subroutine dqag_k2(f,a,b,eps,s,ier) implicit none double precision,intent(in)::a,b,eps double precision,intent(out)::s integer,intent(out)::ier double precision::f ! ier = 0 : success, converged. ! ier > 0 : fail, not converged. integer::neval,limit,lenw,last,key integer,allocatable::iwork(:) double precision,allocatable::work(:) double precision::epsabs,abserr external::f !======== key=2 !======== ier=1 epsabs=-1d0 limit=200 lenw=limit*4 allocate(iwork(1:limit),work(1:lenw)) s=0d0 call dqag(f,a,b,epsabs,eps,key,s,abserr,neval,ier, & limit,lenw,last,iwork,work) deallocate(iwork,work) return end subroutine dqag_k2 subroutine dqag_k6(f,a,b,eps,s,ier) implicit none double precision,intent(in)::a,b,eps double precision,intent(out)::s integer,intent(out)::ier double precision::f ! ier = 0 : success, converged. ! ier > 0 : fail, not converged. integer::neval,limit,lenw,last,key integer,allocatable::iwork(:) double precision,allocatable::work(:) double precision::epsabs,abserr external::f !======== key=6 !======== ier=1 epsabs=-1d0 limit=200 lenw=limit*4 allocate(iwork(1:limit),work(1:lenw)) s=0d0 call dqag(f,a,b,epsabs,eps,key,s,abserr,neval,ier, & limit,lenw,last,iwork,work) deallocate(iwork,work) return end subroutine dqag_k6 ! 2017/09/04 (yyyy/mm/dd) modified by sikino subroutine dqag(f,a,b,epsabs,epsrel,key,result,abserr,neval,ier, & limit,lenw,last,iwork,work) !***begin prologue dqag !***date written 800101 (yymmdd) !***revision date 830518 (yymmdd) !***category no. h2a1a1 !***keywords automatic integrator, general-purpose, ! integrand examinator, globally adaptive, ! gauss-kronrod !***author piessens,robert,appl. math. & progr. div - k.u.leuven ! de doncker,elise,appl. math. & progr. div. - k.u.leuven !***purpose the routine calculates an approximation result to a given ! definite integral i = integral of f over (a,b), ! hopefully satisfying following claim for accuracy ! abs(i-result)le.max(epsabs,epsrel*abs(i)). !***description ! ! computation of a definite integral ! standard fortran subroutine ! double precision version ! ! f - double precision ! function subprogam defining the integrand ! function f(x). the actual name for f needs to be ! declared e x t e r n a l in the driver program. ! ! a - double precision ! lower limit of integration ! ! b - double precision ! upper limit of integration ! ! epsabs - double precision ! absolute accoracy requested ! epsrel - double precision ! relative accuracy requested ! if epsabs.le.0 ! and epsrel.lt.max(50*rel.mach.acc.,0.5d-28), ! the routine will end with ier = 6. ! ! key - integer ! key for choice of local integration rule ! a gauss-kronrod pair is used with ! 7 - 15 points if key.lt.2, ! 10 - 21 points if key = 2, ! 15 - 31 points if key = 3, ! 20 - 41 points if key = 4, ! 25 - 51 points if key = 5, ! 30 - 61 points if key.gt.5. ! ! on return ! result - double precision ! approximation to the integral ! ! abserr - double precision ! estimate of the modulus of the absolute error, ! which should equal or exceed abs(i-result) ! ! neval - integer ! number of integrand evaluations ! ! ier - integer ! ier = 0 normal and reliable termination of the ! routine. it is assumed that the requested ! accuracy has been achieved. ! ier.gt.0 abnormal termination of the routine ! the estimates for result and error are ! less reliable. it is assumed that the ! requested accuracy has not been achieved. ! error messages ! ier = 1 maximum number of subdivisions allowed ! has been achieved. one can allow more ! subdivisions by increasing the value of ! limit (and taking the according dimension ! adjustments into account). however, if ! this yield no improvement it is advised ! to analyze the integrand in order to ! determine the integration difficulaties. ! if the position of a local difficulty can ! be determined (i.e.singularity, ! discontinuity within the interval) one ! will probably gain from splitting up the ! interval at this point and calling the ! integrator on the subranges. if possible, ! an appropriate special-purpose integrator ! should be used which is designed for ! handling the type of difficulty involved. ! = 2 the occurrence of roundoff error is ! detected, which prevents the requested ! tolerance from being achieved. ! = 3 extremely bad integrand behaviour occurs ! at some points of the integration ! interval. ! = 6 the input is invalid, because ! (epsabs.le.0 and ! epsrel.lt.max(50*rel.mach.acc.,0.5d-28)) ! or limit.lt.1 or lenw.lt.limit*4. ! result, abserr, neval, last are set ! to zero. ! except when lenw is invalid, iwork(1), ! work(limit*2+1) and work(limit*3+1) are ! set to zero, work(1) is set to a and ! work(limit+1) to b. ! ! dimensioning parameters ! limit - integer ! dimensioning parameter for iwork ! limit determines the maximum number of subintervals ! in the partition of the given integration interval ! (a,b), limit.ge.1. ! if limit.lt.1, the routine will end with ier = 6. ! ! lenw - integer ! dimensioning parameter for work ! lenw must be at least limit*4. ! if lenw.lt.limit*4, the routine will end with ! ier = 6. ! ! last - integer ! on return, last equals the number of subintervals ! produced in the subdiviosion process, which ! determines the number of significant elements ! actually in the work arrays. ! ! work arrays ! iwork - integer ! vector of dimension at least limit, the first k ! elements of which contain pointers to the error ! estimates over the subintervals, such that ! work(limit*3+iwork(1)),... , work(limit*3+iwork(k)) ! form a decreasing sequence with k = last if ! last.le.(limit/2+2), and k = limit+1-last otherwise ! ! work - double precision ! vector of dimension at least lenw ! on return ! work(1), ..., work(last) contain the left end ! points of the subintervals in the partition of ! (a,b), ! work(limit+1), ..., work(limit+last) contain the ! right end points, ! work(limit*2+1), ..., work(limit*2+last) contain ! the integral approximations over the subintervals, ! work(limit*3+1), ..., work(limit*3+last) contain ! the error estimates. ! !***references (none) !***routines called dqage,xerror !***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 ! dimension iwork(limit),work(lenw) ! external f ! ! check validity of lenw. ! !***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 if(limit.ge.1.and.lenw.ge.limit*4)then ! ! prepare call for dqage. ! l1 = limit+1 l2 = limit+l1 l3 = limit+l2 ! call dqage(f,a,b,epsabs,epsrel,key,limit,result,abserr,neval,& ier,work(1),work(l1),work(l2),work(l3),iwork,last) ! ! call error handler if necessary. ! lvl = 0 endif if(ier.eq.6) lvl = 1 if(ier.ne.0) call xerror("26habnormal return from dqag" ,26,ier,lvl) return end subroutine dqag subroutine dqage(f,a,b,epsabs,epsrel,key,limit,result,abserr, & neval,ier,alist,blist,rlist,elist,iord,last) !***begin prologue dqage !***date written 800101 (yymmdd) !***revision date 830518 (yymmdd) !***category no. h2a1a1 !***keywords automatic integrator, general-purpose, ! integrand examinator, globally adaptive, ! gauss-kronrod !***author piessens,robert,appl. math. & progr. div. - k.u.leuven ! de doncker,elise,appl. math. & progr. div. - k.u.leuven !***purpose the routine calculates an approximation result to a given ! definite integral i = integral of f over (a,b), ! hopefully satisfying following claim for accuracy ! abs(i-reslt).le.max(epsabs,epsrel*abs(i)). !