# qagp.f

          SUBROUTINE QAGP (F, A, B, NPTS2, POINTS, EPSABS, EPSREL, RESULT,
         +   ABSERR, NEVAL, IER, LENIW, LENW, LAST, IWORK, WORK)
    C***BEGIN PROLOGUE  QAGP
    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            break points of the integration interval, where local
    C            difficulties of the integrand may occur(e.g. SINGULARITIES,
    C            DISCONTINUITIES), are provided by the user.
    C***LIBRARY   SLATEC (QUADPACK)
    C***CATEGORY  H2A2A1
    C***TYPE      SINGLE PRECISION (QAGP-S, DQAGP-D)
    C***KEYWORDS  AUTOMATIC INTEGRATOR, EXTRAPOLATION, GENERAL-PURPOSE,
    C             GLOBALLY ADAPTIVE, QUADPACK, QUADRATURE,
    C             SINGULARITIES AT USER SPECIFIED POINTS
    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        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 driver program.
    C
    C            A      - Real
    C                     Lower limit of integration
    C
    C            B      - Real
    C                     Upper limit of integration
    C
    C            NPTS2  - Integer
    C                     Number equal to two more than the number of
    C                     user-supplied break points within the integration
    C                     range, NPTS.GE.2.
    C                     If NPTS2.LT.2, The routine will end with IER = 6.
    C
    C            POINTS - Real
    C                     Vector of dimension NPTS2, the first (NPTS2-2)
    C                     elements of which are the user provided break
    C                     points. If these points do not constitute an
    C                     ascending sequence there will be an automatic
    C                     sorting.
    C
    C            EPSABS - Real
    C                     Absolute accuracy requested
    C            EPSREL - Real
    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 - Real
    C                     Approximation to the integral
    C
    C            ABSERR - Real
    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
    C                             subdivisions by increasing the value of
    C                             LIMIT (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 (i.e. SINGULARITY,
    C                             DISCONTINUITY within the interval), it
    C                             should be supplied to the routine as an
    C                             element of the vector points. If necessary
    C                             an appropriate special-purpose integrator
    C                             must be used, which is designed for
    C                             handling the type of difficulty involved.
    C                         = 2 The occurrence of roundoff error is
    C                             detected, which prevents the requested
    C                             tolerance from being achieved.
    C                             The error may be under-estimated.
    C                         = 3 Extremely bad integrand behaviour occurs
    C                             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.
    C                             It is presumed that the requested
    C                             tolerance cannot be achieved, and that
    C                             the returned RESULT is the best which
    C                             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.GT.0.
    C                         = 6 The input is invalid because
    C                             NPTS2.LT.2 or
    C                             break points are specified outside
    C                             the integration range or
    C                             (EPSABS.LE.0 and
    C                              EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28))
    C                             RESULT, ABSERR, NEVAL, LAST are set to
    C                             zero.  Except when LENIW or LENW or NPTS2
    C                             is invalid, IWORK(1), IWORK(LIMIT+1),
    C                             WORK(LIMIT*2+1) and WORK(LIMIT*3+1)
    C                             are set to zero.
    C                             WORK(1) is set to A and WORK(LIMIT+1)
    C                             to B (where LIMIT = (LENIW-NPTS2)/2).
    C
    C         DIMENSIONING PARAMETERS
    C            LENIW - Integer
    C                    Dimensioning parameter for IWORK
    C                    LENIW determines LIMIT = (LENIW-NPTS2)/2,
    C                    which is the maximum number of subintervals in the
    C                    partition of the given integration interval (A,B),
    C                    LENIW.GE.(3*NPTS2-2).
    C                    If LENIW.LT.(3*NPTS2-2), the routine will end with
    C                    IER = 6.
    C
    C            LENW  - Integer
    C                    Dimensioning parameter for WORK
    C                    LENW must be at least LENIW*2-NPTS2.
    C                    If LENW.LT.LENIW*2-NPTS2, 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, which
    C                    determines the number of significant elements
    C                    actually in the WORK ARRAYS.
    C
    C         WORK ARRAYS
    C            IWORK - Integer
    C                    Vector of dimension at least LENIW. on return,
    C                    the first K elements of which contain
    C                    pointers to the error estimates over the
    C                    subintervals, 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), and
    C                    K = LIMIT+1-LAST otherwise
    C                    IWORK(LIMIT+1), ...,IWORK(LIMIT+LAST) Contain the
    C                     subdivision levels of the subintervals, i.e.
    C                     if (AA,BB) is a subinterval of (P1,P2)
    C                     where P1 as well as P2 is a user-provided
    C                     break point or integration LIMIT, then (AA,BB) has
    C                     level L if ABS(BB-AA) = ABS(P2-P1)*2**(-L),
    C                    IWORK(LIMIT*2+1), ..., IWORK(LIMIT*2+NPTS2) have
    C                     no significance for the user,
    C                    note that LIMIT = (LENIW-NPTS2)/2.
    C
    C            WORK  - Real
    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 corresponding error estimates,
    C                    WORK(LIMIT*4+1), ..., WORK(LIMIT*4+NPTS2)
    C                     contain the integration limits and the
    C                     break points sorted in an ascending sequence.
    C                    note that LIMIT = (LENIW-NPTS2)/2.
    C
    C***REFERENCES  (NONE)
    C***ROUTINES CALLED  QAGPE, 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  QAGP