qagie.f

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      subroutine qagie(f,bound,inf,epsabs,epsrel,limit,result,abserr,
     *   neval,ier,alist,blist,rlist,elist,iord,last)
c***begin prologue  qagie
c***date written   800101   (yymmdd)
c***revision date  830518   (yymmdd)
c***category no.  h2a3a1,h2a4a1
c***keywords  automatic integrator, infinite intervals,
c             general-purpose, transformation, extrapolation,
c             globally adaptive
c***author  piessens,robert,appl. math & progr. div - k.u.leuven
c           de doncker,elise,appl. math & progr. div - k.u.leuven
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***description
c
c integration over infinite intervals
c standard fortran subroutine
c
c            f      - subroutine f(x,ierr,result) 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***end prologue  qagie
c
      real abseps,abserr,alist,area,area1,area12,area2,a1,
     *  a2,blist,boun,bound,b1,b2,correc,defabs,defab1,defab2,
     *  dres,r1mach,elist,epmach,epsabs,epsrel,erlarg,erlast,
     *  errbnd,errmax,error1,error2,erro12,errsum,ertest,oflow,resabs,
     *  reseps,result,res3la,rlist,rlist2,small,uflow
      integer id,ier,ierro,inf,iord,iroff1,iroff2,iroff3,jupbnd,k,ksgn,
     *  ktmin,last,limit,maxerr,neval,nres,nrmax,numrl2
      logical extrap,noext
c
      dimension alist(limit),blist(limit),elist(limit),iord(limit),
     *  res3la(3),rlist(limit),rlist2(52)
c
      external f
c
c            the dimension of rlist2 is determined by the value of
c            limexp in subroutine qelg.
c
c
c            list of major variables
c            -----------------------
c
c           alist     - list of left end points of all subintervals
c                       considered up to now
c           blist     - list of right end points of all subintervals
c                       considered up to now
c           rlist(i)  - approximation to the integral over
c                       (alist(i),blist(i))
c           rlist2    - array of dimension at least (limexp+2),
c                       containing the part of the epsilon table
c                       wich is still needed for further computations
c           elist(i)  - error estimate applying to rlist(i)
c           maxerr    - pointer to the interval with largest error
c                       estimate
c           errmax    - elist(maxerr)
c           erlast    - error on the interval currently subdivided
c                       (before that subdivision has taken place)
c           area      - sum of the integrals over the subintervals
c           errsum    - sum of the errors over the subintervals
c           errbnd    - requested accuracy max(epsabs,epsrel*
c                       abs(result))
c           *****1    - variable for the left subinterval
c           *****2    - variable for the right subinterval
c           last      - index for subdivision
c           nres      - number of calls to the extrapolation routine
c           numrl2    - number of elements currently in rlist2. if an
c                       appropriate approximation to the compounded
c                       integral has been obtained, it is put in
c                       rlist2(numrl2) after numrl2 has been increased
c                       by one.
c           small     - length of the smallest interval considered up
c                       to now, multiplied by 1.5
c           erlarg    - sum of the errors over the intervals larger
c                       than the smallest interval considered up to now
c           extrap    - logical variable denoting that the routine
c                       is attempting to perform extrapolation. i.e.
c                       before subdividing the smallest interval we
c                       try to decrease the value of erlarg.
c           noext     - logical variable denoting that extrapolation
c                       is no longer allowed (true-value)
c
c            machine dependent constants
c            ---------------------------
c
c           epmach is the largest relative spacing.
c           uflow is the smallest positive magnitude.
c           oflow is the largest positive magnitude.
c
       epmach = r1mach(4)
c
c           test on validity of parameters
c           -----------------------------
c
c***first executable statement  qagie
      ier = 0
      neval = 0
      last = 0
      result = 0.0e+00
      abserr = 0.0e+00
      alist(1) = 0.0e+00
      blist(1) = 0.1e+01
      rlist(1) = 0.0e+00
      elist(1) = 0.0e+00
      iord(1) = 0
      if(epsabs.le.0.0e+00.and.epsrel.lt.amax1(0.5e+02*epmach,0.5e-14))
     *  ier = 6
      if(ier.eq.6) go to 999
c
c
c           first approximation to the integral
c           -----------------------------------
c
c           determine the interval to be mapped onto (0,1).
