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subroutine dqagp(f,a,b,npts2,points,epsabs,epsrel,result,abserr,
* neval,ier,leniw,lenw,last,iwork,work)
c***begin prologue dqagp
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),
c hopefully satisfying following claim for accuracy
c break points of the integration interval, where local
c difficulties of the integrand may occur (e.g.
c singularities, discontinuities), are provided by the 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, npts.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 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.
c it is presumed that the requested
c tolerance cannot be achieved, and that
c the returned result is the best which
c 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 result, abserr, neval, last are set to
c zero. exept when leniw or lenw or npts2 is
c invalid, iwork(1), iwork(limit+1),
c work(limit*2+1) and work(limit*3+1)
c are set to zero.
c work(1) is set to a and work(limit+1)
c to b (where limit = (leniw-npts2)/2).
c
c dimensioning parameters
c leniw - integer
c dimensioning parameter for iwork
c leniw determines limit = (leniw-npts2)/2,
c which is the maximum number of subintervals in the
c partition of the given integration interval (a,b),
c leniw.ge.(3*npts2-2).
c if leniw.lt.(3*npts2-2), the routine will end with
c ier = 6.
c
c lenw - integer
c dimensioning parameter for work
c lenw must be at least leniw*2-npts2.
c if lenw.lt.leniw*2-npts2, the routine will end
c with ier = 6.
c
c last - integer
c on return, last equals the number of subintervals
c produced in the subdivision process, which
c determines the number of significant elements
c actually in the work arrays.
c
c work arrays
c iwork - integer
c vector of dimension at least leniw. on return,
c the first k elements of which contain
c pointers to the error estimates over the
c subintervals, such that work(limit*3+iwork(1)),...,
c work(limit*3+iwork(k)) form a decreasing
c sequence, with k = last if last.le.(limit/2+2), and
c k = limit+1-last otherwise
c iwork(limit+1), ...,iwork(limit+last) contain the
c subdivision levels of the subintervals, i.e.
c if (aa,bb) is a subinterval of (p1,p2)
c where p1 as well as p2 is a user-provided
c break point or integration limit, then (aa,bb) has
c level l if abs(bb-aa) = abs(p2-p1)*2**(-l),
c iwork(limit*2+1), ..., iwork(limit*2+npts2) have
c no significance for the user,
c note that limit = (leniw-npts2)/2.
c
c work - double precision
c vector of dimension at least lenw
c on return
c work(1), ..., work(last) contain the left
c end points of the subintervals in the
c partition of (a,b),
c work(limit+1), ..., work(limit+last) contain
c the right end points,
c work(limit*2+1), ..., work(limit*2+last) contain
c the integral approximations over the subintervals,
c work(limit*3+1), ..., work(limit*3+last)
c contain the corresponding error estimates,
c work(limit*4+1), ..., work(limit*4+npts2)
c contain the integration limits and the
c break points sorted in an ascending sequence.
c note that limit = (leniw-npts2)/2.
c
c***references (none)
c***routines called dqagpe,xerror
c***end prologue dqagp
c
double precision a,abserr,b,epsabs,epsrel,f,points,result,work
integer ier,iwork,last,leniw,lenw,limit,lvl,l1,l2,l3,l4,neval,
* npts2
c
dimension iwork(leniw),points(npts2),work(lenw)
c
external f
c
c check validity of limit and lenw.
c
c***first executable statement dqagp
ier = 6
neval = 0
last = 0
result = 0.0d+00
abserr = 0.0d+00
if(leniw.lt.(3*npts2-2).or.lenw.lt.(leniw*2-npts2).or.npts2.lt.2)
* go to 10
c
c prepare call for dqagpe.
c
limit = (leniw-npts2)/2
l1 = limit+1
l2 = limit+l1
l3 = limit+l2
l4 = limit+l3
c
call dqagpe(f,a,b,npts2,points,epsabs,epsrel,limit,result,abserr,
* neval,ier,work(1),work(l1),work(l2),work(l3),work(l4),
* iwork(1),iwork(l1),iwork(l2),last)
c
c call error handler if necessary.
c
lvl = 0
10 if(ier.eq.6) lvl = 1
if(ier.ne.0) call xerror('abnormal return from dqagp',26,ier,lvl)
return
end
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