398 0.3233164884771637 5 0.04128300871886588 0.2429565425267207 6 0.2966592827516153 0.1337845805120262 7 0.5332475568814603 0.05063788256866946 8 0.7356332759809726 0.01126952303045689 9 0.890851998362436 0.0009546834154139908 N = 1 ALPHA = 2.5 BETA = 2.5 0 0 0.9817477042468103 N = 2 ALPHA = 2.5 BETA = 2.5 0 -0.3535533905932737 0.4908738521234051 1 0.3535533905932737 0.4908738521234051 N = 3 ALPHA = 2.5 BETA = 2.5 0 -0.5477225575051663 0.2045307717180854 1 4.336808689942018e-17 0.5726861608106395 2 0.5477225575051661 0.2045307717180856 N = 4 ALPHA = 2.5 BETA = 2.5 0 -0.6660699417556469 0.08700807153148338 1 -0.237383303308445 0.4038657805919218 2 0.237383303308445 0.4038657805919215 3 0.6660699417556469 0.08700807153148342 N = 5 ALPHA = 2.5 BETA = 2.5 0 -0.7434770045212192 0.03928938913176724 1 -0.40190503608921 0.2454174450998076 2 1.245242164985443e-16 0.4123340357836606 3 0.4019050360892101 0.245417445099808 4 0.7434770045212191 0.03928938913176733 N = 6 ALPHA = 2.5 BETA = 2.5 0 -0.7968373566406803 0.01891164336669309 1 -0.5196148456949135 0.1431407318203966 2 -0.1804179569647765 0.3288214769363159 3 0.1804179569647765 0.3288214769363157 4 0.5196148456949131 0.1431407318203972 5 0.7968373566406806 0.01891164336669309 N = 7 ALPHA = 2.5 BETA = 2.5 0 -0.8351564541339109 0.009658525021001006 1 -0.6063141605450342 0.08344467054898488 2 -0.3186902924591671 0.2357822853526958 3 5.104982774491088e-17 0.3239767424014474 4 0.3186902924591672 0.2357822853526957 5 0.6063141605450342 0.08344467054898476 6 0.8351564541339108 0.009658525021001023 N = 8 ALPHA = 2.5 BETA = 2.5 0 -0.8635912403827581 0.005199707422132458 1 -0.6718232507247469 0.04941818815652228 2 -0.4260961495798481 0.1615070338519728 3 -0.1459649294626306 0.2747489226927776 4 0.1459649294626304 0.2747489226927781 5 0.426096149579848 0.161507033851973 6 0.6718232507247468 0.04941818815652241 7 0.8635912403827586 0.0051997074221325 N = 9 ALPHA = 2.5 BETA = 2.5 0 -0.8852662998295733 0.002931900984992426 1 -0.7224309555097358 0.02992095719368029 2 -0.5107538326640052 0.1088013928543096 3 -0.2643695362367074 0.2155149137501429 4 -1.634301058359553e-17 0.26740937468056 5 0.2643695362367075 0.2155149137501428 6 0.5107538326640052 0.1088013928543095 7 0.7224309555097355 0.02992095719368041 8 0.8852662998295734 0.002931900984992388 N = 10 ALPHA = 2.5 BETA = 2.5 0 -0.9021636877146701 0.001721638228893138 1 -0.7622868838395971 0.01855643700599038 2 -0.5784393987065429 0.07316132839714104 3 -0.3610608688185056 0.1622660968500983 4 -0.1227285554678421 0.2351683516412827 5 0.122728555467842 0.2351683516412817 6 0.3610608688185061 0.1622660968500987 7 0.5784393987065428 0.07316132839714092 8 0.7622868838395969 0.01855643700599038 9 0.9021636877146699 0.001721638228893156 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 1 0 1 1 1 1 0 1 2 1 2 1 1 3 1 6 5 1 4 1 24 23 2 0 1 1 0 2 1 1 1 0 2 2 2 2 0 2 3 6 6 0 2 4 20 24 4 2 5 68 120 52 2 6 232 720 488 3 0 1 1 0 3 1 1 1 0 3 2 2 2 4.44089e-16 3 3 6 6 3.55271e-15 3 4 24 24 2.84217e-14 3 5 120 120 2.13163e-13 3 6 684 720 36 3 7 4140 5040 900 3 8 25668 40320 14652 4 0 1 1 4.44089e-16 4 1 1 1 4.44089e-16 4 2 2 2 8.88178e-16 4 3 6 6 1.77636e-15 4 4 24 24 1.06581e-14 4 5 120 120 8.52651e-14 4 6 720 720 9.09495e-13 4 7 5040 5040 1.09139e-11 4 8 39744 40320 576 4 9 339264 362880 23616 4 10 3.03322e+06 3.6288e+06 595584 5 0 1 1 2.22045e-16 5 1 1 1 0 5 2 2 2 1.11022e-15 5 3 6 6 6.21725e-15 5 4 24 24 3.55271e-14 5 5 120 120 2.27374e-13 5 6 720 720 1.36424e-12 5 7 5040 5040 1.00044e-11 5 8 40320 40320 7.27596e-11 5 9 362880 362880 5.82077e-10 5 10 3.6144e+06 3.6288e+06 14400 5 11 3.90384e+07 3.99168e+07 878400 5 12 4.47221e+08 4.79002e+08 3.17808e+07 6 0 1 1 2.22045e-16 6 1 1 1 4.44089e-16 6 2 2 2 2.22045e-15 6 3 6 6 7.99361e-15 6 4 24 24 3.90799e-14 6 5 120 120 1.7053e-13 6 6 720 720 9.09495e-13 6 7 5040 5040 4.54747e-12 6 8 40320 40320 0 6 9 362880 362880 1.74623e-10 6 10 3.6288e+06 3.6288e+06 3.72529e-09 6 11 3.99168e+07 3.99168e+07 5.21541e-08 6 12 4.78483e+08 4.79002e+08 518400 6 13 6.18296e+09 6.22702e+09 4.4064e+07 6 14 8.50124e+10 8.71783e+10 2.16588e+09 7 0 1 1 2.22045e-16 7 1 1 1 2.22045e-16 7 2 2 2 8.88178e-16 7 3 6 6 8.88178e-16 7 4 24 24 7.10543e-15 7 5 120 120 1.42109e-14 7 6 720 720 2.27374e-13 7 7 5040 5040 1.81899e-12 7 8 40320 40320 1.45519e-11 7 9 362880 362880 2.32831e-10 7 10 3.6288e+06 3.6288e+06 3.25963e-09 7 11 3.99168e+07 3.99168e+07 4.47035e-08 7 12 4.79002e+08 4.79002e+08 6.55651e-07 7 13 6.22702e+09 6.22702e+09 9.53674e-06 7 14 8.71529e+10 8.71783e+10 2.54016e+07 7 15 1.3048e+12 1.30767e+12 2.87038e+09 7 16 2.07387e+13 2.09228e+13 1.84085e+11 8 0 1 1 1.11022e-16 8 1 1 1 3.33067e-16 8 2 2 2 8.88178e-16 8 3 6 6 4.44089e-15 8 4 24 24 2.84217e-14 8 5 120 120 1.42109e-13 8 6 720 720 6.82121e-13 8 7 5040 5040 3.63798e-12 8 8 40320 40320 2.91038e-11 8 9 362880 362880 3.49246e-10 8 10 3.6288e+06 3.6288e+06 4.65661e-09 8 11 3.99168e+07 3.99168e+07 6.70552e-08 8 12 4.79002e+08 4.79002e+08 8.9407e-07 8 13 6.22702e+09 6.22702e+09 1.43051e-05 8 14 8.71783e+10 8.71783e+10 0.000213623 8 15 1.30767e+12 1.30767e+12 0.00341797 8 16 2.09212e+13 2.09228e+13 1.6257e+09 8 17 3.55452e+14 3.55687e+14 2.35727e+11 8 18 6.38325e+15 6.40237e+15 1.91215e+13 9 0 1 1 8.88178e-16 9 1 1 1 6.66134e-16 9 2 2 2 1.77636e-15 9 3 6 6 6.21725e-15 9 4 24 24 1.77636e-14 9 5 120 120 8.52651e-14 9 6 720 720 1.13687e-13 9 7 5040 5040 9.09495e-13 9 8 40320 40320 0 9 9 362880 362880 0 9 10 3.6288e+06 3.6288e+06 0 9 11 3.99168e+07 3.99168e+07 0 9 12 4.79002e+08 4.79002e+08 1.19209e-07 9 13 6.22702e+09 6.22702e+09 0 9 14 8.71783e+10 8.71783e+10 0 9 15 1.30767e+12 1.30767e+12 0.000732422 9 16 2.09228e+13 2.09228e+13 0.015625 9 17 3.55687e+14 3.55687e+14 0.4375 9 18 6.40224e+15 6.40237e+15 1.31682e+11 9 19 1.21621e+17 1.21645e+17 2.38344e+13 9 20 2.43052e+18 2.4329e+18 2.38594e+15 10 0 1 1 6.66134e-16 10 1 1 1 6.66134e-16 10 2 2 2 2.22045e-15 10 3 6 6 8.88178e-15 10 4 24 24 3.19744e-14 10 5 120 120 1.27898e-13 10 6 720 720 4.54747e-13 10 7 5040 5040 2.72848e-12 10 8 40320 40320 7.27596e-12 10 9 362880 362880 1.16415e-10 10 10 3.6288e+06 3.6288e+06 2.79397e-09 10 11 3.99168e+07 3.99168e+07 5.21541e-08 10 12 4.79002e+08 4.79002e+08 8.9407e-07 10 13 6.22702e+09 6.22702e+09 1.62125e-05 10 14 8.71783e+10 8.71783e+10 0.000259399 10 15 1.30767e+12 1.30767e+12 0.00439453 10 16 2.09228e+13 2.09228e+13 0.0742188 10 17 3.55687e+14 3.55687e+14 1.0625 10 18 6.40237e+15 6.40237e+15 18 10 19 1.21645e+17 1.21645e+17 256 10 20 2.43289e+18 2.4329e+18 1.31682e+13 10 21 5.1088e+19 5.10909e+19 2.91017e+15 10 22 1.12365e+21 1.124e+21 3.52407e+17 TEST20 LAGUERRE_COMPUTE computes a generalized Gauss-Laguerre rule which is appropriate for integrands of the form Integral ( 0 <= x < +oo ) f(x) exp(-x) dx. Order = 1 0 1 1 Order = 2 0 0.5857864376269051 0.8535533905932737 1 3.414213562373095 0.1464466094067262 Order = 3 0 0.4157745567834789 0.7110930099291729 1 2.294280360279041 0.2785177335692409 2 6.28994508293748 0.01038925650158615 Order = 4 0 0.3225476896193923 0.6031541043416337 1 1.745761101158346 0.3574186924377999 2 4.536620296921128 0.03888790851500538 3 9.395070912301136 0.0005392947055613278 Order = 5 0 0.263560319718141 0.5217556105828089 1 1.413403059106517 0.398666811083176 2 3.596425771040721 0.07594244968170749 3 7.085810005858835 0.00361175867992205 4 12.64080084427578 2.336997238577621e-05 Order = 6 0 0.222846604179261 0.4589646739499636 1 1.188932101672623 0.4170008307721207 2 2.992736326059314 0.1133733820740452 3 5.775143569104512 0.01039919745314908 4 9.837467418382587 0.0002610172028149323 5 15.9828739806017 8.985479064296196e-07 Order = 7 0 0.1930436765603621 0.4093189517012737 1 1.026664895339191 0.4218312778617199 2 2.567876744950745 0.1471263486575055 3 4.900353084526484 0.02063351446871694 4 8.182153444562859 0.001074010143280748 5 12.73418029179781 1.586546434856422e-05 6 19.39572786226254 3.170315478995581e-08 Order = 8 0 0.1702796323051008 0.3691885893416376 1 0.9037017767993794 0.4187867808143426 2 2.251086629866129 0.1757949866371721 3 4.266700170287657 0.03334349226121563 4 7.045905402393464 0.002794536235225677 5 10.75851601018099 9.076508773358235e-05 6 15.740678641278 8.485746716272511e-07 7 22.86313173688927 1.048001174871507e-09 Order = 9 0 0.152322227731808 0.3361264217979629 1 0.8072200227422562 0.4112139804239849 2 2.00513515561935 0.1992875253708856 3 3.783473973331235 0.04746056276565148 4 6.204956777876611 0.005599626610794589 5 9.372985251687574 0.000305249767093211 6 13.46623691109209 6.592123026075368e-06 7 18.83359778899169 4.110769330349564e-08 8 26.37407189092738 3.290874030350713e-11 Order = 10 0 0.1377934705404923 0.3084411157650204 1 0.7294545495031706 0.4011199291552736 2 1.808342901740316 0.2180682876118095 3 3.401433697854898 0.06208745609867788 4 5.552496140063801 0.009501516975181156 5 8.330152746764496 0.0007530083885875383 6 11.84378583790007 2.825923349599563e-05 7 16.27925783137811 4.249313984962688e-07 8 21.99658581198076 1.839564823979634e-09 9 29.92069701227389 9.911827219609e-13 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 2 4.44089e-16 1 1 0 0 0 1 2 0 0.666667 0.666667 1 3 0 0 0 1 4 0 0.4 0.4 2 0 2 2 0 2 1 0 0 0 2 2 0.666667 0.666667 3.33067e-16 2 3 0 0 0 2 4 0.222222 0.4 0.177778 2 5 0 0 0 2 6 0.0740741 0.285714 0.21164 3 0 2 2 4.44089e-16 3 1 -2.22045e-16 0 2.22045e-16 3 2 0.666667 0.666667 3.33067e-16 3 3 -2.22045e-16 0 2.22045e-16 3 4 0.4 0.4 4.44089e-16 3 5 -1.38778e-16 0 1.38778e-16 3 6 0.24 0.285714 0.0457143 3 7 -1.52656e-16 0 1.52656e-16 3 8 0.144 0.222222 0.0782222 4 0 2 2 1.33227e-15 4 1 -7.21645e-16 0 7.21645e-16 4 2 0.666667 0.666667 8.88178e-16 4 3 -4.71845e-16 0 4.71845e-16 4 4 0.4 0.4 6.66134e-16 4 5 -3.88578e-16 0 3.88578e-16 4 6 0.285714 0.285714 6.10623e-16 4 7 -3.05311e-16 0 3.05311e-16 4 8 0.210612 0.222222 0.01161 4 9 -2.63678e-16 0 2.63678e-16 4 10 0.156035 0.181818 0.0257832 5 0 2 2 1.33227e-15 5 1 2.77556e-17 0 2.77556e-17 5 2 0.666667 0.666667 3.33067e-16 5 3 -1.38778e-16 0 1.38778e-16 5 4 0.4 0.4 1.11022e-16 5 5 -2.22045e-16 0 2.22045e-16 5 6 0.285714 0.285714 5.55112e-17 5 7 -2.498e-16 0 2.498e-16 5 8 0.222222 0.222222 5.55112e-17 5 9 -2.498e-16 0 2.498e-16 5 10 0.178886 0.181818 0.00293181 5 11 -2.63678e-16 0 2.63678e-16 5 12 0.145853 0.153846 0.00799357 6 0 2 2 6.66134e-16 6 1 3.05311e-16 0 3.05311e-16 6 2 0.666667 0.666667 5.55112e-16 6 3 1.94289e-16 0 1.94289e-16 6 4 0.4 0.4 5.55112e-16 6 5 1.80411e-16 0 1.80411e-16 6 6 0.285714 0.285714 6.10623e-16 6 7 1.94289e-16 0 1.94289e-16 6 8 0.222222 0.222222 5.55112e-16 6 9 1.66533e-16 0 1.66533e-16 6 10 0.181818 0.181818 5.27356e-16 6 11 1.52656e-16 0 1.52656e-16 6 12 0.153108 0.153846 0.000738079 6 13 1.249e-16 0 1.249e-16 6 14 0.130949 0.133333 0.00238422 7 0 2 2 1.33227e-15 7 1 9.15934e-16 0 9.15934e-16 7 2 0.666667 0.666667 4.44089e-16 7 3 4.30211e-16 0 4.30211e-16 7 4 0.4 0.4 0 7 5 2.08167e-16 0 2.08167e-16 7 6 0.285714 0.285714 5.55112e-17 7 7 9.71445e-17 0 9.71445e-17 7 8 0.222222 0.222222 1.11022e-16 7 9 4.16334e-17 0 4.16334e-17 7 10 0.181818 0.181818 1.66533e-16 7 11 -2.77556e-17 0 2.77556e-17 7 12 0.153846 0.153846 1.94289e-16 7 13 -6.93889e-17 0 6.93889e-17 7 14 0.133148 0.133333 0.000185466 7 15 -6.245e-17 0 6.245e-17 7 16 0.116955 0.117647 0.00069235 8 0 2 2 1.11022e-15 8 1 -4.996e-16 0 4.996e-16 8 2 0.666667 0.666667 5.55112e-16 8 3 -2.498e-16 0 2.498e-16 8 4 0.4 0.4 4.44089e-16 8 5 -2.63678e-16 0 2.63678e-16 8 6 0.285714 0.285714 2.77556e-16 8 7 -2.35922e-16 0 2.35922e-16 8 8 0.222222 0.222222 1.38778e-16 8 9 -2.35922e-16 0 2.35922e-16 8 10 0.181818 0.181818 5.55112e-17 8 11 -2.22045e-16 0 2.22045e-16 8 12 0.153846 0.153846 2.77556e-17 8 13 -2.01228e-16 0 2.01228e-16 8 14 0.133333 0.133333 8.32667e-17 8 15 -1.94289e-16 0 1.94289e-16 8 16 0.117601 0.117647 4.65483e-05 8 17 -1.80411e-16 0 1.80411e-16 8 18 0.105066 0.105263 0.000197136 9 0 2 2 2.22045e-15 9 1 3.33067e-16 0 3.33067e-16 9 2 0.666667 0.666667 1.22125e-15 9 3 2.63678e-16 0 2.63678e-16 9 4 0.4 0.4 7.21645e-16 9 5 -1.38778e-17 0 1.38778e-17 9 6 0.285714 0.285714 4.44089e-16 9 7 -1.52656e-16 0 1.52656e-16 9 8 0.222222 0.222222 1.66533e-16 9 9 -2.28983e-16 0 2.28983e-16 9 10 0.181818 0.181818 5.55112e-17 9 11 -2.91434e-16 0 2.91434e-16 9 12 0.153846 0.153846 8.32667e-17 9 13 -3.05311e-16 0 3.05311e-16 9 14 0.133333 0.133333 1.38778e-16 9 15 -3.1225e-16 0 3.1225e-16 9 16 0.117647 0.117647 2.08167e-16 9 17 -3.1225e-16 0 3.1225e-16 9 18 0.105251 0.105263 1.16731e-05 9 19 -3.1225e-16 0 3.1225e-16 9 20 0.0951828 0.0952381 5.52919e-05 10 0 2 2 1.33227e-15 10 1 2.91434e-16 0 2.91434e-16 10 2 0.666667 0.666667 2.22045e-16 10 3 5.13478e-16 0 5.13478e-16 10 4 0.4 0.4 2.77556e-16 10 5 4.09395e-16 0 4.09395e-16 10 6 0.285714 0.285714 2.77556e-16 10 7 3.05311e-16 0 3.05311e-16 10 8 0.222222 0.222222 2.22045e-16 10 9 2.08167e-16 0 2.08167e-16 10 10 0.181818 0.181818 1.66533e-16 10 11 1.31839e-16 0 1.31839e-16 10 12 0.153846 0.153846 1.11022e-16 10 13 6.245e-17 0 6.245e-17 10 14 0.133333 0.133333 8.32667e-17 10 15 5.55112e-17 0 5.55112e-17 10 16 0.117647 0.117647 2.77556e-17 10 17 0 0 0 10 18 0.105263 0.105263 0 10 19 -1.38778e-17 0 1.38778e-17 10 20 0.0952352 0.0952381 2.92559e-06 10 21 -3.46945e-17 0 3.46945e-17 10 22 0.0869412 0.0869565 1.53242e-05 TEST22 LEGENDRE_COMPUTE computes a Gauss-Legendre rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. Order = 1 0 0 2 Order = 2 0 -0.5773502691896256 1 1 0.5773502691896256 1 Order = 3 0 -0.7745966692414833 0.5555555555555556 1 1.994931997373328e-17 0.8888888888888895 2 0.7745966692414832 0.5555555555555554 Order = 4 0 -0.8611363115940527 0.3478548451374547 1 -0.3399810435848563 0.6521451548625466 2 0.3399810435848563 0.6521451548625458 3 0.8611363115940526 0.3478548451374541 Order = 5 0 -0.9061798459386641 0.2369268850561892 1 -0.538469310105683 0.4786286704993669 2 -1.081853856991421e-16 0.568888888888889 3 0.5384693101056831 0.4786286704993672 4 0.9061798459386639 0.2369268850561891 Order = 6 0 -0.9324695142031522 0.1713244923791705 1 -0.6612093864662647 0.3607615730481384 2 -0.238619186083197 0.4679139345726904 3 0.2386191860831969 0.467913934572691 4 0.6612093864662647 0.3607615730481382 5 0.9324695142031522 0.1713244923791708 Order = 7 0 -0.9491079123427585 0.1294849661688697 1 -0.7415311855993943 0.2797053914892765 2 -0.4058451513773971 0.3818300505051193 3 2.944352847269754e-16 0.4179591836734696 4 0.4058451513773971 0.3818300505051192 5 0.7415311855993943 0.2797053914892776 6 0.9491079123427584 0.1294849661688697 Order = 8 0 -0.9602898564975365 0.101228536290376 1 -0.796666477413627 0.2223810344533743 2 -0.525532409916329 0.3137066458778873 3 -0.1834346424956498 0.3626837833783622 4 0.1834346424956496 0.3626837833783619 5 0.5255324099163292 0.3137066458778869 6 0.7966664774136268 0.2223810344533742 7 0.9602898564975364 0.1012285362903759 Order = 9 0 -0.968160239507626 0.08127438836157462 1 -0.836031107326636 0.1806481606948576 2 -0.61337143270059 0.2606106964029357 3 -0.3242534234038094 0.3123470770400033 4 -3.64953385043433e-16 0.3302393550012602 5 0.3242534234038092 0.3123470770400024 6 0.6133714327005907 0.2606106964029365 7 0.8360311073266359 0.1806481606948582 8 0.9681602395076259 0.08127438836157423 Order = 10 0 -0.9739065285171715 0.06667134430868846 1 -0.8650633666889844 0.14945134915058 2 -0.6794095682990244 0.2190863625159823 3 -0.4333953941292472 0.2692667193099961 4 -0.1488743389816311 0.2955242247147523 5 0.1488743389816314 0.2955242247147531 6 0.4333953941292472 0.2692667193099947 7 0.6794095682990243 0.2190863625159819 8 0.8650633666889843 0.1494513491505811 9 0.9739065285171714 0.06667134430868851 TEST01 CHEBYSHEV1_COMPUTE_NP computes a Gauss-Chebyshev type 1 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) / sqrt ( 1 - x^2 ) dx. CHEBYSHEV1_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 3.14159 3.14159 0 1 1 0 0 0 1 2 0 1.5708 1.5708 1 3 0 0 0 1 4 0 1.1781 1.1781 2 0 3.14159 3.14159 0 2 1 0 0 0 2 2 1.5708 1.5708 0 2 3 2.22045e-16 0 2.22045e-16 2 4 0.785398 1.1781 0.392699 2 5 3.33067e-16 0 3.33067e-16 2 6 0.392699 0.981748 0.589049 3 0 3.14159 3.14159 0 3 1 0 0 0 3 2 1.5708 1.5708 2.22045e-16 3 3 0 0 0 3 4 1.1781 1.1781 4.44089e-16 3 5 0 0 0 3 6 0.883573 0.981748 0.0981748 3 7 0 0 0 3 8 0.66268 0.859029 0.19635 4 0 3.14159 3.14159 0 4 1 -1.11022e-16 0 1.11022e-16 4 2 1.5708 1.5708 0 4 3 1.11022e-16 0 1.11022e-16 4 4 1.1781 1.1781 0 4 5 0 0 0 4 6 0.981748 0.981748 1.11022e-16 4 7 0 0 0 4 8 0.834486 0.859029 0.0245437 4 9 0 0 0 4 10 0.711767 0.773126 0.0613592 5 0 3.14159 3.14159 0 5 1 1.11022e-16 0 1.11022e-16 5 2 1.5708 1.5708 0 5 3 1.11022e-16 0 1.11022e-16 5 4 1.1781 1.1781 4.44089e-16 5 5 5.55112e-17 0 5.55112e-17 5 6 0.981748 0.981748 3.33067e-16 5 7 5.55112e-17 0 5.55112e-17 5 8 0.859029 0.859029 5.55112e-16 5 9 0 0 0 5 10 0.76699 0.773126 0.00613592 5 11 0 0 0 5 12 0.690291 0.708699 0.0184078 6 0 3.14159 3.14159 4.44089e-16 6 1 2.22045e-16 0 2.22045e-16 6 2 1.5708 1.5708 0 6 3 2.77556e-16 0 2.77556e-16 6 4 1.1781 1.1781 0 6 5 3.33067e-16 0 3.33067e-16 6 6 0.981748 0.981748 0 6 7 4.44089e-16 0 4.44089e-16 6 8 0.859029 0.859029 1.11022e-16 6 9 4.44089e-16 0 4.44089e-16 6 10 0.773126 0.773126 0 6 11 5.55112e-16 0 5.55112e-16 6 12 0.707165 0.708699 0.00153398 6 13 5.55112e-16 0 5.55112e-16 6 14 0.652709 0.658078 0.00536893 7 0 3.14159 3.14159 0 7 1 1.11022e-16 0 1.11022e-16 7 2 1.5708 1.5708 2.22045e-16 7 3 1.11022e-16 0 1.11022e-16 7 4 1.1781 1.1781 0 7 5 1.11022e-16 0 1.11022e-16 7 6 0.981748 0.981748 3.33067e-16 7 7 -1.66533e-16 0 1.66533e-16 7 8 0.859029 0.859029 3.33067e-16 7 9 -1.66533e-16 0 1.66533e-16 7 10 0.773126 0.773126 5.55112e-16 7 11 -2.77556e-16 0 2.77556e-16 7 12 0.708699 0.708699 7.77156e-16 7 13 -3.33067e-16 0 3.33067e-16 7 14 0.657694 0.658078 0.000383495 7 15 -4.996e-16 0 4.996e-16 7 16 0.615414 0.616948 0.00153398 8 0 3.14159 3.14159 0 8 1 2.22045e-16 0 2.22045e-16 8 2 1.5708 1.5708 0 8 3 1.11022e-16 0 1.11022e-16 8 4 1.1781 1.1781 0 8 5 5.55112e-17 0 5.55112e-17 8 6 0.981748 0.981748 1.11022e-16 8 7 0 0 0 8 8 0.859029 0.859029 3.33067e-16 8 9 -5.55112e-17 0 5.55112e-17 8 10 0.773126 0.773126 1.11022e-16 8 11 -5.55112e-17 0 5.55112e-17 8 12 0.708699 0.708699 0 8 13 -5.55112e-17 0 5.55112e-17 8 14 0.658078 0.658078 1.11022e-16 8 15 -5.55112e-17 0 5.55112e-17 8 16 0.616852 0.616948 9.58738e-05 8 17 0 0 0 8 18 0.582242 0.582673 0.000431432 9 0 3.14159 3.14159 0 9 1 1.66533e-16 0 1.66533e-16 9 2 1.5708 1.5708 2.22045e-16 9 3 1.11022e-16 0 1.11022e-16 9 4 1.1781 1.1781 0 9 5 5.55112e-17 0 5.55112e-17 9 6 0.981748 0.981748 1.11022e-16 9 7 -5.55112e-17 0 5.55112e-17 9 8 0.859029 0.859029 1.11022e-16 9 9 0 0 0 9 10 0.773126 0.773126 0 9 11 -1.11022e-16 0 1.11022e-16 9 12 0.708699 0.708699 0 9 13 -2.22045e-16 0 2.22045e-16 9 14 0.658078 0.658078 1.11022e-16 9 15 -3.33067e-16 0 3.33067e-16 9 16 0.616948 0.616948 0 9 17 -3.33067e-16 0 3.33067e-16 9 18 0.582649 0.582673 2.39684e-05 9 19 -4.996e-16 0 4.996e-16 9 20 0.55342 0.553539 0.000119842 10 0 3.14159 3.14159 0 10 1 1.66533e-16 0 1.66533e-16 10 2 1.5708 1.5708 2.22045e-16 10 3 3.33067e-16 0 3.33067e-16 10 4 1.1781 1.1781 2.22045e-16 10 5 3.33067e-16 0 3.33067e-16 10 6 0.981748 0.981748 1.11022e-16 10 7 4.996e-16 0 4.996e-16 10 8 0.859029 0.859029 3.33067e-16 10 9 4.996e-16 0 4.996e-16 10 10 0.773126 0.773126 1.11022e-16 10 11 5.55112e-16 0 5.55112e-16 10 12 0.708699 0.708699 2.22045e-16 10 13 5.55112e-16 0 5.55112e-16 10 14 0.658078 0.658078 3.33067e-16 10 15 6.10623e-16 0 6.10623e-16 10 16 0.616948 0.616948 3.33067e-16 10 17 6.10623e-16 0 6.10623e-16 10 18 0.582673 0.582673 4.44089e-16 10 19 6.38378e-16 0 6.38378e-16 10 20 0.553533 0.553539 5.99211e-06 10 21 6.93889e-16 0 6.93889e-16 10 22 0.528346 0.528378 3.29566e-05 TEST02 CHEBYSHEV1_COMPUTE_NP computes a Gauss-Chebyshev type 1 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) / sqrt(1-x^2) dx. Order = 1 0 0 3.141592653589793 Order = 2 0 -0.7071067811865475 1.570796326794897 1 0.7071067811865476 1.570796326794897 Order = 3 0 -0.8660254037844387 1.047197551196598 1 0 1.047197551196598 2 0.8660254037844387 1.047197551196598 Order = 4 0 -0.9238795325112867 0.7853981633974483 1 -0.3826834323650897 0.7853981633974483 2 0.3826834323650898 0.7853981633974483 3 0.9238795325112867 0.7853981633974483 Order = 5 0 -0.9510565162951535 0.6283185307179586 1 -0.587785252292473 0.6283185307179586 2 0 0.6283185307179586 3 0.5877852522924732 0.6283185307179586 4 0.9510565162951535 0.6283185307179586 Order = 6 0 -0.9659258262890682 0.5235987755982988 1 -0.7071067811865475 0.5235987755982988 2 -0.2588190451025206 0.5235987755982988 3 0.2588190451025207 0.5235987755982988 4 0.7071067811865476 0.5235987755982988 5 0.9659258262890683 0.5235987755982988 Order = 7 0 -0.9749279121818237 0.4487989505128276 1 -0.7818314824680295 0.4487989505128276 2 -0.4338837391175581 0.4487989505128276 3 0 0.4487989505128276 4 0.4338837391175582 0.4487989505128276 5 0.7818314824680298 0.4487989505128276 6 0.9749279121818236 0.4487989505128276 Order = 8 0 -0.9807852804032304 0.3926990816987241 1 -0.8314696123025453 0.3926990816987241 2 -0.555570233019602 0.3926990816987241 3 -0.1950903220161282 0.3926990816987241 4 0.1950903220161283 0.3926990816987241 5 0.5555702330196023 0.3926990816987241 6 0.8314696123025452 0.3926990816987241 7 0.9807852804032304 0.3926990816987241 Order = 9 0 -0.9848077530122081 0.3490658503988659 1 -0.8660254037844385 0.3490658503988659 2 -0.6427876096865393 0.3490658503988659 3 -0.3420201433256685 0.3490658503988659 4 0 0.3490658503988659 5 0.3420201433256688 0.3490658503988659 6 0.6427876096865394 0.3490658503988659 7 0.8660254037844387 0.3490658503988659 8 0.984807753012208 0.3490658503988659 Order = 10 0 -0.9876883405951377 0.3141592653589793 1 -0.8910065241883678 0.3141592653589793 2 -0.7071067811865475 0.3141592653589793 3 -0.4539904997395467 0.3141592653589793 4 -0.1564344650402306 0.3141592653589793 5 0.1564344650402309 0.3141592653589793 6 0.4539904997395468 0.3141592653589793 7 0.7071067811865476 0.3141592653589793 8 0.8910065241883679 0.3141592653589793 9 0.9876883405951378 0.3141592653589793 TEST03 CHEBYSHEV2_COMPUTE_NP computes a Gauss-Chebyshev type 2 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) * sqrt ( 1 - x^2 ) dx. CHEBYSHEV2_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 1.5708 1.5708 0 1 1 0 0 0 1 2 0 0.392699 0.392699 1 3 0 0 0 1 4 0 0.19635 0.19635 2 0 1.5708 1.5708 0 2 1 0 0 0 2 2 0.392699 0.392699 5.55112e-17 2 3 1.249e-16 0 1.249e-16 2 4 0.0981748 0.19635 0.0981748 2 5 6.59195e-17 0 6.59195e-17 2 6 0.0245437 0.122718 0.0981748 3 0 1.5708 1.5708 0 3 1 -1.11022e-16 0 1.11022e-16 3 2 0.392699 0.392699 0 3 3 0 0 0 3 4 0.19635 0.19635 0 3 5 2.77556e-17 0 2.77556e-17 3 6 0.0981748 0.122718 0.0245437 3 7 2.77556e-17 0 2.77556e-17 3 8 0.0490874 0.0859029 0.0368155 4 0 1.5708 1.5708 2.22045e-16 4 1 -5.55112e-17 0 5.55112e-17 4 2 0.392699 0.392699 0 4 3 0 0 0 4 4 0.19635 0.19635 2.77556e-17 4 5 -1.38778e-17 0 1.38778e-17 4 6 0.122718 0.122718 1.38778e-17 4 7 0 0 0 4 8 0.079767 0.0859029 0.00613592 4 9 6.93889e-18 0 6.93889e-18 4 10 0.0521553 0.0644272 0.0122718 5 0 1.5708 1.5708 0 5 1 0 0 0 5 2 0.392699 0.392699 5.55112e-17 5 3 6.93889e-17 0 6.93889e-17 5 4 0.19635 0.19635 0 5 5 4.16334e-17 0 4.16334e-17 5 6 0.122718 0.122718 4.16334e-17 5 7 1.38778e-17 0 1.38778e-17 5 8 0.0859029 0.0859029 4.16334e-17 5 9 6.93889e-18 0 6.93889e-18 5 10 0.0628932 0.0644272 0.00153398 5 11 0 0 0 5 12 0.0467864 0.0506214 0.00383495 6 0 1.5708 1.5708 0 6 1 -2.77556e-17 0 2.77556e-17 6 2 0.392699 0.392699 0 6 3 1.38778e-17 0 1.38778e-17 6 4 0.19635 0.19635 2.77556e-17 6 5 3.46945e-17 0 3.46945e-17 6 6 0.122718 0.122718 0 6 7 2.77556e-17 0 2.77556e-17 6 8 0.0859029 0.0859029 1.38778e-17 6 9 2.08167e-17 0 2.08167e-17 6 10 