***description ! ! computation of a definite integral ! standard fortran subroutine ! double precision version ! ! parameters ! on entry ! f - double precision ! function subprogram defining the integrand ! function f(x). the actual name for f needs to be ! declared e x t e r n a l in the driver program. ! ! a - double precision ! lower limit of integration ! ! b - double precision ! upper limit of integration ! ! epsabs - double precision ! absolute accuracy requested ! epsrel - double precision ! relative accuracy requested ! if epsabs.le.0 ! and epsrel.lt.max(50*rel.mach.acc.,0.5d-28), ! the routine will end with ier = 6. ! ! key - integer ! key for choice of local integration rule ! a gauss-kronrod pair is used with ! 7 - 15 points if key.lt.2, ! 10 - 21 points if key = 2, ! 15 - 31 points if key = 3, ! 20 - 41 points if key = 4, ! 25 - 51 points if key = 5, ! 30 - 61 points if key.gt.5. ! ! limit - integer ! gives an upperbound on the number of subintervals ! in the partition of (a,b), limit.ge.1. ! ! on return ! result - double precision ! approximation to the integral ! ! abserr - double precision ! estimate of the modulus of the absolute error, ! which should equal or exceed abs(i-result) ! ! neval - integer ! number of integrand evaluations ! ! ier - integer ! ier = 0 normal and reliable termination of the ! routine. it is assumed that the requested ! accuracy has been achieved. ! ier.gt.0 abnormal termination of the routine ! the estimates for result and error are ! less reliable. it is assumed that the ! requested accuracy has not been achieved. ! error messages ! ier = 1 maximum number of subdivisions allowed ! has been achieved. one can allow more ! subdivisions by increasing the value ! of limit. ! however, if this yields no improvement it ! is rather advised to analyze the integrand ! in order to determine the integration ! difficulties. if the position of a local ! difficulty can be determined(e.g. ! singularity, discontinuity within the ! interval) one will probably gain from ! splitting up the interval at this point ! and calling the integrator on the ! subranges. if possible, an appropriate ! special-purpose integrator should be used ! which is designed for handling the type of ! difficulty involved. ! = 2 the occurrence of roundoff error is ! detected, which prevents the requested ! tolerance from being achieved. ! = 3 extremely bad integrand behaviour occurs ! at some points of the integration ! interval. ! = 6 the input is invalid, because ! (epsabs.le.0 and ! epsrel.lt.max(50*rel.mach.acc.,0.5d-28), ! result, abserr, neval, last, rlist(1) , ! elist(1) and iord(1) are set to zero. ! alist(1) and blist(1) are set to a and b ! respectively. ! ! alist - double precision ! vector of dimension at least limit, the first ! last elements of which are the left ! end points of the subintervals in the partition ! of the given integration range (a,b) ! ! blist - double precision ! vector of dimension at least limit, the first ! last elements of which are the right ! end points of the subintervals in the partition ! of the given integration range (a,b) ! ! rlist - double precision ! vector of dimension at least limit, the first ! last elements of which are the ! integral approximations on the subintervals ! ! elist - double precision ! vector of dimension at least limit, the first ! last elements of which are the moduli of the ! absolute error estimates on the subintervals ! ! iord - integer ! vector of dimension at least limit, the first k ! elements of which are pointers to the ! error estimates over the subintervals, ! such that elist(iord(1)), ..., ! elist(iord(k)) form a decreasing sequence, ! with k = last if last.le.(limit/2+2), and ! k = limit+1-last otherwise ! ! last - integer ! number of subintervals actually produced in the ! subdivision process ! !***references (none) !***routines called d1mach,dqk15,dqk21,dqk31, ! dqk41,dqk51,dqk61,dqpsrt !***end prologue dqage ! double precision a,abserr,alist,area,area1,area12,area2,a1,a2,b,& blist,b1,b2,dabs,defabs,defab1,defab2,dmax1,elist,epmach,& epsabs,epsrel,errbnd,errmax,error1,error2,erro12,errsum,f,& resabs,result,rlist,uflow integer ier,iord,iroff1,iroff2,k,key,keyf,last,limit,maxerr,neval,& nrmax,igt,igk ! dimension alist(limit),blist(limit),elist(limit),iord(limit),& rlist(limit) ! external f ! ! list of major variables ! ----------------------- ! ! alist - list of left end points of all subintervals ! considered up to now ! blist - list of right end points of all subintervals ! considered up to now ! rlist(i) - approximation to the integral over ! (alist(i),blist(i)) ! elist(i) - error estimate applying to rlist(i) ! maxerr - pointer to the interval with largest ! error estimate ! errmax - elist(maxerr) ! area - sum of the integrals over the subintervals ! errsum - sum of the errors over the subintervals ! errbnd - requested accuracy max(epsabs,epsrel* ! abs(result)) ! *****1 - variable for the left subinterval ! *****2 - variable for the right subinterval ! last - index for subdivision ! ! ! machine dependent constants ! --------------------------- ! ! epmach is the largest relative spacing. ! uflow is the smallest positive magnitude. ! !***first executable statement dqage epmach = epsilon(a) uflow = tiny(a) ! ! test on validity of parameters ! ------------------------------ ! ier = 0 neval = 0 last = 0 result = 0.0d+00 abserr = 0.0d+00 alist(1) = a blist(1) = b rlist(1) = 0.0d+00 elist(1) = 0.0d+00 iord(1) = 0 if(epsabs.le.0.0d+00.and.& epsrel.lt.dmax1(0.5d+02*epmach,0.5d-28)) ier = 6 ! if(ier.eq.6) go to 999 if(ier.ne.6)then ! ! first approximation to the integral ! ----------------------------------- ! keyf = key if(key.le.0) keyf = 1 if(key.ge.7) keyf = 6 neval = 0 if(keyf.eq.1) call dqk15(f,a,b,result,abserr,defabs,resabs) if(keyf.eq.2) call dqk21(f,a,b,result,abserr,defabs,resabs) if(keyf.eq.3) call dqk31(f,a,b,result,abserr,defabs,resabs) if(keyf.eq.4) call dqk41(f,a,b,result,abserr,defabs,resabs) if(keyf.eq.5) call dqk51(f,a,b,result,abserr,defabs,resabs) if(keyf.eq.6) call dqk61(f,a,b,result,abserr,defabs,resabs) last = 1 rlist(1) = result elist(1) = abserr iord(1) = 1 ! ! test on accuracy. ! errbnd = dmax1(epsabs,epsrel*dabs(result)) if(abserr.le.0.5d+02*epmach*defabs.and.abserr.gt.errbnd) ier = 2 if(limit.eq.1) ier = 1 ! if(ier.ne.0.or.(abserr.le.errbnd.and.abserr.ne.resabs)& ! .or.abserr.eq.0.0d+00) go to 60 igk=0 if(ier.ne.0.or.(abserr.le.errbnd.and.abserr.ne.resabs)& .or.abserr.eq.0.0d+00)igk=60 if(igk.ne.60)then ! ! initialization ! -------------- ! ! errmax = abserr maxerr = 1 area = result errsum = abserr nrmax = 1 iroff1 = 0 iroff2 = 0 ! ! main do-loop ! ------------ ! do last = 2,limit ! ! bisect the subinterval with the largest error estimate. ! a1 = alist(maxerr) b1 = 0.5d+00*(alist(maxerr)+blist(maxerr)) a2 = b1 b2 = blist(maxerr) if(keyf.eq.1) call dqk15(f,a1,b1,area1,error1,resabs,defab1) if(keyf.eq.2) call dqk21(f,a1,b1,area1,error1,resabs,defab1) if(keyf.eq.3) call dqk31(f,a1,b1,area1,error1,resabs,defab1) if(keyf.eq.4) call dqk41(f,a1,b1,area1,error1,resabs,defab1) if(keyf.eq.5) call dqk51(f,a1,b1,area1,error1,resabs,defab1) if(keyf.eq.6) call dqk61(f,a1,b1,area1,error1,resabs,defab1) if(keyf.eq.1) call dqk15(f,a2,b2,area2,error2,resabs,defab2) if(keyf.eq.2) call dqk21(f,a2,b2,area2,error2,resabs,defab2) if(keyf.eq.3) call dqk31(f,a2,b2,area2,error2,resabs,defab2) if(keyf.eq.4) call dqk41(f,a2,b2,area2,error2,resabs,defab2) if(keyf.eq.5) call dqk51(f,a2,b2,area2,error2,resabs,defab2) if(keyf.eq.6) call dqk61(f,a2,b2,area2,error2,resabs,defab2) ! ! improve previous approximations to integral ! and error and test for accuracy. ! neval = neval+1 area12 = area1+area2 erro12 = error1+error2 errsum = errsum+erro12-errmax area = area+area12-rlist(maxerr) if(defab1.ne.error1.and.defab2.ne.error2)then if(dabs(rlist(maxerr)-area12).le.0.1d-04*dabs(area12)& .and.erro12.ge.0.99d+00*errmax) iroff1 = iroff1+1 if(last.gt.10.and.erro12.gt.errmax) iroff2 = iroff2+1 endif rlist(maxerr) = area1 rlist(last) = area2 errbnd = dmax1(epsabs,epsrel*dabs(area)) ! if(errsum.le.errbnd) go to 8 if(errsum.gt.errbnd)then ! ! test for roundoff error and eventually set error flag. ! if(iroff1.ge.6.or.iroff2.ge.20) ier = 2 ! ! set error flag in the case that the number of subintervals ! equals limit. ! if(last.eq.limit) ier = 1 ! ! set error flag in the case of bad integrand behaviour ! at a point of the integration range. ! if(dmax1(dabs(a1),dabs(b2)).le.