      SUBROUTINE DQK15(F,A,B,RESULT,ABSERR,RESABS,RESASC)
C***BEGIN PROLOGUE  DQK15
C***DATE WRITTEN   800101   (YYMMDD)
C***REVISION DATE  870530   (YYMMDD)
C***CATEGORY NO.  H2A1A2
C***KEYWORDS  15-POINT GAUSS-KRONROD RULES
C***AUTHOR  PIESSENS, ROBERT, AND DE DONCKER, ELISE,
C             APPLIED MATH. AND PROGR. DIV. - K. U. LEUVEN
C***PURPOSE  To compute I = Integral of F over (A,B), with error estimate,
C               and     J = integral of ABS(F) over (A,B)
C***DESCRIPTION
C           Double precision version
C
C           PARAMETERS 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: Lower limit of integration
C
C              B      - Double precision: Upper limit of integration
C
C            PARAMETERS ON RETURN
C              RESULT - Double precision: 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 the7-POINT GAUSS RULE(RESG).
C
C              ABSERR - Double precision: Estimate of the modulus of the
C                       absolute error, which should not exceed ABS(I-RESULT)
C
C              RESABS - Double precision: Approximation to the integral J
C
C              RESASC - Double precision: Approximation to the integral of
C                       ABS(F-I/(B-A))  over (A,B)
C***REFERENCES  (NONE)
C***ROUTINES CALLED  D1MACH
C***END PROLOGUE  DQK15
C
      DOUBLE PRECISION A,ABSC,ABSERR,B,CENTR,DABS,DHLGTH,DMAX1,DMIN1,
     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 KRONRON QUADRATURE ABSCISSAE AND WEIGHTS
C AS EVALUATED WITH 80 DECIMAL DIGIT ARITHMETIC BY L. W. FULLERTON,
C BELL LABS, NOV. 1981.
C
      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 = DABS(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 = DABS(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)*(DABS(FVAL1)+DABS(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)*(DABS(FVAL1)+DABS(FVAL2))
   15 CONTINUE
      RESKH = RESK*0.5D+00
      RESASC = WGK(8)*DABS(FC-RESKH)
      DO 20 J=1,7
        RESASC = RESASC+WGK(J)*(DABS(FV1(J)-RESKH)+DABS(FV2(J)-RESKH))
   20 CONTINUE
      RESULT = RESK*HLGTH
      RESABS = RESABS*DHLGTH
      RESASC = RESASC*DHLGTH
      ABSERR = DABS((RESK-RESG)*HLGTH)
      IF(RESASC.NE.0.0D+00.AND.ABSERR.NE.0.0D+00)
     1  ABSERR = RESASC*DMIN1(0.1D+01,(0.2D+03*ABSERR/RESASC)**1.5D+00)
      IF(RESABS.GT.UFLOW/(0.5D+02*EPMACH)) ABSERR = DMAX1
     1  ((EPMACH*0.5D+02)*RESABS,ABSERR)
      RETURN
      END
