# qaws.f

          SUBROUTINE QAWS (F, A, B, ALFA, BETA, INTEGR, EPSABS, EPSREL,
         +   RESULT, ABSERR, NEVAL, IER, LIMIT, LENW, LAST, IWORK, WORK)
    C***BEGIN PROLOGUE  QAWS
    C***PURPOSE  The routine calculates an approximation result to a given
    C            definite integral I = Integral of F*W over (A,B),
    C            (where W shows a singular behaviour at the end points
    C            see parameter INTEGR).
    C            Hopefully satisfying following claim for accuracy
    C            ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)).
    C***LIBRARY   SLATEC (QUADPACK)
    C***CATEGORY  H2A2A1
    C***TYPE      SINGLE PRECISION (QAWS-S, DQAWS-D)
    C***KEYWORDS  ALGEBRAIC-LOGARITHMIC END POINT SINGULARITIES,
    C             AUTOMATIC INTEGRATOR, CLENSHAW-CURTIS METHOD,
    C             GLOBALLY ADAPTIVE, QUADPACK, QUADRATURE, SPECIAL-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 of functions having algebraico-logarithmic
    C        end point singularities
    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, B.GT.A
    C                     If B.LE.A, the routine will end with IER = 6.
    C
    C            ALFA   - Real
    C                     Parameter in the integrand function, ALFA.GT.(-1)
    C                     If ALFA.LE.(-1), the routine will end with
    C                     IER = 6.
    C
    C            BETA   - Real
    C                     Parameter in the integrand function, BETA.GT.(-1)
    C                     If BETA.LE.(-1), the routine will end with
    C                     IER = 6.
    C
    C            INTEGR - Integer
    C                     Indicates which WEIGHT function is to be used
    C                     = 1  (X-A)**ALFA*(B-X)**BETA
    C                     = 2  (X-A)**ALFA*(B-X)**BETA*LOG(X-A)
    C                     = 3  (X-A)**ALFA*(B-X)**BETA*LOG(B-X)
    C                     = 4  (X-A)**ALFA*(B-X)**BETA*LOG(X-A)*LOG(B-X)
    C                     If INTEGR.LT.1 or INTEGR.GT.4, the routine
    C                     will end with IER = 6.
    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 the integral and error
    C                             are 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
    C                             which prevent the requested tolerance from
    C                             being achieved. In case of a jump
    C                             discontinuity or a local singularity
    C                             of algebraico-logarithmic type at one or
    C                             more interior points of the integration
    C                             range, one should proceed by splitting up
    C                             the interval at these points and calling
    C                             the integrator on the subranges.
    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                             B.LE.A or ALFA.LE.(-1) or BETA.LE.(-1) or
    C                             or INTEGR.LT.1 or INTEGR.GT.4 or
    C                             (EPSABS.LE.0 and
    C                              EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28))
    C                             or LIMIT.LT.2 or LENW.LT.LIMIT*4.
    C                             RESULT, ABSERR, NEVAL, LAST are set to
    C                             zero. Except when LENW or LIMIT is invalid
    C                             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
    C                     subintervals in the partition of the given
    C                     integration interval (A,B), LIMIT.GE.2.
    C                     If LIMIT.LT.2, 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
    C                     subintervals produced in the subdivision process,
    C                     which determines the significant number of
    C                     elements actually in the WORK ARRAYS.
    C
    C         WORK ARRAYS
    C            IWORK  - Integer
    C                     Vector of dimension LIMIT, the first K
    C                     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),
    C                     and K = LIMIT+1-LAST otherwise
    C
    C            WORK   - Real
    C                     Vector of dimension 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)
    C                      contain the integral approximations over
    C                      the subintervals,
    C                     WORK(LIMIT*3+1), ..., WORK(LIMIT*3+LAST)
    C                      contain the error estimates.
    C
    C***REFERENCES  (NONE)
    C***ROUTINES CALLED  QAWSE, 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  QAWS