(0.1d+01+0.1d+03* & epmach)*(dabs(a2)+0.1d+04*uflow)) ier = 3 ! ! append the newly-created intervals to the list. ! endif ! if(error2.gt.error1) go to 10 igt=0 if(error2.le.error1)then alist(last) = a2 blist(maxerr) = b1 blist(last) = b2 elist(maxerr) = error1 elist(last) = error2 !go to 20 igt=20 endif if(igt.ne.20)then !10 alist(maxerr) = a2 alist(maxerr) = a2 alist(last) = a1 blist(last) = b1 rlist(maxerr) = area2 rlist(last) = area1 elist(maxerr) = error2 elist(last) = error1 ! ! call subroutine dqpsrt to maintain the descending ordering ! in the list of error estimates and select the subinterval ! with the largest error estimate (to be bisected next). ! endif !20 call dqpsrt(limit,last,maxerr,errmax,elist,iord,nrmax) call dqpsrt(limit,last,maxerr,errmax,elist,iord,nrmax) ! ***jump out of do-loop !if(ier.ne.0.or.errsum.le.errbnd) go to 40 if(ier.ne.0.or.errsum.le.errbnd)exit ! 30 continue enddo ! ! compute final result. ! --------------------- ! if(last.eq.limit+1)last=last-1 result = 0.0d+00 do k=1,last result = result+rlist(k) enddo abserr = errsum endif if(keyf.ne.1) neval = (10*keyf+1)*(2*neval+1) if(keyf.eq.1) neval = 30*neval+15 end if return end subroutine dqage subroutine dqk15(f,a,b,result,abserr,resabs,resasc) !***begin prologue dqk15 !***date written 800101 (yymmdd) !***revision date 830518 (yymmdd) !***category no. h2a1a2 !***keywords 15-point gauss-kronrod rules !***author piessens,robert,appl. math. & progr. div. - k.u.leuven ! de doncker,elise,appl. math. & progr. div - k.u.leuven !***purpose to compute i = integral of f over (a,b), with error ! estimate ! j = integral of abs(f) over (a,b) !***description ! ! integration rules ! standard fortran subroutine ! double precision version ! ! parameters ! on entry ! f - double precision ! function subprogram defining the integrand ! function f(x). the actual name for f needs to be ! declared e x t e r n a l in the calling program. ! ! a - double precision ! lower limit of integration ! ! b - double precision ! upper limit of integration ! ! on return ! result - double precision ! approximation to the integral i ! result is computed by applying the 15-point ! kronrod rule (resk) obtained by optimal addition ! of abscissae to the7-point gauss rule(resg). ! ! abserr - double precision ! estimate of the modulus of the absolute error, ! which should not exceed abs(i-result) ! ! resabs - double precision ! approximation to the integral j ! ! resasc - double precision ! approximation to the integral of abs(f-i/(b-a)) ! over (a,b) ! !***references (none) !***routines called d1mach !***end prologue dqk15 ! double precision a,absc,abserr,b,centr,dabs,dhlgth,dmax1,dmin1,& epmach,f,fc,fsum,fval1,fval2,fv1,fv2,hlgth,resabs,resasc,& resg,resk,reskh,result,uflow,wg,wgk,xgk integer j,jtw,jtwm1 external f ! dimension fv1(7),fv2(7),wg(4),wgk(8),xgk(8) ! ! the abscissae and weights are given for the interval (-1,1). ! because of symmetry only the positive abscissae and their ! corresponding weights are given. ! ! xgk - abscissae of the 15-point kronrod rule ! xgk(2), xgk(4), ... abscissae of the 7-point ! gauss rule ! xgk(1), xgk(3), ... abscissae which are optimally ! added to the 7-point gauss rule ! ! wgk - weights of the 15-point kronrod rule ! ! wg - weights of the 7-point gauss rule ! ! ! gauss quadrature weights and kronron quadrature abscissae and weights ! as evaluated with 80 decimal digit arithmetic by l. w. fullerton, ! bell labs, nov. 1981. ! data wg ( 1) / 0.129484966168869693270611432679082d0 / data wg ( 2) / 0.279705391489276667901467771423780d0 / data wg ( 3) / 0.381830050505118944950369775488975d0 / data wg ( 4) / 0.417959183673469387755102040816327d0 / ! data xgk ( 1) / 0.991455371120812639206854697526329d0 / data xgk ( 2) / 0.949107912342758524526189684047851d0 / data xgk ( 3) / 0.864864423359769072789712788640926d0 / data xgk ( 4) / 0.741531185599394439863864773280788d0 / data xgk ( 5) / 0.586087235467691130294144838258730d0 / data xgk ( 6) / 0.405845151377397166906606412076961d0 / data xgk ( 7) / 0.207784955007898467600689403773245d0 / data xgk ( 8) / 0.000000000000000000000000000000000d0 / ! data wgk ( 1) / 0.022935322010529224963732008058970d0 / data wgk ( 2) / 0.063092092629978553290700663189204d0 / data wgk ( 3) / 0.104790010322250183839876322541518d0 / data wgk ( 4) / 0.140653259715525918745189590510238d0 / data wgk ( 5) / 0.169004726639267902826583426598550d0 / data wgk ( 6) / 0.190350578064785409913256402421014d0 / data wgk ( 7) / 0.204432940075298892414161999234649d0 / data wgk ( 8) / 0.209482141084727828012999174891714d0 / ! ! ! list of major variables ! ----------------------- ! ! centr - mid point of the interval ! hlgth - half-length of the interval ! absc - abscissa ! fval* - function value ! resg - result of the 7-point gauss formula ! resk - result of the 15-point kronrod formula ! reskh - approximation to the mean value of f over (a,b), ! i.e. to i/(b-a) ! ! machine dependent constants ! --------------------------- ! ! epmach is the largest relative spacing. ! uflow is the smallest positive magnitude. ! !***first executable statement dqk15 epmach = epsilon(a) uflow = tiny(a) ! centr = 0.5d+00*(a+b) hlgth = 0.5d+00*(b-a) dhlgth = dabs(hlgth) ! ! compute the 15-point kronrod approximation to ! the integral, and estimate the absolute error. ! fc = f(centr) resg = fc*wg(4) resk = fc*wgk(8) resabs = dabs(resk) do j=1,3 jtw = j*2 absc = hlgth*xgk(jtw) fval1 = f(centr-absc) fval2 = f(centr+absc) fv1(jtw) = fval1 fv2(jtw) = fval2 fsum = fval1+fval2 resg = resg+wg(j)*fsum resk = resk+wgk(jtw)*fsum resabs = resabs+wgk(jtw)*(dabs(fval1)+dabs(fval2)) enddo do j = 1,4 jtwm1 = j*2-1 absc = hlgth*xgk(jtwm1) fval1 = f(centr-absc) fval2 = f(centr+absc) fv1(jtwm1) = fval1 fv2(jtwm1) = fval2 fsum = fval1+fval2 resk = resk+wgk(jtwm1)*fsum resabs = resabs+wgk(jtwm1)*(dabs(fval1)+dabs(fval2)) enddo reskh = resk*0.5d+00 resasc = wgk(8)*dabs(fc-reskh) do j=1,7 resasc = resasc+wgk(j)*(dabs(fv1(j)-reskh)+dabs(fv2(j)-reskh)) enddo result = resk*hlgth resabs = resabs*dhlgth resasc = resasc*dhlgth abserr = dabs((resk-resg)*hlgth) if(resasc.ne.0.0d+00.and.abserr.ne.0.0d+00) & abserr = resasc*dmin1(0.1d+01,(0.2d+03*abserr/resasc)**1.5d+00) if(resabs.gt.uflow/(0.5d+02*epmach)) abserr = dmax1 & ((epmach*0.5d+02)*resabs,abserr) return end subroutine dqk15 subroutine dqk21(f,a,b,result,abserr,resabs,resasc) !***begin prologue dqk21 !***date written 800101 (yymmdd) !***revision date 830518 (yymmdd) !***category no. h2a1a2 !***keywords 21-point gauss-kronrod rules !***author piessens,robert,appl. math. & progr. div. - k.u.leuven ! de doncker,elise,appl. math. & progr. div. - k.u.leuven !***purpose to compute i = integral of f over (a,b), with error ! estimate ! j = integral of abs(f) over (a,b) !