c           if inf = 2 the integral is computed as i = i1+i2, where
c           i1 = integral of f over (-infinity,0),
c           i2 = integral of f over (0,+infinity).
c
      boun = bound
      if(inf.eq.2) boun = 0.0e+00
      call qk15i(f,boun,inf,0.0e+00,0.1e+01,result,abserr,
     *  defabs,resabs,ier)
      if (ier.lt.0) return
c
c           test on accuracy
c
      last = 1
      rlist(1) = result
      elist(1) = abserr
      iord(1) = 1
      dres = abs(result)
      errbnd = amax1(epsabs,epsrel*dres)
      if(abserr.le.1.0e+02*epmach*defabs.and.abserr.gt.
     *  errbnd) ier = 2
      if(limit.eq.1) ier = 1
      if(ier.ne.0.or.(abserr.le.errbnd.and.abserr.ne.resabs).or.
     *  abserr.eq.0.0e+00) go to 130
c
c           initialization
c           --------------
c
      uflow = r1mach(1)
      oflow = r1mach(2)
      rlist2(1) = result
      errmax = abserr
      maxerr = 1
      area = result
      errsum = abserr
      abserr = oflow
      nrmax = 1
      nres = 0
      ktmin = 0
      numrl2 = 2
      extrap = .false.
      noext = .false.
      ierro = 0
      iroff1 = 0
      iroff2 = 0
      iroff3 = 0
      ksgn = -1
      if(dres.ge.(0.1e+01-0.5e+02*epmach)*defabs) ksgn = 1
c
c           main do-loop
c           ------------
c
      do 90 last = 2,limit
c
c           bisect the subinterval with nrmax-th largest
c           error estimate.
c
        a1 = alist(maxerr)
        b1 = 0.5e+00*(alist(maxerr)+blist(maxerr))
        a2 = b1
        b2 = blist(maxerr)
        erlast = errmax
        call qk15i(f,boun,inf,a1,b1,area1,error1,resabs,defab1,ier)
        if (ier.lt.0) return
        call qk15i(f,boun,inf,a2,b2,area2,error2,resabs,defab2,ier)
        if (ier.lt.0) return
c
c           improve previous approximations to integral
c           and error and test for accuracy.
c
        area12 = area1+area2
        erro12 = error1+error2
        errsum = errsum+erro12-errmax
        area = area+area12-rlist(maxerr)
        if(defab1.eq.error1.or.defab2.eq.error2)go to 15
        if(abs(rlist(maxerr)-area12).gt.0.1e-04*abs(area12)
     *  .or.erro12.lt.0.99e+00*errmax) go to 10
        if(extrap) iroff2 = iroff2+1
        if(.not.extrap) iroff1 = iroff1+1
   10   if(last.gt.10.and.erro12.gt.errmax) iroff3 = iroff3+1
   15   rlist(maxerr) = area1
        rlist(last) = area2
        errbnd = amax1(epsabs,epsrel*abs(area))
c
c           test for roundoff error and eventually
c           set error flag.
c
        if(iroff1+iroff2.ge.10.or.iroff3.ge.20) ier = 2
        if(iroff2.ge.5) ierro = 3
c
c           set error flag in the case that the number of
c           subintervals equals limit.
c
        if(last.eq.limit) ier = 1
c
c           set error flag in the case of bad integrand behaviour
c           at some points of the integration range.
c
        if(amax1(abs(a1),abs(b2)).le.(0.1e+01+0.1e+03*epmach)*
     *  (abs(a2)+0.1e+04*uflow)) ier = 4
c
c           append the newly-created intervals to the list.
