November 2 2009 3:51:23.091 PM QUADRULE_PRB FORTRAN90 version Test the routines in the QUADRULE library. TEST01 BASHFORTH_SET sets up an Adams-Bashforth rule; SUMMER carries it out. The integration interval is [0,1]. Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 1.00000000 0.00000000 0.00000000 0.00000000 0.00000000 2 1.00000000 0.50000000 -0.50000000 0.50000000 -0.50000000 3 1.00000000 0.50000000 0.33333333 -2.00000000 5.33333333 4 1.00000000 0.50000000 0.33333333 0.25000000 -8.16666667 5 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 6 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 7 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 8 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 9 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 10 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.00000000 0.00000000 0.00000000 0.00000000 1.00000000 2 0.50000000 -0.50000000 0.50000000 0.42073549 1.31606028 3 -12.00000000 25.33333333 -52.00000000 0.74308739 1.48255045 4 44.25000000 -177.16666667 625.25000000 0.71970264 1.57726812 5 -39.41666667 366.66666667-2303.08333333 0.45175383 1.63292782 6 0.16666667 -227.08333333 3238.58333333 0.23756477 1.66621912 7 0.16666667 0.14285714-1533.16666667 0.27217635 1.68635290 8 0.16666667 0.14285714 0.12500000 0.48447149 1.69862146 9 0.16666667 0.14285714 0.12500000 0.64391964 1.70613816 10 0.16666667 0.14285714 0.12500000 0.60247512 1.71076244 Order SQRT(|X|) 1 0.00000000 2 -0.50000000 3 -0.74407768 4 -0.92760648 5 -1.08201396 6 -1.21872174 7 -1.34329561 8 -1.45891434 9 -1.56758582 10 -1.67067011 TEST02 BDFC_SET sets up a Backward Difference Corrector rule; BDF_SUM carries it out. The integration interval is [0,1]. Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 2 1.00000000 0.50000000 0.50000000 0.50000000 0.50000000 3 1.00000000 0.50000000 0.33333333 0.50000000 0.33333333 4 1.00000000 0.50000000 0.33333333 0.25000000 0.83333333 5 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 6 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 7 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 8 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 9 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 10 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 1.00000000 1.00000000 1.00000000 0.84147098 2.71828183 2 0.50000000 0.50000000 0.50000000 0.42073549 1.85914091 3 0.50000000 0.33333333 0.50000000 0.42073549 1.76862748 4 -0.75000000 2.83333333 -4.75000000 0.45297068 1.74001977 5 2.41666667 -9.83333333 39.58333333 0.47174071 1.72856688 6 0.16666667 10.41666667 -86.41666667 0.47066572 1.72342294 7 0.16666667 0.14285714 57.41666667 0.46058220 1.72094841 8 0.16666667 0.14285714 1.51153598 0.45396020 1.71973247 9 0.16666667 0.14285714 1.51153598 0.45490363 1.71908412 10 0.16666667 0.14285714 1.51153598 0.45996731 1.71873841 Order SQRT(|X|) 1 1.00000000 2 0.50000000 3 0.33333333 4 0.22559223 5 0.14444121 6 0.07847649 7 0.02236210 8 -0.02563744 9 -0.06968178 10 -0.10974341 TEST03 BDFP_SET sets up a Backward Difference Predictor rule; BDF_SUM carries it out. The integration interval is [0,1]. Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 1.00000000 0.00000000 0.00000000 0.00000000 0.00000000 2 1.00000000 0.50000000 -0.50000000 0.50000000 -0.50000000 3 1.00000000 0.50000000 0.33333333 -2.00000000 5.33333333 4 1.00000000 0.50000000 0.33333333 0.25000000 -8.16666667 5 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 6 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 7 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 8 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 9 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 10 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.00000000 0.00000000 0.00000000 0.00000000 1.00000000 2 0.50000000 -0.50000000 0.50000000 0.42073549 1.31606028 3 -12.00000000 25.33333333 -52.00000000 0.74308739 1.48255045 4 44.25000000 -177.16666667 625.25000000 0.71970264 1.57726812 5 -39.41666667 366.66666667-2303.08333333 0.45175383 1.63292782 6 0.16666667 -227.08333333 3238.58333333 0.23756477 1.66621912 7 0.16666667 0.14285714-1533.16666667 0.27217635 1.68635290 8 0.16666667 0.14285714 0.12500000 0.48447149 1.69862146 9 0.16666667 0.14285714 0.12500000 0.64391964 1.70613816 10 0.16666667 0.14285714 0.12500000 0.60247512 1.71076244 Order SQRT(|X|) 1 0.00000000 2 -0.50000000 3 -0.74407768 4 -0.92760648 5 -1.08201396 6 -1.21872174 7 -1.34329561 8 -1.45891434 9 -1.56758582 10 -1.67067011 TEST04 CHEB_SET sets up a Chebyshev rule; SUM_SUB carries it out. The integration interval is [ 0.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 1.00000000 0.50000000 0.25000000 0.12500000 0.06250000 2 1.00000000 0.50000000 0.33333333 0.25000000 0.19444444 3 1.00000000 0.50000000 0.33333333 0.25000000 0.19791667 4 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 5 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 6 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 7 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 9 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.03125000 0.01562500 0.00781250 0.47942554 1.64872127 2 0.15277778 0.12037037 0.09490741 0.45958781 1.71789638 3 0.16145833 0.13411458 0.11263021 0.45965669 1.71813657 4 0.16666667 0.14259259 0.12407407 0.45969787 1.71828122 5 0.16666667 0.14272280 0.12452980 0.45969778 1.71828152 6 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 7 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 9 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 Order SQRT(|X|) 1 0.70710678 2 0.67388734 3 0.67122325 4 0.66890560 5 0.66839073 6 0.66777528 7 0.66758934 9 0.66724897 TEST05 CHEBYSHEV1_COMPUTE sets a Gauss-Chebyshev type 1 rule, SUMMER carries it out. The integration interval is [ -1.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. The weight function is 1 / sqrt ( 1 - X**2 ) Order 1 X X**2 X**3 X**4 1 3.14159265 0.00000000 0.00000000 0.00000000 0.00000000 2 3.14159265 0.00000000 1.57079633 0.00000000 0.78539816 3 3.14159265 0.00000000 1.57079633 0.00000000 1.17809725 4 3.14159265 -0.00000000 1.57079633 0.00000000 1.17809725 5 3.14159265 0.00000000 1.57079633 0.00000000 1.17809725 6 3.14159265 0.00000000 1.57079633 0.00000000 1.17809725 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.00000000 0.00000000 0.00000000 0.00000000 3.14159265 2 0.00000000 0.39269908 0.00000000 0.00000000 3.96026605 3 0.00000000 0.88357293 0.00000000 0.00000000 3.97732196 4 0.00000000 0.98174770 0.00000000 0.00000000 3.97746263 5 0.00000000 0.98174770 0.00000000 0.00000000 3.97746326 6 0.00000000 0.98174770 0.00000000 -0.00000000 3.97746326 Order SQRT(|X|) 1 0.00000002 2 2.64175400 3 1.94905427 4 2.48154505 5 2.18892706 6 2.44254041 TEST06 CHEBYSHEV2_COMPUTE sets a Gauss-Chebyshev type 2 rule; SUMMER carries it out. The integration interval is [ -1.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. The weight function is sqrt ( 1 - X**2 ) Order 1 X X**2 X**3 X**4 1 1.57079633 0.00000000 0.00000000 0.00000000 0.00000000 2 1.57079633 0.00000000 0.39269908 0.00000000 0.09817477 3 1.57079633 0.00000000 0.39269908 0.00000000 0.19634954 4 1.57079633 -0.00000000 0.39269908 0.00000000 0.19634954 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.00000000 0.00000000 0.00000000 0.00000000 1.57079633 2 0.00000000 0.02454369 0.00000000 0.00000000 1.77127072 3 0.00000000 0.09817477 0.00000000 0.00000000 1.77546468 4 -0.00000000 0.12271846 0.00000000 0.00000000 1.77549953 Order SQRT(|X|) 1 0.00000001 2 1.11072073 3 0.66043851 4 1.02235409 TEST07 CHEBYSHEV3_COMPUTE sets a Gauss-Chebyshev type 3 rule, SUMMER carries it out. The integration interval is [ -1.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. The weight function is 1 / sqrt ( 1 - X**2 ) Order 1 X X**2 X**3 X**4 1 3.14159265 0.00000000 0.00000000 0.00000000 0.00000000 2 3.14159265 0.00000000 3.14159265 0.00000000 3.14159265 3 3.14159265 0.00000000 1.57079633 0.00000000 1.57079633 4 3.14159265 0.00000000 1.57079633 0.00000000 1.17809725 5 3.14159265 0.00000000 1.57079633 -0.00000000 1.17809725 6 3.14159265 0.00000000 1.57079633 0.00000000 1.17809725 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.00000000 0.00000000 0.00000000 0.00000000 3.14159265 2 0.00000000 3.14159265 0.00000000 0.00000000 4.84773079 3 0.00000000 1.57079633 0.00000000 0.00000000 3.99466172 4 0.00000000 1.07992247 0.00000000 0.00000000 3.97760456 5 0.00000000 0.98174770 0.00000000 0.00000000 3.97746389 6 0.00000000 0.98174770 0.00000000 0.00000000 3.97746326 Order SQRT(|X|) 1 0.00000000 2 3.14159265 3 1.57079634 4 2.52815853 5 2.10627517 6 2.45716112 TEST0725 CLENSHAW_CURTIS_COMPUTE computes a Clenshaw-Curtis quadrature rule over [-1,1] of given order. Order W X 1 2.000000000000000 0.0000000000000000 2 1.0000000000000000 -1.0000000000000000 1.0000000000000000 1.0000000000000000 3 0.3333333333333334 -1.0000000000000000 1.333333333333333 0.0000000000000001 0.3333333333333334 1.0000000000000000 4 0.1111111111111111 -1.0000000000000000 0.8888888888888892 -0.4999999999999998 0.8888888888888888 0.5000000000000001 0.1111111111111111 1.0000000000000000 5 0.6666666666666668E-01 -1.0000000000000000 0.5333333333333334 -0.7071067811865475 0.7999999999999999 0.0000000000000001 0.5333333333333333 0.7071067811865475 0.6666666666666668E-01 1.0000000000000000 6 0.4000000000000001E-01 -1.0000000000000000 0.3607430412000113 -0.8090169943749473 0.5992569587999887 -0.3090169943749473 0.5992569587999889 0.3090169943749475 0.3607430412000112 0.8090169943749475 0.4000000000000001E-01 1.0000000000000000 7 0.2857142857142858E-01 -1.0000000000000000 0.2539682539682539 -0.8660254037844387 0.4571428571428573 -0.4999999999999998 0.5206349206349206 0.0000000000000001 0.4571428571428571 0.5000000000000001 0.2539682539682539 0.8660254037844387 0.2857142857142858E-01 1.0000000000000000 8 0.2040816326530613E-01 -1.0000000000000000 0.1901410072182084 -0.9009688679024190 0.3522424237181591 -0.6234898018587335 0.4372084057983264 -0.2225209339563143 0.4372084057983264 0.2225209339563145 0.3522424237181591 0.6234898018587336 0.1901410072182084 0.9009688679024191 0.2040816326530613E-01 1.0000000000000000 9 0.1587301587301588E-01 -1.0000000000000000 0.1462186492160182 -0.9238795325112867 0.2793650793650794 -0.7071067811865475 0.3617178587204898 -0.3826834323650897 0.3936507936507936 0.0000000000000001 0.3617178587204897 0.3826834323650898 0.2793650793650794 0.7071067811865475 0.1462186492160182 0.9238795325112867 0.1587301587301588E-01 1.0000000000000000 10 0.1234567901234569E-01 -1.0000000000000000 0.1165674565720372 -0.9396926207859083 0.2252843233381044 -0.7660444431189779 0.3019400352733687 -0.4999999999999998 0.3438625058041442 -0.1736481776669303 0.3438625058041442 0.1736481776669304 0.3019400352733685 0.5000000000000001 0.2252843233381044 0.7660444431189780 0.1165674565720371 0.9396926207859084 0.1234567901234569E-01 1.0000000000000000 TEST075 CLENSHAW_CURTIS_SET sets up a Clenshaw-Curtis rule; SUMMER carries it out. The integration interval is [-1,1]. Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 2.00000000 0.00000000 0.00000000 0.00000000 0.00000000 2 2.00000000 0.00000000 2.00000000 0.00000000 2.00000000 3 2.00000000 0.00000000 0.66666667 0.00000000 0.66666667 4 2.00000000 -0.00000000 0.66666667 0.00000000 0.33333333 5 2.00000000 0.00000000 0.66666667 0.00000000 0.40000000 6 2.00000000 -0.00000000 0.66666667 -0.00000000 0.40000000 7 2.00000000 -0.00000000 0.66666667 -0.00000000 0.40000000 8 2.00000000 -0.00000000 0.66666667 0.00000000 0.40000000 9 2.00000000 0.00000000 0.66666667 0.00000000 0.40000000 10 2.00000000 -0.00000000 0.66666667 0.00000000 0.40000000 11 2.00000000 -0.00000000 0.66666667 0.00000000 0.40000000 12 2.00000000 -0.00000000 0.66666667 -0.00000000 0.40000000 13 2.00000000 0.00000000 0.66666667 -0.00000000 0.40000000 14 2.00000000 -0.00000000 0.66666667 -0.00000000 0.40000000 15 2.00000000 0.00000000 0.66666667 -0.00000000 0.40000000 16 2.00000000 -0.00000000 0.66666667 -0.00000000 0.40000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.00000000 0.00000000 0.00000000 0.00000000 2.00000000 2 0.00000000 2.00000000 0.00000000 0.00000000 3.08616127 3 0.00000000 0.66666667 0.00000000 0.00000000 2.36205376 4 -0.00000000 0.25000000 0.00000000 -0.00000000 2.34757519 5 0.00000000 0.26666667 -0.00000000 -0.00000000 2.35037538 6 -0.00000000 0.28333333 -0.00000000 -0.00000000 2.35039884 7 -0.00000000 0.28571429 0.00000000 0.00000000 2.35040237 8 0.00000000 0.28571429 0.00000000 -0.00000000 2.35040238 9 0.00000000 0.28571429 0.00000000 -0.00000000 2.35040239 10 0.00000000 0.28571429 0.00000000 0.00000000 2.35040239 11 0.00000000 0.28571429 -0.00000000 -0.00000000 2.35040239 12 0.00000000 0.28571429 0.00000000 -0.00000000 2.35040239 13 0.00000000 0.28571429 0.00000000 0.00000000 2.35040239 14 -0.00000000 0.28571429 -0.00000000 0.00000000 2.35040239 15 0.00000000 0.28571429 -0.00000000 -0.00000000 2.35040239 16 -0.00000000 0.28571429 -0.00000000 0.00000000 2.35040239 Order SQRT(|X|) 1 0.00000000 2 2.00000000 3 0.66666667 4 1.47930094 5 1.03028951 6 1.39518880 7 1.17632867 8 1.37052953 9 1.23019456 10 1.35863300 11 1.26011262 12 1.35202995 13 1.27747460 14 1.34785403 15 1.28912023 16 1.34504148 TEST076 FEJER1_COMPUTE computes a Fejer type 1 rule. FEJER1_SET looks up a Fejer type 1 rule. Compare: (W1,X1) from FEJER1_SET, (W2,X2) from FEJER1_COMPUTE. Order W1 W2 X1 X2 1 2.00000 2.00000 0.00000 