dqage_2d_inner.f
14.4 KB
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! fvn comment :
! Modified version of the dqage quadpack routine from http://www.netlib.org/quadpack
!
! + The external 'f' function is a 2 parameters function f(x,y). The routine
! takes one more parameter 'x' and evaluate the integral of f against y between a and b
! for a given x
subroutine dqage_2d_inner(f,x,a,b,epsabs,epsrel,key, &
limit,result,abserr,neval,ier,alist,blist,rlist, &
elist,iord,last)
!***begin prologue dqage
!***date written 800101 (yymmdd)
!***revision date 830518 (yymmdd)
!***category no. h2a1a1
!***keywords automatic integrator, general-purpose,
! integrand examinator, globally adaptive,
! gauss-kronrod
!***author piessens,robert,appl. math. & progr. div. - k.u.leuven
! de doncker,elise,appl. math. & progr. div. - k.u.leuven
!***purpose the routine calculates an approximation result to a given
! definite integral i = integral of f over (a,b),
! hopefully satisfying following claim for accuracy
! abs(i-reslt).le.max(epsabs,epsrel*abs(i)).
!***description
!
! computation of a definite integral
! standard fortran subroutine
! double precision version
!
! parameters
! on entry
! f - double precision
! function subprogram defining the integrand
! function f(x). the actual name for f needs to be
! declared e x t e r n a l in the driver program.
!
! a - double precision
! lower limit of integration
!
! b - double precision
! upper limit of integration
!
! epsabs - double precision
! absolute accuracy requested
! epsrel - double precision
! relative accuracy requested
! if epsabs.le.0
! and epsrel.lt.max(50*rel.mach.acc.,0.5d-28),
! the routine will end with ier = 6.
!
! key - integer
! key for choice of local integration rule
! a gauss-kronrod pair is used with
! 7 - 15 points if key.lt.2,
! 10 - 21 points if key = 2,
! 15 - 31 points if key = 3,
! 20 - 41 points if key = 4,
! 25 - 51 points if key = 5,
! 30 - 61 points if key.gt.5.
!
! limit - integer
! gives an upperbound on the number of subintervals
! in the partition of (a,b), limit.ge.1.
!
! on return
! result - double precision
! approximation to the integral
!
! abserr - double precision
! estimate of the modulus of the absolute error,
! which should equal or exceed abs(i-result)
!
! neval - integer
! number of integrand evaluations
!
! ier - integer
! ier = 0 normal and reliable termination of the
! routine. it is assumed that the requested
! accuracy has been achieved.
! ier.gt.0 abnormal termination of the routine
! the estimates for result and error are
! less reliable. it is assumed that the
! requested accuracy has not been achieved.
! error messages
! ier = 1 maximum number of subdivisions allowed
! has been achieved. one can allow more
! subdivisions by increasing the value
! of limit.
! however, if this yields no improvement it
! is rather advised to analyze the integrand
! in order to determine the integration
! difficulties. if the position of a local
! difficulty can be determined(e.g.
! singularity, discontinuity within the
! interval) one will probably gain from
! splitting up the interval at this point
! and calling the integrator on the
! subranges. if possible, an appropriate
! special-purpose integrator should be used
! which is designed for handling the type of
! difficulty involved.
! = 2 the occurrence of roundoff error is
! detected, which prevents the requested
! tolerance from being achieved.
! = 3 extremely bad integrand behaviour occurs
! at some points of the integration
! interval.
! = 6 the input is invalid, because
! (epsabs.le.0 and
! epsrel.lt.max(50*rel.mach.acc.,0.5d-28),
! result, abserr, neval, last, rlist(1) ,
! elist(1) and iord(1) are set to zero.
! alist(1) and blist(1) are set to a and b
! respectively.
!
! alist - double precision
! vector of dimension at least limit, the first
! last elements of which are the left
! end points of the subintervals in the partition
! of the given integration range (a,b)
!
