# dqage.f

          SUBROUTINE DQAGE (F, A, B, EPSABS, EPSREL, KEY, LIMIT, RESULT,
         +   ABSERR, NEVAL, IER, ALIST, BLIST, RLIST, ELIST, IORD, LAST)
    C***BEGIN PROLOGUE  DQAGE
    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-RESLT).LE.MAX(EPSABS,EPSREL*ABS(I)).
    C***LIBRARY   SLATEC (QUADPACK)
    C***CATEGORY  H2A1A1
    C***TYPE      DOUBLE PRECISION (QAGE-S, DQAGE-D)
    C***KEYWORDS  AUTOMATIC INTEGRATOR, GAUSS-KRONROD RULES,
    C             GENERAL-PURPOSE, GLOBALLY ADAPTIVE, INTEGRAND EXAMINATOR,
    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        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            KEY    - Integer
    C                     Key for choice of local integration rule
    C                     A Gauss-Kronrod pair is used with
    C                          7 - 15 points if KEY.LT.2,
    C                         10 - 21 points if KEY = 2,
    C                         15 - 31 points if KEY = 3,
    C                         20 - 41 points if KEY = 4,
    C                         25 - 51 points if KEY = 5,
    C                         30 - 61 points if KEY.GT.5.
    C
    C            LIMIT  - Integer
    C                     Gives an upper bound on the number of subintervals
    C                     in the partition of (A,B), LIMIT.GE.1.
    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 result 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
    C                             subdivisions by increasing the value
    C                             of LIMIT.
    C                             However, if this yields no improvement it
    C                             is rather advised to analyze the integrand
    C                             in order to determine the integration
    C                             difficulties. If the position of a local
    C                             difficulty can be determined(e.g.
    C                             SINGULARITY, DISCONTINUITY within the
    C                             interval) one will probably gain from
    C                             splitting up the interval at this point
    C                             and calling the integrator on the
    C                             subranges. If possible, an appropriate
    C                             special-purpose integrator should be used
    C                             which is designed for handling the type of
    C                             difficulty involved.
    C                         = 2 The occurrence of roundoff error is
    C                             detected, which prevents the requested
    C                             tolerance from being achieved.
    C                         = 3 Extremely bad integrand behaviour occurs
    C                             at some points of the integration
    C                             interval.
    C                         = 6 The input is invalid, because
    C                             (EPSABS.LE.0 and
    C                              EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28),
    C                             RESULT, ABSERR, NEVAL, LAST, RLIST(1) ,
    C                             ELIST(1) and IORD(1) are set to zero.
    C                             ALIST(1) and BLIST(1) are set to A and B
    C                             respectively.
    C
    C            ALIST   - Double precision
    C                      Vector of dimension at least LIMIT, the first
    C                       LAST  elements of which are the left
    C                      end points of the subintervals in the partition
    C                      of the given integration range (A,B)
    C
    C            BLIST   - Double precision
    C                      Vector of dimension at least LIMIT, the first
    C                       LAST  elements of which are the right
    C                      end points of the subintervals in the partition
    C                      of the given integration range (A,B)
    C
    C            RLIST   - Double precision
    C                      Vector of dimension at least LIMIT, the first
    C                       LAST  elements of which are the
    C                      integral approximations on the subintervals
    C
    C            ELIST   - Double precision
    C                      Vector of dimension at least LIMIT, the first
    C                       LAST  elements of which are the moduli of the
    C                      absolute error estimates on the subintervals
    C
    C            IORD    - Integer
    C                      Vector of dimension at least LIMIT, the first K
    C                      elements of which are pointers to the
    C                      error estimates over the subintervals,
    C                      such that ELIST(IORD(1)), ...,
    C                      ELIST(IORD(K)) form a decreasing sequence,
    C                      with K = LAST if LAST.LE.(LIMIT/2+2), and
    C                      K = LIMIT+1-LAST otherwise
    C
    C            LAST    - Integer
    C                      Number of subintervals actually produced in the
    C                      subdivision process
    C
    C***REFERENCES  (NONE)
    C***ROUTINES CALLED  D1MACH, DQK15, DQK21, DQK31, DQK41, DQK51, DQK61,
    C                    DQPSRT
    C***REVISION HISTORY  (YYMMDD)
    C   800101  DATE WRITTEN
    C   890531  Changed all specific intrinsics to generic.  (WRB)
    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***END PROLOGUE  DQAGE