7 September 2007 10:34:54.100 AM INT_EXACTNESS_GEN_HERMITE FORTRAN90 version Investigate the polynomial exactness of a generalized Gauss-Hermite quadrature rule by integrating exponentially weighted monomials up to a given degree over the (-oo,oo) interval. INT_EXACTNESS_GEN_HERMITE: User input: Quadrature rule X file = "gen_herm_o16_a1.0_modified_x.txt". Quadrature rule W file = "gen_herm_o16_a1.0_modified_w.txt". Quadrature rule R file = "gen_herm_o16_a1.0_modified_r.txt". Maximum degree to check = 35 Weighting function exponent ALPHA = 1.00000 Spatial dimension = 1 Number of points = 16 The quadrature rule to be tested is a generalized Gauss-Hermite rule ORDER = 16 A = -0.179769 ALPHA = 1.00000 OPTION = 1, modified rule: Integral ( -oo < x < oo ) f(x) dx is to be approximated by sum ( 1 <= I <= ORDER ) w(i) * f(x(i)). Weights W: w( 1) = 0.9313134323506840 w( 2) = 0.7332266104960289 w( 3) = 0.6506939781279630 w( 4) = 0.6043787440156341 w( 5) = 0.5753596025931054 w( 6) = 0.5564355737688126 w( 7) = 0.5437798314152072 w( 8) = 0.5303815727772847 w( 9) = 0.5303815727772847 w(10) = 0.5437798314152072 w(11) = 0.5564355737688126 w(12) = 0.5753596025931054 w(13) = 0.6043787440156341 w(14) = 0.6506939781279630 w(15) = 0.7332266104960289 w(16) = 0.9313134323506840 Abscissas X: x( 1) = -4.781540728352031 x( 2) = -3.967452411973961 x( 3) = -3.280017684431137 x( 4) = -2.654412440144422 x( 5) = -2.065599227896752 x( 6) = -1.500362166233917 x( 7) = -0.9506323036797034 x( 8) = -0.4126495272081394 x( 9) = 0.4126495272081394 x(10) = 0.9506323036797034 x(11) = 1.500362166233917 x(12) = 2.065599227896752 x(13) = 2.654412440144422 x(14) = 3.280017684431137 x(15) = 3.967452411973961 x(16) = 4.781540728352031 Region R: r( 1) = -0.1797693134862000 r( 2) = 0.1797693134862000 A generalized Gauss-Hermite rule would be able to exactly integrate monomials up to and including degree = 31 Error Degree Exponents 0.0000000000000007 0 0 0.0000000000000000 1 1 0.0000000000000004 2 2 0.0000000000000000 3 3 0.0000000000000004 4 4 0.0000000000000001 5 5 0.0000000000000004 6 6 0.0000000000000002 7 7 0.0000000000000001 8 8 0.0000000000000006 9 9 0.0000000000000007 10 10 0.0000000000000010 11 11 0.0000000000000006 12 12 0.0000000000000098 13 13 0.0000000000000011 14 14 0.0000000000000147 15 15 0.0000000000000007 16 16 0.0000000000011596 17 17 0.0000000000000005 18 18 0.0000000000133582 19 19 0.0000000000000001 20 20 0.0000000000674163 21 21 0.0000000000000004 22 22 0.0000000039642600 23 23 0.0000000000000025 24 24 0.0000000272048055 25 25 0.0000000000000028 26 26 0.0000004007015377 27 27 0.0000000000000026 28 28 0.0000020582228899 29 29 0.0000000000000022 30 30 0.0001835227012634 31 31 0.0000777000777021 32 32 0.0038652420043945 33 33 0.0006627359568475 34 34 0.0015563964843750 35 35 INT_EXACTNESS_GEN_HERMITE: Normal end of execution. 7 September 2007 10:34:54.211 AM