June 24 2002 2:12:56.971 PM QUADRULE_PRB Test problems for the QUADRULE library. TEST01 BASHFORTH_SET sets up Adams-Bashforth quadrature; SUMMER carries it out. Integration interval is [0,1]. Quadrature order will vary. Integrand will vary. 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 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 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 TEST02 BDFC_SET sets up Backward Difference Corrector quadrature; BDF_SUM carries it out. Integration interval is [0,1]. Quadrature order will vary. Integrand will vary. 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 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 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 TEST04 BDFP_SET sets up Backward Difference Predictor quadrature; BDF_SUM carries it out. Integration interval is [0,1]. Quadrature order will vary. Integrand will vary. 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 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 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 Chebyshev quadrature; SUM_SUB carries it out. The integration interval is ?ð Number of subintervals is 1 Quadrature order will vary. Integrand will vary. 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 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 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 CHEB_TC_SET sets up Gauss-Chebyshev quadrature, ( closed, first kind ); SUMMER carries it out. Integration interval is -1.00000 1.00000 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. The weight function is 1 / sqrt ( 1 - X**2 ) 1 X X**2 X**3 X**4 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 X**5 X**6 X**7 SIN(X) EXP(X) 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 SQRT(|X|) 2 3.14159265 3 1.57079634 4 2.52815853 5 2.10627517 6 2.45716112 TEST06 CHEB_TO_SET sets up Gauss-Chebyshev quadrature, ( open, first kind ); SUMMER carries it out. Integration interval is -1.00000 1.00000 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. The weight function is 1 / sqrt ( 1 - X**2 ) 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 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 SQRT(|X|) 1 0.00000002 2 2.64175400 3 1.94905427 4 2.48154505 5 2.18892706 6 2.44254041 TEST07 CHEB_U_SET sets up Gauss-Chebyshev quadrature; SUMMER carries it out. Integration interval is -1.00000 1.00000 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. The weight function is sqrt ( 1 - X**2 ) 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 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 SQRT(|X|) 1 0.00000001 2 1.11072073 3 0.66043851 4 1.02235409 TEST08 HERMITE_COM computes a Gauss-Hermite rule; SUMMER carries it out. The integration interval is ( -Infinity, +Infinity ). 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 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 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 TEST09 HERMITE_SET sets up Gauss-Hermite quadrature; SUMMER carries it out. The integration interval is ( -Infinity, +Infinity ). 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 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 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_COM sets up Gauss-Jacobi quadrature; SUM_SUB carries it out. The integration interval is -1.00000 1.00000 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. ALPHA = 0.00000 BETA = 0.00000 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 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 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 1 X X**2 X**3 X**4 1 8.00000000 -2.66666667 0.88888889 -0.29629630 0.09876543 2 8.00000000 -0.53333333 1.81333333 -0.83200000 0.69546667 3 8.00000000 0.26618847 1.94076989 -0.03196347 0.86827331 4 8.00000000 0.73716517 2.02172495 0.29463100 0.97632955 5 8.00000000 1.05231614 2.08794711 0.49992465 1.04479947 6 8.00000000 1.27874269 2.14349717 0.64599339 1.09805385 7 8.00000000 1.44944518 2.19034901 0.75659735 1.14198518 8 8.00000000 1.58277672 2.23011382 0.84370954 1.17908574 9 8.00000000 1.68980621 2.26412507 0.91427306 1.21084898 10 8.00000000 1.77761799 2.29345546 0.97267183 1.23832423 X**5 X**6 X**7 SIN(X) EXP(X) 1 -0.03292181 0.01097394 -0.00365798 -2.61755757 5.73225048 2 -0.44458667 0.31692800 -0.21568853 -0.39832904 8.26034038 3 -0.21946406 0.46343343 -0.21824693 0.26972968 9.26620207 4 0.11248378 0.56074115 -0.00229162 0.68899767 9.83953751 5 0.27967733 0.63505093 0.16125869 0.97129424 10.22639915 6 0.39278036 0.68603859 0.25902103 1.17429938 10.50819875 7 0.47721084 0.72619585 0.32952456 1.32725765 10.72336609 8 0.54356385 0.75953001 0.38426542 1.44661271 10.89325559 9 0.59743531 0.78790349 0.42855358 1.54232186 11.03087264 10 0.64219519 0.81242063 0.46535705 1.62076629 11.14464253 SQRT(|X|) 1 4.61880215 2 5.15800624 3 5.01268212 4 5.02888343 5 5.04564210 6 5.05918486 7 5.08398457 8 5.09602744 9 5.11662301 10 5.12665718 TEST11 KRONROD_SET sets up Kronrod quadrature; LEGENDRE_SET sets up Gauss-Legendre quadrature; SUMMER_GK carries it out. 