SLATEC Routines --- QAGPE ---

```*DECK QAGPE
SUBROUTINE QAGPE (F, A, B, NPTS2, POINTS, EPSABS, EPSREL, LIMIT,
+   RESULT, ABSERR, NEVAL, IER, ALIST, BLIST, RLIST, ELIST, PTS,
+   IORD, LEVEL, NDIN, LAST)
C***BEGIN PROLOGUE  QAGPE
C***PURPOSE  Approximate a given definite integral I = Integral of F
C            over (A,B), hopefully satisfying the accuracy claim:
C                  ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)).
C            Break points of the integration interval, where local
C            difficulties of the integrand may occur (e.g. singularities
C            or discontinuities) are provided by the user.
C***CATEGORY  H2A2A1
C***TYPE      SINGLE PRECISION (QAGPE-S, DQAGPE-D)
C***KEYWORDS  AUTOMATIC INTEGRATOR, EXTRAPOLATION, GENERAL-PURPOSE,
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, NPTS2.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            LIMIT  - Integer
C                     Gives an upper bound on the number of subintervals
C                     in the partition of (A,B), LIMIT.GE.NPTS2
C                     If LIMIT.LT.NPTS2, the routine will end with
C                     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. 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.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                             or LIMIT.LT.NPTS2.
C                             RESULT, ABSERR, NEVAL, LAST, RLIST(1),
C                             and ELIST(1) are set to zero. ALIST(1) and
C                             BLIST(1) are set to A and B respectively.
C
C            ALIST  - Real
C                     Vector of dimension at least LIMIT, the first
C                      LAST  elements of which are the left end points
C                     of the subintervals in the partition of the given
C                     integration range (A,B)
C
C            BLIST  - Real
C                     Vector of dimension at least LIMIT, the first
C                      LAST  elements of which are the right end points
C                     of the subintervals in the partition of the given
C                     integration range (A,B)
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            PTS    - Real
C                     Vector of dimension at least NPTS2, containing the
C                     integration limits and the break points of the
C                     interval in ascending sequence.
C
C            LEVEL  - Integer
C                     Vector of dimension at least LIMIT, containing the
C                     subdivision levels of the subinterval, i.e. if
C                     (AA,BB) is a subinterval of (P1,P2) where P1 as
C                     well as P2 is a user-provided break point or
C                     integration limit, then (AA,BB) has level L if
C                     ABS(BB-AA) = ABS(P2-P1)*2**(-L).
C
C            NDIN   - Integer
C                     Vector of dimension at least NPTS2, after first
C                     integration over the intervals (PTS(I)),PTS(I+1),
C                     I = 0,1, ..., NPTS2-2, the error estimates over
C                     some of the intervals may have been increased
C                     artificially, in order to put their subdivision
C                     forward. If this happens for the subinterval
C                     numbered K, NDIN(K) is put to 1, otherwise
C                     NDIN(K) = 0.
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)), ..., 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 in the
C                     subdivisions process
C
C***REFERENCES  (NONE)
C***ROUTINES CALLED  QELG, QK21, 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  QAGPE
```