0.0644272 0.0644272 1.38778e-17 6 11 2.08167e-17 0 2.08167e-17 6 12 0.0502379 0.0506214 0.000383495 6 13 2.08167e-17 0 2.08167e-17 6 14 0.0399794 0.0411299 0.00115049 7 0 1.5708 1.5708 0 7 1 -1.38778e-17 0 1.38778e-17 7 2 0.392699 0.392699 5.55112e-17 7 3 6.93889e-18 0 6.93889e-18 7 4 0.19635 0.19635 0 7 5 0 0 0 7 6 0.122718 0.122718 1.38778e-17 7 7 0 0 0 7 8 0.0859029 0.0859029 0 7 9 6.93889e-18 0 6.93889e-18 7 10 0.0644272 0.0644272 1.38778e-17 7 11 -3.46945e-18 0 3.46945e-18 7 12 0.0506214 0.0506214 6.93889e-18 7 13 -3.46945e-18 0 3.46945e-18 7 14 0.041034 0.0411299 9.58738e-05 7 15 -3.46945e-18 0 3.46945e-18 7 16 0.0339393 0.0342749 0.000335558 8 0 1.5708 1.5708 2.22045e-16 8 1 6.93889e-18 0 6.93889e-18 8 2 0.392699 0.392699 0 8 3 8.32667e-17 0 8.32667e-17 8 4 0.19635 0.19635 5.55112e-17 8 5 2.77556e-17 0 2.77556e-17 8 6 0.122718 0.122718 1.38778e-17 8 7 3.81639e-17 0 3.81639e-17 8 8 0.0859029 0.0859029 1.38778e-17 8 9 3.81639e-17 0 3.81639e-17 8 10 0.0644272 0.0644272 0 8 11 2.42861e-17 0 2.42861e-17 8 12 0.0506214 0.0506214 1.38778e-17 8 13 2.77556e-17 0 2.77556e-17 8 14 0.0411299 0.0411299 6.93889e-18 8 15 2.77556e-17 0 2.77556e-17 8 16 0.0342509 0.0342749 2.39684e-05 8 17 2.25514e-17 0 2.25514e-17 8 18 0.0290378 0.0291337 9.58738e-05 9 0 1.5708 1.5708 2.22045e-16 9 1 -6.59195e-17 0 6.59195e-17 9 2 0.392699 0.392699 0 9 3 1.73472e-17 0 1.73472e-17 9 4 0.19635 0.19635 2.77556e-17 9 5 0 0 0 9 6 0.122718 0.122718 1.38778e-17 9 7 0 0 0 9 8 0.0859029 0.0859029 0 9 9 0 0 0 9 10 0.0644272 0.0644272 1.38778e-17 9 11 -3.46945e-18 0 3.46945e-18 9 12 0.0506214 0.0506214 1.38778e-17 9 13 -8.67362e-18 0 8.67362e-18 9 14 0.0411299 0.0411299 2.77556e-17 9 15 -6.93889e-18 0 6.93889e-18 9 16 0.0342749 0.0342749 2.08167e-17 9 17 -8.67362e-18 0 8.67362e-18 9 18 0.0291277 0.0291337 5.99211e-06 9 19 -5.20417e-18 0 5.20417e-18 9 20 0.0251339 0.0251609 2.69645e-05 10 0 1.5708 1.5708 2.22045e-16 10 1 -6.59195e-17 0 6.59195e-17 10 2 0.392699 0.392699 5.55112e-17 10 3 2.42861e-17 0 2.42861e-17 10 4 0.19635 0.19635 5.55112e-17 10 5 1.73472e-17 0 1.73472e-17 10 6 0.122718 0.122718 0 10 7 1.73472e-17 0 1.73472e-17 10 8 0.0859029 0.0859029 1.38778e-17 10 9 1.56125e-17 0 1.56125e-17 10 10 0.0644272 0.0644272 0 10 11 1.56125e-17 0 1.56125e-17 10 12 0.0506214 0.0506214 6.93889e-18 10 13 1.38778e-17 0 1.38778e-17 10 14 0.0411299 0.0411299 2.08167e-17 10 15 8.67362e-18 0 8.67362e-18 10 16 0.0342749 0.0342749 1.38778e-17 10 17 6.93889e-18 0 6.93889e-18 10 18 0.0291337 0.0291337 2.08167e-17 10 19 6.93889e-18 0 6.93889e-18 10 20 0.0251594 0.0251609 1.49803e-06 10 21 6.93889e-18 0 6.93889e-18 10 22 0.0220083 0.0220158 7.49014e-06 TEST04 CHEBYSHEV2_COMPUTE_NP computes a Gauss-Chebyshev type 2 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) * sqrt(1-x^2) dx. Order = 1 0 0 1.570796326794897 Order = 2 0 -0.4999999999999998 0.7853981633974486 1 0.5000000000000001 0.7853981633974481 Order = 3 0 -0.7071067811865475 0.3926990816987243 1 0 0.7853981633974483 2 0.7071067811865476 0.392699081698724 Order = 4 0 -0.8090169943749473 0.2170787134227061 1 -0.3090169943749473 0.5683194499747424 2 0.3090169943749475 0.5683194499747423 3 0.8090169943749475 0.217078713422706 Order = 5 0 -0.8660254037844387 0.1308996938995747 1 -0.4999999999999998 0.3926990816987243 2 0 0.5235987755982988 3 0.5000000000000001 0.392699081698724 4 0.8660254037844387 0.1308996938995747 Order = 6 0 -0.900968867902419 0.08448869089158863 1 -0.6234898018587334 0.2743330560697779 2 -0.2225209339563143 0.4265764164360819 3 0.2225209339563144 0.4265764164360819 4 0.6234898018587336 0.2743330560697778 5 0.9009688679024191 0.08448869089158857 Order = 7 0 -0.9238795325112867 0.05750944903191315 1 -0.7071067811865475 0.1963495408493621 2 -0.3826834323650897 0.335189632666811 3 0 0.3926990816987241 4 0.3826834323650898 0.335189632666811 5 0.7071067811865476 0.196349540849362 6 0.9238795325112867 0.05750944903191313 Order = 8 0 -0.9396926207859083 0.04083294770910712 1 -0.7660444431189779 0.1442256007956728 2 -0.4999999999999998 0.2617993877991495 3 -0.1736481776669303 0.338540227093519 4 0.1736481776669304 0.338540227093519 5 0.5000000000000001 0.2617993877991494 6 0.7660444431189781 0.1442256007956727 7 0.9396926207859084 0.04083294770910708 Order = 9 0 -0.9510565162951535 0.02999954037160818 1 -0.8090169943749473 0.108539356711353 2 -0.587785252292473 0.2056199086476263 3 -0.3090169943749473 0.2841597249873712 4 0 0.3141592653589793 5 0.3090169943749475 0.2841597249873711 6 0.5877852522924732 0.2056199086476263 7 0.8090169943749475 0.108539356711353 8 0.9510565162951535 0.02999954037160816 Order = 10 0 -0.9594929736144974 0.02266894250185883 1 -0.8412535328311811 0.08347854093418908 2 -0.654860733945285 0.1631221774548167 3 -0.4154150130018863 0.2363135602034873 4 -0.142314838273285 0.2798149423030966 5 0.1423148382732851 0.2798149423030965 6 0.4154150130018864 0.2363135602034873 7 0.6548607339452852 0.1631221774548166 8 0.8412535328311812 0.08347854093418902 9 0.9594929736144974 0.02266894250185884 TEST05 CLENSHAW_CURTIS_COMPUTE_NP computes a Clenshaw Curtis rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. LEGENDRE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = ORDER+1 if ORDER is odd, or N = ORDER if ORDER is even In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 2 2 0 1 1 0 0 0 1 2 0 0.666667 0.666667 1 3 0 0 0 1 4 0 0.4 0.4 2 0 2 2 0 2 1 0 0 0 2 2 2 0.666667 1.33333 2 3 0 0 0 2 4 2 0.4 1.6 3 0 2 2 0 3 1 0 0 0 3 2 0.666667 0.666667 1.11022e-16 3 3 0 0 0 3 4 0.666667 0.4 0.266667 3 5 0 0 0 3 6 0.666667 0.285714 0.380952 4 0 2 2 4.44089e-16 4 1 4.57967e-16 0 4.57967e-16 4 2 0.666667 0.666667 1.11022e-16 4 3 2.63678e-16 0 2.63678e-16 4 4 0.333333 0.4 0.0666667 4 5 9.71445e-17 0 9.71445e-17 4 6 0.25 0.285714 0.0357143 5 0 2 2 0 5 1 1.38778e-16 0 1.38778e-16 5 2 0.666667 0.666667 1.11022e-16 5 3 1.11022e-16 0 1.11022e-16 5 4 0.4 0.4 0 5 5 1.249e-16 0 1.249e-16 5 6 0.266667 0.285714 0.0190476 5 7 6.93889e-17 0 6.93889e-17 5 8 0.2 0.222222 0.0222222 6 0 2 2 0 6 1 1.38778e-16 0 1.38778e-16 6 2 0.666667 0.666667 1.11022e-16 6 3 1.38778e-16 0 1.38778e-16 6 4 0.4 0.4 0 6 5 8.32667e-17 0 8.32667e-17 6 6 0.283333 0.285714 0.00238095 6 7 6.93889e-17 0 6.93889e-17 6 8 0.2125 0.222222 0.00972222 7 0 2 2 0 7 1 3.57353e-16 0 3.57353e-16 7 2 0.666667 0.666667 1.11022e-16 7 3 1.9082e-16 0 1.9082e-16 7 4 0.4 0.4 1.11022e-16 7 5 1.21431e-16 0 1.21431e-16 7 6 0.285714 0.285714 1.11022e-16 7 7 5.20417e-17 0 5.20417e-17 7 8 0.221429 0.222222 0.000793651 7 9 2.42861e-17 0 2.42861e-17 7 10 0.178571 0.181818 0.00324675 8 0 2 2 0 8 1 1.49186e-16 0 1.49186e-16 8 2 0.666667 0.666667 1.11022e-16 8 3 1.49186e-16 0 1.49186e-16 8 4 0.4 0.4 0 8 5 1.07553e-16 0 1.07553e-16 8 6 0.285714 0.285714 5.55112e-17 8 7 9.36751e-17 0 9.36751e-17 8 8 0.222024 0.222222 0.000198413 8 9 9.36751e-17 0 9.36751e-17 8 10 0.181101 0.181818 0.000716991 9 0 2 2 0 9 1 2.56739e-16 0 2.56739e-16 9 2 0.666667 0.666667 1.11022e-16 9 3 1.8735e-16 0 1.8735e-16 9 4 0.4 0.4 5.55112e-17 9 5 1.45717e-16 0 1.45717e-16 9 6 0.285714 0.285714 5.55112e-17 9 7 1.17961e-16 0 1.17961e-16 9 8 0.222222 0.222222 5.55112e-17 9 9 7.63278e-17 0 7.63278e-17 9 10 0.181746 0.181818 7.21501e-05 9 11 6.93889e-17 0 6.93889e-17 9 12 0.153571 0.153846 0.000274725 10 0 2 2 0 10 1 3.95517e-16 0 3.95517e-16 10 2 0.666667 0.666667 1.11022e-16 10 3 2.56739e-16 0 2.56739e-16 10 4 0.4 0.4 5.55112e-17 10 5 2.56739e-16 0 2.56739e-16 10 6 0.285714 0.285714 0 10 7 1.59595e-16 0 1.59595e-16 10 8 0.222222 0.222222 2.77556e-17 10 9 1.73472e-16 0 1.73472e-16 10 10 0.181796 0.181818 2.25469e-05 10 11 1.249e-16 0 1.249e-16 10 12 0.153757 0.153846 8.87134e-05 TEST06 CLENSHAW_CURTIS_COMPUTE_NP computes a Clenshaw Curtis rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. Order = 1 0 0 2 Order = 2 0 -1 1 1 1 1 Order = 3 0 -1 0.3333333333333334 1 0 1.333333333333333 2 1 0.3333333333333334 Order = 4 0 -1 0.1111111111111111 1 -0.4999999999999998 0.8888888888888888 2 0.5000000000000001 0.8888888888888892 3 1 0.1111111111111111 Order = 5 0 -1 0.06666666666666668 1 -0.7071067811865475 0.5333333333333333 2 0 0.7999999999999999 3 0.7071067811865476 0.5333333333333334 4 1 0.06666666666666668 Order = 6 0 -1 0.04000000000000001 1 -0.8090169943749473 0.3607430412000112 2 -0.3090169943749473 0.5992569587999889 3 0.3090169943749475 0.5992569587999887 4 0.8090169943749475 0.3607430412000113 5 1 0.04000000000000001 Order = 7 0 -1 0.02857142857142858 1 -0.8660254037844387 0.2539682539682539 2 -0.4999999999999998 0.4571428571428571 3 0 0.5206349206349206 4 0.5000000000000001 0.4571428571428573 5 0.8660254037844387 0.253968253968254 6 1 0.02857142857142858 Order = 8 0 -1 0.02040816326530613 1 -0.900968867902419 0.1901410072182084 2 -0.6234898018587334 0.3522424237181591 3 -0.2225209339563143 0.4372084057983264 4 0.2225209339563144 0.4372084057983264 5 0.6234898018587336 0.3522424237181591 6 0.9009688679024191 0.1901410072182084 7 1 0.02040816326530613 Order = 9 0 -1 0.01587301587301588 1 -0.9238795325112867 0.1462186492160181 2 -0.7071067811865475 0.2793650793650794 3 -0.3826834323650897 0.3617178587204897 4 0 0.3936507936507936 5 0.3826834323650898 0.3617178587204898 6 0.7071067811865476 0.2793650793650794 7 0.9238795325112867 0.1462186492160182 8 1 0.01587301587301588 Order = 10 0 -1 0.01234567901234569 1 -0.9396926207859083 0.1165674565720371 2 -0.7660444431189779 0.2252843233381044 3 -0.4999999999999998 0.3019400352733685 4 -0.1736481776669303 0.3438625058041442 5 0.1736481776669304 0.3438625058041442 6 0.5000000000000001 0.3019400352733687 7 0.7660444431189781 0.2252843233381044 8 0.9396926207859084 0.1165674565720372 9 1 0.01234567901234569 TEST07 FEJER2_COMPUTE_NP computes a Fejer Type 2 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. LEGENDRE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = ORDER+1 if ORDER is odd, or N = ORDER if ORDER is even In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 2 2 0 1 1 0 0 0 1 2 0 0.666667 0.666667 1 3 0 0 0 1 4 0 0.4 0.4 2 0 2 2 0 2 1 3.33067e-16 0 3.33067e-16 2 2 0.5 0.666667 0.166667 2 3 2.498e-16 0 2.498e-16 2 4 0.125 0.4 0.275 3 0 2 2 0 3 1 0 0 0 3 2 0.666667 0.666667 1.11022e-16 3 3 8.32667e-17 0 8.32667e-17 3 4 0.333333 0.4 0.0666667 3 5 1.11022e-16 0 1.11022e-16 3 6 0.166667 0.285714 0.119048 4 0 2 2 0 4 1 1.11022e-16 0 1.11022e-16 4 2 0.666667 0.666667 1.11022e-16 4 3 0 0 0 4 4 0.375 0.4 0.025 4 5 0 0 0 4 6 0.239583 0.285714 0.046131 5 0 2 2 0 5 1 0 0 0 5 2 0.666667 0.666667 1.11022e-16 5 3 5.55112e-17 0 5.55112e-17 5 4 0.4 0.4 1.11022e-16 5 5 0 0 0 5 6 0.275 0.285714 0.0107143 5 7 -1.38778e-17 0 1.38778e-17 5 8 0.2 0.222222 0.0222222 6 0 2 2 2.22045e-16 6 1 1.66533e-16 0 1.66533e-16 6 2 0.666667 0.666667 0 6 3 1.66533e-16 0 1.66533e-16 6 4 0.4 0.4 1.11022e-16 6 5 1.11022e-16 0 1.11022e-16 6 6 0.28125 0.285714 0.00446429 6 7 1.11022e-16 0 1.11022e-16 6 8 0.211979 0.222222 0.0102431 7 0 2 2 0 7 1 -8.32667e-17 0 8.32667e-17 7 2 0.666667 0.666667 1.11022e-16 7 3 -8.32667e-17 0 8.32667e-17 7 4 0.4 0.4 0 7 5 -5.55112e-17 0 5.55112e-17 7 6 0.285714 0.285714 5.55112e-17 7 7 -4.16334e-17 0 4.16334e-17 7 8 0.220238 0.222222 0.00198413 7 9 -4.16334e-17 0 4.16334e-17 7 10 0.176786 0.181818 0.00503247 8 0 2 2 4.44089e-16 8 1 0 0 0 8 2 0.666667 0.666667 0 8 3 1.249e-16 0 1.249e-16 8 4 0.4 0.4 1.11022e-16 8 5 1.38778e-16 0 1.38778e-16 8 6 0.285714 0.285714 0 8 7 1.11022e-16 0 1.11022e-16 8 8 0.221354 0.222222 0.000868056 8 9 1.249e-16 0 1.249e-16 8 10 0.179408 0.181818 0.0024097 9 0 2 2 0 9 1 6.93889e-17 0 6.93889e-17 9 2 0.666667 0.666667 0 9 3 1.38778e-17 0 1.38778e-17 9 4 0.4 0.4 5.55112e-17 9 5 0 0 0 9 6 0.285714 0.285714 5.55112e-17 9 7 1.38778e-17 0 1.38778e-17 9 8 0.222222 0.222222 5.55112e-17 9 9 -1.38778e-17 0 1.38778e-17 9 10 0.181424 0.181818 0.000394571 9 11 -1.38778e-17 0 1.38778e-17 9 12 0.152654 0.153846 0.00119238 10 0 2 2 0 10 1 2.77556e-17 0 2.77556e-17 10 2 0.666667 0.666667 1.11022e-16 10 3 1.38778e-16 0 1.38778e-16 10 4 0.4 0.4 1.11022e-16 10 5 6.93889e-17 0 6.93889e-17 10 6 0.285714 0.285714 0 10 7 6.93889e-17 0 6.93889e-17 10 8 0.222222 0.222222 0 10 9 2.77556e-17 0 2.77556e-17 10 10 0.181641 0.181818 0.000177557 10 11 4.16334e-17 0 4.16334e-17 10 12 0.153266 0.153846 0.000580095 TEST08 FEJER2_COMPUTE_NP computes a Fejer Type 2 rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. Order = 1 0 0 2 Order = 2 0 -0.4999999999999998 1 1 0.5000000000000001 1 Order = 3 0 -0.7071067811865475 0.6666666666666667 1 0 0.6666666666666666 2 0.7071067811865476 0.6666666666666666 Order = 4 0 -0.8090169943749473 0.4254644007500071 1 -0.3090169943749473 0.574535599249993 2 0.3090169943749475 0.574535599249993 3 0.8090169943749475 0.425464400750007 Order = 5 0 -0.8660254037844387 0.3111111111111111 1 -0.4999999999999998 0.4000000000000001 2 0 0.5777777777777777 3 0.5000000000000001 0.4 4 0.8660254037844387 0.3111111111111111 Order = 6 0 -0.900968867902419 0.2269152467244296 1 -0.6234898018587334 0.3267938603769863 2 -0.2225209339563143 0.4462908928985842 3 0.2225209339563144 0.4462908928985841 4 0.6234898018587336 0.3267938603769863 5 0.9009688679024191 0.2269152467244296 Order = 7 0 -0.9238795325112867 0.17796468096205 1 -0.7071067811865475 0.2476190476190477 2 -0.3826834323650897 0.3934638904665215 3 0 0.3619047619047619 4 0.3826834323650898 0.3934638904665215 5 0.7071067811865476 0.2476190476190476 6 0.9238795325112867 0.1779646809620499 Order = 8 0 -0.9396926207859083 0.1397697435050226 1 -0.7660444431189779 0.2063696457302284 2 -0.4999999999999998 0.3142857142857144 3 -0.1736481776669303 0.3395748964790348 4 0.1736481776669304 0.3395748964790348 5 0.5000000000000001 0.3142857142857143 6 0.7660444431189781 0.2063696457302284 7 0.9396926207859084 0.1397697435050225 Order = 9 0 -0.9510565162951535 0.1147810750857218 1 -0.8090169943749473 0.1654331942222276 2 -0.587785252292473 0.2737903534857068 3 -0.3090169943749473 0.2790112502222169 4 0 0.3339682539682539 5 0.3090169943749475 0.279011250222217 6 0.5877852522924732 0.2737903534857068 7 0.8090169943749475 0.1654331942222276 8 0.9510565162951535 0.1147810750857217 Order = 10 0 -0.9594929736144974 0.09441954173982806 1 -0.8412535328311811 0.1411354380109716 2 -0.654860733945285 0.2263866903636005 3 -0.4154150130018863 0.2530509772156453 4 -0.142314838273285 0.2850073526699546 5 0.1423148382732851 0.2850073526699544 6 0.4154150130018864 0.2530509772156453 7 0.6548607339452852 0.2263866903636005 8 0.8412535328311812 0.1411354380109716 9 0.9594929736144974 0.09441954173982806 TEST09 GEGENBAUER_COMPUTE_NP computes a generalized Gauss-Gegenbauer rule which is appropriate for integrands of the form Integral ( 0 <= x < +oo ) f(x) (1-x^2)^alpha dx. GEGENBAUER_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Alpha Estimate Exact Error 1 0 0.5 1.5708 1.5708 1.33227e-15 1 1 0.5 0 0 0 1 2 0.5 0 0.392699 0.392699 1 3 0.5 0 0 0 1 4 0.5 0 0.19635 0.19635 2 0 0.5 1.5708 1.5708 1.33227e-15 2 1 0.5 0 0 0 2 2 0.5 0.392699 0.392699 1.11022e-16 2 3 0.5 0 0 0 2 4 0.5 0.0981748 0.19635 0.0981748 2 5 0.5 0 0 0 2 6 0.5 0.0245437 0.122718 0.0981748 3 0 0.5 1.5708 1.5708 1.33227e-15 3 1 0.5 4.996e-16 0 4.996e-16 3 2 0.5 0.392699 0.392699 1.11022e-16 3 3 0.5 3.05311e-16 0 3.05311e-16 3 4 0.5 0.19635 0.19635 8.32667e-17 3 5 0.5 1.94289e-16 0 1.94289e-16 3 6 0.5 0.0981748 0.122718 0.0245437 3 7 0.5 1.04083e-16 0 1.04083e-16 3 8 0.5 0.0490874 0.0859029 0.0368155 4 0 0.5 1.5708 1.5708 1.11022e-15 4 1 0.5 -2.77556e-17 0 2.77556e-17 4 2 0.5 0.392699 0.392699 1.11022e-16 4 3 0.5 -1.38778e-17 0 1.38778e-17 4 4 0.5 0.19635 0.19635 5.55112e-17 4 5 0.5 0 0 0 4 6 0.5 0.122718 0.122718 1.38778e-17 4 7 0.5 0 0 0 4 8 0.5 0.079767 0.0859029 0.00613592 4 9 0.5 0 0 0 4 10 0.5 0.0521553 0.0644272 0.0122718 5 0 0.5 1.5708 1.5708 2.22045e-15 5 1 0.5 4.44089e-16 0 4.44089e-16 5 2 0.5 0.392699 0.392699 7.21645e-16 5 3 0.5 2.91434e-16 0 2.91434e-16 5 4 0.5 0.19635 0.19635 5.55112e-16 5 5 0.5 2.35922e-16 0 2.35922e-16 5 6 0.5 0.122718 0.122718 3.46945e-16 5 7 0.5 1.73472e-16 0 1.73472e-16 5 8 0.5 0.0859029 0.0859029 2.35922e-16 5 9 0.5 1.31839e-16 0 1.31839e-16 5 10 0.5 0.0628932 0.0644272 0.00153398 5 11 0.5 9.71445e-17 0 9.71445e-17 5 12 0.5 0.0467864 0.0506214 0.00383495 6 0 0.5 1.5708 1.5708 1.9984e-15 6 1 0.5 -1.38778e-17 0 1.38778e-17 6 2 0.5 0.392699 0.392699 6.10623e-16 6 3 0.5 1.38778e-17 0 1.38778e-17 6 4 0.5 0.19635 0.19635 4.16334e-16 6 5 0.5 6.93889e-18 0 6.93889e-18 6 6 0.5 0.122718 0.122718 2.498e-16 6 7 0.5 0 0 0 6 8 0.5 0.0859029 0.0859029 1.94289e-16 6 9 0.5 0 0 0 6 10 0.5 0.0644272 0.0644272 1.38778e-17 6 11 0.5 0 0 0 6 12 0.5 0.0502379 0.0506214 0.000383495 6 13 0.5 0 0 0 6 14 0.5 0.0399794 0.0411299 0.00115049 7 0 0.5 1.5708 1.5708 1.11022e-15 7 1 0.5 3.19189e-16 0 3.19189e-16 7 2 0.5 0.392699 0.392699 2.22045e-16 7 3 0.5 2.08167e-16 0 2.08167e-16 7 4 0.5 0.19635 0.19635 1.66533e-16 7 5 0.5 1.17961e-16 0 1.17961e-16 7 6 0.5 0.122718 0.122718 2.08167e-16 7 7 0.5 6.93889e-17 0 6.93889e-17 7 8 0.5 0.0859029 0.0859029 1.66533e-16 7 9 0.5 3.81639e-17 0 3.81639e-17 7 10 0.5 0.0644272 0.0644272 3.05311e-16 7 11 0.5 2.08167e-17 0 2.08167e-17 7 12 0.5 0.0506214 0.0506214 6.03684e-16 7 13 0.5 1.04083e-17 0 1.04083e-17 7 14 0.5 0.041034 0.0411299 9.58738e-05 7 15 0.5 6.93889e-18 0 6.93889e-18 7 16 0.5 0.0339393 0.0342749 0.000335558 8 0 0.5 1.5708 1.5708 6.66134e-16 8 1 0.5 5.6205e-16 0 5.6205e-16 8 2 0.5 0.392699 0.392699 5.55112e-17 8 3 0.5 4.996e-16 0 4.996e-16 8 4 0.5 0.19635 0.19635 1.11022e-16 8 5 0.5 4.4062e-16 0 4.4062e-16 8 6 0.5 0.122718 0.122718 1.66533e-16 8 7 0.5 3.85109e-16 0 3.85109e-16 8 8 0.5 0.0859029 0.0859029 1.52656e-16 8 9 0.5 3.43475e-16 0 3.43475e-16 8 10 0.5 0.0644272 0.0644272 3.19189e-16 8 11 0.5 3.08781e-16 0 3.08781e-16 8 12 0.5 0.0506214 0.0506214 6.17562e-16 8 13 0.5 2.67147e-16 0 2.67147e-16 8 14 0.5 0.0411299 0.0411299 8.32667e-17 8 15 0.5 2.32453e-16 0 2.32453e-16 8 16 0.5 0.0342509 0.0342749 2.39684e-05 8 17 0.5 2.11636e-16 0 2.11636e-16 8 18 0.5 0.0290378 0.0291337 9.58738e-05 9 0 0.5 1.5708 1.5708 4.44089e-16 9 1 0.5 -6.93889e-18 0 6.93889e-18 9 2 0.5 0.392699 0.392699 4.44089e-16 9 3 0.5 -3.46945e-18 0 3.46945e-18 9 4 0.5 0.19635 0.19635 3.33067e-16 9 5 0.5 0 0 0 9 6 0.5 0.122718 0.122718 3.60822e-16 9 7 0.5 0 0 0 9 8 0.5 0.0859029 0.0859029 3.46945e-16 9 9 0.5 0 0 0 9 10 0.5 0.0644272 0.0644272 4.71845e-16 9 11 0.5 0 0 0 9 12 0.5 0.0506214 0.0506214 7.70217e-16 9 13 0.5 0 0 0 9 14 0.5 0.0411299 0.0411299 2.08167e-16 9 15 0.5 1.73472e-18 0 1.73472e-18 9 16 0.5 0.0342749 0.0342749 6.93889e-17 9 17 0.5 0 0 0 9 18 0.5 0.0291277 0.0291337 5.99211e-06 9 19 0.5 0 0 0 9 20 0.5 0.0251339 0.0251609 2.69645e-05 10 0 0.5 1.5708 1.5708 1.55431e-15 10 1 0.5 -2.42861e-17 0 2.42861e-17 10 2 0.5 0.392699 0.392699 5.55112e-17 10 3 0.5 1.73472e-17 0 1.73472e-17 10 4 0.5 0.19635 0.19635 5.55112e-17 10 5 0.5 -3.46945e-18 0 3.46945e-18 10 6 0.5 0.122718 0.122718 1.249e-16 10 7 0.5 0 0 0 10 8 0.5 0.0859029 0.0859029 1.66533e-16 10 9 0.5 0 0 0 10 10 0.5 0.0644272 0.0644272 3.33067e-16 10 11 0.5 -1.73472e-18 0 1.73472e-18 10 12 0.5 0.0506214 0.0506214 6.45317e-16 10 13 0.5 -1.73472e-18 0 1.73472e-18 10 14 0.5 0.0411299 0.0411299 1.04083e-16 10 15 0.5 0 0 0 10 16 0.5 0.0342749 0.0342749 3.46945e-17 10 17 0.5 0 0 0 10 18 0.5 0.0291337 0.0291337 1.38778e-16 10 19 0.5 0 0 0 10 20 0.5 0.0251594 0.0251609 1.49803e-06 10 21 0.5 0 0 0 10 22 0.5 0.0220083 0.0220158 7.49014e-06 1 0 1 1.33333 1.33333 0 1 1 1 0 0 0 1 2 1 0 0.266667 0.266667 1 3 1 0 0 0 1 4 1 0 0.114286 0.114286 2 0 1 1.33333 1.33333 0 2 1 1 1.66533e-16 0 1.66533e-16 2 2 1 0.266667 0.266667 5.55112e-17 2 3 1 4.85723e-17 0 4.85723e-17 2 4 1 0.0533333 0.114286 0.0609524 2 5 1 1.04083e-17 0 1.04083e-17 2 6 1 0.0106667 0.0634921 0.0528254 3 0 1 1.33333 1.33333 4.44089e-16 3 1 1 0 0 0 3 2 1 0.266667 0.266667 2.77556e-16 3 3 1 0 0 0 3 4 1 0.114286 0.114286 1.11022e-16 3 5 1 0 0 0 3 6 1 0.0489796 0.0634921 0.0145125 3 7 1 0 0 0 3 8 1 0.0209913 0.040404 0.0194128 4 0 1 1.33333 1.33333 2.22045e-16 4 1 1 -2.08167e-16 0 2.08167e-16 4 2 1 0.266667 0.266667 5.55112e-17 4 3 1 -1.11022e-16 0 1.11022e-16 4 4 1 0.114286 0.114286 2.77556e-17 4 5 1 -6.245e-17 0 6.245e-17 4 6 1 0.0634921 0.0634921 1.38778e-17 4 7 1 -3.1225e-17 0 3.1225e-17 4 8 1 0.0368859 0.040404 0.00351817 4 9 1 -1.73472e-17 0 1.73472e-17 4 10 1 0.0215671 0.027972 0.00640488 5 0 1 1.33333 1.33333 0 5 1 1 2.91434e-16 0 2.91434e-16 5 2 1 0.266667 0.266667 0 5 3 1 2.22045e-16 0 2.22045e-16 5 4 1 0.114286 0.114286 0 5 5 1 1.59595e-16 0 1.59595e-16 5 6 1 0.0634921 0.0634921 0 5 7 1 1.249e-16 0 1.249e-16 5 8 1 0.040404 0.040404 6.93889e-18 5 9 1 8.67362e-17 0 8.67362e-17 5 10 1 0.0271109 0.027972 0.000861092 5 11 1 6.76542e-17 0 6.76542e-17 5 12 1 0.0185245 0.0205128 0.00198834 6 0 1 1.33333 1.33333 0 6 1 1 -2.08167e-16 0 2.08167e-16 6 2 1 0.266667 0.266667 5.55112e-17 6 3 1 -1.38778e-16 0 1.38778e-16 6 4 1 0.114286 0.114286 1.38778e-17 6 5 1 -9.71445e-17 0 9.71445e-17 6 6 1 0.0634921 0.0634921 1.38778e-17 6 7 1 -6.93889e-17 0 6.93889e-17 6 8 1 0.040404 0.040404 2.08167e-17 6 9 1 -4.85723e-17 0 4.85723e-17 6 10 1 0.027972 0.027972 1.49186e-16 6 11 1 -3.29597e-17 0 3.29597e-17 6 12 1 0.0203009 0.0205128 0.000211961 6 13 1 -2.42861e-17 0 2.42861e-17 6 14 1 0.0150926 0.0156863 0.000593683 7 0 1 1.33333 1.33333 2.22045e-16 7 1 1 -3.81639e-17 0 3.81639e-17 7 2 1 0.266667 0.266667 5.55112e-17 7 3 1 -3.46945e-18 0 3.46945e-18 7 4 1 0.114286 0.114286 5.55112e-17 7 5 1 0 0 0 7 6 1 0.0634921 0.0634921 2.77556e-17 7 7 1 1.73472e-18 0 1.73472e-18 7 8 1 0.040404 0.040404 2.08167e-17 7 9 1 0 0 0 7 10 1 0.027972 0.027972 1.21431e-16 7 11 1 0 0 0 7 12 1 0.0205128 0.0205128 8.32667e-17 7 13 1 0 0 0 7 14 1 0.0156339 0.0156863 5.23668e-05 7 15 1 0 0 0 7 16 1 0.0122114 0.0123839 0.000172535 8 0 1 1.33333 1.33333 1.33227e-15 8 1 1 5.20417e-17 0 5.20417e-17 8 2 1 0.266667 0.266667 3.88578e-16 8 3 1 1.38778e-17 0 1.38778e-17 8 4 1 0.114286 0.114286 2.22045e-16 8 5 1 0 0 0 8 6 1 0.0634921 0.0634921 1.38778e-16 8 7 1 1.73472e-18 0 1.73472e-18 8 8 1 0.040404 0.040404 1.04083e-16 8 9 1 1.73472e-18 0 1.73472e-18 8 10 1 0.027972 0.027972 2.11636e-16 8 11 1 0 0 0 8 12 1 0.0205128 0.0205128 1.04083e-17 8 13 1 8.67362e-19 0 8.67362e-19 8 14 1 0.0156863 0.0156863 2.42861e-17 8 15 1 0 0 0 8 16 1 0.0123709 0.0123839 1.29701e-05 8 17 1 0 0 0 8 18 1 0.00997591 0.0100251 4.91557e-05 9 0 1 1.33333 1.33333 2.22045e-16 9 1 1 6.93889e-18 0 6.93889e-18 9 2 1 0.266667 0.266667 5.55112e-17 9 3 1 0 0 0 9 4 1 0.114286 0.114286 5.55112e-17 9 5 1 3.46945e-18 0 3.46945e-18 9 6 1 0.0634921 0.0634921 2.77556e-17 9 7 1 -3.46945e-18 0 3.46945e-18 9 8 1 0.040404 0.040404 2.08167e-17 9 9 1 0 0 0 9 10 1 0.027972 0.027972 1.52656e-16 9 11 1 8.67362e-19 0 8.67362e-19 9 12 1 0.0205128 0.0205128 4.85723e-17 9 13 1 0 0 0 9 14 1 0.0156863 0.0156863 6.245e-17 9 15 1 0 0 0 9 16 1 0.0123839 0.0123839 5.0307e-17 9 17 1 8.67362e-19 0 8.67362e-19 9 18 1 0.0100218 0.0100251 3.21815e-06 9 19 1 0 0 0 9 20 1 0.00826778 0.00828157 1.37931e-05 10 0 1 1.33333 1.33333 4.44089e-16 10 1 1 -3.40006e-16 0 3.40006e-16 10 2 1 0.266667 0.266667 2.22045e-16 10 3 1 -2.56739e-16 0 2.56739e-16 10 4 1 0.114286 0.114286 1.52656e-16 10 5 1 -1.94289e-16 0 1.94289e-16 10 6 1 0.0634921 0.0634921 1.11022e-16 10 7 1 -1.66533e-16 0 1.66533e-16 10 8 1 0.040404 0.040404 8.32667e-17 10 9 1 -1.48319e-16 0 1.48319e-16 10 10 1 0.027972 0.027972 1.83881e-16 10 11 1 -1.30104e-16 0 1.30104e-16 10 12 1 0.0205128 0.0205128 3.46945e-17 10 13 1 -1.17961e-16 0 1.17961e-16 10 14 1 0.0156863 0.0156863 5.20417e-17 10 15 1 -1.04083e-16 0 1.04083e-16 10 16 1 0.0123839 0.0123839 5.72459e-17 10 17 1 -9.28077e-17 0 9.28077e-17 10 18 1 0.0100251 0.0100251 1.73472e-18 10 19 1 -8.41341e-17 0 8.41341e-17 10 20 1 0.00828077 0.00828157 7.9954e-07 10 21 1 -7.54605e-17 0 