# SLATEC Routines --- QAGIE ---

```*DECK QAGIE
SUBROUTINE QAGIE (F, BOUND, INF, EPSABS, EPSREL, LIMIT, RESULT,
+   ABSERR, NEVAL, IER, ALIST, BLIST, RLIST, ELIST, IORD, LAST)
C***BEGIN PROLOGUE  QAGIE
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***CATEGORY  H2A3A1, H2A4A1
C***TYPE      SINGLE PRECISION (QAGIE-S, DQAGIE-D)
C***KEYWORDS  AUTOMATIC INTEGRATOR, EXTRAPOLATION, GENERAL-PURPOSE,
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            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    - Real
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            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 - 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.
C                             If the position of a local difficulty can
C                             be 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                             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 0
C                             and 1 respectively.
C
C            ALIST  - Real
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 transformed integration range (0,1).
C
C            BLIST  - Real
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 transformed integration range (0,1).
C
C            RLIST  - Real
C                     Vector of dimension at least LIMIT, the first
C                      LAST  elements of which are the integral
C                     approximations on the subintervals
C
C            ELIST  - Real
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 LIMIT, the first K
C                     elements of which are pointers to the
C                     error estimates over the subintervals,
C                     such that ELIST(IORD(1)), ..., ELIST(IORD(K))
C                     form a decreasing sequence, with K = LAST
C                     If LAST.LE.(LIMIT/2+2), and K = LIMIT+1-LAST
C                     otherwise
C
C            LAST   - Integer
C                     Number of subintervals actually produced
C                     in the subdivision process
C
C***REFERENCES  (NONE)
C***ROUTINES CALLED  QELG, QK15I, QPSRT, R1MACH
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  QAGIE
```