***description ! ! integration rules ! standard fortran subroutine ! double precision version ! ! parameters ! on entry ! f - double precision ! function subprogram defining the integrand ! function f(x). the actual name for f needs to be ! declared e x t e r n a l in the driver program. ! ! a - double precision ! lower limit of integration ! ! b - double precision ! upper limit of integration ! ! on return ! result - double precision ! approximation to the integral i ! result is computed by applying the 21-point ! kronrod rule (resk) obtained by optimal addition ! of abscissae to the 10-point gauss rule (resg). ! ! abserr - double precision ! estimate of the modulus of the absolute error, ! which should not exceed abs(i-result) ! ! resabs - double precision ! approximation to the integral j ! ! resasc - double precision ! approximation to the integral of abs(f-i/(b-a)) ! over (a,b) ! !***references (none) !***routines called d1mach !***end prologue dqk21 ! double precision a,absc,abserr,b,centr,dabs,dhlgth,dmax1,dmin1, & epmach,f,fc,fsum,fval1,fval2,fv1,fv2,hlgth,resabs,resasc, & resg,resk,reskh,result,uflow,wg,wgk,xgk integer j,jtw,jtwm1 external f ! dimension fv1(10),fv2(10),wg(5),wgk(11),xgk(11) ! ! the abscissae and weights are given for the interval (-1,1). ! because of symmetry only the positive abscissae and their ! corresponding weights are given. ! ! xgk - abscissae of the 21-point kronrod rule ! xgk(2), xgk(4), ... abscissae of the 10-point ! gauss rule ! xgk(1), xgk(3), ... abscissae which are optimally ! added to the 10-point gauss rule ! ! wgk - weights of the 21-point kronrod rule ! ! wg - weights of the 10-point gauss rule ! ! ! gauss quadrature weights and kronron quadrature abscissae and weights ! as evaluated with 80 decimal digit arithmetic by l. w. fullerton, ! bell labs, nov. 1981. ! data wg ( 1) / 0.066671344308688137593568809893332d0 / data wg ( 2) / 0.149451349150580593145776339657697d0 / data wg ( 3) / 0.219086362515982043995534934228163d0 / data wg ( 4) / 0.269266719309996355091226921569469d0 / data wg ( 5) / 0.295524224714752870173892994651338d0 / ! data xgk ( 1) / 0.995657163025808080735527280689003d0 / data xgk ( 2) / 0.973906528517171720077964012084452d0 / data xgk ( 3) / 0.930157491355708226001207180059508d0 / data xgk ( 4) / 0.865063366688984510732096688423493d0 / data xgk ( 5) / 0.780817726586416897063717578345042d0 / data xgk ( 6) / 0.679409568299024406234327365114874d0 / data xgk ( 7) / 0.562757134668604683339000099272694d0 / data xgk ( 8) / 0.433395394129247190799265943165784d0 / data xgk ( 9) / 0.294392862701460198131126603103866d0 / data xgk ( 10) / 0.148874338981631210884826001129720d0 / data xgk ( 11) / 0.000000000000000000000000000000000d0 / ! data wgk ( 1) / 0.011694638867371874278064396062192d0 / data wgk ( 2) / 0.032558162307964727478818972459390d0 / data wgk ( 3) / 0.054755896574351996031381300244580d0 / data wgk ( 4) / 0.075039674810919952767043140916190d0 / data wgk ( 5) / 0.093125454583697605535065465083366d0 / data wgk ( 6) / 0.109387158802297641899210590325805d0 / data wgk ( 7) / 0.123491976262065851077958109831074d0 / data wgk ( 8) / 0.134709217311473325928054001771707d0 / data wgk ( 9) / 0.142775938577060080797094273138717d0 / data wgk ( 10) / 0.147739104901338491374841515972068d0 / data wgk ( 11) / 0.149445554002916905664936468389821d0 / ! ! ! list of major variables ! ----------------------- ! ! centr - mid point of the interval ! hlgth - half-length of the interval ! absc - abscissa ! fval* - function value ! resg - result of the 10-point gauss formula ! resk - result of the 21-point kronrod formula ! reskh - approximation to the mean value of f over (a,b), ! i.e. to i/(b-a) ! ! ! machine dependent constants ! --------------------------- ! ! epmach is the largest relative spacing. ! uflow is the smallest positive magnitude. ! !***first executable statement dqk21 epmach = epsilon(a) uflow = tiny(a) ! centr = 0.5d+00*(a+b) hlgth = 0.5d+00*(b-a) dhlgth = dabs(hlgth) ! ! compute the 21-point kronrod approximation to ! the integral, and estimate the absolute error. ! resg = 0.0d+00 fc = f(centr) resk = wgk(11)*fc resabs = dabs(resk) do j=1,5 jtw = 2*j absc = hlgth*xgk(jtw) fval1 = f(centr-absc) fval2 = f(centr+absc) fv1(jtw) = fval1 fv2(jtw) = fval2 fsum = fval1+fval2 resg = resg+wg(j)*fsum resk = resk+wgk(jtw)*fsum resabs = resabs+wgk(jtw)*(dabs(fval1)+dabs(fval2)) enddo do j = 1,5 jtwm1 = 2*j-1 absc = hlgth*xgk(jtwm1) fval1 = f(centr-absc) fval2 = f(centr+absc) fv1(jtwm1) = fval1 fv2(jtwm1) = fval2 fsum = fval1+fval2 resk = resk+wgk(jtwm1)*fsum resabs = resabs+wgk(jtwm1)*(dabs(fval1)+dabs(fval2)) enddo reskh = resk*0.5d+00 resasc = wgk(11)*dabs(fc-reskh) do j=1,10 resasc = resasc+wgk(j)*(dabs(fv1(j)-reskh)+dabs(fv2(j)-reskh)) enddo result = resk*hlgth resabs = resabs*dhlgth resasc = resasc*dhlgth abserr = dabs((resk-resg)*hlgth) if(resasc.ne.0.0d+00.and.abserr.ne.0.0d+00) & abserr = resasc*dmin1(0.1d+01,(0.2d+03*abserr/resasc)**1.5d+00) if(resabs.gt.uflow/(0.5d+02*epmach)) abserr = dmax1 & ((epmach*0.5d+02)*resabs,abserr) return end subroutine dqk21 subroutine dqk31(f,a,b,result,abserr,resabs,resasc) !***begin prologue dqk31 !***date written 800101 (yymmdd) !***revision date 830518 (yymmdd) !***category no. h2a1a2 !***keywords 31-point gauss-kronrod rules !***author piessens,robert,appl. math. & progr. div. - k.u.leuven ! de doncker,elise,appl. math. & progr. div. - k.u.leuven !***purpose to compute i = integral of f over (a,b) with error ! estimate ! j = integral of abs(f) over (a,b) !***description ! ! integration rules ! standard fortran subroutine ! double precision version ! ! parameters ! on entry ! f - double precision ! function subprogram defining the integrand ! function f(x). the actual name for f needs to be ! declared e x t e r n a l in the calling program. ! ! a - double precision ! lower limit of integration ! ! b - double precision ! upper limit of integration ! ! on return ! result - double precision ! approximation to the integral i ! result is computed by applying the 31-point ! gauss-kronrod rule (resk), obtained by optimal ! addition of abscissae to the 15-point gauss ! rule (resg). ! ! abserr - double precison ! estimate of the modulus of the modulus, ! which should not exceed abs(i-result) ! ! resabs - double precision ! approximation to the integral j ! ! resasc - double precision ! approximation to the integral of abs(f-i/(b-a)) ! over (a,b) ! !***references (none) !***routines called d1mach !***end prologue dqk31 double precision a,absc,abserr,b,centr,dabs,dhlgth,dmax1,dmin1,& epmach,f,fc,fsum,fval1,fval2,fv1,fv2,hlgth,resabs,resasc,& resg,resk,reskh,result,uflow,wg,wgk,xgk integer j,jtw,jtwm1 external f ! dimension fv1(15),fv2(15),xgk(16),wgk(16),wg(8) ! ! the abscissae and weights are given for the interval (-1,1). ! because of symmetry only the positive abscissae and their ! corresponding weights are given. ! ! xgk - abscissae of the 31-point kronrod rule ! xgk(2), xgk(4), ... abscissae of the 15-point ! gauss rule ! xgk(1), xgk(3), ... abscissae which are optimally ! added to the 15-point gauss rule ! ! wgk - weights of the 31-point kronrod rule ! ! wg - weights of the 15-point gauss rule ! ! ! gauss quadrature weights and kronron quadrature abscissae and weights ! as evaluated with 80 decimal digit arithmetic by l. w. fullerton, ! bell labs, nov. 1981. ! data wg ( 1) / 0.030753241996117268354628393577204d0 / data wg ( 2) / 0.070366047488108124709267416450667d0 / data wg ( 3) / 0.107159220467171935011869546685869d0 / data wg ( 4) / 0.139570677926154314447804794511028d0 / data wg ( 5) / 0.166269205816993933553200860481209d0 / data wg ( 6) / 0.186161000015562211026800561866423d0 / data wg ( 7) / 0.198431485327111576456118326443839d0 / data wg ( 8) / 0.202578241925561272880620199967519d0 / ! data xgk ( 1) / 0.998002298693397060285172840152271d0 / data xgk ( 2) / 0.987992518020485428489565718586613d0 / data xgk ( 3) / 0.967739075679139134257347978784337d0 / data xgk ( 4) / 0.937273392400705904307758947710209d0 / data xgk ( 5) / 0.897264532344081900882509656454496d0 / data xgk ( 6) / 0.848206583410427216200648320774217d0 / data xgk ( 7) / 0.790418501442465932967649294817947d0 / data xgk ( 8) / 0.724417731360170047416186054613938d0 / data xgk ( 9) / 0.650996741297416970533735895313275d0 / data xgk ( 10) / 0.570972172608538847537226737253911d0 / data xgk ( 11) / 0.485081863640239680693655740232351d0 / data xgk ( 12) / 0.394151347077563369897207370981045d0 / data xgk ( 13) / 0.299180007153168812166780024266389d0 / data xgk ( 14) / 0.201194093997434522300628303394596d0 / data xgk ( 15) / 0.101142066918717499027074231447392d0 / data xgk ( 16) / 0.000000000000000000000000000000000d0 / ! data wgk ( 1) / 0.005377479872923348987792051430128d0 / data wgk ( 2) / 0.015007947329316122538374763075807d0 / data wgk ( 3) / 0.025460847326715320186874001019653d0 / data wgk ( 4) / 0.035346360791375846222037948478360d0 / data wgk ( 5) / 0.044589751324764876608227299373280d0 / data wgk ( 6) / 0.053481524690928087265343147239430d0 / data wgk ( 7) / 0.062009567800670640285139230960803d0 / data wgk ( 8) / 0.069854121318728258709520077099147d0 / data wgk ( 9) / 0.076849680757720378894432777482659d0 / data wgk ( 10) / 0.083080502823133021038289247286104d0 / data wgk ( 11) / 0.088564443056211770647275443693774d0 / data wgk ( 12) / 0.093126598170825321225486872747346d0 / data wgk ( 13) / 0.096642726983623678505179907627589d0 / data wgk ( 14) / 0.099173598721791959332393173484603d0 / data wgk ( 15) / 0.100769845523875595044946662617570d0 / data wgk ( 16) / 0.101330007014791549017374792767493d0 / ! ! ! list of major variables ! ----------------------- ! centr - mid point of the interval ! hlgth - half-length of the interval ! absc - abscissa ! fval* - function value ! resg - result of the 15-point gauss formula ! resk - result of the 31-point kronrod formula ! reskh - approximation to the mean value of f over (a,b), ! i.e. to i/(b-a) ! ! machine dependent constants ! --------------------------- ! epmach is the largest relative spacing. ! uflow is the smallest positive magnitude. !***first executable statement dqk31 epmach = epsilon(a) uflow = tiny(a) ! centr = 0.5d+00*(a+b) hlgth = 0.5d+00*(b-a) dhlgth = dabs(hlgth) ! ! compute the 31-point kronrod approximation to ! the integral, and estimate the absolute error. ! fc = f(centr) resg = wg(8)*fc resk = wgk(16)*fc resabs = dabs(resk) do j=1,7 jtw = j*2 absc = hlgth*xgk(jtw) fval1 = f(centr-absc) fval2 = f(centr+absc) fv1(jtw) = fval1 fv2(jtw) = fval2 fsum = fval1+fval2 resg = resg+wg(j)*fsum resk = resk+wgk(jtw)*fsum resabs = resabs+wgk(jtw)*(dabs(fval1)+dabs(fval2)) enddo do j = 1,8 jtwm1 = j*2-1 absc = hlgth*xgk(jtwm1) fval1 = f(centr-absc) fval2 = f(centr+absc) fv1(jtwm1) = fval1 fv2(jtwm1) = fval2 fsum = fval1+fval2 resk = resk+wgk(jtwm1)*fsum resabs = resabs+wgk(jtwm1)*(dabs(fval1)+dabs(fval2)) enddo reskh = resk*0.5d+00 resasc = wgk(16)*dabs(fc-reskh) do j=1,15 resasc = resasc+wgk(j)*(dabs(fv1(j)-reskh)+dabs(fv2(j)-reskh)) enddo result = resk*hlgth resabs = resabs*dhlgth resasc = resasc*dhlgth abserr = dabs((resk-resg)*hlgth) if(resasc.ne.0.0d+00.and.abserr.ne.0.0d+00)& abserr = resasc*dmin1(0.1d+01,(0.2d+03*abserr/resasc)**1.5d+00) if(resabs.gt.uflow/(0.5d+02*epmach)) abserr = dmax1& ((epmach*0.5d+02)*resabs,abserr) return end subroutine dqk31 subroutine dqk41(f,a,b,result,abserr,resabs,resasc) !***begin prologue dqk41 !***date written 800101 (yymmdd) !***revision date 830518 (yymmdd) !***category no. h2a1a2 !***keywords 41-point gauss-kronrod rules !***author piessens,robert,appl. math. & progr. div. - k.u.leuven ! de doncker,elise,appl. math. & progr. div. - k.u.leuven !***purpose to compute i = integral of f over (a,b), with error ! estimate ! j = integral of abs(f) over (a,b) !***description ! ! integration rules ! standard fortran subroutine ! double precision version ! ! parameters ! on entry ! f - double precision ! function subprogram defining the integrand ! function f(x). the actual name for f needs to be ! declared e x t e r n a l in the calling program. ! ! a - double precision ! lower limit of integration ! ! b - double precision ! upper limit of integration ! ! on return ! result - double precision ! approximation to the integral i ! result is computed by applying the 41-point ! gauss-kronrod rule (resk) obtained by optimal ! addition of abscissae to the 20-point gauss ! rule (resg). ! ! abserr - double precision ! estimate of the modulus of the absolute error, ! which should not exceed abs(i-result) ! ! resabs - double precision ! approximation to the integral j ! ! resasc - double precision ! approximation to the integal of abs(f-i/(b-a)) ! over (a,b) ! !***references (none) !***routines called d1mach !***end prologue dqk41 ! double precision a,absc,abserr,b,centr,dabs,dhlgth,dmax1,dmin1,& epmach,f,fc,fsum,fval1,fval2,fv1,fv2,hlgth,resabs,resasc,& resg,resk,reskh,result,uflow,wg,wgk,xgk integer j,jtw,jtwm1 external f ! dimension fv1(20),fv2(20),xgk(21),wgk(21),wg(10) ! ! the abscissae and weights are given for the interval (-1,1). ! because of symmetry only the positive abscissae and their ! corresponding weights are given. ! ! xgk - abscissae of the 41-point gauss-kronrod rule ! xgk(2), xgk(4), ... abscissae of the 20-point ! gauss rule ! xgk(1), xgk(3), ... abscissae which are optimally ! added to the 20-point gauss rule ! ! wgk - weights of the 41-point gauss-kronrod rule ! ! wg - weights of the 20-point gauss rule ! ! ! gauss quadrature weights and kronron quadrature abscissae and weights ! as evaluated with 80 decimal digit arithmetic by l. w. fullerton, ! bell labs, nov. 1981. ! data wg ( 1) / 0.017614007139152118311861962351853d0 / data wg ( 2) / 0.040601429800386941331039952274932d0 / data wg ( 3) / 0.062672048334109063569506535187042d0 / data wg ( 4) / 0.083276741576704748724758143222046d0 / data wg ( 5) / 0.101930119817240435036750135480350d0 / data wg ( 6) / 0.118194531961518417312377377711382d0 / data wg ( 7) / 0.131688638449176626898494499748163d0 / data wg ( 8) / 0.142096109318382051329298325067165d0 / data wg ( 9) / 0.149172986472603746787828737001969d0 / data wg ( 10) / 0.152753387130725850698084331955098d0 / ! data xgk ( 1) / 0.998859031588277663838315576545863d0 / data xgk ( 2) / 0.993128599185094924786122388471320d0 / data xgk ( 3) / 0.981507877450250259193342994720217d0 / data xgk ( 4) / 0.963971927277913791267666131197277d0 / data xgk ( 5) / 0.940822633831754753519982722212443d0 / data xgk ( 6) / 0.912234428251325905867752441203298d0 / data xgk ( 7) / 0.878276811252281976077442995113078d0 / data xgk ( 8) / 0.839116971822218823394529061701521d0 / data xgk ( 9) / 0.795041428837551198350638833272788d0 / data xgk ( 10) / 0.746331906460150792614305070355642d0 / data xgk ( 11) / 0.693237656334751384805490711845932d0 / data xgk ( 12) / 0.636053680726515025452836696226286d0 / data xgk ( 13) / 0.575140446819710315342946036586425d0 / data xgk ( 14) / 0.510867001950827098004364050955251d0 / data xgk ( 15) / 0.443593175238725103199992213492640d0 / data xgk ( 16) / 0.373706088715419560672548177024927d0 / data xgk ( 17) / 0.301627868114913004320555356858592d0 / data xgk ( 18) / 0.227785851141645078080496195368575d0 / data xgk ( 19) / 0.152605465240922675505220241022678d0 / data xgk ( 20) / 0.076526521133497333754640409398838d0 / data xgk ( 21) / 0.000000000000000000000000000000000d0 / ! data wgk ( 1) / 0.003073583718520531501218293246031d0 / data wgk ( 2) / 0.008600269855642942198661787950102d0 / data wgk ( 3) / 0.014626169256971252983787960308868d0 / data wgk ( 4) / 0.020388373461266523598010231432755d0 / data wgk ( 5) / 0.025882133604951158834505067096153d0 / data wgk ( 6) / 0.031287306777032798958543119323801d0 / data wgk ( 7) / 0.036600169758200798030557240707211d0 / data wgk ( 8) / 0.041668873327973686263788305936895d0 / data wgk ( 9) / 0.046434821867497674720231880926108d0 / data wgk ( 10) / 0.050944573923728691932707670050345d0 / data wgk ( 11) / 0.055195105348285994744832372419777d0 / data wgk ( 12) / 0.059111400880639572374967220648594d0 / data wgk ( 13) / 0.062653237554781168025870122174255d0 / data wgk ( 14) / 0.065834597133618422111563556969398d0 / data wgk ( 15) / 0.068648672928521619345623411885368d0 / data wgk ( 16) / 0.071054423553444068305790361723210d0 / data wgk ( 17) / 0.073030690332786667495189417658913d0 / data wgk ( 18) / 0.074582875400499188986581418362488d0 / data wgk ( 19) / 0.075704497684556674659542775376617d0 / data wgk ( 20) / 0.076377867672080736705502835038061d0 / data wgk ( 21) / 0.076600711917999656445049901530102d0 / ! ! ! list of major variables ! ----------------------- ! ! centr - mid point of the interval ! hlgth - half-length of the interval ! absc - abscissa ! fval* - function value ! resg - result of the 20-point gauss formula ! resk - result of the 41-point kronrod formula ! reskh - approximation to mean value of f over (a,b), i.e. ! to i/(b-a) ! ! machine dependent constants ! --------------------------- ! ! epmach is the largest relative spacing. ! uflow is the smallest positive magnitude. ! !***first executable statement dqk41 epmach = epsilon(a) uflow = tiny(a) ! centr = 0.5d+00*(a+b) hlgth = 0.5d+00*(b-a) dhlgth = dabs(hlgth) ! ! compute the 41-point gauss-kronrod approximation to ! the integral, and estimate the absolute error. ! resg = 0.0d+00 fc = f(centr) resk = wgk(21)*fc resabs = dabs(resk) do j=1,10 jtw = j*2 absc = hlgth*xgk(jtw) fval1 = f(centr-absc) fval2 = f(centr+absc) fv1(jtw) = fval1 fv2(jtw) = fval2 fsum = fval1+fval2 resg = resg+wg(j)*fsum resk = resk+wgk(jtw)*fsum resabs = resabs+wgk(jtw)*(dabs(fval1)+dabs(fval2)) enddo do j = 1,10 jtwm1 = j*2-1 absc = hlgth*xgk(jtwm1) fval1 = f(centr-absc) fval2 = f(centr+absc) fv1(jtwm1) = fval1 fv2(jtwm1) = fval2 fsum = fval1+fval2 resk = resk+wgk(jtwm1)*fsum resabs = resabs+wgk(jtwm1)*(dabs(fval1)+dabs(fval2)) enddo reskh = resk*0.5d+00 resasc = wgk(21)*dabs(fc-reskh) do j=1,20 resasc = resasc+wgk(j)*(dabs(fv1(j)-reskh)+dabs(fv2(j)-reskh)) enddo result = resk*hlgth resabs = resabs*dhlgth resasc = resasc*dhlgth abserr = dabs((resk-resg)*hlgth) if(resasc.ne.0.0d+00.and.abserr.ne.0.d+00)& abserr = resasc*dmin1(0.1d+01,(0.2d+03*abserr/resasc)**1.5d+00) if(resabs.gt.uflow/(0.5d+02*epmach)) abserr = dmax1& ((epmach*0.5d+02)*resabs,abserr) return end subroutine dqk41 subroutine dqk51(f,a,b,result,abserr,resabs,resasc) !***begin prologue dqk51 !***date written 800101 (yymmdd) !***revision date 830518 (yymmdd) !***category no. h2a1a2 !***keywords 51-point gauss-kronrod rules !***author piessens,robert,appl. math. & progr. div. - k.u.leuven ! de doncker,elise,appl. math & progr. div. - k.u.leuven !***purpose to compute i = integral of f over (a,b) with error ! estimate ! j = integral of abs(f) over (a,b) !***description ! ! integration rules ! standard fortran subroutine ! double precision version ! ! parameters ! on entry ! f - double precision ! function subroutine defining the integrand ! function f(x). the actual name for f needs to be ! declared e x t e r n a l in the calling program. ! ! a - double precision ! lower limit of integration ! ! b - double precision ! upper limit of integration ! ! on return ! result - double precision ! approximation to the integral i ! result is computed by applying the 51-point ! kronrod rule (resk) obtained by optimal addition ! of abscissae to the 25-point gauss rule (resg). ! ! abserr - double precision ! estimate of the modulus of the absolute error, ! which should not exceed abs(i-result) ! ! resabs - double precision ! approximation to the integral j ! ! resasc - double precision ! approximation to the integral of abs(f-i/(b-a)) ! over (a,b) ! !***references (none) !***routines called d1mach !***end prologue dqk51 ! double precision a,absc,abserr,b,centr,dabs,dhlgth,dmax1,dmin1,& epmach,f,fc,fsum,fval1,fval2,fv1,fv2,hlgth,resabs,resasc,& resg,resk,reskh,result,uflow,wg,wgk,xgk integer j,jtw,jtwm1 external f ! dimension fv1(25),fv2(25),xgk(26),wgk(26),wg(13) ! ! the abscissae and weights are given for the interval (-1,1). ! because of symmetry only the positive abscissae and their ! corresponding weights are given. ! ! xgk - abscissae of the 51-point kronrod rule ! xgk(2), xgk(4), ... abscissae of the 25-point ! gauss rule ! xgk(1), xgk(3), ... abscissae which are optimally ! added to the 25-point gauss rule ! ! wgk - weights of the 51-point kronrod rule ! ! wg - weights of the 25-point gauss rule ! ! ! gauss quadrature weights and kronron quadrature abscissae and weights ! as evaluated with 80 decimal digit arithmetic by l. w. fullerton, ! bell labs, nov. 1981. ! data wg ( 1) / 0.011393798501026287947902964113235d0 / data wg ( 2) / 0.026354986615032137261901815295299d0 / data wg ( 3) / 0.040939156701306312655623487711646d0 / data wg ( 4) / 0.054904695975835191925936891540473d0 / data wg ( 5) / 0.068038333812356917207187185656708d0 / data wg ( 6) / 0.080140700335001018013234959669111d0 / data wg ( 7) / 0.091028261982963649811497220702892d0 / data wg ( 8) / 0.100535949067050644202206890392686d0 / data wg ( 9) / 0.108519624474263653116093957050117d0 / data wg ( 10) / 0.114858259145711648339325545869556d0 / data wg ( 11) / 0.119455763535784772228178126512901d0 / data wg ( 12) / 0.122242442990310041688959518945852d0 / data wg ( 13) / 0.123176053726715451203902873079050d0 / ! data xgk ( 1) / 0.999262104992609834193457486540341d0 / data xgk ( 2) / 0.995556969790498097908784946893902d0 / data xgk ( 3) / 0.988035794534077247637331014577406d0 / data xgk ( 4) / 0.976663921459517511498315386479594d0 / data xgk ( 5) / 0.961614986425842512418130033660167d0 / data xgk ( 6) / 0.942974571228974339414011169658471d0 / data xgk ( 7) / 0.920747115281701561746346084546331d0 / data xgk ( 8) / 0.894991997878275368851042006782805d0 / data xgk ( 9) / 0.865847065293275595448996969588340d0 / data xgk ( 10) / 0.833442628760834001421021108693570d0 / data xgk ( 11) / 0.797873797998500059410410904994307d0 / data xgk ( 12) / 0.759259263037357630577282865204361d0 / data xgk ( 13) / 0.717766406813084388186654079773298d0 / data xgk ( 14) / 0.673566368473468364485120633247622d0 / data xgk ( 15) / 0.626810099010317412788122681624518d0 / data xgk ( 16) / 0.577662930241222967723689841612654d0 / data xgk ( 17) / 0.526325284334719182599623778158010d0 / data xgk ( 18) / 0.473002731445714960522182115009192d0 / data xgk ( 19) / 0.417885382193037748851814394594572d0 / data xgk ( 20) / 0.361172305809387837735821730127641d0 / data xgk ( 21) / 0.303089538931107830167478909980339d0 / data xgk ( 22) / 0.243866883720988432045190362797452d0 / data xgk ( 23) / 0.183718939421048892015969888759528d0 / data xgk ( 24) / 0.122864692610710396387359818808037d0 / data xgk ( 25) / 0.061544483005685078886546392366797d0 / data xgk ( 26) / 0.000000000000000000000000000000000d0 / ! data wgk ( 1) / 0.001987383892330315926507851882843d0 / data wgk ( 2) / 0.005561932135356713758040236901066d0 / data wgk ( 3) / 0.009473973386174151607207710523655d0 / data wgk ( 4) / 0.013236229195571674813656405846976d0 / data wgk ( 5) / 0.016847817709128298231516667536336d0 / data wgk ( 6) / 0.020435371145882835456568292235939d0 / data wgk ( 7) / 0.024009945606953216220092489164881d0 / data wgk ( 8) / 0.027475317587851737802948455517811d0 / data wgk ( 9) / 0.030792300167387488891109020215229d0 / data wgk ( 10) / 0.034002130274329337836748795229551d0 / data wgk ( 11) / 0.037116271483415543560330625367620d0 / data wgk ( 12) / 0.040083825504032382074839284467076d0 / data wgk ( 13) / 0.042872845020170049476895792439495d0 / data wgk ( 14) / 0.045502913049921788909870584752660d0 / data wgk ( 15) / 0.047982537138836713906392255756915d0 / data wgk ( 16) / 0.050277679080715671963325259433440d0 / data wgk ( 17) / 0.052362885806407475864366712137873d0 / data wgk ( 18) / 0.054251129888545490144543370459876d0 / data wgk ( 19) / 0.055950811220412317308240686382747d0 / data wgk ( 20) / 0.057437116361567832853582693939506d0 / data wgk ( 21) / 0.058689680022394207961974175856788d0 / data wgk ( 22) / 0.059720340324174059979099291932562d0 / data wgk ( 23) / 0.060539455376045862945360267517565d0 / data wgk ( 24) / 0.061128509717053048305859030416293d0 / data wgk ( 25) / 0.061471189871425316661544131965264d0 / ! note: wgk (26) was calculated from the values of wgk(1..25) data wgk ( 26) / 0.061580818067832935078759824240066d0 / ! ! ! list of major variables ! ----------------------- ! ! centr - mid point of the interval ! hlgth - half-length of the interval ! absc - abscissa ! fval* - function value ! resg - result of the 25-point gauss formula ! resk - result of the 51-point kronrod formula ! reskh - approximation to the mean value of f over (a,b), ! i.e. to i/(b-a) ! ! machine dependent constants ! --------------------------- ! ! epmach is the largest relative spacing. ! uflow is the smallest positive magnitude. ! !***first executable statement dqk51 epmach = epsilon(a) uflow = tiny(a) ! centr = 0.5d+00*(a+b) hlgth = 0.5d+00*(b-a) dhlgth = dabs(hlgth) ! ! compute the 51-point kronrod approximation to ! the integral, and estimate the absolute error. ! fc = f(centr) resg = wg(13)*fc resk = wgk(26)*fc resabs = dabs(resk) do j=1,12 jtw = j*2 absc = hlgth*xgk(jtw) fval1 = f(centr-absc) fval2 = f(centr+absc) fv1(jtw) = fval1 fv2(jtw) = fval2 fsum = fval1+fval2 resg = resg+wg(j)*fsum resk = resk+wgk(jtw)*fsum resabs = resabs+wgk(jtw)*(dabs(fval1)+dabs(fval2)) enddo do j = 1,13 jtwm1 = j*2-1 absc = hlgth*xgk(jtwm1) fval1 = f(centr-absc) fval2 = f(centr+absc) fv1(jtwm1) = fval1 fv2(jtwm1) = fval2 fsum = fval1+fval2 resk = resk+wgk(jtwm1)*fsum resabs = resabs+wgk(jtwm1)*(dabs(fval1)+dabs(fval2)) enddo reskh = resk*0.5d+00 resasc = wgk(26)*dabs(fc-reskh) do j=1,25 resasc = resasc+wgk(j)*(dabs(fv1(j)-reskh)+dabs(fv2(j)-reskh)) enddo result = resk*hlgth resabs = resabs*dhlgth resasc = resasc*dhlgth abserr = dabs((resk-resg)*hlgth) if(resasc.ne.0.0d+00.and.abserr.ne.0.0d+00)& abserr = resasc*dmin1(0.1d+01,(0.2d+03*abserr/resasc)**1.5d+00) if(resabs.gt.uflow/(0.5d+02*epmach)) abserr = dmax1& ((epmach*0.5d+02)*resabs,abserr) return end subroutine dqk51 subroutine dqk61(f,a,b,result,abserr,resabs,resasc) !***begin prologue dqk61 !***date written 800101 (yymmdd) !***revision date 830518 (yymmdd) !***category no. h2a1a2 !***keywords 61-point gauss-kronrod rules !***author piessens,robert,appl. math. & progr. div. - k.u.leuven ! de doncker,elise,appl. math. & progr. div. - k.u.leuven !***purpose to compute i = integral of f over (a,b) with error ! estimate ! j = integral of dabs(f) over (a,b) !***description ! ! integration rule ! standard fortran subroutine ! double precision version ! ! ! parameters ! on entry ! f - double precision ! function subprogram defining the integrand ! function f(x). the actual name for f needs to be ! declared e x t e r n a l in the calling program. ! ! a - double precision ! lower limit of integration ! ! b - double precision ! upper limit of integration ! ! on return ! result - double precision ! approximation to the integral i ! result is computed by applying the 61-point ! kronrod rule (resk) obtained by optimal addition of ! abscissae to the 30-point gauss rule (resg). ! ! abserr - double precision ! estimate of the modulus of the absolute error, ! which should equal or exceed dabs(i-result) ! ! resabs - double precision ! approximation to the integral j ! ! resasc - double precision ! approximation to the integral of dabs(f-i/(b-a)) ! ! !***references (none) !***routines called d1mach !***end prologue dqk61 ! double precision a,dabsc,abserr,b,centr,dabs,dhlgth,dmax1,dmin1,& epmach,f,fc,fsum,fval1,fval2,fv1,fv2,hlgth,resabs,resasc,& resg,resk,reskh,result,uflow,wg,wgk,xgk integer j,jtw,jtwm1 external f ! dimension fv1(30),fv2(30),xgk(31),wgk(31),wg(15) ! ! the abscissae and weights are given for the ! interval (-1,1). because of symmetry only the positive ! abscissae and their corresponding weights are given. ! ! xgk - abscissae of the 61-point kronrod rule ! xgk(2), xgk(4) ... abscissae of the 30-point ! gauss rule ! xgk(1), xgk(3) ... optimally added abscissae ! to the 30-point gauss rule ! ! wgk - weights of the 61-point kronrod rule ! ! wg - weigths of the 30-point gauss rule ! ! ! gauss quadrature weights and kronron quadrature abscissae and weights ! as evaluated with 80 decimal digit arithmetic by l. w. fullerton, ! bell labs, nov. 1981. ! data wg ( 1) / 0.007968192496166605615465883474674d0 / data wg ( 2) / 0.018466468311090959142302131912047d0 / data wg ( 3) / 0.028784707883323369349719179611292d0 / data wg ( 4) / 0.038799192569627049596801936446348d0 / data wg ( 5) / 0.048402672830594052902938140422808d0 / data wg ( 6) / 0.057493156217619066481721689402056d0 / data wg ( 7) / 0.065974229882180495128128515115962d0 / data wg ( 8) / 0.073755974737705206268243850022191d0 / data wg ( 9) / 0.080755895229420215354694938460530d0 / data wg ( 10) / 0.086899787201082979802387530715126d0 / data wg ( 11) / 0.092122522237786128717632707087619d0 / data wg ( 12) / 0.096368737174644259639468626351810d0 / data wg ( 13) / 0.099593420586795267062780282103569d0 / data wg ( 14) / 0.101762389748405504596428952168554d0 / data wg ( 15) / 0.102852652893558840341285636705415d0 / ! data xgk ( 1) / 0.999484410050490637571325895705811d0 / data xgk ( 2) / 0.996893484074649540271630050918695d0 / data xgk ( 3) / 0.991630996870404594858628366109486d0 / data xgk ( 4) / 0.983668123279747209970032581605663d0 / data xgk ( 5) / 0.973116322501126268374693868423707d0 / data xgk ( 6) / 0.960021864968307512216871025581798d0 / data xgk ( 7) / 0.944374444748559979415831324037439d0 / data xgk ( 8) / 0.926200047429274325879324277080474d0 / data xgk ( 9) / 0.905573307699907798546522558925958d0 / data xgk ( 10) / 0.882560535792052681543116462530226d0 / data xgk ( 11) / 0.857205233546061098958658510658944d0 / data xgk ( 12) / 0.829565762382768397442898119732502d0 / data xgk ( 13) / 0.799727835821839083013668942322683d0 / data xgk ( 14) / 0.767777432104826194917977340974503d0 / data xgk ( 15) / 0.733790062453226804726171131369528d0 / data xgk ( 16) / 0.697850494793315796932292388026640d0 / data xgk ( 17) / 0.660061064126626961370053668149271d0 / data xgk ( 18) / 0.620526182989242861140477556431189d0 / data xgk ( 19) / 0.579345235826361691756024932172540d0 / data xgk ( 20) / 0.536624148142019899264169793311073d0 / data xgk ( 21) / 0.492480467861778574993693061207709d0 / data xgk ( 22) / 0.447033769538089176780609900322854d0 / data xgk ( 23) / 0.400401254830394392535476211542661d0 / data xgk ( 24) / 0.352704725530878113471037207089374d0 / data xgk ( 25) / 0.304073202273625077372677107199257d0 / data xgk ( 26) / 0.254636926167889846439805129817805d0 / data xgk ( 27) / 0.204525116682309891438957671002025d0 / data xgk ( 28) / 0.153869913608583546963794672743256d0 / data xgk ( 29) / 0.102806937966737030147096751318001d0 / data xgk ( 30) / 0.051471842555317695833025213166723d0 / data