7.54605e-17 10 22 1 0.0069527 0.00695652 3.82409e-06 1 0 2.5 0.981748 0.981748 1.11022e-16 1 1 2.5 0 0 0 1 2 2.5 0 0.122718 0.122718 1 3 2.5 0 0 0 1 4 2.5 0 0.0368155 0.0368155 2 0 2.5 0.981748 0.981748 1.11022e-16 2 1 2.5 0 0 0 2 2 2.5 0.122718 0.122718 1.249e-16 2 3 2.5 0 0 0 2 4 2.5 0.0153398 0.0368155 0.0214757 2 5 2.5 0 0 0 2 6 2.5 0.00191748 0.0153398 0.0134223 3 0 2.5 0.981748 0.981748 4.44089e-16 3 1 2.5 0 0 0 3 2 2.5 0.122718 0.122718 1.38778e-17 3 3 2.5 0 0 0 3 4 2.5 0.0368155 0.0368155 2.77556e-17 3 5 2.5 0 0 0 3 6 2.5 0.0110447 0.0153398 0.00429515 3 7 2.5 0 0 0 3 8 2.5 0.0033134 0.0076699 0.00435651 4 0 2.5 0.981748 0.981748 0 4 1 2.5 1.45717e-16 0 1.45717e-16 4 2 2.5 0.122718 0.122718 6.93889e-17 4 3 2.5 6.59195e-17 0 6.59195e-17 4 4 2.5 0.0368155 0.0368155 2.77556e-17 4 5 2.5 2.94903e-17 0 2.94903e-17 4 6 2.5 0.0153398 0.0153398 3.46945e-18 4 7 2.5 1.30104e-17 0 1.30104e-17 4 8 2.5 0.00674952 0.0076699 0.000920388 4 9 2.5 5.63785e-18 0 5.63785e-18 4 10 2.5 0.00299126 0.00431432 0.00132306 5 0 2.5 0.981748 0.981748 3.33067e-16 5 1 2.5 -3.46945e-17 0 3.46945e-17 5 2 2.5 0.122718 0.122718 1.52656e-16 5 3 2.5 -6.93889e-18 0 6.93889e-18 5 4 2.5 0.0368155 0.0368155 6.245e-17 5 5 2.5 -1.73472e-18 0 1.73472e-18 5 6 2.5 0.0153398 0.0153398 1.56125e-17 5 7 2.5 0 0 0 5 8 2.5 0.0076699 0.0076699 1.04083e-17 5 9 2.5 0 0 0 5 10 2.5 0.00410888 0.00431432 0.000205444 5 11 2.5 0 0 0 5 12 2.5 0.0022501 0.00263653 0.00038643 6 0 2.5 0.981748 0.981748 2.22045e-16 6 1 2.5 1.83881e-16 0 1.83881e-16 6 2 2.5 0.122718 0.122718 6.93889e-17 6 3 2.5 4.16334e-17 0 4.16334e-17 6 4 2.5 0.0368155 0.0368155 3.46945e-17 6 5 2.5 1.30104e-17 0 1.30104e-17 6 6 2.5 0.0153398 0.0153398 0 6 7 2.5 3.90313e-18 0 3.90313e-18 6 8 2.5 0.0076699 0.0076699 1.99493e-17 6 9 2.5 8.67362e-19 0 8.67362e-19 6 10 2.5 0.00431432 0.00431432 1.99493e-17 6 11 2.5 2.1684e-19 0 2.1684e-19 6 12 2.5 0.00258945 0.00263653 4.70809e-05 6 13 2.5 0 0 0 6 14 2.5 0.00160369 0.00171374 0.000110052 7 0 2.5 0.981748 0.981748 1.11022e-16 7 1 2.5 1.78677e-16 0 1.78677e-16 7 2 2.5 0.122718 0.122718 2.08167e-16 7 3 2.5 1.04083e-16 0 1.04083e-16 7 4 2.5 0.0368155 0.0368155 1.31839e-16 7 5 2.5 7.02563e-17 0 7.02563e-17 7 6 2.5 0.0153398 0.0153398 6.245e-17 7 7 2.5 4.85723e-17 0 4.85723e-17 7 8 2.5 0.0076699 0.0076699 2.60209e-17 7 9 2.5 3.40439e-17 0 3.40439e-17 7 10 2.5 0.00431432 0.00431432 1.21431e-17 7 11 2.5 2.38524e-17 0 2.38524e-17 7 12 2.5 0.00263653 0.00263653 1.69136e-17 7 13 2.5 1.64799e-17 0 1.64799e-17 7 14 2.5 0.00170276 0.00171374 1.09855e-05 7 15 2.5 1.1601e-17 0 1.1601e-17 7 16 2.5 0.00113767 0.00116846 3.07928e-05 8 0 2.5 0.981748 0.981748 1.11022e-16 8 1 2.5 2.60209e-17 0 2.60209e-17 8 2 2.5 0.122718 0.122718 9.71445e-17 8 3 2.5 4.33681e-19 0 4.33681e-19 8 4 2.5 0.0368155 0.0368155 3.46945e-17 8 5 2.5 -4.33681e-19 0 4.33681e-19 8 6 2.5 0.0153398 0.0153398 0 8 7 2.5 4.33681e-19 0 4.33681e-19 8 8 2.5 0.0076699 0.0076699 2.08167e-17 8 9 2.5 2.1684e-19 0 2.1684e-19 8 10 2.5 0.00431432 0.00431432 2.1684e-17 8 11 2.5 0 0 0 8 12 2.5 0.00263653 0.00263653 6.07153e-18 8 13 2.5 0 0 0 8 14 2.5 0.00171374 0.00171374 2.1684e-19 8 15 2.5 0 0 0 8 16 2.5 0.00116587 0.00116846 2.59658e-06 8 17 2.5 0 0 0 8 18 2.5 0.000819157 0.000827661 8.50381e-06 9 0 2.5 0.981748 0.981748 1.11022e-16 9 1 2.5 1.64799e-17 0 1.64799e-17 9 2 2.5 0.122718 0.122718 2.77556e-17 9 3 2.5 1.73472e-18 0 1.73472e-18 9 4 2.5 0.0368155 0.0368155 0 9 5 2.5 -8.67362e-19 0 8.67362e-19 9 6 2.5 0.0153398 0.0153398 2.60209e-17 9 7 2.5 -2.1684e-19 0 2.1684e-19 9 8 2.5 0.0076699 0.0076699 3.55618e-17 9 9 2.5 0 0 0 9 10 2.5 0.00431432 0.00431432 2.94903e-17 9 11 2.5 -1.0842e-19 0 1.0842e-19 9 12 2.5 0.00263653 0.00263653 1.12757e-17 9 13 2.5 0 0 0 9 14 2.5 0.00171374 0.00171374 3.68629e-18 9 15 2.5 -5.42101e-20 0 5.42101e-20 9 16 2.5 0.00116846 0.00116846 6.50521e-19 9 17 2.5 0 0 0 9 18 2.5 0.000827041 0.000827661 6.19639e-07 9 19 2.5 0 0 0 9 20 2.5 0.000602504 0.000604829 2.32473e-06 10 0 2.5 0.981748 0.981748 4.44089e-16 10 1 2.5 -1.32706e-16 0 1.32706e-16 10 2 2.5 0.122718 0.122718 1.66533e-16 10 3 2.5 -3.49113e-17 0 3.49113e-17 10 4 2.5 0.0368155 0.0368155 6.93889e-17 10 5 2.5 -1.12757e-17 0 1.12757e-17 10 6 2.5 0.0153398 0.0153398 1.21431e-17 10 7 2.5 -4.01155e-18 0 4.01155e-18 10 8 2.5 0.0076699 0.0076699 1.47451e-17 10 9 2.5 -1.40946e-18 0 1.40946e-18 10 10 2.5 0.00431432 0.00431432 1.82146e-17 10 11 2.5 -3.25261e-19 0 3.25261e-19 10 12 2.5 0.00263653 0.00263653 6.50521e-18 10 13 2.5 -3.25261e-19 0 3.25261e-19 10 14 2.5 0.00171374 0.00171374 0 10 15 2.5 -1.0842e-19 0 1.0842e-19 10 16 2.5 0.00116846 0.00116846 2.81893e-18 10 17 2.5 0 0 0 10 18 2.5 0.000827661 0.000827661 4.22839e-18 10 19 2.5 2.71051e-20 0 2.71051e-20 10 20 2.5 0.00060468 0.000604829 1.48952e-07 10 21 2.5 2.71051e-20 0 2.71051e-20 10 22 2.5 0.000452991 0.000453622 6.30385e-07 TEST10 GEGENBAUER_COMPUTE_NP computes a generalized Gauss-Gegenbauer rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) (1-x^2)^alpha dx. Order = 1 ALPHA = 0.5 0 0 1.570796326794897 Order = 2 ALPHA = 0.5 0 -0.5 0.7853981633974484 1 0.5 0.7853981633974484 Order = 3 ALPHA = 0.5 0 -0.7071067811865475 0.3926990816987239 1 0 0.7853981633974484 2 0.7071067811865476 0.3926990816987245 Order = 4 ALPHA = 0.5 0 -0.8090169943749475 0.217078713422706 1 -0.3090169943749475 0.5683194499747424 2 0.3090169943749474 0.5683194499747424 3 0.8090169943749475 0.217078713422706 Order = 5 ALPHA = 0.5 0 -0.8660254037844387 0.130899693899574 1 -0.5 0.3926990816987242 2 0 0.5235987755982989 3 0.5 0.3926990816987242 4 0.8660254037844387 0.1308996938995745 Order = 6 ALPHA = 0.5 0 -0.9009688679024191 0.08448869089158841 1 -0.6234898018587335 0.2743330560697777 2 -0.2225209339563144 0.4265764164360819 3 0.2225209339563144 0.4265764164360819 4 0.6234898018587335 0.2743330560697777 5 0.9009688679024191 0.08448869089158841 Order = 7 ALPHA = 0.5 0 -0.9238795325112867 0.05750944903191331 1 -0.7071067811865475 0.1963495408493619 2 -0.3826834323650898 0.3351896326668111 3 0 0.3926990816987242 4 0.3826834323650898 0.3351896326668108 5 0.7071067811865476 0.1963495408493624 6 0.9238795325112867 0.05750944903191331 Order = 8 ALPHA = 0.5 0 -0.9396926207859084 0.04083294770910693 1 -0.766044443118978 0.1442256007956728 2 -0.5 0.2617993877991495 3 -0.1736481776669303 0.3385402270935191 4 0.1736481776669303 0.3385402270935191 5 0.5 0.2617993877991495 6 0.766044443118978 0.1442256007956728 7 0.9396926207859084 0.04083294770910754 Order = 9 ALPHA = 0.5 0 -0.9510565162951536 0.02999954037160841 1 -0.8090169943749475 0.108539356711353 2 -0.5877852522924731 0.2056199086476264 3 -0.3090169943749475 0.2841597249873712 4 0 0.3141592653589794 5 0.3090169943749475 0.2841597249873712 6 0.5877852522924731 0.2056199086476264 7 0.8090169943749475 0.108539356711353 8 0.9510565162951536 0.02999954037160841 Order = 10 ALPHA = 0.5 0 -0.9594929736144974 0.02266894250185901 1 -0.8412535328311812 0.08347854093418892 2 -0.6548607339452851 0.1631221774548165 3 -0.4154150130018864 0.2363135602034873 4 -0.1423148382732851 0.2798149423030965 5 0.1423148382732851 0.2798149423030966 6 0.4154150130018864 0.2363135602034873 7 0.6548607339452851 0.1631221774548165 8 0.8412535328311812 0.08347854093418892 9 0.9594929736144974 0.02266894250185901 Order = 1 ALPHA = 1 0 0 1.333333333333333 Order = 2 ALPHA = 1 0 -0.4472135954999579 0.6666666666666665 1 0.447213595499958 0.6666666666666667 Order = 3 ALPHA = 1 0 -0.6546536707079772 0.3111111111111113 1 0 0.711111111111111 2 0.6546536707079772 0.3111111111111113 Order = 4 ALPHA = 1 0 -0.7650553239294646 0.1569499125956942 1 -0.2852315164806451 0.5097167540709727 2 0.2852315164806451 0.5097167540709727 3 0.7650553239294647 0.1569499125956939 Order = 5 ALPHA = 1 0 -0.8302238962785669 0.08601768212280729 1 -0.4688487934707142 0.3368394607343354 2 -2.407412430484045e-35 0.4876190476190476 3 0.4688487934707142 0.3368394607343354 4 0.830223896278567 0.08601768212280764 Order = 6 ALPHA = 1 0 -0.8717401485096066 0.05058357701608194 1 -0.5917001814331423 0.2216925320225178 2 -0.2092992179024789 0.3943905576280671 3 0.2092992179024789 0.3943905576280671 4 0.5917001814331423 0.2216925320225178 5 0.8717401485096067 0.05058357701608171 Order = 7 ALPHA = 1 0 -0.8997579954114602 0.03151620015049876 1 -0.6771862795107377 0.1486404045834101 2 -0.3631174638261782 0.3007504247445493 3 0 0.3715192743764172 4 0.3631174638261782 0.3007504247445493 5 0.6771862795107377 0.1486404045834101 6 0.8997579954114602 0.03151620015049876 Order = 8 ALPHA = 1 0 -0.9195339081664589 0.02059009564912187 1 -0.738773865105505 0.1021477023603581 2 -0.4779249498104445 0.2253365549698578 3 -0.165278957666387 0.3185923136873282 4 0.165278957666387 0.3185923136873282 5 0.4779249498104445 0.225336554969858 6 0.738773865105505 0.1021477023603581 7 0.9195339081664589 0.02059009564912187 Order = 9 ALPHA = 1 0 -0.9340014304080592 0.01399105612442671 1 -0.7844834736631444 0.07198285617566075 2 -0.565235326996205 0.1687989736144601 3 -0.2957581355869394 0.2617849830242735 4 -3.673419846319648e-40 0.3002175954556907 5 0.2957581355869394 0.2617849830242735 6 0.565235326996205 0.1687989736144601 7 0.7844834736631444 0.07198285617566075 8 0.9340014304080592 0.01399105612442671 Order = 10 ALPHA = 1 0 -0.9448992722228823 0.009825404805768293 1 -0.8192793216440066 0.05193914377420781 2 -0.6328761530318606 0.1273919483477328 3 -0.399530940965349 0.2111657418760753 4 -0.1365529328549276 0.2663444278628824 5 0.1365529328549276 0.2663444278628824 6 0.399530940965349 0.2111657418760753 7 0.6328761530318606 0.1273919483477326 8 0.8192793216440066 0.05193914377420781 9 0.9448992722228822 0.009825404805768065 Order = 1 ALPHA = 2.5 0 0 0.9817477042468103 Order = 2 ALPHA = 2.5 0 -0.3535533905932737 0.4908738521234051 1 0.3535533905932737 0.4908738521234051 Order = 3 ALPHA = 2.5 0 -0.5477225575051661 0.2045307717180856 1 7.703719777548943e-34 0.5726861608106394 2 0.5477225575051661 0.2045307717180856 Order = 4 ALPHA = 2.5 0 -0.6660699417556469 0.08700807153148325 1 -0.2373833033084449 0.4038657805919217 2 0.2373833033084449 0.4038657805919217 3 0.6660699417556469 0.08700807153148347 Order = 5 ALPHA = 2.5 0 -0.7434770045212192 0.03928938913176718 1 -0.4019050360892101 0.2454174450998077 2 -1.540743955509789e-33 0.4123340357836603 3 0.4019050360892101 0.2454174450998076 4 0.7434770045212192 0.03928938913176718 Order = 6 ALPHA = 2.5 0 -0.7968373566406807 0.01891164336669311 1 -0.5196148456949133 0.1431407318203965 2 -0.1804179569647766 0.3288214769363154 3 0.1804179569647766 0.3288214769363154 4 0.5196148456949133 0.1431407318203969 5 0.7968373566406807 0.01891164336669311 Order = 7 ALPHA = 2.5 0 -0.8351564541339112 0.009658525021000812 1 -0.6063141605450342 0.08344467054898472 2 -0.3186902924591673 0.2357822853526957 3 -1.469367938527859e-39 0.3239767424014473 4 0.3186902924591672 0.2357822853526958 5 0.6063141605450342 0.08344467054898472 6 0.8351564541339112 0.009658525021000983 Order = 8 ALPHA = 2.5 0 -0.8635912403827584 0.005199707422132487 1 -0.6718232507247468 0.04941818815652229 2 -0.4260961495798479 0.1615070338519726 3 -0.1459649294626305 0.2747489226927776 4 0.1459649294626305 0.2747489226927777 5 0.4260961495798479 0.1615070338519726 6 0.6718232507247468 0.04941818815652229 7 0.8635912403827584 0.005199707422132487 Order = 9 ALPHA = 2.5 0 -0.8852662998295736 0.002931900984992394 1 -0.7224309555097356 0.02992095719368033 2 -0.5107538326640052 0.1088013928543096 3 -0.2643695362367075 0.215514913750143 4 0 0.2674093746805597 5 0.2643695362367075 0.215514913750143 6 0.5107538326640052 0.1088013928543096 7 0.7224309555097356 0.02992095719368033 8 0.8852662998295736 0.002931900984992394 Order = 10 ALPHA = 2.5 0 -0.9021636877146698 0.00172163822889316 1 -0.7622868838395971 0.01855643700599032 2 -0.5784393987065429 0.07316132839714104 3 -0.3610608688185057 0.1622660968500981 4 -0.1227285554678421 0.2351683516412825 5 0.1227285554678421 0.2351683516412824 6 0.3610608688185057 0.1622660968500981 7 0.5784393987065429 0.07316132839714087 8 0.7622868838395971 0.01855643700599032 9 0.9021636877146698 0.00172163822889316 TEST11 GEN_HERMITE_COMPUTE_NP computes a generalized Gauss-Hermite rule which is appropriate for integrands of the form Integral ( -oo < x < +oo ) f(x) x^alpha exp(-x*x) dx. GEN_HERMITE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Alpha Estimate Exact Error 1 0 0.5 1.22542 1.22542 0 1 1 0.5 0 0 0 1 2 0.5 0 0.919063 0.919063 1 3 0.5 0 0 0 1 4 0.5 0 1.60836 1.60836 2 0 0.5 1.22542 1.22542 0 2 1 0.5 0 0 0 2 2 0.5 0.919063 0.919063 4.44089e-16 2 3 0.5 0 0 0 2 4 0.5 0.689297 1.60836 0.919063 2 5 0.5 0 0 0 2 6 0.5 0.516973 4.42299 3.90602 3 0 0.5 1.22542 1.22542 0 3 1 0.5 -3.33067e-16 0 3.33067e-16 3 2 0.5 0.919063 0.919063 6.66134e-16 3 3 0.5 -9.99201e-16 0 9.99201e-16 3 4 0.5 1.60836 1.60836 1.9984e-15 3 5 0.5 -1.9984e-15 0 1.9984e-15 3 6 0.5 2.81463 4.42299 1.60836 3 7 0.5 -4.44089e-15 0 4.44089e-15 3 8 0.5 4.9256 16.5862 11.6606 4 0 0.5 1.22542 1.22542 0 4 1 0.5 2.498e-16 0 2.498e-16 4 2 0.5 0.919063 0.919063 5.55112e-16 4 3 0.5 2.22045e-16 0 2.22045e-16 4 4 0.5 1.60836 1.60836 1.33227e-15 4 5 0.5 6.66134e-16 0 6.66134e-16 4 6 0.5 4.42299 4.42299 3.55271e-15 4 7 0.5 1.33227e-15 0 1.33227e-15 4 8 0.5 13.3695 16.5862 3.21672 4 9 0.5 3.55271e-15 0 3.55271e-15 4 10 0.5 40.988 78.7845 37.7964 5 0 0.5 1.22542 1.22542 1.11022e-15 5 1 0.5 -5.48173e-16 0 5.48173e-16 5 2 0.5 0.919063 0.919063 4.44089e-16 5 3 0.5 -6.66134e-16 0 6.66134e-16 5 4 0.5 1.60836 1.60836 2.22045e-16 5 5 0.5 -4.10783e-15 0 4.10783e-15 5 6 0.5 4.42299 4.42299 8.88178e-16 5 7 0.5 -2.53131e-14 0 2.53131e-14 5 8 0.5 16.5862 16.5862 3.55271e-15 5 9 0.5 -1.38556e-13 0 1.38556e-13 5 10 0.5 69.9385 78.7845 8.84598 5 11 0.5 -6.96332e-13 0 6.96332e-13 5 12 0.5 304.841 453.011 148.17 6 0 0.5 1.22542 1.22542 4.44089e-16 6 1 0.5 3.98986e-16 0 3.98986e-16 6 2 0.5 0.919063 0.919063 9.99201e-16 6 3 0.5 6.52256e-16 0 6.52256e-16 6 4 0.5 1.60836 1.60836 1.9984e-15 6 5 0.5 1.38778e-15 0 1.38778e-15 6 6 0.5 4.42299 4.42299 8.88178e-15 6 7 0.5 1.77636e-15 0 1.77636e-15 6 8 0.5 16.5862 16.5862 4.9738e-14 6 9 0.5 8.88178e-15 0 8.88178e-15 6 10 0.5 78.7845 78.7845 2.70006e-13 6 11 0.5 5.68434e-14 0 5.68434e-14 6 12 0.5 426.473 453.011 26.5379 6 13 0.5 8.52651e-13 0 8.52651e-13 6 14 0.5 2440.82 3057.82 617.007 7 0 0.5 1.22542 1.22542 2.22045e-16 7 1 0.5 5.39499e-16 0 5.39499e-16 7 2 0.5 0.919063 0.919063 6.66134e-16 7 3 0.5 1.04777e-15 0 1.04777e-15 7 4 0.5 1.60836 1.60836 1.11022e-15 7 5 0.5 2.91434e-15 0 2.91434e-15 7 6 0.5 4.42299 4.42299 1.77636e-15 7 7 0.5 8.88178e-15 0 8.88178e-15 7 8 0.5 16.5862 16.5862 3.55271e-15 7 9 0.5 3.01981e-14 0 3.01981e-14 7 10 0.5 78.7845 78.7845 1.27898e-13 7 11 0.5 1.13687e-13 0 1.13687e-13 7 12 0.5 453.011 453.011 1.36424e-12 7 13 0.5 2.84217e-13 0 2.84217e-13 7 14 0.5 2958.31 3057.82 99.5172 7 15 0.5 0 0 0 7 16 0.5 20687.7 23698.1 3010.4 8 0 0.5 1.22542 1.22542 1.55431e-15 8 1 0.5 6.95841e-16 0 6.95841e-16 8 2 0.5 0.919063 0.919063 2.22045e-16 8 3 0.5 2.02789e-15 0 2.02789e-15 8 4 0.5 1.60836 1.60836 2.22045e-16 8 5 0.5 6.16868e-15 0 6.16868e-15 8 6 0.5 4.42299 4.42299 4.44089e-15 8 7 0.5 2.53686e-14 0 2.53686e-14 8 8 0.5 16.5862 16.5862 3.55271e-14 8 9 0.5 1.06581e-13 0 1.06581e-13 8 10 0.5 78.7845 78.7845 2.70006e-13 8 11 0.5 5.47118e-13 0 5.47118e-13 8 12 0.5 453.011 453.011 2.10321e-12 8 13 0.5 2.44427e-12 0 2.44427e-12 8 14 0.5 3057.82 3057.82 1.68257e-11 8 15 0.5 1.27329e-11 0 1.27329e-11 8 16 0.5 23300.1 23698.1 398.069 8 17 0.5 5.09317e-11 0 5.09317e-11 8 18 0.5 191933 207359 15425.2 9 0 0.5 1.22542 1.22542 2.22045e-16 9 1 0.5 -1.59703e-16 0 1.59703e-16 9 2 0.5 0.919063 0.919063 8.88178e-16 9 3 0.5 -7.80626e-16 0 7.80626e-16 9 4 0.5 1.60836 1.60836 2.22045e-15 9 5 0.5 -3.93088e-15 0 3.93088e-15 9 6 0.5 4.42299 4.42299 9.76996e-15 9 7 0.5 -1.94844e-14 0 1.94844e-14 9 8 0.5 16.5862 16.5862 4.9738e-14 9 9 0.5 -1.10134e-13 0 1.10134e-13 9 10 0.5 78.7845 78.7845 3.12639e-13 9 11 0.5 -6.82121e-13 0 6.82121e-13 9 12 0.5 453.011 453.011 2.04636e-12 9 13 0.5 -4.718e-12 0 4.718e-12 9 14 0.5 3057.82 3057.82 1.40972e-11 9 15 0.5 -3.50155e-11 0 3.50155e-11 9 16 0.5 23698.1 23698.1 1.09139e-10 9 17 0.5 -2.98314e-10 0 2.98314e-10 9 18 0.5 205468 207359 1890.83 9 19 0.5 -2.85218e-09 0 2.85218e-09 9 20 0.5 1.93146e+06 2.02175e+06 90287 10 0 0.5 1.22542 1.22542 1.11022e-15 10 1 0.5 7.70027e-16 0 7.70027e-16 10 2 0.5 0.919063 0.919063 9.99201e-16 10 3 0.5 9.876e-16 0 9.876e-16 10 4 0.5 1.60836 1.60836 2.44249e-15 10 5 0.5 2.52402e-15 0 2.52402e-15 10 6 0.5 4.42299 4.42299 9.76996e-15 10 7 0.5 8.25034e-15 0 8.25034e-15 10 8 0.5 16.5862 16.5862 4.61853e-14 10 9 0.5 3.4639e-14 0 3.4639e-14 10 10 0.5 78.7845 78.7845 2.55795e-13 10 11 0.5 1.3145e-13 0 1.3145e-13 10 12 0.5 453.011 453.011 1.47793e-12 10 13 0.5 7.95808e-13 0 7.95808e-13 10 14 0.5 3057.82 3057.82 9.09495e-12 10 15 0.5 7.27596e-12 0 7.27596e-12 10 16 0.5 23698.1 23698.1 6.18456e-11 10 17 0.5 9.45874e-11 0 9.45874e-11 10 18 0.5 207359 207359 4.07454e-10 10 19 0.5 1.36788e-09 0 1.36788e-09 10 20 0.5 2.01229e+06 2.02175e+06 9454.14 10 21 0.5 1.81608e-08 0 1.81608e-08 10 22 0.5 2.11831e+07 2.17338e+07 550704 1 0 1 1 1 0 1 1 1 0 0 0 1 2 1 0 1 1 1 3 1 0 0 0 1 4 1 0 2 2 2 0 1 1 1 2.22045e-16 2 1 1 0 0 0 2 2 1 1 1 6.66134e-16 2 3 1 0 0 0 2 4 1 1 2 1 2 5 1 0 0 0 2 6 1 1 6 5 3 0 1 1 1 2.22045e-16 3 1 1 -3.88578e-16 0 3.88578e-16 3 2 1 1 1 0 3 3 1 -1.11022e-15 0 1.11022e-15 3 4 1 2 2 6.66134e-16 3 5 1 -2.44249e-15 0 2.44249e-15 3 6 1 4 6 2 3 7 1 -5.77316e-15 0 5.77316e-15 3 8 1 8 24 16 4 0 1 1 1 4.44089e-16 4 1 1 -1.94289e-16 0 1.94289e-16 4 2 1 1 1 1.11022e-16 4 3 1 -1.11022e-16 0 1.11022e-16 4 4 1 2 2 8.88178e-16 4 5 1 0 0 0 4 6 1 6 6 7.10543e-15 4 7 1 0 0 0 4 8 1 20 24 4 4 9 1 0 0 0 4 10 1 68 120 52 5 0 1 1 1 0 5 1 1 -7.35523e-16 0 7.35523e-16 5 2 1 1 1 7.77156e-16 5 3 1 -7.49401e-16 0 7.49401e-16 5 4 1 2 2 2.88658e-15 5 5 1 -4.44089e-16 0 4.44089e-16 5 6 1 6 6 1.06581e-14 5 7 1 2.66454e-15 0 2.66454e-15 5 8 1 24 24 4.9738e-14 5 9 1 2.4869e-14 0 2.4869e-14 5 10 1 108 120 12 5 11 1 1.27898e-13 0 1.27898e-13 5 12 1 504 720 216 6 0 1 1 1 4.44089e-16 6 1 1 7.33788e-16 0 7.33788e-16 6 2 1 1 1 4.44089e-16 6 3 1 1.3739e-15 0 1.3739e-15 6 4 1 2 2 8.88178e-16 6 5 1 2.9976e-15 0 2.9976e-15 6 6 1 6 6 7.10543e-15 6 7 1 3.55271e-15 0 3.55271e-15 6 8 1 24 24 4.9738e-14 6 9 1 -4.26326e-14 0 4.26326e-14 6 10 1 120 120 3.69482e-13 6 11 1 -5.40012e-13 0 5.40012e-13 6 12 1 684 720 36 6 13 1 -4.54747e-12 0 4.54747e-12 6 14 1 4140 5040 900 7 0 1 1 1 1.11022e-16 7 1 1 3.20924e-17 0 3.20924e-17 7 2 1 1 1 4.44089e-16 7 3 1 -2.32453e-16 0 2.32453e-16 7 4 1 2 2 8.88178e-16 7 5 1 -2.22045e-16 0 2.22045e-16 7 6 1 6 6 1.77636e-15 7 7 1 5.10703e-15 0 5.10703e-15 7 8 1 24 24 3.55271e-15 7 9 1 6.21725e-14 0 6.21725e-14 7 10 1 120 120 7.10543e-14 7 11 1 5.40012e-13 0 5.40012e-13 7 12 1 720 720 1.02318e-12 7 13 1 4.20641e-12 0 4.20641e-12 7 14 1 4896 5040 144 7 15 1 3.27418e-11 0 3.27418e-11 7 16 1 35712 40320 4608 8 0 1 1 1 3.33067e-16 8 1 1 -7.84962e-17 0 7.84962e-17 8 2 1 1 1 6.66134e-16 8 3 1 -1.82146e-17 0 1.82146e-17 8 4 1 2 2 1.55431e-15 8 5 1 1.23512e-15 0 1.23512e-15 8 6 1 6 6 5.32907e-15 8 7 1 1.37668e-14 0 1.37668e-14 8 8 1 24 24 1.06581e-14 8 9 1 7.81597e-14 0 7.81597e-14 8 10 1 120 120 5.68434e-14 8 11 1 3.76588e-13 0 3.76588e-13 8 12 1 720 720 4.54747e-13 8 13 1 1.25056e-12 0 1.25056e-12 8 14 1 5040 5040 5.45697e-12 8 15 1 0 0 0 8 16 1 39744 40320 576 8 17 1 -8.00355e-11 0 8.00355e-11 8 18 1 339264 362880 23616 9 0 1 1 1 8.88178e-16 9 1 1 9.23605e-16 0 9.23605e-16 9 2 1 1 1 1.33227e-15 9 3 1 2.23259e-15 0 2.23259e-15 9 4 1 2 2 1.33227e-15 9 5 1 8.3579e-15 0 8.3579e-15 9 6 1 6 6 8.88178e-16 9 7 1 3.9746e-14 0 3.9746e-14 9 8 1 24 24 1.77636e-14 9 9 1 2.05613e-13 0 2.05613e-13 9 10 1 120 120 1.27898e-13 9 11 1 1.09068e-12 0 1.09068e-12 9 12 1 720 720 5.68434e-13 9 13 1 4.94538e-12 0 4.94538e-12 9 14 1 5040 5040 0 9 15 1 1.00044e-11 0 1.00044e-11 9 16 1 40320 40320 5.09317e-11 9 17 1 -1.67347e-10 0 1.67347e-10 9 18 1 360000 362880 2880 9 19 1 -3.49246e-09 0 3.49246e-09 9 20 1 3.4848e+06 3.6288e+06 144000 10 0 1 1 1 3.33067e-16 10 1 1 6.07905e-16 0 6.07905e-16 10 2 1 1 1 4.44089e-16 10 3 1 9.83588e-16 0 9.83588e-16 10 4 1 2 2 3.10862e-15 10 5 1 2.25167e-15 0 2.25167e-15 10 6 1 6 6 1.77636e-14 10 7 1 5.67602e-15 0 5.67602e-15 10 8 1 24 24 1.10134e-13 10 9 1 1.35447e-14 0 1.35447e-14 10 10 1 120 120 6.96332e-13 10 11 1 8.17124e-14 0 8.17124e-14 10 12 1 720 720 4.88853e-12 10 13 1 3.12639e-13 0 3.12639e-13 10 14 1 5040 5040 3.72893e-11 10 15 1 -1.81899e-12 0 1.81899e-12 10 16 1 40320 40320 2.98314e-10 10 17 1 -7.27596e-11 0 7.27596e-11 10 18 1 362880 362880 2.61934e-09 10 19 1 -1.62981e-09 0 1.62981e-09 10 20 1 3.6144e+06 3.6288e+06 14400 10 21 1 -2.8871e-08 0 2.8871e-08 10 22 1 3.90384e+07 3.99168e+07 878400 1 0 2.5 0.919063 0.919063 0 1 1 2.5 0 0 0 1 2 2.5 0 1.60836 1.60836 1 3 2.5 0 0 0 1 4 2.5 0 4.42299 4.42299 2 0 2.5 0.919063 0.919063 2.22045e-16 2 1 2.5 0 0 0 2 2 2.5 1.60836 1.60836 8.88178e-16 2 3 2.5 0 0 0 2 4 2.5 2.81463 4.42299 1.60836 2 5 2.5 0 0 0 2 6 2.5 4.9256 16.5862 11.6606 3 0 2.5 0.919063 0.919063 0 3 1 2.5 0 0 0 3 2 2.5 1.60836 1.60836 2.22045e-16 3 3 2.5 0 0 0 3 4 2.5 4.42299 4.42299 1.77636e-15 3 5 2.5 0 0 0 3 6 2.5 12.1632 16.5862 4.42299 3 7 2.5 0 0 0 3 8 2.5 33.4488 78.7845 45.3356 4 0 2.5 0.919063 0.919063 3.33067e-16 4 1 2.5 -7.77156e-16 0 7.77156e-16 4 2 2.5 1.60836 1.60836 6.66134e-16 4 3 2.5 -1.44329e-15 0 1.44329e-15 4 4 2.5 4.42299 4.42299 8.88178e-16 4 5 2.5 -3.10862e-15 0 3.10862e-15 4 6 2.5 16.5862 16.5862 7.10543e-15 4 7 2.5 -1.06581e-14 0 1.06581e-14 4 8 2.5 69.9385 78.7845 8.84598 4 9 2.5 -1.42109e-14 0 1.42109e-14 4 10 2.5 304.841 453.011 148.17 5 0 2.5 0.919063 0.919063 3.33067e-16 5 1 2.5 -5.55112e-17 0 5.55112e-17 5 2 2.5 1.60836 1.60836 2.22045e-16 5 3 2.5 -2.22045e-16 0 2.22045e-16 5 4 2.5 4.42299 4.42299 1.77636e-15 5 5 2.5 8.88178e-16 0 8.88178e-16 5 6 2.5 16.5862 16.5862 1.77636e-14 5 7 2.5 1.59872e-14 0 1.59872e-14 5 8 2.5 78.7845 78.7845 1.27898e-13 5 9 2.5 1.13687e-13 0 1.13687e-13 5 10 2.5 419.838 453.011 33.1724 5 11 2.5 8.52651e-13 0 8.52651e-13 5 12 2.5 2336.32 3057.82 721.5 6 0 2.5 0.919063 0.919063 2.22045e-16 6 1 2.5 8.18789e-16 0 8.18789e-16 6 2 2.5 1.60836 1.60836 1.11022e-15 6 3 2.5 2.44249e-15 0 2.44249e-15 6 4 2.5 4.42299 4.42299 8.88178e-16 6 5 2.5 1.06581e-14 0 1.06581e-14 6 6 2.5 16.5862 16.5862 7.10543e-15 6 7 2.5 5.32907e-14 0 5.32907e-14 6 8 2.5 78.7845 78.7845 8.52651e-14 6 9 2.5 3.2685e-13 0 3.2685e-13 6 10 2.5 453.011 453.011 8.52651e-13 6 11 2.5 2.27374e-12 0 2.27374e-12 6 12 2.5 2958.31 3057.82 99.5172 6 13 2.5 1.59162e-11 0 1.59162e-11 6 14 2.5 20687.7 23698.1 3010.4 7 0 2.5 0.919063 0.919063 4.44089e-16 7 1 2.5 -5.44703e-16 0 5.44703e-16 7 2 2.5 1.60836 1.60836 1.55431e-15 7 3 2.5 -5.13478e-16 0 5.13478e-16 7 4 2.5 4.42299 4.42299 1.77636e-15 7 5 2.5 3.77476e-15 0 3.77476e-15 7 6 2.5 16.5862 16.5862 1.42109e-14 7 7 2.5 3.4639e-14 0 3.4639e-14 7 8 2.5 78.7845 78.7845 1.56319e-13 7 9 2.5 1.27898e-13 0 1.27898e-13 7 10 2.5 453.011 453.011 1.36424e-12 7 11 2.5 -1.13687e-13 0 1.13687e-13 7 12 2.5 3057.82 3057.82 1.18234e-11 7 13 2.5 -8.18545e-12 0 8.18545e-12 7 14 2.5 23225.4 23698.1 472.707 7 15 2.5 -1.09139e-10 0 1.09139e-10 7 16 2.5 189750 207359 17608.3 8 0 2.5 0.919063 0.919063 1.11022e-16 8 1 2.5 5.35596e-17 0 5.35596e-17 8 2 2.5 1.60836 1.60836 8.88178e-16 8 3 2.5 -9.12465e-16 0 9.12465e-16 8 4 2.5 4.42299 4.42299 8.88178e-16 8 5 2.5 -8.04912e-15 0 8.04912e-15 8 6 2.5 16.5862 16.5862 3.55271e-15 8 7 2.5 -5.39568e-14 0 5.39568e-14 8 8 2.5 78.7845 78.7845 2.84217e-14 8 9 2.5 -3.97904e-13 0 3.97904e-13 8 10 2.5 453.011 453.011 4.54747e-13 8 11 2.5 -3.15481e-12 0 3.15481e-12 8 12 2.5 3057.82 3057.82 4.54747e-12 8 13 2.5 -2.95586e-11 0 