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subroutine dqawse(f,a,b,alfa,beta,integr,epsabs,epsrel,limit,
* result,abserr,neval,ier,alist,blist,rlist,elist,iord,last)
c***begin prologue dqawse
c***date written 800101 (yymmdd)
c***revision date 830518 (yymmdd)
c***category no. h2a2a1
c***keywords automatic integrator, special-purpose,
c algebraico-logarithmic end point singularities,
c clenshaw-curtis method
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*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***description
c
c integration of functions having algebraico-logarithmic
c end point singularities
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, b.gt.a
c if b.le.a, the routine will end with ier = 6.
c
c alfa - double precision
c parameter in the weight function, alfa.gt.(-1)
c if alfa.le.(-1), the routine will end with
c ier = 6.
c
c beta - double precision
c parameter in the weight 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 - 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.2
c if limit.lt.2, the routine will end with 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 the integral and error
c are less reliable. it is assumed that the
c requested accuracy has not been achieved.
c error messages
c = 1 maximum number of subdivisions allowed
c has been achieved. one can allow more
c subdivisions by increasing the value of
c limit. however, if this yields no
c improvement, it is advised to analyze the
c integrand in order to determine the
c integration difficulties which prevent the
c requested tolerance from being achieved.
c in case of a jump discontinuity or a local
c singularity of algebraico-logarithmic type
c at one or more interior points of the
c integration range, one should proceed by
c splitting up the interval at these
c points and calling the integrator on the
c 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 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.
c result, abserr, neval, rlist(1), elist(1),
c iord(1) and last are set to zero. alist(1)
c and blist(1) are set to a and b
c respectively.
c
c alist - double precision
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 given 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
c end points of the subintervals in the partition
c of the given 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 iord - integer
c vector of dimension at least limit, the first k
c of which are pointers to the error
c estimates over the subintervals, so that
c elist(iord(1)), ..., elist(iord(k)) with k = last
c if last.le.(limit/2+2), and k = limit+1-last
c otherwise form a decreasing sequence
c
c last - integer
c number of subintervals actually produced in
c the subdivision process
c
c***references (none)
c***routines called d1mach,dqc25s,dqmomo,dqpsrt
c***end prologue dqawse
c
double precision a,abserr,alfa,alist,area,area1,area12,area2,a1,
* a2,b,beta,blist,b1,b2,centre,dabs,dmax1,d1mach,elist,epmach,
* epsabs,epsrel,errbnd,errmax,error1,erro12,error2,errsum,f,
* resas1,resas2,result,rg,rh,ri,rj,rlist,uflow
integer ier,integr,iord,iroff1,iroff2,k,last,limit,maxerr,nev,
* neval,nrmax
c
external f
c
dimension alist(limit),blist(limit),rlist(limit),elist(limit),
* iord(limit),ri(25),rj(25),rh(25),rg(25)
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 elist(i) - error estimate applying to rlist(i)
c maxerr - pointer to the interval with largest
c error estimate
c errmax - elist(maxerr)
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
c
c machine dependent constants
c ---------------------------
c
c epmach is the largest relative spacing.
c uflow is the smallest positive magnitude.
c
c***first executable statement dqawse
epmach = d1mach(4)
uflow = d1mach(1)
c
c test on validity of parameters
c ------------------------------
c
ier = 6
neval = 0
last = 0
rlist(1) = 0.0d+00
elist(1) = 0.0d+00
iord(1) = 0
result = 0.0d+00
abserr = 0.0d+00
if(b.le.a.or.(epsabs.eq.0.0d+00.and.
* epsrel.lt.dmax1(0.5d+02*epmach,0.5d-28)).or.alfa.le.(-0.1d+01).
* or.beta.le.(-0.1d+01).or.integr.lt.1.or.integr.gt.4.or.
* limit.lt.2) go to 999
ier = 0
c
c compute the modified chebyshev moments.
c
call dqmomo(alfa,beta,ri,rj,rg,rh,integr)
c
c integrate over the intervals (a,(a+b)/2) and ((a+b)/2,b).
c
centre = 0.5d+00*(b+a)
call dqc25s(f,a,b,a,centre,alfa,beta,ri,rj,rg,rh,area1,
* error1,resas1,integr,nev)
neval = nev
call dqc25s(f,a,b,centre,b,alfa,beta,ri,rj,rg,rh,area2,
* error2,resas2,integr,nev)
last = 2
neval = neval+nev
result = area1+area2
abserr = error1+error2
c
c test on accuracy.
c
errbnd = dmax1(epsabs,epsrel*dabs(result))
c
c initialization
c --------------
c
if(error2.gt.error1) go to 10
alist(1) = a
alist(2) = centre
blist(1) = centre
blist(2) = b
rlist(1) = area1
rlist(2) = area2
elist(1) = error1
elist(2) = error2
go to 20
10 alist(1) = centre
alist(2) = a
blist(1) = b
blist(2) = centre
rlist(1) = area2
rlist(2) = area1
elist(1) = error2
elist(2) = error1
20 iord(1) = 1
iord(2) = 2
if(limit.eq.2) ier = 1
if(abserr.le.errbnd.or.ier.eq.1) go to 999
errmax = elist(1)
maxerr = 1
nrmax = 1
area = result
errsum = abserr
iroff1 = 0
iroff2 = 0
c
c main do-loop
c ------------
c
do 60 last = 3,limit
c
c bisect the subinterval with largest error estimate.
c
a1 = alist(maxerr)
b1 = 0.5d+00*(alist(maxerr)+blist(maxerr))
a2 = b1
b2 = blist(maxerr)
c
call dqc25s(f,a,b,a1,b1,alfa,beta,ri,rj,rg,rh,area1,
* error1,resas1,integr,nev)
neval = neval+nev
call dqc25s(f,a,b,a2,b2,alfa,beta,ri,rj,rg,rh,area2,
* error2,resas2,integr,nev)
neval = neval+nev
c
c improve previous approximations integral and error
c and test for accuracy.
c
area12 = area1+area2
erro12 = error1+error2
errsum = errsum+erro12-errmax
area = area+area12-rlist(maxerr)
if(a.eq.a1.or.b.eq.b2) go to 30
if(resas1.eq.error1.or.resas2.eq.error2) go to 30
c
c test for roundoff error.
c
if(dabs(rlist(maxerr)-area12).lt.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
30 rlist(maxerr) = area1
rlist(last) = area2
c
c test on accuracy.
c
errbnd = dmax1(epsabs,epsrel*dabs(area))
if(errsum.le.errbnd) go to 35
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
c set error flag in the case of roundoff error.
c
if(iroff1.ge.6.or.iroff2.ge.20) ier = 2
c
c set error flag in the case of bad integrand behaviour
c at interior points of integration range.
c
if(dmax1(dabs(a1),dabs(b2)).le.(0.1d+01+0.1d+03*epmach)*
* (dabs(a2)+0.1d+04*uflow)) ier = 3
c
c append the newly-created intervals to the list.
c
35 if(error2.gt.error1) go to 40
alist(last) = a2
blist(maxerr) = b1
blist(last) = b2
elist(maxerr) = error1
elist(last) = error2
go to 50
40 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 largest error estimate (to be bisected next).
c
50 call dqpsrt(limit,last,maxerr,errmax,elist,iord,nrmax)
c ***jump out of do-loop
if (ier.ne.0.or.errsum.le.errbnd) go to 70
60 continue
c
c compute final result.
c ---------------------
c
70 result = 0.0d+00
do 80 k=1,last
result = result+rlist(k)
80 continue
abserr = errsum
999 return
end
|
