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subroutine dqags(f, a, b, epsabs, epsrel, alist, blist,elist,
* rlist, limit, iord, liord, result, abserr, ier)
c
c based on quadpack routine dqags (formerly qags)
c **********************************************************
c
c purpose
c the routine calculates an approximation
c /result/ to a given definite integral i =
c integral of /f/ over (a,b), hopefully
c satisfying following claim for accuracy .
c abs(i-result) .le. max(epsabs,epsrel*abs(i)).
c
c calling sequence
c call dqags (f,a,b,epsabs,epsrel,alist,blist,elist,
c rlist,limit,iord,liord,result,abserr,ier)
c
c parameters
c f - function subprogram defining the integrand
c function f(x). the actual name for f
c needs to be declared e x t e r n a l
c in the driver program
c
c a - lower limit of integration
c
c b - upper limit of integration
c
c epsabs - absolute accuracy requested
c
c epsrel - relative accuracy requested
c
c alist,blist,elist,rlist
c - work arrays (functions described below)
c
c limit - upper bound for number of subintervals
c
c iord - work array
c
c liord - length of iord (at least limit/2 + 2)
c
c result - approximation to the integral
c
c abserr - estimate of the modulus of the absolute error,
c which should equal or exceed abs(i-result)
c
c ier - ier = 0 normal and reliable
c termination of the routine.
c it is assumed that the
c requested accuracy has been
c achieved.
c - ier .ne. 0 abnormal termination of
c the routine. the estimates
c for integral and error are
c less reliable. it is assumed
c that the requested accuracy
c has not been achieved.
c = 1 maximum number of subdivisions allowed
c has been achieved. the user can
c allow more sub divisions by
c increasing the dimensions of the
c work arrays work and iwork.
c however, this may
c yield no improvement, and it
c is rather advised to have a
c close look at the integrand,
c in order to determine the
c integration difficulties. if
c the position of a local
c difficulty can be determined
c (i.e. singularity,
c discontinuity within the
c interval) one will probably
c gain from splitting up the
c interval at this point and
c calling the integrator on the
c sub-ranges. if possible, an
c appropriate special-purpose
c integrator should be used
c which is designed for
c handling the type of
c difficulty involved.
c = 2 the occurrence of roundoff
c error is detected which
c prevents the requested
c tolerance from being
c achieved. the error may be
c under-estimated.
c = 3 extremely bad integrand behaviour
c occurs at some interior points of the
c integration interval.
c = 4 it is presumed that the requested
c tolerance cannot be achieved,
c and that the returned result
c is the best which can be
c obtained.
c = 5 the integral is probably divergent, or
c slowly convergent. it must be noted
c that divergency can occur
c with any other value of ier.
c = -1 an error occurs during the evaluation of f
c **********************************************************
c .. scalar arguments ..
double precision a, abserr, b, epsabs, epsrel, result
integer ier, limit, liord
c .. array arguments ..
double precision alist(limit), blist(limit), elist(limit),
* rlist(limit)
integer iord(liord)
c .. function arguments ..
double precision f
c ..
c .. scalars in common ..
integer jupbnd
integer iero
common/ierajf/iero
c ..
c .. local scalars ..
double precision a1, a2, abseps, area12, area1, area2, area, b1,
* b2,correc, defab1, defab2, defabs, dres, epmach, erlarg,erlast,
* errbnd, errmax, erro12, error1, error2, errsum,ertest, oflow,
* resabs, reseps, small, uflow
integer id, ierro, iroff1, iroff2, iroff3, k, ksgn, ktmin,last1,
* last, maxerr, nres, nrmax, numrl2
logical extrap, noext
c .. local arrays ..
double precision res3la(3), rlist2(52)
c .. function references ..
double precision dlamch
c .. subroutine references ..
c order, epsalg, quarul
c ..
external f
common /dqa001/ jupbnd
c
c the dimension of /rlist2/ is determined by
c data /limexp/ in subroutine epsalg (/rlist2/
c should be of dimension (limexp+2) at least).
c
epmach=dlamch('p')
uflow=dlamch('u')
oflow=dlamch('o')
iero=0
c
c list of major variables
c -----------------------
c
c alist - list of left end-points of all subintervals
c considered up to now
c
c blist - list of right end-points of all subintervals
c considered up to now
c
c rlist(i) - approximation to the integral over
c (alist(i),blist(i))
c
c rlist2 - array of dimension at least limexp+2
c containing the part of the epsilon table
c which is still needed for further
c computations
c
c elist(i) - error estimate applying to rlist(i)
c
c maxerr - pointer to the interval with largest error
c estimate
c
c errmax - elist(maxerr)
c
c erlast - error on the interval currently subdivided
c (before that subdivision has taken place)
c
c area - sum of the integrals over the subintervals
c
c errsum - sum of the errors over the subintervals
c
c errbnd - requested accuracy max(epsabs,epsrel*
c abs(result))
c
c *****1 - variable for the left interval
c
c *****2 - variable for the right interval
c
c last - index for subdivision
c
c nres - number of calls to the extrapolation routine
c
c numrl2 - number of elements currently in
c rlist2. if an appropriate
c approximation to the compounded
c integral has been obtained it is
c put in rlist2(numrl2) after numrl2
c has been increased by one.