2.95586e-11 8 14 2.5 23698.1 23698.1 4.36557e-11 8 15 2.5 -2.76486e-10 0 2.76486e-10 8 16 2.5 205468 207359 1890.83 8 17 2.5 -2.82307e-09 0 2.82307e-09 8 18 2.5 1.93146e+06 2.02175e+06 90287 9 0 2.5 0.919063 0.919063 1.11022e-16 9 1 2.5 8.48713e-16 0 8.48713e-16 9 2 2.5 1.60836 1.60836 1.11022e-15 9 3 2.5 1.99927e-15 0 1.99927e-15 9 4 2.5 4.42299 4.42299 6.21725e-15 9 5 2.5 5.80092e-15 0 5.80092e-15 9 6 2.5 16.5862 16.5862 3.19744e-14 9 7 2.5 2.9976e-14 0 2.9976e-14 9 8 2.5 78.7845 78.7845 1.84741e-13 9 9 2.5 1.91847e-13 0 1.91847e-13 9 10 2.5 453.011 453.011 1.13687e-12 9 11 2.5 1.50635e-12 0 1.50635e-12 9 12 2.5 3057.82 3057.82 5.91172e-12 9 13 2.5 1.25056e-11 0 1.25056e-11 9 14 2.5 23698.1 23698.1 2.18279e-11 9 15 2.5 1.09139e-10 0 1.09139e-10 9 16 2.5 207359 207359 1.45519e-10 9 17 2.5 1.22236e-09 0 1.22236e-09 9 18 2.5 2.01087e+06 2.02175e+06 10872.3 9 19 2.5 1.44355e-08 0 1.44355e-08 9 20 2.5 2.11168e+07 2.17338e+07 617001 10 0 2.5 0.919063 0.919063 5.55112e-16 10 1 2.5 3.22279e-17 0 3.22279e-17 10 2 2.5 1.60836 1.60836 1.11022e-15 10 3 2.5 -1.60896e-16 0 1.60896e-16 10 4 2.5 4.42299 4.42299 1.77636e-15 10 5 2.5 -1.06165e-15 0 1.06165e-15 10 6 2.5 16.5862 16.5862 1.06581e-14 10 7 2.5 4.21885e-15 0 4.21885e-15 10 8 2.5 78.7845 78.7845 1.42109e-13 10 9 2.5 1.63869e-13 0 1.63869e-13 10 10 2.5 453.011 453.011 1.3074e-12 10 11 2.5 2.01084e-12 0 2.01084e-12 10 12 2.5 3057.82 3057.82 1.18234e-11 10 13 2.5 2.22826e-11 0 2.22826e-11 10 14 2.5 23698.1 23698.1 1.05501e-10 10 15 2.5 2.21917e-10 0 2.21917e-10 10 16 2.5 207359 207359 9.31323e-10 10 17 2.5 1.96451e-09 0 1.96451e-09 10 18 2.5 2.02175e+06 2.02175e+06 8.14907e-09 10 19 2.5 1.6531e-08 0 1.6531e-08 10 20 2.5 2.16794e+07 2.17338e+07 54361.3 10 21 2.5 1.19209e-07 0 1.19209e-07 10 22 2.5 2.51607e+08 2.55372e+08 3.76452e+06 TEST12 GEN_HERMITE_COMPUTE_NP computes a generalized Gauss-Hermite rule which is appropriate for integrands of the form Integral ( -oo < x < +oo ) f(x) x^alpha exp(-x*x) dx. Order = 1 ALPHA = 0.5 0 0 1.225416702465178 Order = 2 ALPHA = 0.5 0 -0.8660254037844385 0.6127083512325888 1 0.8660254037844385 0.6127083512325888 Order = 3 ALPHA = 0.5 0 -1.322875655532295 0.2625892933853953 1 9.974659986866641e-17 0.7002381156943873 2 1.322875655532295 0.262589293385395 Order = 4 ALPHA = 0.5 0 -1.752961966367866 0.07477218653431637 1 -0.6535475074298001 0.5379361646982722 2 0.6535475074298001 0.5379361646982725 3 1.752961966367866 0.0747721865343164 Order = 5 ALPHA = 0.5 0 -2.099598150879759 0.0206908527402406 1 -1.044838554429487 0.337385456421662 2 -2.563766590867021e-16 0.5092640841413719 3 1.044838554429487 0.3373854564216615 4 2.099598150879757 0.02069085274024055 Order = 6 ALPHA = 0.5 0 -2.431196006814871 0.004758432285876815 1 -1.428264330850235 0.1432946705182553 2 -0.5471261076464521 0.4646552484284568 3 0.5471261076464523 0.4646552484284568 4 1.428264330850235 0.1432946705182556 5 2.431196006814872 0.004758432285876798 Order = 7 ALPHA = 0.5 0 -2.719880088556291 0.00110628940196846 1 -1.747360778896522 0.05564733125066088 2 -0.8938582730216025 0.3522490969234104 3 1.730010352426829e-16 0.4074112673130981 4 0.8938582730216027 0.3522490969234107 5 1.747360778896522 0.05564733125066092 6 2.719880088556291 0.00110628940196846 Order = 8 ALPHA = 0.5 0 -2.999078968343317 0.0002288084584739132 1 -2.057439418477469 0.01787577463926715 2 -1.241738340943189 0.1866121206001916 3 -0.4801606747408061 0.4079916475346554 4 0.4801606747408064 0.4079916475346549 5 1.241738340943189 0.1866121206001921 6 2.05743941847747 0.01787577463926723 7 2.999078968343317 0.0002288084584739144 Order = 9 ALPHA = 0.5 0 -3.251152326134132 4.824428349517051e-05 1 -2.331322119300713 0.005575754103643749 2 -1.537416408684744 0.0887579748998607 3 -0.7945417010067838 0.3467847917084952 4 2.193238645380863e-16 0.3430831724741884 5 0.7945417010067843 0.3467847917084951 6 1.537416408684744 0.08875797489986054 7 2.331322119300712 0.005575754103643714 8 3.251152326134132 4.824428349517031e-05 Order = 10 ALPHA = 0.5 0 -3.496605880747678 9.347334083394694e-06 1 -2.598397149544625 0.001536356442402549 2 -1.827991812365275 0.03517634314374581 3 -1.114905370566644 0.2117439807373518 4 -0.4330259998733385 0.3642423235750056 5 0.433025999873339 0.3642423235750059 6 1.114905370566644 0.2117439807373522 7 1.827991812365276 0.03517634314374581 8 2.598397149544625 0.00153635644240255 9 3.496605880747678 9.347334083394745e-06 Order = 1 ALPHA = 1 0 0 1 Order = 2 ALPHA = 1 0 -0.9999999999999998 0.4999999999999999 1 0.9999999999999998 0.4999999999999999 Order = 3 ALPHA = 1 0 -1.414213562373095 0.2500000000000002 1 1.318389841742373e-16 0.5000000000000001 2 1.414213562373095 0.2499999999999999 Order = 4 ALPHA = 1 0 -1.847759065022574 0.07322330470336304 1 -0.7653668647301797 0.4267766952966369 2 0.7653668647301797 0.4267766952966366 3 1.847759065022574 0.07322330470336304 Order = 5 ALPHA = 1 0 -2.175327747161075 0.0223290993692602 1 -1.126032500610494 0.3110042339640734 2 -1.496023277898329e-16 0.3333333333333333 3 1.126032500610494 0.3110042339640728 4 2.175327747161075 0.02232909936926018 Order = 6 ALPHA = 1 0 -2.5079762923396 0.005194628250793071 1 -1.514688205631456 0.1392588667846203 2 -0.6448058287449634 0.3555465049645866 3 0.6448058287449638 0.3555465049645871 4 1.514688205631457 0.1392588667846204 5 2.507976292339599 0.005194628250793074 Order = 7 ALPHA = 1 0 -2.785456961280075 0.001295466838573473 1 -1.818077910688174 0.05917819272755037 2 -0.9673790505919011 0.3145263404338762 3 3.998727457032224e-17 0.2499999999999997 4 0.967379050591901 0.3145263404338764 5 1.818077910688175 0.05917819272755026 6 2.785456961280075 0.00129546683857348 Order = 8 ALPHA = 1 0 -3.065137992375079 0.0002696473527806644 1 -2.129934340988268 0.01944395425750265 2 -1.321272530993643 0.1787093462188998 3 -0.5679328213965031 0.3015770521708169 4 0.5679328213965031 0.3015770521708167 5 1.321272530993642 0.1787093462188998 6 2.129934340988268 0.01944395425750272 7 3.065137992375078 0.0002696473527806649 Order = 9 ALPHA = 1 0 -3.309666797833764 6.006309942116473e-05 1 -2.393988043347146 0.00647142481022687 2 -1.603631817982631 0.09286616703842253 3 -0.8621437977399313 0.3006023450519296 4 5.341886282576138e-16 0.2000000000000001 5 0.8621437977399317 0.3006023450519298 6 1.603631817982631 0.09286616703842276 7 2.393988043347147 0.006471424810226918 8 3.309666797833763 6.006309942116449e-05 Order = 10 ALPHA = 1 0 -3.555390392667984 1.168498619288802e-05 1 -2.661918482196411 0.001805879339961025 2 -1.896424470165033 0.03797122484085379 3 -1.188866291517476 0.1993334055415881 4 -0.5133812615572766 0.2608778052914038 5 0.5133812615572765 0.2608778052914043 6 1.188866291517476 0.1993334055415882 7 1.896424470165033 0.03797122484085385 8 2.661918482196412 0.001805879339961023 9 3.555390392667981 1.168498619288812e-05 Order = 1 ALPHA = 2.5 0 0 0.9190625268488832 Order = 2 ALPHA = 2.5 0 -1.322875655532295 0.4595312634244415 1 1.322875655532295 0.4595312634244415 Order = 3 ALPHA = 2.5 0 -1.6583123951777 0.2924289858155537 1 3.122502256758253e-17 0.3342045552177758 2 1.6583123951777 0.2924289858155537 Order = 4 ALPHA = 2.5 0 -2.099598150879758 0.09121174260159903 1 -1.044838554429487 0.368319520822843 2 1.044838554429487 0.3683195208228425 3 2.099598150879758 0.09121174260159901 Order = 5 ALPHA = 2.5 0 -2.384636591412559 0.03419534541492211 1 -1.346665632923144 0.3362147032847794 2 1.170800878753932e-16 0.1782424294494806 3 1.346665632923144 0.3362147032847794 4 2.38463659141256 0.03419534541492211 Order = 6 ALPHA = 2.5 0 -2.719880088556291 0.008184049874659632 1 -1.747360778896522 0.1699063099275072 2 -0.8938582730216031 0.2814409036222746 3 0.8938582730216029 0.2814409036222749 4 1.747360778896522 0.1699063099275075 5 2.719880088556291 0.008184049874659666 Order = 7 ALPHA = 2.5 0 -2.970220362508424 0.002409813135674279 1 -2.015021402749417 0.08718962060778231 2 -1.169392895737827 0.3136447466969385 3 7.8959598226121e-17 0.1125741659680929 4 1.169392895737826 0.3136447466969383 5 2.015021402749418 0.08718962060778225 6 2.970220362508423 0.002409813135674278 Order = 8 ALPHA = 2.5 0 -3.251152326134132 0.0005099416639456907 1 -2.331322119300714 0.03030457384414924 2 -1.537416408684745 0.2097927175810473 3 -0.7945417010067843 0.2189240303352995 4 0.794541701006785 0.2189240303352991 5 1.537416408684744 0.2097927175810477 6 2.331322119300714 0.03030457384414915 7 3.251152326134132 0.0005099416639456854 Order = 9 ALPHA = 2.5 0 -3.476534136593392 0.0001303236536236376 1 -2.571291374597262 0.0117476928703869 2 -1.789145352000116 0.1310178979856828 3 -1.049347403598514 0.2774791172736721 4 -3.404766541673904e-16 0.07831246328215158 5 1.049347403598514 0.2774791172736726 6 1.789145352000115 0.131017897985683 7 2.571291374597262 0.01174769287038695 8 3.476534136593393 0.0001303236536236379 Order = 10 ALPHA = 2.5 0 -3.722979319411836 2.563250655784923e-05 1 -2.841454909638193 0.003388190430560018 2 -2.087540034875418 0.05794395491107242 3 -1.391239401265924 0.2236074413923169 4 -0.7226261238582253 0.1745660441839348 5 0.7226261238582262 0.1745660441839347 6 1.391239401265924 0.2236074413923169 7 2.087540034875418 0.05794395491107234 8 2.841454909638194 0.00338819043056003 9 3.722979319411834 2.563250655784949e-05 TEST13 GEN_LAGUERRE_COMPUTE_NP computes a generalized Gauss-Laguerre rule which is appropriate for integrands of the form Integral ( 0 <= x < +oo ) f(x) x^alpha exp(-x) dx. GEN_LAGUERRE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Alpha Estimate Exact Error 1 0 0.5 0.886227 0.886227 0 1 1 0.5 1.32934 1.32934 0 1 2 0.5 1.99401 3.32335 1.32934 1 3 0.5 2.99102 11.6317 8.64071 1 4 0.5 4.48652 52.3428 47.8563 2 0 0.5 0.886227 0.886227 3.33067e-16 2 1 0.5 1.32934 1.32934 4.44089e-16 2 2 0.5 3.32335 3.32335 2.22045e-15 2 3 0.5 11.6317 11.6317 1.06581e-14 2 4 0.5 45.6961 52.3428 6.6467 2 5 0.5 184.861 287.885 103.024 2 6 0.5 752.947 1871.25 1118.31 3 0 0.5 0.886227 0.886227 2.22045e-16 3 1 0.5 1.32934 1.32934 0 3 2 0.5 3.32335 3.32335 8.88178e-16 3 3 0.5 11.6317 11.6317 8.88178e-15 3 4 0.5 52.3428 52.3428 4.9738e-14 3 5 0.5 287.885 287.885 2.84217e-13 3 6 0.5 1801.46 1871.25 69.7904 3 7 0.5 12045.4 14034.4 1989.03 3 8 0.5 82966.6 119292 36325.9 4 0 0.5 0.886227 0.886227 6.66134e-16 4 1 0.5 1.32934 1.32934 1.33227e-15 4 2 0.5 3.32335 3.32335 2.66454e-15 4 3 0.5 11.6317 11.6317 7.10543e-15 4 4 0.5 52.3428 52.3428 2.13163e-14 4 5 0.5 287.885 287.885 1.7053e-13 4 6 0.5 1871.25 1871.25 2.04636e-12 4 7 0.5 14034.4 14034.4 2.18279e-11 4 8 0.5 118036 119292 1256.23 4 9 0.5 1.07612e+06 1.13328e+06 57158.3 4 10 0.5 1.03156e+07 1.18994e+07 1.58379e+06 5 0 0.5 0.886227 0.886227 2.22045e-16 5 1 0.5 1.32934 1.32934 4.44089e-16 5 2 0.5 3.32335 3.32335 1.33227e-15 5 3 0.5 11.6317 11.6317 3.55271e-15 5 4 0.5 52.3428 52.3428 2.84217e-14 5 5 0.5 287.885 287.885 1.7053e-13 5 6 0.5 1871.25 1871.25 1.36424e-12 5 7 0.5 14034.4 14034.4 9.09495e-12 5 8 0.5 119292 119292 8.73115e-11 5 9 0.5 1.13328e+06 1.13328e+06 4.65661e-10 5 10 0.5 1.18649e+07 1.18994e+07 34546.2 5 11 0.5 1.34546e+08 1.36843e+08 2.29732e+06 5 12 0.5 1.62055e+09 1.71054e+09 8.99929e+07 6 0 0.5 0.886227 0.886227 3.33067e-16 6 1 0.5 1.32934 1.32934 6.66134e-16 6 2 0.5 3.32335 3.32335 2.66454e-15 6 3 0.5 11.6317 11.6317 8.88178e-15 6 4 0.5 52.3428 52.3428 3.55271e-14 6 5 0.5 287.885 287.885 1.13687e-13 6 6 0.5 1871.25 1871.25 0 6 7 0.5 14034.4 14034.4 5.45697e-12 6 8 0.5 119292 119292 8.73115e-11 6 9 0.5 1.13328e+06 1.13328e+06 1.39698e-09 6 10 0.5 1.18994e+07 1.18994e+07 1.67638e-08 6 11 0.5 1.36843e+08 1.36843e+08 5.66244e-07 6 12 0.5 1.70919e+09 1.71054e+09 1.3473e+06 6 13 0.5 2.2969e+10 2.30923e+10 1.23278e+08 6 14 0.5 3.28348e+11 3.34839e+11 6.4903e+09 7 0 0.5 0.886227 0.886227 1.11022e-16 7 1 0.5 1.32934 1.32934 8.88178e-16 7 2 0.5 3.32335 3.32335 2.66454e-15 7 3 0.5 11.6317 11.6317 1.42109e-14 7 4 0.5 52.3428 52.3428 9.9476e-14 7 5 0.5 287.885 287.885 5.68434e-13 7 6 0.5 1871.25 1871.25 3.63798e-12 7 7 0.5 14034.4 14034.4 2.18279e-11 7 8 0.5 119292 119292 1.45519e-10 7 9 0.5 1.13328e+06 1.13328e+06 1.16415e-09 7 10 0.5 1.18994e+07 1.18994e+07 7.45058e-09 7 11 0.5 1.36843e+08 1.36843e+08 3.27826e-07 7 12 0.5 1.71054e+09 1.71054e+09 1.90735e-06 7 13 0.5 2.30923e+10 2.30923e+10 0.000106812 7 14 0.5 3.34768e+11 3.34839e+11 7.07334e+07 7 15 0.5 5.18148e+12 5.19e+12 8.52338e+09 7 16 0.5 8.50543e+13 8.5635e+13 5.80686e+11 8 0 0.5 0.886227 0.886227 0 8 1 0.5 1.32934 1.32934 0 8 2 0.5 3.32335 3.32335 8.88178e-16 8 3 0.5 11.6317 11.6317 1.77636e-15 8 4 0.5 52.3428 52.3428 3.55271e-14 8 5 0.5 287.885 287.885 2.84217e-13 8 6 0.5 1871.25 1871.25 2.95586e-12 8 7 0.5 14034.4 14034.4 2.72848e-11 8 8 0.5 119292 119292 2.47383e-10 8 9 0.5 1.13328e+06 1.13328e+06 2.56114e-09 8 10 0.5 1.18994e+07 1.18994e+07 3.1665e-08 8 11 0.5 1.36843e+08 1.36843e+08 7.15256e-07 8 12 0.5 1.71054e+09 1.71054e+09 3.33786e-06 8 13 0.5 2.30923e+10 2.30923e+10 0.000175476 8 14 0.5 3.34839e+11 3.34839e+11 0.000366211 8 15 0.5 5.19e+12 5.19e+12 0.00195312 8 16 0.5 8.56302e+13 8.5635e+13 4.80987e+09 8 17 0.5 1.49787e+15 1.49861e+15 7.38315e+11 8 18 0.5 2.76611e+16 2.77243e+16 6.32101e+13 9 0 0.5 0.886227 0.886227 1.11022e-16 9 1 0.5 1.32934 1.32934 2.22045e-16 9 2 0.5 3.32335 3.32335 0 9 3 0.5 11.6317 11.6317 7.10543e-15 9 4 0.5 52.3428 52.3428 4.26326e-14 9 5 0.5 287.885 287.885 4.54747e-13 9 6 0.5 1871.25 1871.25 3.18323e-12 9 7 0.5 14034.4 14034.4 2.36469e-11 9 8 0.5 119292 119292 2.03727e-10 9 9 0.5 1.13328e+06 1.13328e+06 1.86265e-09 9 10 0.5 1.18994e+07 1.18994e+07 1.49012e-08 9 11 0.5 1.36843e+08 1.36843e+08 4.76837e-07 9 12 0.5 1.71054e+09 1.71054e+09 9.53674e-07 9 13 0.5 2.30923e+10 2.30923e+10 9.53674e-05 9 14 0.5 3.34839e+11 3.34839e+11 0.00164795 9 15 0.5 5.19e+12 5.19e+12 0.0234375 9 16 0.5 8.5635e+13 8.5635e+13 0.015625 9 17 0.5 1.49861e+15 1.49861e+15 6.25 9 18 0.5 2.77239e+16 2.77243e+16 4.11244e+11 9 19 0.5 5.40546e+17 5.40624e+17 7.8342e+13 9 20 0.5 1.10746e+19 1.10828e+19 8.23393e+15 10 0 0.5 0.886227 0.886227 0 10 1 0.5 1.32934 1.32934 2.22045e-16 10 2 0.5 3.32335 3.32335 1.77636e-15 10 3 0.5 11.6317 11.6317 1.06581e-14 10 4 0.5 52.3428 52.3428 5.68434e-14 10 5 0.5 287.885 287.885 2.27374e-13 10 6 0.5 1871.25 1871.25 1.36424e-12 10 7 0.5 14034.4 14034.4 5.45697e-12 10 8 0.5 119292 119292 0 10 9 0.5 1.13328e+06 1.13328e+06 0 10 10 0.5 1.18994e+07 1.18994e+07 0 10 11 0.5 1.36843e+08 1.36843e+08 3.57628e-07 10 12 0.5 1.71054e+09 1.71054e+09 1.19209e-06 10 13 0.5 2.30923e+10 2.30923e+10 0.000114441 10 14 0.5 3.34839e+11 3.34839e+11 0.00109863 10 15 0.5 5.19e+12 5.19e+12 0.0126953 10 16 0.5 8.5635e+13 8.5635e+13 0.25 10 17 0.5 1.49861e+15 1.49861e+15 1.5 10 18 0.5 2.77243e+16 2.77243e+16 40 10 19 0.5 5.40624e+17 5.40624e+17 896 10 20 0.5 1.10828e+19 1.10828e+19 4.31806e+13 10 21 0.5 2.3827e+20 2.3828e+20 9.99632e+15 10 22 0.5 5.36004e+21 5.3613e+21 1.26546e+18 1 0 1 1 1 0 1 1 1 2 2 0 1 2 1 4 6 2 1 3 1 8 24 16 1 4 1 16 120 104 2 0 1 1 1 2.22045e-16 2 1 1 2 2 0 2 2 1 6 6 0 2 3 1 24 24 3.55271e-15 2 4 1 108 120 12 2 5 1 504 720 216 2 6 1 2376 5040 2664 3 0 1 1 1 4.44089e-16 3 1 1 2 2 1.33227e-15 3 2 1 6 6 5.32907e-15 3 3 1 24 24 2.84217e-14 3 4 1 120 120 1.7053e-13 3 5 1 720 720 1.13687e-12 3 6 1 4896 5040 144 3 7 1 35712 40320 4608 3 8 1 269568 362880 93312 4 0 1 1 1 0 4 1 1 2 2 0 4 2 1 6 6 1.77636e-15 4 3 1 24 24 1.06581e-14 4 4 1 120 120 9.9476e-14 4 5 1 720 720 9.09495e-13 4 6 1 5040 5040 8.18545e-12 4 7 1 40320 40320 8.00355e-11 4 8 1 360000 362880 2880 4 9 1 3.4848e+06 3.6288e+06 144000 4 10 1 3.5568e+07 3.99168e+07 4.3488e+06 5 0 1 1 1 0 5 1 1 2 2 2.22045e-16 5 2 1 6 6 2.66454e-15 5 3 1 24 24 2.4869e-14 5 4 1 120 120 1.84741e-13 5 5 1 720 720 1.81899e-12 5 6 1 5040 5040 1.54614e-11 5 7 1 40320 40320 1.52795e-10 5 8 1 362880 362880 1.39698e-09 5 9 1 3.6288e+06 3.6288e+06 1.21072e-08 5 10 1 3.98304e+07 3.99168e+07 86400 5 11 1 4.72781e+08 4.79002e+08 6.2208e+06 5 12 1 5.96471e+09 6.22702e+09 2.6231e+08 6 0 1 1 1 2.22045e-16 6 1 1 2 2 8.88178e-16 6 2 1 6 6 6.21725e-15 6 3 1 24 24 3.19744e-14 6 4 1 120 120 1.56319e-13 6 5 1 720 720 1.02318e-12 6 6 1 5040 5040 7.27596e-12 6 7 1 40320 40320 4.36557e-11 6 8 1 362880 362880 3.49246e-10 6 9 1 3.6288e+06 3.6288e+06 2.79397e-09 6 10 1 3.99168e+07 3.99168e+07 7.45058e-09 6 11 1 4.79002e+08 4.79002e+08 2.6226e-06 6 12 1 6.22339e+09 6.22702e+09 3.6288e+06 6 13 1 8.68227e+10 8.71783e+10 3.55622e+08 6 14 1 1.28771e+12 1.30767e+12 1.99657e+10 7 0 1 1 1 1.11022e-16 7 1 1 2 2 0 7 2 1 6 6 1.77636e-15 7 3 1 24 24 7.10543e-15 7 4 1 120 120 7.10543e-14 7 5 1 720 720 6.82121e-13 7 6 1 5040 5040 5.45697e-12 7 7 1 40320 40320 5.82077e-11 7 8 1 362880 362880 5.82077e-10 7 9 1 3.6288e+06 3.6288e+06 6.98492e-09 7 10 1 3.99168e+07 3.99168e+07 5.21541e-08 7 11 1 4.79002e+08 4.79002e+08 3.21865e-06 7 12 1 6.22702e+09 6.22702e+09 2.09808e-05 7 13 1 8.71783e+10 8.71783e+10 0.000274658 7 14 1 1.30747e+12 1.30767e+12 2.03213e+08 7 15 1 2.08968e+13 2.09228e+13 2.60112e+10 7 16 1 3.53811e+14 3.55687e+14 1.87606e+12 8 0 1 1 1 3.33067e-16 8 1 1 2 2 6.66134e-16 8 2 1 6 6 8.88178e-16 8 3 1 24 24 1.06581e-14 8 4 1 120 120 7.10543e-14 8 5 1 720 720 1.02318e-12 8 6 1 5040 5040 1.00044e-11 8 7 1 40320 40320 1.01863e-10 8 8 1 362880 362880 9.8953e-10 8 9 1 3.6288e+06 3.6288e+06 1.02445e-08 8 10 1 3.99168e+07 3.99168e+07 1.2666e-07 8 11 1 4.79002e+08 4.79002e+08 1.49012e-06 8 12 1 6.22702e+09 6.22702e+09 6.67572e-06 8 13 1 8.71783e+10 8.71783e+10 0.000167847 8 14 1 1.30767e+12 1.30767e+12 0.00366211 8 15 1 2.09228e+13 2.09228e+13 0 8 16 1 3.55673e+14 3.55687e+14 1.46313e+10 8 17 1 6.4e+15 6.40237e+15 2.37027e+12 8 18 1 1.21432e+17 1.21645e+17 2.13588e+14 9 0 1 1 1 1.11022e-16 9 1 1 2 2 8.88178e-16 9 2 1 6 6 0 9 3 1 24 24 3.55271e-15 9 4 1 120 120 2.84217e-14 9 5 1 720 720 2.27374e-13 9 6 1 5040 5040 1.81899e-12 9 7 1 40320 40320 7.27596e-12 9 8 1 362880 362880 5.82077e-11 9 9 1 3.6288e+06 3.6288e+06 1.39698e-09 9 10 1 3.99168e+07 3.99168e+07 7.45058e-09 9 11 1 4.79002e+08 4.79002e+08 2.74181e-06 9 12 1 6.22702e+09 6.22702e+09 1.81198e-05 9 13 1 8.71783e+10 8.71783e+10 0.000244141 9 14 1 1.30767e+12 1.30767e+12 0.00317383 9 15 1 2.09228e+13 2.09228e+13 0.0390625 9 16 1 3.55687e+14 3.55687e+14 0.5625 9 17 1 6.40237e+15 6.40237e+15 5 9 18 1 1.21644e+17 1.21645e+17 1.31682e+12 9 19 1 2.43264e+18 2.4329e+18 2.63364e+14 9 20 1 5.10619e+19 5.10909e+19 2.89964e+16 10 0 1 1 1 4.44089e-16 10 1 1 2 2 4.44089e-16 10 2 1 6 6 1.77636e-15 10 3 1 24 24 1.42109e-14 10 4 1 120 120 8.52651e-14 10 5 1 720 720 6.82121e-13 10 6 1 5040 5040 6.36646e-12 10 7 1 40320 40320 7.27596e-11 10 8 1 362880 362880 6.40284e-10 10 9 1 3.6288e+06 3.6288e+06 6.0536e-09 10 10 1 3.99168e+07 3.99168e+07 2.23517e-08 10 11 1 4.79002e+08 4.79002e+08 2.86102e-06 10 12 1 6.22702e+09 6.22702e+09 1.62125e-05 10 13 1 8.71783e+10 8.71783e+10 0.000183105 10 14 1 1.30767e+12 1.30767e+12 0.00170898 10 15 1 2.09228e+13 2.09228e+13 0.0664062 10 16 1 3.55687e+14 3.55687e+14 1.125 10 17 1 6.40237e+15 6.40237e+15 6 10 18 1 1.21645e+17 1.21645e+17 432 10 19 1 2.4329e+18 2.4329e+18 9216 10 20 1 5.10908e+19 5.10909e+19 1.4485e+14 10 21 1 1.12397e+21 1.124e+21 3.50537e+16 10 22 1 2.58474e+22 2.5852e+22 4.63028e+18 1 0 2.5 3.32335 3.32335 4.44089e-16 1 1 2.5 11.6317 11.6317 1.77636e-15 1 2 2.5 40.711 52.3428 11.6317 1 3 2.5 142.489 287.885 145.397 1 4 2.5 498.71 1871.25 1372.54 2 0 2.5 3.32335 3.32335 8.88178e-16 2 1 2.5 11.6317 11.6317 3.55271e-15 2 2 2.5 52.3428 52.3428 2.84217e-14 2 3 2.5 287.885 287.885 2.27374e-13 2 4 2.5 1766.57 1871.25 104.686 2 5 2.5 11364.9 14034.4 2669.48 2 6 2.5 74460.9 119292 44831.6 3 0 2.5 3.32335 3.32335 1.33227e-15 3 1 2.5 11.6317 11.6317 3.55271e-15 3 2 2.5 52.3428 52.3428 2.84217e-14 3 3 2.5 287.885 287.885 1.7053e-13 3 4 2.5 1871.25 1871.25 1.36424e-12 3 5 2.5 14034.4 14034.4 1.27329e-11 3 6 2.5 117565 119292 1727.31 3 7 2.5 1.05987e+06 1.13328e+06 73410.7 3 8 2.5 9.97433e+06 1.18994e+07 1.92509e+06 4 0 2.5 3.32335 3.32335 1.33227e-15 4 1 2.5 11.6317 11.6317 1.77636e-15 4 2 2.5 52.3428 52.3428 0 4 3 2.5 287.885 287.885 1.13687e-13 4 4 2.5 1871.25 1871.25 1.13687e-12 4 5 2.5 14034.4 14034.4 1.27329e-11 4 6 2.5 119292 119292 1.74623e-10 4 7 2.5 1.13328e+06 1.13328e+06 2.32831e-09 4 8 2.5 1.18545e+07 1.18994e+07 44910.1 4 9 2.5 1.33992e+08 1.36843e+08 2.85179e+06 4 10 2.5 1.60308e+09 1.71054e+09 1.07459e+08 5 0 2.5 3.32335 3.32335 1.33227e-15 5 1 2.5 11.6317 11.6317 5.32907e-15 5 2 2.5 52.3428 52.3428 1.42109e-14 5 3 2.5 287.885 287.885 1.7053e-13 5 4 2.5 1871.25 1871.25 9.09495e-13 5 5 2.5 14034.4 14034.4 1.09139e-11 5 6 2.5 119292 119292 1.16415e-10 5 7 2.5 1.13328e+06 1.13328e+06 1.39698e-09 5 8 2.5 1.18994e+07 1.18994e+07 2.04891e-08 5 9 2.5 1.36843e+08 1.36843e+08 5.96046e-08 5 10 2.5 1.70886e+09 1.71054e+09 1.68413e+06 5 11 2.5 2.29433e+10 2.30923e+10 1.49045e+08 5 12 2.5 3.27215e+11 3.34839e+11 7.62405e+09 6 0 2.5 3.32335 3.32335 4.44089e-16 6 1 2.5 11.6317 11.6317 1.77636e-15 6 2 2.5 52.3428 52.3428 0 6 3 2.5 287.885 287.885 5.68434e-14 6 4 2.5 1871.25 1871.25 2.27374e-13 6 5 2.5 14034.4 14034.4 3.63798e-12 6 6 2.5 119292 119292 4.36557e-11 6 7 2.5 1.13328e+06 1.13328e+06 6.98492e-10 6 8 2.5 1.18994e+07 1.18994e+07 1.30385e-08 6 9 2.5 1.36843e+08 1.36843e+08 1.78814e-07 6 10 2.5 1.71054e+09 1.71054e+09 3.57628e-06 6 11 2.5 2.30923e+10 2.30923e+10 7.24792e-05 6 12 2.5 3.34753e+11 3.34839e+11 8.58906e+07 6 13 2.5 5.17991e+12 5.19e+12 1.00921e+10 6 14 2.5 8.49625e+13 8.5635e+13 6.72459e+11 7 0 2.5 3.32335 3.32335 1.77636e-15 7 1 2.5 11.6317 11.6317 1.06581e-14 7 2 2.5 52.3428 52.3428 4.26326e-14 7 3 2.5 287.885 287.885 2.84217e-13 7 4 2.5 1871.25 1871.25 1.59162e-12 7 5 2.5 14034.4 14034.4 5.45697e-12 7 6 2.5 119292 119292 1.45519e-11 7 7 2.5 1.13328e+06 1.13328e+06 4.65661e-10 7 8 2.5 1.18994e+07 1.18994e+07 1.30385e-08 7 9 2.5 1.36843e+08 1.36843e+08 8.9407e-08 7 10 2.5 1.71054e+09 1.71054e+09 5.72205e-06 7 11 2.5 2.30923e+10 2.30923e+10 3.43323e-05 7 12 2.5 3.34839e+11 3.34839e+11 0.00250244 7 13 2.5 5.19e+12 5.19e+12 0.0371094 7 14 2.5 8.56293e+13 8.5635e+13 5.71172e+09 7 15 2.5 1.49775e+15 1.49861e+15 8.59614e+11 7 16 2.5 2.7652e+16 2.77243e+16 7.23076e+13 8 0 2.5 3.32335 3.32335 1.33227e-15 8 1 2.5 11.6317 11.6317 3.55271e-15 8 2 2.5 52.3428 52.3428 1.42109e-14 8 3 2.5 287.885 287.885 5.68434e-14 8 4 2.5 1871.25 1871.25 2.27374e-13 8 5 2.5 14034.4 14034.4 0 8 6 2.5 119292 119292 1.45519e-11 8 7 2.5 1.13328e+06 1.13328e+06 0 8 8 2.5 1.18994e+07 1.18994e+07 1.86265e-09 8 9 2.5 1.36843e+08 1.36843e+08 2.98023e-07 8 10 2.5 1.71054e+09 1.71054e+09 2.38419e-06 8 11 2.5 2.30923e+10 2.30923e+10 8.7738e-05 8 12 2.5 3.34839e+11 3.34839e+11 0.00164795 8 13 2.5 5.19e+12 5.19e+12 0.0224609 8 14 2.5 8.5635e+13 8.5635e+13 0.015625 8 15 2.5 1.49861e+15 1.49861e+15 6.25 8 16 2.5 2.77238e+16 2.77243e+16 4.79785e+11 8 17 2.5 5.40534e+17 5.40624e+17 8.99596e+13 8 18 2.5 1.10735e+19 1.10828e+19 9.32018e+15 9 0 2.5 3.32335 3.32335 1.33227e-15 9 1 2.5 11.6317 11.6317 7.10543e-15 9 2 2.5 52.3428 52.3428 1.42109e-14 9 3 2.5 287.885 287.885 5.68434e-14 9 4 2.5 1871.25 1871.25 2.27374e-13 9 5 2.5 14034.4 14034.4 5.45697e-12 9 6 2.5 119292 119292 4.36557e-11 9 7 2.5 1.13328e+06 1.13328e+06 0 9 8 2.5 1.18994e+07 1.18994e+07 1.86265e-09 9 9 2.5 1.36843e+08 1.36843e+08 3.57628e-07 9 10 2.5 1.71054e+09 1.71054e+09 1.43051e-06 9 11 2.5 2.30923e+10 2.30923e+10 0.000102997 9 12 2.5 3.34839e+11 3.34839e+11 0.00146484 9 13 2.5 5.19e+12 5.19e+12 0.0195312 9 14 2.5 8.5635e+13 8.5635e+13 0.03125 9 15 2.5 1.49861e+15 1.49861e+15 6.75 9 16 2.5 2.77243e+16 2.77243e+16 164 9 17 2.5 5.40624e+17 5.40624e+17 2304 9 18 2.5 1.10827e+19 1.10828e+19 4.96577e+13 9 19 2.5 2.38269e+20 2.3828e+20 1.13468e+16 9 20 2.5 5.35988e+21 5.3613e+21 1.41942e+18 10 0 2.5 3.32335 3.32335 1.77636e-15 10 1 2.5 11.6317 11.6317 7.10543e-15 10 2 