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 Kronrod quadrature; LEGENDRE_SET sets up Gauss-Legendre quadrature; SUM_SUB_GK carries it out. Integration interval is -1.00000 1.00000 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_COM 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 , +Infinity ). ALPHA = 0.00000 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 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 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_COM 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 ). ALPHA = 0.00000 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 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 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 TEST15 LAGUERRE_COM 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 ). ALPHA = 1.00000 1 X X**2 X**3 X**4 1 1.00000 2.00000 4.00000 8.00000 16.0000 2 1.00000 2.00000 6.00000 24.0000 108.000 3 1.00000 2.00000 6.00000 24.0000 120.000 4 1.00000 2.00000 6.00000 24.0000 120.000 5 1.00000 2.00000 6.00000 24.0000 120.000 6 1.00000 2.00000 6.00000 24.0000 120.000 7 1.00000 2.00000 6.00000 24.0000 120.000 8 1.00000 2.00000 6.00000 24.0000 120.000 9 1.00000 2.00000 6.00000 24.0000 120.000 10 1.00000 2.00000 6.00000 24.0000 120.000 X**5 X**6 X**7 SIN(X) EXP(X) 1 32.0000 64.0000 128.000 0.909297 7.38906 2 504.000 2376.00 11232.0 0.541499 26.7939 3 720.000 4896.00 35712.0 0.430151 59.2456 4 720.000 5040.00 40320.0 0.519921 105.297 5 720.000 5040.00 40320.0 0.498951 165.302 6 720.000 5040.00 40320.0 0.498970 239.507 7 720.000 5040.00 40320.0 0.500370 328.096 8 720.000 5040.00 40320.0 0.499954 431.213 9 720.000 5040.00 40320.0 0.499992 548.974 10 720.000 5040.00 40320.0 0.500005 681.476 SQRT(|X|) 1 1.41421 2 1.34777 3 1.33673 4 1.33315 5 1.33161 6 1.33081 7 1.33036 8 1.33008 9 1.32990 10 1.32977 TEST16 LAGUERRE_COM 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 ). ALPHA = 2.00000 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 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 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 TEST17 LAGUERRE_SET sets up Gauss-Laguerre quadrature; 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 ). 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 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 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 TEST16 LEGENDRE_COM computes Gauss-Legendre data; LEGENDRE_SET looks up the same data. Compare the data for NORDER = 19 The computed and stored abscissas are identical. Maximum weight difference is 0.138777878078144568E-15 for index I = 9 Computed: 0.158968843393954479 Stored: 0.158968843393954340 TEST19 LEGENDRE_COM computes a Gauss-Legendre rule; SUM_SUB carries it out over subintervals. The integration interval is 0.00000 1.00000 Here, we use a fixed order NORDER = 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_COM sets up Gauss-Legendre quadrature; SUM_SUB carries it out. The integration interval is 0.00000 1.00000 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. 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 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 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 TEST21 LEGENDRE_SET sets up Gauss-Legendre quadrature; SUM_SUB carries it out. The integration interval is 0.00000 1.00000 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. 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 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 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 Gauss-Legendre quadrature for integrating F(X) over [-1,1]; RULE_ADJUST adjusts a rule for a new interval. LEGENDRE_SET_X0_01 sets up Gauss-Legendre quadrature 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_SET_X0_01. 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_SET_COS sets up Gauss-Legendre quadrature over [-PI/2,PI/2] with weight function COS(X); SUM_SUB carries it out. The integration interval is -1.57080 1.57080 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. 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 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 SQRT(|X|) 1 0.00000000 2 1.65368363 4 1.51755614 8 1.45413652 16 1.42628067 TEST235 LEGENDRE_SET_SQRTX_01 sets up Gauss-Legendre quadrature over [0,1] with weight function SQRT(X); SUM_SUB carries it out. The integration interval is 0.00000 1.00000 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. 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 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 SQRT(|X|) 1 0.51639778 2 0.50205963 4 0.50020905 8 0.50001755 TEST236 LEGENDRE_SET_SQRTX2_01 sets up Gauss-Legendre quadrature over [0,1] with weight function 1/SQRT(X); SUM_SUB carries it out. The integration interval is 0.00000 1.00000 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. 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 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 SQRT(|X|) 1 0.86066297 2 0.94485044 4 0.98168976 8 0.99462159 TEST24 LEGENDRE_SET_COS2 sets up Gauss-Legendre quadrature over [0,PI/2] with weight function COS(X); SUM_SUB carries it out. The integration interval is -1.57080 1.57080 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. 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 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 SQRT(|X|) 2 0.71452828 4 0.70593166 8 0.70433678 16 0.70408047 TEST25 LEGENDRE_SET_LOG sets up Gauss-Legendre quadrature over [0,1] with weight function -LOG(X); SUM_SUB carries it out. The integration interval is 0.00000 1.00000 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. 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 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 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 TEST26 LEGENDRE_SET_X0_01 sets up Gauss-Legendre quadrature for integrating F(X) over [0,1]; SUM_SUB carries it out. The integration interval is 0.00000 1.00000 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. 