# qagi.f

          SUBROUTINE QAGI (F, BOUND, INF, EPSABS, EPSREL, RESULT, ABSERR,
         +   NEVAL, IER, LIMIT, LENW, LAST, IWORK, WORK)
    C***BEGIN PROLOGUE  QAGI
    C***PURPOSE  The routine calculates an approximation result to a given
    C            INTEGRAL   I = Integral of F over (BOUND,+INFINITY)
    C                    OR I = Integral of F over (-INFINITY,BOUND)
    C                    OR I = Integral of F over (-INFINITY,+INFINITY)
    C            Hopefully satisfying following claim for accuracy
    C            ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)).
    C***LIBRARY   SLATEC (QUADPACK)
    C***CATEGORY  H2A3A1, H2A4A1
    C***TYPE      SINGLE PRECISION (QAGI-S, DQAGI-D)
    C***KEYWORDS  AUTOMATIC INTEGRATOR, EXTRAPOLATION, GENERAL-PURPOSE,
    C             GLOBALLY ADAPTIVE, INFINITE INTERVALS, QUADPACK,
    C             QUADRATURE, TRANSFORMATION
    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 over infinite intervals
    C        Standard fortran subroutine
    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            BOUND  - Real
    C                     Finite bound of integration range
    C                     (has no meaning if interval is doubly-infinite)
    C
    C            INF    - Integer
    C                     indicating the kind of integration range involved
    C                     INF = 1 corresponds to  (BOUND,+INFINITY),
    C                     INF = -1            to  (-INFINITY,BOUND),
    C                     INF = 2             to (-INFINITY,+INFINITY).
    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
    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. The
    C                             estimates for result and error are less
    C                             reliable. It is assumed that the requested
    C                             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 (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
    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 assumed that the requested tolerance
    C                             cannot be achieved, and that the returned
    C                             RESULT is 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 LENIW.LT.LIMIT*4.
    C                             RESULT, ABSERR, NEVAL, LAST are set to
    C                             zero.  Except when LIMIT or LENIW 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, 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 LIMIT, the first
    C                    K 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), and
    C                    K = LIMIT+1-LAST otherwise
    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 the
    C                     integral approximations over the subintervals,
    C                    WORK(LIMIT*3+1), ..., WORK(LIMIT*3)
    C                     contain the error estimates.
    C
    C***REFERENCES  (NONE)
    C***ROUTINES CALLED  QAGIE, 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  QAGI