0.00000 2 1.00000 1.00000 -0.707107 -0.707107 1.00000 1.00000 0.707107 0.707107 3 0.444444 0.444444 -0.866025 -0.866025 1.11111 1.11111 0.00000 0.612323E-16 0.444444 0.444444 0.866025 0.866025 4 0.264298 0.264298 -0.923880 -0.923880 0.735702 0.735702 -0.382683 -0.382683 0.735702 0.735702 0.382683 0.382683 0.264298 0.264298 0.923880 0.923880 5 0.167781 0.167781 -0.951057 -0.951057 0.525552 0.525552 -0.587785 -0.587785 0.613333 0.613333 0.00000 0.612323E-16 0.525552 0.525552 0.587785 0.587785 0.167781 0.167781 0.951057 0.951057 6 0.118661 0.118661 -0.965926 -0.965926 0.377778 0.377778 -0.707107 -0.707107 0.503561 0.503561 -0.258819 -0.258819 0.503561 0.503561 0.258819 0.258819 0.377778 0.377778 0.707107 0.707107 0.118661 0.118661 0.965926 0.965926 7 0.867162E-01 0.867162E-01 -0.974928 -0.974928 0.287831 0.287831 -0.781831 -0.781831 0.398242 0.398242 -0.433884 -0.433884 0.454422 0.454422 0.00000 0.612323E-16 0.398242 0.398242 0.433884 0.433884 0.287831 0.287831 0.781831 0.781831 0.867162E-01 0.867162E-01 0.974928 0.974928 8 0.669829E-01 0.669829E-01 -0.980785 -0.980785 0.222988 0.222988 -0.831470 -0.831470 0.324153 0.324153 -0.555570 -0.555570 0.385877 0.385877 -0.195090 -0.195090 0.385877 0.385877 0.195090 0.195090 0.324153 0.324153 0.555570 0.555570 0.222988 0.222988 0.831470 0.831470 0.669829E-01 0.669829E-01 0.980785 0.980785 9 0.527366E-01 0.527366E-01 -0.984808 -0.984808 0.179189 0.179189 -0.866025 -0.866025 0.264037 0.264037 -0.642788 -0.642788 0.330845 0.330845 -0.342020 -0.342020 0.346384 0.346384 0.00000 0.612323E-16 0.330845 0.330845 0.342020 0.342020 0.264037 0.264037 0.642788 0.642788 0.179189 0.179189 0.866025 0.866025 0.527366E-01 0.527366E-01 0.984808 0.984808 10 0.429391E-01 0.429391E-01 -0.987688 -0.987688 0.145875 0.145875 -0.891007 -0.891007 0.220317 0.220317 -0.707107 -0.707107 0.280879 0.280879 -0.453990 -0.453990 0.309989 0.309989 -0.156434 -0.156434 0.309989 0.309989 0.156434 0.156434 0.280879 0.280879 0.453990 0.453990 0.220317 0.220317 0.707107 0.707107 0.145875 0.145875 0.891007 0.891007 0.429391E-01 0.429391E-01 0.987688 0.987688 TEST078 FEJER2_COMPUTE computes a Fejer type 2 rule. FEJER2_SET looks up a Fejer type 2 rule. Compare: (W1,X1) from FEJER2_SET, (W2,X2) from FEJER2_COMPUTE. Order W1 W2 X1 X2 1 2.00000 2.00000 0.00000 0.00000 2 1.00000 1.00000 -0.500000 -0.500000 1.00000 1.00000 0.500000 0.500000 3 0.666667 0.666667 -0.707107 -0.707107 0.666667 0.666667 0.00000 0.612323E-16 0.666667 0.666667 0.707107 0.707107 4 0.425464 0.425464 -0.809017 -0.809017 0.574536 0.574536 -0.309017 -0.309017 0.574536 0.574536 0.309017 0.309017 0.425464 0.425464 0.809017 0.809017 5 0.311111 0.311111 -0.866025 -0.866025 0.400000 0.400000 -0.500000 -0.500000 0.577778 0.577778 0.00000 0.612323E-16 0.400000 0.400000 0.500000 0.500000 0.311111 0.311111 0.866025 0.866025 6 0.226915 0.226915 -0.900969 -0.900969 0.326794 0.326794 -0.623490 -0.623490 0.446291 0.446291 -0.222521 -0.222521 0.446291 0.446291 0.222521 0.222521 0.326794 0.326794 0.623490 0.623490 0.226915 0.226915 0.900969 0.900969 7 0.177965 0.177965 -0.923880 -0.923880 0.247619 0.247619 -0.707107 -0.707107 0.393464 0.393464 -0.382683 -0.382683 0.361905 0.361905 0.00000 0.612323E-16 0.393464 0.393464 0.382683 0.382683 0.247619 0.247619 0.707107 0.707107 0.177965 0.177965 0.923880 0.923880 8 0.139770 0.139770 -0.939693 -0.939693 0.206370 0.206370 -0.766044 -0.766044 0.314286 0.314286 -0.500000 -0.500000 0.339575 0.339575 -0.173648 -0.173648 0.339575 0.339575 0.173648 0.173648 0.314286 0.314286 0.500000 0.500000 0.206370 0.206370 0.766044 0.766044 0.139770 0.139770 0.939693 0.939693 9 0.114781 0.114781 -0.951057 -0.951057 0.165433 0.165433 -0.809017 -0.809017 0.273790 0.273790 -0.587785 -0.587785 0.279011 0.279011 -0.309017 -0.309017 0.333968 0.333968 0.00000 0.612323E-16 0.279011 0.279011 0.309017 0.309017 0.273790 0.273790 0.587785 0.587785 0.165433 0.165433 0.809017 0.809017 0.114781 0.114781 0.951057 0.951057 10 0.944195E-01 0.944195E-01 -0.959493 -0.959493 0.141135 0.141135 -0.841254 -0.841254 0.226387 0.226387 -0.654861 -0.654861 0.253051 0.253051 -0.415415 -0.415415 0.285007 0.285007 -0.142315 -0.142315 0.285007 0.285007 0.142315 0.142315 0.253051 0.253051 0.415415 0.415415 0.226387 0.226387 0.654861 0.654861 0.141135 0.141135 0.841254 0.841254 0.944195E-01 0.944195E-01 0.959493 0.959493 TEST079 GEGENBAUER_COMPUTE computes a Gauss-Gegenbauer rule; The printed output of this test can be inserted into a FORTRAN program. ! ! Abscissas X and weights W for a Gauss Gegenbauer rule ! of order = 5 ! with ALPHA = 0.500000 ! x( 1) = -0.8660254037844387 x( 2) = -0.5000000000000000 x( 3) = 0.000000000000000 x( 4) = 0.5000000000000000 x( 5) = 0.8660254037844387 w( 1) = 0.1308996938995740 w( 2) = 0.3926990816987242 w( 3) = 0.5235987755982989 w( 4) = 0.3926990816987242 w( 5) = 0.1308996938995745 TEST079 GEGENBAUER_COMPUTE computes a Gauss-Gegenbauer rule; The printed output of this test can be inserted into a FORTRAN program. ! ! Abscissas X and weights W for a Gauss Gegenbauer rule ! of order = 10 ! with ALPHA = -0.500000 ! x( 1) = -0.9876883405951378 x( 2) = -0.8910065241883679 x( 3) = -0.7071067811865476 x( 4) = -0.4539904997395468 x( 5) = -0.1564344650402309 x( 6) = 0.1564344650402309 x( 7) = 0.4539904997395468 x( 8) = 0.7071067811865476 x( 9) = 0.8910065241883679 x( 10) = 0.9876883405951378 w( 1) = 0.3141592653589749 w( 2) = 0.3141592653589789 w( 3) = 0.3141592653589792 w( 4) = 0.3141592653589794 w( 5) = 0.3141592653589794 w( 6) = 0.3141592653589794 w( 7) = 0.3141592653589789 w( 8) = 0.3141592653589792 w( 9) = 0.3141592653589789 w( 10) = 0.3141592653589749 TEST08 HERMITE_COMPUTE computes a Gauss-Hermite rule; SUMMER carries it out. The integration interval is ( -Infinity, +Infinity ). The weight function is exp ( - X**2 ) Order 1 X X**2 X**3 X**4 1 1.77245385 0.00000000 0.00000000 0.00000000 0.00000000 2 1.77245385 0.00000000 0.88622693 0.00000000 0.44311346 3 1.77245385 0.00000000 0.88622693 0.00000000 1.32934039 4 1.77245385 0.00000000 0.88622693 0.00000000 1.32934039 5 1.77245385 -0.00000000 0.88622693 0.00000000 1.32934039 6 1.77245385 -0.00000000 0.88622693 -0.00000000 1.32934039 7 1.77245385 0.00000000 0.88622693 0.00000000 1.32934039 8 1.77245385 -0.00000000 0.88622693 0.00000000 1.32934039 9 1.77245385 -0.00000000 0.88622693 0.00000000 1.32934039 10 1.77245385 0.00000000 0.88622693 0.00000000 1.32934039 11 1.77245385 -0.00000000 0.88622693 -0.00000000 1.32934039 12 1.77245385 -0.00000000 0.88622693 0.00000000 1.32934039 13 1.77245385 0.00000000 0.88622693 -0.00000000 1.32934039 14 1.77245385 0.00000000 0.88622693 0.00000000 1.32934039 15 1.77245385 0.00000000 0.88622693 0.00000000 1.32934039 16 1.77245385 0.00000000 0.88622693 0.00000000 1.32934039 17 1.77245385 0.00000000 0.88622693 0.00000000 1.32934039 18 1.77245385 0.00000000 0.88622693 -0.00000000 1.32934039 19 1.77245385 0.00000000 0.88622693 0.00000000 1.32934039 20 1.77245385 0.00000000 0.88622693 -0.00000000 1.32934039 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.00000000 0.00000000 0.00000000 0.00000000 1.77245385 2 0.00000000 0.22155673 0.00000000 0.00000000 2.23434086 3 0.00000000 1.99401058 0.00000000 0.00000000 2.27380139 4 0.00000000 3.32335097 0.00000000 0.00000000 2.27580168 5 0.00000000 3.32335097 0.00000000 -0.00000000 2.27587373 6 -0.00000000 3.32335097 0.00000000 -0.00000000 2.27587575 7 -0.00000000 3.32335097 -0.00000000 -0.00000000 2.27587579 8 -0.00000000 3.32335097 0.00000000 0.00000000 2.27587579 9 0.00000000 3.32335097 0.00000000 -0.00000000 2.27587579 10 -0.00000000 3.32335097 0.00000000 -0.00000000 2.27587579 11 -0.00000000 3.32335097 -0.00000000 -0.00000000 2.27587579 12 0.00000000 3.32335097 0.00000000 -0.00000000 2.27587579 13 0.00000000 3.32335097 -0.00000000 -0.00000000 2.27587579 14 0.00000000 3.32335097 -0.00000000 0.00000000 2.27587579 15 -0.00000000 3.32335097 0.00000000 0.00000000 2.27587579 16 -0.00000000 3.32335097 -0.00000000 0.00000000 2.27587579 17 -0.00000000 3.32335097 0.00000000 -0.00000000 2.27587579 18 0.00000000 3.32335097 0.00000000 0.00000000 2.27587579 19 -0.00000000 3.32335097 -0.00000000 0.00000000 2.27587579 20 0.00000000 3.32335097 -0.00000000 0.00000000 2.27587579 Order SQRT(|X|) 1 0.00000000 2 1.49045009 3 0.65384754 4 1.37497932 5 0.82747983 6 1.33399910 7 0.91327495 8 1.31223221 9 0.96550308 10 1.29850427 11 1.00104727 12 1.28896093 13 1.02699402 14 1.28189459 15 1.04687269 16 1.27642527 17 1.06265103 18 1.27205059 19 1.07551821 20 1.26846158 TEST085 HERMITE_COMPUTE computes a Gauss-Hermite rule which is appropriate for integrands of the form f(x) * exp(-x**2) from -infinity to infinity. HERMITE_INTEGRAL determines the exact value of this integal when f(x) = x^n. N Order Estimate Exact Error 0 1 1.77245 1.77245 0.00000 0 2 1.77245 1.77245 0.222045E-15 0 3 1.77245 1.77245 0.222045E-15 0 4 1.77245 1.77245 0.00000 0 5 1.77245 1.77245 0.222045E-15 0 6 1.77245 1.77245 0.444089E-15 0 7 1.77245 1.77245 0.00000 0 8 1.77245 1.77245 0.444089E-15 0 9 1.77245 1.77245 0.00000 0 10 1.77245 1.77245 0.222045E-15 2 1 0.00000 0.886227 0.886227 2 2 0.886227 0.886227 0.333067E-15 2 3 0.886227 0.886227 0.333067E-15 2 4 0.886227 0.886227 0.00000 2 5 0.886227 0.886227 0.222045E-15 2 6 0.886227 0.886227 0.444089E-15 2 7 0.886227 0.886227 0.555112E-15 2 8 0.886227 0.886227 0.777156E-15 2 9 0.886227 0.886227 0.00000 2 10 0.886227 0.886227 0.111022E-15 4 1 0.00000 1.32934 1.32934 4 2 0.443113 1.32934 0.886227 4 3 1.32934 1.32934 0.222045E-15 4 4 1.32934 1.32934 0.444089E-15 4 5 1.32934 1.32934 0.222045E-15 4 6 1.32934 1.32934 0.666134E-15 4 7 1.32934 1.32934 0.133227E-14 4 8 1.32934 1.32934 0.155431E-14 4 9 1.32934 1.32934 0.222045E-15 4 10 1.32934 1.32934 0.444089E-15 6 1 0.00000 3.32335 3.32335 6 2 0.221557 3.32335 3.10179 6 3 1.99401 3.32335 1.32934 6 4 3.32335 3.32335 0.00000 6 5 3.32335 3.32335 0.444089E-15 6 6 3.32335 3.32335 0.444089E-15 6 7 3.32335 3.32335 0.310862E-14 6 8 3.32335 3.32335 0.133227E-14 6 9 3.32335 3.32335 0.133227E-14 6 10 3.32335 3.32335 0.444089E-15 8 1 0.00000 11.6317 11.6317 8 2 0.110778 11.6317 11.5210 8 3 2.99102 11.6317 8.64071 8 4 8.97305 11.6317 2.65868 8 5 11.6317 11.6317 0.355271E-14 8 6 11.6317 11.6317 0.177636E-14 8 7 11.6317 11.6317 0.710543E-14 8 8 11.6317 11.6317 0.355271E-14 8 9 11.6317 11.6317 0.888178E-14 8 10 11.6317 11.6317 0.177636E-14 10 1 0.00000 52.3428 52.3428 10 2 0.553892E-01 52.3428 52.2874 10 3 4.48652 52.3428 47.8563 10 4 24.4266 52.3428 27.9161 10 5 45.6961 52.3428 6.64670 10 6 52.3428 52.3428 0.00000 10 7 52.3428 52.3428 0.00000 10 8 52.3428 52.3428 0.00000 10 9 52.3428 52.3428 0.639488E-13 10 10 52.3428 52.3428 0.284217E-13 TEST087 HERMITE_COMPUTE computes a Gauss-Hermite rule; Compute the data for ORDER = 31 xtab( 1) = -6.9956801237185404218621442851145 xtab( 2) = -6.2750787049428602415446221129969 xtab( 3) = -5.6739614446185884233386786945630 xtab( 4) = -5.1335955771123806812283874023706 xtab( 5) = -4.6315595063128602859592319873627 xtab( 6) = -4.1562717558181452304211234149989 xtab( 7) = -3.7007434032314692196052874351153 xtab( 8) = -3.2603207323135405815150988928508 xtab( 9) = -2.8316804533902053542249177553458 xtab( 10) = -2.4123177054804201269178065558663 xtab( 11) = -2.0002585489356388137593967257999 xtab( 12) = -1.5938858604721397593806386794313 xtab( 13) = -1.1918269983500464626047232741257 xtab( 14) = -0.79287697691530889532174342093640 xtab( 15) = -0.39594273647142314009883534708933 xtab( 16) = 0.23509887016445750159374730744445E-37 xtab( 17) = 0.39594273647142314009883534708933 xtab( 18) = 0.79287697691530889532174342093640 xtab( 19) = 1.1918269983500464626047232741257 xtab( 20) = 1.5938858604721397593806386794313 xtab( 21) = 2.0002585489356388137593967257999 xtab( 22) = 2.4123177054804201269178065558663 xtab( 23) = 2.8316804533902053542249177553458 xtab( 24) = 3.2603207323135405815150988928508 xtab( 25) = 3.7007434032314692196052874351153 xtab( 26) = 4.1562717558181452304211234149989 xtab( 27) = 4.6315595063128602859592319873627 xtab( 28) = 5.1335955771123806812283874023706 xtab( 29) = 5.6739614446185884233386786945630 xtab( 30) = 6.2750787049428602415446221129969 xtab( 31) = 6.9956801237185404218621442851145 weight( 1) = 0.46189683944642099497340397411543E-21 weight( 2) = 0.51106090079271837164946072151953E-17 weight( 3) = 0.58995564987539064401447923138504E-14 weight( 4) = 0.18603735214521610549773408705562E-11 weight( 5) = 0.23524920032086438601945290091373E-09 weight( 6) = 0.14611988344910416428744870875282E-07 weight( 7) = 0.50437125589398166310844962012538E-06 weight( 8) = 0.10498602757675659015337669976997E-04 weight( 9) = 0.13952090395047151566432219738090E-03 weight( 10) = 0.12336833073068973355007615921863E-02 weight( 11) = 0.74827999140352685278343436436899E-02 weight( 12) = 0.31847230731300614492429446045207E-01 weight( 13) = 0.96717948160871244911440669511649E-01 weight( 14) = 0.21213278866876639217053934771684 weight( 15) = 0.33877265789411031837730092775018 weight( 16) = 0.39577855609861262164983486400160 weight( 17) = 0.33877265789411031837730092775018 weight( 18) = 0.21213278866876639217053934771684 weight( 19) = 0.96717948160871244911440669511649E-01 weight( 20) = 0.31847230731300614492429446045207E-01 weight( 21) = 0.74827999140352685278343436436899E-02 weight( 22) = 0.12336833073068973355007615921863E-02 weight( 23) = 0.13952090395047151566432219738090E-03 weight( 24) = 0.10498602757675659015337669976997E-04 weight( 25) = 