2.5 52.3428 52.3428 5.68434e-14 10 3 2.5 287.885 287.885 3.41061e-13 10 4 2.5 1871.25 1871.25 2.50111e-12 10 5 2.5 14034.4 14034.4 1.63709e-11 10 6 2.5 119292 119292 1.45519e-10 10 7 2.5 1.13328e+06 1.13328e+06 1.62981e-09 10 8 2.5 1.18994e+07 1.18994e+07 1.49012e-08 10 9 2.5 1.36843e+08 1.36843e+08 1.78814e-07 10 10 2.5 1.71054e+09 1.71054e+09 4.52995e-06 10 11 2.5 2.30923e+10 2.30923e+10 5.34058e-05 10 12 2.5 3.34839e+11 3.34839e+11 0.0022583 10 13 2.5 5.19e+12 5.19e+12 0.0322266 10 14 2.5 8.5635e+13 8.5635e+13 0.171875 10 15 2.5 1.49861e+15 1.49861e+15 10 10 16 2.5 2.77243e+16 2.77243e+16 212 10 17 2.5 5.40624e+17 5.40624e+17 2944 10 18 2.5 1.10828e+19 1.10828e+19 6144 10 19 2.5 2.3828e+20 2.3828e+20 360448 10 20 2.5 5.3613e+21 5.3613e+21 6.20722e+15 10 21 2.5 1.25989e+23 1.25991e+23 1.69767e+18 10 22 2.5 3.08652e+24 3.08677e+24 2.5225e+20 TEST14 GEN_LAGUERRE_COMPUTE_NP computes a generalized Gauss-Laguerre rule which is appropriate for integrands of the form Integral ( 0 <= x < +oo ) f(x) x^alpha exp(-x) dx. Order = 1 ALPHA = 0.5 0 1.5 0.8862269254527581 Order = 2 ALPHA = 0.5 0 0.9188611699158105 0.7233630235462752 1 4.081138830084189 0.1628639019064825 Order = 3 ALPHA = 0.5 0 0.666325907702371 0.5671862778403116 1 2.800775054150256 0.3053717688445466 2 7.032899038147372 0.01366887876790012 Order = 4 ALPHA = 0.5 0 0.5235260767382686 0.4530087465586073 1 2.156648763269094 0.3816169601718006 2 5.137387546176712 0.05079462757224074 3 10.18243761381593 0.0008065911501100318 Order = 5 ALPHA = 0.5 0 0.4313988071478522 0.3704505700074587 1 1.759753698423698 0.4125843737694527 2 4.104465362828316 0.09777982005318063 3 7.746703779542558 0.005373415341171988 4 13.45767835205758 3.874628149393578e-05 Order = 6 ALPHA = 0.5 0 0.3669498773083705 0.3094240968362603 1 1.488534292310453 0.4177521497070222 2 3.434007968424071 0.1432858732209771 3 6.349067925680379 0.01533249102263385 4 10.54046985844834 0.0004306911960439409 5 16.82097007782838 1.623469821074067e-06 Order = 7 ALPHA = 0.5 0 0.3193036339206293 0.2631245143958913 1 1.290758622959152 0.4091418694141027 2 2.958374458696651 0.1821177320927163 3 5.409031597244436 0.03005332430127098 4 8.804079578056774 0.001760894117540066 5 13.46853574325148 2.852947122115974e-05 6 20.24991636587088 6.166001541039146e-08 Order = 8 ALPHA = 0.5 0 0.2826336481165983 0.227139361952471 1 1.139873801581612 0.3935945428036152 2 2.601524843406029 0.2129089708672288 3 4.72411453752779 0.04787748320313817 4 7.605256299231614 0.004542517474762631 5 11.41718207654583 0.0001624046001853252 6 16.49941079765581 1.642377413806097e-06 7 23.73000399593471 2.173943126630911e-09 Order = 9 ALPHA = 0.5 0 0.2535325549744193 0.1985712548680196 1 1.02084427772039 0.3749207846631705 2 2.323096077022465 0.2360748210008252 3 4.199350600657293 0.06709610500320433 4 6.71397431661503 0.009008508896644308 5 9.972009159539351 0.0005426607386359281 6 14.15405367127805 1.270536687910834e-05 7 19.61190281916595 8.484309239668581e-08 8 27.25123652302706 7.228647164396506e-11 Order = 10 ALPHA = 0.5 0 0.2298729805186566 0.1754708150466599 1 0.9244815469866562 0.355223388802072 2 2.099410462708798 0.2526835596756785 3 3.782880873707289 0.08635610269533257 4 6.019918027701461 0.01510977803486079 5 8.880347597996709 0.001328215628363565 6 12.47483240483621 5.418780021170343e-05 7 16.99084729354256 8.737475869187116e-07 8 22.79100289494895 4.019699886939779e-09 9 30.80640591705272 2.292221530204704e-12 Order = 1 ALPHA = 1 0 2 1 Order = 2 ALPHA = 1 0 1.267949192431123 0.7886751345948131 1 4.732050807568878 0.2113248654051871 Order = 3 ALPHA = 1 0 0.9358222275240875 0.5886814810396593 1 3.305407289332279 0.3912160592223105 2 7.758770483143635 0.02010245973803056 Order = 4 ALPHA = 1 0 0.7432919279814317 0.4468705932187766 1 2.571635007646279 0.477635772363868 2 5.731178751689098 0.07417778473105212 3 10.95389431268319 0.00131584968630324 Order = 5 ALPHA = 1 0 0.6170308532782706 0.3480145400233489 1 2.112965958578525 0.502280674132493 2 4.610833151017531 0.1409159194944725 3 8.399066971204835 0.00871989302609999 4 14.26010306592084 6.89733235856401e-05 Order = 6 ALPHA = 1 0 0.5276681217111289 0.2776501420298752 1 1.796299809643409 0.4939105830503541 2 3.876641520476912 0.2030042967437302 3 6.918816566704723 0.0246688203691898 4 11.23461042908311 0.000763042767463531 5 17.64596355238071 3.115039387527555e-06 Order = 7 ALPHA = 1 0 0.4610242198049947 0.226210596383189 1 1.563586189654263 0.4697087077412661 2 3.352050502536734 0.2531225162324008 3 5.916297249020539 0.04780310888423712 4 9.420699383021594 0.003100459756025395 5 14.19416554800746 5.448466889341361e-05 6 21.09217690795441 1.26333988153628e-07 Order = 8 ALPHA = 1 0 0.409383573203186 0.187632541405724 1 1.384963184803141 0.4389853607311423 2 2.956254556168864 0.289996070781313 3 5.181943101040073 0.07514138461669727 4 8.161709688145816 0.007932646648707327 5 12.07005512683716 0.0003086421368133028 6 17.249735526149 3.348958209797085e-06 7 24.58595524365278 4.721392823193254e-09 Order = 9 ALPHA = 1 0 0.3681784529417419 0.1580309235007823 1 1.243357962140484 0.4065825794555729 2 2.646033841384209 0.314982483665097 3 4.616882514635062 0.1037755199516432 4 7.221786539396571 0.01557763772427843 5 10.56732080774187 0.001024871969850758 6 14.83591451526093 2.580021138717233e-05 7 20.38218198544925 1.833559550172429e-07 8 28.11834338104989 1.654329455969927e-10 Order = 10 ALPHA = 1 0 0.3345286763247526 0.1348565545313779 1 1.128253355876639 0.3749243411054598 2 2.395869924747309 0.3302471308042158 3 4.166840987928768 0.1315288846413064 4 6.487353031380811 0.02584206421171715 5 9.428354813335606 0.002489645795895956 6 13.1017235803678 0.0001094884764266961 7 17.69648756684622 1.881275414181711e-06 8 23.57778708836015 9.152685028575581e-09 9 31.68280097483193 5.501303496178695e-12 Order = 1 ALPHA = 2.5 0 3.5 3.323350970447842 Order = 2 ALPHA = 2.5 0 2.378679656440358 2.444996821046109 1 6.621320343559642 0.8783541494017333 Order = 3 ALPHA = 2.5 0 1.824819803075874 1.640491690990948 1 4.813746226487698 1.576420610283191 2 9.861433970436428 0.1064386691737022 Order = 4 ALPHA = 2.5 0 1.48624114266535 1.109022578585149 1 3.8376839879001 1.820324124907821 2 7.482058366898981 0.3853099204909315 3 13.19401650253557 0.008694346463943117 Order = 5 ALPHA = 2.5 0 1.255846140918443 0.769346187647318 1 3.207040626614909 1.789143657137866 2 6.123908223328411 0.7075497394015858 3 10.31612586639135 0.05675567145898618 4 16.59707914274689 0.000555714802087273 Order = 6 ALPHA = 2.5 0 1.088263537814163 0.5488342282458472 1 2.76085557138551 1.635828855251358 2 5.213082470955716 0.97570188767536 3 8.608834500537 0.1568919688876181 4 13.27367226930421 0.006064045733978006 5 20.05529165000341 2.99846536813126e-05 Order = 7 ALPHA = 2.5 0 0.9606221295119317 0.401864465233434 1 2.426677101427741 1.44402357612933 2 4.550654967582769 1.157819445450442 3 7.432417772624608 0.2949071223080977 4 11.24467756580187 0.02421750114397002 5 16.32710951627492 0.0005174322318932411 6 23.55784094677616 1.427950674360879e-06 Order = 8 ALPHA = 2.5 0 0.8600416237603872 0.3011867253184345 1 2.166243528992247 1.25296378034376 2 4.043742677349125 1.258345796079632 3 6.558922538427902 0.4473442363684211 4 9.817255505225706 0.06058255027905918 5 13.99947975071506 0.002890481769915709 6 19.45744743181463 3.733852866568666e-05 7 27.09686694371495 6.175995605353352e-08 Order = 9 ALPHA = 2.5 0 0.7786800215607389 0.2304206696494764 1 1.957190223504775 1.078733434868177 2 3.641788795064597 1.294498383186512 3 5.87932972203498 0.5941550811279991 4 8.740028894917545 0.1158251816851087 5 12.33467558588798 0.00943162869193187 6 16.85079612749538 0.00028422378718938 7 22.65102569838607 2.364977643120573e-06 8 30.66648493114793 2.473803674819386e-09 Order = 10 ALPHA = 2.5 0 0.711476069898304 0.179502944206614 1 1.785477806129485 0.9263170003438098 2 3.314481217602208 1.284767518101653 3 5.333049252470586 0.7226374668786899 4 7.890763402531441 0.1864748527585193 5 11.06167397628529 0.02243790620506889 6 14.96102232439852 0.001189142723014832 7 19.78221807877333 2.400423791594193e-05 8 25.89766445973611 1.348994981145529e-07 9 34.26217341217473 9.306038978295321e-11 TEST15 HERMITE_COMPUTE_NP computes a Gauss-Hermite rule which is appropriate for integrands of the form Integral ( -oo < x < +oo ) f(x) exp(-x*x) dx. HERMITE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 1.77245 1.77245 2.22045e-16 1 1 0 0 0 1 2 0 0.886227 0.886227 1 3 0 0 0 1 4 0 1.32934 1.32934 2 0 1.77245 1.77245 2.22045e-16 2 1 0 0 0 2 2 0.886227 0.886227 3.33067e-16 2 3 0 0 0 2 4 0.443113 1.32934 0.886227 2 5 0 0 0 2 6 0.221557 3.32335 3.10179 3 0 1.77245 1.77245 2.22045e-16 3 1 5.55112e-17 0 5.55112e-17 3 2 0.886227 0.886227 1.22125e-15 3 3 3.33067e-16 0 3.33067e-16 3 4 1.32934 1.32934 2.66454e-15 3 5 4.44089e-16 0 4.44089e-16 3 6 1.99401 3.32335 1.32934 3 7 6.66134e-16 0 6.66134e-16 3 8 2.99102 11.6317 8.64071 4 0 1.77245 1.77245 4.44089e-16 4 1 -3.33067e-16 0 3.33067e-16 4 2 0.886227 0.886227 3.33067e-16 4 3 7.21645e-16 0 7.21645e-16 4 4 1.32934 1.32934 1.33227e-15 4 5 2.88658e-15 0 2.88658e-15 4 6 3.32335 3.32335 5.32907e-15 4 7 1.06581e-14 0 1.06581e-14 4 8 8.97305 11.6317 2.65868 4 9 3.28626e-14 0 3.28626e-14 4 10 24.4266 52.3428 27.9161 5 0 1.77245 1.77245 1.33227e-15 5 1 -2.77556e-17 0 2.77556e-17 5 2 0.886227 0.886227 4.44089e-16 5 3 -4.71845e-16 0 4.71845e-16 5 4 1.32934 1.32934 4.44089e-16 5 5 -2.55351e-15 0 2.55351e-15 5 6 3.32335 3.32335 1.33227e-15 5 7 -1.15463e-14 0 1.15463e-14 5 8 11.6317 11.6317 1.77636e-15 5 9 -4.79616e-14 0 4.79616e-14 5 10 45.6961 52.3428 6.6467 5 11 -1.98952e-13 0 1.98952e-13 5 12 184.861 287.885 103.024 6 0 1.77245 1.77245 6.66134e-16 6 1 -7.45931e-16 0 7.45931e-16 6 2 0.886227 0.886227 7.77156e-16 6 3 -1.19349e-15 0 1.19349e-15 6 4 1.32934 1.32934 1.77636e-15 6 5 -3.16414e-15 0 3.16414e-15 6 6 3.32335 3.32335 3.10862e-15 6 7 -1.26565e-14 0 1.26565e-14 6 8 11.6317 11.6317 0 6 9 -6.03961e-14 0 6.03961e-14 6 10 52.3428 52.3428 6.39488e-14 6 11 -3.12639e-13 0 3.12639e-13 6 12 267.945 287.885 19.9401 6 13 -1.53477e-12 0 1.53477e-12 6 14 1442.54 1871.25 428.712 7 0 1.77245 1.77245 1.33227e-15 7 1 3.47378e-16 0 3.47378e-16 7 2 0.886227 0.886227 6.66134e-16 7 3 5.44703e-16 0 5.44703e-16 7 4 1.32934 1.32934 1.33227e-15 7 5 1.7486e-15 0 1.7486e-15 7 6 3.32335 3.32335 3.9968e-15 7 7 6.43929e-15 0 6.43929e-15 7 8 11.6317 11.6317 1.24345e-14 7 9 2.93099e-14 0 2.93099e-14 7 10 52.3428 52.3428 5.68434e-14 7 11 1.42109e-13 0 1.42109e-13 7 12 287.885 287.885 3.41061e-13 7 13 7.38964e-13 0 7.38964e-13 7 14 1801.46 1871.25 69.7904 7 15 5.00222e-12 0 5.00222e-12 7 16 12045.4 14034.4 1989.03 8 0 1.77245 1.77245 8.88178e-16 8 1 -6.66893e-16 0 6.66893e-16 8 2 0.886227 0.886227 4.44089e-16 8 3 -3.66027e-16 0 3.66027e-16 8 4 1.32934 1.32934 1.33227e-15 8 5 2.56739e-16 0 2.56739e-16 8 6 3.32335 3.32335 7.99361e-15 8 7 3.83027e-15 0 3.83027e-15 8 8 11.6317 11.6317 4.44089e-14 8 9 1.95399e-14 0 1.95399e-14 8 10 52.3428 52.3428 2.27374e-13 8 11 7.81597e-14 0 7.81597e-14 8 12 287.885 287.885 1.25056e-12 8 13 1.7053e-13 0 1.7053e-13 8 14 1871.25 1871.25 8.18545e-12 8 15 0 0 0 8 16 13755.2 14034.4 279.161 8 17 -1.09139e-11 0 1.09139e-11 8 18 109103 119292 10189.4 9 0 1.77245 1.77245 1.33227e-15 9 1 4.80871e-16 0 4.80871e-16 9 2 0.886227 0.886227 1.55431e-15 9 3 6.39462e-16 0 6.39462e-16 9 4 1.32934 1.32934 3.77476e-15 9 5 8.50015e-16 0 8.50015e-16 9 6 3.32335 3.32335 1.15463e-14 9 7 2.69229e-15 0 2.69229e-15 9 8 11.6317 11.6317 3.90799e-14 9 9 1.33227e-14 0 1.33227e-14 9 10 52.3428 52.3428 1.63425e-13 9 11 7.4607e-14 0 7.4607e-14 9 12 287.885 287.885 6.82121e-13 9 13 1.7053e-13 0 1.7053e-13 9 14 1871.25 1871.25 2.95586e-12 9 15 -4.54747e-13 0 4.54747e-13 9 16 14034.4 14034.4 3.63798e-12 9 17 -2.72848e-11 0 2.72848e-11 9 18 118036 119292 1256.23 9 19 -4.65661e-10 0 4.65661e-10 9 20 1.07612e+06 1.13328e+06 57158.3 10 0 1.77245 1.77245 6.66134e-16 10 1 -7.40117e-16 0 7.40117e-16 10 2 0.886227 0.886227 1.11022e-16 10 3 -1.93498e-15 0 1.93498e-15 10 4 1.32934 1.32934 2.22045e-15 10 5 -4.02109e-15 0 4.02109e-15 10 6 3.32335 3.32335 1.11022e-14 10 7 -5.72459e-15 0 5.72459e-15 10 8 11.6317 11.6317 6.03961e-14 10 9 2.40918e-14 0 2.40918e-14 10 10 52.3428 52.3428 3.2685e-13 10 11 3.78364e-13 0 3.78364e-13 10 12 287.885 287.885 1.81899e-12 10 13 3.36797e-12 0 3.36797e-12 10 14 1871.25 1871.25 1.11413e-11 10 15 2.55795e-11 0 2.55795e-11 10 16 14034.4 14034.4 6.18456e-11 10 17 1.81899e-10 0 1.81899e-10 10 18 119292 119292 3.49246e-10 10 19 1.19326e-09 0 1.19326e-09 10 20 1.127e+06 1.13328e+06 6281.13 10 21 6.98492e-09 0 6.98492e-09 10 22 1.15508e+07 1.18994e+07 348603 TEST16 HERMITE_COMPUTE_NP computes a Gauss-Hermite rule which is appropriate for integrands of the form Integral ( -oo < x < +oo ) f(x) exp(-x*x) dx. Order = 1 0 0 1.772453850905516 Order = 2 0 -0.7071067811865475 0.8862269254527578 1 0.7071067811865475 0.8862269254527578 Order = 3 0 -1.224744871391589 0.2954089751509195 1 -9.862844991098402e-17 1.181635900603677 2 1.224744871391589 0.2954089751509196 Order = 4 0 -1.650680123885784 0.08131283544724513 1 -0.5246476232752904 0.8049140900055129 2 0.5246476232752902 0.8049140900055121 3 1.650680123885784 0.08131283544724525 Order = 5 0 -2.020182870456086 0.01995324205904597 1 -0.9585724646138184 0.3936193231522413 2 2.402579402021668e-16 0.945308720482943 3 0.9585724646138187 0.3936193231522411 4 2.020182870456086 0.01995324205904589 Order = 6 0 -2.350604973674493 0.004530009905508857 1 -1.335849074013696 0.1570673203228569 2 -0.4360774119276162 0.7246295952243929 3 0.436077411927616 0.7246295952243927 4 1.335849074013696 0.1570673203228566 5 2.350604973674493 0.004530009905508823 Order = 7 0 -2.651961356835234 0.0009717812450995172 1 -1.673551628767471 0.05451558281912706 2 -0.8162878828589647 0.4256072526101277 3 -1.059786349168287e-16 0.8102646175568082 4 0.8162878828589646 0.4256072526101282 5 1.673551628767472 0.05451558281912713 6 2.651961356835234 0.0009717812450995191 Order = 8 0 -2.930637420257243 0.0001996040722113679 1 -1.981656756695842 0.01707798300741343 2 -1.15719371244678 0.2078023258148922 3 -0.3811869902073223 0.6611470125582418 4 0.381186990207322 0.661147012558241 5 1.15719371244678 0.2078023258148921 6 1.981656756695842 0.01707798300741346 7 2.930637420257243 0.0001996040722113677 Order = 9 0 -3.190993201781527 3.960697726326444e-05 1 -2.266580584531843 0.004943624275536957 2 -1.468553289216668 0.08847452739437681 3 -0.7235510187528373 0.4326515590025557 4 -2.361558093483325e-16 0.7202352156060512 5 0.7235510187528379 0.4326515590025563 6 1.468553289216668 0.08847452739437676 7 2.266580584531844 0.004943624275536958 8 3.190993201781527 3.960697726326444e-05 Order = 10 0 -3.436159118837738 7.640432855232631e-06 1 -2.532731674232788 0.001343645746781224 2 -1.756683649299881 0.03387439445548121 3 -1.036610829789514 0.240138611082315 4 -0.3429013272237045 0.6108626337353257 5 0.3429013272237046 0.6108626337353257 6 1.036610829789512 0.2401386110823148 7 1.75668364929988 0.03387439445548109 8 2.532731674232791 0.001343645746781228 9 3.436159118837738 7.640432855232614e-06 TEST17 JACOBI_COMPUTE_NP computes a Gauss-Jacobi rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) (1-x)^alpha (1+x)^beta dx. JACOBI_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Alpha Beta Estimate Exact Error 1 0 0.5 0.5 1.5708 1.5708 1.33227e-15 1 1 0.5 0.5 0 0 0 1 2 0.5 0.5 0 0.392699 0.392699 1 3 0.5 0.5 0 0 0 1 4 0.5 0.5 0 0.19635 0.19635 2 0 0.5 0.5 1.5708 1.5708 1.33227e-15 2 1 0.5 0.5 0 0 0 2 2 0.5 0.5 0.392699 0.392699 2.77556e-16 2 3 0.5 0.5 0 0 0 2 4 0.5 0.5 0.0981748 0.19635 0.0981748 2 5 0.5 0.5 0 0 0 2 6 0.5 0.5 0.0245437 0.122718 0.0981748 3 0 0.5 0.5 1.5708 1.5708 1.11022e-15 3 1 0.5 0.5 -3.88578e-16 0 3.88578e-16 3 2 0.5 0.5 0.392699 0.392699 2.77556e-16 3 3 0.5 0.5 -2.498e-16 0 2.498e-16 3 4 0.5 0.5 0.19635 0.19635 2.22045e-16 3 5 0.5 0.5 -1.38778e-16 0 1.38778e-16 3 6 0.5 0.5 0.0981748 0.122718 0.0245437 3 7 0.5 0.5 -8.32667e-17 0 8.32667e-17 3 8 0.5 0.5 0.0490874 0.0859029 0.0368155 4 0 0.5 0.5 1.5708 1.5708 2.22045e-16 4 1 0.5 0.5 3.60822e-16 0 3.60822e-16 4 2 0.5 0.5 0.392699 0.392699 1.66533e-16 4 3 0.5 0.5 1.80411e-16 0 1.80411e-16 4 4 0.5 0.5 0.19635 0.19635 1.66533e-16 4 5 0.5 0.5 1.66533e-16 0 1.66533e-16 4 6 0.5 0.5 0.122718 0.122718 1.80411e-16 4 7 0.5 0.5 1.45717e-16 0 1.45717e-16 4 8 0.5 0.5 0.079767 0.0859029 0.00613592 4 9 0.5 0.5 1.17961e-16 0 1.17961e-16 4 10 0.5 0.5 0.0521553 0.0644272 0.0122718 5 0 0.5 0.5 1.5708 1.5708 1.55431e-15 5 1 0.5 0.5 -2.35922e-16 0 2.35922e-16 5 2 0.5 0.5 0.392699 0.392699 5.55112e-17 5 3 0.5 0.5 -1.80411e-16 0 1.80411e-16 5 4 0.5 0.5 0.19635 0.19635 1.11022e-16 5 5 0.5 0.5 -1.249e-16 0 1.249e-16 5 6 0.5 0.5 0.122718 0.122718 1.66533e-16 5 7 0.5 0.5 -9.71445e-17 0 9.71445e-17 5 8 0.5 0.5 0.0859029 0.0859029 1.66533e-16 5 9 0.5 0.5 -9.02056e-17 0 9.02056e-17 5 10 0.5 0.5 0.0628932 0.0644272 0.00153398 5 11 0.5 0.5 -6.59195e-17 0 6.59195e-17 5 12 0.5 0.5 0.0467864 0.0506214 0.00383495 6 0 0.5 0.5 1.5708 1.5708 1.11022e-15 6 1 0.5 0.5 -1.38778e-17 0 1.38778e-17 6 2 0.5 0.5 0.392699 0.392699 5.55112e-17 6 3 0.5 0.5 -1.52656e-16 0 1.52656e-16 6 4 0.5 0.5 0.19635 0.19635 2.22045e-16 6 5 0.5 0.5 -1.38778e-16 0 1.38778e-16 6 6 0.5 0.5 0.122718 0.122718 2.08167e-16 6 7 0.5 0.5 -1.04083e-16 0 1.04083e-16 6 8 0.5 0.5 0.0859029 0.0859029 2.22045e-16 6 9 0.5 0.5 -7.63278e-17 0 7.63278e-17 6 10 0.5 0.5 0.0644272 0.0644272 4.16334e-17 6 11 0.5 0.5 -5.89806e-17 0 5.89806e-17 6 12 0.5 0.5 0.0502379 0.0506214 0.000383495 6 13 0.5 0.5 -4.85723e-17 0 4.85723e-17 6 14 0.5 0.5 0.0399794 0.0411299 0.00115049 7 0 0.5 0.5 1.5708 1.5708 0 7 1 0.5 0.5 7.63278e-17 0 7.63278e-17 7 2 0.5 0.5 0.392699 0.392699 2.22045e-16 7 3 0.5 0.5 -5.55112e-17 0 5.55112e-17 7 4 0.5 0.5 0.19635 0.19635 1.11022e-16 7 5 0.5 0.5 -1.59595e-16 0 1.59595e-16 7 6 0.5 0.5 0.122718 0.122718 8.32667e-17 7 7 0.5 0.5 -1.8735e-16 0 1.8735e-16 7 8 0.5 0.5 0.0859029 0.0859029 2.77556e-17 7 9 0.5 0.5 -1.94289e-16 0 1.94289e-16 7 10 0.5 0.5 0.0644272 0.0644272 1.80411e-16 7 11 0.5 0.5 -1.9082e-16 0 1.9082e-16 7 12 0.5 0.5 0.0506214 0.0506214 4.51028e-16 7 13 0.5 0.5 -1.8735e-16 0 1.8735e-16 7 14 0.5 0.5 0.041034 0.0411299 9.58738e-05 7 15 0.5 0.5 -1.76942e-16 0 1.76942e-16 7 16 0.5 0.5 0.0339393 0.0342749 0.000335558 8 0 0.5 0.5 1.5708 1.5708 0 8 1 0.5 0.5 -1.80411e-16 0 1.80411e-16 8 2 0.5 0.5 0.392699 0.392699 2.22045e-16 8 3 0.5 0.5 -2.01228e-16 0 2.01228e-16 8 4 0.5 0.5 0.19635 0.19635 8.32667e-17 8 5 0.5 0.5 -1.70003e-16 0 1.70003e-16 8 6 0.5 0.5 0.122718 0.122718 8.32667e-17 8 7 0.5 0.5 -1.14492e-16 0 1.14492e-16 8 8 0.5 0.5 0.0859029 0.0859029 9.71445e-17 8 9 0.5 0.5 -5.89806e-17 0 5.89806e-17 8 10 0.5 0.5 0.0644272 0.0644272 2.63678e-16 8 11 0.5 0.5 -1.73472e-17 0 1.73472e-17 8 12 0.5 0.5 0.0506214 0.0506214 5.75928e-16 8 13 0.5 0.5 6.93889e-18 0 6.93889e-18 8 14 0.5 0.5 0.0411299 0.0411299 5.55112e-17 8 15 0.5 0.5 2.08167e-17 0 2.08167e-17 8 16 0.5 0.5 0.0342509 0.0342749 2.39684e-05 8 17 0.5 0.5 3.81639e-17 0 3.81639e-17 8 18 0.5 0.5 0.0290378 0.0291337 9.58738e-05 9 0 0.5 0.5 1.5708 1.5708 0 9 1 0.5 0.5 2.87964e-16 0 2.87964e-16 9 2 0.5 0.5 0.392699 0.392699 3.33067e-16 9 3 0.5 0.5 -3.46945e-17 0 3.46945e-17 9 4 0.5 0.5 0.19635 0.19635 1.11022e-16 9 5 0.5 0.5 -7.28584e-17 0 7.28584e-17 9 6 0.5 0.5 0.122718 0.122718 6.93889e-17 9 7 0.5 0.5 -7.63278e-17 0 7.63278e-17 9 8 0.5 0.5 0.0859029 0.0859029 1.38778e-17 9 9 0.5 0.5 -7.28584e-17 0 7.28584e-17 9 10 0.5 0.5 0.0644272 0.0644272 1.66533e-16 9 11 0.5 0.5 -6.93889e-17 0 6.93889e-17 9 12 0.5 0.5 0.0506214 0.0506214 4.64906e-16 9 13 0.5 0.5 -6.76542e-17 0 6.76542e-17 9 14 0.5 0.5 0.0411299 0.0411299 6.93889e-17 9 15 0.5 0.5 -6.07153e-17 0 6.07153e-17 9 16 0.5 0.5 0.0342749 0.0342749 1.94289e-16 9 17 0.5 0.5 -5.89806e-17 0 5.89806e-17 9 18 0.5 0.5 0.0291277 0.0291337 5.99211e-06 9 19 0.5 0.5 -5.20417e-17 0 5.20417e-17 9 20 0.5 0.5 0.0251339 0.0251609 2.69645e-05 10 0 0.5 0.5 1.5708 1.5708 4.44089e-16 10 1 0.5 0.5 -2.22045e-16 0 2.22045e-16 10 2 0.5 0.5 0.392699 0.392699 2.22045e-16 10 3 0.5 0.5 -1.21431e-16 0 1.21431e-16 10 4 0.5 0.5 0.19635 0.19635 2.77556e-17 10 5 0.5 0.5 -7.63278e-17 0 7.63278e-17 10 6 0.5 0.5 0.122718 0.122718 0 10 7 0.5 0.5 -5.20417e-17 0 5.20417e-17 10 8 0.5 0.5 0.0859029 0.0859029 0 10 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0.0079482 0.139352 0.131404 2 0 2.5 1 1.43666 1.43666 0 2 1 2.5 1 -0.391817 -0.391817 0 2 2 2.5 1 0.311444 0.311444 1.66533e-16 2 3 2.5 1 -0.166773 -0.166773 2.77556e-17 2 4 2.5 1 0.100249 0.139352 0.0391026 2 5 2.5 1 -0.05806 -0.0922233 0.0341633 2 6 2.5 1 0.0340201 0.0795329 0.0455128 3 0 2.5 1 1.43666 1.43666 1.11022e-15 3 1 2.5 1 -0.391817 -0.391817 4.44089e-16 3 2 2.5 1 0.311444 0.311444 4.44089e-16 3 3 2.5 1 -0.166773 -0.166773 1.66533e-16 3 4 2.5 1 0.139352 0.139352 1.38778e-16 3 5 2.5 1 -0.0922233 -0.0922233 1.38778e-16 3 6 2.5 1 0.0712024 0.0795329 0.00833043 3 7 2.5 1 -0.050198 -0.0584903 0.0082923 3 8 2.5 1 0.0374173 0.0515572 0.01414 4 0 2.5 1 1.43666 1.43666 0 4 1 2.5 1 -0.391817 -0.391817 1.11022e-16 4 2 2.5 1 0.311444 0.311444 0 4 3 2.5 1 -0.166773 -0.166773 5.55112e-17 4 4 2.5 1 0.139352 0.139352 5.55112e-17 4 5 2.5 1 -0.0922233 -0.0922233 1.38778e-17 4 6 2.5 1 0.0795329 0.0795329 5.55112e-17 4 7 2.5 1 -0.0584903 -0.0584903 2.77556e-17 4 8 2.5 1 0.0496855 0.0515572 0.0018717 4 9 2.5 1 -0.0383732 -0.0403895 0.00201638 4 10 2.5 1 0.032008 0.0361793 0.00417126 5 0 2.5 1 1.43666 1.43666 0 5 1 2.5 1 -0.391817 -0.391817 5.55112e-17 5 2 2.5 1 0.311444 0.311444 5.55112e-17 5 3 2.5 1 -0.166773 -0.166773 8.32667e-17 5 4 2.5 1 0.139352 0.139352 2.77556e-17 5 5 2.5 1 -0.0922233 -0.0922233 0 5 6 2.5 1 0.0795329 0.0795329 2.77556e-17 5 7 2.5 1 -0.0584903 -0.0584903 0 5 8 2.5 1 0.0515572 0.0515572 6.93889e-18 5 9 2.5 1 -0.0403895 -0.0403895 6.93889e-18 5 10 2.5 1 0.0357458 0.0361793 0.000433464 5 11 2.5 1 -0.0290665 -0.029559 0.000492502 5 12 2.5 1 0.0256123 0.0268067 0.00119435 6 0 2.5 1 1.43666 1.43666 6.66134e-16 6 1 2.5 1 -0.391817 -0.391817 2.22045e-16 6 2 2.5 1 0.311444 0.311444 2.77556e-16 6 3 2.5 1 -0.166773 -0.166773 2.77556e-17 6 4 2.5 1 0.139352 0.139352 8.32667e-17 6 5 2.5 1 -0.0922233 -0.0922233 8.32667e-17 6 6 2.5 1 0.0795329 0.0795329 6.93889e-17 6 7 2.5 1 -0.0584903 -0.0584903 4.16334e-17 6 8 2.5 1 0.0515572 0.0515572 7.63278e-17 6 9 2.5 1 -0.0403895 -0.0403895 4.85723e-17 6 10 2.5 1 0.0361793 0.0361793 2.56739e-16 6 11 2.5 1 -0.029559 -0.029559 1.38778e-16 6 12 2.5 1 0.0267044 0.0268067 0.000102303 6 13 2.5 1 -0.0224459 -0.0225667 0.000120784 6 14 2.5 1 0.0203315 0.0206669 0.000335346 7 0 2.5 1 1.43666 1.43666 2.22045e-16 7 1 2.5 1 -0.391817 -0.391817 5.55112e-17 7 2 2.5 1 0.311444 0.311444 1.11022e-16 7 3 2.5 1 -0.166773 -0.166773 5.55112e-17 7 4 2.5 1 0.139352 0.139352 5.55112e-17 7 5 2.5 1 -0.0922233 -0.0922233 2.77556e-17 7 6 2.5 1 0.0795329 0.0795329 1.11022e-16 7 7 2.5 1 -0.0584903 -0.0584903 7.63278e-17 7 8 2.5 1 0.0515572 0.0515572 7.63278e-17 7 9 2.5 1 -0.0403895 -0.0403895 7.63278e-17 7 10 2.5 1 0.0361793 0.0361793 1.11022e-16 7 11 2.5 1 -0.029559 -0.029559 1.04083e-17 7 12 2.5 1 0.0268067 0.0268067 1.49186e-16 7 13 2.5 1 -0.0225667 -0.0225667 6.59195e-17 7 14 2.5 1 0.0206424 0.0206669 2.44522e-05 7 15 2.5 1 -0.0177618 -0.0177915 2.97189e-05 7 16 2.5 1 0.0163311 0.0164239 9.28462e-05 8 0 2.5 1 1.43666 1.43666 8.88178e-16 8 1 2.5 1 -0.391817 -0.391817 1.11022e-16 8 2 2.5 1 0.311444 0.311444 0 8 3 2.5 1 -0.166773 -0.166773 8.32667e-17 8 4 2.5 1 0.139352 0.139352 1.38778e-16 8 5 2.5 1 -0.0922233 -0.0922233 1.11022e-16 8 6 2.5 1 0.0795329 0.0795329 1.66533e-16 8 7 2.5 1 -0.0584903 -0.0584903 1.249e-16 8 8 2.5 1 0.0515572 0.0515572 1.11022e-16 8 9 2.5 1 -0.0403895 -0.0403895 