subroutine qawfe(f,a,omega,integr,epsabs,limlst,limit,maxp1, * result,abserr,neval,ier,rslst,erlst,ierlst,lst,alist,blist, * rlist,elist,iord,nnlog,chebmo) c***begin prologue qawfe c***date written 800101 (yymmdd) c***revision date 830518 (yymmdd) c***category no. h2a3a1 c***keywords automatic integrator, special-purpose, c fourier integrals, c integration between zeros with dqawoe, c convergence acceleration with dqelg c***author piessens,robert,appl. math. & progr. div. - k.u.leuven c dedoncker,elise,appl. math. & progr. div. - k.u.leuven c***purpose the routine calculates an approximation result to a c given fourier integal c i = integral of f(x)*w(x) over (a,infinity) c where w(x) = cos(omega*x) or w(x) = sin(omega*x), c hopefully satisfying following claim for accuracy c abs(i-result).le.epsabs. c***description c c computation of fourier integrals 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 c be declared e x t e r n a l in the driver program. c c a - real c lower limit of integration c c omega - real c parameter in the weight function c c integr - integer c indicates which weight function is used c integr = 1 w(x) = cos(omega*x) c integr = 2 w(x) = sin(omega*x) c if integr.ne.1.and.integr.ne.2, the routine will c end with ier = 6. c c epsabs - real c absolute accuracy requested, epsabs.gt.0 c if epsabs.le.0, the routine will end with ier = 6. c c limlst - integer c limlst gives an upper bound on the number of c cycles, limlst.ge.1. c if limlst.lt.3, the routine will end with ier = 6. c c limit - integer c gives an upper bound on the number of subintervals c allowed in the partition of each cycle, limit.ge.1 c each cycle, limit.ge.1. c c maxp1 - integer c gives an upper bound on the number of c chebyshev moments which can be stored, i.e. c for the intervals of lengths abs(b-a)*2**(-l), c l=0,1, ..., maxp1-2, maxp1.ge.1 c c on return c result - real c approximation to the integral x 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 - ier = 0 normal and reliable termination of c the routine. it is assumed that the c requested accuracy has been achieved. c ier.gt.0 abnormal termination of the routine. the c estimates for integral and error are less c reliable. it is assumed that the requested c accuracy has not been achieved. c error messages c if omega.ne.0 c ier = 1 maximum number of cycles allowed c has been achieved., i.e. of subintervals c (a+(k-1)c,a+kc) where c c = (2*int(abs(omega))+1)*pi/abs(omega), c for k = 1, 2, ..., lst. c one can allow more cycles by increasing c the value of limlst (and taking the c according dimension adjustments into c account). c examine the array iwork which contains c the error flags on the cycles, in order to c look for eventual local integration c difficulties. if the position of a local c difficulty can be determined (e.g. c singularity, discontinuity within the c interval) one will probably gain from c splitting up the interval at this point c and calling appropriate integrators on c the subranges. c = 4 the extrapolation table constructed for c convergence acceleration of the series c formed by the integral contributions over c the cycles, does not converge to within c the requested accuracy. as in the case of c ier = 1, it is advised to examine the c array iwork which contains the error c flags on the cycles. c = 6 the input is invalid because c (integr.ne.1 and integr.ne.2) or c epsabs.le.0 or limlst.lt.3. c result, abserr, neval, lst are set c to zero. c = 7 bad integrand behaviour occurs within one c or more of the cycles. location and type c of the difficulty involved can be c determined from the vector ierlst. here c lst is the number of cycles actually c needed (see below). c ierlst(k) = 1 the maximum number of c subdivisions (= limit) has c been achieved on the k th c cycle. c = 2 occurrence of roundoff error c is detected and prevents the c tolerance imposed on the c k th cycle, from being c achieved. c = 3 extremely bad integrand c behaviour occurs at some c points of the k th cycle. c = 4 the integration procedure c over the k th cycle does c not converge (to within the c required accuracy) due to c roundoff in the c extrapolation procedure c invoked on this cycle. it c is assumed that the result c on this interval is the c best which can be obtained. c = 5 the integral over the k th c cycle is probably divergent c or slowly convergent. it c must be noted that c divergence can occur with c any other value of c ierlst(k). c if omega = 0 and integr = 1, c the integral is calculated by means of dqagie c and ier = ierlst(1) (with meaning as described c for ierlst(k), k = 1). c c rslst - real c vector of dimension at least limlst c rslst(k) contains the integral contribution c over the interval (a+(k-1)c,a+kc) where c c = (2*int(abs(omega))+1)*pi/abs(omega), c k = 1, 2, ..., lst. c note that, if omega = 0, rslst(1) contains c the value of the integral over (a,infinity). c c erlst - real c vector of dimension at least