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subroutine dqagpe(f,a,b,npts2,points,epsabs,epsrel,limit,result,
* abserr,neval,ier,alist,blist,rlist,elist,pts,iord,level,ndin,
* last)
c***begin prologue dqagpe
c***date written 800101 (yymmdd)
c***revision date 830518 (yymmdd)
c***category no. h2a2a1
c***keywords automatic integrator, general-purpose,
c singularities at user specified points,
c extrapolation, 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 definite integral i = integral of f over (a,b), hopefully
c satisfying following claim for accuracy abs(i-result).le.
c max(epsabs,epsrel*abs(i)). break points of the integration
c interval, where local difficulties of the integrand may
c occur(e.g. singularities,discontinuities),provided by user.
c***description
c
c computation of a definite integral
c standard fortran subroutine
c double precision version
c
c parameters
c on entry
c f - double precision
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 - double precision
c lower limit of integration
c
c b - double precision
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 - double precision
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 - double precision
c absolute accuracy requested
c epsrel - double precision
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 - double precision
c approximation to the integral
c
c abserr - double precision
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 - double precision
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 - double precision
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 - double precision
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 - double precision
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 - double precision
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 d1mach,dqelg,dqk21,dqpsrt
c***end prologue dqagpe
double precision a,abseps,abserr,alist,area,area1,area12,area2,a1,
* a2,b,blist,b1,b2,correc,dabs,defabs,defab1,defab2,dmax1,dmin1,
* dres,d1mach,elist,epmach,epsabs,epsrel,erlarg,erlast,errbnd,
* errmax,error1,erro12,error2,errsum,ertest,f,oflow,points,pts,
* resa,resabs,reseps,result,res3la,rlist,rlist2,sign,temp,uflow
integer i,id,ier,ierro,ind1,ind2,iord,ip1,iroff1,iroff2,iroff3,j,
* jlow,jupbnd,k,ksgn,ktmin,last,levcur,level,levmax,limit,maxerr,
* ndin,neval,nint,nintp1,npts,npts2,nres,nrmax,numrl2
logical extrap,noext
c
c
dimension alist(limit),blist(limit),elist(limit),iord(limit),
* level(limit),ndin(npts2),points(npts2),pts(npts2),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 epsalg (rlist2 should be of dimension
c (limexp+2) at least).
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 which
c 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 in rlist2. if an appropriate
c approximation to the compounded integral has
c been obtained, it is put in rlist2(numrl2) after
c numrl2 has been increased by one.
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 is
c 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
c***first executable statement dqagpe
epmach = d1mach(4)
c
c test on validity of parameters
c -----------------------------
c
ier = 0
neval = 0
last = 0
result = 0.0d+00
abserr = 0.0d+00
alist(1) = a
blist(1) = b
rlist(1) = 0.0d+00
elist(1) = 0.0d+00
iord(1) = 0
level(1) = 0
npts = npts2-2
if(npts2.lt.2.or.limit.le.npts.or.(epsabs.le.0.0d+00.and.
* epsrel.lt.dmax1(0.5d+02*epmach,0.5d-28))) ier = 6
if(ier.eq.6) go to 999
c
c if any break points are provided, sort them into an
c ascending sequence.
c
sign = 1.0d+00
if(a.gt.b) sign = -1.0d+00
pts(1) = dmin1(a,b)
if(npts.eq.0) go to 15
do 10 i = 1,npts
pts(i+1) = points(i)
10 continue
15 pts(npts+2) = dmax1(a,b)
nint = npts+1
a1 = pts(1)
if(npts.eq.0) go to 40
nintp1 = nint+1
do 20 i = 1,nint
ip1 = i+1
do 20 j = ip1,nintp1
if(pts(i).le.pts(j)) go to 20
temp = pts(i)
pts(i) = pts(j)
pts(j) = temp
20 continue
if(pts(1).ne.dmin1(a,b).or.pts(nintp1).ne.dmax1(a,b)) ier = 6
if(ier.eq.6) go to 999
c
c compute first integral and error approximations.
c ------------------------------------------------
c
40 resabs = 0.0d+00
do 50 i = 1,nint
b1 = pts(i+1)
call dqk21(f,a1,b1,area1,error1,defabs,resa)
abserr = abserr+error1
result = result+area1
ndin(i) = 0
if(error1.eq.resa.and.error1.ne.0.0d+00) ndin(i) = 1
resabs = resabs+defabs
level(i) = 0
elist(i) = error1
alist(i) = a1
blist(i) = b1
rlist(i) = area1
iord(i) = i
a1 = b1
50 continue
errsum = 0.0d+00
do 55 i = 1,nint
if(ndin(i).eq.1) elist(i) = abserr
errsum = errsum+elist(i)
55 continue
c
c test on accuracy.
c
last = nint
neval = 21*nint
dres = dabs(result)
errbnd = dmax1(epsabs,epsrel*dres)
if(abserr.le.0.1d+03*epmach*resabs.and.abserr.gt.errbnd) ier = 2
if(nint.eq.1) go to 80
do 70 i = 1,npts
jlow = i+1
ind1 = iord(i)
do 60 j = jlow,nint
ind2 = iord(j)
if(elist(ind1).gt.elist(ind2)) go to 60
ind1 = ind2
k = j
60 continue
if(ind1.eq.iord(i)) go to 70
iord(k) = iord(i)
iord(i) = ind1
70 continue
if(limit.lt.npts2) ier = 1
80 if(ier.ne.0.or.abserr.le.errbnd) go to 210
c
c initialization
c --------------
c
rlist2(1) = result
maxerr = iord(1)
errmax = elist(maxerr)
area = result
nrmax = 1
nres = 0
numrl2 = 1
ktmin = 0
extrap = .false.