! blist - double precision
! vector of dimension at least limit, the first
! last elements of which are the right
! end points of the subintervals in the partition
! of the given integration range (a,b)
!
! rlist - double precision
! vector of dimension at least limit, the first
! last elements of which are the
! integral approximations on the subintervals
!
! elist - double precision
! vector of dimension at least limit, the first
! last elements of which are the moduli of the
! absolute error estimates on the subintervals
!
! iord - integer
! vector of dimension at least limit, the first k
! elements of which are pointers to the
! error estimates over the subintervals,
! such that elist(iord(1)), ...,
! elist(iord(k)) form a decreasing sequence,
! with k = last if last.le.(limit/2+2), and
! k = limit+1-last otherwise
!
! last - integer
! number of subintervals actually produced in the
! subdivision process
!
!***references (none)
!***routines called d1mach,dqk15,dqk21,dqk31,
! dqk41,dqk51,dqk61,dqpsrt
!***end prologue dqage
!
double precision a,abserr,alist,area,area1,area12,area2,a1,a2,b, &
blist,b1,b2,dabs,defabs,defab1,defab2,dmax1,elist,epmach, &
epsabs,epsrel,errbnd,errmax,error1,error2,erro12,errsum,f, &
resabs,result,rlist,uflow,x
integer ier,iord,iroff1,iroff2,k,key,keyf,last,limit,maxerr,neval, &
nrmax
!
dimension alist(limit),blist(limit),elist(limit),iord(limit), &
rlist(limit)
!
external f
!
! list of major variables
! -----------------------
!
! alist - list of left end points of all subintervals
! considered up to now
! blist - list of right end points of all subintervals
! considered up to now
! rlist(i) - approximation to the integral over
! (alist(i),blist(i))
! elist(i) - error estimate applying to rlist(i)
! maxerr - pointer to the interval with largest
! error estimate
! errmax - elist(maxerr)
! area - sum of the integrals over the subintervals
! errsum - sum of the errors over the subintervals
! errbnd - requested accuracy max(epsabs,epsrel*
! abs(result))
! *****1 - variable for the left subinterval
! *****2 - variable for the right subinterval
! last - index for subdivision
!
!
! machine dependent constants
! ---------------------------
!
! epmach is the largest relative spacing.
! uflow is the smallest positive magnitude.
!
!***first executable statement dqage
epmach = d1mach(4)
uflow = d1mach(1)
!
! test on validity of parameters
! ------------------------------
!
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
if(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
! first approximation to the integral
! -----------------------------------
!
keyf = key
if(key.le.0) keyf = 1
if(key.ge.7) keyf = 6
neval = 0
if(keyf.eq.1) then
call dqk15_2d_inner(f,x,a,b,result,abserr,defabs,resabs)
end if
if(keyf.eq.2) then
call dqk21_2d_inner(f,x,a,b,result,abserr,defabs,resabs)
end if
if(keyf.eq.3) then
call dqk31_2d_inner(f,x,a,b,result,abserr,defabs,resabs)
end if
if(keyf.eq.4) then
call dqk41_2d_inner(f,x,a,b,result,abserr,defabs,resabs)
end if
if(keyf.eq.5) then
call dqk51_2d_inner(f,x,a,b,result,abserr,defabs,resabs)
end if
if(keyf.eq.6) then
call dqk61_2d_inner(f,x,a,b,result,abserr,defabs,resabs)
end if
last = 1
rlist(1) = result
elist(1) = abserr
iord(1) = 1
!
! test on accuracy.
!
errbnd = dmax1(epsabs,epsrel*dabs(result))
if(abserr.le.0.5d+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.0.0d+00) go to 60
!
! initialization
! --------------
!
!
errmax = abserr
maxerr = 1
area = result
errsum = abserr
nrmax = 1
iroff1 = 0
iroff2 = 0
!
! main do-loop
! ------------
do 30 last = 2,limit
! bisect the subinterval with the largest error estimate.