0.50437125589398166310844962012538E-06 weight( 26) = 0.14611988344910416428744870875282E-07 weight( 27) = 0.23524920032086438601945290091373E-09 weight( 28) = 0.18603735214521610549773408705562E-11 weight( 29) = 0.58995564987539064401447923138504E-14 weight( 30) = 0.51106090079271837164946072151953E-17 weight( 31) = 0.46189683944642099497340397411543E-21 TEST087 HERMITE_COMPUTE computes a Gauss-Hermite rule; Compute the data for ORDER = 63 xtab( 1) = -10.435499877854168104818199935835 xtab( 2) = -9.8028759912974958723452800768428 xtab( 3) = -9.2792019543050390240068736602552 xtab( 4) = -8.8118581437284539958909590495750 xtab( 5) = -8.3807683451863219659117021365091 xtab( 6) = -7.9755950801420372187067187041976 xtab( 7) = -7.5901395198641070649614448484499 xtab( 8) = -7.2203167078889674002084575477056 xtab( 9) = -6.8632544331795370950999313208740 xtab( 10) = -6.5168348106821163412405439885333 xtab( 11) = -6.1794379922705973129382073238958 xtab( 12) = -5.8497884000810671523140626959503 xtab( 13) = -5.5268572526403030309438690892421 xtab( 14) = -5.2097979830408354118276292865630 xtab( 15) = -4.8979018644975740315317125350703 xtab( 16) = -4.5905665744435193431627340032719 xtab( 17) = -4.2872733352824399233327312686015 xtab( 18) = -3.9875699104197157573992171819555 xtab( 19) = -3.6910577000963464833205307513708 xtab( 20) = -3.3973817713303913201627892703982 xtab( 21) = -3.1062230279282565525988957233494 xtab( 22) = -2.8172919672837979376822659105528 xtab( 23) = -2.5303236304712011239814728469355 xtab( 24) = -2.2450734604812065953183264355175 xtab( 25) = -1.9613138583081486387982295127586 xtab( 26) = -1.6788312791720136907258620340144 xtab( 27) = -1.3974237486049625456274725365802 xtab( 28) = -1.1168987050996461718455066147726 xtab( 29) = -0.83707109558947612182322473017848 xtab( 30) = -0.55776166427908224765275235768058 xtab( 31) = -0.27879538567115219604986009471759 xtab( 32) = 0.0000000000000000000000000000000 xtab( 33) = 0.27879538567115219604986009471759 xtab( 34) = 0.55776166427908224765275235768058 xtab( 35) = 0.83707109558947612182322473017848 xtab( 36) = 1.1168987050996461718455066147726 xtab( 37) = 1.3974237486049625456274725365802 xtab( 38) = 1.6788312791720136907258620340144 xtab( 39) = 1.9613138583081486387982295127586 xtab( 40) = 2.2450734604812065953183264355175 xtab( 41) = 2.5303236304712011239814728469355 xtab( 42) = 2.8172919672837979376822659105528 xtab( 43) = 3.1062230279282565525988957233494 xtab( 44) = 3.3973817713303913201627892703982 xtab( 45) = 3.6910577000963464833205307513708 xtab( 46) = 3.9875699104197157573992171819555 xtab( 47) = 4.2872733352824399233327312686015 xtab( 48) = 4.5905665744435193431627340032719 xtab( 49) = 4.8979018644975740315317125350703 xtab( 50) = 5.2097979830408354118276292865630 xtab( 51) = 5.5268572526403030309438690892421 xtab( 52) = 5.8497884000810671523140626959503 xtab( 53) = 6.1794379922705973129382073238958 xtab( 54) = 6.5168348106821163412405439885333 xtab( 55) = 6.8632544331795370950999313208740 xtab( 56) = 7.2203167078889674002084575477056 xtab( 57) = 7.5901395198641070649614448484499 xtab( 58) = 7.9755950801420372187067187041976 xtab( 59) = 8.3807683451863219659117021365091 xtab( 60) = 8.8118581437284539958909590495750 xtab( 61) = 9.2792019543050390240068736602552 xtab( 62) = 9.8028759912974958723452800768428 xtab( 63) = 10.435499877854168104818199935835 weight( 1) = 0.37099206434902295280580538302007E-47 weight( 2) = 0.10400778615224689994701507366390E-41 weight( 3) = 0.19796804708319766706772828841736E-37 weight( 4) = 0.84687478191904030916243193619612E-34 weight( 5) = 0.13071305930819667718756471743980E-30 weight( 6) = 0.93437837175657111926953190858057E-28 weight( 7) = 0.36027426635284283637656485510562E-25 weight( 8) = 0.82963863116207885949558055037853E-23 weight( 9) = 0.12266629909143226010129960636013E-20 weight( 10) = 0.12288435628835049190918292492783E-18 weight( 11) = 0.86925536958456573133355438472530E-17 weight( 12) = 0.44857058689315425593168220392976E-15 weight( 13) = 0.17335817955788915844393159209596E-13 weight( 14) = 0.51265062385197196052394320538762E-12 weight( 15) = 0.11808921844569562902500534629730E-10 weight( 16) = 0.21508698297874489676337427884284E-09 weight( 17) = 0.31371929535382395402978493941610E-08 weight( 18) = 0.37041625984896193197382947967164E-07 weight( 19) = 0.35734732949990376356253353289094E-06 weight( 20) = 0.28393114498468796279772533713848E-05 weight( 21) = 0.18709113003788475860979939602302E-04 weight( 22) = 0.10284880800685466375934268290493E-03 weight( 23) = 0.47411702610319954526510444203780E-03 weight( 24) = 0.18409222622441843118801241274696E-02 weight( 25) = 0.60436044551374747471617077110295E-02 weight( 26) = 0.16829299199651866908444830528424E-01 weight( 27) = 0.39858264027816399377446288099236E-01 weight( 28) = 0.80467087994199576472809098959260E-01 weight( 29) = 0.13871950817658254551112406716129 weight( 30) = 0.20448695346897091673632473884936 weight( 31) = 0.25799889943137954695373537106207 weight( 32) = 0.27876694884924768658152061107103 weight( 33) = 0.25799889943137954695373537106207 weight( 34) = 0.20448695346897091673632473884936 weight( 35) = 0.13871950817658254551112406716129 weight( 36) = 0.80467087994199576472809098959260E-01 weight( 37) = 0.39858264027816399377446288099236E-01 weight( 38) = 0.16829299199651866908444830528424E-01 weight( 39) = 0.60436044551374747471617077110295E-02 weight( 40) = 0.18409222622441843118801241274696E-02 weight( 41) = 0.47411702610319954526510444203780E-03 weight( 42) = 0.10284880800685466375934268290493E-03 weight( 43) = 0.18709113003788475860979939602302E-04 weight( 44) = 0.28393114498468796279772533713848E-05 weight( 45) = 0.35734732949990376356253353289094E-06 weight( 46) = 0.37041625984896193197382947967164E-07 weight( 47) = 0.31371929535382395402978493941610E-08 weight( 48) = 0.21508698297874489676337427884284E-09 weight( 49) = 0.11808921844569562902500534629730E-10 weight( 50) = 0.51265062385197196052394320538762E-12 weight( 51) = 0.17335817955788915844393159209596E-13 weight( 52) = 0.44857058689315425593168220392976E-15 weight( 53) = 0.86925536958456573133355438472530E-17 weight( 54) = 0.12288435628835049190918292492783E-18 weight( 55) = 0.12266629909143226010129960636013E-20 weight( 56) = 0.82963863116207885949558055037853E-23 weight( 57) = 0.36027426635284283637656485510562E-25 weight( 58) = 0.93437837175657111926953190858057E-28 weight( 59) = 0.13071305930819667718756471743980E-30 weight( 60) = 0.84687478191904030916243193619612E-34 weight( 61) = 0.19796804708319766706772828841736E-37 weight( 62) = 0.10400778615224689994701507366390E-41 weight( 63) = 0.37099206434902295280580538302007E-47 TEST09 HERMITE_SET sets up a Gauss-Hermite rule; SUMMER carries it out. The integration interval is ( -Infinity, +Infinity ). The weight function is exp ( - X**2 ) Order 1 X X**2 X**3 X**4 1 1.77245385 0.00000000 0.00000000 0.00000000 0.00000000 2 1.77245385 0.00000000 0.88622693 0.00000000 0.44311346 3 1.77245385 0.00000000 0.88622693 0.00000000 1.32934039 4 1.77245385 0.00000000 0.88622693 0.00000000 1.32934039 5 1.77245385 0.00000000 0.88622693 0.00000000 1.32934039 6 1.77245385 -0.00000000 0.88622693 -0.00000000 1.32934039 7 1.77245385 0.00000000 0.88622693 -0.00000000 1.32934039 8 1.77245385 0.00000000 0.88622693 -0.00000000 1.32934039 9 1.77245385 -0.00000000 0.88622693 -0.00000000 1.32934039 10 1.77245385 0.00000000 0.88622693 0.00000000 1.32934039 11 1.77245385 -0.00000000 0.88622693 0.00000000 1.32934039 12 1.77245385 0.00000000 0.88622693 -0.00000000 1.32934039 13 1.77245385 0.00000000 0.88622693 -0.00000000 1.32934039 14 1.77245385 0.00000000 0.88622693 -0.00000000 1.32934039 15 1.77245385 0.00000000 0.88622693 0.00000000 1.32934039 16 1.77245385 0.00000000 0.88622693 0.00000000 1.32934039 17 1.77245385 -0.00000000 0.88622693 -0.00000000 1.32934039 18 1.77245385 -0.00000000 0.88622693 0.00000000 1.32934039 19 1.77245385 0.00000000 0.88622693 0.00000000 1.32934039 20 1.77245385 -0.00000000 0.88622693 0.00000000 1.32934039 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.00000000 0.00000000 0.00000000 0.00000000 1.77245385 2 0.00000000 0.22155673 0.00000000 0.00000000 2.23434086 3 0.00000000 1.99401058 0.00000000 0.00000000 2.27380139 4 0.00000000 3.32335097 0.00000000 -0.00000000 2.27580168 5 0.00000000 3.32335097 0.00000000 -0.00000000 2.27587373 6 0.00000000 3.32335097 0.00000000 0.00000000 2.27587575 7 -0.00000000 3.32335097 0.00000000 0.00000000 2.27587579 8 -0.00000000 3.32335097 0.00000000 0.00000000 2.27587579 9 0.00000000 3.32335097 0.00000000 -0.00000000 2.27587579 10 -0.00000000 3.32335097 0.00000000 -0.00000000 2.27587579 11 0.00000000 3.32335097 0.00000000 0.00000000 2.27587579 12 0.00000000 3.32335097 -0.00000000 -0.00000000 2.27587579 13 0.00000000 3.32335097 -0.00000000 -0.00000000 2.27587579 14 0.00000000 3.32335097 0.00000000 -0.00000000 2.27587579 15 -0.00000000 3.32335097 -0.00000000 0.00000000 2.27587579 16 -0.00000000 3.32335097 0.00000000 -0.00000000 2.27587579 17 0.00000000 3.32335097 -0.00000000 0.00000000 2.27587579 18 0.00000000 3.32335097 -0.00000000 0.00000000 2.27587579 19 -0.00000000 3.32335097 -0.00000000 0.00000000 2.27587579 20 0.00000000 3.32335097 0.00000000 0.00000000 2.27587579 Order SQRT(|X|) 1 0.00000000 2 1.49045009 3 0.65384754 4 1.37497932 5 0.82747983 6 1.33399910 7 0.91327495 8 1.31223221 9 0.96550308 10 1.29850427 11 1.00104727 12 1.28896093 13 1.02699402 14 1.28189459 15 1.04687269 16 1.27642527 17 1.06265103 18 1.27205059 19 1.07551821 20 1.26846158 TEST10 JACOBI_COMPUTE computes a Gauss-Jacobi rule; SUM_SUB carries it out. The integration interval is [ -1.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. ALPHA = 0.00000 BETA = 0.00000 Order 1 X X**2 X**3 X**4 1 2.00000000 0.00000000 0.00000000 0.00000000 0.00000000 2 2.00000000 0.00000000 0.66666667 0.00000000 0.22222222 3 2.00000000 0.00000000 0.66666667 0.00000000 0.40000000 4 2.00000000 0.00000000 0.66666667 0.00000000 0.40000000 5 2.00000000 0.00000000 0.66666667 -0.00000000 0.40000000 6 2.00000000 -0.00000000 0.66666667 -0.00000000 0.40000000 7 2.00000000 -0.00000000 0.66666667 -0.00000000 0.40000000 8 2.00000000 0.00000000 0.66666667 0.00000000 0.40000000 9 2.00000000 -0.00000000 0.66666667 0.00000000 0.40000000 10 2.00000000 0.00000000 0.66666667 -0.00000000 0.40000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.00000000 0.00000000 0.00000000 0.00000000 2.00000000 2 0.00000000 0.07407407 0.00000000 0.00000000 2.34269609 3 0.00000000 0.24000000 0.00000000 0.00000000 2.35033693 4 0.00000000 0.28571429 0.00000000 -0.00000000 2.35040209 5 0.00000000 0.28571429 -0.00000000 0.00000000 2.35040239 6 -0.00000000 0.28571429 -0.00000000 -0.00000000 2.35040239 7 -0.00000000 0.28571429 -0.00000000 -0.00000000 2.35040239 8 0.00000000 0.28571429 0.00000000 0.00000000 2.35040239 9 -0.00000000 0.28571429 0.00000000 0.00000000 2.35040239 10 -0.00000000 0.28571429 0.00000000 0.00000000 2.35040239 Order SQRT(|X|) 1 0.00000000 2 1.51967137 3 0.97790193 4 1.40610492 5 1.15351781 6 1.37472152 7 1.22051222 8 1.36087837 9 1.25422195 10 1.35334988 ALPHA = 1.00000 BETA = 0.00000 Order 1 X X**2 X**3 X**4 1 2.00000000 -0.66666667 0.22222222 -0.07407407 0.02469136 2 2.00000000 -0.66666667 0.66666667 -0.40000000 0.29333333 3 2.00000000 -0.66666667 0.66666667 -0.40000000 0.40000000 4 2.00000000 -0.66666667 0.66666667 -0.40000000 0.40000000 5 2.00000000 -0.66666667 0.66666667 -0.40000000 0.40000000 6 2.00000000 -0.66666667 0.66666667 -0.40000000 0.40000000 7 2.00000000 -0.66666667 0.66666667 -0.40000000 0.40000000 8 2.00000000 -0.66666667 0.66666667 -0.40000000 0.40000000 9 2.00000000 -0.66666667 0.66666667 -0.40000000 0.40000000 10 2.00000000 -0.66666667 0.66666667 -0.40000000 0.40000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 -0.00823045 0.00274348 -0.00091449 -0.65438939 1.43306262 2 -0.19733333 0.13760000 -0.09450667 -0.60162582 1.61075164 3 -0.28571429 0.25959184 -0.19941691 -0.60234176 1.61461062 4 -0.28571429 0.28571429 -0.22222222 -0.60233734 1.61464336 5 -0.28571429 0.28571429 -0.22222222 -0.60233736 1.61464350 6 -0.28571429 0.28571429 -0.22222222 -0.60233736 1.61464350 7 -0.28571429 0.28571429 -0.22222222 -0.60233736 1.61464350 8 -0.28571429 0.28571429 -0.22222222 -0.60233736 1.61464350 9 -0.28571429 0.28571429 -0.22222222 -0.60233736 1.61464350 10 -0.28571429 0.28571429 -0.22222222 -0.60233736 1.61464350 Order SQRT(|X|) 1 1.15470054 2 1.44854379 3 1.33109862 4 1.37208362 5 1.33860220 6 1.35355790 7 1.33865476 8 1.34606066 9 1.33792425 10 1.34222470 TEST105 JACOBI_COMPUTE computes a Gauss-Jacobi rule; Here, we simply compute a single rule and print it, to check for accuracy. The integration interval is [ -1.0000, 1.0000] N = 20 ALPHA = 0.000000 BETA = 1.000000 I X(I) W(I) 1 -0.98339991 0.00046153 2 -0.94471389 0.00273225 3 -0.88493108 0.00804583 4 -0.80539174 0.01729288 5 -0.70787351 0.03091067 6 -0.59455502 0.04881618 7 -0.46796772 0.07038827 8 -0.33093940 0.09450118 9 -0.18653107 0.11960697 10 -0.03796858 0.14385968 11 0.11142943 0.16527083 12 0.25832564 0.18188236 13 0.39943865 0.19194185 14 0.53161623 0.19406442 15 0.65190577 0.18736677 16 0.75762027 0.17156118 17 0.84639833 0.14700096 18 0.91625714 0.11467295 19 0.96563756 0.07613988 20 0.99344779 0.03348338 TEST108 JACOBI_COMPUTE computes a Gauss-Jacobi rule; The printed output of this test can be inserted into a FORTRAN program. ! ! Abscissas X and weights W for a Gauss Jacobi rule ! of order = 10 ! with ALPHA = 0.500000 ! and BETA = 2.00000 ! x( 1) = -0.9057833107479930 x( 2) = -0.7536615689945177 x( 3) = -0.5486368932258150 x( 4) = -0.3051523832352315 x( 5) = -0.4048417093085124E-01 x( 6) = 0.2265685497214919 x( 7) = 0.4770321235676097 x( 8) = 0.6931099037348964 x( 9) = 0.8594479437211198 x(10) = 0.9642264730559573 w( 1) = 0.1493406975095294E-02 w( 2) = 0.1451910366053505E-01 w( 3) = 0.5753614967752346E-01 w( 4) = 0.1418263934072605 w( 5) = 0.2526743544928464 w( 6) = 0.3464772296000080 w( 7) = 0.3724465193703724 w( 8) = 0.3072940130036499 w( 9) = 0.1778276475883610 w(10) = 0.5189885826012010E-01 TEST11 KRONROD_SET sets up a Kronrod rule; LEGENDRE_SET sets up a Gauss-Legendre rule; SUMMER_GK carries it out. The integration interval is [-1, 1]. Integrand is X**2 / SQRT ( 1.1 - X**2 ). 