1.249e-16 8 10 2.5 1 0.0361793 0.0361793 6.93889e-17 8 11 2.5 1 -0.029559 -0.029559 2.42861e-17 8 12 2.5 1 0.0268067 0.0268067 1.73472e-16 8 13 2.5 1 -0.0225667 -0.0225667 7.97973e-17 8 14 2.5 1 0.0206669 0.0206669 1.73472e-16 8 15 2.5 1 -0.0177915 -0.0177915 1.249e-16 8 16 2.5 1 0.016418 0.0164239 5.89662e-06 8 17 2.5 1 -0.0143787 -0.0143861 7.33122e-06 8 18 2.5 1 0.0133428 0.0133683 2.54338e-05 9 0 2.5 1 1.43666 1.43666 6.66134e-16 9 1 2.5 1 -0.391817 -0.391817 3.33067e-16 9 2 2.5 1 0.311444 0.311444 1.11022e-16 9 3 2.5 1 -0.166773 -0.166773 1.66533e-16 9 4 2.5 1 0.139352 0.139352 1.11022e-16 9 5 2.5 1 -0.0922233 -0.0922233 6.93889e-17 9 6 2.5 1 0.0795329 0.0795329 4.16334e-17 9 7 2.5 1 -0.0584903 -0.0584903 5.55112e-17 9 8 2.5 1 0.0515572 0.0515572 6.93889e-18 9 9 2.5 1 -0.0403895 -0.0403895 2.08167e-17 9 10 2.5 1 0.0361793 0.0361793 2.08167e-16 9 11 2.5 1 -0.029559 -0.029559 8.67362e-17 9 12 2.5 1 0.0268067 0.0268067 4.51028e-17 9 13 2.5 1 -0.0225667 -0.0225667 3.46945e-17 9 14 2.5 1 0.0206669 0.0206669 4.51028e-17 9 15 2.5 1 -0.0177915 -0.0177915 3.46945e-18 9 16 2.5 1 0.0164239 0.0164239 1.31839e-16 9 17 2.5 1 -0.0143861 -0.0143861 7.80626e-17 9 18 2.5 1 0.0133668 0.0133683 1.4312e-06 9 19 2.5 1 -0.0118706 -0.0118724 1.81219e-06 9 20 2.5 1 0.0110872 0.0110941 6.90871e-06 10 0 2.5 1 1.43666 1.43666 6.66134e-16 10 1 2.5 1 -0.391817 -0.391817 2.77556e-16 10 2 2.5 1 0.311444 0.311444 1.66533e-16 10 3 2.5 1 -0.166773 -0.166773 8.32667e-17 10 4 2.5 1 0.139352 0.139352 0 10 5 2.5 1 -0.0922233 -0.0922233 1.38778e-17 10 6 2.5 1 0.0795329 0.0795329 2.77556e-17 10 7 2.5 1 -0.0584903 -0.0584903 3.46945e-17 10 8 2.5 1 0.0515572 0.0515572 4.16334e-17 10 9 2.5 1 -0.0403895 -0.0403895 6.93889e-17 10 10 2.5 1 0.0361793 0.0361793 1.17961e-16 10 11 2.5 1 -0.029559 -0.029559 1.04083e-17 10 12 2.5 1 0.0268067 0.0268067 1.42247e-16 10 13 2.5 1 -0.0225667 -0.0225667 6.59195e-17 10 14 2.5 1 0.0206669 0.0206669 1.59595e-16 10 15 2.5 1 -0.0177915 -0.0177915 1.14492e-16 10 16 2.5 1 0.0164239 0.0164239 3.46945e-18 10 17 2.5 1 -0.0143861 -0.0143861 4.85723e-17 10 18 2.5 1 0.0133683 0.0133683 7.63278e-17 10 19 2.5 1 -0.0118724 -0.0118724 5.20417e-17 10 20 2.5 1 0.0110937 0.0110941 3.4907e-07 10 21 2.5 1 -0.00996383 -0.00996428 4.48679e-07 10 22 2.5 1 0.0093537 0.00935557 1.86378e-06 1 0 2.5 2.5 0.981748 0.981748 1.11022e-16 1 1 2.5 2.5 0 0 0 1 2 2.5 2.5 0 0.122718 0.122718 1 3 2.5 2.5 0 0 0 1 4 2.5 2.5 0 0.0368155 0.0368155 2 0 2.5 2.5 0.981748 0.981748 1.11022e-16 2 1 2.5 2.5 0 0 0 2 2 2.5 2.5 0.122718 0.122718 1.249e-16 2 3 2.5 2.5 0 0 0 2 4 2.5 2.5 0.0153398 0.0368155 0.0214757 2 5 2.5 2.5 0 0 0 2 6 2.5 2.5 0.00191748 0.0153398 0.0134223 3 0 2.5 2.5 0.981748 0.981748 3.33067e-16 3 1 2.5 2.5 1.11022e-16 0 1.11022e-16 3 2 2.5 2.5 0.122718 0.122718 1.38778e-17 3 3 2.5 2.5 -6.93889e-18 0 6.93889e-18 3 4 2.5 2.5 0.0368155 0.0368155 6.93889e-18 3 5 2.5 2.5 -1.04083e-17 0 1.04083e-17 3 6 2.5 2.5 0.0110447 0.0153398 0.00429515 3 7 2.5 2.5 -6.50521e-18 0 6.50521e-18 3 8 2.5 2.5 0.0033134 0.0076699 0.00435651 4 0 2.5 2.5 0.981748 0.981748 1.11022e-16 4 1 2.5 2.5 -6.93889e-17 0 6.93889e-17 4 2 2.5 2.5 0.122718 0.122718 4.16334e-17 4 3 2.5 2.5 1.04083e-17 0 1.04083e-17 4 4 2.5 2.5 0.0368155 0.0368155 6.93889e-18 4 5 2.5 2.5 5.20417e-18 0 5.20417e-18 4 6 2.5 2.5 0.0153398 0.0153398 1.38778e-17 4 7 2.5 2.5 2.60209e-18 0 2.60209e-18 4 8 2.5 2.5 0.00674952 0.0076699 0.000920388 4 9 2.5 2.5 1.30104e-18 0 1.30104e-18 4 10 2.5 2.5 0.00299126 0.00431432 0.00132306 5 0 2.5 2.5 0.981748 0.981748 5.55112e-16 5 1 2.5 2.5 2.91434e-16 0 2.91434e-16 5 2 2.5 2.5 0.122718 0.122718 0 5 3 2.5 2.5 5.89806e-17 0 5.89806e-17 5 4 2.5 2.5 0.0368155 0.0368155 0 5 5 2.5 2.5 2.42861e-17 0 2.42861e-17 5 6 2.5 2.5 0.0153398 0.0153398 1.73472e-17 5 7 2.5 2.5 1.04083e-17 0 1.04083e-17 5 8 2.5 2.5 0.0076699 0.0076699 2.68882e-17 5 9 2.5 2.5 3.46945e-18 0 3.46945e-18 5 10 2.5 2.5 0.00410888 0.00431432 0.000205444 5 11 2.5 2.5 2.1684e-18 0 2.1684e-18 5 12 2.5 2.5 0.0022501 0.00263653 0.00038643 6 0 2.5 2.5 0.981748 0.981748 1.33227e-15 6 1 2.5 2.5 2.28983e-16 0 2.28983e-16 6 2 2.5 2.5 0.122718 0.122718 2.77556e-17 6 3 2.5 2.5 5.0307e-17 0 5.0307e-17 6 4 2.5 2.5 0.0368155 0.0368155 4.16334e-17 6 5 2.5 2.5 1.38778e-17 0 1.38778e-17 6 6 2.5 2.5 0.0153398 0.0153398 1.04083e-17 6 7 2.5 2.5 7.37257e-18 0 7.37257e-18 6 8 2.5 2.5 0.0076699 0.0076699 5.20417e-18 6 9 2.5 2.5 6.07153e-18 0 6.07153e-18 6 10 2.5 2.5 0.00431432 0.00431432 8.67362e-18 6 11 2.5 2.5 4.77049e-18 0 4.77049e-18 6 12 2.5 2.5 0.00258945 0.00263653 4.70809e-05 6 13 2.5 2.5 3.25261e-18 0 3.25261e-18 6 14 2.5 2.5 0.00160369 0.00171374 0.000110052 7 0 2.5 2.5 0.981748 0.981748 3.33067e-16 7 1 2.5 2.5 -1.21431e-17 0 1.21431e-17 7 2 2.5 2.5 0.122718 0.122718 0 7 3 2.5 2.5 -1.47451e-17 0 1.47451e-17 7 4 2.5 2.5 0.0368155 0.0368155 0 7 5 2.5 2.5 -3.46945e-18 0 3.46945e-18 7 6 2.5 2.5 0.0153398 0.0153398 1.04083e-17 7 7 2.5 2.5 -8.67362e-19 0 8.67362e-19 7 8 2.5 2.5 0.0076699 0.0076699 2.08167e-17 7 9 2.5 2.5 1.0842e-18 0 1.0842e-18 7 10 2.5 2.5 0.00431432 0.00431432 1.64799e-17 7 11 2.5 2.5 6.50521e-19 0 6.50521e-19 7 12 2.5 2.5 0.00263653 0.00263653 8.67362e-19 7 13 2.5 2.5 4.33681e-19 0 4.33681e-19 7 14 2.5 2.5 0.00170276 0.00171374 1.09855e-05 7 15 2.5 2.5 4.33681e-19 0 4.33681e-19 7 16 2.5 2.5 0.00113767 0.00116846 3.07928e-05 8 0 2.5 2.5 0.981748 0.981748 9.99201e-16 8 1 2.5 2.5 1.96891e-16 0 1.96891e-16 8 2 2.5 2.5 0.122718 0.122718 9.71445e-17 8 3 2.5 2.5 8.0231e-17 0 8.0231e-17 8 4 2.5 2.5 0.0368155 0.0368155 2.08167e-17 8 5 2.5 2.5 3.98986e-17 0 3.98986e-17 8 6 2.5 2.5 0.0153398 0.0153398 1.73472e-17 8 7 2.5 2.5 2.77556e-17 0 2.77556e-17 8 8 2.5 2.5 0.0076699 0.0076699 2.42861e-17 8 9 2.5 2.5 1.92988e-17 0 1.92988e-17 8 10 2.5 2.5 0.00431432 0.00431432 2.08167e-17 8 11 2.5 2.5 1.45283e-17 0 1.45283e-17 8 12 2.5 2.5 0.00263653 0.00263653 4.77049e-18 8 13 2.5 2.5 1.11673e-17 0 1.11673e-17 8 14 2.5 2.5 0.00171374 0.00171374 1.0842e-18 8 15 2.5 2.5 8.5652e-18 0 8.5652e-18 8 16 2.5 2.5 0.00116587 0.00116846 2.59658e-06 8 17 2.5 2.5 6.83047e-18 0 6.83047e-18 8 18 2.5 2.5 0.000819157 0.000827661 8.50381e-06 9 0 2.5 2.5 0.981748 0.981748 2.22045e-16 9 1 2.5 2.5 -4.81386e-17 0 4.81386e-17 9 2 2.5 2.5 0.122718 0.122718 1.38778e-17 9 3 2.5 2.5 -1.73472e-17 0 1.73472e-17 9 4 2.5 2.5 0.0368155 0.0368155 3.46945e-17 9 5 2.5 2.5 -1.77809e-17 0 1.77809e-17 9 6 2.5 2.5 0.0153398 0.0153398 4.68375e-17 9 7 2.5 2.5 -1.60462e-17 0 1.60462e-17 9 8 2.5 2.5 0.0076699 0.0076699 4.59702e-17 9 9 2.5 2.5 -1.36609e-17 0 1.36609e-17 9 10 2.5 2.5 0.00431432 0.00431432 3.64292e-17 9 11 2.5 2.5 -1.05168e-17 0 1.05168e-17 9 12 2.5 2.5 0.00263653 0.00263653 1.51788e-17 9 13 2.5 2.5 -8.13152e-18 0 8.13152e-18 9 14 2.5 2.5 0.00171374 0.00171374 5.85469e-18 9 15 2.5 2.5 -6.07153e-18 0 6.07153e-18 9 16 2.5 2.5 0.00116846 0.00116846 4.33681e-19 9 17 2.5 2.5 -4.66207e-18 0 4.66207e-18 9 18 2.5 2.5 0.000827041 0.000827661 6.19639e-07 9 19 2.5 2.5 -3.52366e-18 0 3.52366e-18 9 20 2.5 2.5 0.000602504 0.000604829 2.32473e-06 10 0 2.5 2.5 0.981748 0.981748 1.11022e-16 10 1 2.5 2.5 1.47451e-17 0 1.47451e-17 10 2 2.5 2.5 0.122718 0.122718 1.38778e-17 10 3 2.5 2.5 1.60462e-17 0 1.60462e-17 10 4 2.5 2.5 0.0368155 0.0368155 2.08167e-17 10 5 2.5 2.5 -1.73472e-18 0 1.73472e-18 10 6 2.5 2.5 0.0153398 0.0153398 0 10 7 2.5 2.5 -1.51788e-18 0 1.51788e-18 10 8 2.5 2.5 0.0076699 0.0076699 1.56125e-17 10 9 2.5 2.5 3.25261e-19 0 3.25261e-19 10 10 2.5 2.5 0.00431432 0.00431432 1.64799e-17 10 11 2.5 2.5 8.67362e-19 0 8.67362e-19 10 12 2.5 2.5 0.00263653 0.00263653 2.60209e-18 10 13 2.5 2.5 1.30104e-18 0 1.30104e-18 10 14 2.5 2.5 0.00171374 0.00171374 2.81893e-18 10 15 2.5 2.5 1.19262e-18 0 1.19262e-18 10 16 2.5 2.5 0.00116846 0.00116846 4.98733e-18 10 17 2.5 2.5 1.19262e-18 0 1.19262e-18 10 18 2.5 2.5 0.000827661 0.000827661 2.81893e-18 10 19 2.5 2.5 9.48677e-19 0 9.48677e-19 10 20 2.5 2.5 0.00060468 0.000604829 1.48952e-07 10 21 2.5 2.5 7.86047e-19 0 7.86047e-19 10 22 2.5 2.5 0.000452991 0.000453622 6.30385e-07 TEST18 JACOBI_COMPUTE_NP computes a Gauss-Jacobi rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) (1-x)^alpha (1+x)^beta dx. Order = 1 ALPHA = 0.5 BETA = 0.5 0 0 1.570796326794897 Order = 2 ALPHA = 0.5 BETA = 0.5 0 -0.4999999999999999 0.7853981633974484 1 0.4999999999999999 0.7853981633974484 Order = 3 ALPHA = 0.5 BETA = 0.5 0 -0.7071067811865475 0.3926990816987245 1 6.591949208711867e-17 0.7853981633974486 2 0.7071067811865474 0.3926990816987239 Order = 4 ALPHA = 0.5 BETA = 0.5 0 -0.8090169943749475 0.2170787134227061 1 -0.3090169943749473 0.5683194499747424 2 0.3090169943749472 0.5683194499747432 3 0.8090169943749477 0.2170787134227062 Order = 5 ALPHA = 0.5 BETA = 0.5 0 -0.8660254037844389 0.1308996938995749 1 -0.4999999999999998 0.3926990816987244 2 5.952490290336006e-17 0.5235987755982987 3 0.4999999999999998 0.3926990816987242 4 0.8660254037844388 0.1308996938995747 Order = 6 ALPHA = 0.5 BETA = 0.5 0 -0.9009688679024188 0.0844886908915887 1 -0.6234898018587335 0.2743330560697781 2 -0.2225209339563142 0.4265764164360816 3 0.2225209339563143 0.4265764164360817 4 0.6234898018587332 0.2743330560697784 5 0.9009688679024188 0.08448869089158853 Order = 7 ALPHA = 0.5 BETA = 0.5 0 -0.9238795325112868 0.05750944903191328 1 -0.7071067811865476 0.1963495408493622 2 -0.3826834323650896 0.3351896326668111 3 7.901929723605659e-18 0.3926990816987249 4 0.3826834323650899 0.335189632666811 5 0.7071067811865475 0.1963495408493624 6 0.9238795325112863 0.0575094490319132 Order = 8 ALPHA = 0.5 BETA = 0.5 0 -0.9396926207859084 0.04083294770910714 1 -0.7660444431189782 0.144225600795673 2 -0.4999999999999999 0.2617993877991496 3 -0.1736481776669302 0.3385402270935193 4 0.1736481776669302 0.338540227093519 5 0.5 0.2617993877991501 6 0.7660444431189779 0.1442256007956725 7 0.9396926207859086 0.04083294770910712 Order = 9 ALPHA = 0.5 BETA = 0.5 0 -0.9510565162951536 0.02999954037160819 1 -0.8090169943749472 0.1085393567113534 2 -0.587785252292473 0.2056199086476264 3 -0.3090169943749472 0.2841597249873707 4 5.567534423109432e-17 0.3141592653589796 5 0.3090169943749471 0.2841597249873716 6 0.5877852522924728 0.2056199086476266 7 0.8090169943749472 0.1085393567113536 8 0.9510565162951536 0.02999954037160805 Order = 10 ALPHA = 0.5 BETA = 0.5 0 -0.9594929736144974 0.02266894250185894 1 -0.8412535328311809 0.08347854093418919 2 -0.6548607339452849 0.1631221774548168 3 -0.4154150130018863 0.2363135602034877 4 -0.1423148382732851 0.2798149423030964 5 0.1423148382732851 0.2798149423030961 6 0.4154150130018863 0.2363135602034874 7 0.6548607339452848 0.1631221774548172 8 0.8412535328311812 0.08347854093418883 9 0.9594929736144973 0.02266894250185892 Order = 1 ALPHA = 0.5 BETA = 1 0 0.1428571428571428 1.508494466531301 Order = 2 ALPHA = 0.5 BETA = 1 0 -0.3785434359039294 0.6707847483124826 1 0.5603616177221111 0.8377097182188189 Order = 3 ALPHA = 0.5 BETA = 1 0 -0.6191289006136985 0.2886624303931599 1 0.08283748568481 0.7711842727220678 2 0.7362914149288884 0.4486477634160737 Order = 4 ALPHA = 0.5 BETA = 1 0 -0.744717344008601 0.1380862915234591 1 -0.2308286409011867 0.5011891464890383 2 0.3610180502309863 0.6101454555318188 3 0.8250542504682752 0.2590735729869861 Order = 5 ALPHA = 0.5 BETA = 1 0 -0.8175993274098119 0.07295559350758933 1 -0.4326084003843105 0.3103351805927456 2 0.05829076979846798 0.5176668424406614 3 0.5335962093870672 0.4466360854739412 4 0.8757120529564133 0.1609007645163651 Order = 6 ALPHA = 0.5 BETA = 1 0 -0.8633993972645073 0.04177873920985499 1 -0.5667404500118454 0.1951823756778568 2 -0.1659246338882321 0.3864855437976809 3 0.264914101715074 0.4544246037858506 4 0.646127253853395 0.3245589588523247 5 0.9072453478183379 0.1060642452077325 Order = 7 ALPHA = 0.5 BETA = 1 0 -0.8939724663908035 0.02551501467247916 1 -0.6594047339183104 0.1266052232513152 2 -0.330720748514499 0.2785906683424275 3 0.04495789586532556 0.3894994287201929 4 0.413803698784373 0.3759820326953337 5 0.7229723820004327 0.2389626799471576 6 0.928170423786385 0.07333941890239537 Order = 8 ALPHA = 0.5 BETA = 1 0 -0.9153619779765985 0.0164120900902138 1 -0.7257160980084201 0.08485547335632405 2 -0.4533780733947833 0.2002825508076703 3 -0.1294870729403755 0.312349320032714 4 0.2089491097631813 0.3577748934099902 5 0.5232643292074005 0.3048404655795939 6 0.7775490329551161 0.1792709060668855 7 0.9427521789659081 0.05270876718791008 Order = 9 ALPHA = 0.5 BETA = 1 0 -0.930896636109472 0.01101439194199641 1 -0.7746386880679893 0.0586443046921294 2 -0.5463185334791455 0.145393304746393 3 -0.2672381326388558 0.2442199669465208 4 0.03658634644395028 0.3121710958691362 5 0.3368351070820351 0.3145887170547924 6 0.6055223420849569 0.2462380941383865 7 0.8176042531316312 0.1371247634867023 8 0.9533131723221203 0.03909982765524507 Order = 10 ALPHA = 0.5 BETA = 1 0 -0.9425275499488472 0.00765699873374967 1 -0.8116864125172604 0.04166302380858723 2 -0.6180513952640334 0.1070695285744134 3 -0.3766418843493625 0.1896495013540221 4 -0.1061646285278262 0.2614695484772913 5 0.172423767431727 0.2937514863341338 6 0.4375388507874067 0.2709981593343714 7 0.6686402341966643 0.1996303237647672 8 0.8478228797503714 0.1068280415109269 9 0.961204277976044 0.02977785463903795 Order = 1 ALPHA = 0.5 BETA = 2.5 0 0.4 1.963495408493621 Order = 2 ALPHA = 0.5 BETA = 2.5 0 -0.1055161125036901 0.6949608437695366 1 0.6769446839322615 1.268534564724084 Order = 3 ALPHA = 0.5 BETA = 2.5 0 -0.3949726057630552 0.2155740450619748 1 0.264766277445141 0.9816213720815014 2 0.7968729949845808 0.7662999913501441 Order = 4 ALPHA = 0.5 BETA = 2.5 0 -0.5673928432278891 0.07352704993487792 1 -0.04381297095741903 0.4997101315556031 2 0.4782610448380588 0.9082548004475428 3 0.8602174966199764 0.482003426555598 Order = 5 ALPHA = 0.5 BETA = 2.5 0 -0.6766083904771948 0.02810427105392544 1 -0.2615276527389885 0.2379108911216861 2 0.1979083288674991 0.6351035815608072 3 0.611609816996844 0.7438254266897733 4 0.8978486665826094 0.3185512380674297 Order = 6 ALPHA = 0.5 BETA = 2.5 0 -0.7496281758956908 0.01189334443853825 1 -0.4162703993964169 0.1152628351592501 2 -0.02409923314933786 0.3830143850747024 3 0.3678575869061303 0.6464461334770024 4 0.700084306553539 0.5868443910764994 5 0.9220559149817764 0.2200343192676292 Order = 7 ALPHA = 0.5 BETA = 2.5 0 -0.8006672147724485 0.005485816372248914 1 -0.528679239032316 0.05815886499830619 2 -0.1959098000069027 0.2221608406878784 3 0.1580287890648373 0.4638578133115197 4 0.4905660544916725 0.5963414735975121 5 0.7616344330284672 0.4597291058720924 6 0.9385563889913959 0.1577614936540629 Order = 8 ALPHA = 0.5 BETA = 2.5 0 -0.837664685678683 0.002719896845113349 1 -0.6123079224732046 0.03070567363123355 2 -0.3290736457267182 0.1289926195321413 3 -0.01526789516983224 0.3092909316152767 4 0.2984865079144489 0.4881539875384042 5 0.581509661195563 0.5250491653932975 6 0.8061111752639282 0.3618821131546536 7 0.950312067832393 0.1167010207835007 Order = 9 ALPHA = 0.5 BETA = 2.5 0 -0.8653022034444988 0.001433142315199593 1 -0.6759411199023829 0.01693724138624853 2 -0.4333019473215539 0.07616801324207163 3 -0.1566993909441266 0.2010801278076632 4 0.1315346053479577 0.3633349438080313 5 0.4080752758535235 0.4761564190243807 6 0.6505301580495418 0.4521871712839325 7 0.8392624591522133 0.2875734496942869 8 0.9589850203521831 0.08862489993180625 Order = 10 ALPHA = 0.5 BETA = 2.5 0 -0.8864732152715593 0.00079518424116648 1 -0.7253515462755796 0.009727000011821089 2 -0.5159038845260621 0.04601457728802532 3 -0.2721183637403177 0.130301269175584 4 -0.01054505175311357 0.2590099865557985 5 0.2510118405448967 0.3871376587183351 6 0.494736845160108 0.4448595186308594 7 0.7040250190649779 0.3857617289382688 8 0.8646161230141249 0.2310686902877489 9 0.965567451173829 0.06881979464601419 Order = 1 ALPHA = 1 BETA = 0.5 0 -0.1428571428571428 1.508494466531301 Order = 2 ALPHA = 1 BETA = 0.5 0 -0.5603616177221111 0.8377097182188189 1 0.3785434359039294 0.6707847483124826 Order = 3 ALPHA = 1 BETA = 0.5 0 -0.7362914149288884 0.4486477634160737 1 -0.08283748568481 0.7711842727220678 2 0.6191289006136985 0.2886624303931599 Order = 4 ALPHA = 1 BETA = 0.5 0 -0.8250542504682752 0.2590735729869861 1 -0.3610180502309863 0.6101454555318188 2 0.2308286409011867 0.5011891464890383 3 0.744717344008601 0.1380862915234591 Order = 5 ALPHA = 1 BETA = 0.5 0 -0.8757120529564133 0.1609007645163651 1 -0.5335962093870672 0.4466360854739412 2 -0.05829076979846798 0.5176668424406614 3 0.4326084003843105 0.3103351805927456 4 0.8175993274098119 0.07295559350758933 Order = 6 ALPHA = 1 BETA = 0.5 0 -0.9072453478183379 0.1060642452077325 1 -0.646127253853395 0.3245589588523247 2 -0.264914101715074 0.4544246037858506 3 0.1659246338882321 0.3864855437976809 4 0.5667404500118454 0.1951823756778568 5 0.8633993972645073 0.04177873920985499 Order = 7 ALPHA = 1 BETA = 0.5 0 -0.928170423786385 0.07333941890239537 1 -0.7229723820004327 0.2389626799471576 2 -0.413803698784373 0.3759820326953337 3 -0.04495789586532556 0.3894994287201929 4 0.330720748514499 0.2785906683424275 5 0.6594047339183104 0.1266052232513152 6 0.8939724663908035 0.02551501467247916 Order = 8 ALPHA = 1 BETA = 0.5 0 -0.9427521789659081 0.05270876718791008 1 -0.7775490329551161 0.1792709060668855 2 -0.5232643292074005 0.3048404655795939 3 -0.2089491097631813 0.3577748934099902 4 0.1294870729403755 0.312349320032714 5 0.4533780733947833 0.2002825508076703 6 0.7257160980084201 0.08485547335632405 7 0.9153619779765985 0.0164120900902138 Order = 9 ALPHA = 1 BETA = 0.5 0 -0.9533131723221203 0.03909982765524507 1 -0.8176042531316312 0.1371247634867023 2 -0.6055223420849569 0.2462380941383865 3 -0.3368351070820351 0.3145887170547924 4 -0.03658634644395028 0.3121710958691362 5 0.2672381326388558 0.2442199669465208 6 0.5463185334791455 0.145393304746393 7 0.7746386880679893 0.0586443046921294 8 0.930896636109472 0.01101439194199641 Order = 10 ALPHA = 1 BETA = 0.5 0 -0.961204277976044 0.02977785463903795 1 -0.8478228797503714 0.1068280415109269 2 -0.6686402341966643 0.1996303237647672 3 -0.4375388507874067 0.2709981593343714 4 -0.172423767431727 0.2937514863341338 5 0.1061646285278262 0.2614695484772913 6 0.3766418843493625 0.1896495013540221 7 0.6180513952640334 0.1070695285744134 8 0.8116864125172604 0.04166302380858723 9 0.9425275499488472 0.00765699873374967 Order = 1 ALPHA = 1 BETA = 1 0 0 1.333333333333333 Order = 2 ALPHA = 1 BETA = 1 0 -0.4472135954999578 0.6666666666666665 1 0.4472135954999578 0.6666666666666665 Order = 3 ALPHA = 1 BETA = 1 0 -0.6546536707079771 0.3111111111111109 1 6.982261990806649e-17 0.7111111111111111 2 0.654653670707977 0.3111111111111108 Order = 4 ALPHA = 1 BETA = 1 0 -0.7650553239294646 0.1569499125956939 1 -0.285231516480645 0.5097167540709726 2 0.2852315164806451 0.5097167540709729 3 0.7650553239294645 0.1569499125956942 Order = 5 ALPHA = 1 BETA = 1 0 -0.8302238962785671 0.08601768212280743 1 -0.4688487934707142 0.3368394607343353 2 1.630695671821295e-16 0.4876190476190478 3 0.4688487934707142 0.3368394607343351 4 0.8302238962785669 0.08601768212280747 Order = 6 ALPHA = 1 BETA = 1 0 -0.8717401485096067 0.05058357701608177 1 -0.5917001814331422 0.2216925320225175 2 -0.2092992179024789 0.3943905576280672 3 0.2092992179024789 0.3943905576280671 4 0.5917001814331422 0.2216925320225176 5 0.8717401485096065 0.05058357701608172 Order = 7 ALPHA = 1 BETA = 1 0 -0.8997579954114602 0.03151620015049866 1 -0.6771862795107376 0.1486404045834097 2 -0.3631174638261783 0.3007504247445495 3 2.454652867869912e-16 0.3715192743764173 4 0.3631174638261781 0.3007504247445493 5 0.6771862795107377 0.1486404045834098 6 0.8997579954114601 0.03151620015049871 Order = 8 ALPHA = 1 BETA = 1 0 -0.9195339081664586 0.02059009564912185 1 -0.7387738651055049 0.1021477023603584 2 -0.4779249498104446 0.2253365549698581 3 -0.165278957666387 0.3185923136873287 4 0.1652789576663869 0.3185923136873286 5 0.4779249498104445 0.2253365549698582 6 0.7387738651055049 0.1021477023603586 7 0.9195339081664586 0.02059009564912179 Order = 9 ALPHA = 1 BETA = 1 0 -0.9340014304080593 0.01399105612442672 1 -0.7844834736631444 0.07198285617566094 2 -0.5652353269962052 0.1687989736144601 3 -0.2957581355869395 0.2617849830242734 4 2.636198166955511e-16 0.3002175954556909 5 0.2957581355869395 0.2617849830242732 6 0.5652353269962052 0.1687989736144599 7 0.7844834736631442 0.07198285617566089 8 0.9340014304080595 0.01399105612442669 Order = 10 ALPHA = 1 BETA = 1 0 -0.9448992722228819 0.00982540480576815 1 -0.819279321644007 0.05193914377420777 2 -0.6328761530318606 0.1273919483477327 3 -0.399530940965349 0.2111657418760753 4 -0.1365529328549277 0.2663444278628823 5 0.1365529328549273 0.2663444278628825 6 0.3995309409653491 0.2111657418760754 7 0.6328761530318608 0.1273919483477327 8 0.8192793216440072 0.05193914377420812 9 0.9448992722228821 0.009825404805768231 Order = 1 ALPHA = 1 BETA = 2.5 0 0.2727272727272727 1.436661396696478 Order = 2 ALPHA = 1 BETA = 2.5 0 -0.1843075691322091 0.5823920991551214 1 0.584307569132209 0.8542692975413564 Order = 3 ALPHA = 1 BETA = 2.5 0 -0.4423027360564286 0.2004355575312489 1 0.1860296262211932 0.775695309158561 2 0.7299573203615513 0.4605305300066668 Order = 4 ALPHA = 1 BETA = 2.5 0 -0.5972505982938984 0.07353804976128515 1 -0.101125968523521 0.4428338893708899 2 0.4098198869416554 0.6621803747350364 3 0.8102958103105465 0.2581090828292663 Order = 5 ALPHA = 1 BETA = 2.5 0 -0.6964392762006608 0.02963015201149462 1 -0.3026155820854005 0.2285362779908424 2 0.1412598214162332 0.5280679973185715 3 0.5539710608669561 0.4975219906041121 4 0.8593795315584275 0.1529049787714574 Order = 6 ALPHA = 1 BETA = 2.5 0 -0.7633972293043557 0.0130439061802296 1 -0.446152924876019 0.1174255136957449 2 -0.06867842657056338 0.3490370430142557 3 0.3155805295665461 0.5014050892168358 4 0.6517131087948295 0.36034844895849 5 0.8915801036798848 0.09540139563092104 Order = 7 ALPHA = 1 BETA = 2.5 0 -0.8105874821854354 0.00620279036570164 1 -0.5508759997785407 0.06193824188895962 2 -0.2306894365569856 0.2164618775946896 3 0.1138876346036921 0.3988213674613816 4 0.443600343147574 0.4310114591490306 5 0.7208146570340379 0.2599688822120861 6 0.9138502837356578 0.06225677802462855 Order = 8 ALPHA = 1 BETA = 2.5 0 -0.845035277918601 0.003150650566260482 1 -0.6291538133028906 0.03384819480025154 2 -0.3563555020358466 0.132215777052771 3 -0.05166907474429276 0.2867379491637258 4 0.2565958341049685 0.3952631184069375 5 0.5397363757841801 0.3540223921913581 6 0.7713710662668611 0.1892132923890619 7 0.9298950072302367 0.04221002212611247 Order = 9 ALPHA = 1 BETA = 2.5 0 -0.8709215211401555 0.001692975804662645 1 -0.6889839677544535 0.01919029629201514 2 -0.4549257546478145 0.08121761104756811 3 -0.1865538397099543 0.1974051443898488 4 0.09541224177708327 0.3197898022964819 5 0.3691513199896206 0.3631243013752991 6 0.6134647393951399 0.2850633375254082 7 0.8094259398841321 0.1396134790482666 8 0.9418378189505883 0.02956444891692782 Order = 10 ALPHA = 1 BETA = 2.5 0 -0.890851998362436 0.0009546834154139908 1 -0.7356332759809726 0.01126952303045689 2 -0.5332475568814603 0.05063788256866946 3 -0.2966592827516153 0.1337845805120262 4 -0.04128300871886588 0.2429565425267207 5 0.2161975251464398 0.3233164884771637 6 0.4589489303958385 0.3198849047578982 7 0.6710962948352331 0.2279739859839845 8 0.8387626487347724 0.1045909447918315 9 0.9509675959234919 0.02129186063231215 Order = 1 ALPHA = 2.5 BETA = 0.5 0 -0.4 1.963495408493621 Order = 2 ALPHA = 2.5 BETA = 0.5 0 -0.6769446839322615 1.268534564724084 1 0.1055161125036901 0.6949608437695366 Order = 3 ALPHA = 2.5 BETA = 0.5 0 -0.7968729949845808 0.7662999913501441 1 -0.264766277445141 0.9816213720815014 2 0.3949726057630552 0.2155740450619748 Order = 4 ALPHA = 