!
a1 = alist(maxerr)
b1 = 0.5d+00*(alist(maxerr)+blist(maxerr))
a2 = b1
b2 = blist(maxerr)
if(keyf.eq.1) then
call dqk15_2d_inner(f,x,a1,b1,area1,error1,resabs,defab1)
end if
if(keyf.eq.2) then
call dqk21_2d_inner(f,x,a1,b1,area1,error1,resabs,defab1)
end if
if(keyf.eq.3) then
call dqk31_2d_inner(f,x,a1,b1,area1,error1,resabs,defab1)
end if
if(keyf.eq.4) then
call dqk41_2d_inner(f,x,a1,b1,area1,error1,resabs,defab1)
end if
if(keyf.eq.5) then
call dqk51_2d_inner(f,x,a1,b1,area1,error1,resabs,defab1)
end if
if(keyf.eq.6) then
call dqk61_2d_inner(f,x,a1,b1,area1,error1,resabs,defab1)
end if
if(keyf.eq.1) then
call dqk15_2d_inner(f,x,a2,b2,area2,error2,resabs,defab2)
end if
if(keyf.eq.2) then
call dqk21_2d_inner(f,x,a2,b2,area2,error2,resabs,defab2)
end if
if(keyf.eq.3) then
call dqk31_2d_inner(f,x,a2,b2,area2,error2,resabs,defab2)
end if
if(keyf.eq.4) then
call dqk41_2d_inner(f,x,a2,b2,area2,error2,resabs,defab2)
end if
if(keyf.eq.5) then
call dqk51_2d_inner(f,x,a2,b2,area2,error2,resabs,defab2)
end if
if(keyf.eq.6) then
call dqk61_2d_inner(f,x,a2,b2,area2,error2,resabs,defab2)
end if
!
! improve previous approximations to integral
! and error and test for accuracy.
!
neval = neval+1
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 5
if(dabs(rlist(maxerr)-area12).le.0.1d-04*dabs(area12) &
.and.erro12.ge.0.99d+00*errmax) iroff1 = iroff1+1
if(last.gt.10.and.erro12.gt.errmax) iroff2 = iroff2+1
5 rlist(maxerr) = area1
rlist(last) = area2
errbnd = dmax1(epsabs,epsrel*dabs(area))
if(errsum.le.errbnd) go to 8
!
! test for roundoff error and eventually set error flag.
!
if(iroff1.ge.6.or.iroff2.ge.20) ier = 2
!
! set error flag in the case that the number of subintervals
! equals limit.
!
if(last.eq.limit) ier = 1
!
! set error flag in the case of bad integrand behaviour
! at a point of the integration range.
!
if(dmax1(dabs(a1),dabs(b2)).le.(0.1d+01+0.1d+03* &
epmach)*(dabs(a2)+0.1d+04*uflow)) ier = 3
!
! append the newly-created intervals to the list.
!
8 if(error2.gt.error1) go to 10
alist(last) = a2
blist(maxerr) = b1
blist(last) = b2
elist(maxerr) = error1
elist(last) = error2
go to 20
10 alist(maxerr) = a2
alist(last) = a1
blist(last) = b1
rlist(maxerr) = area2
rlist(last) = area1
elist(maxerr) = error2
elist(last) = error1
!
! call subroutine dqpsrt to maintain the descending ordering
! in the list of error estimates and select the subinterval
! with the largest error estimate (to be bisected next).
!
20 call dqpsrt(limit,last,maxerr,errmax,elist,iord,nrmax)
! ***jump out of do-loop
if(ier.ne.0.or.errsum.le.errbnd) go to 40
30 continue
!
! compute final result.
! ---------------------
!
40 result = 0.0d+00
do 50 k=1,last
result = result+rlist(k)
50 continue
abserr = errsum
60 if(keyf.ne.1) neval = (10*keyf+1)*(2*neval+1)
if(keyf.eq.1) neval = 30*neval+15
999 return
end subroutine