10 1.073789659 21 1.074744084 -.9544251929E-03 TEST12 KRONROD_SET sets up a Kronrod rule; LEGENDRE_SET sets up a Gauss-Legendre rule; SUM_SUB_GK carries it out. The integration interval is [ -1.0000, 1.0000] Number of subintervals is 5 Integrand is X**2 / SQRT ( 1.1 - X**2 ). 7 1.074724759 15 1.074743089 0.1832906306E-04 TEST13 LAGUERRE_COMPUTE computes a Gauss-Laguerre rule; LAGUERRE_SUM carries it out. Quadrature order will vary. Integrand will vary. The weight function is EXP ( - X ) The integration interval is [ 1.00000 , +oo ). Order 1 X X**2 X**3 X**4 1 0.36787944 0.73575888 1.47151776 2.94303553 5.88607106 2 0.36787944 0.73575888 1.83939721 5.88607106 22.44064591 3 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 4 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 5 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 6 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 7 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 8 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 9 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 10 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 11 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 12 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 13 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 14 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 15 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 16 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 17 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 18 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 19 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 20 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 11.77214212 23.54428423 47.08856847 0.33451183 2.71828183 2 93.44137806 403.56374697 1767.29283539 0.26247204 5.98428337 3 119.92869782 706.69640649 4616.15122782 0.24610565 9.44093045 4 119.92869782 719.94006637 5039.94834405 0.25590478 12.99913295 5 119.92869782 719.94006637 5039.94834405 0.25411170 16.62186841 6 119.92869782 719.94006637 5039.94834405 0.25409157 20.28984002 7 119.92869782 719.94006637 5039.94834405 0.25418388 23.99157889 8 119.92869782 719.94006637 5039.94834405 0.25416093 27.71965323 9 119.92869782 719.94006637 5039.94834405 0.25416257 31.46894199 10 119.92869782 719.94006637 5039.94834405 0.25416319 35.23575011 11 119.92869782 719.94006637 5039.94834405 0.25416296 39.01731425 12 119.92869782 719.94006637 5039.94834405 0.25416299 42.81150767 13 119.92869782 719.94006637 5039.94834405 0.25416299 46.61665469 14 119.92869782 719.94006637 5039.94834405 0.25416299 50.43140886 15 119.92869782 719.94006637 5039.94834405 0.25416299 54.25467016 16 119.92869782 719.94006637 5039.94834405 0.25416299 58.08552697 17 119.92869782 719.94006637 5039.94834405 0.25416299 61.92321439 18 119.92869782 719.94006637 5039.94834405 0.25416299 65.76708368 19 119.92869782 719.94006637 5039.94834405 0.25416299 69.61657927 20 119.92869782 719.94006637 5039.94834405 0.25416299 73.47122131 Order SQRT(|X|) 1 0.52026010 2 0.50861083 3 0.50755159 4 0.50735604 5 0.50730644 6 0.50729121 7 0.50728588 8 0.50728382 9 0.50728297 10 0.50728259 11 0.50728241 12 0.50728233 13 0.50728228 14 0.50728226 15 0.50728225 16 0.50728224 17 0.50728224 18 0.50728224 19 0.50728224 20 0.50728223 TEST14 LAGUERRE_COMPUTE computes a Gauss-Laguerre rule; LAGUERRE_SUM carries it out. Quadrature order will vary. Integrand will vary. The weight function is EXP ( - X ) The integration interval is [ 0.00000 , +Infinity ). Order 1 X X**2 X**3 X**4 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 2 1.00000000 1.00000000 2.00000000 6.00000000 20.00000000 3 1.00000000 1.00000000 2.00000000 6.00000000 24.00000000 4 1.00000000 1.00000000 2.00000000 6.00000000 24.00000000 5 1.00000000 1.00000000 2.00000000 6.00000000 24.00000000 6 1.00000000 1.00000000 2.00000000 6.00000000 24.00000000 7 1.00000000 1.00000000 2.00000000 6.00000000 24.00000000 8 1.00000000 1.00000000 2.00000000 6.00000000 24.00000000 9 1.00000000 1.00000000 2.00000000 6.00000000 24.00000000 10 1.00000000 1.00000000 2.00000000 6.00000000 24.00000000 11 1.00000000 1.00000000 2.00000000 6.00000000 24.00000000 12 1.00000000 1.00000000 2.00000000 6.00000000 24.00000000 13 1.00000000 1.00000000 2.00000000 6.00000000 24.00000000 14 1.00000000 1.00000000 2.00000000 6.00000000 24.00000000 15 1.00000000 1.00000000 2.00000000 6.00000000 24.00000000 16 1.00000000 1.00000000 2.00000000 6.00000000 24.00000000 17 1.00000000 1.00000000 2.00000000 6.00000000 24.00000000 18 1.00000000 1.00000000 2.00000000 6.00000000 24.00000000 19 1.00000000 1.00000000 2.00000000 6.00000000 24.00000000 20 1.00000000 1.00000000 2.00000000 6.00000000 24.00000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 1.00000000 1.00000000 1.00000000 0.84147098 2.71828183 2 68.00000000 232.00000000 792.00000000 0.43245945 5.98428337 3 120.00000000 684.00000000 4140.00000000 0.49602983 9.44093045 4 120.00000000 720.00000000 5040.00000000 0.50487928 12.99913295 5 120.00000000 720.00000000 5040.00000000 0.49890332 16.62186841 6 120.00000000 720.00000000 5040.00000000 0.50004947 20.28984002 7 120.00000000 720.00000000 5040.00000000 0.50003891 23.99157889 8 120.00000000 720.00000000 5040.00000000 0.49998775 27.71965323 9 120.00000000 720.00000000 5040.00000000 0.50000135 31.46894199 10 120.00000000 720.00000000 5040.00000000 0.50000020 35.23575011 11 120.00000000 720.00000000 5040.00000000 0.49999989 39.01731425 12 120.00000000 720.00000000 5040.00000000 0.50000002 42.81150767 13 120.00000000 720.00000000 5040.00000000 0.50000000 46.61665469 14 120.00000000 720.00000000 5040.00000000 0.50000000 50.43140886 15 120.00000000 720.00000000 5040.00000000 0.50000000 54.25467016 16 120.00000000 720.00000000 5040.00000000 0.50000000 58.08552697 17 120.00000000 720.00000000 5040.00000000 0.50000000 61.92321439 18 120.00000000 720.00000000 5040.00000000 0.50000000 65.76708368 19 120.00000000 720.00000000 5040.00000000 0.50000000 69.61657927 20 120.00000000 720.00000000 5040.00000000 0.50000000 73.47122131 Order SQRT(|X|) 1 1.00000000 2 0.92387953 3 0.90644045 4 0.89928022 5 0.89553750 6 0.89329552 7 0.89182852 8 0.89080711 9 0.89006239 10 0.88949970 11 0.88906231 12 0.88871436 13 0.88843218 14 0.88819960 15 0.88800521 16 0.88784077 17 0.88770020 18 0.88757892 19 0.88747341 20 0.88738094 TEST15 LAGUERRE_COMPUTE computes a Gauss-Laguerre rule; LAGUERRE_SUM carries it out. Quadrature order will vary. Integrand will vary. The integration interval is [ 0.00000 , +Infinity ). The weight function is EXP ( - X ) Order 1 X X**2 X**3 X**4 1 1.00000 1.00000 1.00000 1.00000 1.00000 2 1.00000 1.00000 2.00000 6.00000 20.0000 3 1.00000 1.00000 2.00000 6.00000 24.0000 4 1.00000 1.00000 2.00000 6.00000 24.0000 5 1.00000 1.00000 2.00000 6.00000 24.0000 6 1.00000 1.00000 2.00000 6.00000 24.0000 7 1.00000 1.00000 2.00000 6.00000 24.0000 8 1.00000 1.00000 2.00000 6.00000 24.0000 9 1.00000 1.00000 2.00000 6.00000 24.0000 10 1.00000 1.00000 2.00000 6.00000 24.0000 11 1.00000 1.00000 2.00000 6.00000 24.0000 12 1.00000 1.00000 2.00000 6.00000 24.0000 13 1.00000 1.00000 2.00000 6.00000 24.0000 14 1.00000 1.00000 2.00000 6.00000 24.0000 15 1.00000 1.00000 2.00000 6.00000 24.0000 16 1.00000 1.00000 2.00000 6.00000 24.0000 17 1.00000 1.00000 2.00000 6.00000 24.0000 18 1.00000 1.00000 2.00000 6.00000 24.0000 19 1.00000 1.00000 2.00000 6.00000 24.0000 20 1.00000 1.00000 2.00000 6.00000 24.0000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 1.00000 1.00000 1.00000 0.841471 2.71828 2 68.0000 232.000 792.000 0.432459 5.98428 3 120.000 684.000 4140.00 0.496030 9.44093 4 120.000 720.000 5040.00 0.504879 12.9991 5 120.000 720.000 5040.00 0.498903 16.6219 6 120.000 720.000 5040.00 0.500049 20.2898 7 120.000 720.000 5040.00 0.500039 23.9916 8 120.000 720.000 5040.00 0.499988 27.7197 9 120.000 720.000 5040.00 0.500001 31.4689 10 120.000 720.000 5040.00 0.500000 35.2358 11 120.000 720.000 5040.00 0.500000 39.0173 12 120.000 720.000 5040.00 0.500000 42.8115 13 120.000 720.000 5040.00 0.500000 46.6167 14 120.000 720.000 5040.00 0.500000 50.4314 15 120.000 720.000 5040.00 0.500000 54.2547 16 120.000 720.000 5040.00 0.500000 58.0855 17 120.000 720.000 5040.00 0.500000 61.9232 18 120.000 720.000 5040.00 0.500000 65.7671 19 120.000 720.000 5040.00 0.500000 69.6166 20 120.000 720.000 5040.00 0.500000 73.4712 Order SQRT(|X|) 1 1.00000 2 0.923880 3 0.906440 4 0.899280 5 0.895538 6 0.893296 7 0.891829 8 0.890807 9 0.890062 10 0.889500 11 0.889062 12 0.888714 13 0.888432 14 0.888200 15 0.888005 16 0.887841 17 0.887700 18 0.887579 19 0.887473 20 0.887381 TEST16 GEN_LAGUERRE_COMPUTE computes a Gauss-Laguerre rule; LAGUERRE_SUM carries it out. Quadrature order will vary. Integrand will vary. The weight function is EXP ( - X ) * X ^ 2.00000 The integration interval is [ 0.00000 , +Infinity ). Order 1 X X**2 X**3 X**4 1 2.00000 6.00000 18.0000 54.0000 162.000 2 2.00000 6.00000 24.0000 120.000 672.000 3 2.00000 6.00000 24.0000 120.000 720.000 4 2.00000 6.00000 24.0000 120.000 720.000 5 2.00000 6.00000 24.0000 120.000 720.000 6 2.00000 6.00000 24.0000 120.000 720.000 7 2.00000 6.00000 24.0000 120.000 720.000 8 2.00000 6.00000 24.0000 120.000 720.000 9 2.00000 6.00000 24.0000 120.000 720.000 10 2.00000 6.00000 24.0000 120.000 720.000 11 2.00000 6.00000 24.0000 120.000 720.000 12 2.00000 6.00000 24.0000 120.000 720.000 13 2.00000 6.00000 24.0000 120.000 720.000 14 2.00000 6.00000 24.0000 120.000 720.000 15 2.00000 6.00000 24.0000 120.000 720.000 16 2.00000 6.00000 24.0000 120.000 720.000 17 2.00000 6.00000 24.0000 120.000 720.000 18 2.00000 6.00000 24.0000 120.000 720.000 19 2.00000 6.00000 24.0000 120.000 720.000 20 2.00000 6.00000 24.0000 120.000 720.000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 486.000 1458.00 4374.00 0.282240 40.1711 2 3936.00 23424.0 140160. 1.22424 212.798 3 5040.00 39600.0 334800. 0.216297 617.935 4 5040.00 40320.0 362880. 0.517969 1360.87 5 5040.00 40320.0 362880. 0.523213 2550.25 6 5040.00 40320.0 362880. 0.490280 4297.07 7 5040.00 40320.0 362880. 0.501370 6714.10 8 5040.00 40320.0 362880. 0.500277 9915.46 9 5040.00 40320.0 362880. 0.499820 14016.4 10 5040.00 40320.0 362880. 0.500037 19133.0 11 5040.00 40320.0 362880. 0.500000 25382.2 12 5040.00 40320.0 362880. 0.499998 32881.6 13 5040.00 40320.0 362880. 0.500001 41749.2 14 5040.00 40320.0 362880. 0.500000 52103.7 15 5040.00 40320.0 362880. 0.500000 64064.1 16 5040.00 40320.0 362880. 0.500000 77749.9 17 5040.00 40320.0 362880. 0.500000 93280.8 18 5040.00 40320.0 362880. 0.500000 110777. 19 5040.00 40320.0 362880. 0.500000 130359. 20 5040.00 40320.0 362880. 