noext = .false.
erlarg = errsum
ertest = errbnd
levmax = 1
iroff1 = 0
iroff2 = 0
iroff3 = 0
ierro = 0
uflow = d1mach(1)
oflow = d1mach(2)
abserr = oflow
ksgn = -1
if(dres.ge.(0.1d+01-0.5d+02*epmach)*resabs) ksgn = 1
c
c main do-loop
c ------------
c
do 160 last = npts2,limit
c
c bisect the subinterval with the nrmax-th largest error
c estimate.
c
levcur = level(maxerr)+1
a1 = alist(maxerr)
b1 = 0.5d+00*(alist(maxerr)+blist(maxerr))
a2 = b1
b2 = blist(maxerr)
erlast = errmax
call dqk21(f,a1,b1,area1,error1,resa,defab1)
call dqk21(f,a2,b2,area2,error2,resa,defab2)
c
c improve previous approximations to integral
c and error and test for accuracy.
c
neval = neval+42
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 95
if(dabs(rlist(maxerr)-area12).gt.0.1d-04*dabs(area12)
* .or.erro12.lt.0.99d+00*errmax) go to 90
if(extrap) iroff2 = iroff2+1
if(.not.extrap) iroff1 = iroff1+1
90 if(last.gt.10.and.erro12.gt.errmax) iroff3 = iroff3+1
95 level(maxerr) = levcur
level(last) = levcur
rlist(maxerr) = area1
rlist(last) = area2
errbnd = dmax1(epsabs,epsrel*dabs(area))
c
c test for roundoff error and eventually 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 a point of the integration range
c
if(dmax1(dabs(a1),dabs(b2)).le.(0.1d+01+0.1d+03*epmach)*
* (dabs(a2)+0.1d+04*uflow)) ier = 4
c
c append the newly-created intervals to the list.
c
if(error2.gt.error1) go to 100
alist(last) = a2
blist(maxerr) = b1
blist(last) = b2
elist(maxerr) = error1
elist(last) = error2
go to 110
100 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 dqpsrt to maintain the descending ordering
c in the list of error estimates and select the subinterval
c with nrmax-th largest error estimate (to be bisected next).
c
110 call dqpsrt(limit,last,maxerr,errmax,elist,iord,nrmax)
c ***jump out of do-loop
if(errsum.le.errbnd) go to 190
c ***jump out of do-loop
if(ier.ne.0) go to 170
if(noext) go to 160
erlarg = erlarg-erlast
if(levcur+1.le.levmax) erlarg = erlarg+erro12
if(extrap) go to 120
c
c test whether the interval to be bisected next is the
c smallest interval.
c
if(level(maxerr)+1.le.levmax) go to 160
extrap = .true.
nrmax = 2
120 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 over
c the larger intervals (erlarg) and perform extrapolation.
c
id = nrmax
jupbnd = last
if(last.gt.(2+limit/2)) jupbnd = limit+3-last
do 130 k = id,jupbnd
maxerr = iord(nrmax)
errmax = elist(maxerr)
c ***jump out of do-loop
if(level(maxerr)+1.le.levmax) go to 160
nrmax = nrmax+1
130 continue
c
c perform extrapolation.
c
140 numrl2 = numrl2+1
rlist2(numrl2) = area
if(numrl2.le.2) go to 155
call dqelg(numrl2,rlist2,reseps,abseps,res3la,nres)
ktmin = ktmin+1
if(ktmin.gt.5.and.abserr.lt.0.1d-02*errsum) ier = 5
if(abseps.ge.abserr) go to 150
ktmin = 0
abserr = abseps
result = reseps
correc = erlarg
ertest = dmax1(epsabs,epsrel*dabs(reseps))
c ***jump out of do-loop
if(abserr.lt.ertest) go to 170
c
c prepare bisection of the smallest interval.
c
150 if(numrl2.eq.1) noext = .true.
if(ier.ge.5) go to 170
155 maxerr = iord(1)
errmax = elist(maxerr)
nrmax = 1
extrap = .false.
levmax = levmax+1
erlarg = errsum
160 continue
c
c set the final result.
c ---------------------
c
c
170 if(abserr.eq.oflow) go to 190
if((ier+ierro).eq.0) go to 180
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 175
if(abserr.gt.errsum)go to 190
if(area.eq.0.0d+00) go to 210
go to 180
175 if(abserr/dabs(result).gt.errsum/dabs(area))go to 190
c
c test on divergence.
c
180 if(ksgn.eq.(-1).and.dmax1(dabs(result),dabs(area)).le.
* resabs*0.1d-01) go to 210
if(0.1d-01.gt.(result/area).or.(result/area).gt.0.1d+03.or.
* errsum.gt.dabs(area)) ier = 6
go to 210
c
c compute global integral sum.
c
190 result = 0.0d+00
do 200 k = 1,last
result = result+rlist(k)
200 continue
abserr = errsum
210 if(ier.gt.2) ier = ier-1
result = result*sign
999 return
end
|
