*DECK QK21 SUBROUTINE QK21 (F, A, B, RESULT, ABSERR, RESABS, RESASC) C***BEGIN PROLOGUE QK21 C***PURPOSE To compute I = Integral of F over (A,B), with error C estimate C J = Integral of ABS(F) over (A,B) C***LIBRARY SLATEC (QUADPACK) C***CATEGORY H2A1A2 C***TYPE SINGLE PRECISION (QK21-S, DQK21-D) C***KEYWORDS 21-POINT GAUSS-KRONROD RULES, QUADPACK, QUADRATURE C***AUTHOR Piessens, Robert C Applied Mathematics and Programming Division C K. U. Leuven C de Doncker, Elise C Applied Mathematics and Programming Division C K. U. Leuven C***DESCRIPTION C C Integration rules C Standard fortran subroutine C Real version C C PARAMETERS C ON ENTRY C F - Real C Function subprogram defining the integrand C FUNCTION F(X). The actual name for F needs to be C Declared E X T E R N A L in the driver program. C C A - Real C Lower limit of integration C C B - Real C Upper limit of integration C C ON RETURN C RESULT - Real C Approximation to the integral I C RESULT is computed by applying the 21-POINT C KRONROD RULE (RESK) obtained by optimal addition C of abscissae to the 10-POINT GAUSS RULE (RESG). C C ABSERR - Real C Estimate of the modulus of the absolute error, C which should not exceed ABS(I-RESULT) C C RESABS - Real C Approximation to the integral J C C RESASC - Real C Approximation to the integral of ABS(F-I/(B-A)) C over (A,B) C C***REFERENCES (NONE) C***ROUTINES CALLED R1MACH C***REVISION HISTORY (YYMMDD) C 800101 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 890531 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C***END PROLOGUE QK21 C REAL A,ABSC,ABSERR,B,CENTR,DHLGTH,EPMACH,F,FC,FSUM,FVAL1,FVAL2, 1 FV1,FV2,HLGTH,RESABS,RESG,RESK,RESKH,RESULT,R1MACH,UFLOW,WG,WGK, 2 XGK INTEGER J,JTW,JTWM1 EXTERNAL F C DIMENSION FV1(10),FV2(10),WG(5),WGK(11),XGK(11) C C THE ABSCISSAE AND WEIGHTS ARE GIVEN FOR THE INTERVAL (-1,1). C BECAUSE OF SYMMETRY ONLY THE POSITIVE ABSCISSAE AND THEIR C CORRESPONDING WEIGHTS ARE GIVEN. C C XGK - ABSCISSAE OF THE 21-POINT KRONROD RULE C XGK(2), XGK(4), ... ABSCISSAE OF THE 10-POINT C GAUSS RULE C XGK(1), XGK(3), ... ABSCISSAE WHICH ARE OPTIMALLY C ADDED TO THE 10-POINT GAUSS RULE C C WGK - WEIGHTS OF THE 21-POINT KRONROD RULE C C WG - WEIGHTS OF THE 10-POINT GAUSS RULE C SAVE XGK, WGK, WG DATA XGK(1),XGK(2),XGK(3),XGK(4),XGK(5),XGK(6),XGK(7), 1 XGK(8),XGK(9),XGK(10),XGK(11)/ 2 0.9956571630258081E+00, 0.9739065285171717E+00, 3 0.9301574913557082E+00, 0.8650633666889845E+00, 4 