24 January 2008 12:01:00 PM INT_EXACTNESS_GEN_HERMITE C++ 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 Power of |X|, ALPHA = 1 OPTION = 1, integrate f(x) Spatial dimension = 1 Number of points = 16 The quadrature rule to be tested is a generalized Gauss-Hermite rule ORDER = 16 ALPHA = 1 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[ 0] = 0.931313432350684 w[ 1] = 0.7332266104960289 w[ 2] = 0.650693978127963 w[ 3] = 0.6043787440156341 w[ 4] = 0.5753596025931054 w[ 5] = 0.5564355737688126 w[ 6] = 0.5437798314152072 w[ 7] = 0.5303815727772847 w[ 8] = 0.5303815727772847 w[ 9] = 0.5437798314152072 w[10] = 0.5564355737688126 w[11] = 0.5753596025931054 w[12] = 0.6043787440156341 w[13] = 0.650693978127963 w[14] = 0.7332266104960289 w[15] = 0.931313432350684 Abscissas X: x[ 0] = -4.781540728352031 x[ 1] = -3.967452411973961 x[ 2] = -3.280017684431137 x[ 3] = -2.654412440144422 x[ 4] = -2.065599227896752 x[ 5] = -1.500362166233917 x[ 6] = -0.9506323036797034 x[ 7] = -0.4126495272081394 x[ 8] = 0.4126495272081394 x[ 9] = 0.9506323036797034 x[10] = 1.500362166233917 x[11] = 2.065599227896752 x[12] = 2.654412440144422 x[13] = 3.280017684431137 x[14] = 3.967452411973961 x[15] = 4.781540728352031 Region R: r[ 0] = -1e+30 r[ 1] = 1e+30 A generalized Gauss-Hermite rule would be able to exactly integrate monomials up to and including degree = 31 Error Degree 6.661338147750939e-16 0 4.309842271100544e-17 1 4.440892098500626e-16 2 3.388125171572301e-18 3 2.220446049250313e-16 4 6.744648073278136e-17 5 1.480297366166875e-16 6 4.819448063287518e-16 7 1.480297366166875e-16 8 1.364793694724753e-15 9 5.921189464667501e-16 10 8.004014118156988e-15 11 9.473903143468002e-16 12 7.349676423018536e-14 13 1.082731787824915e-15 14 2.220446049250313e-13 15 1.082731787824915e-15 16 2.501110429875553e-12 17 6.41618837229579e-16 18 7.003109203651547e-11 19 5.132950697836632e-16 20 5.820766091346741e-10 21 7.466110105944187e-16 22 4.656612873077393e-10 23 5.350712242593365e-15 24 4.470348358154297e-08 25 2.450415624515021e-15 26 7.152557373046875e-07 27 2.62544531198038e-15 28 7.62939453125e-06 29 2.24037999955659e-15 30 0.0003662109375 31 7.770007770213118e-05 32 0.001953125 33 0.0006627359568543054 34 0 35 INT_EXACTNESS_GEN_HERMITE: Normal end of execution. 24 January 2008 12:01:00 PM