0.500000 152147. Order SQRT(|X|) 1 3.46410 2 3.34607 3 3.33062 4 3.32647 5 3.32494 6 3.32425 7 3.32391 8 3.32371 9 3.32360 10 3.32353 11 3.32348 12 3.32345 13 3.32343 14 3.32341 15 3.32340 16 3.32339 17 3.32338 18 3.32338 19 3.32337 20 3.32337 TEST165 GEN_LAGUERRE_COMPUTE computes a generalized Gauss-Laguerre rule; The printed output of this test can be inserted into a FORTRAN program. ! ! Abscissas X and weights W for a Gauss Laguerre rule ! of order = 11 ! with ALPHA = 0.00000 ! x( 1) = 0.1257964421879677 x( 2) = 0.6654182558392281 x( 3) = 1.647150545872170 x( 4) = 3.091138143035255 x( 5) = 5.029284401579833 x( 6) = 7.509887863806616 x( 7) = 10.60595099954697 x( 8) = 14.43161375806419 x( 9) = 19.17885740321468 x( 10) = 25.21770933967756 x( 11) = 33.49719284717554 w( 1) = 0.2849332128941960 w( 2) = 0.3897208895278507 w( 3) = 0.2327818318489907 w( 4) = 0.7656445354619686E-01 w( 5) = 0.1439328276735069E-01 w( 6) = 0.1518880846484874E-02 w( 7) = 0.8513122435471922E-04 w( 8) = 0.2292403879574503E-05 w( 9) = 0.2486353702767795E-07 w( 10) = 0.7712626933691313E-10 w( 11) = 0.2883775868323608E-13 TEST165 GEN_LAGUERRE_COMPUTE computes a generalized Gauss-Laguerre rule; The printed output of this test can be inserted into a FORTRAN program. ! ! Abscissas X and weights W for a Gauss Laguerre rule ! of order = 11 ! with ALPHA = 0.500000 ! x( 1) = 0.2102574184831799 x( 2) = 0.8448394164252121 x( 3) = 1.915565574736173 x( 4) = 3.443537268140795 x( 5) = 5.461644772694418 x( 6) = 8.019112688782045 x( 7) = 11.18987571617644 x( 8) = 15.08912822002115 x( 9) = 19.91025764276847 x( 10) = 26.02565538189142 x( 11) = 34.39012589988069 w( 1) = 0.1564919998464248 w( 2) = 0.3356460803810054 w( 3) = 0.2639277337279414 w( 4) = 0.1047323210183777 w( 5) = 0.2260648913203013E-01 w( 6) = 0.2655315844729840E-02 w( 7) = 0.1622374407044610E-03 w( 8) = 0.4693742912624425E-05 w( 9) = 0.5414115327653742E-07 w( 10) = 0.1774072162971696E-09 w( 11) = 0.6988338125898187E-13 TEST165 GEN_LAGUERRE_COMPUTE computes a generalized Gauss-Laguerre rule; The printed output of this test can be inserted into a FORTRAN program. ! ! Abscissas X and weights W for a Gauss Laguerre rule ! of order = 11 ! with ALPHA = 2.00000 ! x( 1) = 0.5297880635681294 x( 2) = 1.431834248511049 x( 3) = 2.753324209989985 x( 4) = 4.518949913023012 x( 5) = 6.764325462912580 x( 6) = 9.541162512224671 x( 7) = 12.92592983812984 x( 8) = 17.03674989811998 x( 9) = 22.07099934899301 x( 10) = 28.40788699071937 x( 11) = 37.01904951380837 w( 1) = 0.1154861791808032 w( 2) = 0.5427759748685629 w( 3) = 0.7432051572620280 w( 4) = 0.4448454978860337 w( 5) = 0.1320746211103724 w( 6) = 0.2004515682275461E-01 w( 7) = 0.1514234411685173E-02 w( 8) = 0.5246733328213365E-04 w( 9) = 0.7084465340665184E-06 w( 10) = 0.2676735958607183E-08 w( 11) = 0.1209708239787269E-11 TEST17 LAGUERRE_SET sets up a Gauss-Laguerre rule; LAGUERRE_SUM carries it out. The integration interval is [ 1.00000 , +Infinity ). Quadrature order will vary. Integrand will vary. The weight function is EXP ( - X ). Order 1 X X**2 X**3 X**4 1 0.36787944 0.73575888 1.47151776 2.94303553 5.88607106 2 0.36787944 0.73575888 1.83939721 5.88607106 22.44064591 3 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 4 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 5 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 6 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 7 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 8 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 9 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 10 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 11 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 12 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 13 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 14 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 15 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 16 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 17 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 18 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 19 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 20 0.36787944 0.73575888 1.83939721 5.88607106 23.91216368 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 11.77214212 23.54428423 47.08856847 0.33451183 2.71828183 2 93.44137806 403.56374697 1767.29283539 0.26247204 5.98428337 3 119.92869782 706.69640649 4616.15122782 0.24610565 9.44093045 4 119.92869782 719.94006637 5039.94834405 0.25590478 12.99913295 5 119.92869782 719.94006637 5039.94834405 0.25411170 16.62186841 6 119.92869782 719.94006637 5039.94834405 0.25409157 20.28984002 7 119.92869782 719.94006637 5039.94834405 0.25418388 23.99157889 8 119.92869782 719.94006637 5039.94834405 0.25416093 27.71965323 9 119.92869782 719.94006637 5039.94834405 0.25416257 31.46894199 10 119.92869782 719.94006637 5039.94834405 0.25416319 35.23575011 11 119.92869782 719.94006637 5039.94834405 0.25416296 39.01731425 12 119.92869782 719.94006637 5039.94834405 0.25416299 42.81150767 13 119.92869782 719.94006637 5039.94834405 0.25416299 46.61665469 14 119.92869782 719.94006637 5039.94834405 0.25416299 50.43140886 15 119.92869782 719.94006637 5039.94834405 0.25416299 54.25467016 16 119.92869782 719.94006637 5039.94834405 0.25416299 58.08552697 17 119.92869782 719.94006637 5039.94834405 0.25416299 61.92321439 18 119.92869782 719.94006637 5039.94834405 0.25416299 65.76708368 19 119.92869782 719.94006637 5039.94834405 0.25416299 69.61657927 20 119.92869782 719.94006637 5039.94834405 0.25416299 73.47122131 Order SQRT(|X|) 1 0.52026010 2 0.50861083 3 0.50755159 4 0.50735604 5 0.50730644 6 0.50729121 7 0.50728588 8 0.50728382 9 0.50728297 10 0.50728259 11 0.50728241 12 0.50728233 13 0.50728228 14 0.50728226 15 0.50728225 16 0.50728224 17 0.50728224 18 0.50728224 19 0.50728224 20 0.50728223 TEST18 LEGENDRE_COMPUTE_DR computes a Gauss-Legendre rule; LEGENDRE_SET sets up a Gauss-Legendre rule. Compare the data for N = 31 Maximum abscissa difference is 0.111022302462515654E-15 for index I = 7 Computed X1:-0.781733148416625001 Stored X2:-0.781733148416624890 Computed P(X1):-0.630320168819443745E-15 Stored P(X2): 0.515716501761363073E-15 Maximum weight difference is 0.111022302462515654E-15 for index I = 8 Computed: 0.696285832354102546E-01 Stored: 0.696285832354103656E-01 TEST185 LEGENDRE_COMPUTE_DR computes a Gauss-Legendre rule; Compute the data for ORDER = 31 xtab( 1) = -0.9970874818194770 xtab( 2) = -0.9846859096651525 xtab( 3) = -0.9625039250929497 xtab( 4) = -0.9307569978966481 xtab( 5) = -0.8897600299482711 xtab( 6) = -0.8399203201462674 xtab( 7) = -0.7817331484166250 xtab( 8) = -0.7157767845868532 xtab( 9) = -0.6427067229242603 xtab( 10) = -0.5632491614071492 xtab( 11) = -0.4781937820449025 xtab( 12) = -0.3883859016082329 xtab( 13) = -0.2947180699817016 xtab( 14) = -0.1981211993355706 xtab( 15) = -0.9955531215234151E-01 xtab( 16) = 0.000000000000000 xtab( 17) = 0.9955531215234151E-01 xtab( 18) = 0.1981211993355706 xtab( 19) = 0.2947180699817016 xtab( 20) = 0.3883859016082329 xtab( 21) = 0.4781937820449025 xtab( 22) = 0.5632491614071492 xtab( 23) = 0.6427067229242603 xtab( 24) = 0.7157767845868532 xtab( 25) = 0.7817331484166250 xtab( 26) = 0.8399203201462674 xtab( 27) = 0.8897600299482711 xtab( 28) = 0.9307569978966481 xtab( 29) = 0.9625039250929497 xtab( 30) = 0.9846859096651525 xtab( 31) = 0.9970874818194770 weight( 1) = 0.7470831579248850E-02 weight( 2) = 0.1731862079031060E-01 weight( 3) = 0.2700901918497942E-01 weight( 4) = 0.3643227391238549E-01 weight( 5) = 0.4549370752720111E-01 weight( 6) = 0.5410308242491679E-01 weight( 7) = 0.6217478656102853E-01 weight( 8) = 0.6962858323541025E-01 weight( 9) = 0.7639038659877659E-01 weight( 10) = 0.8239299176158929E-01 weight( 11) = 0.8757674060847785E-01 weight( 12) = 0.9189011389364142E-01 weight( 13) = 0.9529024291231956E-01 weight( 14) = 0.9774333538632875E-01 weight( 15) = 0.9922501122667235E-01 weight( 16) = 0.9972054479342644E-01 weight( 17) = 0.9922501122667235E-01 weight( 18) = 0.9774333538632875E-01 weight( 19) = 0.9529024291231956E-01 weight( 20) = 0.9189011389364142E-01 weight( 21) = 0.8757674060847785E-01 weight( 22) = 0.8239299176158929E-01 weight( 23) = 0.7639038659877659E-01 weight( 24) = 0.6962858323541025E-01 weight( 25) = 0.6217478656102853E-01 weight( 26) = 0.5410308242491679E-01 weight( 27) = 0.4549370752720111E-01 weight( 28) = 0.3643227391238549E-01 weight( 29) = 0.2700901918497942E-01 weight( 30) = 0.1731862079031060E-01 weight( 31) = 0.7470831579248850E-02 TEST19 LEGENDRE_COMPUTE_DR computes a Gauss-Legendre rule; SUM_SUB carries it out over subintervals. The integration interval is [ 0.0000, 1.0000] Here, we use a fixed order ORDER = 2 and use more and more subintervals. NSUB Integral 1 0.47157381 2 0.51475612 4 0.53156894 8 0.53634334 16 0.53724874 32 0.53736100 64 0.53737080 128 0.53737150 256 0.53737154 512 0.53737154 TEST20 LEGENDRE_COMPUTE_DR computes a Gauss-Legendre rule; SUM_SUB carries it out. The integration interval is [ 0.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 1.00000000 0.50000000 0.25000000 0.12500000 0.06250000 2 1.00000000 0.50000000 0.33333333 0.25000000 0.19444444 3 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 4 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 5 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 6 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 7 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 8 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 9 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 10 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.03125000 0.01562500 0.00781250 0.47942554 1.64872127 2 0.15277778 0.12037037 0.09490741 0.45958781 1.71789638 3 0.16666667 0.14250000 0.12375000 0.45969793 1.71828100 4 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 5 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 6 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 7 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 8 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 9 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 10 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 Order SQRT(|X|) 1 0.70710678 2 0.67388734 3 0.66917963 4 0.66782765 5 0.66729679 6 0.66704644 7 0.66691309 8 0.66683558 9 0.66678747 10 0.66675604 TEST202 LEGENDRE_COMPUTE_GW computes a Gauss-Legendre rule; LEGENDRE_SET sets up a Gauss-Legendre rule. Compare the data for N = 31 Maximum abscissa difference is 0.666133814775093924E-15 for index I = 21 Computed X1: 0.478193782044903148 Stored X2: 0.478193782044902482 Computed P(X1): 0.355271367880050093E-14 Stored P(X2): 0.429763751467802520E-16 Maximum weight difference is 0.212330153459561188E-14 for index I = 21 Computed: 0.875767406084800026E-01 Stored: 0.875767406084778793E-01 TEST204 LEGENDRE_COMPUTE_SS computes a Gauss-Legendre rule; Compute the data for N = 31 x( 1) = -0.9970874818194770 x( 2) = -0.9846859096651525 x( 3) = -0.9625039250929497 x( 4) = -0.9307569978966481 x( 5) = -0.8897600299482710 x( 6) = -0.8399203201462674 x( 7) = -0.7817331484166249 x( 8) = -0.7157767845868532 x( 9) = -0.6427067229242603 x( 10) = -0.5632491614071492 x( 11) = -0.4781937820449025 x( 12) = -0.3883859016082329 x( 13) = -0.2947180699817016 x( 14) = -0.1981211993355706 x( 15) = -0.9955531215234152E-01 x( 16) = 0.000000000000000 x( 17) = 0.9955531215234152E-01 x( 18) = 0.1981211993355706 x( 19) = 0.2947180699817016 x( 20) = 0.3883859016082329 x( 21) = 0.4781937820449025 x( 22) = 0.5632491614071492 x( 23) = 0.6427067229242603 x( 24) = 0.7157767845868532 x( 25) = 0.7817331484166249 x( 26) = 0.8399203201462674 x( 27) = 0.8897600299482710 x( 28) = 0.9307569978966481 x( 29) = 0.9625039250929497 x( 30) = 0.9846859096651525 x( 31) = 0.9970874818194770 w( 1) = 0.7470831579248542E-02 w( 2) = 0.1731862079031078E-01 w( 3) = 0.2700901918497909E-01 w( 4) = 0.3643227391238587E-01 w( 5) = 0.4549370752720149E-01 w( 6) = 0.5410308242491675E-01 w( 7) = 0.6217478656102860E-01 w( 8) = 0.6962858323541052E-01 w( 9) = 0.7639038659877666E-01 w( 10) = 0.8239299176158935E-01 w( 11) = 0.8757674060847792E-01 w( 12) = 0.9189011389364150E-01 w( 13) = 0.9529024291231951E-01 w( 14) = 0.9774333538632873E-01 w( 15) = 0.9922501122667231E-01 w( 16) = 0.9972054479342643E-01 w( 17) = 0.9922501122667231E-01 w( 18) = 0.9774333538632873E-01 w( 19) = 0.9529024291231961E-01 w( 20) = 0.9189011389364150E-01 w( 21) = 0.8757674060847792E-01 w( 22) = 0.8239299176158910E-01 w( 23) = 0.7639038659877666E-01 w( 24) = 0.6962858323541052E-01 w( 25) = 0.6217478656102860E-01 w( 26) = 0.5410308242491675E-01 w( 27) = 0.4549370752720066E-01 w( 28) = 0.3643227391238587E-01 w( 29) = 0.2700901918497909E-01 w( 30) = 0.1731862079031078E-01 w( 31) = 0.7470831579248542E-02 TEST205 Compare timings and accuracy of LEGENDRE_COMPUTE_DR, LEGENDRE_COMPUTE_GW, LEGENDRE_COMPUTE_SS, LEGENDRE_SET. Compare the data for N = 127 TImings in seconds: LEGENDRE_COMPUTE_DR: 0.513000E-03 LEGENDRE_COMPUTE_GW: 0.772360E-01 LEGENDRE_COMPUTE_SS: 0.133200E-02 LEGENDRE_SET: 0.780000E-04 Accuracy: Max ( Abs ( L(N,X) ): LEGENDRE_COMPUTE_DR: 0.167320E-12 LEGENDRE_COMPUTE_GW: 0.253867E-11 LEGENDRE_COMPUTE_SS: 0.167320E-12 LEGENDRE_SET: 0.167320E-12 Accuracy: Abs ( 2 - sum ( W ): LEGENDRE_COMPUTE_DR: 0.888178E-15 LEGENDRE_COMPUTE_GW: 0.222045E-15 LEGENDRE_COMPUTE_SS: 0.124345E-13 LEGENDRE_SET: 0.00000 TEST205 Compare timings and accuracy of LEGENDRE_COMPUTE_DR, LEGENDRE_COMPUTE_GW, LEGENDRE_COMPUTE_SS, LEGENDRE_SET. Compare the data for N = 255 TImings in seconds: LEGENDRE_COMPUTE_DR: 0.191900E-02 LEGENDRE_COMPUTE_GW: 0.553497 LEGENDRE_COMPUTE_SS: 0.519100E-02 LEGENDRE_SET: 0.700000E-04 Accuracy: Max ( Abs ( L(N,X) ): LEGENDRE_COMPUTE_DR: 0.333342E-12 LEGENDRE_COMPUTE_GW: 0.246796E-10 LEGENDRE_COMPUTE_SS: 0.333342E-12 LEGENDRE_SET: 0.333342E-12 Accuracy: Abs ( 2 - sum ( W ): LEGENDRE_COMPUTE_DR: 0.888178E-15 LEGENDRE_COMPUTE_GW: 0.177636E-14 LEGENDRE_COMPUTE_SS: 0.182077E-13 LEGENDRE_SET: 0.888178E-15 TEST21 LEGENDRE_SET sets up a Gauss-Legendre rule; SUM_SUB carries it out. The integration interval is [ 0.