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 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 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 TEST27 LEGENDRE_SET_X1 sets up Gauss-Legendre quadrature for integrating ( 1 + X ) * F(X) over [-1,1]; SUM_SUB carries it out. The integration interval is 0.00000 1.00000 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. 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 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 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 TEST28 LEGENDRE_SET_X1 sets up Gauss-Legendre quadrature for integrating ( 1 + X ) * F(X) over [-1,1]; RULE_ADJUST adjusts a rule for a new interval. LEGENDRE_SET_X1_01 sets up Gauss-Legendre quadrature for integrating X * F(X) over [0,1]; We will use LEGENDRE_SET_X1 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_SET_X1_01. 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 0.146388870100 4 0.668698552377 0.167174638094 0.167174638100 5 0.387126360907 0.096781590227 0.096781590200 TEST29 LEGENDRE_SET_X1_01 sets up Gauss-Legendre quadrature for integrating X * F(X) over [0,1]; SUM_SUB carries it out. The integration interval is 0.00000 1.00000 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. 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 0.50000188 0.33333439 0.25000059 0.20000033 0.16666685 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 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 0.14285725 0.12500006 0.11111114 0.30116968 1.00000330 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 SQRT(|X|) 1 0.40824829 2 0.40076931 3 0.40016502 4 0.40005174 5 0.40002173 6 0.40000927 7 0.40000471 8 0.40000260 TEST30 LEGENDRE_SET_X2 sets up Gauss-Legendre quadrature for integrating (1+X)**2 * F(X) over [-1,1]; SUM_SUB carries it out. The integration interval is 0.00000 1.00000 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. 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 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 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 TEST31 LEGENDRE_SET_X2 sets up Gauss-Legendre quadrature for integrating ( 1 + X )^2 * F(X) over [-1,1]; RULE_ADJUST adjusts a rule for a new interval. LEGENDRE_SET_X2_01 sets up Gauss-Legendre quadrature for integrating X^2 * F(X) over [0,1]; We will use LEGENDRE_SET_X2 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_SET_X2_01. 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 TEST32 LEGENDRE_SET_X2_01 sets up Gauss-Legendre quadrature for integrating X*X * F(X) over [0,1]; SUM_SUB carries it out. The integration interval is 0.00000 1.00000 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. 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 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 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 TEST33 LOBATTO_SET sets up Lobatto quadrature; SUM_SUB carries it out. Integration interval is -1.00000 1.00000 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. 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 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 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 TEST34 MOULTON_SET sets up Adams-Moulton quadrature; SUMMER carries it out. Integration interval is [0,1]. Quadrature order will vary. Integrand will vary. 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 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 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 TEST35 NCC_SET sets up a closed Newton Cotes rule; SUM_SUB carries it out. Integration interval is 0.00000 1.00000 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. 1 X X**2 X**3 X**4 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 X**5 X**6 X**7 SIN(X) EXP(X) 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 SQRT(|X|) 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 0.66551148 15 0.66570768 16 0.66578077 17 0.66591226 18 0.66596337 19 0.66605610 20 0.66609334 TEST36 NCC_COM computes a closed Newton Cotes rule; SUM_SUB carries it out. Integration interval is 0.00000 1.00000 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. 1 X X**2 X**3 X**4 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 X**5 X**6 X**7 SIN(X) EXP(X) 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 SQRT(|X|) 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 TEST37 NCO_SET sets up an open Newton-Cotes rule; SUM_SUB carries it out. Integration interval is 0.00000 1.00000 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. 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 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 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 TEST38 NCO_COM sets up an open Newton-Cotes rule; SUM_SUB carries it out. Integration interval is 0.00000 1.00000 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. 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 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 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 TEST39 RADAU_SET sets up Radau quadrature; SUM_SUB carries it out. The integration interval is 0.00000 1.00000 Number of subintervals is 1 Quadrature order will vary. Integrand will vary. 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 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 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.