OpenLibm/slatec/qk15.f

*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