2.5 BETA = 0.5 0 -0.8602174966199764 0.482003426555598 1 -0.4782610448380588 0.9082548004475428 2 0.04381297095741903 0.4997101315556031 3 0.5673928432278891 0.07352704993487792 Order = 5 ALPHA = 2.5 BETA = 0.5 0 -0.8978486665826094 0.3185512380674297 1 -0.611609816996844 0.7438254266897733 2 -0.1979083288674991 0.6351035815608072 3 0.2615276527389885 0.2379108911216861 4 0.6766083904771948 0.02810427105392544 Order = 6 ALPHA = 2.5 BETA = 0.5 0 -0.9220559149817764 0.2200343192676292 1 -0.700084306553539 0.5868443910764994 2 -0.3678575869061303 0.6464461334770024 3 0.02409923314933786 0.3830143850747024 4 0.4162703993964169 0.1152628351592501 5 0.7496281758956908 0.01189334443853825 Order = 7 ALPHA = 2.5 BETA = 0.5 0 -0.9385563889913959 0.1577614936540629 1 -0.7616344330284672 0.4597291058720924 2 -0.4905660544916725 0.5963414735975121 3 -0.1580287890648373 0.4638578133115197 4 0.1959098000069027 0.2221608406878784 5 0.528679239032316 0.05815886499830619 6 0.8006672147724485 0.005485816372248914 Order = 8 ALPHA = 2.5 BETA = 0.5 0 -0.950312067832393 0.1167010207835007 1 -0.8061111752639282 0.3618821131546536 2 -0.581509661195563 0.5250491653932975 3 -0.2984865079144489 0.4881539875384042 4 0.01526789516983224 0.3092909316152767 5 0.3290736457267182 0.1289926195321413 6 0.6123079224732046 0.03070567363123355 7 0.837664685678683 0.002719896845113349 Order = 9 ALPHA = 2.5 BETA = 0.5 0 -0.9589850203521831 0.08862489993180625 1 -0.8392624591522133 0.2875734496942869 2 -0.6505301580495418 0.4521871712839325 3 -0.4080752758535235 0.4761564190243807 4 -0.1315346053479577 0.3633349438080313 5 0.1566993909441266 0.2010801278076632 6 0.4333019473215539 0.07616801324207163 7 0.6759411199023829 0.01693724138624853 8 0.8653022034444988 0.001433142315199593 Order = 10 ALPHA = 2.5 BETA = 0.5 0 -0.965567451173829 0.06881979464601419 1 -0.8646161230141249 0.2310686902877489 2 -0.7040250190649779 0.3857617289382688 3 -0.494736845160108 0.4448595186308594 4 -0.2510118405448967 0.3871376587183351 5 0.01054505175311357 0.2590099865557985 6 0.2721183637403177 0.130301269175584 7 0.5159038845260621 0.04601457728802532 8 0.7253515462755796 0.009727000011821089 9 0.8864732152715593 0.00079518424116648 Order = 1 ALPHA = 2.5 BETA = 1 0 -0.2727272727272727 1.436661396696478 Order = 2 ALPHA = 2.5 BETA = 1 0 -0.584307569132209 0.8542692975413564 1 0.1843075691322091 0.5823920991551214 Order = 3 ALPHA = 2.5 BETA = 1 0 -0.7299573203615513 0.4605305300066668 1 -0.1860296262211932 0.775695309158561 2 0.4423027360564286 0.2004355575312489 Order = 4 ALPHA = 2.5 BETA = 1 0 -0.8102958103105465 0.2581090828292663 1 -0.4098198869416554 0.6621803747350364 2 0.101125968523521 0.4428338893708899 3 0.5972505982938984 0.07353804976128515 Order = 5 ALPHA = 2.5 BETA = 1 0 -0.8593795315584275 0.1529049787714574 1 -0.5539710608669561 0.4975219906041121 2 -0.1412598214162332 0.5280679973185715 3 0.3026155820854005 0.2285362779908424 4 0.6964392762006608 0.02963015201149462 Order = 6 ALPHA = 2.5 BETA = 1 0 -0.8915801036798848 0.09540139563092104 1 -0.6517131087948295 0.36034844895849 2 -0.3155805295665461 0.5014050892168358 3 0.06867842657056338 0.3490370430142557 4 0.446152924876019 0.1174255136957449 5 0.7633972293043557 0.0130439061802296 Order = 7 ALPHA = 2.5 BETA = 1 0 -0.9138502837356578 0.06225677802462855 1 -0.7208146570340379 0.2599688822120861 2 -0.443600343147574 0.4310114591490306 3 -0.1138876346036921 0.3988213674613816 4 0.2306894365569856 0.2164618775946896 5 0.5508759997785407 0.06193824188895962 6 0.8105874821854354 0.00620279036570164 Order = 8 ALPHA = 2.5 BETA = 1 0 -0.9298950072302367 0.04221002212611247 1 -0.7713710662668611 0.1892132923890619 2 -0.5397363757841801 0.3540223921913581 3 -0.2565958341049685 0.3952631184069375 4 0.05166907474429276 0.2867379491637258 5 0.3563555020358466 0.132215777052771 6 0.6291538133028906 0.03384819480025154 7 0.845035277918601 0.003150650566260482 Order = 9 ALPHA = 2.5 BETA = 1 0 -0.9418378189505883 0.02956444891692782 1 -0.8094259398841321 0.1396134790482666 2 -0.6134647393951399 0.2850633375254082 3 -0.3691513199896206 0.3631243013752991 4 -0.09541224177708327 0.3197898022964819 5 0.1865538397099543 0.1974051443898488 6 0.4549257546478145 0.08121761104756811 7 0.6889839677544535 0.01919029629201514 8 0.8709215211401555 0.001692975804662645 Order = 10 ALPHA = 2.5 BETA = 1 0 -0.9509675959234919 0.02129186063231215 1 -0.8387626487347724 0.1045909447918315 2 -0.6710962948352331 0.2279739859839845 3 -0.4589489303958385 0.3198849047578982 4 -0.2161975251464398 0.3233164884771637 5 0.04128300871886588 0.2429565425267207 6 0.2966592827516153 0.1337845805120262 7 0.5332475568814603 0.05063788256866946 8 0.7356332759809726 0.01126952303045689 9 0.890851998362436 0.0009546834154139908 Order = 1 ALPHA = 2.5 BETA = 2.5 0 0 0.9817477042468103 Order = 2 ALPHA = 2.5 BETA = 2.5 0 -0.3535533905932737 0.4908738521234051 1 0.3535533905932737 0.4908738521234051 Order = 3 ALPHA = 2.5 BETA = 2.5 0 -0.5477225575051663 0.2045307717180854 1 4.336808689942018e-17 0.5726861608106395 2 0.5477225575051661 0.2045307717180856 Order = 4 ALPHA = 2.5 BETA = 2.5 0 -0.6660699417556469 0.08700807153148338 1 -0.237383303308445 0.4038657805919218 2 0.237383303308445 0.4038657805919215 3 0.6660699417556469 0.08700807153148342 Order = 5 ALPHA = 2.5 BETA = 2.5 0 -0.7434770045212192 0.03928938913176724 1 -0.40190503608921 0.2454174450998076 2 1.245242164985443e-16 0.4123340357836606 3 0.4019050360892101 0.245417445099808 4 0.7434770045212191 0.03928938913176733 Order = 6 ALPHA = 2.5 BETA = 2.5 0 -0.7968373566406803 0.01891164336669309 1 -0.5196148456949135 0.1431407318203966 2 -0.1804179569647765 0.3288214769363159 3 0.1804179569647765 0.3288214769363157 4 0.5196148456949131 0.1431407318203972 5 0.7968373566406806 0.01891164336669309 Order = 7 ALPHA = 2.5 BETA = 2.5 0 -0.8351564541339109 0.009658525021001006 1 -0.6063141605450342 0.08344467054898488 2 -0.3186902924591671 0.2357822853526958 3 5.104982774491088e-17 0.3239767424014474 4 0.3186902924591672 0.2357822853526957 5 0.6063141605450342 0.08344467054898476 6 0.8351564541339108 0.009658525021001023 Order = 8 ALPHA = 2.5 BETA = 2.5 0 -0.8635912403827581 0.005199707422132458 1 -0.6718232507247469 0.04941818815652228 2 -0.4260961495798481 0.1615070338519728 3 -0.1459649294626306 0.2747489226927776 4 0.1459649294626304 0.2747489226927781 5 0.426096149579848 0.161507033851973 6 0.6718232507247468 0.04941818815652241 7 0.8635912403827586 0.0051997074221325 Order = 9 ALPHA = 2.5 BETA = 2.5 0 -0.8852662998295733 0.002931900984992426 1 -0.7224309555097358 0.02992095719368029 2 -0.5107538326640052 0.1088013928543096 3 -0.2643695362367074 0.2155149137501429 4 -1.634301058359553e-17 0.26740937468056 5 0.2643695362367075 0.2155149137501428 6 0.5107538326640052 0.1088013928543095 7 0.7224309555097355 0.02992095719368041 8 0.8852662998295734 0.002931900984992388 Order = 10 ALPHA = 2.5 BETA = 2.5 0 -0.9021636877146701 0.001721638228893138 1 -0.7622868838395971 0.01855643700599038 2 -0.5784393987065429 0.07316132839714104 3 -0.3610608688185056 0.1622660968500983 4 -0.1227285554678421 0.2351683516412827 5 0.122728555467842 0.2351683516412817 6 0.3610608688185061 0.1622660968500987 7 0.5784393987065428 0.07316132839714092 8 0.7622868838395969 0.01855643700599038 9 0.9021636877146699 0.001721638228893156 TEST19 LAGUERRE_COMPUTE_NP computes a Gauss-Laguerre rule which is appropriate for integrands of the form Integral ( 0 <= x < +oo ) f(x) exp(-x) dx. LAGUERRE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 1 1 0 1 1 1 1 0 1 2 1 2 1 1 3 1 6 5 1 4 1 24 23 2 0 1 1 0 2 1 1 1 0 2 2 2 2 0 2 3 6 6 0 2 4 20 24 4 2 5 68 120 52 2 6 232 720 488 3 0 1 1 0 3 1 1 1 0 3 2 2 2 4.44089e-16 3 3 6 6 3.55271e-15 3 4 24 24 2.84217e-14 3 5 120 120 2.13163e-13 3 6 684 720 36 3 7 4140 5040 900 3 8 25668 40320 14652 4 0 1 1 4.44089e-16 4 1 1 1 4.44089e-16 4 2 2 2 8.88178e-16 4 3 6 6 1.77636e-15 4 4 24 24 1.06581e-14 4 5 120 120 8.52651e-14 4 6 720 720 9.09495e-13 4 7 5040 5040 1.09139e-11 4 8 39744 40320 576 4 9 339264 362880 23616 4 10 3.03322e+06 3.6288e+06 595584 5 0 1 1 2.22045e-16 5 1 1 1 0 5 2 2 2 1.11022e-15 5 3 6 6 6.21725e-15 5 4 24 24 3.55271e-14 5 5 120 120 2.27374e-13 5 6 720 720 1.36424e-12 5 7 5040 5040 1.00044e-11 5 8 40320 40320 7.27596e-11 5 9 362880 362880 5.82077e-10 5 10 3.6144e+06 3.6288e+06 14400 5 11 3.90384e+07 3.99168e+07 878400 5 12 4.47221e+08 4.79002e+08 3.17808e+07 6 0 1 1 2.22045e-16 6 1 1 1 4.44089e-16 6 2 2 2 2.22045e-15 6 3 6 6 7.99361e-15 6 4 24 24 3.90799e-14 6 5 120 120 1.7053e-13 6 6 720 720 9.09495e-13 6 7 5040 5040 4.54747e-12 6 8 40320 40320 0 6 9 362880 362880 1.74623e-10 6 10 3.6288e+06 3.6288e+06 3.72529e-09 6 11 3.99168e+07 3.99168e+07 5.21541e-08 6 12 4.78483e+08 4.79002e+08 518400 6 13 6.18296e+09 6.22702e+09 4.4064e+07 6 14 8.50124e+10 8.71783e+10 2.16588e+09 7 0 1 1 2.22045e-16 7 1 1 1 2.22045e-16 7 2 2 2 8.88178e-16 7 3 6 6 8.88178e-16 7 4 24 24 7.10543e-15 7 5 120 120 1.42109e-14 7 6 720 720 2.27374e-13 7 7 5040 5040 1.81899e-12 7 8 40320 40320 1.45519e-11 7 9 362880 362880 2.32831e-10 7 10 3.6288e+06 3.6288e+06 3.25963e-09 7 11 3.99168e+07 3.99168e+07 4.47035e-08 7 12 4.79002e+08 4.79002e+08 6.55651e-07 7 13 6.22702e+09 6.22702e+09 9.53674e-06 7 14 8.71529e+10 8.71783e+10 2.54016e+07 7 15 1.3048e+12 1.30767e+12 2.87038e+09 7 16 2.07387e+13 2.09228e+13 1.84085e+11 8 0 1 1 1.11022e-16 8 1 1 1 3.33067e-16 8 2 2 2 8.88178e-16 8 3 6 6 4.44089e-15 8 4 24 24 2.84217e-14 8 5 120 120 1.42109e-13 8 6 720 720 6.82121e-13 8 7 5040 5040 3.63798e-12 8 8 40320 40320 2.91038e-11 8 9 362880 362880 3.49246e-10 8 10 3.6288e+06 3.6288e+06 4.65661e-09 8 11 3.99168e+07 3.99168e+07 6.70552e-08 8 12 4.79002e+08 4.79002e+08 8.9407e-07 8 13 6.22702e+09 6.22702e+09 1.43051e-05 8 14 8.71783e+10 8.71783e+10 0.000213623 8 15 1.30767e+12 1.30767e+12 0.00341797 8 16 2.09212e+13 2.09228e+13 1.6257e+09 8 17 3.55452e+14 3.55687e+14 2.35727e+11 8 18 6.38325e+15 6.40237e+15 1.91215e+13 9 0 1 1 8.88178e-16 9 1 1 1 6.66134e-16 9 2 2 2 1.77636e-15 9 3 6 6 6.21725e-15 9 4 24 24 1.77636e-14 9 5 120 120 8.52651e-14 9 6 720 720 1.13687e-13 9 7 5040 5040 9.09495e-13 9 8 40320 40320 0 9 9 362880 362880 0 9 10 3.6288e+06 3.6288e+06 0 9 11 3.99168e+07 3.99168e+07 0 9 12 4.79002e+08 4.79002e+08 1.19209e-07 9 13 6.22702e+09 6.22702e+09 0 9 14 8.71783e+10 8.71783e+10 0 9 15 1.30767e+12 1.30767e+12 0.000732422 9 16 2.09228e+13 2.09228e+13 0.015625 9 17 3.55687e+14 3.55687e+14 0.4375 9 18 6.40224e+15 6.40237e+15 1.31682e+11 9 19 1.21621e+17 1.21645e+17 2.38344e+13 9 20 2.43052e+18 2.4329e+18 2.38594e+15 10 0 1 1 6.66134e-16 10 1 1 1 6.66134e-16 10 2 2 2 2.22045e-15 10 3 6 6 8.88178e-15 10 4 24 24 3.19744e-14 10 5 120 120 1.27898e-13 10 6 720 720 4.54747e-13 10 7 5040 5040 2.72848e-12 10 8 40320 40320 7.27596e-12 10 9 362880 362880 1.16415e-10 10 10 3.6288e+06 3.6288e+06 2.79397e-09 10 11 3.99168e+07 3.99168e+07 5.21541e-08 10 12 4.79002e+08 4.79002e+08 8.9407e-07 10 13 6.22702e+09 6.22702e+09 1.62125e-05 10 14 8.71783e+10 8.71783e+10 0.000259399 10 15 1.30767e+12 1.30767e+12 0.00439453 10 16 2.09228e+13 2.09228e+13 0.0742188 10 17 3.55687e+14 3.55687e+14 1.0625 10 18 6.40237e+15 6.40237e+15 18 10 19 1.21645e+17 1.21645e+17 256 10 20 2.43289e+18 2.4329e+18 1.31682e+13 10 21 5.1088e+19 5.10909e+19 2.91017e+15 10 22 1.12365e+21 1.124e+21 3.52407e+17 TEST20 LAGUERRE_COMPUTE_NP computes a generalized Gauss-Laguerre rule which is appropriate for integrands of the form Integral ( 0 <= x < +oo ) f(x) exp(-x) dx. Order = 1 0 1 1 Order = 2 0 0.5857864376269051 0.8535533905932737 1 3.414213562373095 0.1464466094067262 Order = 3 0 0.4157745567834789 0.7110930099291729 1 2.294280360279041 0.2785177335692409 2 6.28994508293748 0.01038925650158615 Order = 4 0 0.3225476896193923 0.6031541043416337 1 1.745761101158346 0.3574186924377999 2 4.536620296921128 0.03888790851500538 3 9.395070912301136 0.0005392947055613278 Order = 5 0 0.263560319718141 0.5217556105828089 1 1.413403059106517 0.398666811083176 2 3.596425771040721 0.07594244968170749 3 7.085810005858835 0.00361175867992205 4 12.64080084427578 2.336997238577621e-05 Order = 6 0 0.222846604179261 0.4589646739499636 1 1.188932101672623 0.4170008307721207 2 2.992736326059314 0.1133733820740452 3 5.775143569104512 0.01039919745314908 4 9.837467418382587 0.0002610172028149323 5 15.9828739806017 8.985479064296196e-07 Order = 7 0 0.1930436765603621 0.4093189517012737 1 1.026664895339191 0.4218312778617199 2 2.567876744950745 0.1471263486575055 3 4.900353084526484 0.02063351446871694 4 8.182153444562859 0.001074010143280748 5 12.73418029179781 1.586546434856422e-05 6 19.39572786226254 3.170315478995581e-08 Order = 8 0 0.1702796323051008 0.3691885893416376 1 0.9037017767993794 0.4187867808143426 2 2.251086629866129 0.1757949866371721 3 4.266700170287657 0.03334349226121563 4 7.045905402393464 0.002794536235225677 5 10.75851601018099 9.076508773358235e-05 6 15.740678641278 8.485746716272511e-07 7 22.86313173688927 1.048001174871507e-09 Order = 9 0 0.152322227731808 0.3361264217979629 1 0.8072200227422562 0.4112139804239849 2 2.00513515561935 0.1992875253708856 3 3.783473973331235 0.04746056276565148 4 6.204956777876611 0.005599626610794589 5 9.372985251687574 0.000305249767093211 6 13.46623691109209 6.592123026075368e-06 7 18.83359778899169 4.110769330349564e-08 8 26.37407189092738 3.290874030350713e-11 Order = 10 0 0.1377934705404923 0.3084411157650204 1 0.7294545495031706 0.4011199291552736 2 1.808342901740316 0.2180682876118095 3 3.401433697854898 0.06208745609867788 4 5.552496140063801 0.009501516975181156 5 8.330152746764496 0.0007530083885875383 6 11.84378583790007 2.825923349599563e-05 7 16.27925783137811 4.249313984962688e-07 8 21.99658581198076 1.839564823979634e-09 9 29.92069701227389 9.911827219609e-13 TEST21 LEGENDRE_COMPUTE_NP computes a Gauss-Legendre rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. LEGENDRE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = 2*ORDER-1 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 2 2 4.44089e-16 1 1 0 0 0 1 2 0 0.666667 0.666667 1 3 0 0 0 1 4 0 0.4 0.4 2 0 2 2 0 2 1 0 0 0 2 2 0.666667 0.666667 3.33067e-16 2 3 0 0 0 2 4 0.222222 0.4 0.177778 2 5 0 0 0 2 6 0.0740741 0.285714 0.21164 3 0 2 2 4.44089e-16 3 1 -2.22045e-16 0 2.22045e-16 3 2 0.666667 0.666667 3.33067e-16 3 3 -2.22045e-16 0 2.22045e-16 3 4 0.4 0.4 4.44089e-16 3 5 -1.38778e-16 0 1.38778e-16 3 6 0.24 0.285714 0.0457143 3 7 -1.52656e-16 0 1.52656e-16 3 8 0.144 0.222222 0.0782222 4 0 2 2 1.33227e-15 4 1 -7.21645e-16 0 7.21645e-16 4 2 0.666667 0.666667 8.88178e-16 4 3 -4.71845e-16 0 4.71845e-16 4 4 0.4 0.4 6.66134e-16 4 5 -3.88578e-16 0 3.88578e-16 4 6 0.285714 0.285714 6.10623e-16 4 7 -3.05311e-16 0 3.05311e-16 4 8 0.210612 0.222222 0.01161 4 9 -2.63678e-16 0 2.63678e-16 4 10 0.156035 0.181818 0.0257832 5 0 2 2 1.33227e-15 5 1 2.77556e-17 0 2.77556e-17 5 2 0.666667 0.666667 3.33067e-16 5 3 -1.38778e-16 0 1.38778e-16 5 4 0.4 0.4 1.11022e-16 5 5 -2.22045e-16 0 2.22045e-16 5 6 0.285714 0.285714 5.55112e-17 5 7 -2.498e-16 0 2.498e-16 5 8 0.222222 0.222222 5.55112e-17 5 9 -2.498e-16 0 2.498e-16 5 10 0.178886 0.181818 0.00293181 5 11 -2.63678e-16 0 2.63678e-16 5 12 0.145853 0.153846 0.00799357 6 0 2 2 6.66134e-16 6 1 3.05311e-16 0 3.05311e-16 6 2 0.666667 0.666667 5.55112e-16 6 3 1.94289e-16 0 1.94289e-16 6 4 0.4 0.4 5.55112e-16 6 5 1.80411e-16 0 1.80411e-16 6 6 0.285714 0.285714 6.10623e-16 6 7 1.94289e-16 0 1.94289e-16 6 8 0.222222 0.222222 5.55112e-16 6 9 1.66533e-16 0 1.66533e-16 6 10 0.181818 0.181818 5.27356e-16 6 11 1.52656e-16 0 1.52656e-16 6 12 0.153108 0.153846 0.000738079 6 13 1.249e-16 0 1.249e-16 6 14 0.130949 0.133333 0.00238422 7 0 2 2 1.33227e-15 7 1 9.15934e-16 0 9.15934e-16 7 2 0.666667 0.666667 4.44089e-16 7 3 4.30211e-16 0 4.30211e-16 7 4 0.4 0.4 0 7 5 2.08167e-16 0 2.08167e-16 7 6 0.285714 0.285714 5.55112e-17 7 7 9.71445e-17 0 9.71445e-17 7 8 0.222222 0.222222 1.11022e-16 7 9 4.16334e-17 0 4.16334e-17 7 10 0.181818 0.181818 1.66533e-16 7 11 -2.77556e-17 0 2.77556e-17 7 12 0.153846 0.153846 1.94289e-16 7 13 -6.93889e-17 0 6.93889e-17 7 14 0.133148 0.133333 0.000185466 7 15 -6.245e-17 0 6.245e-17 7 16 0.116955 0.117647 0.00069235 8 0 2 2 1.11022e-15 8 1 -4.996e-16 0 4.996e-16 8 2 0.666667 0.666667 5.55112e-16 8 3 -2.498e-16 0 2.498e-16 8 4 0.4 0.4 4.44089e-16 8 5 -2.63678e-16 0 2.63678e-16 8 6 0.285714 0.285714 2.77556e-16 8 7 -2.35922e-16 0 2.35922e-16 8 8 0.222222 0.222222 1.38778e-16 8 9 -2.35922e-16 0 2.35922e-16 8 10 0.181818 0.181818 5.55112e-17 8 11 -2.22045e-16 0 2.22045e-16 8 12 0.153846 0.153846 2.77556e-17 8 13 -2.01228e-16 0 2.01228e-16 8 14 0.133333 0.133333 8.32667e-17 8 15 -1.94289e-16 0 1.94289e-16 8 16 0.117601 0.117647 4.65483e-05 8 17 -1.80411e-16 0 1.80411e-16 8 18 0.105066 0.105263 0.000197136 9 0 2 2 2.22045e-15 9 1 3.33067e-16 0 3.33067e-16 9 2 0.666667 0.666667 1.22125e-15 9 3 2.63678e-16 0 2.63678e-16 9 4 0.4 0.4 7.21645e-16 9 5 -1.38778e-17 0 1.38778e-17 9 6 0.285714 0.285714 4.44089e-16 9 7 -1.52656e-16 0 1.52656e-16 9 8 0.222222 0.222222 1.66533e-16 9 9 -2.28983e-16 0 2.28983e-16 9 10 0.181818 0.181818 5.55112e-17 9 11 -2.91434e-16 0 2.91434e-16 9 12 0.153846 0.153846 8.32667e-17 9 13 -3.05311e-16 0 3.05311e-16 9 14 0.133333 0.133333 1.38778e-16 9 15 -3.1225e-16 0 3.1225e-16 9 16 0.117647 0.117647 2.08167e-16 9 17 -3.1225e-16 0 3.1225e-16 9 18 0.105251 0.105263 1.16731e-05 9 19 -3.1225e-16 0 3.1225e-16 9 20 0.0951828 0.0952381 5.52919e-05 10 0 2 2 1.33227e-15 10 1 2.91434e-16 0 2.91434e-16 10 2 0.666667 0.666667 2.22045e-16 10 3 5.13478e-16 0 5.13478e-16 10 4 0.4 0.4 2.77556e-16 10 5 4.09395e-16 0 4.09395e-16 10 6 0.285714 0.285714 2.77556e-16 10 7 3.05311e-16 0 3.05311e-16 10 8 0.222222 0.222222 2.22045e-16 10 9 2.08167e-16 0 2.08167e-16 10 10 0.181818 0.181818 1.66533e-16 10 11 1.31839e-16 0 1.31839e-16 10 12 0.153846 0.153846 1.11022e-16 10 13 6.245e-17 0 6.245e-17 10 14 0.133333 0.133333 8.32667e-17 10 15 5.55112e-17 0 5.55112e-17 10 16 0.117647 0.117647 2.77556e-17 10 17 0 0 0 10 18 0.105263 0.105263 0 10 19 -1.38778e-17 0 1.38778e-17 10 20 0.0952352 0.0952381 2.92559e-06 10 21 -3.46945e-17 0 3.46945e-17 10 22 0.0869412 0.0869565 1.53242e-05 TEST22 LEGENDRE_COMPUTE_NP computes a Gauss-Legendre rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. Order = 1 0 0 2 Order = 2 0 -0.5773502691896256 1 1 0.5773502691896256 1 Order = 3 0 -0.7745966692414833 0.5555555555555556 1 1.994931997373328e-17 0.8888888888888895 2 0.7745966692414832 0.5555555555555554 Order = 4 0 -0.8611363115940527 0.3478548451374547 1 -0.3399810435848563 0.6521451548625466 2 0.3399810435848563 0.6521451548625458 3 0.8611363115940526 0.3478548451374541 Order = 5 0 -0.9061798459386641 0.2369268850561892 1 -0.538469310105683 0.4786286704993669 2 -1.081853856991421e-16 0.568888888888889 3 0.5384693101056831 0.4786286704993672 4 0.9061798459386639 0.2369268850561891 Order = 6 0 -0.9324695142031522 0.1713244923791705 1 -0.6612093864662647 0.3607615730481384 2 -0.238619186083197 0.4679139345726904 3 0.2386191860831969 0.467913934572691 4 0.6612093864662647 0.3607615730481382 5 0.9324695142031522 0.1713244923791708 Order = 7 0 -0.9491079123427585 0.1294849661688697 1 -0.7415311855993943 0.2797053914892765 2 -0.4058451513773971 0.3818300505051193 3 2.944352847269754e-16 0.4179591836734696 4 0.4058451513773971 0.3818300505051192 5 0.7415311855993943 0.2797053914892776 6 0.9491079123427584 0.1294849661688697 Order = 8 0 -0.9602898564975365 0.101228536290376 1 -0.796666477413627 0.2223810344533743 2 -0.525532409916329 0.3137066458778873 3 -0.1834346424956498 0.3626837833783622 4 0.1834346424956496 0.3626837833783619 5 0.5255324099163292 0.3137066458778869 6 0.7966664774136268 0.2223810344533742 7 0.9602898564975364 0.1012285362903759 Order = 9 0 -0.968160239507626 0.08127438836157462 1 -0.836031107326636 0.1806481606948576 2 -0.61337143270059 0.2606106964029357 3 -0.3242534234038094 0.3123470770400033 4 -3.64953385043433e-16 0.3302393550012602 5 0.3242534234038092 0.3123470770400024 6 0.6133714327005907 0.2606106964029365 7 0.8360311073266359 0.1806481606948582 8 0.9681602395076259 0.08127438836157423 Order = 10 0 -0.9739065285171715 0.06667134430868846 1 -0.8650633666889844 0.14945134915058 2 -0.6794095682990244 0.2190863625159823 3 -0.4333953941292472 0.2692667193099961 4 -0.1488743389816311 0.2955242247147523 5 0.1488743389816314 0.2955242247147531 6 0.4333953941292472 0.2692667193099947 7 0.6794095682990243 0.2190863625159819 8 0.8650633666889843 0.1494513491505811 9 0.9739065285171714 0.06667134430868851 TEST23 LEVEL_GROWTH_TO_ORDER uses Level, Growth and Rule to determine the orders of each entry of a vector of 1D rules. Here we examine rule 1. LEVEL:0 1 2 3 4 5 6 7 8 9 10 GROWTH 0 1 5 9 17 17 33 33 33 33 65 65 1 1 2 3 4 5 6 7 8 9 10 11 2 1 3 3 5 5 7 7 9 9 11 11 3 1 3 5 7 9 11 13 15 17 19 21 4 1 3 5 9 9 17 17 17 17 33 33 5 1 5 9 17 17 33 33 33 33 65 65 6 1 3 5 9 17 33 65 129 257 513 1025 TEST23 LEVEL_GROWTH_TO_ORDER uses Level, Growth and Rule to determine the orders of each entry of a vector of 1D rules. Here we examine rule 3. LEVEL:0 1 2 3 4 5 6 7 8 9 10 GROWTH 0 1 3 7 15 15 15 31 31 31 31 31 4 1 3 3 7 7 7 15 15 15 15 15 5 1 3 7 15 15 15 31 31 31 31 31 6 1 3 7 15 31 63 127 255 511 1023 2047 TEST23 LEVEL_GROWTH_TO_ORDER uses Level, Growth and Rule to determine the orders of each entry of a vector of 1D rules. Here we examine rule 4. LEVEL:0 1 2 3 4 5 6 7 8 9 10 GROWTH 0 1 3 5 7 9 11 13 15 17 19 21 1 1 2 3 4 5 6 7 8 9 10 11 2 1 3 3 5 5 7 7 9 9 11 11 3 1 3 5 7 9 11 13 15 17 19 21 4 1 3 3 7 7 7 7 15 15 15 15 5 1 3 7 7 15 15 15 15 31 31 31 6 1 3 7 15 31 63 127 255 511 1023 2047 TEST23 LEVEL_GROWTH_TO_ORDER uses Level, Growth and Rule to determine the orders of each entry of a vector of 1D rules. Here we examine rule 11. LEVEL:0 1 2 3 4 5 6 7 8 9 10 GROWTH 0 1 3 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 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 TEST25 LEVEL_TO_ORDER_EXPONENTIAL uses an exponential rule to determine the order of a rule from its level. RULE/LEVEL 0 1 2 3 4 5 6 7 8 9 10 1 1 3 5 9 17 33 65 129 257 513 1025 2 1 3 7 15 31 63 127 255 511 1023 2047 3 1 3 7 15 31 63 127 255 511 1023 2047 4 1 3 7 15 31 63 127 255 511 1023 2047 5 1 3 7 15 31 63 127 255 511 1023 2047 6 1 3 7 15 31 63 127 255 511 1023 2047 7 1 3 7 15 31 63 127 255 511 1023 2047 8 1 3 7 15 31 63 127 255 511 1023 2047 9 1 3 7 15 31 63 127 255 511 1023 2047 10 1 3 7 15 31 63 127 255 511 1023 2047 TEST26 LEVEL_TO_ORDER_EXPONENTIAL_SLOW uses a slow exponential rule to determine the order of a rule from its level. Since it is really only useful for fully nested rules, we only consider rules 11, 12 and 13. RULE/LEVEL 0 1 2 3 4 5 6 7 8 9 10 11 1 3 5 9 9 17 17 17 17 33 33 12 1 3 7 7 15 15 15 15 31 31 31 13 1 3 3 7 7 7 15 15 15 15 15 TEST27 LEVEL_TO_ORDER_LINEAR uses a linear rule to determine the order of a rule from its level. RULE/LEVEL 0 1 2 3 4 5 6 7 8 9 10 1 1 3 5 7 9 11 13 15 17 19 21 2 1 3 5 7 9 11 13 15 17 19 21 3 1 3 5 7 9 11 13 15 17 19 21 4 1 3 5 7 9 11 13 15 17 19 21 5 1 3 5 7 9 11 13 15 17 19 21 6 1 3 5 7 9 11 13 15 17 19 21 7 1 3 5 7 9 11 13 15 17 19 21 8 1 3 5 7 9 11 13 15 17 19 21 9 1 3 5 7 9 11 13 15 17 19 21 10 1 3 5 7 9 11 13 15 17 19 21 TEST28 PATTERSON_LOOKUP computes a Gauss-Patterson rule which is appropriate for integrands of the form Integral ( -1 <= x <= +1 ) f(x) dx. LEGENDRE_INTEGRAL determines the exact value of this integal when f(x) = x^n. A rule of order ORDER should be exact for monomials X^N up to N = (3*ORDER+1)/2 In the following table, for each order, the LAST THREE estimates are made on monomials that exceed the exactness limit for the rule. Order N Estimate Exact Error 1 0 2 2 0 1 1 0 0 0 1 2 0 0.666667 