0.7808177265864169E+00, 0.6794095682990244E+00, 5 0.5627571346686047E+00, 0.4333953941292472E+00, 6 0.2943928627014602E+00, 0.1488743389816312E+00, 7 0.0000000000000000E+00/ C DATA WGK(1),WGK(2),WGK(3),WGK(4),WGK(5),WGK(6),WGK(7), 1 WGK(8),WGK(9),WGK(10),WGK(11)/ 2 0.1169463886737187E-01, 0.3255816230796473E-01, 3 0.5475589657435200E-01, 0.7503967481091995E-01, 4 0.9312545458369761E-01, 0.1093871588022976E+00, 5 0.1234919762620659E+00, 0.1347092173114733E+00, 6 0.1427759385770601E+00, 0.1477391049013385E+00, 7 0.1494455540029169E+00/ C DATA WG(1),WG(2),WG(3),WG(4),WG(5)/ 1 0.6667134430868814E-01, 0.1494513491505806E+00, 2 0.2190863625159820E+00, 0.2692667193099964E+00, 3 0.2955242247147529E+00/ C C C LIST OF MAJOR VARIABLES C ----------------------- C C CENTR - MID POINT OF THE INTERVAL C HLGTH - HALF-LENGTH OF THE INTERVAL C ABSC - ABSCISSA C FVAL* - FUNCTION VALUE C RESG - RESULT OF THE 10-POINT GAUSS FORMULA C RESK - RESULT OF THE 21-POINT KRONROD FORMULA C RESKH - APPROXIMATION TO THE MEAN VALUE OF F OVER (A,B), C I.E. TO I/(B-A) C C C MACHINE DEPENDENT CONSTANTS C --------------------------- C C EPMACH IS THE LARGEST RELATIVE SPACING. C UFLOW IS THE SMALLEST POSITIVE MAGNITUDE. C C***FIRST EXECUTABLE STATEMENT QK21 EPMACH = R1MACH(4) UFLOW = R1MACH(1) C CENTR = 0.5E+00*(A+B) HLGTH = 0.5E+00*(B-A) DHLGTH = ABS(HLGTH) C C COMPUTE THE 21-POINT KRONROD APPROXIMATION TO C THE INTEGRAL, AND ESTIMATE THE ABSOLUTE ERROR. C RESG = 0.0E+00 FC = F(CENTR) RESK = WGK(11)*FC RESABS = ABS(RESK) DO 10 J=1,5 JTW = 2*J ABSC = HLGTH*XGK(JTW) FVAL1 = F(CENTR-ABSC) FVAL2 = F(CENTR+ABSC) FV1(JTW) = FVAL1 FV2(JTW) = FVAL2 FSUM = FVAL1+FVAL2 RESG = RESG+WG(J)*FSUM RESK = RESK+WGK(JTW)*FSUM RESABS = RESABS+WGK(JTW)*(ABS(FVAL1)+ABS(FVAL2)) 10 CONTINUE DO 15 J = 1,5 JTWM1 = 2*J-1 ABSC = HLGTH*XGK(JTWM1) FVAL1 = F(CENTR-ABSC) FVAL2 = F(CENTR+ABSC) FV1(JTWM1) = FVAL1 FV2(JTWM1) = FVAL2 FSUM = FVAL1+FVAL2 RESK = RESK+WGK(JTWM1)*FSUM RESABS = RESABS+WGK(JTWM1)*(ABS(FVAL1)+ABS(FVAL2)) 15 CONTINUE RESKH = RESK*0.5E+00 RESASC = WGK(11)*ABS(FC-RESKH) DO 20 J=1,10 RESASC = RESASC+WGK(J)*(ABS(FV1(J)-RESKH)+ABS(FV2(J)-RESKH)) 20 CONTINUE RESULT = RESK*HLGTH RESABS = RESABS*DHLGTH RESASC = RESASC*DHLGTH ABSERR = ABS((RESK-RESG)*HLGTH) IF(RESASC.NE.0.0E+00.AND.ABSERR.NE.0.0E+00) 1 ABSERR = RESASC*MIN(0.1E+01, 2 (0.2E+03*ABSERR/RESASC)**1.5E+00) IF(RESABS.GT.UFLOW/(0.5E+02*EPMACH)) ABSERR = MAX 1 ((EPMACH*0.5E+02)*RESABS,ABSERR) RETURN END