subroutine dqag(f,a,b,epsabs,epsrel,key,result,abserr,neval,ier, * limit,lenw,last,iwork,work) c***begin prologue dqag c***date written 800101 (yymmdd) c***revision date 830518 (yymmdd) c***category no. h2a1a1 c***keywords automatic integrator, general-purpose, c integrand examinator, globally adaptive, c gauss-kronrod 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 abs(i-result)le.max(epsabs,epsrel*abs(i)). c***description c c computation of a definite integral c standard fortran subroutine c double precision version c c f - double precision c function subprogam 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 epsabs - double precision c absolute accoracy 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 key - integer c key for choice of local integration rule c a gauss-kronrod pair is used with c 7 - 15 points if key.lt.2, c 10 - 21 points if key = 2, c 15 - 31 points if key = 3, c 20 - 41 points if key = 4, c 25 - 51 points if key = 5, c 30 - 61 points if key.gt.5. 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 result 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 yield no improvement it is advised c to analyze the integrand in order to c determine the integration difficulaties. c if the position of a local difficulty can c be determined (i.e.singularity, c discontinuity within the interval) one c will probably gain from splitting up the c interval at this point and calling the c integrator on the subranges. if possible, c an appropriate special-purpose integrator c should 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 = 3 extremely bad integrand behaviour occurs c at some points of the integration c interval. c = 6 the input is invalid, because c (epsabs.le.0 and c epsrel.lt.max(50*rel.mach.acc.,0.5d-28)) c or limit.lt.1 or lenw.lt.limit*4. c result, abserr, neval, last are set c to zero. c except when lenw is invalid, iwork(1), c work(limit*2+1) and work(limit*3+1) are c set to zero, work(1) is set to a and c work(limit+1) to b. c c dimensioning parameters c limit - integer c dimensioning parameter for iwork c limit determines the maximum number of subintervals c in the partition of the given integration interval c (a,b), limit.ge.1. c if limit.lt.1, the routine will end with ier = 6. c c lenw - integer c dimensioning parameter for work c lenw must be at least limit*4. c if lenw.lt.limit*4, the routine will end with c ier = 6. c c last - integer c on return, last equals the number of subintervals c produced in the subdiviosion 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 limit, the first k c elements of which contain pointers to the error c estimates over the subintervals, such that c work(limit*3+iwork(1)),... , work(limit*3+iwork(k)) c form a decreasing sequence with k = last if c last.le.(limit/2+2), and k = limit+1-last otherwise c c work - double precision c vector of dimension at least lenw c on return c work(1), ..., work(last) contain the left end c points of the subintervals in the partition of c (a,b), c work(limit+1), ..., work(limit+last) contain the c 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) contain c the error estimates. c c***references (none) c***routines called dqage,xerror c***end prologue dqag double precision a,abserr,b,epsabs,epsrel,f,result,work integer ier,iwork,key,last,lenw,limit,lvl,l1,l2,l3,neval c dimension iwork(limit),work(lenw) c external f c c check validity of lenw. c c***first executable statement dqag ier = 6 neval = 0 last = 0 result = 0.0d+00 abserr = 0.0d+00 if(limit.lt.1.or.lenw.lt.limit*4) go to 10 c c prepare call for dqage. c l1 = limit+1 l2 = limit+l1 l3 = limit+l2 c call dqage(f,a,b,epsabs,epsrel,key,limit,result,abserr,neval, * ier,work(1),work(l1),work(l2),work(l3),iwork,last) c c call error handler if necessary. c lvl = 0 10 if(ier.eq.6) lvl = 1 if(ier.ne.0) call xerror(26habnormal return from dqag ,26,ier,lvl) return end