0.666667 1 3 0 0 0 1 4 0 0.4 0.4 3 0 2 2 0 3 1 0 0 0 3 2 0.666667 0.666667 1.11022e-16 3 3 0 0 0 3 4 0.4 0.4 1.11022e-16 3 5 0 0 0 3 6 0.24 0.285714 0.0457143 3 7 0 0 0 3 8 0.144 0.222222 0.0782222 7 0 2 2 0 7 1 -1.38778e-17 0 1.38778e-17 7 2 0.666667 0.666667 1.11022e-16 7 3 -2.77556e-17 0 2.77556e-17 7 4 0.4 0.4 0 7 5 -1.38778e-17 0 1.38778e-17 7 6 0.285714 0.285714 5.55112e-17 7 7 2.77556e-17 0 2.77556e-17 7 8 0.222222 0.222222 2.77556e-17 7 9 0 0 0 7 10 0.181818 0.181818 2.77556e-17 7 11 0 0 0 7 12 0.154127 0.153846 0.000280652 7 13 0 0 0 7 14 0.134081 0.133333 0.000748122 15 0 2 2 0 15 1 2.42861e-17 0 2.42861e-17 15 2 0.666667 0.666667 0 15 3 5.89806e-17 0 5.89806e-17 15 4 0.4 0.4 1.11022e-16 15 5 0 0 0 15 6 0.285714 0.285714 0 15 7 -6.93889e-18 0 6.93889e-18 15 8 0.222222 0.222222 2.77556e-17 15 9 1.04083e-17 0 1.04083e-17 15 10 0.181818 0.181818 0 15 11 0 0 0 15 12 0.153846 0.153846 2.77556e-17 15 13 -6.93889e-18 0 6.93889e-18 15 14 0.133333 0.133333 0 15 15 -1.04083e-17 0 1.04083e-17 15 16 0.117647 0.117647 1.38778e-17 15 17 -3.46945e-18 0 3.46945e-18 15 18 0.105263 0.105263 0 15 19 1.73472e-18 0 1.73472e-18 15 20 0.0952381 0.0952381 2.77556e-17 15 21 1.73472e-18 0 1.73472e-18 15 22 0.0869565 0.0869565 0 15 23 1.73472e-18 0 1.73472e-18 15 24 0.08 0.08 5.39491e-09 15 25 -1.73472e-18 0 1.73472e-18 15 26 0.0740741 0.0740741 1.75247e-08 31 0 2 2 0 31 1 5.63785e-18 0 5.63785e-18 31 2 0.666667 0.666667 1.11022e-16 31 3 1.64799e-17 0 1.64799e-17 31 4 0.4 0.4 5.55112e-17 31 5 1.21431e-17 0 1.21431e-17 31 6 0.285714 0.285714 0 31 7 -1.82146e-17 0 1.82146e-17 31 8 0.222222 0.222222 2.77556e-17 31 9 -7.80626e-18 0 7.80626e-18 31 10 0.181818 0.181818 2.77556e-17 31 11 -1.73472e-18 0 1.73472e-18 31 12 0.153846 0.153846 2.77556e-17 31 13 1.30104e-18 0 1.30104e-18 31 14 0.133333 0.133333 0 31 15 -8.67362e-19 0 8.67362e-19 31 16 0.117647 0.117647 1.38778e-17 31 17 -1.73472e-18 0 1.73472e-18 31 18 0.105263 0.105263 0 31 19 -2.60209e-18 0 2.60209e-18 31 20 0.0952381 0.0952381 1.38778e-17 31 21 6.07153e-18 0 6.07153e-18 31 22 0.0869565 0.0869565 0 31 23 -4.33681e-19 0 4.33681e-19 31 24 0.08 0.08 1.38778e-17 31 25 -2.1684e-18 0 2.1684e-18 31 26 0.0740741 0.0740741 4.16334e-17 31 27 1.73472e-18 0 1.73472e-18 31 28 0.0689655 0.0689655 2.77556e-17 31 29 8.23994e-18 0 8.23994e-18 31 30 0.0645161 0.0645161 4.16334e-17 31 31 2.1684e-18 0 2.1684e-18 31 32 0.0606061 0.0606061 3.46945e-17 31 33 8.67362e-19 0 8.67362e-19 31 34 0.0571429 0.0571429 2.08167e-17 31 35 2.1684e-18 0 2.1684e-18 31 36 0.0540541 0.0540541 4.85723e-17 31 37 -4.33681e-18 0 4.33681e-18 31 38 0.0512821 0.0512821 2.08167e-17 31 39 3.90313e-18 0 3.90313e-18 31 40 0.0487805 0.0487805 4.16334e-17 31 41 8.67362e-19 0 8.67362e-19 31 42 0.0465116 0.0465116 3.46945e-17 31 43 -8.67362e-19 0 8.67362e-19 31 44 0.0444444 0.0444444 4.85723e-17 31 45 8.67362e-19 0 8.67362e-19 31 46 0.0425532 0.0425532 2.77556e-17 31 47 1.30104e-18 0 1.30104e-18 31 48 0.0408163 0.0408163 2.77556e-17 31 49 -4.33681e-19 0 4.33681e-19 31 50 0.0392157 0.0392157 3.46945e-17 63 0 2 2 2.22045e-16 63 1 -9.26993e-18 0 9.26993e-18 63 2 0.666667 0.666667 1.11022e-16 63 3 -9.75782e-19 0 9.75782e-19 63 4 0.4 0.4 5.55112e-17 63 5 2.54788e-18 0 2.54788e-18 63 6 0.285714 0.285714 1.11022e-16 63 7 -1.02999e-17 0 1.02999e-17 63 8 0.222222 0.222222 0 63 9 -2.49366e-18 0 2.49366e-18 63 10 0.181818 0.181818 8.32667e-17 63 11 -5.74627e-18 0 5.74627e-18 63 12 0.153846 0.153846 2.77556e-17 63 13 1.0842e-19 0 1.0842e-19 63 14 0.133333 0.133333 0 63 15 5.80048e-18 0 5.80048e-18 63 16 0.117647 0.117647 1.38778e-17 63 17 3.1984e-18 0 3.1984e-18 63 18 0.105263 0.105263 2.77556e-17 63 19 -9.75782e-18 0 9.75782e-18 63 20 0.0952381 0.0952381 2.77556e-17 63 21 8.13152e-18 0 8.13152e-18 63 22 0.0869565 0.0869565 0 63 23 -2.00577e-18 0 2.00577e-18 63 24 0.08 0.08 1.38778e-17 63 25 -2.05998e-18 0 2.05998e-18 63 26 0.0740741 0.0740741 1.38778e-17 63 27 4.66207e-18 0 4.66207e-18 63 28 0.0689655 0.0689655 1.38778e-17 63 29 -3.7405e-18 0 3.7405e-18 63 30 0.0645161 0.0645161 1.38778e-17 63 31 1.0842e-19 0 1.0842e-19 63 32 0.0606061 0.0606061 2.08167e-17 63 33 -6.50521e-19 0 6.50521e-19 63 34 0.0571429 0.0571429 6.93889e-18 63 35 3.84892e-18 0 3.84892e-18 63 36 0.0540541 0.0540541 6.93889e-18 63 37 9.75782e-19 0 9.75782e-19 63 38 0.0512821 0.0512821 6.93889e-18 63 39 4.87891e-19 0 4.87891e-19 63 40 0.0487805 0.0487805 1.38778e-17 63 41 -4.33681e-19 0 4.33681e-19 63 42 0.0465116 0.0465116 2.08167e-17 63 43 1.46367e-18 0 1.46367e-18 63 44 0.0444444 0.0444444 0 63 45 2.76472e-18 0 2.76472e-18 63 46 0.0425532 0.0425532 2.77556e-17 63 47 2.54788e-18 0 2.54788e-18 63 48 0.0408163 0.0408163 3.46945e-17 63 49 1.24683e-18 0 1.24683e-18 63 50 0.0392157 0.0392157 2.77556e-17 63 51 1.6263e-19 0 1.6263e-19 63 52 0.0377358 0.0377358 6.93889e-18 63 53 -1.24683e-18 0 1.24683e-18 127 0 2 2 0 127 1 -3.74185e-17 0 3.74185e-17 127 2 0.666667 0.666667 0 127 3 1.67035e-17 0 1.67035e-17 127 4 0.4 0.4 4.44089e-16 127 5 1.42505e-17 0 1.42505e-17 127 6 0.285714 0.285714 5.55112e-17 127 7 4.92634e-18 0 4.92634e-18 127 8 0.222222 0.222222 2.77556e-17 127 9 2.69899e-17 0 2.69899e-17 127 10 0.181818 0.181818 2.77556e-17 127 11 -1.3417e-17 0 1.3417e-17 127 12 0.153846 0.153846 5.55112e-17 127 13 -4.64174e-18 0 4.64174e-18 127 14 0.133333 0.133333 2.77556e-17 127 15 -2.62919e-18 0 2.62919e-18 127 16 0.117647 0.117647 1.38778e-17 127 17 -2.59531e-18 0 2.59531e-18 127 18 0.105263 0.105263 4.16334e-17 127 19 9.29703e-18 0 9.29703e-18 127 20 0.0952381 0.0952381 4.16334e-17 127 21 -3.42201e-18 0 3.42201e-18 127 22 0.0869565 0.0869565 2.77556e-17 127 23 2.12775e-18 0 2.12775e-18 127 24 0.08 0.08 5.55112e-17 127 25 3.38813e-19 0 3.38813e-19 127 26 0.0740741 0.0740741 1.38778e-17 127 27 -2.12097e-18 0 2.12097e-18 127 28 0.0689655 0.0689655 8.32667e-17 127 29 -4.74338e-19 0 4.74338e-19 127 30 0.0645161 0.0645161 1.38778e-17 127 31 3.38813e-18 0 3.38813e-18 127 32 0.0606061 0.0606061 3.46945e-17 127 33 1.06387e-18 0 1.06387e-18 127 34 0.0571429 0.0571429 6.93889e-18 127 35 -5.55654e-19 0 5.55654e-19 127 36 0.0540541 0.0540541 2.08167e-17 127 37 2.75116e-18 0 2.75116e-18 127 38 0.0512821 0.0512821 1.38778e-17 127 39 4.20128e-18 0 4.20128e-18 127 40 0.0487805 0.0487805 4.16334e-17 127 41 3.01544e-18 0 3.01544e-18 127 42 0.0465116 0.0465116 4.85723e-17 127 43 4.73661e-18 0 4.73661e-18 127 44 0.0444444 0.0444444 2.08167e-17 127 45 1.49078e-19 0 1.49078e-19 127 46 0.0425532 0.0425532 2.77556e-17 127 47 9.28348e-19 0 9.28348e-19 127 48 0.0408163 0.0408163 1.38778e-17 127 49 3.08998e-18 0 3.08998e-18 127 50 0.0392157 0.0392157 2.77556e-17 127 51 2.54788e-18 0 2.54788e-18 127 52 0.0377358 0.0377358 2.77556e-17 127 53 -5.42101e-19 0 5.42101e-19 255 0 2 2 8.88178e-16 255 1 8.0358e-18 0 8.0358e-18 255 2 0.666667 0.666667 0 255 3 -2.6934e-17 0 2.6934e-17 255 4 0.4 0.4 4.996e-16 255 5 1.00416e-17 0 1.00416e-17 255 6 0.285714 0.285714 1.66533e-16 255 7 2.77996e-18 0 2.77996e-18 255 8 0.222222 0.222222 5.55112e-17 255 9 1.0034e-17 0 1.0034e-17 255 10 0.181818 0.181818 1.38778e-16 255 11 8.05444e-18 0 8.05444e-18 255 12 0.153846 0.153846 8.32667e-17 255 13 -6.57636e-18 0 6.57636e-18 255 14 0.133333 0.133333 8.32667e-17 255 15 1.67916e-17 0 1.67916e-17 255 16 0.117647 0.117647 2.77556e-17 255 17 1.21465e-18 0 1.21465e-18 255 18 0.105263 0.105263 5.55112e-17 255 19 1.02576e-18 0 1.02576e-18 255 20 0.0952381 0.0952381 6.93889e-17 255 21 1.59496e-18 0 1.59496e-18 255 22 0.0869565 0.0869565 2.77556e-17 255 23 3.65071e-19 0 3.65071e-19 255 24 0.08 0.08 1.38778e-17 255 25 5.18892e-18 0 5.18892e-18 255 26 0.0740741 0.0740741 1.38778e-17 255 27 6.08424e-18 0 6.08424e-18 255 28 0.0689655 0.0689655 5.55112e-17 255 29 -1.44843e-19 0 1.44843e-19 255 30 0.0645161 0.0645161 0 255 31 -1.88465e-18 0 1.88465e-18 255 32 0.0606061 0.0606061 1.38778e-17 255 33 1.7415e-18 0 1.7415e-18 255 34 0.0571429 0.0571429 6.93889e-18 255 35 1.16891e-19 0 1.16891e-19 255 36 0.0540541 0.0540541 0 255 37 1.21295e-18 0 1.21295e-18 255 38 0.0512821 0.0512821 6.93889e-18 255 39 -1.25361e-19 0 1.25361e-19 255 40 0.0487805 0.0487805 2.77556e-17 255 41 -1.34255e-18 0 1.34255e-18 255 42 0.0465116 0.0465116 2.08167e-17 255 43 3.79725e-18 0 3.79725e-18 255 44 0.0444444 0.0444444 1.38778e-17 255 45 8.69903e-19 0 8.69903e-19 255 46 0.0425532 0.0425532 6.93889e-18 255 47 -6.03934e-19 0 6.03934e-19 255 48 0.0408163 0.0408163 1.38778e-17 255 49 -1.15027e-18 0 1.15027e-18 255 50 0.0392157 0.0392157 6.93889e-18 255 51 1.18585e-18 0 1.18585e-18 255 52 0.0377358 0.0377358 0 255 53 -2.91972e-18 0 2.91972e-18 511 0 2 2 1.77636e-15 511 1 -2.94069e-17 0 2.94069e-17 511 2 0.666667 0.666667 2.22045e-16 511 3 5.38949e-17 0 5.38949e-17 511 4 0.4 0.4 5.55112e-17 511 5 1.3295e-17 0 1.3295e-17 511 6 0.285714 0.285714 1.66533e-16 511 7 -1.16071e-17 0 1.16071e-17 511 8 0.222222 0.222222 8.32667e-17 511 9 6.21479e-18 0 6.21479e-18 511 10 0.181818 0.181818 0 511 11 -2.54766e-18 0 2.54766e-18 511 12 0.153846 0.153846 5.55112e-17 511 13 -4.27328e-18 0 4.27328e-18 511 14 0.133333 0.133333 1.66533e-16 511 15 1.08505e-18 0 1.08505e-18 511 16 0.117647 0.117647 1.38778e-17 511 17 1.29903e-18 0 1.29903e-18 511 18 0.105263 0.105263 2.77556e-17 511 19 1.44027e-18 0 1.44027e-18 511 20 0.0952381 0.0952381 8.32667e-17 511 21 -1.90752e-18 0 1.90752e-18 511 22 0.0869565 0.0869565 1.38778e-16 511 23 -2.06718e-18 0 2.06718e-18 511 24 0.08 0.08 2.77556e-17 511 25 -7.90811e-18 0 7.90811e-18 511 26 0.0740741 0.0740741 2.77556e-17 511 27 -1.07351e-18 0 1.07351e-18 511 28 0.0689655 0.0689655 1.11022e-16 511 29 -3.85199e-18 0 3.85199e-18 511 30 0.0645161 0.0645161 2.77556e-17 511 31 -1.17134e-18 0 1.17134e-18 511 32 0.0606061 0.0606061 6.93889e-18 511 33 1.64388e-18 0 1.64388e-18 511 34 0.0571429 0.0571429 1.38778e-17 511 35 -1.28749e-18 0 1.28749e-18 511 36 0.0540541 0.0540541 1.38778e-17 511 37 6.3591e-19 0 6.3591e-19 511 38 0.0512821 0.0512821 2.08167e-17 511 39 1.0482e-19 0 1.0482e-19 511 40 0.0487805 0.0487805 6.93889e-18 511 41 -1.36859e-18 0 1.36859e-18 511 42 0.0465116 0.0465116 2.08167e-17 511 43 -2.17095e-18 0 2.17095e-18 511 44 0.0444444 0.0444444 2.08167e-17 511 45 3.94823e-19 0 3.94823e-19 511 46 0.0425532 0.0425532 1.38778e-17 511 47 -1.63266e-19 0 1.63266e-19 511 48 0.0408163 0.0408163 3.46945e-17 511 49 2.23161e-18 0 2.23161e-18 511 50 0.0392157 0.0392157 6.245e-17 511 51 -2.46995e-18 0 2.46995e-18 511 52 0.0377358 0.0377358 6.93889e-18 511 53 -1.06229e-18 0 1.06229e-18 TEST285 POINT_RADIAL_TOL_UNIQUE_INDEX_INC1 can index unique points in a "permanent" point set; POINT_RADIAL_TOL_UNIQUE_INDEX_INC2 can incremented by "temporary" points. POINT_RADIAL_TOL_UNIQUE_INDEX_INC3 can merge permanent and temporary points. Using tolerance TOL = 1.49012e-08 UNIQUE_NUM1 = 9 Expected = 9 Item I1, unique index XDNU1[I1], representative location UNDX1[XDNU1[I1]]: I1 XDNU1 UNDX1 0 7 6 1 0 5 2 5 2 3 8 3 4 6 4 5 0 5 6 7 6 7 4 7 8 2 8 9 1 9 10 3 10 Unique item I1, location UNDX1(I1), value A1(:,UNDX1(I1)): I1 UNDX1 --A1(1,*)--- --A1(2,*)--- 0 5 0.5 0.5 1 9 1 0.5 2 8 0.5 0 3 10 0.5 1 4 7 0 0.5 5 2 1 0 6 4 1 1 7 6 0 0 8 3 0 1 Unique item I1, location UNDX1(I1), value A1(:,UNDX1(I1)): I1 UNIQUE1 R1 --A1(1,I1)-- --A1(2,I1)-- 0 0 0.748166 0 0 1 0 0.0956059 0.5 0.5 2 1 0.619486 1 0 3 1 0.796566 0 1 4 1 0.677147 1 1 5 1 0.0956059 0.5 0.5 6 1 0.748166 0 0 7 1 0.589182 0 0.5 8 1 0.470912 0.5 0 9 1 0.413698 1 0.5 10 1 0.544539 0.5 1 I1 INDX1 R1(INDX1) 0 5 0.0956059 1 1 0.0956059 2 9 0.413698 3 8 0.470912 4 10 0.544539 5 7 0.589182 6 2 0.619486 7 4 0.677147 8 6 0.748166 9 0 0.748166 10 3 0.796566 UNIQUE_NUM2 = 3 Expected = 3 Item I2, unique index XDNU2[I2], representative location UNDX2[XDNU2[I2]]: I2 XDNU2 UNDX2 (Temporary data) 0 11 0 5 1 12 9 16 2 13 3 10 3 14 2 8 4 15 11 17 5 16 9 16 6 17 11 17 7 18 10 18 Unique item I2, location UNDX2(I2), value A2(:,UNDX2(I2)): I2 UNDX2 --A2(1,*)--- --A2(2,*)--- (Temporary data) 0 9 16 0.75 0.25 1 10 18 0.75 0.75 2 11 17 0.25 0.75 Merge the temporary data with the permanent data. UNIQUE_NUM3 = 12 Expected = 12 Item I3, unique index XDNU3[I3], representative location UNDX3[XDNU3[I3]]: I3 XDNU3 UNDX3 0 7 6 1 0 5 2 5 2 3 8 3 4 6 4 5 0 5 6 7 6 7 4 7 8 2 8 9 1 9 10 3 10 11 0 5 12 9 16 13 3 10 14 2 8 15 11 17 16 9 16 17 11 17 18 10 18 Unique item I3, location UNDX3(I3), value A3(:,UNDX3(I3)): I3 UNDX3 --A3(1,*)--- --A3(2,*)--- 0 5 0.5 0.5 1 9 1 0.5 2 8 0.5 0 3 10 0.5 1 4 7 0 0.5 5 2 1 0 6 4 1 1 7 6 0 0 8 3 0 1 9 16 0.75 0.25 10 18 0.75 0.75 11 17 0.25 0.75 Unique item I3, location UNDX3(I3), value A3(:,UNDX3(I3)): I3 UNIQUE3 R3 --A3(1,I3)-- --A3(2,I3)-- 0 0 0.748166 0 0 1 0 0.0956059 0.5 0.5 2 1 0.619486 1 0 3 1 0.796566 0 1 4 1 0.677147 1 1 5 1 0.0956059 0.5 0.5 6 1 0.748166 0 0 7 1 0.589182 0 0.5 8 1 0.470912 0.5 0 9 1 0.413698 1 0.5 10 1 0.544539 0.5 1 11 0 0.0956059 0.5 0.5 12 0 0.267305 0.75 0.25 13 0 0.544539 0.5 1 14 0 0.470912 0.5 1e-10 15 0 0.443654 0.25 0.75 16 1 0.267305 0.75 0.25 17 1 0.443654 0.25 0.75 18 1 0.329901 0.75 0.75 I3 INDX3 R3(INDX3) 0 5 0.0956059 1 1 0.0956059 2 11 0.0956059 3 16 0.267305 4 12 0.267305 5 18 0.329901 6 9 0.413698 7 17 0.443654 8 15 0.443654 9 14 0.470912 10 8 0.470912 11 13 0.544539 12 10 0.544539 13 7 0.589182 14 2 0.619486 15 4 0.677147 16 6 0.748166 17 0 0.748166 18 3 0.796566 TEST29 R8COL_TOL_UNDEX produces index vectors which create a sorted list of the tolerably unique columns of an R8COL, and a map from the original R8COL to the (implicit) R8COL of sorted tolerably unique elements. The unsorted R8COL (transposed): Row: 0 1 2 Col 0: 1.9 0 10 1: 2 6 10 2: 4 8 12 3: 1 5 9 4: 3 7 11 5: 2 6 0 6: 2 0 10.1 7: 2 0.1 10 8: 3 4 18 9: 1.9 8 10 10: 0 0 0 11: 0 6 10 12: 2.1 0 10 13: 2 6 10 14: 3 7 11 15: 2 0 10 16: 2 0 10 17: 2 6 10 18: 1 5 9 19: 2 0 10.1 20: 1 5 9.1 21: 1 5.1 9 Using tolerance = 0.25 Number of tolerably unique columns is 10 XDNU points to the representative for each item. UNDX selects the representatives. I XDNU UNDX 0 3 10 1 6 11 2 9 18 3 2 0 4 8 9 5 5 5 6 3 13 7 3 8 8 7 14 9 4 2 10 0 11 1 12 3 13 6 14 8 15 3 16 3 17 6 18 2 19 3 20 2 21 2 The tolerably unique R8COL (transposed): Row: 0 1 2 Col 0: 0 0 0 1: 0 6 10 2: 1 5 9 3: 1.9 0 10 4: 1.9 8 10 5: 2 6 0 6: 2 6 10 7: 3 4 18 8: 3 7 11 9: 4 8 12 TEST30 R8VEC_SORT_HEAP_INDEX_A_NEW creates an ascending sort index for a R8VEC. The unsorted array: 0: 0.218418 1: 0.956318 2: 0.829509 3: 0.561695 4: 0.415307 5: 0.0661187 6: 0.257578 7: 0.109957 8: 0.043829 9: 0.633966 10: 0.0617272 11: 0.449539 12: 0.401306 13: 0.754673 14: 0.797287 15: 0.00183837 16: 0.897504 17: 0.350752 18: 0.0945448 19: 0.0136169 The index vector: 0: 15 1: 19 2: 8 3: 10 4: 5 5: 18 6: 7 7: 0 8: 6 9: 17 10: 12 11: 4 12: 11 13: 3 14: 9 15: 13 16: 14 17: 2 18: 16 19: 1 The sorted array A(INDX(:)): 0: 0.00183837 1: 0.0136169 2: 0.043829 3: 0.0617272 4: 0.0661187 5: 0.0945448 6: 0.109957 7: 0.218418 8: 0.257578 9: 0.350752 10: 0.401306 11: 0.415307 12: 0.449539 13: 0.561695 14: 0.633966 15: 0.754673 16: 0.797287 17: 0.829509 18: 0.897504 19: 0.956318 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) 0 0 0.05 1 0 0.000833333 2 0.1 0.1 3 0.1 0 4 0.2 0.1 5 0.2 -4.62593e-19 6 0.3 0.1 7 0.3 9.25186e-19 8 0.4 0.1 9 0.4 -9.25186e-19 10 0.5 0.1 11 0.5 0 12 0.6 0.1 13 0.6 0 14 0.7 0.1 15 0.7 1.85037e-18 16 0.8 0.1 17 0.8 -1.85037e-18 18 0.9 0.1 19 0.9 0 20 1 0.05 21 1 -0.000833333 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) 0 -1 0.0244717 1 -1 0.000199622 2 -0.951057 0.0954915 3 -0.951057 0.00148165 4 -0.809017 0.181636 5 -0.809017 0.00239735 6 -0.587785 0.25 7 -0.587785 0.00239735 8 -0.309017 0.293893 9 -0.309017 0.00148165 10 0 0.309017 11 0 5.71796e-18 12 0.309017 0.293893 13 0.309017 -0.00148165 14 0.587785 0.25 15 0.587785 -0.00239735 16 0.809017 0.181636 17 0.809017 -0.00239735 18 0.951057 0.0954915 19 0.951057 -0.00148165 20 1 0.0244717 21 1 -0.000199622 TEST33: HC_COMPUTE_WEIGHTS_FROM_POINTS returns quadrature weights given the points. Trial #1: Random spacing Number of points N = 11 Interval = [0.218418, 4.15442] Q = -6.75338 Q (exact) = -6.75338 Trial #2: Random spacing Number of points N = 11 Interval = [0.449539, 5.46101] Q = -9.35882 Q (exact) = -9.35882 Trial #3: Random spacing Number of points N = 11 Interval = [0.123104, 4.97366] Q = -10.483 Q (exact) = -10.483 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)) | 3 0.530048 5 0.223403 7 0.390862 9 1.13792 11 3.83581 13 13.9642 15 53.4249 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 <= X <= 1 I X W(F(X)) W(F'(X)) 0 0 -6.66667 -1.5 1 0.5 5.33333 -2.66667 2 1 2.33333 -0.333333 I X W(F(X)) W(F'(X)) 0 0 -951.678 -64.4722 1 0.25 -3269.21 -656.356 2 0.5 1361.6 -1009.6 3 0.75 2571.38 -333.511 4 1 288.915 -16.5056 I X W(F(X)) W(F'(X)) 0 0 -128378 -4699.38 1 0.166667 -1.66705e+06 -122484 2 0.333333 -3.20273e+06 -582226 3 0.5 1.03003e+06 -823715 4 0.666667 3.08218e+06 -382601 5 0.833333 848082 -52005.7 6 1 37868.7 -1254.21 I X W(F(X)) W(F'(X)) 0 0 -1.79706e+07 -435044 1 0.125 -5.10033e+08 -2.16079e+07 2 0.25 -2.89076e+09 -2.12806e+08 3 0.375 -4.09812e+09 -7.05643e+08 4 0.5 1.13487e+09 -9.37409e+08 5 0.625 4.30251e+09 -5.20585e+08 6 0.75 1.85366e+09 -1.14797e+08 7 0.875 2.20612e+08 -8.37644e+06 8 1 5.23433e+06 -118268 I X W(F(X)) W(F'(X)) 0 0 -2.60271e+09 -4.61959e+07 1 0.1 -1.2973e+11 -3.74681e+09 2 0.2 -1.43117e+12 -6.31919e+10 3 0.3 -5.24034e+12 -3.8283e+11 4 0.4 -6.11016e+12 -1.01787e+12 5 0.5 1.53665e+12 -1.29255e+12 6 0.6 6.73925e+12 -8.01827e+11 7 0.7 3.82127e+12 -2.36405e+11 8 0.8 7.64733e+11 -3.02886e+10 9 0.9 5.13565e+10 -1.37316e+09 10 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 <= X <= 5 I X W(F(X)) W(F'(X)) 0 -5 61.2405 10.2063 1 -4 4469.25 1177.08 2 -3 67539.6 26832.2 3 -2 253916 187109 4 -1 -31806.8 372881 5 0 -588350 3.95812e-09 6 1 -31806.8 -372881 7 2 253916 -187109 8 3 67539.6 -26832.2 9 4 4469.25 -1177.08 10 5 61.2405 -10.2063 Estimate integral of 1 = 10 Estimate integral of X = -6.72418e-08 Estimate integral of X^2 = 83.3333 Estimate integral of 1/(1+x^2) = -68509.5 Use the rule with N = 11 to estimate integrals. Points are Chebyshev spaced. We are working in -5 <= X <= 5 I X W(F(X)) W(F'(X)) 0 -5 114869 8510.62 1 -4.75528 -66202.1 34839.6 2 -4.04508 -14974.8 36848.8 3 -2.93893 -9014.04 37252 4 -1.54508 -14785.4 26848 5 0 -19775 -2.5575e-09 6 1.54508 -14785.4 -26848 7 2.93893 -9014.04 -37252 8 4.04508 -14974.8 -36848.8 9 4.75528 -66202.1 -34839.6 10 5 114869 -8510.62 Estimate integral of 1 = 10 Estimate integral of X = -1.7646e-06 Estimate integral of X^2 = 83.3333 Estimate integral of 1/(1+x^2) = -6276.92 Use the rule with N = 11 to estimate integrals. Points are Legendre spaced. We are working in -5 <= X <= 5 I X W(F(X)) W(F'(X)) 0 -4.89114 7380.63 802.295 1 -4.43531 16678.5 9418.7 2 -3.65076 22800.7 32858.7 3 -2.59548 12958.6 64690.5 4 -1.34772 -29369 65052.6 5 0 -60888.7 6.97764e-09 6 1.34772 -29369 -65052.6 7 2.59548 12958.6 -64690.5 8 3.65076 22800.7 -32858.7 9 4.43531 16678.5 -9418.7 10 4.89114 7380.63 -802.295 Estimate integral of 1 = 10 Estimate integral of X = -8.44634e-08 Estimate integral of X^2 = 83.3333 Estimate integral of 1/(1+x^2) = -14818.5 Approximate integral of 1/(1+x^2) with increasing N. Points are Chebyshev spaced. We are working in -5 <= X <= 5 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 2.67564e-14 X^ 5 0 -2.70894e-14 2.70894e-14 X^ 6 0.285714 2.77556e-14 0.285714 X^ 7 0 -2.75335e-14 2.75335e-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 -3.4639e-14 3.4639e-14 1/(1+x^2) 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 3.72324e-12 3.72324e-12 X^ 6 0.285714 0.285714 1.94486e-11 X^ 7 0 4.37979e-11 4.37979e-11 X^ 8 0.222222 0.222222 5.7235e-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 -1.17325e-10 0.153846 X^ 13 0 1.33241e-10 1.33241e-10 X^ 14 0.133333 -1.41711e-10 0.133333 X^ 15 0 1.55978e-10 1.55978e-10 1/(1+x^2) 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 4.53031e-06 X^ 5 0 4.45541e-06 4.45541e-06 X^ 6 0.285714 0.28571 4.40477e-06 X^ 7 0 4.33392e-06 4.33392e-06 X^ 8 0.222222 0.222218 4.28545e-06 X^ 9 0 4.21632e-06 4.21632e-06 X^ 10 0.181818 0.181814 4.17129e-06 X^ 11 0 4.10362e-06 4.10362e-06 X^ 12 0.153846 0.153842 4.06307e-06 X^ 13 0 3.99714e-06 3.99714e-06 X^ 14 0.133333 -3.96212e-06 0.133337 X^ 15 0 3.89834e-06 3.89834e-06 1/(1+x^2) 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 0.0173781 0.0173781 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 0.0135533 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 1/(1+x^2) 1.5708 1.61447 0.0436754 Estimates are made using N = 21 F(X) Integral Estimate Error X^ 0 2 465.228 463.228 X^ 1 0 -452.235 452.235 X^ 2 0.666667 445.483 444.817 X^ 3 0 -436.799 436.799 X^ 4 0.4 431.798 431.398 X^ 5 0 -424.78 424.78 X^ 6 0.285714 420.806 420.52 X^ 7 0 -414.724 414.724 X^ 8 0.222222 411.458 411.236 X^ 9 0 -405.984 405.984 X^ 10 0.181818 403.239 403.057 X^ 11 0 -398.189 398.189 X^ 12 0.153846 395.842 395.688 X^ 13 0 -391.101 391.101 X^ 14 0.133333 389.064 388.93 X^ 15 0 -384.552 384.552 1/(1+x^2) 1.5708 238.683 237.112 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 0 1 1.77245 1 3 1.77245 2 9 1.77245 3 19 1.77245 4 35 1.77245 5 37 1.77245 6 41 1.77245 7 43 1.77245 Correct sum: 1.77245 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)) 0 0 0 0 1 1 1 0.25 0.247404 0.247407 0.968912 0.968921 2 0.5 0.479426 0.479428 0.877583 0.877575 3 0.75 0.681639 0.681639 0.731689 0.731683 4 1 0.841471 0.841471 0.540302 0.540302 5 1.25 0.948985 0.948985 0.315322 0.315324 6 1.5 0.997495 0.997495 0.0707372 0.0707368 7 1.75 0.983986 0.983986 -0.178246 -0.178247 8 2 0.909297 0.909297 -0.416147 -0.416147 9 2.25 0.778073 0.778073 -0.628174 -0.628172 10 2.5 0.598472 0.598473 -0.801144 -0.801143 11 2.75 0.381661 0.381661 -0.924302 -0.924304 12 3 0.14112 0.14112 -0.989992 -0.989992 13 3.25 -0.108195 -0.108194 -0.99413 -0.994124 14 3.5 -0.350783 -0.350781 -0.936457 -0.936451 15 3.75 -0.571561 -0.571559 -0.820559 -0.820567 16 4 -0.756802 -0.756802 -0.653644 -0.653644 TEST39: Test the various LEVEL_TO_ORDER_** functions, which, for a given type of quadrature rule, accept LEVEL, the index of the rule in the family, and GROWTH (0=slow, 1=moderate, 2=full), a growth rate, and return the appropriate corresponding order for the rule. LEVEL_TO_ORDER_EXP_CC: Slow/moderate/full exponential growth typical of the Clenshaw Curtis rule. LEVEL ORDER ORDER ORDER G = 0 G = 1 G = 2 0 1 1 1 1 3 5 3 2 5 9 5 3 9 17 9 4 9 17 17 5 17 33 33 6 17 33 65 7 17 33 129 8 17 33 257 LEVEL_TO_ORDER_EXP_F2: Slow/moderate/full exponential growth typical of the Fejer Type 2 rule. LEVEL ORDER ORDER ORDER G = 0 G = 1 G = 2 0 1 1 1 1 3 7 3 2 7 15 7 3 7 15 15 4 15 31 31 5 15 31 63 6 15 31 127 7 15 31 255 8 31 63 511 LEVEL_TO_ORDER_EXP_GAUSS: Slow/moderate/full exponential growth typical of a Gauss rule. LEVEL ORDER ORDER ORDER G = 0 G = 1 G = 2 0 1 1 1 1 3 3 3 2 3 7 7 3 7 7 15 4 7 15 31 5 7 15 63 6 7 15 127 7 15 15 255 8 15 31 511 LEVEL_TO_ORDER_EXP_GP: Slow/moderate/full exponential growth typical of a Gauss-Patterson rule. LEVEL ORDER ORDER ORDER G = 0 G = 1 G = 2 0 1 1 1 1 3 3 3 2 3 7 7 3 7 15 15 4 7 15 31 5 7 15 63 6 15 31 127 7 15 31 255 8 15 31 511 LEVEL_TO_ORDER_EXP_HGK: Slow/moderate/full exponential growth typical of a Hermite Genz-Keister rule. LEVEL ORDER ORDER ORDER G = 0 G = 1 G = 2 0 1 1 1 1 3 3 3 2 3 9 9 3 9 9 19 4 9 19 35 5 9 19 43 6 9 19 7 9 19 8 19 35 LEVEL_TO_ORDER_LINEAR_NN: Slow/moderate linear growth typical of a non-nested Gauss rule. LEVEL ORDER ORDER G = 0 G = 1 0 1 1 1 2 3 2 3 5 3 4 7 4 5 9 5 6 11 6 7 13 7 8 15 8 9 17 LEVEL_TO_ORDER_LINEAR_WN: Slow/moderate linear growth typical of a weakly-nested Gauss rule. LEVEL ORDER ORDER G = 0 G = 1 0 1 1 1 3 3 2 3 5 3 5 7 4 5 9 5 7 11 6 7 13 7 9 15 8 9 17 SANDIA_RULES_PRB Normal end of execution. 26 January 2012 04:06:25 PM