xgk ( 31) / 0.000000000000000000000000000000000d0 / ! data wgk ( 1) / 0.001389013698677007624551591226760d0 / data wgk ( 2) / 0.003890461127099884051267201844516d0 / data wgk ( 3) / 0.006630703915931292173319826369750d0 / data wgk ( 4) / 0.009273279659517763428441146892024d0 / data wgk ( 5) / 0.011823015253496341742232898853251d0 / data wgk ( 6) / 0.014369729507045804812451432443580d0 / data wgk ( 7) / 0.016920889189053272627572289420322d0 / data wgk ( 8) / 0.019414141193942381173408951050128d0 / data wgk ( 9) / 0.021828035821609192297167485738339d0 / data wgk ( 10) / 0.024191162078080601365686370725232d0 / data wgk ( 11) / 0.026509954882333101610601709335075d0 / data wgk ( 12) / 0.028754048765041292843978785354334d0 / data wgk ( 13) / 0.030907257562387762472884252943092d0 / data wgk ( 14) / 0.032981447057483726031814191016854d0 / data wgk ( 15) / 0.034979338028060024137499670731468d0 / data wgk ( 16) / 0.036882364651821229223911065617136d0 / data wgk ( 17) / 0.038678945624727592950348651532281d0 / data wgk ( 18) / 0.040374538951535959111995279752468d0 / data wgk ( 19) / 0.041969810215164246147147541285970d0 / data wgk ( 20) / 0.043452539701356069316831728117073d0 / data wgk ( 21) / 0.044814800133162663192355551616723d0 / data wgk ( 22) / 0.046059238271006988116271735559374d0 / data wgk ( 23) / 0.047185546569299153945261478181099d0 / data wgk ( 24) / 0.048185861757087129140779492298305d0 / data wgk ( 25) / 0.049055434555029778887528165367238d0 / data wgk ( 26) / 0.049795683427074206357811569379942d0 / data wgk ( 27) / 0.050405921402782346840893085653585d0 / data wgk ( 28) / 0.050881795898749606492297473049805d0 / data wgk ( 29) / 0.051221547849258772170656282604944d0 / data wgk ( 30) / 0.051426128537459025933862879215781d0 / data wgk ( 31) / 0.051494729429451567558340433647099d0 / ! ! list of major variables ! ----------------------- ! ! centr - mid point of the interval ! hlgth - half-length of the interval ! dabsc - abscissa ! fval* - function value ! resg - result of the 30-point gauss rule ! resk - result of the 61-point kronrod rule ! reskh - approximation to the mean value of f ! over (a,b), i.e. to i/(b-a) ! ! machine dependent constants ! --------------------------- ! ! epmach is the largest relative spacing. ! uflow is the smallest positive magnitude. ! epmach = epsilon(a) uflow = tiny(a) ! centr = 0.5d+00*(b+a) hlgth = 0.5d+00*(b-a) dhlgth = dabs(hlgth) ! ! compute the 61-point kronrod approximation to the ! integral, and estimate the absolute error. ! !***first executable statement dqk61 resg = 0.0d+00 fc = f(centr) resk = wgk(31)*fc resabs = dabs(resk) do j=1,15 jtw = j*2 dabsc = hlgth*xgk(jtw) fval1 = f(centr-dabsc) fval2 = f(centr+dabsc) fv1(jtw) = fval1 fv2(jtw) = fval2 fsum = fval1+fval2 resg = resg+wg(j)*fsum resk = resk+wgk(jtw)*fsum resabs = resabs+wgk(jtw)*(dabs(fval1)+dabs(fval2)) enddo do j=1,15 jtwm1 = j*2-1 dabsc = hlgth*xgk(jtwm1) fval1 = f(centr-dabsc) fval2 = f(centr+dabsc) fv1(jtwm1) = fval1 fv2(jtwm1) = fval2 fsum = fval1+fval2 resk = resk+wgk(jtwm1)*fsum resabs = resabs+wgk(jtwm1)*(dabs(fval1)+dabs(fval2)) enddo reskh = resk*0.5d+00 resasc = wgk(31)*dabs(fc-reskh) do j=1,30 resasc = resasc+wgk(j)*(dabs(fv1(j)-reskh)+dabs(fv2(j)-reskh)) enddo result = resk*hlgth resabs = resabs*dhlgth resasc = resasc*dhlgth abserr = dabs((resk-resg)*hlgth) if(resasc.ne.0.0d+00.and.abserr.ne.0.0d+00)& abserr = resasc*dmin1(0.1d+01,(0.2d+03*abserr/resasc)**1.5d+00) if(resabs.gt.uflow/(0.5d+02*epmach)) abserr = dmax1& ((epmach*0.5d+02)*resabs,abserr) return end subroutine dqk61 subroutine dqpsrt(limit,last,maxerr,ermax,elist,iord,nrmax) !***begin prologue dqpsrt !***refer to dqage,dqagie,dqagpe,dqawse !***routines called (none) !***revision date 810101 (yymmdd) !***keywords sequential sorting !***author piessens,robert,appl. math. & progr. div. - k.u.leuven ! de doncker,elise,appl. math. & progr. div. - k.u.leuven !***purpose this routine maintains the descending ordering in the ! list of the local error estimated resulting from the ! interval subdivision process. at each call two error ! estimates are inserted using the sequential search ! method, top-down for the largest error estimate and ! bottom-up for the smallest error estimate. !***description ! ! ordering routine ! standard fortran subroutine ! double precision version ! ! parameters (meaning at output) ! limit - integer ! maximum number of error estimates the list ! can contain ! ! last - integer ! number of error estimates currently in the list ! ! maxerr - integer ! maxerr points to the nrmax-th largest error ! estimate currently in the list ! ! ermax - double precision ! nrmax-th largest error estimate ! ermax = elist(maxerr) ! ! elist - double precision ! vector of dimension last containing ! the error estimates ! ! iord - integer ! vector of dimension last, the first k elements ! of which contain pointers to the error ! estimates, such that ! elist(iord(1)),..., elist(iord(k)) ! form a decreasing sequence, with ! k = last if last.le.(limit/2+2), and ! k = limit+1-last otherwise ! ! nrmax - integer ! maxerr = iord(nrmax) ! !***end prologue dqpsrt ! double precision elist,ermax,errmax,errmin integer i,ibeg,ido,iord,isucc,j,jbnd,jupbn,k,last,limit,maxerr,& nrmax,igt,igt1,igt2 dimension elist(last),iord(last) ! ! check whether the list contains more than ! two error estimates. ! !***first executable statement dqpsrt !if(last.gt.2) go to 10 igt=0 if(last.le.2)then iord(1) = 1 iord(2) = 2 ! go to 90 igt=90 ! ! this part of the routine is only executed if, due to a ! difficult integrand, subdivision increased the error ! estimate. in the normal case the insert procedure should ! start after the nrmax-th largest error estimate. ! endif if(igt.ne.90)then errmax = elist(maxerr) ! if(nrmax.eq.1) go to 30 if(nrmax.ne.1)then ido = nrmax-1 do i = 1,ido isucc = iord(nrmax-1) ! ***jump out of do-loop if(errmax.le.elist(isucc))exit iord(nrmax) = isucc nrmax = nrmax-1 enddo ! ! compute the number of elements in the list to be maintained ! in descending order. this number depends on the number of ! subdivisions still allowed. ! endif !30 jupbn = last jupbn = last if(last.gt.(limit/2+2)) jupbn = limit+3-last errmin = elist(last) ! ! insert errmax by traversing the list top-down, ! starting comparison from the element elist(iord(nrmax+1)). ! jbnd = jupbn-1 ibeg = nrmax+1 !if(ibeg.gt.jbnd) go to 50 igt1=0 if(ibeg.le.jbnd)then do i=ibeg,jbnd isucc = iord(i) ! ***jump out of do-loop !if(errmax.ge.elist(isucc)) go to 60 if(errmax.ge.elist(isucc))then igt1=60 exit endif iord(i-1) = isucc enddo endif if(igt1.ne.60)then !50 iord(jbnd) = maxerr iord(jbnd) = maxerr iord(jupbn) = last !go to 90 igt=90 ! ! insert errmin by traversing the list bottom-up. ! endif igt2=0 if(igt.ne.90)then !60 iord(i-1) = maxerr iord(i-1) = maxerr k = jbnd do j=i,jbnd isucc = iord(k) ! ***jump out of do-loop !if(errmin.lt.elist(isucc)) go to 80 if(errmin.lt.elist(isucc))then igt2=80 exit endif iord(k+1) = isucc k = k-1 enddo if(igt2.ne.80)then iord(i) = last !go to 90 igt=90 endif if(igt.ne.90)then iord(k+1) = last endif ! ! set maxerr and ermax. ! endif endif maxerr = iord(nrmax) ermax = elist(maxerr) return end subroutine dqpsrt subroutine xerror ( xmess, nmess, nerr, level ) !*****************************************************************************80 ! !! XERROR replaces the SLATEC XERROR routine. ! ! Modified: ! ! 12 September 2015 ! implicit none integer::level,nerr,nmess character ( len = * ) xmess if ( 1 <= LEVEL ) then WRITE ( *,'(1X,A)') XMESS(1:NMESS) WRITE ( *,'('' ERROR NUMBER = '',I5,'', MESSAGE LEVEL = '',I5)') & NERR,LEVEL end if return end subroutine xerror