*DECK DQK15 SUBROUTINE DQK15 (F, A, B, RESULT, ABSERR, RESABS, RESASC) C***BEGIN PROLOGUE DQK15 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 DOUBLE 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 Double precision version C C PARAMETERS C ON ENTRY C F - Double precision 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 - Double precision C Lower limit of integration C C B - Double precision C Upper limit of integration C C ON RETURN C RESULT - Double precision 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 - Double precision C Estimate of the modulus of the absolute error, C which should not exceed ABS(I-RESULT) C C RESABS - Double precision C Approximation to the integral J C C RESASC - Double precision C Approximation to the integral of ABS(F-I/(B-A)) C over (A,B) C C***REFERENCES (NONE) C***ROUTINES CALLED D1MACH 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 DQK15 C DOUBLE PRECISION A,ABSC,ABSERR,B,CENTR,DHLGTH, 1 D1MACH,EPMACH,F,FC,FSUM,FVAL1,FVAL2,FV1,FV2,HLGTH,RESABS,RESASC, 2 RESG,RESK,RESKH,RESULT,UFLOW,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 C C GAUSS QUADRATURE WEIGHTS AND KRONROD QUADRATURE ABSCISSAE AND WEIGHTS C AS EVALUATED WITH 80 DECIMAL DIGIT ARITHMETIC BY L. W. FULLERTON, C BELL LABS, NOV. 1981. C SAVE WG, XGK, WGK DATA WG ( 1) / 0.1294849661 6886969327 0611432679 082 D0 / DATA WG ( 2) / 0.2797053914 8927666790 1467771423 780 D0 / DATA WG ( 3) / 0.3818300505 0511894495 0369775488 975 D0 / DATA WG ( 4) / 0.4179591836 7346938775 5102040816 327 D0 / C DATA XGK ( 1) / 0.9914553711 2081263920 6854697526 329 D0 / DATA XGK ( 2) / 0.9491079123 4275852452 6189684047 851 D0 / DATA XGK ( 3) / 0.8648644233 5976907278 9712788640 926 D0 / DATA XGK ( 4) / 0.7415311855 9939443986 3864773280 788 D0 / DATA XGK ( 5) / 0.5860872354 6769113029 4144838258 730 D0 / DATA XGK ( 6) / 0.4058451513 7739716690 6606412076 961 D0 / DATA XGK ( 7) / 0.2077849550 0789846760 0689403773 245 D0 / DATA XGK ( 8) / 0.0000000000 0000000000 0000000000 000 D0 / C DATA WGK ( 1) / 0.0229353220 1052922496 3732008058 970 D0 / DATA WGK ( 2) / 0.0630920926 2997855329 0700663189 204 D0 / DATA WGK ( 3) / 0.1047900103 2225018383 9876322541 518 D0 / DATA WGK ( 4) / 0.1406532597 1552591874 5189590510 238 D0 / DATA WGK ( 5) / 0.1690047266 3926790282 6583426598 550 D0 / DATA WGK ( 6) / 0.1903505780 6478540991 3256402421 014 D0 / DATA WGK ( 7) / 0.2044329400 7529889241 4161999234 649 D0 / DATA WGK ( 8) / 0.2094821410 8472782801 2999174891 714 D0 / 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 DQK15 EPMACH = D1MACH(4) UFLOW = D1MACH(1) C CENTR = 0.5D+00*(A+B) HLGTH = 0.5D+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*0.5D+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.0.0D+00.AND.ABSERR.NE.0.0D+00) 1 ABSERR = RESASC*MIN(0.1D+01,(0.2D+03*ABSERR/RESASC)**1.5D+00) IF(RESABS.GT.UFLOW/(0.5D+02*EPMACH)) ABSERR = MAX 1 ((EPMACH*0.5D+02)*RESABS,ABSERR) RETURN END