# dqags.f

          SUBROUTINE DQAGS (F, A, B, EPSABS, EPSREL, RESULT, ABSERR, NEVAL,
         +   IER, LIMIT, LENW, LAST, IWORK, WORK)
    C***BEGIN PROLOGUE  DQAGS
    C***PURPOSE  The routine calculates an approximation result to a given
    C            Definite integral  I = Integral of F over (A,B),
    C            Hopefully satisfying following claim for accuracy
    C            ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)).
    C***LIBRARY   SLATEC (QUADPACK)
    C***CATEGORY  H2A1A1
    C***TYPE      DOUBLE PRECISION (QAGS-S, DQAGS-D)
    C***KEYWORDS  AUTOMATIC INTEGRATOR, END POINT SINGULARITIES,
    C             EXTRAPOLATION, GENERAL-PURPOSE, GLOBALLY ADAPTIVE,
    C             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        Computation of a definite integral
    C        Standard fortran subroutine
    C        Double precision version
    C
    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 driver program.
    C
    C            A      - Double precision
    C                     Lower limit of integration
    C
    C            B      - Double precision
    C                     Upper limit of integration
    C
    C            EPSABS - Double precision
    C                     Absolute accuracy requested
    C            EPSREL - Double precision
    C                     Relative accuracy requested
    C                     If  EPSABS.LE.0
    C                     And EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28),
    C                     The routine will end with IER = 6.
    C
    C         ON RETURN
    C            RESULT - Double precision
    C                     Approximation to the integral
    C
    C            ABSERR - Double precision
    C                     Estimate of the modulus of the absolute error,
    C                     which should equal or exceed ABS(I-RESULT)
    C
    C            NEVAL  - Integer
    C                     Number of integrand evaluations
    C
    C            IER    - Integer
    C                     IER = 0 Normal and reliable termination of the
    C                             routine. It is assumed that the requested
    C                             accuracy has been achieved.
    C                     IER.GT.0 Abnormal termination of the routine
    C                             The estimates for integral and error are
    C                             less reliable. It is assumed that the
    C                             requested accuracy has not been achieved.
    C            ERROR MESSAGES
    C                     IER = 1 Maximum number of subdivisions allowed
    C                             has been achieved. One can allow more sub-
    C                             divisions by increasing the value of LIMIT
    C                             (and taking the according dimension
    C                             adjustments into account. However, if
    C                             this yields no improvement it is advised
    C                             to analyze the integrand in order to
    C                             determine the integration difficulties. If
    C                             the position of a local difficulty can be
    C                             determined (E.G. SINGULARITY,
    C                             DISCONTINUITY WITHIN THE INTERVAL) one
    C                             will probably gain from splitting up the
    C                             interval at this point and calling the
    C                             integrator on the subranges. If possible,
    C                             an appropriate special-purpose integrator
    C                             should be used, which is designed for
    C                             handling the type of difficulty involved.
    C                         = 2 The occurrence of roundoff error is detec-
    C                             ted, which prevents the requested
    C                             tolerance from being achieved.
    C                             The error may be under-estimated.
    C                         = 3 Extremely bad integrand behaviour
    C                             occurs at some points of the integration
    C                             interval.
    C                         = 4 The algorithm does not converge.
    C                             Roundoff error is detected in the
    C                             Extrapolation table. It is presumed that
    C                             the requested tolerance cannot be
    C                             achieved, and that the returned result is
    C                             the best which can be obtained.
    C                         = 5 The integral is probably divergent, or
    C                             slowly convergent. It must be noted that
    C                             divergence can occur with any other value
    C                             of IER.
    C                         = 6 The input is invalid, because
    C                             (EPSABS.LE.0 AND
    C                              EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28)
    C                             OR LIMIT.LT.1 OR LENW.LT.LIMIT*4.
    C                             RESULT, ABSERR, NEVAL, LAST are set to
    C                             zero.  Except when LIMIT or LENW is
    C                             invalid, IWORK(1), WORK(LIMIT*2+1) and
    C                             WORK(LIMIT*3+1) are set to zero, WORK(1)
    C                             is set to A and WORK(LIMIT+1) TO B.
    C
    C         DIMENSIONING PARAMETERS
    C            LIMIT - Integer
    C                    DIMENSIONING PARAMETER FOR IWORK
    C                    LIMIT determines the maximum number of subintervals
    C                    in the partition of the given integration interval
    C                    (A,B), LIMIT.GE.1.
    C                    IF LIMIT.LT.1, the routine will end with IER = 6.
    C
    C            LENW  - Integer
    C                    DIMENSIONING PARAMETER FOR WORK
    C                    LENW must be at least LIMIT*4.
    C                    If LENW.LT.LIMIT*4, the routine will end
    C                    with IER = 6.
    C
    C            LAST  - Integer
    C                    On return, LAST equals the number of subintervals
    C                    produced in the subdivision process, determines the
    C                    number of significant elements actually in the WORK
    C                    Arrays.
    C
    C         WORK ARRAYS
    C            IWORK - Integer
    C                    Vector of dimension at least LIMIT, the first K
    C                    elements of which contain pointers
    C                    to the error estimates over the subintervals
    C                    such that WORK(LIMIT*3+IWORK(1)),... ,
    C                    WORK(LIMIT*3+IWORK(K)) form a decreasing
    C                    sequence, with K = LAST IF LAST.LE.(LIMIT/2+2),
    C                    and K = LIMIT+1-LAST otherwise
    C
    C            WORK  - Double precision
    C                    Vector of dimension at least LENW
    C                    on return
    C                    WORK(1), ..., WORK(LAST) contain the left
    C                     end-points of the subintervals in the
    C                     partition of (A,B),
    C                    WORK(LIMIT+1), ..., WORK(LIMIT+LAST) contain
    C                     the right end-points,
    C                    WORK(LIMIT*2+1), ..., WORK(LIMIT*2+LAST) contain
    C                     the integral approximations over the subintervals,
    C                    WORK(LIMIT*3+1), ..., WORK(LIMIT*3+LAST)
    C                     contain the error estimates.
    C
    C***REFERENCES  (NONE)
    C***ROUTINES CALLED  DQAGSE, XERMSG
    C***REVISION HISTORY  (YYMMDD)
    C   800101  DATE WRITTEN
    C   890831  Modified array declarations.  (WRB)
    C   890831  REVISION DATE from Version 3.2
    C   891214  Prologue converted to Version 4.0 format.  (BAB)
    C   900315  CALLs to XERROR changed to CALLs to XERMSG.  (THJ)
    C***END PROLOGUE  DQAGS