subroutine qags(f,a,b,epsabs,epsrel,result,abserr,neval,ier, * limit,lenw,last,iwork,work) c***begin prologue qags c***date written 800101 (yymmdd) c***revision date 830518 (yymmdd) c***category no. h2a1a1 c***keywords automatic integrator, general-purpose, c (end-point) singularities, extrapolation, c globally adaptive c***author piessens,robert,appl. math. & progr. div. - k.u.leuven c de doncker,elise,appl. math. & prog. 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 real version c c c parameters c on entry c f - real 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 - real c lower limit of integration c c b - real c upper limit of integration c c epsabs - real c absolute accuracy requested c epsrel - real 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 - real c approximation to the integral c c abserr - real 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 sub- c divisions by increasing the value of limit c (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 (e.g. 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 detec- c ted, which prevents the requested c tolerance from being achieved. c the error may be under-estimated. c = 3 extremely bad integrand behaviour c occurs 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. 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 to c zero.except when limit or lenw is invalid, c iwork(1), work(limit*2+1) and c work(limit*3+1) are set to zero, work(1) c is set to a and 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 c with ier = 6. c c last - integer c on return, last equals the number of subintervals c produced in the subdivision process, detemines the c number of significant elements actually in the work c arrays. c c work arrays c iwork - integer c vector of dimension at least limit, the first k c elements of which contain pointers c to the error estimates over the subintervals c 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), c and k = limit+1-last otherwise c c work - real 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 error estimates. c c c c***references (none) c***routines called qagse,xerror c***end prologue qags c c real a,abserr,b,epsabs,epsrel,f,result,work integer ier,iwork,lenw,limit,lvl,l1,l2,l3,neval c dimension iwork(limit),work(lenw) c external f c c check validity of limit and lenw. c c***first executable statement qags ier = 6 neval = 0 last = 0 result = 0.0e+00 abserr = 0.0e+00 if(limit.lt.1.or.lenw.lt.limit*4) go to 10 c c prepare call for qagse. c l1 = limit+1 l2 = limit+l1 l3 = limit+l2 c call qagse(f,a,b,epsabs,epsrel,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 qags, * 26,ier,lvl) return end