22 January 2009 9:24:58.977 AM 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 = "leg_o16_x.txt". Quadrature rule W file = "leg_o16_w.txt". Quadrature rule R file = "leg_o16_r.txt". Maximum degree to check = 70 Spatial dimension = 1 Number of points = 16 The quadrature rule to be tested is a Gauss-Legendre rule ORDER = 16 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) = 0.2715245941175407E-01 w( 2) = 0.6225352393864787E-01 w( 3) = 0.9515851168249276E-01 w( 4) = 0.1246289712555339 w( 5) = 0.1495959888165767 w( 6) = 0.1691565193950026 w( 7) = 0.1826034150449236 w( 8) = 0.1894506104550686 w( 9) = 0.1894506104550686 w(10) = 0.1826034150449236 w(11) = 0.1691565193950026 w(12) = 0.1495959888165767 w(13) = 0.1246289712555339 w(14) = 0.9515851168249276E-01 w(15) = 0.6225352393864787E-01 w(16) = 0.2715245941175407E-01 Abscissas X: x( 1) = -0.9894009349916500 x( 2) = -0.9445750230732326 x( 3) = -0.8656312023878318 x( 4) = -0.7554044083550030 x( 5) = -0.6178762444026438 x( 6) = -0.4580167776572274 x( 7) = -0.2816035507792589 x( 8) = -0.9501250983763745E-01 x( 9) = 0.9501250983763745E-01 x(10) = 0.2816035507792589 x(11) = 0.4580167776572274 x(12) = 0.6178762444026438 x(13) = 0.7554044083550030 x(14) = 0.8656312023878318 x(15) = 0.9445750230732326 x(16) = 0.9894009349916500 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 = 31 Error Error Degree (This rule) (Trapezoid) 0.0000000000000002 0.0000000000000001 0 0.0000000000000001 0.0000000000000000 1 0.0000000000000000 0.0088888888888890 2 0.0000000000000000 0.0000000000000000 3 0.0000000000000003 0.0295769547325100 4 0.0000000000000000 0.0000000000000000 5 0.0000000000000002 0.0618544343850021 6 0.0000000000000000 0.0000000000000000 7 0.0000000000000001 0.1053504666044810 8 0.0000000000000000 0.0000000000000000 9 0.0000000000000000 0.1595476770658492 10 0.0000000000000000 0.0000000000000000 11 0.0000000000000002 0.2238039162511361 12 0.0000000000000000 0.0000000000000000 13 0.0000000000000004 0.2973793952229118 14 0.0000000000000000 0.0000000000000000 15 0.0000000000000005 0.3794669678437371 16 0.0000000000000000 0.0000000000000000 17 0.0000000000000009 0.4692231145177214 18 0.0000000000000000 0.0000000000000000 19 0.0000000000000010 0.5657973148522170 20 0.0000000000000000 0.0000000000000000 21 0.0000000000000013 0.6683579124582246 22 0.0000000000000000 0.0000000000000000 23 0.0000000000000017 0.7761131791404083 24 0.0000000000000000 0.0000000000000000 25 0.0000000000000021 0.8883269563474357 26 0.0000000000000000 0.0000000000000000 27 0.0000000000000020 1.0043288728096322 28 0.0000000000000000 0.0000000000000000 29 0.0000000000000024 1.1235196242289249 30 0.0000000000000000 0.0000000000000000 31 0.0000000118876126 1.2453721135051228 32 0.0000000000000000 0.0000000000000000 33 0.0000001039207631 1.3694293903219130 34 0.0000000000000000 0.0000000000000000 35 0.0000004939099590 1.4953003285729984 36 0.0000000000000000 0.0000000000000000 37 0.0000016928215931 1.6226538840027360 38 0.0000000000000000 0.0000000000000000 39 0.0000046852412668 1.7512126270560364 40 0.0000000000000000 0.0000000000000000 41 0.0000111233193625 1.8807460830422646 42 0.0000000000000000 0.0000000000000000 43 0.0000235083252644 2.0110642574599247 44 0.0000000000000000 0.0000000000000000 45 0.0000453374340064 2.1420115920358720 46 0.0000000000000000 0.0000000000000000 47 0.0000812002970256 2.2734614917898845 48 0.0000000000000000 0.0000000000000000 49 0.0001368178659326 2.4053114849608526 50 0.0000000000000000 0.0000000000000000 51 0.0002190231266343 2.5374790227125779 52 0.0000000000000000 0.0000000000000000 53 0.0003356889478670 2.6698978897753332 54 0.0000000000000000 0.0000000000000000 55 0.0004956118797938 2.8025151760698224 56 0.0000000000000000 0.0000000000000000 57 0.0007083625904348 2.9352887488721309 58 0.0000000000000000 0.0000000000000000 59 0.0009841140424736 3.0681851618905709 60 0.0000000000000000 0.0000000000000000 61 0.0013334578951639 3.2011779391601132 62 0.0000000000000000 0.0000000000000000 63 0.0017672183467015 3.3342461760112290 64 0.0000000000000000 0.0000000000000000 65 0.0022962710268585 3.4673734051890763 66 0.0000000000000000 0.0000000000000000 67 0.0029313728448769 3.6005466825740156 68 0.0000000000000000 0.0000000000000000 69 0.0036830070559619 3.7337558533004409 70 INT_EXACTNESS_LEGENDRE: Normal end of execution. 22 January 2009 9:24:58.984 AM