limlst c erlst(k) contains the error estimate corresponding c with rslst(k). c c ierlst - integer c vector of dimension at least limlst c ierlst(k) contains the error flag corresponding c with rslst(k). for the meaning of the local error c flags see description of output parameter ier. c c lst - integer c number of subintervals needed for the integration c if omega = 0 then lst is set to 1. c c alist, blist, rlist, elist - real c vector of dimension at least limit, c c iord, nnlog - integer c vector of dimension at least limit, providing c space for the quantities needed in the subdivision c process of each cycle c c chebmo - real c array of dimension at least (maxp1,25), providing c space for the chebyshev moments needed within the c cycles c c***references (none) c***routines called qagie,qawoe,qelg,r1mach c***end prologue qawfe c real a,abseps,abserr,alist,blist,chebmo,correc,cycle, * c1,c2,dl,dla,drl,elist,ep,eps,epsa,epsabs,erlst, * errsum,fact,omega,p,pi,p1,psum,reseps,result,res3la,rlist,rslst * ,r1mach,uflow integer ier,ierlst,integr,iord,ktmin,l,lst,limit,ll,maxp1, * nev,neval,nnlog,nres,numrl2 c dimension alist(limit),blist(limit),chebmo(maxp1,25),elist(limit), * erlst(limlst),ierlst(limlst),iord(limit),nnlog(limit),psum(52), * res3la(3),rlist(limit),rslst(limlst) c external f c c c the dimension of psum is determined by the value of c limexp in subroutine qelg (psum must be c of dimension (limexp+2) at least). c c list of major variables c ----------------------- c c c1, c2 - end points of subinterval (of length c cycle) c cycle - (2*int(abs(omega))+1)*pi/abs(omega) c psum - vector of dimension at least (limexp+2) c (see routine qelg) c psum contains the part of the epsilon c table which is still needed for further c computations. c each element of psum is a partial sum of c the series which should sum to the value of c the integral. c errsum - sum of error estimates over the c subintervals, calculated cumulatively c epsa - absolute tolerance requested over current c subinterval c chebmo - array containing the modified chebyshev c moments (see also routine qc25f) c data p/0.9e+00/,pi/0.31415926535897932e+01/ c c test on validity of parameters c ------------------------------ c c***first executable statement qawfe result = 0.0e+00 abserr = 0.0e+00 neval = 0 lst = 0 ier = 0 if((integr.ne.1.and.integr.ne.2).or.epsabs.le.0.0e+00.or. * limlst.lt.3) ier = 6 if(ier.eq.6) go to 999 if(omega.ne.0.0e+00) go to 10 c c integration by qagie if omega is zero c -------------------------------------- c if(integr.eq.1) call qagie(f,0.0e+00,1,epsabs,0.0e+00,limit, * result,abserr,neval,ier,alist,blist,rlist,elist,iord,last) rslst(1) = result erlst(1) = abserr ierlst(1) = ier lst = 1 go to 999 c c initializations c --------------- c 10 l = abs(omega) dl = 2*l+1 cycle = dl*pi/abs(omega) ier = 0 ktmin = 0 neval = 0 numrl2 = 0 nres = 0 c1 = a c2 = cycle+a p1 = 0.1e+01-p eps = epsabs uflow = r1mach(1) if(epsabs.gt.uflow/p1) eps = epsabs*p1 ep = eps fact = 0.1e+01 correc = 0.0e+00 abserr = 0.0e+00 errsum = 0.0e+00 c c main do-loop c ------------ c do 50 lst = 1,limlst c c integrate over current subinterval. c dla = lst epsa = eps*fact call qawoe(f,c1,c2,omega,integr,epsa,0.0e+00,limit,lst,maxp1, * rslst(lst),erlst(lst),nev,ierlst(lst),last,alist,blist,rlist, * elist,iord,nnlog,momcom,chebmo) neval = neval+nev fact = fact*p errsum = errsum+erlst(lst) drl = 0.5e+02*abs(rslst(lst)) c c test on accuracy with partial sum c if(errsum+drl.le.epsabs.and.lst.ge.6) go to 80 correc = amax1(correc,erlst(lst)) if(ierlst(lst).ne.0) eps = amax1(ep,correc*p1) if(ierlst(lst).ne.0) ier = 7 if(ier.eq.7.and.(errsum+drl).le.correc*0.1e+02.and. * lst.gt.5) go to 80 numrl2 = numrl2+1 if(lst.gt.1) go to 20 psum(1) = rslst(1) go to 40 20 psum(numrl2) = psum(ll)+rslst(lst) if(lst.eq.2) go to 40 c c test on maximum number of subintervals c if(lst.eq.limlst) ier = 1 c c perform new extrapolation c call qelg(numrl2,psum,reseps,abseps,res3la,nres) c c test whether extrapolated result is influenced by c roundoff c ktmin = ktmin+1 if(ktmin.ge.15.and.abserr.le.0.1e-02*(errsum+drl)) ier = 4 if(abseps.gt.abserr.and.lst.ne.3) go to 30 abserr = abseps result = reseps ktmin = 0 c c if ier is not 0, check whether direct result (partial c sum) or extrapolated result yields the best integral c approximation c if((abserr+0.1e+02*correc).le.epsabs.or. * (abserr.le.epsabs.and.0.1e+02*correc.ge.epsabs)) go to 60 30 if(ier.ne.0.and.ier.ne.7) go to 60 40 ll = numrl2 c1 = c2 c2 = c2+cycle 50 continue c c set final result and error estimate c ----------------------------------- c 60 abserr = abserr+0.1e+02*correc if(ier.eq.0) go to 999 if(result.ne.0.0e+00.and.psum(numrl2).ne.0.0e+00) go to 70 if(abserr.gt.errsum) go to 80 if(psum(numrl2).eq.0.0e+00) go to 999 70 if(abserr/abs(result).gt.(errsum+drl)/abs(psum(numrl2))) * go to 80 if(ier.ge.1.and.ier.ne.7) abserr = abserr+drl go to 999 80 result = psum(numrl2) abserr = errsum+drl 999 return end