09 February 2008 05:13:06 PM INT_EXACTNESS_JACOBI C++ version Investigate the polynomial exactness of a Gauss-Jacobi quadrature rule by integrating exponentially weighted monomials up to a given degree over the (-oo,+oo) interval. INT_EXACTNESS_JACOBI: User input: Quadrature rule X file = "jac_o16_a0.5_b1.5_x.txt". Quadrature rule W file = "jac_o16_a0.5_b1.5_w.txt". Quadrature rule R file = "jac_o16_a0.5_b1.5_r.txt". Maximum degree to check = 35 Exponent of (1-x), ALPHA = 0.5 Exponent of (1+x), BETA = 1.5 Spatial dimension = 1 Number of points = 16 The quadrature rule to be tested is a Gauss-Jacobi rule ORDER = 16 ALPHA = 0.5 BETA = 1.5 Standard rule: Integral ( -1 <= x <= +1 ) (1-x)^alplha (1+x)^beta f(x) dx is to be approximated by sum ( 1 <= I <= ORDER ) w(i) * f(x(i)). Weights W: w[ 0] = 0.0003988966638936207 w[ 1] = 0.003198240298907031 w[ 2] = 0.01159952258657551 w[ 3] = 0.02865636008049065 w[ 4] = 0.05581657557108236 w[ 5] = 0.09183919887761288 w[ 6] = 0.132474782714899 w[ 7] = 0.1710802448681879 w[ 8] = 0.2000637064771614 w[ 9] = 0.2127999546344757 w[10] = 0.2054979111101689 w[11] = 0.1784862462090924 w[12] = 0.1365200311515025 w[13] = 0.0879649206719887 w[14] = 0.04301699747981774 w[15] = 0.01138273739868221 Abscissas X: x[ 0] = -0.9671984819405668 x[ 1] = -0.9040845839929046 x[ 2] = -0.811977966578046 x[ 3] = -0.6938290260300457 x[ 4] = -0.5534333596595263 x[ 5] = -0.3953028602480263 x[ 6] = -0.2245197622496786 x[ 7] = -0.04657307825358586 x[ 8] = 0.1328178890268011 x[ 9] = 0.3078873946321201 x[10] = 0.4730085776933708 x[11] = 0.6228743183404623 x[12] = 0.7526678154351841 x[13] = 0.8582174038957007 x[14] = 0.9361306362220502 x[15] = 0.9839033190008089 Region R: r[ 0] = -1 r[ 1] = 1 A Gauss-Jacobi rule would be able to exactly integrate monomials up to and including degree = 31 Error Degree 2.277277151927879e-13 0 2.278690731786307e-13 1 2.278690731786307e-13 2 2.274449992211023e-13 3 2.273036412352595e-13 4 2.276429004012821e-13 5 2.278690731786307e-13 6 2.273036412352594e-13 7 2.274651932190798e-13 8 2.274651932190796e-13 9 2.250957641230482e-13 10 2.279547446852018e-13 11 2.183595359492063e-13 12 2.280918190957159e-13 13 2.28091819095716e-13 14 2.224907477983962e-13 15 2.318033723650241e-13 16 2.310293051056494e-13 17 2.257894651960381e-13 18 2.200732762037344e-13 19 2.22417415110666e-13 20 2.256676917463271e-13 21 2.201520707888414e-13 22 2.385351900794428e-13 23 2.289153989933436e-13 24 2.192742305937193e-13 25 2.254593999735226e-13 26 2.132664137409353e-13 27 2.223557307506822e-13 28 2.347168961037184e-13 29 2.345945840161189e-13 30 2.153087296875078e-13 31 1.497324946488494e-08 32 1.540031964129023e-08 33 1.343179508994477e-07 34 1.37940033955194e-07 35 INT_EXACTNESS_JACOBI: Normal end of execution. 09 February 2008 05:13:06 PM