subroutine qawse(f,a,b,alfa,beta,integr,epsabs,epsrel,limit, * result,abserr,neval,ier,alist,blist,rlist,elist,iord,last) c***begin prologue qawse 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 real version 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, b.gt.a c if b.le.a, the routine will end with ier = 6. c c alfa - real 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 - real 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 - 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 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 - 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 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 - real 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 - real 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 - real 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 - real 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 qc25s,qmomo,qpsrt,r1mach c***end prologue qawse c real a,abserr,alfa,alist,area,area1,area12, * area2,a1,a2,b,beta,blist,b1,b2,centre, * r1mach,elist,epmach,epsabs,epsrel,errbnd,errmax, * error1,erro12,error2,errsum,f,resas1,resas2,result,rg,rh,ri,rj, * rlist,uflow integer ier,int,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 qawse epmach = r1mach(4) uflow = r1mach(1) c c test on validity of parameters c ------------------------------ c ier = 6 neval = 0 last = 0 rlist(1) = 0.0e+00 elist(1) = 0.0e+00 iord(1) = 0 result = 0.0e+00 abserr = 0.0e+00 if(b.le.a.or.(epsabs.eq.0.0e+00.and. * epsrel.lt.amax1(0.5e+02*epmach,0.5e-14)).or.alfa.le.(-0.1e+01). * or.beta.le.(-0.1e+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 qmomo(alfa,beta,ri,rj,rg,rh,integr) c c integrate over the intervals (a,(a+b)/2) c and ((a+b)/2,b). c centre = 0.5e+00*(b+a) call qc25s(f,a,b,a,centre,alfa,beta,ri,rj,rg,rh,area1, * error1,resas1,integr,nev) neval = nev call qc25s(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 = amax1(epsabs,epsrel*abs(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 = 0.5e+00*(alist(maxerr)+blist(maxerr)) a2 = b1 b2 = blist(maxerr) c call qc25s(f,a,b,a1,b1,alfa,beta,ri,rj,rg,rh,area1, * error1,resas1,integr,nev) neval = neval+nev call qc25s(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(abs(rlist(maxerr)-area12).lt.0.1e-04*abs(area12) * .and.erro12.ge.0.99e+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 = amax1(epsabs,epsrel*abs(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(amax1(abs(a1),abs(b2)).le.(0.1e+01+0.1e+03*epmach)* * (abs(a2)+0.1e+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 qpsrt to maintain the descending ordering c in the list of error estimates and select the c subinterval with largest error estimate (to be c bisected next). c 50 call qpsrt(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 = 0.0e+00 do 80 k=1,last result = result+rlist(k) 80 continue abserr = errsum 999 return end