21 January 2009 4:00:23.837 PM INT_EXACTNESS_LEGENDRE FORTRAN90 version Investigate the polynomial exactness of a Gauss-Legendre quadrature rule by integrating weighted monomials up to a given degree over the [-1,+1] interval. INT_EXACTNESS_LEGENDRE: User input: Quadrature rule X file = "gp_o31_x.txt". Quadrature rule W file = "gp_o31_w.txt". Quadrature rule R file = "gp_o31_r.txt". Maximum degree to check = 60 Spatial dimension = 1 Number of points = 31 The quadrature rule to be tested is a Gauss-Legendre rule ORDER = 31 Standard rule: Integral ( -1 <= x <= +1 ) f(x) dx is to be approximated by sum ( 1 <= I <= ORDER ) w(i) * f(x(i)). Weights W: w( 1) = 0.2544780791561875E-02 w( 2) = 0.8434565739321106E-02 w( 3) = 0.1644604985438781E-01 w( 4) = 0.2580759809617665E-01 w( 5) = 0.3595710330712933E-01 w( 6) = 0.4646289326175800E-01 w( 7) = 0.5697950949412336E-01 w( 8) = 0.6720775429599070E-01 w( 9) = 0.7687962049900354E-01 w(10) = 0.8575592004999034E-01 w(11) = 0.9362710998126449E-01 w(12) = 0.1003142786117956 w(13) = 0.1056698935802348 w(14) = 0.1095784210559246 w(15) = 0.1119568730209535 w(16) = 0.1127552567207687 w(17) = 0.1119568730209535 w(18) = 0.1095784210559246 w(19) = 0.1056698935802348 w(20) = 0.1003142786117956 w(21) = 0.9362710998126449E-01 w(22) = 0.8575592004999034E-01 w(23) = 0.7687962049900354E-01 w(24) = 0.6720775429599070E-01 w(25) = 0.5697950949412336E-01 w(26) = 0.4646289326175800E-01 w(27) = 0.3595710330712933E-01 w(28) = 0.2580759809617665E-01 w(29) = 0.1644604985438781E-01 w(30) = 0.8434565739321106E-02 w(31) = 0.2544780791561875E-02 Abscissas X: x( 1) = -0.9990981249676676 x( 2) = -0.9938319632127550 x( 3) = -0.9815311495537400 x( 4) = -0.9604912687080204 x( 5) = -0.9296548574297400 x( 6) = -0.8884592328722570 x( 7) = -0.8367259381688688 x( 8) = -0.7745966692414834 x( 9) = -0.7024962064915271 x(10) = -0.6211029467372264 x(11) = -0.5313197436443756 x(12) = -0.4342437493468025 x(13) = -0.3311353932579768 x(14) = -0.2233866864289669 x(15) = -0.1124889431331866 x(16) = 0.000000000000000 x(17) = 0.1124889431331866 x(18) = 0.2233866864289669 x(19) = 0.3311353932579768 x(20) = 0.4342437493468025 x(21) = 0.5313197436443756 x(22) = 0.6211029467372264 x(23) = 0.7024962064915271 x(24) = 0.7745966692414834 x(25) = 0.8367259381688688 x(26) = 0.8884592328722570 x(27) = 0.9296548574297400 x(28) = 0.9604912687080204 x(29) = 0.9815311495537400 x(30) = 0.9938319632127550 x(31) = 0.9990981249676676 Region R: r( 1) = -1.0000000000000000 r( 2) = 1.0000000000000000 A Gauss-Legendre rule would be able to exactly integrate monomials up to and including degree = 61 Error Degree 0.0000000000000000 0 0.0000000000000000 1 0.0000000000000002 2 0.0000000000000000 3 0.0000000000000000 4 0.0000000000000000 5 0.0000000000000000 6 0.0000000000000000 7 0.0000000000000001 8 0.0000000000000000 9 0.0000000000000000 10 0.0000000000000000 11 0.0000000000000002 12 0.0000000000000000 13 0.0000000000000002 14 0.0000000000000000 15 0.0000000000000001 16 0.0000000000000000 17 0.0000000000000004 18 0.0000000000000000 19 0.0000000000000003 20 0.0000000000000000 21 0.0000000000000003 22 0.0000000000000000 23 0.0000000000000005 24 0.0000000000000000 25 0.0000000000000009 26 0.0000000000000000 27 0.0000000000000004 28 0.0000000000000000 29 0.0000000000000009 30 0.0000000000000000 31 0.0000000000000008 32 0.0000000000000000 33 0.0000000000000007 34 0.0000000000000000 35 0.0000000000000013 36 0.0000000000000000 37 0.0000000000000008 38 0.0000000000000000 39 0.0000000000000011 40 0.0000000000000000 41 0.0000000000000010 42 0.0000000000000000 43 0.0000000000000016 44 0.0000000000000000 45 0.0000000000000011 46 0.0000000000000000 47 0.0000000000000012 48 0.0000000000000000 49 0.0000000000000016 50 0.0000000000000000 51 0.0000000000000026 52 0.0000000000000000 53 0.0000000000000078 54 0.0000000000000000 55 0.0000000000000196 56 0.0000000000000000 57 0.0000000000000291 58 0.0000000000000000 59 0.0000000000000150 60 INT_EXACTNESS_LEGENDRE: Normal end of execution. 21 January 2009 4:00:23.844 PM