c
        if(error2.gt.error1) go to 20
        alist(last) = a2
        blist(maxerr) = b1
        blist(last) = b2
        elist(maxerr) = error1
        elist(last) = error2
        go to 30
   20   alist(maxerr) = a2
        alist(last) = a1
        blist(last) = b1
        rlist(maxerr) = area2
        rlist(last) = area1
        elist(maxerr) = error2
        elist(last) = error1
c
c           call subroutine qpsrt to maintain the descending ordering
c           in the list of error estimates and select the
c           subinterval with nrmax-th largest error estimate (to be
c           bisected next).
c
   30   call qpsrt(limit,last,maxerr,errmax,elist,iord,nrmax)
        if(errsum.le.errbnd) go to 115
        if(ier.ne.0) go to 100
        if(last.eq.2) go to 80
        if(noext) go to 90
        erlarg = erlarg-erlast
        if(abs(b1-a1).gt.small) erlarg = erlarg+erro12
        if(extrap) go to 40
c
c           test whether the interval to be bisected next is the
c           smallest interval.
c
        if(abs(blist(maxerr)-alist(maxerr)).gt.small) go to 90
        extrap = .true.
        nrmax = 2
   40   if(ierro.eq.3.or.erlarg.le.ertest) go to 60
c
c           the smallest interval has the largest error.
c           before bisecting decrease the sum of the errors
c           over the larger intervals (erlarg) and perform
c           extrapolation.
c
        id = nrmax
        jupbnd = last
        if(last.gt.(2+limit/2)) jupbnd = limit+3-last
        do 50 k = id,jupbnd
          maxerr = iord(nrmax)
          errmax = elist(maxerr)
          if(abs(blist(maxerr)-alist(maxerr)).gt.small) go to 90
          nrmax = nrmax+1
   50   continue
c
c           perform extrapolation.
c
   60   numrl2 = numrl2+1
        rlist2(numrl2) = area
        call qelg(numrl2,rlist2,reseps,abseps,res3la,nres)
        ktmin = ktmin+1
        if(ktmin.gt.5.and.abserr.lt.0.1e-02*errsum) ier = 5
        if(abseps.ge.abserr) go to 70
        ktmin = 0
        abserr = abseps
        result = reseps
        correc = erlarg
        ertest = amax1(epsabs,epsrel*abs(reseps))
        if(abserr.le.ertest) go to 100
c
c            prepare bisection of the smallest interval.
c
   70   if(numrl2.eq.1) noext = .true.
        if(ier.eq.5) go to 100
        maxerr = iord(1)
        errmax = elist(maxerr)
        nrmax = 1
        extrap = .false.
        small = small*0.5e+00
        erlarg = errsum
        go to 90
   80   small = 0.375e+00
        erlarg = errsum
        ertest = errbnd
        rlist2(2) = area
   90 continue
c
c           set final result and error estimate.
c           ------------------------------------
c
  100 if(abserr.eq.oflow) go to 115
      if((ier+ierro).eq.0) go to 110
      if(ierro.eq.3) abserr = abserr+correc
      if(ier.eq.0) ier = 3
      if(result.ne.0.0e+00.and.area.ne.0.0e+00)go to 105
      if(abserr.gt.errsum)go to 115
      if(area.eq.0.0e+00) go to 130
      go to 110
  105 if(abserr/abs(result).gt.errsum/abs(area))go to 115
c
c           test on divergence
c
  110 if(ksgn.eq.(-1).and.amax1(abs(result),abs(area)).le.
     * defabs*0.1e-01) go to 130
      if(0.1e-01.gt.(result/area).or.(result/area).gt.0.1e+03.
     *or.errsum.gt.abs(area)) ier = 6
      go to 130
c
c           compute global integral sum.
c
  115 result = 0.0e+00
      do 120 k = 1,last
        result = result+rlist(k)
  120 continue
      abserr = errsum
  130 neval = 30*last-15
      if(inf.eq.2) neval = 2*neval
      if(ier.gt.2) ier=ier-1
  999 return
      end