c
c small - length of the smallest interval considered
c up to now, multiplied by 1.5
c
c erlarg - sum of the errors over the intervals larger
c than the smallest interval
c considered up to now
c extrap - logical variable denoting that the
c routine is attempting to perform
c extrapolation. i.e. before
c subdividing the smallest interval
c we try to decrease the value of
c erlarg
c noext - logical variable denoting that extrapolation
c is no longer allowed(/true/ value)
c
c first approximation to the integral
c -----------------------------------
c
last1 = 1
ier = 0
ierro = 0
call quarul(f, a, b, result, abserr, defabs, resabs)
if(iero.gt.0) then
ier=6
return
endif
c
c test on accuracy
c
dres = abs(result)
errbnd = max(epsabs,epsrel*dres)
if (abserr.le.1.0d+02*epmach*defabs .and. abserr.gt.errbnd)ier = 2
if (limit.lt.2 .and. abserr.gt.errbnd) ier = 1
if (ier.ne.0 .or. abserr.le.errbnd) go to 320
c
c initialization
c --------------
c
alist(1) = a
blist(1) = b
rlist(1) = result
rlist2(1) = result
errmax = abserr
maxerr = 1
area = result
errsum = abserr
abserr = oflow
nrmax = 1
nres = 0
numrl2 = 2
ktmin = 0
extrap = .false.
noext = .false.
iroff1 = 0
iroff2 = 0
iroff3 = 0
ksgn = -1
if (dres.ge.(0.10d+01-0.50d+02*epmach)*defabs) ksgn = 1
c
c main do-loop
c ------------
c
if (limit.lt.2) go to 220
do 200 last=2,limit
c
c bisect the subinterval with the nrmax-th largest
c error estimate
c
last1 = last
a1 = alist(maxerr)
b1 = 0.50d+00*(alist(maxerr)+blist(maxerr))
a2 = b1
b2 = blist(maxerr)
erlast = errmax
call quarul(f, a1, b1, area1, error1, resabs, defab1)
if(iero.gt.0) then
ier=6
return
endif
call quarul(f, a2, b2, area2, error2, resabs, defab2)
if(iero.gt.0) then
ier=6
return
endif
c
c improve previous approximation of 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 40
if (abs(rlist(maxerr)-area12).gt.0.10d-04*abs(area12) .or.
* erro12.lt.0.990d+00*errmax) go to 20
if (extrap) iroff2 = iroff2 + 1
if (.not.extrap) iroff1 = iroff1 + 1
20 if (last.gt.10 .and. erro12.gt.errmax) iroff3 = iroff3 + 1
40 rlist(maxerr) = area1
rlist(last) = area2
errbnd = max(epsabs,epsrel*abs(area))
if (errsum.le.errbnd) go to 280
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 interval
c bisections exceeds /limit/
c
if (last.eq.limit) ier = 1
c
c set error flag in the case of bad integrand behaviour
c at interior points of integration range
c
if (max(abs(a1),abs(b2)).le.(0.10d+01+0.10d+03*epmach)*
* (abs(a2)+0.10d+04*uflow)) ier = 4
if (ier.ne.0) go to 220
c
c append the newly-created intervals to the list
c
if (error2.gt.error1) go to 60
alist(last) = a2
blist(maxerr) = b1
blist(last) = b2
elist(maxerr) = error1
elist(last) = error2
go to 80
60 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 order to maintain the
c descending ordering in the list of error
c estimates and select the subinterval with
c nrmax-th largest error estimate (to be bisected
c next)
c
80 call order(limit, last, maxerr, errmax, elist, iord,liord,
* nrmax)
if (last.eq.2) go to 180
if (noext) go to 200
erlarg = erlarg - erlast
if (abs(b1-a1).gt.small) erlarg = erlarg + erro12
if (extrap) go to 100
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 200
extrap = .true.
nrmax = 2
100 if (ierro.eq.3 .or. erlarg.le.ertest) go to 140
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
do 120 k=id,jupbnd
maxerr = iord(nrmax)
errmax = elist(maxerr)
if (abs(blist(maxerr)-alist(maxerr)).gt.small) go to 200
nrmax = nrmax + 1
120 continue
c
c perform extrapolation
c
140 numrl2 = numrl2 + 1
rlist2(numrl2) = area
call epsalg(numrl2, rlist2, reseps, abseps, res3la, nres)
ktmin = ktmin + 1
if (ktmin.gt.5 .and. abserr.lt.0.10d-02*errsum) ier = 5
if (abseps.ge.abserr) go to 160
ktmin = 0
abserr = abseps
result = reseps
correc = erlarg
ertest = max(epsabs,epsrel*abs(reseps))
if (abserr.le.ertest) go to 220
c
c prepare bisection of the smallest interval
c
160 if (numrl2.eq.1) noext = .true.
if (ier.eq.5) go to 220
maxerr = iord(1)
errmax = elist(maxerr)
nrmax = 1
extrap = .false.
small = small*0.50d+00
erlarg = errsum
go to 200
180 small = abs(b-a)*0.3750d+00
erlarg = errsum
ertest = errbnd
rlist2(2) = area
200 continue
c
c set final result and error estimate
c ------------------------------------
c
220 if (abserr.eq.oflow) go to 280
if (ier+ierro.eq.0) go to 260
if (ierro.eq.3) abserr = abserr + correc
if (ier.eq.0) ier = 3
if (result.ne.0.0d+00.and .area. ne.0.0d+00) go to 240
if (abserr.gt.errsum) go to 280
if (area.eq.0.0d+00) go to 320
go to 260
240 if (abserr/abs(result).gt.errsum/abs(area)) go to 280
c
c test on divergency
c
260 if (ksgn.eq.-1 .and. max(abs(result),abs(area)).le.defabs*
*0.10d-01) go to 320
if (0.10d-01.gt.(result/area) .or. (result/area).gt.0.10d+03.or.
* errsum.gt.abs(area)) ier = 6
go to 320
c
c compute global integral sum
c
280 result = 0.0d+00
do 300 k=1,last
result = result + rlist(k)
300 continue
abserr = errsum
320 if (ier.gt.2) ier = ier - 1
iord(1) = 4*last1
return
end
|
