*DECK QK15 SUBROUTINE QK15 (F, A, B, RESULT, ABSERR, RESABS, RESASC) C***BEGIN PROLOGUE QK15 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 (QK15-S, DQK15-D) C***KEYWORDS 15-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 calling 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 15-POINT C KRONROD RULE (RESK) obtained by optimal addition C of abscissae to the 7-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 QK15 C REAL A,ABSC,ABSERR,B,CENTR,DHLGTH,EPMACH,F,FC,FSUM,FVAL1,FVAL2, 1 FV1,FV2,HLGTH,RESABS,RESASC,RESG,RESK,RESKH,RESULT,R1MACH,UFLOW, 2 WG,WGK,XGK INTEGER J,JTW,JTWM1 EXTERNAL F C DIMENSION FV1(7),FV2(7),WG(4),WGK(8),XGK(8) 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 15-POINT KRONROD RULE C XGK(2), XGK(4), ... ABSCISSAE OF THE 7-POINT C GAUSS RULE C XGK(1), XGK(3), ... ABSCISSAE WHICH ARE OPTIMALLY C ADDED TO THE 7-POINT GAUSS RULE C C WGK - WEIGHTS OF THE 15-POINT KRONROD RULE C C WG - WEIGHTS OF THE 7-POINT GAUSS RULE C SAVE XGK, WGK, WG DATA XGK(1),XGK(2),XGK(3),XGK(4),XGK(5),XGK(6),XGK(7),XGK(8)/ 1 0.9914553711208126E+00, 0.9491079123427585E+00, 2 0.8648644233597691E+00, 0.7415311855993944E+00, 3 0.5860872354676911E+00, 0.4058451513773972E+00, 4 0.2077849550078985E+00, 0.0E+00 / DATA WGK(1),WGK(2),WGK(3),WGK(4),WGK(5),WGK(6),WGK(7),WGK(8)/ 1 0.2293532201052922E-01, 0.6309209262997855E-01, 2 0.1047900103222502E+00, 0.1406532597155259E+00, 3 0.1690047266392679E+00, 0.1903505780647854E+00, 4 0.2044329400752989E+00, 0.2094821410847278E+00/ DATA WG(1),WG(2),WG(3),WG(4)/ 1 0.1294849661688697E+00, 0.2797053914892767E+00, 2 0.3818300505051189E+00, 0.4179591836734694E+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 7-POINT GAUSS FORMULA C RESK - RESULT OF THE 15-POINT KRONROD FORMULA C RESKH - APPROXIMATION TO THE MEAN VALUE OF F OVER (A,B), C I.E. TO I/(B-A) 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 QK15 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 15-POINT KRONROD APPROXIMATION TO C THE INTEGRAL, AND ESTIMATE THE ABSOLUTE ERROR. C FC = F(CENTR) RESG = FC*WG(4) RESK = FC*WGK(8) RESABS = ABS(RESK) DO 10 J=1,3 JTW = J*2 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 = 1,4 JTWM1 = J*2-1 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(8)*ABS(FC-RESKH) DO 20 J=1,7 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