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 1.00000000 0.50000000 0.25000000 0.12500000 0.06250000 2 1.00000000 0.50000000 0.33333333 0.25000000 0.19444444 3 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 4 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 5 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 6 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 7 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 8 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 9 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 10 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 11 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 12 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 13 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 14 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 15 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 16 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 17 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 18 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 19 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 20 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.03125000 0.01562500 0.00781250 0.47942554 1.64872127 2 0.15277778 0.12037037 0.09490741 0.45958781 1.71789638 3 0.16666667 0.14250000 0.12375000 0.45969793 1.71828100 4 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 5 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 6 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 7 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 8 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 9 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 10 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 11 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 12 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 13 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 14 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 15 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 16 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 17 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 18 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 19 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 20 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 Order SQRT(|X|) 1 0.70710678 2 0.67388734 3 0.66917963 4 0.66782765 5 0.66729679 6 0.66704644 7 0.66691309 8 0.66683558 9 0.66678747 10 0.66675604 11 0.66673464 12 0.66671956 13 0.66670864 14 0.66670053 15 0.66669438 16 0.66668963 17 0.66668591 18 0.66668295 19 0.66668057 20 0.66667863 TEST22 LEGENDRE_SET sets up a Gauss-Legendre rule for integrating F(X) over [-1,1]; RULE_ADJUST adjusts a rule for a new interval. LEGENDRE_X0_01_SET sets up a Gauss-Legendre rule for integrating F(X) over [0,1]; We will use LEGENDRE_SET to get a rule for [-1,1], adjust it to [0,1] using RULE_ADJUST, and compare the results of LEGENDRE_X0_01_SET. Abscissas: Original Adjusted Stored 1 -0.906179845939 0.046910077031 0.046910077000 2 -0.538469310106 0.230765344947 0.230765344900 3 0.000000000000 0.500000000000 0.500000000000 4 0.538469310106 0.769234655053 0.769234655100 5 0.906179845939 0.953089922969 0.953089923000 Weights: Original Adjusted Stored 1 0.236926885056 0.118463442528 0.118463442500 2 0.478628670499 0.239314335250 0.239314335200 3 0.568888888889 0.284444444444 0.284444444400 4 0.478628670499 0.239314335250 0.239314335200 5 0.236926885056 0.118463442528 0.118463442500 TEST23 LEGENDRE_COS_SET sets up a Gauss-Legendre rule over [-PI/2,PI/2] with weight function COS(X); SUM_SUB carries it out. The integration interval is [ -1.5708, 1.5708] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 2.00000000 0.00000000 0.00000000 0.00000000 0.00000000 2 2.00000000 0.00000000 0.93480220 0.00000000 0.43692758 4 2.00000000 -0.00000000 0.93480220 0.00000000 0.95850997 8 2.00000000 -0.00000000 0.93480220 0.00000000 0.95850997 16 2.00000000 -0.00000000 0.93480220 0.00000000 0.95850997 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.00000000 0.00000000 0.00000000 0.00000000 2.00000000 2 0.00000000 0.20422043 0.00000000 0.00000000 2.48589244 4 0.00000000 1.28811312 0.00000000 -0.00000000 2.50917373 8 -0.00000000 1.28811312 0.00000000 -0.00000000 2.50917848 16 0.00000000 1.28811312 -0.00000000 0.00000000 2.50917848 Order SQRT(|X|) 1 0.00000000 2 1.65368363 4 1.51755614 8 1.45413652 16 1.42628067 TEST24 LEGENDRE_SQRTX_01_SET sets up a Gauss-Legendre rule over [0,1] with weight function SQRT(X); SUMMER carries it out. The integration interval is [ 0.0000, 1.0000] Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 0.66666667 0.40000000 0.24000000 0.14400000 0.08640000 2 0.66666667 0.40000000 0.28571429 0.22222222 0.17888637 4 0.66666667 0.40000000 0.28571429 0.22222222 0.18181818 8 0.66666667 0.40000000 0.28571429 0.22222222 0.18181818 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.05184000 0.03110400 0.01866240 0.37642832 1.21474587 2 0.14585258 0.11946643 0.09801367 0.36415913 1.25541745 4 0.15384615 0.13333333 0.11764706 0.36422193 1.25563008 8 0.15384615 0.13333333 0.11764706 0.36422193 1.25563008 Order SQRT(|X|) 1 0.51639778 2 0.50205963 4 0.50020905 8 0.50001755 TEST25 LEGENDRE_SQRTX2_01_SET sets up a Gauss-Legendre rule over [0,1] with weight function 1/SQRT(X); SUMMER carries it out. The integration interval is [ 0.0000, 1.0000] Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 1.11111111 0.66666667 0.40000000 0.24000000 0.14400000 2 1.43111111 0.66666667 0.40000000 0.28571429 0.22222222 4 1.66976064 0.66666667 0.40000000 0.28571429 0.22222222 8 1.82055353 0.66666667 0.40000000 0.28571429 0.22222222 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.08640000 0.05184000 0.03110400 0.62738053 2.02457644 2 0.17888637 0.14585258 0.11946643 0.62051486 2.35637575 4 0.18181818 0.15384615 0.13333333 0.62053660 2.59506414 8 0.18181818 0.15384615 0.13333333 0.62053660 2.74585702 Order SQRT(|X|) 1 0.86066297 2 0.94485044 4 0.98168976 8 0.99462159 TEST26 LEGENDRE_COS2_SET sets up a Gauss-Legendre rule over [0,PI/2] with weight function COS(X); SUM_SUB carries it out. The integration interval is [ -1.5708, 1.5708] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 2 1.00000000 0.57079633 0.46740110 0.45100662 0.45813342 4 1.00000000 0.57079633 0.46740110 0.45100662 0.47925499 8 1.00000000 0.57079633 0.46740110 0.45100662 0.47925499 16 1.00000000 0.57079633 0.46740110 0.45100662 0.47925499 Order X**5 X**6 X**7 SIN(X) EXP(X) 2 0.47191765 0.48788986 0.50487644 0.49946251 1.90347861 4 0.54298266 0.64405656 0.79076895 0.49999999 1.90523867 8 0.54298266 0.64405656 0.79076895 0.50000000 1.90523869 16 0.54298266 0.64405656 0.79076895 0.50000000 1.90523869 Order SQRT(|X|) 2 0.71452828 4 0.70593166 8 0.70433678 16 0.70408047 TEST27 LEGENDRE_LOG_SET sets up a Gauss-Legendre rule to integrate -LOG(X)*F(X) over [0,1]; SUM_SUB carries it out. The integration interval is [ 0.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 1.00000000 0.25000000 0.06250000 0.01562500 0.00390625 2 1.00000000 0.25000000 0.11111111 0.06250000 0.03714727 3 1.00000000 0.25000000 0.11111111 0.06250000 0.04000000 4 1.00000000 0.25000000 0.11111111 0.06250000 0.04000000 5 1.00000000 0.25000000 0.11111111 0.06250000 0.04000000 6 1.00000000 0.25000000 0.11111111 0.06250000 0.04000000 7 1.00000000 0.25000000 0.11111111 0.06250000 0.04000000 8 1.00000000 0.25000000 0.11111111 0.06250000 0.04000000 16 1.00000000 0.25000000 0.11111111 0.06250000 0.04000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.00097656 0.00024414 0.00006104 0.24740396 1.28402542 2 0.02231749 0.01343510 0.00809095 0.23976772 1.31772640 3 0.02777778 0.02023492 0.01512405 0.23981184 1.31790179 4 0.02777778 0.02040816 0.01562500 0.23981174 1.31790215 5 0.02777778 0.02040816 0.01562500 0.23981174 1.31790215 6 0.02777778 0.02040816 0.01562500 0.23981174 1.31790215 7 0.02777778 0.02040816 0.01562500 0.23981174 1.31790215 8 0.02777778 0.02040816 0.01562500 0.23981174 1.31790215 16 0.02777778 0.02040816 0.01562500 0.23981174 1.31790215 Order SQRT(|X|) 1 0.50000000 2 0.45891049 3 0.45073646 4 0.44783230 5 0.44650553 6 0.44580318 7 0.44539290 8 0.44513560 16 0.44456943 TEST275 LEGENDRE_LOG_COMPUTE computes a Gauss-Legendre rule to integrate -LOG(X)*F(X) over [0,1]; SUM_SUB carries it out. The integration interval is [ 0.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 1.00000000 0.25000000 0.06250000 0.01562500 0.00390625 2 1.00000000 0.25000000 0.11111111 0.06250000 0.03714727 3 1.00000000 0.25000000 0.11111111 0.06250000 0.04000000 4 1.00000000 0.25000000 0.11111111 0.06250000 0.04000000 5 1.00000000 0.25000000 0.11111111 0.06250000 0.04000000 6 1.00000000 0.25000000 0.11111111 0.06250000 0.04000000 7 1.00000000 0.25000000 0.11111111 0.06250000 0.04000000 8 1.00000000 0.25000000 0.11111111 0.06250000 0.04000000 9 1.00000000 0.25000000 0.11111111 0.06250000 0.04000000 10 1.00000000 0.25000000 0.11111111 0.06250000 0.04000000 11 1.00000000 0.25000000 0.11111111 0.06250000 0.04000000 12 1.00000000 0.25000000 0.11111111 0.06250000 0.04000000 13 1.00000000 0.25000000 0.11111111 0.06250000 0.04000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.00097656 0.00024414 0.00006104 0.24740396 1.28402542 2 0.02231749 0.01343510 0.00809095 0.23976772 1.31772640 3 0.02777778 0.02023492 0.01512405 0.23981184 1.31790179 4 0.02777778 0.02040816 0.01562500 0.23981174 1.31790215 5 0.02777778 0.02040816 0.01562500 0.23981174 1.31790215 6 0.02777778 0.02040816 0.01562500 0.23981174 1.31790215 7 0.02777778 0.02040816 0.01562500 0.23981174 1.31790215 8 0.02777778 0.02040816 0.01562500 0.23981174 1.31790215 9 0.02777778 0.02040816 0.01562500 0.23981174 1.31790215 10 0.02777778 0.02040816 0.01562500 0.23981174 1.31790215 11 0.02777778 0.02040816 0.01562500 0.23981174 1.31790215 12 0.02777778 0.02040816 0.01562500 0.23981174 1.31790215 13 0.02777778 0.02040816 0.01562500 0.23981174 1.31790215 Order SQRT(|X|) 1 0.50000000 2 0.45891049 3 0.45073646 4 0.44783230 5 0.44650553 6 0.44580318 7 0.44539290 8 0.44513560 9 0.44496530 10 0.44484770 11 0.44476368 12 0.44470291 13 0.44467371 Rule for order 1 1 0.250000 1.00000 Rule for order 2 1 0.112009 0.718539 2 0.602277 0.281461 Rule for order 3 1 0.638908E-01 0.513405 2 0.368997 0.391980 3 0.766880 0.946154E-01 Rule for order 4 1 0.414485E-01 0.383464 2 0.245275 0.386875 3 0.556165 0.190435 4 0.848982 0.392255E-01 Rule for order 5 1 0.291345E-01 0.297893 2 0.173977 0.349776 3 0.411703 0.234488 4 0.677314 0.989305E-01 5 0.894771 0.189116E-01 Rule for order 6 1 0.216340E-01 0.238764 2 0.129583 0.308287 3 0.314020 0.245317 4 0.538657 0.142009 5 0.756915 0.554546E-01 6 0.922669 0.101690E-01 Rule for order 7 1 0.167194E-01 0.196169 2 0.100186 0.270303 3 0.246294 0.239682 4 0.433463 0.165776 5 0.632351 0.889432E-01 6 0.811119 0.331943E-01 7 0.940848 0.593279E-02 Rule for order 8 1 0.133202E-01 0.164417 2 0.797504E-01 0.237526 3 0.197871 0.226842 4 0.354154 0.175754 5 0.529459 0.112924 6 0.701815 0.578722E-01 7 0.849379 0.209791E-01 8 0.953326 0.368641E-02 Rule for order 9 1 0.108693E-01 0.140068 2 0.649837E-01 0.209772 3 0.162229 0.211427 4 0.293750 0.177156 5 0.446632 0.127799 6 0.605482 0.784789E-01 7 0.754110 0.390225E-01 8 0.877266 0.138673E-01 9 0.962251 0.240804E-02 Rule for order 10 1 0.904267E-02 0.120956 2 0.539715E-01 0.186364 3 0.135312 0.195661 4 0.247053 0.173577 5 0.380213 0.135695 6 0.523793 0.936465E-01 7 0.665776 0.557875E-01 8 0.794191 0.271596E-01 9 0.898161 0.951511E-02 10 0.968848 0.163814E-02 Rule for order 11 1 0.764437E-02 0.105657 2 0.455444E-01 0.166578 3 0.114528 0.180568 4 0.210389 0.167281 5 0.326711 0.138696 6 0.455473 0.103830 7 0.587670 0.695322E-01 8 0.713985 0.405371E-01 9 0.825470 0.194322E-01 10 0.914204 0.673596E-02 11 0.973864 0.115219E-02 Rule for order 12 1 0.656779E-02 0.934137E-01 2 0.390616E-01 0.150056 3 0.984358E-01 0.166818 4 0.181648 0.159780 5 0.283970 0.138429 6 0.399395 0.109892 7 0.521042 0.797536E-01 8 0.641607 0.521782E-01 9 0.753829 0.298796E-01 10 0.850954 0.141283E-01 11 0.927156 0.484853E-02 12 0.977893 0.823991E-03 Rule for order 13 1 0.602311E-02 0.870255E-01 2 0.357478E-01 0.140934 3 0.900019E-01 0.158269 4 0.166003 0.153645 5 0.259516 0.135685 6 0.365318 0.110825 7 0.477677 0.839926E-01 8 0.590759 0.587043E-01 9 0.698884 0.371939E-01 10 0.796638 0.206562E-01 11 0.879036 0.944317E-02 12 0.941946 0.311714E-02 13 0.982659 0.509736E-03 TEST28 LEGENDRE_X0_01_SET sets up a Gauss-Legendre rule for integrating F(X) over [0,1]; SUM_SUB carries it out. The integration interval is [ 0.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 1.00000000 0.50000000 0.25000000 0.12500000 0.06250000 2 1.00000000 0.50000000 0.33333333 0.25000000 0.19444444 3 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 4 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 5 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 6 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 7 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 8 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.03125000 0.01562500 0.00781250 0.47942554 1.64872127 2 0.15277778 0.12037037 0.09490741 0.45958781 1.71789638 3 0.16666667 0.14250000 0.12375000 0.45969793 1.71828100 4 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 5 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 6 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 7 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 8 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 Order SQRT(|X|) 1 0.70710678 2 0.67388734 3 0.66917963 4 0.66782765 5 0.66729679 6 0.66704644 7 0.66691309 8 0.66683558 TEST29 LEGENDRE_X1_SET sets up a Gauss-Legendre rule for integrating ( 1 + X ) * F(X) over [-1,1]; SUM_SUB carries it out. The integration interval is [ 0.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 1.00000000 0.66666667 0.44444444 0.29629630 0.19753086 2 1.00000000 0.66666667 0.50000000 0.40000000 0.33000000 3 1.00000000 0.66666667 0.50000000 0.40000000 0.33333333 4 1.00000000 0.66666667 0.50000000 0.40000000 0.33333333 5 1.00000000 0.66666667 0.50000000 0.40000000 0.33333333 6 1.00000000 0.66666667 0.50000000 0.40000000 0.33333333 7 1.00000000 0.66666667 0.50000000 0.40000000 0.33333333 8 1.00000000 0.66666667 0.50000000 0.40000000 0.33333333 9 1.00000000 0.66666667 0.50000000 0.40000000 0.33333333 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.13168724 0.08779150 0.05852766 0.61836980 1.94773404 2 0.27600000 0.23220000 0.19584000 0.60226153 1.99974921 3 0.28571429 0.24979592 0.22141885 0.60233751 1.99999950 4 0.28571429 0.25000000 0.22222222 0.60233736 2.00000000 5 0.28571429 0.25000000 0.22222222 0.60233736 2.00000000 6 0.28571429 0.25000000 0.22222222 0.60233736 2.00000000 7 0.28571429 0.25000000 0.22222222 0.60233736 2.00000000 8 0.28571429 0.25000000 0.22222222 0.60233736 2.00000000 9 0.28571429 0.25000000 0.22222222 0.60233736 2.00000000 Order SQRT(|X|) 1 0.81649658 2 0.80153861 3 0.80033005 4 0.80010349 5 0.80004063 6 0.80001854 7 0.80000942 8 0.80000520 9 0.80000306 TEST30 LEGENDRE_X1_SET sets up a Gauss-Legendre rule for integrating ( 1 + X ) * F(X) over [-1,1]; RULE_ADJUST adjusts a rule for a new interval. LEGENDRE_X1_01_SET sets up a Gauss-Legendre rule for integrating X * F(X) over [0,1]; We will use LEGENDRE_X1_SET to get a rule for [-1,1], adjust it to [0,1] using RULE_ADJUST, make further adjustments because the weight function is not 1, and compare the results of LEGENDRE_X1_01_SET. Abscissas: Original Adjusted Stored 1 -0.802929828402 0.098535085799 0.098535085800 2 -0.390928546707 0.304535726646 0.304535726600 3 0.124050379505 0.562025189753 0.562025189800 4 0.603973164253 0.801986582126 0.801986582100 5 0.920380285897 0.960190142949 0.960190142900 Weights: Original Adjusted Stored 1 0.062991658087 0.015747914522 0.015747914500 2 0.295635480290 0.073908870073 0.073908870100 3 0.585547948339 0.146386987085 0.146386987100 4 0.668698552377 0.167174638094 0.167174638100 5 0.387126360907 0.096781590227 0.096781590200 TEST31 LEGENDRE_X1_01_SET sets up a Gauss-Legendre rule for integrating X * F(X) over [0,1]; SUM_SUB carries it out. The integration interval is [ 0.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 0.50000000 0.33333333 0.22222222 0.14814815 0.09876543 2 0.50000000 0.33333333 0.25000000 0.20000000 0.16500000 3 0.50000000 0.33333333 0.25000000 0.20000000 0.16666667 4 0.50000000 0.33333333 0.25000000 0.20000000 0.16666667 5 0.50000000 0.33333333 0.25000000 0.20000000 0.16666667 6 0.50000001 0.33333333 0.25000000 0.20000000 0.16666667 7 0.50000000 0.33333333 0.25000000 0.20000000 0.16666667 8 0.50000000 0.33333333 0.25000000 0.20000000 0.16666667 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.06584362 0.04389575 0.02926383 0.30918490 0.97386702 2 0.13800000 0.11610000 0.09792000 0.30113076 0.99987461 3 0.14285714 0.12489796 0.11070943 0.30116875 0.99999975 4 0.14285714 0.12500000 0.11111111 0.30116868 1.00000000 5 0.14285714 0.12500000 0.11111111 0.30116868 1.00000000 6 0.14285714 0.12500000 0.11111111 0.30116868 1.00000001 7 0.14285714 0.12500000 0.11111111 0.30116868 1.00000000 8 0.14285714 0.12500000 0.11111111 0.30116868 1.00000000 Order SQRT(|X|) 1 0.40824829 2 0.40076931 3 0.40016502 4 0.40005174 5 0.40002032 6 0.40000927 7 0.40000471 8 0.40000260 TEST32 LEGENDRE_X2_SET sets up a Gauss-Legendre rule for integrating (1+X)**2 * F(X) over [-1,1]; SUM_SUB carries it out. The integration interval is [ 0.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 1.33333333 1.00000000 0.75000000 0.56250000 0.42187500 2 1.33333333 1.00000000 0.80000000 0.66666667 0.56888889 3 1.33333333 1.00000000 0.80000000 0.66666667 0.57142857 4 1.33333333 1.00000000 0.80000000 0.66666667 0.57142857 5 1.33333333 1.00000000 0.80000000 0.66666667 0.57142857 6 1.33333333 1.00000000 0.80000000 0.66666667 0.57142857 7 1.33333333 1.00000000 0.80000000 0.66666667 0.57142857 8 1.33333333 1.00000000 0.80000000 0.66666667 0.57142857 9 1.33333333 1.00000000 0.80000000 0.66666667 0.57142857 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.31640625 0.23730469 0.17797852 0.90885168 2.82266669 2 0.49185185 0.42824691 0.37425514 0.89291418 2.87292495 3 0.50000000 0.44430272 0.39939413 0.89297721 2.87312695 4 0.50000000 0.44444444 0.40000000 0.89297710 2.87312731 5 0.50000000 0.44444444 0.40000000 0.89297710 2.87312731 6 0.50000000 0.44444444 0.40000000 0.89297710 2.87312731 7 0.50000000 0.44444444 0.40000000 0.89297710 2.87312731 8 0.50000000 0.44444444 0.40000000 0.89297710 2.87312731 9 0.50000000 0.44444444 0.40000000 0.89297710 2.87312731 Order SQRT(|X|) 1 1.15470054 2 1.14353617 3 1.14295593 4 1.14287954 5 1.14286380 6 1.14285952 7 1.14285812 8 1.14285758 9 1.14285736 TEST33 LEGENDRE_X2_SET sets up a Gauss-Legendre rule for integrating ( 1 + X )^2 * F(X) over [-1,1]; RULE_ADJUST adjusts a rule for a new interval. LEGENDRE_X2_01_SET sets up a Gauss-Legendre rule for integrating X^2 * F(X) over [0,1]; We will use LEGENDRE_X2_SET to get a rule for [-1,1], adjust it to [0,1] using RULE_ADJUST, make further adjustments because the weight function is not 1, and compare the results of LEGENDRE_X2_01_SET. Abscissas: Original Adjusted Stored 1 -0.702108425894 0.148945787053 0.148945787100 2 -0.268666945262 0.365666527369 0.365666527400 3 0.220227225869 0.610113612934 0.610113612900 4 0.653039358457 0.826519679228 0.826519679200 5 0.930842120164 0.965421060082 0.965421060100 Weights: Original Adjusted Stored 1 0.032910601625 0.004113825203 0.004113825200 2 0.256444805784 0.032055600723 0.032055600700 3 0.713601289773 0.089200161222 0.089200161200 4 1.009591695199 0.126198961900 0.126198961900 5 0.654118274286 0.081764784286 0.081764784300 TEST34 LEGENDRE_X2_01_SET sets up a Gauss-Legendre rule for integrating X*X * F(X) over [0,1]; SUM_SUB carries it out. The integration interval is [ 0.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 0.33333333 0.25000000 0.18750000 0.14062500 0.10546875 2 0.33333333 0.25000000 0.20000000 0.16666667 0.14222222 3 0.33333333 0.25000000 0.20000000 0.16666667 0.14285714 4 0.33333333 0.25000000 0.20000000 0.16666667 0.14285714 5 0.33333333 0.25000000 0.20000000 0.16666667 0.14285714 6 0.33333333 0.25000000 0.20000000 0.16666667 0.14285714 7 0.33333333 0.25000000 0.20000000 0.16666667 0.14285714 8 0.33333333 0.25000000 0.20000000 0.16666667 0.14285714 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.07910156 0.05932617 0.04449463 0.22721292 0.70566667 2 0.12296296 0.10706173 0.09356379 0.22322855 0.71823124 3 0.12500000 0.11107568 0.09984853 0.22324430 0.71828174 4 0.12500000 0.11111111 0.10000000 0.22324428 0.71828183 5 0.12500000 0.11111111 0.10000000 0.22324428 0.71828183 6 0.12500000 0.11111111 0.10000000 0.22324428 0.71828183 7 0.12500000 0.11111111 0.10000000 0.22324428 0.71828183 8 0.12500000 0.11111111 0.10000000 0.22324428 0.71828183 Order SQRT(|X|) 1 0.28867513 2 0.28588404 3 0.28573898 4 0.28571989 5 0.28571595 6 0.28571488 7 0.28571453 8 0.28571440 TEST345 LOBATTO_COMPUTE computes a Lobatto rule; LOBATTO_SET sets a rule from a table. I X1 X2 W1 W2 1 -1.00000000 -1.00000000 0.16666667 0.16666667 2 -0.44721360 -0.44721360 0.83333333 0.83333333 3 0.44721360 0.44721360 0.83333333 0.83333333 4 1.00000000 1.00000000 0.16666667 0.16666667 1 -1.00000000 -1.00000000 0.04761905 0.04761905 2 -0.83022390 -0.83022390 0.27682605 0.27682605 3 -0.46884879 -0.46884879 0.43174538 0.43174538 4 0.00000000 0.00000000 0.48761905 0.48761905 5 0.46884879 0.46884879 0.43174538 0.43174538 6 0.83022390 0.83022390 0.27682605 0.27682605 7 1.00000000 1.00000000 0.04761905 0.04761905 1 -1.00000000 -1.00000000 0.02222222 0.02222222 2 -0.91953391 -0.91953391 0.13330599 0.13330599 3 -0.73877387 -0.73877387 0.22488934 0.22488934 4 -0.47792495 -0.47792495 0.29204268 0.29204268 5 -0.16527896 -0.16527896 0.32753976 0.32753976 6 0.16527896 0.16527896 0.32753976 0.32753976 7 0.47792495 0.47792495 0.29204268 0.29204268 8 0.73877387 0.73877387 0.22488934 0.22488934 9 0.91953391 0.91953391 0.13330599 0.13330599 10 1.00000000 1.00000000 0.02222222 0.02222222 TEST35 LOBATTO_SET sets up a Lobatto rule; SUM_SUB carries it out. The integration interval is [ -1.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 2 2.00000000 0.00000000 2.00000000 0.00000000 2.00000000 3 2.00000000 0.00000000 0.66666667 0.00000000 0.66666667 4 2.00000000 0.00000000 0.66666667 0.00000000 0.40000000 5 2.00000000 -0.00000000 0.66666667 0.00000000 0.40000000 6 2.00000000 -0.00000000 0.66666667 0.00000000 0.40000000 7 2.00000000 0.00000000 0.66666667 0.00000000 0.40000000 8 2.00000000 -0.00000000 0.66666667 -0.00000000 0.40000000 9 2.00000000 -0.00000000 0.66666667 0.00000000 0.40000000 10 2.00000000 0.00000000 0.66666667 0.00000000 0.40000000 11 2.00000000 0.00000000 0.66666667 0.00000000 0.40000000 12 2.00000000 -0.00000000 0.66666667 0.00000000 0.40000000 13 2.00000000 0.00000000 0.66666667 0.00000000 0.40000000 14 2.00000000 -0.00000000 0.66666667 0.00000000 0.40000000 15 2.00000000 0.00000000 0.66666667 -0.00000000 0.40000000 16 2.00000000 -0.00000000 0.66666667 0.00000000 0.40000000 17 2.00000000 -0.00000000 0.66666667 0.00000000 0.40000000 18 2.00000000 0.00000000 0.66666667 0.00000000 0.40000000 19 2.00000000 0.00000000 0.66666667 0.00000000 0.40000000 20 2.00000000 -0.00000000 0.66666667 0.00000000 0.40000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 2 0.00000000 2.00000000 0.00000000 0.00000000 3.08616127 3 0.00000000 0.66666667 0.00000000 0.00000000 2.36205376 4 0.00000000 0.34666667 0.00000000 -0.00000000 2.35048991 5 0.00000000 0.28571429 0.00000000 0.00000000 2.35040276 6 0.00000000 0.28571429 -0.00000000 -0.00000000 2.35040239 7 0.00000000 0.28571429 -0.00000000 0.00000000 2.35040239 8 -0.00000000 0.28571429 0.00000000 0.00000000 2.35040239 9 0.00000000 0.28571429 0.00000000 -0.00000000 2.35040239 10 -0.00000000 0.28571429 0.00000000 0.00000000 2.35040239 11 -0.00000000 0.28571429 0.00000000 -0.00000000 2.35040239 12 0.00000000 0.28571429 -0.00000000 0.00000000 2.35040239 13 0.00000000 0.28571429 0.00000000 -0.00000000 2.35040239 14 0.00000000 0.28571429 0.00000000 -0.00000000 2.35040239 15 0.00000000 0.28571429 -0.00000000 0.00000000 2.35040239 16 0.00000000 0.28571429 0.00000000 0.00000000 2.35040239 17 -0.00000000 0.28571429 -0.00000000 0.00000000 2.35040239 18 -0.00000000 0.28571429 -0.00000000 -0.00000000 2.35040239 19 0.00000000 0.28571429 -0.00000000 0.00000000 2.35040239 20 0.00000000 0.28571429 -0.00000000 0.00000000 2.35040239 Order SQRT(|X|) 2 2.00000000 3 0.66666667 4 1.44790051 5 1.08102731 6 1.38808526 7 1.19096119 8 1.36707571 9 1.23887211 10 1.35680844 11 1.26479359 12 1.35088305 13 1.28067889 14 1.34709626 15 1.29124343 16 1.34450218 17 1.29868931 18 1.34263308 19 1.30417007 20 1.34123353 TEST36 MOULTON_SET sets up an Adams-Moulton rule; SUMMER carries it out. The integration interval is [0,1]. Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 2 1.00000000 0.50000000 0.50000000 0.50000000 0.50000000 3 1.00000000 0.50000000 0.33333333 0.50000000 0.33333333 4 1.00000000 0.50000000 0.33333333 0.25000000 0.83333333 5 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 6 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 7 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 8 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 9 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 10 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 1.00000000 1.00000000 1.00000000 0.84147098 2.71828183 2 0.50000000 0.50000000 0.50000000 0.42073549 1.85914091 3 0.50000000 0.33333333 0.50000000 0.42073549 1.76862748 4 -0.75000000 2.83333333 -4.75000000 0.45297068 1.74001977 5 2.41666667 -9.83333333 39.58333333 0.47174071 1.72856688 6 0.16666667 10.41666667 -86.41666667 0.47066572 1.72342294 7 0.16666667 0.14285714 57.41666667 0.46058220 1.72094841 8 0.16666667 0.14285714 0.12500000 0.45379596 1.71970231 9 0.16666667 0.14285714 0.12500000 0.45473940 1.71905396 10 0.16666667 0.14285714 0.12500000 0.45980307 1.71870825 Order SQRT(|X|) 1 1.00000000 2 0.50000000 3 0.33333333 4 0.22559223 5 0.14444121 6 0.07847649 7 0.02236210 8 -0.02682791 9 -0.07087225 10 -0.11093388 TEST37 NCC_SET sets up a closed Newton-Cotes rule; SUM_SUB carries it out. The integration interval is [ 0.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 1.00000000 0.50000000 0.25000000 0.12500000 0.06250000 2 1.00000000 0.50000000 0.50000000 0.50000000 0.50000000 3 1.00000000 0.50000000 0.33333333 0.25000000 0.20833333 4 1.00000000 0.50000000 0.33333333 0.25000000 0.20370370 5 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 6 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 7 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 8 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 9 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 10 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 11 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 12 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 13 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 14 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 15 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 16 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 17 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 18 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 19 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 20 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 21 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.03125000 0.01562500 0.00781250 0.47942554 1.64872127 2 0.50000000 0.50000000 0.50000000 0.42073549 1.85914091 3 0.18750000 0.17708333 0.17187500 0.45986219 1.71886115 4 0.17592593 0.15843621 0.14711934 0.45977056 1.71854015 5 0.16666667 0.14322917 0.12630208 0.45969745 1.71828269 6 0.16666667 0.14306667 0.12573333 0.45969756 1.71828231 7 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 8 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 9 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 10 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 11 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 12 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 13 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 14 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 15 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 16 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 17 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 18 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 19 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 20 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 21 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 Order SQRT(|X|) 1 0.70710678 2 0.50000000 3 0.63807119 4 0.64769257 5 0.65775660 6 0.65963774 7 0.66229602 8 0.66297345 9 0.66404880 10 0.66437037 11 0.66491152 12 0.66509039 13 0.66540158 14 0.66551186 15 0.66570768 16 0.66578077 17 0.66591226 18 0.66596337 19 0.66605610 20 0.66609334 21 0.66616129 TEST38 NCC_COMPUTE computes a closed Newton-Cotes rule; SUM_SUB carries it out. The integration interval is [ 0.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 1.00000000 0.50000000 0.25000000 0.12500000 0.06250000 2 1.00000000 0.50000000 0.50000000 0.50000000 0.50000000 3 1.00000000 0.50000000 0.33333333 0.25000000 0.20833333 4 1.00000000 0.50000000 0.33333333 0.25000000 0.20370370 5 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 6 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 7 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 8 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 9 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 10 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 11 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 12 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 13 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 14 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 15 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 16 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 17 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 18 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 19 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 20 0.99999996 0.49999999 0.33333333 0.25000000 0.20000000 21 0.99999990 0.49999995 0.33333330 0.24999998 0.19999999 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.03125000 0.01562500 0.00781250 0.47942554 1.64872127 2 0.50000000 0.50000000 0.50000000 0.42073549 1.85914091 3 0.18750000 0.17708333 0.17187500 0.45986219 1.71886115 4 0.17592593 0.15843621 0.14711934 0.45977056 1.71854015 5 0.16666667 0.14322917 0.12630208 0.45969745 1.71828269 6 0.16666667 0.14306667 0.12573333 0.45969756 1.71828231 7 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 8 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 9 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 10 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 11 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 12 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 13 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 14 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 15 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 16 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 17 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 18 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 19 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 20 0.16666667 0.14285714 0.12500000 0.45969768 1.71828177 21 0.16666666 0.14285714 0.12500000 0.45969764 1.71828166 Order SQRT(|X|) 1 0.70710678 2 0.50000000 3 0.63807119 4 0.64769257 5 0.65775660 6 0.65963774 7 0.66229602 8 0.66297345 9 0.66404880 10 0.66437037 11 0.66491152 12 0.66509039 13 0.66540158 14 0.66551186 15 0.66570768 16 0.66578077 17 0.66591226 18 0.66596337 19 0.66605610 20 0.66609332 21 0.66616122 TEST39 NCO_SET sets up an open Newton-Cotes rule; SUM_SUB carries it out. The integration interval is [ 0.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 1.00000000 0.50000000 0.25000000 0.12500000 0.06250000 2 1.00000000 0.50000000 0.27777778 0.16666667 0.10493827 3 1.00000000 0.50000000 0.33333333 0.25000000 0.19270833 4 1.00000000 0.50000000 0.33333333 0.25000000 0.19493333 5 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 6 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 7 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 9 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.03125000 0.01562500 0.00781250 0.47942554 1.64872127 2 0.06790123 0.04458162 0.02949246 0.47278225 1.67167323 3 0.14843750 0.11360677 0.08642578 0.45955330 1.71777653 4 0.15400000 0.12229333 0.09736000 0.45959752 1.71793017 5 0.16666667 0.14210391 0.12236368 0.45969819 1.71828009 6 0.16666667 0.14232519 0.12313818 0.45969805 1.71828060 7 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 9 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 Order SQRT(|X|) 1 0.70710678 2 0.69692343 3 0.67498134 4 0.67354587 5 0.67016159 6 0.66969878 7 0.66859434 9 0.66789365 TEST40 NCO_COMPUTE computes an open Newton-Cotes rule; SUM_SUB carries it out. The integration interval is [ 0.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 1.00000000 0.50000000 0.25000000 0.12500000 0.06250000 2 1.00000000 0.50000000 0.27777778 0.16666667 0.10493827 3 1.00000000 0.50000000 0.33333333 0.25000000 0.19270833 4 1.00000000 0.50000000 0.33333333 0.25000000 0.19493333 5 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 6 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 7 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 8 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 9 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.03125000 0.01562500 0.00781250 0.47942554 1.64872127 2 0.06790123 0.04458162 0.02949246 0.47278225 1.67167323 3 0.14843750 0.11360677 0.08642578 0.45955330 1.71777653 4 0.15400000 0.12229333 0.09736000 0.45959752 1.71793017 5 0.16666667 0.14210391 0.12236368 0.45969819 1.71828009 6 0.16666667 0.14232519 0.12313818 0.45969805 1.71828060 7 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 8 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 9 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 Order SQRT(|X|) 1 0.70710678 2 0.69692343 3 0.67498134 4 0.67354587 5 0.67016159 6 0.66969878 7 0.66859434 8 0.66838671 9 0.66789365 TEST401 NCOH_SET sets up an open half Newton-Cotes rule; SUM_SUB carries it out. The integration interval is [ 0.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 1.00000000 0.50000000 0.25000000 0.12500000 0.06250000 2 1.00000000 0.50000000 0.31250000 0.21875000 0.16015625 3 1.00000000 0.50000000 0.33333333 0.25000000 0.19675926 4 1.00000000 0.50000000 0.33333333 0.25000000 0.19832357 5 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 6 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 7 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 8 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 9 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 10 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.03125000 0.01562500 0.00781250 0.47942554 1.64872127 2 0.11914063 0.08911133 0.06677246 0.46452136 1.70051272 3 0.15856481 0.12950103 0.10661008 0.45963375 1.71805643 4 0.16247559 0.13584646 0.11513138 0.45966469 1.71816500 5 0.16666667 0.14259167 0.12407083 0.45969787 1.71828122 6 0.16666667 0.14269768 0.12444189 0.45969780 1.71828146 7 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 8 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 9 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 10 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 Order SQRT(|X|) 1 0.70710678 2 0.68301270 3 0.67219640 4 0.67060294 5 0.66887650 6 0.66845061 7 0.66787450 8 0.66769842 9 0.66743641 10 0.66734589 TEST402 NCOH_COMPUTE computes an open half Newton-Cotes rule; SUM_SUB carries it out. The integration interval is [ 0.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 1.00000000 0.50000000 0.25000000 0.12500000 0.06250000 2 1.00000000 0.50000000 0.31250000 0.21875000 0.16015625 3 1.00000000 0.50000000 0.33333333 0.25000000 0.19675926 4 1.00000000 0.50000000 0.33333333 0.25000000 0.19832357 5 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 6 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 7 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 8 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 9 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 10 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.03125000 0.01562500 0.00781250 0.47942554 1.64872127 2 0.11914063 0.08911133 0.06677246 0.46452136 1.70051272 3 0.15856481 0.12950103 0.10661008 0.45963375 1.71805643 4 0.16247559 0.13584646 0.11513138 0.45966469 1.71816500 5 0.16666667 0.14259167 0.12407083 0.45969787 1.71828122 6 0.16666667 0.14269768 0.12444189 0.45969780 1.71828146 7 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 8 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 9 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 10 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 Order SQRT(|X|) 1 0.70710678 2 0.68301270 3 0.67219640 4 0.67060294 5 0.66887650 6 0.66845061 7 0.66787450 8 0.66769842 9 0.66743641 10 0.66734589 TEST403 PATTERSON_SET sets up a Gauss-Patterson rule; SUM_SUB carries it out. The integration interval is [ 0.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 1.00000000 0.50000000 0.25000000 0.12500000 0.06250000 3 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 7 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 15 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 31 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 63 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 127 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.03125000 0.01562500 0.00781250 0.47942554 1.64872127 3 0.16666667 0.14250000 0.12375000 0.45969793 1.71828100 7 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 15 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 31 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 63 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 127 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 Order SQRT(|X|) 1 0.70710678 3 0.66917963 7 0.66680883 15 0.66667372 31 0.66666700 63 0.66666668 127 0.66666667 TEST404 RADAU_COMPUTE computes a Radau rule; RADAU_SET sets a rule from a table. I X1 X2 W1 W2 1 -1.00000000 -1.00000000 0.12500000 0.12500000 2 -0.57531892 -0.57531892 0.65768864 0.65768864 3 0.18106627 0.18106627 0.77638694 0.77638694 4 0.82282408 0.82282408 0.44092442 0.44092442 1 -1.00000000 -1.00000000 0.04081633 0.04081633 2 -0.85389134 -0.85389134 0.23922749 0.23922749 3 -0.53846772 -0.53846772 0.38094987 0.38094987 4 -0.11734304 -0.11734304 0.44710983 0.44710983 5 0.32603062 0.32603062 0.42470378 0.42470378 6 0.70384280 0.70384280 0.31820423 0.31820423 7 0.94136715 0.94136715 0.14898847 0.14898847 1 -1.00000000 -1.00000000 0.02000000 0.02000000 2 -0.92748437 -0.92748437 0.12029667 0.12029667 3 -0.76384204 -0.76384204 0.20427013 0.20427013 4 -0.52564603 -0.52564603 0.26819484 0.26819484 5 -0.23623447 -0.23623447 0.30585929 0.30585929 6 0.07605920 0.07605920 0.31358246 0.31358246 7 0.38066484 0.38066484 0.29061016 0.29061016 8 0.64776669 0.64776669 0.23919343 0.23919343 9 0.85122522 0.85122522 0.16437601 0.16437601 10 0.97117518 0.97117518 0.07361701 0.07361701 TEST41 RADAU_SET sets up a Radau rule; SUM_SUB carries it out. The integration interval is [ 0.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X**2 X**3 X**4 1 1.00000000 0.00000000 0.00000000 0.00000000 0.00000000 2 1.00000000 0.50000000 0.33333333 0.22222222 0.14814815 3 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 4 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 5 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 6 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 7 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 8 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 9 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 10 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 11 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 12 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 13 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 14 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 15 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 Order X**5 X**6 X**7 SIN(X) EXP(X) 1 0.00000000 0.00000000 0.00000000 0.00000000 1.00000000 2 0.09876543 0.06584362 0.04389575 0.46377735 1.71080053 3 0.16500000 0.13800000 0.11610000 0.45968552 1.71825905 4 0.16666667 0.14285714 0.12489796 0.45969771 1.71828180 5 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 6 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 7 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 8 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 9 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 10 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 11 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 12 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 13 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 14 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 15 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 Order SQRT(|X|) 1 0.00000000 2 0.61237244 3 0.65136447 4 0.66031525 5 0.66343850 6 0.66480585 7 0.66549760 8 0.66588467 9 0.66611802 10 0.66626700 11 0.66636655 12 0.66643560 13 0.66648499 14 0.66652124 15 0.66654845 QUADRULE_PRB Normal end of execution. November 2 2009 3:51:23.877 PM